module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 136,
"column": 19
} | {
"line": 136,
"column": 30
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝³ : FunLike F ℍ ℂ\nk : ℤ\ninst✝² : 𝒢.IsFiniteRelIndex ℋ\ninst✝¹ : ℋ.HasDetPlusMinusOne\ninst✝ : ModularFormClass F 𝒢 k\nhf : ⇑f = 0\nτ : ℍ\n⊢ 1 ∈ ℋ ∧ (⇑f ∣[k] 1⁻¹) τ = 0",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Rea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 66
} | [
{
"pp": "𝒢 : Subgroup (GL (Fin 2) ℝ)\ninst✝¹ : 𝒢.IsArithmetic\ninst✝ : 𝒢.HasDetOne\nf : ModularForm 𝒢 0\nthis : ModularFormClass\n (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range\n (0 *\n ↑(Nat.card\n (↥(Matrix.SpecialLinearGroup.mapGL ℝ).range ⧸ 𝒢.subgroupOf (Matrix.SpecialLinearG... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 13
} | [
{
"pp": "hi : 0 < 1\n⊢ (PowerSeries.coeff 0) (qExpansion 1 Δ) = 0",
"usedConstants": [
"Eq.mpr",
"Real",
"Semiring.toModule",
"UpperHalfPlane.qExpansion",
"congrArg",
"LinearMap.instFunLike",
"RingHom",
"id",
"PowerSeries.coeff",
"instOfNatNat",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 50
} | [
{
"pp": "⊢ Module.rank ℂ (CuspForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 12) = 1",
"usedConstants": [
"ModularForm",
"Eq.mpr",
"Subgroup.instHasDetOneRangeSpecialLinearGroupGeneralLinearGroupMapGL",
"MonoidHom.range",
"Real",
"Matrix.SpecialLinearGroup",
"Cardin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula | {
"line": 245,
"column": 2
} | {
"line": 246,
"column": 9
} | [
{
"pp": "⊢ Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range 2) = 0",
"usedConstants": [
"ModularForm",
"Eq.mpr",
"Subgroup.instHasDetOneRangeSpecialLinearGroupGeneralLinearGroupMapGL",
"MonoidHom.range",
"Real",
"Matrix.SpecialLinearGroup",
"Card... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula | {
"line": 257,
"column": 6
} | {
"line": 257,
"column": 17
} | [
{
"pp": "case h.inl.«0»\nk : ℕ\nihn :\n ∀ m < 0,\n Even m →\n Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range ↑m) =\n ↑(if m ≡ 2 [MOD 12] then m / 12 else m / 12 + 1)\nhk2 : Even 0\nhk : 0 < 3\n⊢ Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range ↑0) =\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.LevelOne.DimensionFormula | {
"line": 259,
"column": 6
} | {
"line": 259,
"column": 29
} | [
{
"pp": "case h.inl.«2»\nk : ℕ\nihn :\n ∀ m < 2,\n Even m →\n Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range ↑m) =\n ↑(if m ≡ 2 [MOD 12] then m / 12 else m / 12 + 1)\nhk2 : Even 2\nhk : 2 < 3\n⊢ Module.rank ℂ (ModularForm (Matrix.SpecialLinearGroup.mapGL ℝ).range ↑2) =\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 18
} | [
{
"pp": "m : ℤ\neq : (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1))\n⊢ 8 ∣ (2 * m + 1) ^ 2 - 1",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"HSub.hSub",
"id",
"Distrib.toAdd",
"AddMonoidWithOne.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 266,
"column": 2
} | {
"line": 266,
"column": 18
} | [
{
"pp": "m : ℕ\neq : (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1))\n⊢ 8 ∣ (2 * m + 1) ^ 2 - 1",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"HSub.hSub",
"id",
"Distrib.toAdd",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 272,
"column": 28
} | {
"line": 272,
"column": 39
} | [
{
"pp": "x y : ℤ\nhx : ¬2 ∣ x\nhxy : 4 ∣ x - y\ni : ℕ\nhx_odd : Odd x\nhxy_even : Even (x - y)\n⊢ Odd y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 295,
"column": 28
} | {
"line": 295,
"column": 39
} | [
{
"pp": "x y : ℤ\nn : ℕ\nhxy : 4 ∣ x - y\nhx : ¬2 ∣ x\nhx_odd : Odd x\nhxy_even : Even (x - y)\n⊢ Odd y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 306,
"column": 4
} | {
"line": 306,
"column": 39
} | [
{
"pp": "case succ.hxy\nx y : ℤ\nhxy : 4 ∣ x - y\nhx : ¬2 ∣ x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : ℕ\nh : FiniteMultiplicity 2 (n + 1)\nhpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1\nk : ℕ\nhk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k\n⊢ 2 ∣ x ^ 2 ^ multiplicity 2 (n + 1) - y ^ 2 ^ mul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfiniteAdeleRing | {
"line": 125,
"column": 60
} | {
"line": 125,
"column": 92
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : InfiniteAdeleRing K\nv : InfinitePlace K\nhv : ¬IsUnit (x v)\n⊢ ‖x v‖ ^ v.mult = 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 63
} | [
{
"pp": "case succ.hx\nx y : ℤ\nhxy : 4 ∣ x - y\nhx : ¬2 ∣ x\nhx_odd : Odd x\nhxy_even : Even (x - y)\nhy_odd : Odd y\nn : ℕ\nh : FiniteMultiplicity 2 (n + 1)\nhpn : ¬2 ^ (multiplicity 2 (n + 1) + 1) ∣ n + 1\nk : ℕ\nhk : n + 1 = 2 ^ multiplicity 2 (n + 1) * k\n⊢ ¬2 ∣ x ^ 2 ^ multiplicity 2 (n + 1)",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfiniteAdeleRing | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 39
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K\n⊢ ‖(algebraMap K (InfiniteAdeleRing K)) x‖ = ↑|(Algebra.norm ℚ) x|",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real",
"MonoidHom.instFunLike... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 277,
"column": 6
} | {
"line": 277,
"column": 67
} | [
{
"pp": "case refine_2\nι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\ninst✝¹ : (i : ι) → TopologicalSpace (R i)\nS : Set ι\ninst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)\nhS : Sᶜ.Finite\nhAcompact : ∀ i ∈ S, IsCompact (A i)\nx : Πʳ (i : ι), [R i, A i]_[𝓟 S]\nK : (i : ι) → Set (R i)\nK_compact :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 376,
"column": 2
} | {
"line": 376,
"column": 13
} | [
{
"pp": "x n : ℕ\nh1x : 1 < x\nhx : ¬2 ∣ x\nhn : n ≠ 0\nhneven : Even n\n⊢ padicValNat 2 (x ^ n - 1) + 1 = padicValNat 2 (x + 1) + padicValNat 2 (x - 1) + padicValNat 2 n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.InfinitePlace | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 15
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nx : WithAbs ↑v\n⊢ ‖(v.embedding.comp (WithAbs.equiv ↑v).toRingHom) x‖ = ‖x‖",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real",
"Wit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.InfinitePlace | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 15
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nv : InfinitePlace K\nhv : v.IsReal\nx : WithAbs ↑v\n⊢ ‖((embedding_of_isReal hv).comp (WithAbs.equiv ↑v).toRingHom) x‖ = ‖x‖",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrde... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 637,
"column": 6
} | {
"line": 637,
"column": 17
} | [
{
"pp": "ι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\n𝓕✝ 𝓖 : Filter ι\nS : ι → Type u_3\ninst✝⁴ : (i : ι) → SetLike (S i) (R i)\nB : (i : ι) → S i\nT : Set ι\n𝓕 : Filter ι\ninst✝³ : (i : ι) → TopologicalSpace (R i)\nhBopen : Fact (∀ (i : ι), IsOpen[inst✝³ i] ↑(B i))\ninst✝² : (i : ι) → Group (R i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.InfinitePlace | {
"line": 254,
"column": 6
} | {
"line": 254,
"column": 17
} | [
{
"pp": "case a.hp\nK : Type u_1\ninst✝⁶ : Field K\nL : Type u_2\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\nw : InfinitePlace L\nv : InfinitePlace K\ninst✝³ : Algebra v.Completion w.Completion\ninst✝² : IsScalarTower K v.Completion w.Completion\ninst✝¹ : ContinuousSMul v.Completion w.Completion\ninst✝ : ComplexEm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.InfinitePlace | {
"line": 277,
"column": 4
} | {
"line": 277,
"column": 45
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\nL : Type u_2\ninst✝² : Field L\ninst✝¹ : Algebra K L\nw : InfinitePlace L\nv : InfinitePlace K\ninst✝ : (↑w).LiesOver ↑v\nx : WithAbs ↑v\n⊢ ‖(algebraMap (WithAbs ↑v) (WithAbs ↑w)) x‖ = ‖x‖",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.FiniteAdeleRing | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\na : FiniteAdeleRing R K\n⊢ (∀ (i : HeightOneSpectrum R), a i ≠ 0) ∧\n ({a_1 | a a_1 ∉ adicCompletionIntegers K a_1} ∪ {a_1 | ¬Valued.v (a a_1) = 1}).Fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 17
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\n⊢ regOfFamily u ≠ 0 → IsMaxRank u",
"usedConstants": [
"Real",
"Real.instZero",
"id",
"Ne",
"Zero.toOfNat0",
"NumberField.Units.IsMaxRank",
"OfNat.ofNat",
"NumberField.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 377,
"column": 19
} | {
"line": 377,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A\nh : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w\na : mixedSpace K\nha : a ∈ A\n⊢ normAtComplexPlaces a ∈ Set.univ.pi fun w ↦ if w.IsReal then Set.u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 58
} | [
{
"pp": "case h.refine_2\nK : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A\nh : ∀ (x : mixedSpace K) (w : InfinitePlace K), w.IsComplex → 0 ≤ normAtComplexPlaces x w\nx : realSpace K\nx✝ : x ∈ ⇑mixedSpaceOfRealSpace ⁻¹' A ∩ Set.univ.pi fun w ↦ if w... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 211,
"column": 17
} | {
"line": 211,
"column": 59
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx y : mixedSpace K\nhx : x ∈ fundamentalCone K\nhy : ∀ (w : InfinitePlace K), (normAtPlace w) y = (normAtPlace w) x\n⊢ y ∉ {x | mixedEmbedding.norm x = 0}",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 387,
"column": 63
} | {
"line": 387,
"column": 74
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na b : ↑(integerSet K)\nx✝ : ∃ u, (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b\nu : (↥(𝓞 K)⁰)ˣ\nh : (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b\n⊢ ↑⟨unitsNonZeroDivisorsEquiv u, ?m.50⟩ • ↑a = ↑b",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 393,
"column": 92
} | {
"line": 411,
"column": 100
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nA : Set (mixedSpace K)\ninst✝ : NumberField K\nhA : normAtComplexPlaces ⁻¹' normAtComplexPlaces '' A = A\nhm : MeasurableSet A\n⊢ volume A =\n ENNReal.ofReal (2 * π) ^ nrComplexPlaces K *\n ∫⁻ (x : realSpace K) in normAtComplexPlaces '' A, ∏ w, ENNReal.ofReal (x ... | by
have hA' {x} : (A.indicator 1 x : ℝ≥0∞) =
(normAtComplexPlaces '' A).indicator 1 (normAtComplexPlaces x) := by
simp_rw [← Set.indicator_comp_right, Function.comp_def, Pi.one_def, hA]
rw [← lintegral_indicator_one hm, ← lintegral_comp_polarSpaceCoord_symm, polarSpaceCoord_target',
Measure.volume_eq_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 388,
"column": 60
} | {
"line": 388,
"column": 71
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na b : ↑(integerSet K)\nx✝ : ∃ ζ, ↑ζ • ↑a = ↑b\nu : (𝓞 K)ˣ\nproperty✝ : u ∈ torsion K\nh : ↑⟨u, property✝⟩ • ↑a = ↑b\n⊢ (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑(unitsNonZeroDivisorsEquiv.symm u)) * ↑a = ↑b",
"usedConstants": [
"Units.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 389,
"column": 58
} | {
"line": 389,
"column": 73
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\na b : ↑(integerSet K)\nx✝ : ∃ u, (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b\nu : (↥(𝓞 K)⁰)ˣ\nh : (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b\n⊢ unitsNonZeroDivisorsEquiv u • ↑a ∈ fundamentalCone K",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 429,
"column": 8
} | {
"line": 430,
"column": 15
} | [
{
"pp": "case h.refine_2.refine_1.inl\nK : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtAllPlaces ⁻¹' normAtAllPlaces '' A = A\na : mixedSpace K\nha₁ : a ∈ A\nha₂ : a ∈ {x | ∀ (w : { w // w.IsReal }), 0 < x.1 w}\nw : InfinitePlace K\nhw : w.IsReal\n⊢ normAtAllPlaces a w = normAtComplexPlaces a ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 433,
"column": 6
} | {
"line": 433,
"column": 70
} | [
{
"pp": "case h.refine_2.refine_2\nK : Type u_1\ninst✝ : Field K\nA : Set (mixedSpace K)\nhA : normAtAllPlaces ⁻¹' normAtAllPlaces '' A = A\na : mixedSpace K\nha₁ : a ∈ A\nha₂ : a ∈ {x | ∀ (w : { w // w.IsReal }), 0 < x.1 w}\nw : { w // w.IsReal }\n⊢ normAtComplexPlaces a ∈ {x | x ↑w ≠ 0}",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.Ramification | {
"line": 118,
"column": 51
} | {
"line": 118,
"column": 62
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nv : InfinitePlace K\ninst✝¹ : NumberField K\ninst✝ : NumberField L\nx✝ : InfinitePlace L\nh : x✝ ∈ (ramifiedPlacesOver L v).toFinset\n⊢ x✝ ∈ ramifiedPlacesOver L v",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.Ramification | {
"line": 119,
"column": 51
} | {
"line": 119,
"column": 62
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nv : InfinitePlace K\ninst✝¹ : NumberField K\ninst✝ : NumberField L\nx✝ : InfinitePlace L\nh : x✝ ∈ (unramifiedPlacesOver L v).toFinset\n⊢ x✝ ∈ unramifiedPlacesOver L v",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 39
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nw : InfinitePlace K\nx : ℝ\n⊢ HasDerivAt (↑(expMap_single w)) (deriv_expMap_single w x) x",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"NormedCommRing.toSeminor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 98
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ expMap.target = Set.univ.pi fun x ↦ Set.Ioi 0",
"usedConstants": [
"Real",
"Set.Ioi",
"NumberField.mixedEmbedding.realSpace",
"Pi.topologicalSpace",
"Real.instZero",
"PartialEquiv.target",
"Set.univ",... | simp_rw [expMap, OpenPartialHomeomorph.pi_toPartialEquiv, PartialEquiv.pi_target, expMap_single] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 98
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ expMap.target = Set.univ.pi fun x ↦ Set.Ioi 0",
"usedConstants": [
"Real",
"Set.Ioi",
"NumberField.mixedEmbedding.realSpace",
"Pi.topologicalSpace",
"Real.instZero",
"PartialEquiv.target",
"Set.univ",... | simp_rw [expMap, OpenPartialHomeomorph.pi_toPartialEquiv, PartialEquiv.pi_target, expMap_single] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 98
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ expMap.target = Set.univ.pi fun x ↦ Set.Ioi 0",
"usedConstants": [
"Real",
"Set.Ioi",
"NumberField.mixedEmbedding.realSpace",
"Pi.topologicalSpace",
"Real.instZero",
"PartialEquiv.target",
"Set.univ",... | simp_rw [expMap, OpenPartialHomeomorph.pi_toPartialEquiv, PartialEquiv.pi_target, expMap_single] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 315,
"column": 2
} | {
"line": 316,
"column": 38
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\n⊢ HasFDerivAt (↑expMap) (fderiv_expMap x) x",
"usedConstants": [
"ContinuousLinearMap.comp",
"HasFDerivAt",
"Eq.mpr",
"Pi.Function.module",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 523,
"column": 2
} | {
"line": 524,
"column": 69
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\n⊢ mixedEmbedding.norm (mixedSpaceOfRealSpace (↑expMapBasis x)) = Real.exp (x w₀) ^ finrank ℚ K",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"Pi.Function.module",
"InnerProductSpace.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 467,
"column": 6
} | {
"line": 467,
"column": 21
} | [
{
"pp": "case refine_2\nK : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : CharZero K\ninst✝³ : Algebra.IsIntegral ℚ K\ninst✝² : IsTotallyComplex K\nE : Subfield K\ninst✝¹ : IsTotallyReal ↥E\ninst✝ : IsQuadraticExtension (↥E) K\nh : ¬maximalRealSubfield K ≤ E\nL : IntermediateField (↥E) K := (E ⊔ maximalRealSubfield K).t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 39,
"column": 50
} | {
"line": 39,
"column": 66
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\np f : ℕ\nhK : Fintype.card K = p ^ f\nh0 : f = 0\n⊢ Fintype.card K ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 54,
"column": 8
} | {
"line": 54,
"column": 19
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\np f n : ℕ\nP : K[X]\nhK : Fintype.card K = p ^ f\nhn : p.Coprime n\nhp : Fact (Nat.Prime p)\nhP : P ∣ cyclotomic n K\nhPirr : Irreducible P\nhPmo : P.Monic\nthis✝² : Fact (Irreducible P)\nthis✝¹ : Module.Finite K (AdjoinRoot P)\nthis✝ : Finite (AdjoinR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 41
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\np f n : ℕ\nP : K[X]\nhK : Fintype.card K = p ^ f\nhn : p.Coprime n\nhp : Fact (Nat.Prime p)\nhP : P ∣ cyclotomic n K\nhPirr : Irreducible P\nhPmo : P.Monic\nthis✝² : Fact (Irreducible P)\nthis✝¹ : Module.Finite K (AdjoinRoot P)\nthis✝ : Finite (AdjoinR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 60,
"column": 10
} | {
"line": 60,
"column": 21
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\np f n : ℕ\nP : K[X]\nhK : Fintype.card K = p ^ f\nhn : p.Coprime n\nhp : Fact (Nat.Prime p)\nhP : P ∣ cyclotomic n K\nhPirr : Irreducible P\nhPmo : P.Monic\nthis✝² : Fact (Irreducible P)\nthis✝¹ : Module.Finite K (AdjoinRoot P)\nthis✝ : Finite (AdjoinR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 15
} | [
{
"pp": "case inr\np : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : IsPrimitiveRoot ζ p\nhp : Nat.Prime p\nthis✝ : NeZero p\ni : ℕ\nhi : i < p\nhη₁ : ζ ^ i ∈ ↑(nthRootsFinset p 1)\nj : ℕ\nhj : j < p\nhη₂ : ζ ^ j ∈ ↑(nthRootsFinset p 1)\ne : ζ ^ i ≠ ζ ^ j\nthis :\n ∀ {p : ℕ} {A : Type u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 499,
"column": 2
} | {
"line": 499,
"column": 13
} | [
{
"pp": "F : Type u_1\nK : Type u_2\ninst✝⁷ : Field F\ninst✝⁶ : IsTotallyReal F\ninst✝⁵ : Field K\ninst✝⁴ : CharZero K\ninst✝³ : Algebra.IsIntegral ℚ K\ninst✝² : IsTotallyComplex K\ninst✝¹ : Algebra F K\ninst✝ : IsQuadraticExtension F K\nx : ↥(maximalRealSubfield K)\n⊢ (algebraMap F K) ((equivMaximalRealSubfiel... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 74,
"column": 4
} | {
"line": 75,
"column": 37
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\np f n : ℕ\nP : K[X]\nhK : Fintype.card K = p ^ f\nhn : p.Coprime n\nhp : Fact (Nat.Prime p)\nhP : P ∣ cyclotomic n K\nhPirr : Irreducible P\nhPmo : P.Monic\nthis✝¹ : Fact (Irreducible P)\nthis✝ : Module.Finite K (AdjoinRoot P)\nthis : Fi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 664,
"column": 56
} | {
"line": 664,
"column": 72
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : ∀ (w : InfinitePlace K), 0 < x w\nh : ∀ (i : InfinitePlace K), ↑expMapBasis.symm x i ∈ if i = w₀ then Set.Iic 0 else Set.Ico 0 1\nw : InfinitePlace K\nhw : ¬w = w₀\n⊢ ↑expMapBasis.symm x w ∈ Set.Ico 0 1",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 664,
"column": 81
} | {
"line": 664,
"column": 92
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : ∀ (w : InfinitePlace K), 0 < x w\nh : ∀ (i : InfinitePlace K), ↑expMapBasis.symm x i ∈ if i = w₀ then Set.Iic 0 else Set.Ico 0 1\n⊢ ↑expMapBasis.symm x w₀ ≤ 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 575,
"column": 4
} | {
"line": 575,
"column": 44
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : CharZero K\nS : Set ℕ\ninst✝ : IsCyclotomicExtension S ℚ K\nthis✝² : Algebra.IsIntegral ℚ K\nn : ℕ\nhn₁ : n ∈ S\nhn₂ : 2 < n\nthis✝¹ : NeZero n\nζ : K\nhζ : IsPrimitiveRoot ζ n\nthis✝ : IsCyclotomicExtension {n} ℚ ↥ℚ⟮ζ⟯\nthis : IsTotallyComplex ↥ℚ⟮ζ⟯\n⊢ IsTotall... | exact isTotallyComplex_of_algebra ℚ⟮ζ⟯ K | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 659,
"column": 4
} | {
"line": 667,
"column": 21
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : ∀ (w : InfinitePlace K), 0 < x w\n⊢ x ∈ normAtAllPlaces '' normLeOne K ↔ x ∈ ↑expMapBasis '' paramSet K",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Nontrivial",
"Iff.mpr",
"N... | rw [← expMapBasis.right_inv (Set.mem_univ_pi.mpr hx), (injective_expMapBasis K).mem_set_image]
simp only [normAtAllPlaces_normLeOne, Set.mem_inter_iff, Set.mem_setOf_eq, expMapBasis_nonneg,
Set.mem_preimage, logMap_expMapBasis, implies_true, and_true, norm_expMapBasis,
pow_le_one_iff_of_nonneg (Real.exp... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 659,
"column": 4
} | {
"line": 667,
"column": 21
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : ∀ (w : InfinitePlace K), 0 < x w\n⊢ x ∈ normAtAllPlaces '' normLeOne K ↔ x ∈ ↑expMapBasis '' paramSet K",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Nontrivial",
"Iff.mpr",
"N... | rw [← expMapBasis.right_inv (Set.mem_univ_pi.mpr hx), (injective_expMapBasis K).mem_set_image]
simp only [normAtAllPlaces_normLeOne, Set.mem_inter_iff, Set.mem_setOf_eq, expMapBasis_nonneg,
Set.mem_preimage, logMap_expMapBasis, implies_true, and_true, norm_expMapBasis,
pow_le_one_iff_of_nonneg (Real.exp... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 55
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\np f n : ℕ\nP : K[X]\nhK : Fintype.card K = p ^ f\nhn : p.Coprime n\nhp : Fact (Nat.Prime p)\nhPirr : Irreducible P\nA : K[X]\nhA : cyclotomic n K = P * A\nhQ : P * C P.leadingCoeff⁻¹ ∣ cyclotomic n K\n⊢ P.natDegree = orderOf (unitOfCoprime (p ^ f) ⋯)",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 127,
"column": 8
} | {
"line": 127,
"column": 19
} | [
{
"pp": "p n : ℕ\nhp : Fact (Nat.Prime p)\nP : (ZMod p)[X]\nhpn : ¬p ∣ n\nhP : P ∣ cyclotomic n (ZMod p)\nhPdeg : P.natDegree = orderOf (unitOfCoprime p ⋯)\n⊢ p.Coprime n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Ideal.Basic | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 37
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nI : Ideal (𝓞 K)\ninst✝¹ : NumberField K\nn : ℕ\ninst✝ : NeZero n\nhI₁ : absNorm I ≠ 1\nhI₂ : (absNorm I).Coprime n\nx✝ : ↥(rootsOfUnity n (𝓞 K))\nζ : (𝓞 K)ˣ\nhζ✝ : ζ ∈ rootsOfUnity n (𝓞 K)\nh : (I.rootsOfUnityMapQuot n) ⟨ζ, hζ✝⟩ = 1\nt : ℕ\nht₀ : t ≠ 0\nht : t ∣ n\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : HasFiniteQuotients R\nx : R\nhx : x ≠ 0\nthis✝ : Finite (R ⧸ Ideal.span {x})\nthis : {I | Ideal.comap (Ideal.Quotient.mk (Ideal.span {x})) ⊥ ≤ I}.Finite\n⊢ {I | x ∈ I}.Finite",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 754,
"column": 6
} | {
"line": 754,
"column": 21
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nc : ℝ\nhc : c ∈ Set.Icc 0 1\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then {0} else Set.Icc 0 1\nhx₀ : (fun x1 x2 ↦ x1 • x2) c (↑expMapBasis y) ≠ 0\nw : InfinitePlace K\nh : ¬w = w₀\n⊢ y w ∈ Set.Icc 0 1",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 761,
"column": 6
} | {
"line": 761,
"column": 30
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nc : ℝ\nhc : c ∈ Set.Icc 0 1\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then {0} else Set.Icc 0 1\nhx₀ : (fun x1 x2 ↦ x1 • x2) c (↑expMapBasis y) ≠ 0\nhc' : 0 < c\nw : InfinitePlace K\nh : w = w₀\n⊢ 0 = y w",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 774,
"column": 38
} | {
"line": 774,
"column": 49
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then Set.Iic 0 else Set.Icc 0 1\nhx₀ : ↑expMapBasis y ≠ 0\n⊢ y w₀ ≤ 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients | {
"line": 158,
"column": 27
} | {
"line": 158,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn : ℤ\nhI : ℤ ∙ n ≠ ⊥\n⊢ n ≠ 0",
"usedConstants": [
"id",
"Ne",
"Int",
"Zero.toOfNat0",
"OfNat.ofNat",
"MulZeroClass.toZero",
"Int.instSemiring",
"instMulZeroClassOfSemiring"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 777,
"column": 6
} | {
"line": 777,
"column": 21
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then Set.Iic 0 else Set.Icc 0 1\nhx₀ : ↑expMapBasis y ≠ 0\nw : InfinitePlace K\nh : ¬w = w₀\n⊢ y w ∈ Set.Icc 0 1",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 775,
"column": 4
} | {
"line": 777,
"column": 43
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then Set.Iic 0 else Set.Icc 0 1\nhx₀ : ↑expMapBasis y ≠ 0\nw : InfinitePlace K\n⊢ (if w = w₀ then 0 else y w) ∈ if w = w₀ then {0} else Set.Icc 0 1",
"usedConstants": [
... | split_ifs with h
· rfl
· simpa [h] using hy w (Set.mem_univ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 775,
"column": 4
} | {
"line": 777,
"column": 43
} | [
{
"pp": "case refine_2\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ny : realSpace K\nhy : y ∈ Set.univ.pi fun w ↦ if w = w₀ then Set.Iic 0 else Set.Icc 0 1\nhx₀ : ↑expMapBasis y ≠ 0\nw : InfinitePlace K\n⊢ (if w = w₀ then 0 else y w) ∈ if w = w₀ then {0} else Set.Icc 0 1",
"usedConstants": [
... | split_ifs with h
· rfl
· simpa [h] using hy w (Set.mem_univ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 784,
"column": 4
} | {
"line": 784,
"column": 21
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx₀ : x = 0\n⊢ x ∈ compactSet K ↔ x ∈ ↑expMapBasis '' closure (paramSet K) ∪ {0}",
"usedConstants": [
"Eq.mpr",
"Real",
"NumberField.mixedEmbedding.realSpace",
"Pi.topologicalSpace",
"Rea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 787,
"column": 4
} | {
"line": 787,
"column": 41
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx₀ : ¬x = 0\nhx : x ∈ ↑expMapBasis '' closure (paramSet K)\n⊢ x ∈ compactSet K",
"usedConstants": [
"NumberField.mixedEmbedding.fundamentalCone.compactSet_eq_union_aux₂"
]
}
] | exact compactSet_eq_union_aux₂ hx₀ hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 823,
"column": 8
} | {
"line": 823,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : IsBounded (↑expMapBasis '' paramSet K)\nC : ℝ\nhC : ∀ x ∈ ↑expMapBasis '' paramSet K, ‖x‖ ≤ C\nx : mixedSpace K\nhx : x ∈ normAtAllPlaces ⁻¹' ↑expMapBasis '' paramSet K\n⊢ ‖x‖ ≤ C",
"usedConstants": [
"NormedCommRing.toNormedRing",... | norm_eq_sup'_normAtPlace | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 825,
"column": 4
} | {
"line": 826,
"column": 11
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nthis : IsBounded (↑expMapBasis '' paramSet K)\nC : ℝ\nhC : ∀ x ∈ ↑expMapBasis '' paramSet K, ‖x‖ ≤ C\nx : mixedSpace K\nhx : x ∈ normAtAllPlaces ⁻¹' ↑expMapBasis '' paramSet K\nw : InfinitePlace K\nx✝ : w ∈ univ\n⊢ (normAtPlace w) x ≤ C",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 13
} | [
{
"pp": "p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\n⊢ absNorm (span {hζ.toInteger - 1}) = p",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Mo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 43
} | [
{
"pp": "case h.e'_5.h.e'_3\np k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\n⊢ p = absNorm (span {hζ.toInteger - 1})",
"usedConstants": [
"Nat.instMulZeroOneClass",
... | exact (absNorm_span_zeta_sub_one ..).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | {
"line": 124,
"column": 10
} | {
"line": 124,
"column": 21
} | [
{
"pp": "n : ℕ\ninst✝⁴ : NeZero n\nK : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\nhK : IsCyclotomicExtension {n} ℚ K\np : ℕ\nhp : Fact (Nat.Prime p)\nP : Ideal (𝓞 K)\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver (span {↑p})\nhn : p.Coprime n\nσ : Gal(K/ℚ)\nhσ : σ ∈ stabilizer Gal(K/ℚ) P\nhζ : IsPrimitiveR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 122,
"column": 35
} | {
"line": 122,
"column": 46
} | [
{
"pp": "p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nh : (span {hζ.toInteger - 1}).IsPrime\n⊢ 𝒑 ≠ ⊥",
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 130,
"column": 21
} | {
"line": 130,
"column": 32
} | [
{
"pp": "p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nthis : IsGalois ℚ K\n⊢ 𝒑 ≠ ⊥",
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"congr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | {
"line": 123,
"column": 4
} | {
"line": 124,
"column": 56
} | [
{
"pp": "n : ℕ\ninst✝⁴ : NeZero n\nK : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\nhK : IsCyclotomicExtension {n} ℚ K\np : ℕ\nhp : Fact (Nat.Prime p)\nP : Ideal (𝓞 K)\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver (span {↑p})\nhn : p.Coprime n\nσ : Gal(K/ℚ)\nhσ : σ ∈ stabilizer Gal(K/ℚ) P\nhζ : IsPrimitiveR... | refine hζ.toInteger_isPrimitiveRoot.idealQuotient_mk
(by simpa using IsMaximal.ne_top inferInstance) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 15
} | [
{
"pp": "m p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\nP : Ideal (𝓞 K)\nhP₁ : P.IsPrime\nhP₂ : P.LiesOver 𝒑\ninst✝ : NeZero m\nhK : IsCyclotomicExtension {m} ℚ K\nhm : p.Coprime m\nζ : 𝓞 K := ⋯.toInteger\nh₁ : ¬p ∣ exponent ζ\nh₂ :\n Irreducible ↑((primesOverSpanE... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.ExistsRamified | {
"line": 37,
"column": 69
} | {
"line": 37,
"column": 83
} | [
{
"pp": "K : Type u_1\n𝒪 : Type u_2\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : CommRing 𝒪\ninst✝¹ : Algebra 𝒪 K\ninst✝ : IsIntegralClosure 𝒪 ℤ K\nH : Module.finrank ℚ K ≠ 1\nthis✝³ : IsDomain 𝒪\nthis✝² : IsDedekindDomain 𝒪\nthis✝¹ : CharZero 𝒪\nthis✝ : 0 < Module.finrank ℚ K\nthis : 2 < |discr K... | zify; linarith | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.ExistsRamified | {
"line": 37,
"column": 69
} | {
"line": 37,
"column": 83
} | [
{
"pp": "K : Type u_1\n𝒪 : Type u_2\ninst✝⁴ : Field K\ninst✝³ : NumberField K\ninst✝² : CommRing 𝒪\ninst✝¹ : Algebra 𝒪 K\ninst✝ : IsIntegralClosure 𝒪 ℤ K\nH : Module.finrank ℚ K ≠ 1\nthis✝³ : IsDomain 𝒪\nthis✝² : IsDedekindDomain 𝒪\nthis✝¹ : CharZero 𝒪\nthis✝ : 0 < Module.finrank ℚ K\nthis : 2 < |discr K... | zify; linarith | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 9
} | [
{
"pp": "m p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\nP : Ideal (𝓞 K)\nhP₁ : P.IsPrime\nhP₂ : P.LiesOver 𝒑\ninst✝ : NeZero m\nhK : IsCyclotomicExtension {m} ℚ K\nhm : ¬p ∣ m\nζ : 𝓞 K := ⋯.toInteger\nh₁ : ¬p ∣ exponent ζ\nh₂ :\n Irreducible ↑((primesOverSpanEquivM... | rw [h₃] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | {
"line": 149,
"column": 4
} | {
"line": 152,
"column": 49
} | [
{
"pp": "case h.refine_2\nn : ℕ\ninst✝⁴ : NeZero n\nK : Type u_1\ninst✝³ : Field K\ninst✝² : NumberField K\nhK : IsCyclotomicExtension {n} ℚ K\np : ℕ\nhp : Fact (Nat.Prime p)\nP : Ideal (𝓞 K)\ninst✝¹ : P.IsMaximal\ninst✝ : P.LiesOver (span {↑p})\nhn✝ : p.Coprime n\nthis : IsGalois ℚ K\nhn : ¬p ∣ n\n⊢ orderOf (... | rw [Nat.card_image_equiv, SetLike.coe_sort_coe, Ideal.card_stabilizer_eq (span {(p : ℤ)})
(by simp [hp.out.ne_zero]), inertiaDegIn_eq_of_not_dvd p K hn,
ramificationIdxIn_eq_of_not_dvd p K hn, one_mul, ← orderOf_injective _ Units.coeHom_injective,
Units.coeHom_apply, ZMod.coe_unitOfCoprime] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 312,
"column": 25
} | {
"line": 312,
"column": 36
} | [
{
"pp": "n m p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : IsCyclotomicExtension {n} ℚ K\nhn : n = p ^ (k + 1) * m\nhm : ¬p ∣ m\nthis✝¹ : IsAbelianGalois ℚ K\nthis✝ : NeZero m\nthis : NeZero n\n⊢ 𝒑 ≠ ⊥",
"usedConstants": [
"Submodule.span_eq_bot._si... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.ExistsRamified | {
"line": 83,
"column": 64
} | {
"line": 83,
"column": 75
} | [
{
"pp": "K : Type u_1\n𝒪 : Type u_2\ninst✝⁵ : Field K\ninst✝⁴ : NumberField K\ninst✝³ : CommRing 𝒪\ninst✝² : Algebra 𝒪 K\ninst✝¹ : IsIntegralClosure 𝒪 ℤ K\ninst✝ : IsGalois ℚ K\nH : 1 < Module.finrank ℚ K\nthis✝⁶ : IsDomain 𝒪\nthis✝⁵ : IsDedekindDomain 𝒪\nthis✝⁴ : IsFractionRing 𝒪 K\nthis✝³ : Module.Fini... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.ProperSpace | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 55
} | [
{
"pp": "case refine_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nε : ℝ\nhε : ε > 0\nk : ℕ\nhk : ↑p ^ (-↑k) < ε\nz : ℤ_[p]\nx✝ : z ∈ Set.univ\n⊢ z.appr k ∈ ↑(Finset.range (p ^ k))",
"usedConstants": [
"Eq.mpr",
"Finset",
"Nat.instMonoid",
"_private.Mathlib.NumberTheory.Padics.ProperSpace.0.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.ProperSpace | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 47
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nthis : {y | ‖y‖ ≤ 1} ∈ 𝓝 0\n⊢ IsCompact {y | ‖y‖ ≤ 1}",
"usedConstants": [
"_private.Mathlib.NumberTheory.Padics.ProperSpace.0.Padic.instProperSpace._simp_1",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Nonarchimedean | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 13
} | [
{
"pp": "case h.a.h\nα : Type u_1\nG : Type u_2\ninst✝³ : CommGroup G\ninst✝² : UniformSpace G\ninst✝¹ : IsUniformGroup G\ninst✝ : NonarchimedeanGroup G\nf : α → G\nhf : Tendsto f cofinite (𝓝 1)\nU : Set G\nhU : U ∈ 𝓝 1\nV : OpenSubgroup G\nhV : ↑V ⊆ U\nt : Finset α\nht : Disjoint t (Set.Finite.toFinset ⋯)\ni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Nonarchimedean | {
"line": 76,
"column": 4
} | {
"line": 76,
"column": 15
} | [
{
"pp": "G : Type u_2\ninst✝³ : CommGroup G\ninst✝² : UniformSpace G\ninst✝¹ : IsUniformGroup G\ninst✝ : NonarchimedeanGroup G\nf : ℕ → G\nhf : Tendsto (fun n ↦ f (n + 1) / f n) atTop (𝓝 1)\ns : Set G\nhs : s ∈ 𝓝 1\nt : OpenSubgroup G\nht : ↑t ⊆ s\n⊢ ∃ N, ∀ (b : ℕ), N ≤ b → f (b + 1) / f b ∈ t",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Nonarchimedean | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 25
} | [
{
"pp": "case a.inr\nG : Type u_2\ninst✝³ : CommGroup G\ninst✝² : UniformSpace G\ninst✝¹ : IsUniformGroup G\ninst✝ : NonarchimedeanGroup G\nf : ℕ → G\nhf : Tendsto (fun n ↦ f (n + 1) / f n) atTop (𝓝 1)\ns : Set G\nhs : s ∈ 𝓝 1\nt : OpenSubgroup G\nht : ↑t ⊆ s\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → f (b + 1) / f b ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Nonarchimedean | {
"line": 86,
"column": 17
} | {
"line": 86,
"column": 28
} | [
{
"pp": "case succ\nG : Type u_2\ninst✝³ : CommGroup G\ninst✝² : UniformSpace G\ninst✝¹ : IsUniformGroup G\ninst✝ : NonarchimedeanGroup G\nf : ℕ → G\nhf : Tendsto (fun n ↦ f (n + 1) / f n) atTop (𝓝 1)\ns : Set G\nhs : s ∈ 𝓝 1\nt : OpenSubgroup G\nht : ↑t ⊆ s\nN : ℕ\nhN : ∀ (b : ℕ), N ≤ b → f (b + 1) / f b ∈ t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.House | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 31
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ns : Finset K\n⊢ house (∏ x ∈ s, x) ≤ ∏ x ∈ s, house x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.instLE",
"Real",
"Complex.commRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.House | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 35
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nα : K\ni : ℕ\n⊢ house (α ^ i) ≤ house α ^ i",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.instLE",
"Real",
"congrArg",
"Complex.instN... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.House | {
"line": 132,
"column": 38
} | {
"line": 134,
"column": 12
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : DecidableEq (K →+* ℂ)\n⊢ 0 ≤ c K",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"mul_nonneg",
"NormedCommRing.toSeminormedCommRing",
"_private.Mathlib.NumberTheory.NumberField.... | by
rw [c]
positivity | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Ostrowski | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 81
} | [
{
"pp": "f g : AbsoluteValue ℚ ℝ\n⊢ (∃ c, 0 < c ∧ ∀ (n : ℕ), f ↑n ^ c = g ↑n) ↔ ∃ c, 0 < c ∧ (fun x ↦ f x ^ c) = ⇑g",
"usedConstants": [
"Real.instPow",
"Real.partialOrder",
"Real",
"Real.instZero",
"Rat",
"Real.instLT",
"Exists",
"Real.semiring",
"Nat.c... | refine ⟨fun ⟨c, hc, h⟩ ↦ ⟨c, hc, ?_⟩, fun ⟨c, hc, h⟩ ↦ ⟨c, hc, (congrFun h ·)⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Ostrowski | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 60
} | [
{
"pp": "case refine_2\nf : AbsoluteValue ℚ ℝ\nhf_nontriv : f.IsNontrivial\nbdd : ∀ (n : ℕ), f ↑n ≤ 1\np : ℕ\nhfp : 0 < f ↑p ∧ f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nhprime : Nat.Prime p\nhprime_fact : Fact (Nat.Prime p)\nt : ℝ\nh : 0 < t ∧ f ↑p = ↑p ^ (-t)\nq : ℕ\nx✝ : (fun p ↦ ∃ (h : Fact (N... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Ostrowski | {
"line": 275,
"column": 20
} | {
"line": 275,
"column": 31
} | [
{
"pp": "f g : AbsoluteValue ℚ ℝ\nx y : ℚ\n⊢ |↑(x + y)| ≤ |↑x| + |↑y|",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"Real.lattice",
"DivisionRing.toRatCast",
"Real.instRCLike",
"abs",
"congrArg",
"Real.instRatCast",
"Rat",
"Part... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.AddChar | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 53
} | [
{
"pp": "p : ℕ\ninst✝⁴ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝³ : NormedRing R\ninst✝² : Algebra ℤ_[p] R\ninst✝¹ : IsBoundedSMul ℤ_[p] R\ninst✝ : IsUltrametricDist R\nκ : AddChar ℤ_[p] R\nhκ : Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑κ\nn : ℕ\n⊢ Δ_[1]^[n] (⇑{ toFun := ⇑κ, continuous... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 42
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : ℕ\nx : ℤ_[p]\nf : ℤ_[p] → ℤ_[p] := fun x ↦ Polynomial.eval x (ascPochhammer ℤ_[p] k)\nhC : ↑k.factorial ≠ 0\nhf : ContinuousAt f x\nδ : ℝ\nhδp : δ > 0\nhδ : ∀ ⦃x_1 : ℤ_[p]⦄, dist x_1 x < δ → dist (f x_1) (f x) < ‖↑k.factorial‖\nn : ℕ\nhn' : dist x ↑n < δ\n⊢ ∃ n, ‖f x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 81,
"column": 33
} | {
"line": 81,
"column": 44
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nk : ℕ\n⊢ 0 < ‖↑k.factorial‖",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"Padic.instCharZero",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 82
} | [
{
"pp": "M : Type u_1\nG : Type u_2\ninst✝¹ : AddCommMonoidWithOne M\ninst✝ : AddCommGroup G\nf : M → G\nn R : ℕ\nhR : 1 ≤ R\naux : Δ_[1]^[n + R] f 0 = R.choose (R - 1 + 1) • Δ_[1]^[n + R] f 0\n⊢ ∑ k ∈ range (R + 1), R.choose k • Δ_[1]^[n + k] f 0 = Δ_[1]^[n] f ↑R",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.AddChar | {
"line": 106,
"column": 31
} | {
"line": 106,
"column": 42
} | [
{
"pp": "p : ℕ\ninst✝⁵ : Fact (Nat.Prime p)\nR : Type u_1\ninst✝⁴ : NormedRing R\ninst✝³ : Algebra ℤ_[p] R\ninst✝² : IsBoundedSMul ℤ_[p] R\ninst✝¹ : IsUltrametricDist R\ninst✝ : CompleteSpace R\nx✝ : { κ // Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑κ }\nκ : AddChar ℤ_[p] R\nhκ : Contin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 193,
"column": 6
} | {
"line": 193,
"column": 38
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : Module ℤ_[p] E\ninst✝¹ : IsBoundedSMul ℤ_[p] E\ninst✝ : IsUltrametricDist E\nf : C(ℤ_[p], E)\ns t : ℕ\nhst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s\nn i : ℕ\nx✝ : i ∈ range (n + 1)\n⊢ ‖↑(-1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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