module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 59,
"column": 4
} | {
"line": 60,
"column": 11
} | {
"line": 60,
"column": 12
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : SlashInvariantFormClass F 𝒢 k\ninst✝ : 𝒢.IsFiniteRelIndex ℋ\nh : GL (Fin 2) ℝ\nhh : h ∈ ℋ\nthis : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)\n⊢ (let this := Fintype.ofFinite (↥ℋ... | [
"𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : SlashInvariantFormClass F 𝒢 k\ninst✝ : 𝒢.IsFiniteRelIndex ℋ\nh : GL (Fin 2) ℝ\nhh : h ∈ ℋ\nthis : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)\n⊢ ∑ x, quotientFunc f (⟨h, hh⟩⁻¹ • x) = ∑ q, qu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 68,
"column": 4
} | {
"line": 70,
"column": 53
} | {
"line": 70,
"column": 54
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝³ : FunLike F ℍ ℂ\nk : ℤ\ninst✝² : SlashInvariantFormClass F 𝒢 k\ninst✝¹ : 𝒢.IsFiniteRelIndex ℋ\ninst✝ : ℋ.HasDetPlusMinusOne\nh : GL (Fin 2) ℝ\nhh : h ∈ ℋ\nthis : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)\n⊢ (l... | [
"𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝³ : FunLike F ℍ ℂ\nk : ℤ\ninst✝² : SlashInvariantFormClass F 𝒢 k\ninst✝¹ : 𝒢.IsFiniteRelIndex ℋ\ninst✝ : ℋ.HasDetPlusMinusOne\nh : GL (Fin 2) ℝ\nhh : h ∈ ℋ\nthis : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)\n⊢ ∏ x, quotientF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 15
} | {
"line": 90,
"column": 16
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : 𝒢.IsFiniteRelIndex ℋ\ninst✝ : ModularFormClass F 𝒢 k\nγ : GL (Fin 2) ℝ\nh : IsCusp (γ • OnePoint.infty) ℋ\nx✝¹ : ↥ℋ\nr : GL (Fin 2) ℝ\nhr : r ∈ ℋ\nx✝ : ⟦⟨r, hr⟩⟧ ∈ Finset.univ\n⊢ IsCusp (γ • OnePoint.infty) (... | [
"𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : 𝒢.IsFiniteRelIndex ℋ\ninst✝ : ModularFormClass F 𝒢 k\nγ : GL (Fin 2) ℝ\nh : IsCusp (γ • OnePoint.infty) ℋ\nx✝¹ : ↥ℋ\nr : GL (Fin 2) ℝ\nhr : r ∈ ℋ\nx✝ : ⟦⟨r, hr⟩⟧ ∈ Finset.univ\n⊢ IsCusp (γ • OnePoint.infty) (ConjAct.toCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 15
} | {
"line": 104,
"column": 16
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : 𝒢.IsFiniteRelIndex ℋ\ninst✝ : CuspFormClass F 𝒢 k\nγ : GL (Fin 2) ℝ\nh : IsCusp (γ • OnePoint.infty) ℋ\nthis : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)\nx✝¹ : ↥ℋ\nr : GL (Fin ... | [
"𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝² : FunLike F ℍ ℂ\nk : ℤ\ninst✝¹ : 𝒢.IsFiniteRelIndex ℋ\ninst✝ : CuspFormClass F 𝒢 k\nγ : GL (Fin 2) ℝ\nh : IsCusp (γ • OnePoint.infty) ℋ\nthis : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)\nx✝¹ : ↥ℋ\nr : GL (Fin 2) ℝ\nhr : r... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 15
} | {
"line": 120,
"column": 16
} | [
{
"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\nγ : GL (Fin 2) ℝ\nh : IsCusp (γ • OnePoint.infty) ℋ\nthis : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.sub... | [
"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\nγ : GL (Fin 2) ℝ\nh : IsCusp (γ • OnePoint.infty) ℋ\nthis : Fintype (↥ℋ ⧸ 𝒢.subgroupOf ℋ) := Fintype.ofFinite (↥ℋ ⧸ 𝒢.subgroupOf ℋ)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 41
} | {
"line": 127,
"column": 42
} | [
{
"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 : ∃ i ∈ Finset.univ, quotientFunc f i = 0\n⊢ ⇑f = 0",
"ppTerm": "?m.84",
"assigned": false,
"usedConstants":... | [
"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 : ∃ i ∈ Finset.univ, quotientFunc f i = 0\n⊢ ⇑f = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 136,
"column": 4
} | {
"line": 137,
"column": 11
} | {
"line": 137,
"column": 12
} | [
{
"pp": "case refine_2\n𝒢 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⊢ (ModularForm.norm ℋ f) τ = 0 τ",
"ppTerm": "?refine_2",
"assigned": true,
... | [
"case refine_2\n𝒢 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⊢ ∃ a ∈ ℋ, (⇑f ∣[k] a⁻¹) τ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 137,
"column": 19
} | {
"line": 137,
"column": 30
} | {
"line": 137,
"column": 31
} | [
{
"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",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
... | [
"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⊢ f τ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 66
} | {
"line": 158,
"column": 67
} | [
{
"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... | [
"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.SpecialLinearGroup.mapGL ℝ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 47,
"column": 53
} | {
"line": 47,
"column": 64
} | {
"line": 47,
"column": 65
} | [
{
"pp": "R : Type u_1\nn : ℕ\ninst✝ : CommRing R\nx y p : R\nh : p ∣ x - y\n⊢ p ∣ y - x",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nn : ℕ\ninst✝ : CommRing R\nx y p : R\nh : p ∣ x - y\n⊢ p ∣ y - x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfiniteAdeleRing | {
"line": 124,
"column": 60
} | {
"line": 124,
"column": 92
} | {
"line": 124,
"column": 93
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K∞\nv : InfinitePlace K\nhv : ¬IsUnit (x v)\n⊢ ‖x v‖ ^ v.mult = 0",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr"... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : K∞\nv : InfinitePlace K\nhv : ¬IsUnit (x v)\n⊢ x v = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.LFunction | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 15
} | {
"line": 123,
"column": 16
} | [
{
"pp": "Γ : Subgroup (GL (Fin 2) ℝ)\ninst✝² : Γ.IsArithmetic\nk : ℤ\nhk : 0 < k\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nf : F\ns : ℂ\ninst✝ : ModularFormClass F Γ k\nr : ℝ\nhpos : 0 < s.re\nhs : r + 1 < s.re\nhΛ : Λ hk f s = mellin (fun t ↦ (weakFEPair hk f).f t - (weakFEPair hk f).f₀) s\nhcoeff : (fun n ↦ (Pow... | [
"Γ : Subgroup (GL (Fin 2) ℝ)\ninst✝² : Γ.IsArithmetic\nk : ℤ\nhk : 0 < k\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nf : F\ns : ℂ\ninst✝ : ModularFormClass F Γ k\nr : ℝ\nhpos : 0 < s.re\nhs : r + 1 < s.re\nhΛ : Λ hk f s = mellin (fun t ↦ (weakFEPair hk f).f t - (weakFEPair hk f).f₀) s\nhcoeff : (fun n ↦ (PowerSeries.coe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.LFunction | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 15
} | {
"line": 131,
"column": 16
} | [
{
"pp": "case refine_2\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝² : Γ.IsArithmetic\nk : ℤ\nhk : 0 < k\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nf : F\ns : ℂ\ninst✝ : ModularFormClass F Γ k\nhs : ↑k + 1 < s.re\n⊢ (fun n ↦ (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f)) =O[atTop] fun n ↦ ↑n ^ ↑k",
"ppTerm": "?refine_2",
... | [
"case refine_2\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝² : Γ.IsArithmetic\nk : ℤ\nhk : 0 < k\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nf : F\ns : ℂ\ninst✝ : ModularFormClass F Γ k\nhs : ↑k + 1 < s.re\n⊢ (fun n ↦ (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f)) =O[atTop] fun n ↦ ↑n ^ k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.InfinitePlace | {
"line": 404,
"column": 4
} | {
"line": 404,
"column": 45
} | {
"line": 405,
"column": 6
} | [
{
"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‖",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Norm.norm... | [
"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⊢ ↑w ((algebraMap (WithAbs ↑v) (WithAbs ↑w)) x).ofAbs = ↑v x.ofAbs"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ModularForms.LFunction | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 15
} | {
"line": 187,
"column": 16
} | [
{
"pp": "case refine_2\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝² : Γ.IsArithmetic\nk : ℤ\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nf : F\ns : ℂ\ninst✝ : CuspFormClass F Γ k\nhk : 0 < k\nhs : ↑k / 2 + 1 < s.re\n⊢ (fun n ↦ (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f)) =O[atTop] fun n ↦ ↑n ^ (↑k / 2)",
"ppTerm": "?refi... | [
"case refine_2\nΓ : Subgroup (GL (Fin 2) ℝ)\ninst✝² : Γ.IsArithmetic\nk : ℤ\nF : Type u_1\ninst✝¹ : FunLike F ℍ ℂ\nf : F\ns : ℂ\ninst✝ : CuspFormClass F Γ k\nhk : 0 < k\nhs : ↑k / 2 + 1 < s.re\n⊢ (fun n ↦ (PowerSeries.coeff n) (qExpansion (h Γ) ⇑f)) =O[atTop] fun n ↦ ↑n ^ (↑k / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 277,
"column": 6
} | {
"line": 277,
"column": 67
} | {
"line": 277,
"column": 68
} | [
{
"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 :... | [
"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 : ∀ (i : ι), ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 18
} | {
"line": 261,
"column": 19
} | [
{
"pp": "m : ℤ\neq : (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1))\n⊢ 8 ∣ (2 * m + 1) ^ 2 - 1",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"HSub.hSub",
"id",
... | [
"m : ℤ\neq : (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1))\n⊢ 8 ∣ 4 * (m * (m + 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 266,
"column": 2
} | {
"line": 266,
"column": 18
} | {
"line": 266,
"column": 19
} | [
{
"pp": "m : ℕ\neq : (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1))\n⊢ 8 ∣ (2 * m + 1) ^ 2 - 1",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"HSub... | [
"m : ℕ\neq : (2 * m + 1) ^ 2 - 1 = 4 * (m * (m + 1))\n⊢ 8 ∣ 4 * (m * (m + 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 272,
"column": 28
} | {
"line": 272,
"column": 39
} | {
"line": 272,
"column": 40
} | [
{
"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",
"ppTerm": "?m.95",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℤ\nhx : ¬2 ∣ x\nhxy : 4 ∣ x - y\ni : ℕ\nhx_odd : Odd x\nhxy_even : Even (x - y)\n⊢ Odd y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 295,
"column": 28
} | {
"line": 295,
"column": 39
} | {
"line": 295,
"column": 40
} | [
{
"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",
"ppTerm": "?m.79",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : ℤ\nn : ℕ\nhxy : 4 ∣ x - y\nhx : ¬2 ∣ x\nhx_odd : Odd x\nhxy_even : Even (x - y)\n⊢ Odd y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 306,
"column": 4
} | {
"line": 306,
"column": 39
} | {
"line": 306,
"column": 40
} | [
{
"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... | [
"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 ^ multiplicity 2 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 63
} | {
"line": 307,
"column": 64
} | [
{
"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)",
"ppTerm... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 376,
"column": 2
} | {
"line": 376,
"column": 13
} | {
"line": 376,
"column": 14
} | [
{
"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",
"ppTerm": "?m.59",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 637,
"column": 6
} | {
"line": 637,
"column": 17
} | {
"line": 637,
"column": 18
} | [
{
"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... | [
"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)\ninst✝¹ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 377,
"column": 19
} | {
"line": 377,
"column": 30
} | {
"line": 377,
"column": 31
} | [
{
"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... | [
"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⊢ ∀ (i : InfinitePlace K), i.IsComplex → 0 ≤ normAtComplexPlaces a i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 210,
"column": 17
} | {
"line": 210,
"column": 59
} | {
"line": 210,
"column": 60
} | [
{
"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}",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"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⊢ ¬mixedEmbedding.norm x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 58
} | {
"line": 382,
"column": 59
} | [
{
"pp": "case 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.I... | [
"case 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.IsReal then S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 17
} | {
"line": 167,
"column": 18
} | [
{
"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",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"id",
"Ne",
"Zero.toOfNat0",
"NumberField.Units.IsMaxRa... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nu : Fin (rank K) → (𝓞 K)ˣ\n⊢ ¬regOfFamily u = 0 → IsMaxRank u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 393,
"column": 92
} | {
"line": 411,
"column": 100
} | {
"line": 413,
"column": 0
} | [
{
"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.PolarCoord | {
"line": 429,
"column": 8
} | {
"line": 430,
"column": 15
} | {
"line": 430,
"column": 16
} | [
{
"pp": "case 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 w"... | [
"case 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⊢ 0 ≤ a.1 ⟨w, hw⟩"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 433,
"column": 6
} | {
"line": 433,
"column": 71
} | {
"line": 433,
"column": 72
} | [
{
"pp": "case 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}",
"ppTerm": "?refine... | [
"case 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⊢ ¬a.1 w = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.PolarCoord | {
"line": 433,
"column": 4
} | {
"line": 433,
"column": 83
} | {
"line": 435,
"column": 0
} | [
{
"pp": "case 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}",
"ppTerm": "?refine... | [] | · simpa [Set.mem_ofPred_eq, normAtComplexPlaces_apply_isReal] using (ha₂ w).ne' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 387,
"column": 63
} | {
"line": 387,
"column": 74
} | {
"line": 387,
"column": 75
} | [
{
"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",
"ppTerm": "?m.51",
... | [
"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⊢ (mixedEmbedding K) ((algebraMap (𝓞 K) K) ↑↑u) * ↑a = ↑b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 388,
"column": 60
} | {
"line": 388,
"column": 71
} | {
"line": 388,
"column": 72
} | [
{
"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",
"ppTerm": "?m.85",
"assigned... | [
"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) ↑u) * ↑a = ↑b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 389,
"column": 58
} | {
"line": 389,
"column": 73
} | {
"line": 389,
"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 • ↑a ∈ fundamentalCone K",
"ppTerm": "?m.130... | [
"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⊢ ↑b ∈ fundamentalCone K"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.Ramification | {
"line": 129,
"column": 51
} | {
"line": 129,
"column": 62
} | {
"line": 129,
"column": 63
} | [
{
"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",
"ppTerm": "?m.92",
"assigned": false,
"... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 467,
"column": 6
} | {
"line": 467,
"column": 21
} | {
"line": 467,
"column": 22
} | [
{
"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... | [
"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).toIntermediat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.Ramification | {
"line": 130,
"column": 51
} | {
"line": 130,
"column": 62
} | {
"line": 130,
"column": 63
} | [
{
"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",
"ppTerm": "?m.136",
"assigned": false,
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CMField | {
"line": 575,
"column": 4
} | {
"line": 575,
"column": 44
} | {
"line": 576,
"column": 2
} | [
{
"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.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 39,
"column": 50
} | {
"line": 39,
"column": 66
} | {
"line": 39,
"column": 67
} | [
{
"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",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 527,
"column": 2
} | {
"line": 528,
"column": 69
} | {
"line": 528,
"column": 70
} | [
{
"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",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"P... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\n⊢ ∏ x_1, ↑expMapBasis x x_1 ^ x_1.mult = Real.exp (x w₀) ^ finrank ℚ K"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 55
} | {
"line": 98,
"column": 4
} | [
{
"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) ⋯)",... | [
"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 * C P.leadingCoeff⁻¹).natDegree = orderOf (unitOfCoprime (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 127,
"column": 8
} | {
"line": 127,
"column": 19
} | {
"line": 127,
"column": 20
} | [
{
"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",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 53,
"column": 35
} | {
"line": 53,
"column": 46
} | {
"line": 53,
"column": 47
} | [
{
"pp": "n : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : ζ ^ n = 1\nhζ1 : ζ ≠ 1\nkey : ↑n = ∑ i ∈ range n, (1 - ζ ^ i)\ni : ℕ\nx✝ : i ∈ range n\n⊢ ζ - 1 ∣ ζ ^ i - 1",
"ppTerm": "?m.148",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nhζ : ζ ^ n = 1\nhζ1 : ζ ≠ 1\nkey : ↑n = ∑ i ∈ range n, (1 - ζ ^ i)\ni : ℕ\nx✝ : i ∈ range n\n⊢ ζ - 1 ∣ ζ ^ i - 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 15
} | {
"line": 143,
"column": 16
} | [
{
"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... | [
"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_1} {ζ : A} ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Ideal.Basic | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 37
} | {
"line": 81,
"column": 38
} | [
{
"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... | [
"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\nhζ : IsPrimit... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 670,
"column": 56
} | {
"line": 670,
"column": 72
} | {
"line": 670,
"column": 73
} | [
{
"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",
"ppTerm": "?m.87"... | [
"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⊢ 0 ≤ ↑expMapBasis.symm x w ∧ ↑expMapBasis.symm x w < 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 670,
"column": 81
} | {
"line": 670,
"column": 92
} | {
"line": 670,
"column": 93
} | [
{
"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",
"ppTerm": "?m.89",
"assigned": false,
"usedConstants"... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 13
} | {
"line": 84,
"column": 14
} | [
{
"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",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"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⊢ ((Algebra.norm ℤ) (hζ.toInteger - 1)).natAbs = p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 43
} | {
"line": 91,
"column": 0
} | [
{
"pp": "case e'_5\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})",
"ppTerm": "?e'_5",
"assigned": true,
"usedConstants": [
... | [] | exact (absNorm_span_zeta_sub_one ..).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 127,
"column": 25
} | {
"line": 127,
"column": 36
} | {
"line": 127,
"column": 37
} | [
{
"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 ≠ ⊥",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
... | [
"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 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | {
"line": 124,
"column": 10
} | {
"line": 124,
"column": 21
} | {
"line": 124,
"column": 22
} | [
{
"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... | [
"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 : IsPrimitiveRoot (zeta n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Galois | {
"line": 123,
"column": 4
} | {
"line": 124,
"column": 56
} | {
"line": 125,
"column": 4
} | [
{
"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... | [
"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 : IsPrimitiveRoot (zeta n ... | refine hζ.toInteger_isPrimitiveRoot.idealQuotient_mk
(by simpa using IsMaximal.ne_top inferInstance) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 762,
"column": 6
} | {
"line": 762,
"column": 21
} | {
"line": 762,
"column": 22
} | [
{
"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",
"ppTerm": "?ne... | [
"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⊢ 0 ≤ y w ∧ y w ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 769,
"column": 6
} | {
"line": 769,
"column": 30
} | {
"line": 769,
"column": 31
} | [
{
"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",
"ppTerm": "?... | [
"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 w₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 81
} | {
"line": 205,
"column": 0
} | [
{
"pp": "p : ℕ\nK : Type u_1\ninst✝¹ : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ p\ninst✝ : NeZero p\nη : K\nhη : IsPrimitiveRoot η p\ni : ℕ\nhi : i.Coprime p\nhζη : ζ ^ i = η\n⊢ Associated (hζ.toInteger - 1) (hζ.toInteger ^ i - 1)",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants": [
"IsPri... | [] | exact hζ.toInteger_isPrimitiveRoot.associated_sub_one_pow_sub_one_of_coprime hi | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 13
} | {
"line": 219,
"column": 14
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝ : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ p\nη : 𝓞 K\nhη : η ∈ primitiveRoots p (𝓞 K)\nhη' : IsPrimitiveRoot (↑η) p\n⊢ Associated (hζ.toInteger - 1) (1 - η)",
"ppTerm": "?m.121",
"assigned": false,
"usedConstants": [],
"usedFVars": []... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝ : Field K\nζ : K\nhζ : IsPrimitiveRoot ζ p\nη : 𝓞 K\nhη : η ∈ primitiveRoots p (𝓞 K)\nhη' : IsPrimitiveRoot (↑η) p\n⊢ Associated (hζ.toInteger - 1) (1 - η)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 782,
"column": 38
} | {
"line": 782,
"column": 49
} | {
"line": 782,
"column": 50
} | [
{
"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",
"ppTerm": "?m.96",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 785,
"column": 6
} | {
"line": 785,
"column": 21
} | {
"line": 785,
"column": 22
} | [
{
"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",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [... | [
"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⊢ 0 ≤ y w ∧ y w ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 783,
"column": 4
} | {
"line": 785,
"column": 43
} | {
"line": 787,
"column": 0
} | [
{
"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",
"ppTerm": "?refine_2",
... | [] | 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": 783,
"column": 4
} | {
"line": 785,
"column": 43
} | {
"line": 787,
"column": 0
} | [
{
"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",
"ppTerm": "?refine_2",
... | [] | 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": 791,
"column": 4
} | {
"line": 791,
"column": 21
} | {
"line": 791,
"column": 22
} | [
{
"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}",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NumberField.mixedEmbedding.realSp... | [
"case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx₀ : x = 0\n⊢ 0 ∈ compactSet K"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 794,
"column": 4
} | {
"line": 794,
"column": 41
} | {
"line": 796,
"column": 0
} | [
{
"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",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"NumberField.mixedEmbedding.fundamentalCone.compactSet_eq_union_a... | [] | exact compactSet_eq_union_aux₂ hx₀ hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 830,
"column": 8
} | {
"line": 830,
"column": 32
} | {
"line": 830,
"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",
"ppTerm": "?m.49",
"assigned": true,
"usedConst... | [
"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⊢ (univ.sup' ⋯ fun w ↦ (normAtPlace w) x) ≤ C"
] | norm_eq_sup'_normAtPlace | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 832,
"column": 4
} | {
"line": 833,
"column": 11
} | {
"line": 833,
"column": 12
} | [
{
"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",
"ppTe... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 305,
"column": 4
} | {
"line": 305,
"column": 15
} | {
"line": 305,
"column": 16
} | [
{
"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... | [
"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 ↑((primesOverSpanEquivMonicFac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 9
} | {
"line": 323,
"column": 2
} | [
{
"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... | [
"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 ↑((primesOverSpanEquivMonicFactorsM... | rw [h₃] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.ExistsRamified | {
"line": 90,
"column": 64
} | {
"line": 90,
"column": 75
} | {
"line": 90,
"column": 76
} | [
{
"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... | [
"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.Finite ℤ 𝒪\nthi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.House | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 31
} | {
"line": 56,
"column": 32
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ns : Finset K\n⊢ house (∏ x ∈ s, x) ≤ ∏ x ∈ s, house x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.instLE... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\ns : Finset K\n⊢ ‖∏ x ∈ s, (canonicalEmbedding K) x‖ ≤ ∏ x ∈ s, ‖(canonicalEmbedding K) x‖"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.House | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 35
} | {
"line": 62,
"column": 36
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nα : K\ni : ℕ\n⊢ house (α ^ i) ≤ house α ^ i",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"RingHom.instRingHomClass",
"Real.instLE",
"... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nα : K\ni : ℕ\n⊢ ‖(canonicalEmbedding K) α ^ i‖ ≤ ‖(canonicalEmbedding K) α‖ ^ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.House | {
"line": 133,
"column": 38
} | {
"line": 135,
"column": 12
} | {
"line": 137,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : NumberField K\ninst✝ : DecidableEq (K →+* ℂ)\n⊢ 0 ≤ c K",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"mul_nonneg",
"NormedCommRing.toSeminormedCommRing",
... | [] | by
rw [c]
positivity | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Ostrowski | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 81
} | {
"line": 98,
"column": 2
} | [
{
"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",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real.partialOrder",
"Real",
"Real.instZero",
"Rat",
"Real.instLT",
... | [
"f g : AbsoluteValue ℚ ℝ\nx✝ : ∃ c, 0 < c ∧ ∀ (n : ℕ), f ↑n ^ c = g ↑n\nc : ℝ\nhc : 0 < c\nh : ∀ (n : ℕ), f ↑n ^ c = g ↑n\n⊢ (fun x ↦ f x ^ c) = ⇑g"
] | 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": 202,
"column": 15
} | {
"line": 202,
"column": 51
} | {
"line": 202,
"column": 52
} | [
{
"pp": "case hbc\nf : AbsoluteValue ℚ ℝ\nbdd : ∀ (n : ℕ), f ↑n ≤ 1\np : ℕ\nhp0 : 0 < f ↑p\nhp1 : f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nm : ℕ\nhpm : ¬p ∣ m\nhm : f ↑m < 1\nM : ℝ := ⋯\nhM : M = max (f ↑p) (f ↑m)\nk : ℕ := ⋯\nhk : k = ⌈logb M (1 / 2)⌉₊ + 1\na b : ℤ\nbezout : a * ↑p ^ k + b * ↑m... | [
"case hbc\nf : AbsoluteValue ℚ ℝ\nbdd : ∀ (n : ℕ), f ↑n ≤ 1\np : ℕ\nhp0 : 0 < f ↑p\nhp1 : f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nm : ℕ\nhpm : ¬p ∣ m\nhm : f ↑m < 1\nM : ℝ := max (f ↑p) (f ↑m)\nhM : M = max (f ↑p) (f ↑m)\nk : ℕ := ⌈logb M (1 / 2)⌉₊ + 1\nhk : k = ⌈logb M (1 / 2)⌉₊ + 1\na b : ℤ\nbezo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Ostrowski | {
"line": 202,
"column": 15
} | {
"line": 202,
"column": 51
} | {
"line": 202,
"column": 52
} | [
{
"pp": "case hbc\nf : AbsoluteValue ℚ ℝ\nbdd : ∀ (n : ℕ), f ↑n ≤ 1\np : ℕ\nhp0 : 0 < f ↑p\nhp1 : f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nm : ℕ\nhpm : ¬p ∣ m\nhm : f ↑m < 1\nM : ℝ := ⋯\nhM : M = max (f ↑p) (f ↑m)\nk : ℕ := ⋯\nhk : k = ⌈logb M (1 / 2)⌉₊ + 1\na b : ℤ\nbezout : a * ↑p ^ k + b * ↑m... | [
"case hbc\nf : AbsoluteValue ℚ ℝ\nbdd : ∀ (n : ℕ), f ↑n ≤ 1\np : ℕ\nhp0 : 0 < f ↑p\nhp1 : f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nm : ℕ\nhpm : ¬p ∣ m\nhm : f ↑m < 1\nM : ℝ := max (f ↑p) (f ↑m)\nhM : M = max (f ↑p) (f ↑m)\nk : ℕ := ⌈logb M (1 / 2)⌉₊ + 1\nhk : k = ⌈logb M (1 / 2)⌉₊ + 1\na b : ℤ\nbezo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.ProperSpace | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 55
} | {
"line": 51,
"column": 56
} | [
{
"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))",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset",
"Nat.instMonoid",
"_p... | [
"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 < p ^ k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Ostrowski | {
"line": 215,
"column": 4
} | {
"line": 215,
"column": 15
} | {
"line": 215,
"column": 16
} | [
{
"pp": "f : AbsoluteValue ℚ ℝ\np : ℕ\nhp0 : 0 < f ↑p\nhp1 : f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nhp : 1 < ↑p\n⊢ f ↑p = ↑p ^ (- -logb (↑p) (f ↑p))",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.partialOrder",
"Rea... | [
"f : AbsoluteValue ℚ ℝ\np : ℕ\nhp0 : 0 < f ↑p\nhp1 : f ↑p < 1\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nhp : 1 < ↑p\n⊢ f ↑p = ↑p ^ logb (↑p) (f ↑p)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Nonarchimedean | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 13
} | {
"line": 61,
"column": 14
} | [
{
"pp": "case 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 : α... | [
"case 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 : α\nhi : i ∈ t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Ostrowski | {
"line": 239,
"column": 4
} | {
"line": 239,
"column": 56
} | {
"line": 239,
"column": 57
} | [
{
"pp": "case refine_2\nf : AbsoluteValue ℚ ℝ\nhf_nontriv : f.IsNontrivial\nbdd : ∀ (n : ℕ), f ↑n ≤ 1\np : ℕ\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nhp0 : 0 < f ↑p\nhp1 : f ↑p < 1\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nt : ℝ\nht : 0 < t\nhpt : f ↑p = ↑p ^ (-t)\nq : ℕ\nx✝ : (fun p ↦ ∃ (h : Fact (N... | [
"case refine_2\nf : AbsoluteValue ℚ ℝ\nhf_nontriv : f.IsNontrivial\nbdd : ∀ (n : ℕ), f ↑n ≤ 1\np : ℕ\nhmin : ∀ (m : ℕ), 0 < f ↑m ∧ f ↑m < 1 → p ≤ m\nhp0 : 0 < f ↑p\nhp1 : f ↑p < 1\nhp : Nat.Prime p\nthis : Fact (Nat.Prime p)\nt : ℝ\nht : 0 < t\nhpt : f ↑p = ↑p ^ (-t)\nq : ℕ\nx✝ : (fun p ↦ ∃ (h : Fact (Nat.Prime p))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Ostrowski | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 13
} | {
"line": 336,
"column": 14
} | [
{
"pp": "case inr.inr\nf : AbsoluteValue ℚ ℝ\nn₀ : ℕ\nhn₀ : 1 < n₀\nh : f ↑n₀ ≤ 1\nn : ℕ\nh_ineq1 : ∀ {m : ℕ}, 1 ≤ m → f ↑m ≤ ↑n₀ * (logb ↑n₀ ↑m + 1)\nh₀ : n ≠ 0\nh_ineq2 : ∀ (k : ℕ), 0 < k → f ↑n ≤ (↑n₀ * (logb ↑n₀ ↑n + 1)) ^ (↑k)⁻¹ * ↑k ^ (↑k)⁻¹\nh₁ : n ≠ 1\nthis : 0 < logb ↑n₀ ↑n\n⊢ Tendsto (fun b ↦ (↑n₀ * (... | [
"case inr.inr\nf : AbsoluteValue ℚ ℝ\nn₀ : ℕ\nhn₀ : 1 < n₀\nh : f ↑n₀ ≤ 1\nn : ℕ\nh_ineq1 : ∀ {m : ℕ}, 1 ≤ m → f ↑m ≤ ↑n₀ * (logb ↑n₀ ↑m + 1)\nh₀ : n ≠ 0\nh_ineq2 : ∀ (k : ℕ), 0 < k → f ↑n ≤ (↑n₀ * (logb ↑n₀ ↑n + 1)) ^ (↑k)⁻¹ * ↑k ^ (↑k)⁻¹\nh₁ : n ≠ 1\nthis : 0 < logb ↑n₀ ↑n\n⊢ Tendsto (fun b ↦ (↑n₀ * (logb ↑n₀ ↑n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.AddChar | {
"line": 106,
"column": 31
} | {
"line": 106,
"column": 42
} | {
"line": 106,
"column": 43
} | [
{
"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... | [
"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 : Continuous[_, Pseu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Ostrowski | {
"line": 436,
"column": 2
} | {
"line": 438,
"column": 45
} | {
"line": 440,
"column": 0
} | [
{
"pp": "f : AbsoluteValue ℚ ℝ\nhf_nontriv : f.IsNontrivial\n⊢ f ≈ real ∨ ∃! p, ∃ (x : Fact (Nat.Prime p)), f ≈ padic p",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real.instLE",
"Real",
"Nat.Prime",
"Rat",
"Rat.AbsoluteValue.equ... | [] | by_cases bdd : ∀ n : ℕ, f n ≤ 1
· exact .inr <| equiv_padic_of_bounded hf_nontriv bdd
· exact .inl <| equiv_real_of_unbounded bdd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Ostrowski | {
"line": 436,
"column": 2
} | {
"line": 438,
"column": 45
} | {
"line": 440,
"column": 0
} | [
{
"pp": "f : AbsoluteValue ℚ ℝ\nhf_nontriv : f.IsNontrivial\n⊢ f ≈ real ∨ ∃! p, ∃ (x : Fact (Nat.Prime p)), f ≈ padic p",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real.instLE",
"Real",
"Nat.Prime",
"Rat",
"Rat.AbsoluteValue.equ... | [] | by_cases bdd : ∀ n : ℕ, f n ≤ 1
· exact .inr <| equiv_padic_of_bounded hf_nontriv bdd
· exact .inl <| equiv_real_of_unbounded bdd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 42
} | {
"line": 65,
"column": 43
} | [
{
"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... | [
"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, dist (f ↑n) (f x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 81,
"column": 33
} | {
"line": 81,
"column": 44
} | {
"line": 81,
"column": 45
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nk : ℕ\n⊢ 0 < ‖↑k.factorial‖",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"Real.instZero",
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℤ_[p]\nk : ℕ\n⊢ ¬k.factorial = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 82
} | {
"line": 143,
"column": 6
} | [
{
"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",
"ppTerm": "?m.287",
"assigned": false... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 193,
"column": 6
} | {
"line": 193,
"column": 38
} | {
"line": 193,
"column": 39
} | [
{
"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... | [
"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) ^ (n - i) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 210,
"column": 4
} | {
"line": 210,
"column": 60
} | {
"line": 210,
"column": 61
} | [
{
"pp": "case zero\np : 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 : ℕ\nhk : 0 ≤ s\n⊢ ‖Δ_[1]^[... | [
"case zero\np : 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 : ℕ\nhk : 0 ≤ s\n⊢ ‖Δ_[1]^[n] (⇑f) 0‖ ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 239,
"column": 32
} | {
"line": 239,
"column": 67
} | {
"line": 239,
"column": 68
} | [
{
"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)\nε : ℝ\nhε : ε > 0\nthis✝ : Tendsto (fun s ↦ ‖f‖ / ↑p ^ s) atTop (𝓝 0)\ns : ℕ\nhs : ‖f‖ / ↑p ^ s < ε\nhf : f ≠ 0\nthis : 0 ... | [
"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)\nε : ℝ\nhε : ε > 0\nthis✝ : Tendsto (fun s ↦ ‖f‖ / ↑p ^ s) atTop (𝓝 0)\ns : ℕ\nhs : ‖f‖ / ↑p ^ s < ε\nhf : f ≠ 0\nthis : 0 < ‖f‖ / ↑p ^... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.WithVal | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 58
} | {
"line": 72,
"column": 10
} | [
{
"pp": "case neg\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp0' : 0 < ↑p\nhp0 : 0 < (↑p)⁻¹\nhp1' : 1 < ↑p\nhp1 : (↑p)⁻¹ < 1\nn : ℕ\nhn : Valued.v (↑p ^ n) = exp (-↑n)\nx y : WithVal (Rat.padicValuation p)\nx' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x\nhx : x' = (WithVal.equiv (Rat.padicValuation p)) x\ny' : ℚ ... | [
"case neg\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp0' : 0 < ↑p\nhp0 : 0 < (↑p)⁻¹\nhp1' : 1 < ↑p\nhp1 : (↑p)⁻¹ < 1\nn : ℕ\nhn : Valued.v (↑p ^ n) = exp (-↑n)\nx y : WithVal (Rat.padicValuation p)\nx' : ℚ := (WithVal.equiv (Rat.padicValuation p)) x\nhx : x' = (WithVal.equiv (Rat.padicValuation p)) x\ny' : ℚ := (WithVal.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 42
} | {
"line": 241,
"column": 43
} | [
{
"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)\nε : ℝ\nhε : ε > 0\nthis : Tendsto (fun s ↦ ‖f‖ / ↑p ^ s) atTop (𝓝 0)\ns : ℕ\nhs : ‖f‖ / ↑p ^ s < ε\nt : ℕ\nht : ∀ (x y : ℤ... | [
"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)\nε : ℝ\nhε : ε > 0\nthis : Tendsto (fun s ↦ ‖f‖ / ↑p ^ s) atTop (𝓝 0)\ns : ℕ\nhs : ‖f‖ / ↑p ^ s < ε\nt : ℕ\nht : ∀ (x y : ℤ_[p]), ‖x - ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 59
} | {
"line": 289,
"column": 60
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : Module ℤ_[p] E\ninst✝² : IsBoundedSMul ℤ_[p] E\ninst✝¹ : IsUltrametricDist E\ninst✝ : CompleteSpace E\na : ℕ → E\nha : Tendsto (fun x ↦ ‖a x‖) atTop (𝓝 0)\n⊢ Tendsto (fun x ↦ ‖mahlerTerm (a x) x‖) cofinite (𝓝 0)",
... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : Module ℤ_[p] E\ninst✝² : IsBoundedSMul ℤ_[p] E\ninst✝¹ : IsUltrametricDist E\ninst✝ : CompleteSpace E\na : ℕ → E\nha : Tendsto (fun x ↦ ‖a x‖) atTop (𝓝 0)\n⊢ Tendsto (fun x ↦ ‖a x‖) atTop (𝓝 0)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 305,
"column": 4
} | {
"line": 305,
"column": 39
} | {
"line": 305,
"column": 40
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : Module ℤ_[p] E\ninst✝² : IsBoundedSMul ℤ_[p] E\ninst✝¹ : IsUltrametricDist E\ninst✝ : CompleteSpace E\na : ℕ → E\nha : Tendsto a atTop (𝓝 0)\nm n : ℕ\nhmn : m ≤ n\nh_van : ∀ (i : ℕ), m.choose (i + (n + 1)) = 0\n⊢ Sum... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : Module ℤ_[p] E\ninst✝² : IsBoundedSMul ℤ_[p] E\ninst✝¹ : IsUltrametricDist E\ninst✝ : CompleteSpace E\na : ℕ → E\nha : Tendsto a atTop (𝓝 0)\nm n : ℕ\nhmn : m ≤ n\nh_van : ∀ (i : ℕ), m.choose (i + (n + 1)) = 0\n⊢ Summable fun i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 348,
"column": 2
} | {
"line": 349,
"column": 9
} | {
"line": 349,
"column": 10
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : Module ℤ_[p] E\ninst✝² : IsBoundedSMul ℤ_[p] E\ninst✝¹ : IsUltrametricDist E\ninst✝ : CompleteSpace E\nf : C(ℤ_[p], E)\nthis : HasSum (fun n ↦ mahlerTerm (Δ_[1]^[n] (⇑f) 0) n) (mahlerSeries fun x ↦ Δ_[1]^[x] (⇑f) 0)\n... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : Module ℤ_[p] E\ninst✝² : IsBoundedSMul ℤ_[p] E\ninst✝¹ : IsUltrametricDist E\ninst✝ : CompleteSpace E\nf : C(ℤ_[p], E)\nthis : HasSum (fun n ↦ mahlerTerm (Δ_[1]^[n] (⇑f) 0) n) (mahlerSeries fun x ↦ Δ_[1]^[x] (⇑f) 0)\nn : ℕ\n⊢ f ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.WithVal | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 30
} | {
"line": 98,
"column": 31
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx✝ : ℚ_[p]\n⊢ x✝ ∈ closure (Set.range ⇑((Rat.castHom ℚ_[p]).comp (WithVal.equiv (Rat.padicValuation p)).toRingHom))",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DivisionRing.t... | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx✝ : ℚ_[p]\n⊢ x✝ ∈ closure (Set.range Rat.cast)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.DisjointCover | {
"line": 68,
"column": 25
} | {
"line": 68,
"column": 88
} | {
"line": 68,
"column": 89
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TotallyDisconnectedSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\nU : ι → Opens X\nhU : IsOpenCover U\nn : ℕ\nV : Fin n → Clopens X\nhVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)\nhVun : univ ⊆ ⋃ j, ↑(V j)\nW : Fin n → Clopens X\nhWle... | [
"ι : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TotallyDisconnectedSpace X\ninst✝¹ : T2Space X\ninst✝ : CompactSpace X\nU : ι → Opens X\nhU : IsOpenCover U\nn : ℕ\nV : Fin n → Clopens X\nhVle : ∀ (j : Fin n), ∃ i, ↑(V j) ⊆ ↑(U i)\nhVun : univ ⊆ ⋃ j, ↑(V j)\nW : Fin n → Clopens X\nhWle : W ≤ V\nhW... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.WithVal | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 54
} | {
"line": 173,
"column": 55
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℚ\nh : ‖↑x‖ ≤ 1\n⊢ ¬p ∣ x.den",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\ninst✝ : Fact (Nat.Prime p)\nx : ℚ\nh : ‖↑x‖ ≤ 1\n⊢ ¬p ∣ x.den"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Padics.WithVal | {
"line": 180,
"column": 8
} | {
"line": 180,
"column": 32
} | {
"line": 180,
"column": 32
} | [
{
"pp": "case hp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ IsClosed {a | ‖withValUniformEquiv a‖ ≤ 1 ↔ Valued.v a ≤ 1}",
"ppTerm": "?hp",
"assigned": true,
"usedConstants": [
"Set.ext",
"Norm.norm",
"Eq.mpr",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
... | [
"case hp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ IsClosed {x | Valued.v x ≤ 1 ↔ ‖withValUniformEquiv x‖ ≤ 1}"
] | Set.ext fun _ ↦ Iff.comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.WithVal | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 35
} | {
"line": 184,
"column": 36
} | [
{
"pp": "case hp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ IsClopen {y | ‖y‖ ≤ 1}",
"ppTerm": "?hp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case hp\np : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ IsClopen {y | ‖y‖ ≤ 1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Separation.DisjointCover | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 49
} | {
"line": 98,
"column": 50
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nS : Set (X × X)\ninst✝ : CompactSpace X\nhS : S ∈ 𝓝ˢ (diagonal X)\nU : X → Set X\nhUo : ∀ (x : X), IsOpen[inst✝¹] (U x)\nhUx : ∀ (x : X), x ∈ U x\nhUp : ∀ (x : X), U x ×ˢ U x ⊆ S\nt : Finset X\nht : univ ⊆ ⋃ i ∈ t, U i\n⊢ ⋃ i, U ↑i = univ",
"ppTerm": "?m.... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\nS : Set (X × X)\ninst✝ : CompactSpace X\nhS : S ∈ 𝓝ˢ (diagonal X)\nU : X → Set X\nhUo : ∀ (x : X), IsOpen[inst✝¹] (U x)\nhUx : ∀ (x : X), x ∈ U x\nhUp : ∀ (x : X), U x ×ˢ U x ⊆ S\nt : Finset X\nht : univ ⊆ ⋃ i ∈ t, U i\n⊢ univ ⊆ ⋃ x ∈ t, U x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.ProdApproximation | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 13
} | {
"line": 110,
"column": 14
} | [
{
"pp": "X : Type u_5\nY : Type u_6\nR : Type u_7\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : IsTopologicalRing R\ninst✝³ : CompactSpace X\ninst✝² : T2Space X\ninst✝¹ : CompactSpace Y\ninst✝ : TotallyDisconnectedSpace X\nthis✝¹ : UniformS... | [
"X : Type u_5\nY : Type u_6\nR : Type u_7\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : CommRing R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : IsTopologicalRing R\ninst✝³ : CompactSpace X\ninst✝² : T2Space X\ninst✝¹ : CompactSpace Y\ninst✝ : TotallyDisconnectedSpace X\nthis✝¹ : UniformSpace R := Is... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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