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.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 63,
"column": 67
} | {
"line": 63,
"column": 75
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : Set X\nhs : MeasurableSet s\nP : Finset (Set X)\nhP₁ : ∀ t ∈ P, t ⊆ s\nhP₂ : (↑P).PairwiseDisjoint id\nQ : Finpartition (P.sup id) := Finpartit... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 63,
"column": 67
} | {
"line": 63,
"column": 75
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : Set X\nhs : MeasurableSet s\nP : Finset (Set X)\nhP₁ : ∀ t ∈ P, t ⊆ s\nhP₂ : (↑P).PairwiseDisjoint id\nQ : Finpartition (P.sup id) := Finpartit... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 72,
"column": 8
} | {
"line": 72,
"column": 16
} | [
{
"pp": "case hP.inl\nX : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : Set X\nhs : MeasurableSet s\nP : Finset (Set X)\nhP₁ : ∀ t ∈ P, t ⊆ s\nhP₂ : (↑P).PairwiseDisjoint id\nQ : Finpartition (P.sup id)... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 72,
"column": 8
} | {
"line": 72,
"column": 16
} | [
{
"pp": "case hP.inl\nX : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : Set X\nhs : MeasurableSet s\nP : Finset (Set X)\nhP₁ : ∀ t ∈ P, t ⊆ s\nhP₂ : (↑P).PairwiseDisjoint id\nQ : Finpartition (P.sup id)... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 72,
"column": 8
} | {
"line": 72,
"column": 16
} | [
{
"pp": "case hP.inl\nX : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : Set X\nhs : MeasurableSet s\nP : Finset (Set X)\nhP₁ : ∀ t ∈ P, t ⊆ s\nhP₂ : (↑P).PairwiseDisjoint id\nQ : Finpartition (P.sup id)... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 12
} | [
{
"pp": "case pos\nX : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\nE : Set X\nhE : MeasurableSet E\nhE' : ⟨E, hE⟩ = ⊥\n⊢ ‖↑μ E‖ₑ ≤ μ.variation E",
"usedConstants": [
"MeasureTheory.VectorMeasure... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 12
} | [
{
"pp": "case pos\nX : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\nE : Set X\nhE : MeasurableSet E\nhE' : ⟨E, hE⟩ = ⊥\n⊢ ‖↑μ E‖ₑ ≤ μ.variation E",
"usedConstants": [
"MeasureTheory.VectorMeasure... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 12
} | [
{
"pp": "case pos\nX : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\nE : Set X\nhE : MeasurableSet E\nhE' : ⟨E, hE⟩ = ⊥\n⊢ ‖↑μ E‖ₑ ≤ μ.variation E",
"usedConstants": [
"MeasureTheory.VectorMeasure... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 95,
"column": 27
} | {
"line": 95,
"column": 35
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : Set X\nhs : ↑μ.ennrealVariation s = 0\nhsm : MeasurableSet s\nthis : ‖↑μ s‖ₑ ≤ 0\n⊢ ↑μ s = 0",
"usedConstants": [
"ENNReal.instCanoni... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 95,
"column": 27
} | {
"line": 95,
"column": 35
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : Set X\nhs : ↑μ.ennrealVariation s = 0\nhsm : MeasurableSet s\nthis : ‖↑μ s‖ₑ ≤ 0\n⊢ ↑μ s = 0",
"usedConstants": [
"ENNReal.instCanoni... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 95,
"column": 27
} | {
"line": 95,
"column": 35
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\ns : Set X\nhs : ↑μ.ennrealVariation s = 0\nhsm : MeasurableSet s\nthis : ‖↑μ s‖ₑ ≤ 0\n⊢ ↑μ s = 0",
"usedConstants": [
"ENNReal.instCanoni... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 101,
"column": 2
} | {
"line": 114,
"column": 91
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\nm : Measure X\nh : ∀ (E : Set X), MeasurableSet E → ‖↑μ E‖ₑ ≤ m E\n⊢ μ.variation ≤ m",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
... | refine Measure.le_intro fun s hs _ => ?_
simp only [variation_apply, preVariation, ennrealToMeasure_apply hs, ennrealPreVariation_apply,
preVariationFun, hs, dite_true, iSup_le_iff]
intro i
calc
∑ x ∈ i.parts, ‖μ x‖ₑ ≤ ∑ x ∈ i.parts, m x := Finset.sum_le_sum (fun s hs => h s s.property)
_ = m (i.parts... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 101,
"column": 2
} | {
"line": 114,
"column": 91
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝² : TopologicalSpace V\ninst✝¹ : ENormedAddCommMonoid V\ninst✝ : T2Space V\nμ : VectorMeasure X V\nm : Measure X\nh : ∀ (E : Set X), MeasurableSet E → ‖↑μ E‖ₑ ≤ m E\n⊢ μ.variation ≤ m",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
... | refine Measure.le_intro fun s hs _ => ?_
simp only [variation_apply, preVariation, ennrealToMeasure_apply hs, ennrealPreVariation_apply,
preVariationFun, hs, dite_true, iSup_le_iff]
intro i
calc
∑ x ∈ i.parts, ‖μ x‖ₑ ≤ ∑ x ∈ i.parts, m x := Finset.sum_le_sum (fun s hs => h s s.property)
_ = m (i.parts... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Arithmetic.Presburger.Definability | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 25
} | [
{
"pp": "α : Type u_1\ns : Set (α → ℕ)\nA : Set ℕ\ninst✝ : Finite α\nhs : IsSemilinearSet s\n⊢ A.Definable presburger s",
"usedConstants": [
"Pi.addCommMonoid",
"congrArg",
"Finset",
"Set.sUnion",
"Membership.mem",
"Exists",
"Eq.mp",
"IsLinearSet",
"And"... | isSemilinearSet_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Definability | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 14
} | [
{
"pp": "case func.inl.add\nα : Type u_1\nA : Set ℕ\ninst✝ : Fintype α\nts : Fin 2 → presburger[[↑A]].Term α\nk : Fin 2 → ℕ\nu : Fin 2 → α → ℕ\nih : ∀ (a : Fin 2) (v : α → ℕ), Term.realize v (ts a) = k a + u a ⬝ᵥ v\n⊢ ∃ k u, ∀ (v : α → ℕ), (Structure.funMap (Sum.inl presburgerFunc.add) fun i ↦ Term.realize v (t... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Definability | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 84
} | [
{
"pp": "case equal\nα : Type u_1\nA : Set ℕ\ninst✝ : Finite α\nn : ℕ\nthis : Fintype α\nn✝ : ℕ\nt₁ t₂ : presburger[[↑A]].Term (α ⊕ Fin n✝)\nk₁ : ℕ\nu₁ : α ⊕ Fin n✝ → ℕ\nht₁ : ∀ (v : α ⊕ Fin n✝ → ℕ), Term.realize v t₁ = k₁ + u₁ ⬝ᵥ v\nk₂ : ℕ\nu₂ : α ⊕ Fin n✝ → ℕ\nht₂ : ∀ (v : α ⊕ Fin n✝ → ℕ), Term.realize v t₂ =... | convert! Nat.isSemilinearSet_setOf_mulVec_eq ![k₁] ![k₂] (.of ![u₁]) (.of ![u₂]) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.ModelTheory.Equivalence | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 30
} | [
{
"pp": "L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ ψ : L.BoundedFormula α n\nh : φ ⇔[T] ψ\n⊢ ∀ (M : T.ModelType) (v : α → ↑M) (xs : Fin n → ↑M), (∼φ).Realize v xs ↔ (∼ψ).Realize v xs",
"usedConstants": [
"Eq.mpr",
"FirstOrder.Language.Theory.ModelType",
"FirstOrder.Language.Theory.M... | BoundedFormula.realize_not | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 364,
"column": 10
} | {
"line": 364,
"column": 19
} | [
{
"pp": "case h.e'_5\nM : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : IsCancelAdd M\na : M\nt : Finset M\nih : ∀ m < t.card, ∀ (a : M) (t : Finset M), t.card = m → IsProperSemilinearSet (a +ᵥ ↑(closure ↑t))\nt' : Finset M\nht' : t' ⊆ t\nf : M → ℕ\ni : M\nhi : i ∈ t'\nhfi : 0 < f i\nheq : ∑ x ∈ t', f x • x = ∑ x... | ite_smul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 83,
"column": 52
} | {
"line": 83,
"column": 64
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : AddCommMonoid M\ninst✝³ : PartialOrder M\ninst✝² : WellQuasiOrderedLE M\ninst✝¹ : IsOrderedCancelAddMonoid M\ninst✝ : CanonicallyOrderedAdd M\ns : Set M\nhs : IsSlice s\nhs' : s.Nonempty\nf : M → AddSubmonoid M :=\n fun x ↦ if hx : x ∈ s then { carrier := {y | x + y ∈ s}, add_me... | simp_all [f] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.ModelTheory.Complexity | {
"line": 229,
"column": 8
} | {
"line": 229,
"column": 18
} | [
{
"pp": "case ex\nL : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nn : ℕ\ninst✝ : Nonempty M\nψ : L.BoundedFormula α n\nv : α → M\nn✝ : ℕ\nφ✝ : L.BoundedFormula α (n✝ + 1)\nh✝ : φ✝.IsPrenex\nih :\n ∀ {φ : L.BoundedFormula α (n✝ + 1)},\n φ.IsQF → ∀ {xs : Fin (n✝ + 1) → M}, (φ.toPrenexImpRight φ... | realize_ex | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Complexity | {
"line": 252,
"column": 8
} | {
"line": 252,
"column": 18
} | [
{
"pp": "case all\nL : Language\nM : Type w\ninst✝¹ : L.Structure M\nα : Type u'\nn : ℕ\ninst✝ : Nonempty M\nφ : L.BoundedFormula α n\nv : α → M\nn✝ : ℕ\nφ✝ : L.BoundedFormula α (n✝ + 1)\nh✝ : φ✝.IsPrenex\nih :\n ∀ {xs : Fin (n✝ + 1) → M} {ψ : L.BoundedFormula α (n✝ + 1)},\n ψ.IsPrenex → ((φ✝.toPrenexImp ψ)... | realize_ex | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 412,
"column": 6
} | {
"line": 423,
"column": 41
} | [
{
"pp": "case inr\nS : Finset (Set ℕ)\nhS : ∀ t ∈ S, IsProperLinearSet t\na b : ℕ\nht : LinearIndepOn ℕ id ↑{b}\n⊢ ∃ k, ∃ p > 0, ∀ x ≥ k, x ∈ a +ᵥ ↑(closure ↑{b}) ↔ x + p ∈ a +ᵥ ↑(closure ↑{b})",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Finset.coe_singleton",
"HMul.hMul",
"in... | · have hb : b ≠ 0 := by simpa [ne_comm] using ht.zero_notMem_image
rw [Nat.ne_zero_iff_zero_lt] at hb
refine ⟨a, b, hb, fun x hx => ?_⟩
simp only [Finset.coe_singleton, mem_vadd_set, SetLike.mem_coe,
AddSubmonoid.mem_closure_singleton, smul_eq_mul, vadd_eq_add, exists_exists_eq_and]
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 435,
"column": 63
} | {
"line": 436,
"column": 14
} | [
{
"pp": "ι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\ni : ↑hs.basisSet\n⊢ hs.basis i = toRatVec ↑i",
"usedConstants": [
"Rat.addCommMonoid",
"Pi.Function.module",
"Semiring.toModule",
"Pi.addCommMonoid",
"congrArg",
"_private.Mathlib.ModelThe... | by
simp [basis] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 449,
"column": 2
} | {
"line": 449,
"column": 25
} | [
{
"pp": "ι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nthis : Fintype ι\nx : ι → ℕ\nhx : x ∈ hs.fundamentalDomain\ni : ↑hs.basisSet\nhi : i ∈ Finset.univ\n⊢ (hs.basis.repr (toRatVec x - toRatVec hs.base)) i • hs.basis i ≤ toRatVec ↑i",
"usedConstants": [
"Rat.addCommMonoid... | rw [← hs.basis_apply i] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.Fraisse | {
"line": 172,
"column": 91
} | {
"line": 173,
"column": 64
} | [
{
"pp": "L : Language\nM : Type w\ninst✝ : L.Structure M\nS : L.Substructure M\nfg : S.FG\n⊢ { α := ↥S, str := inferInstance } ∈ L.age M",
"usedConstants": [
"FirstOrder.Language.Substructure.fg_iff_structure_fg",
"FirstOrder.Language.Substructure.inducedStructure",
"Membership.mem",
... | by
exact ⟨(Substructure.fg_iff_structure_fg _).1 fg, ⟨subtype _⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Graph | {
"line": 113,
"column": 10
} | {
"line": 113,
"column": 19
} | [
{
"pp": "V : Type u\nS : Language.graph.Structure V\ninst✝ : V ⊨ Theory.simpleGraph\nn : ℕ\nr : Language.graph.Relations n\nxs : Fin 2 → V\n⊢ RelMap graphRel.adj xs ↔ RelMap graphRel.adj xs",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"FirstOrder.Language.graphRel.adj",
"in... | iff_eq_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.CountableDenseLinearOrder | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 23
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : LinearOrder β\ninst✝³ : DenselyOrdered β\ninst✝² : NoMinOrder β\ninst✝¹ : NoMaxOrder β\ninst✝ : Nonempty β\nf : PartialIso α β\na : α\nh : ∃ b, (a, b) ∈ ↑f\n⊢ ∃ b, ∀ p ∈ ↑f, cmp p.1 a = cmp p.2 b",
"usedConstants": []
}
] | obtain ⟨b, hb⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 324,
"column": 6
} | {
"line": 324,
"column": 14
} | [
{
"pp": "case neg\nι : Type u_1\nX : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : X → E\nB : E →L[ℝ] F →L[ℝ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 324,
"column": 6
} | {
"line": 324,
"column": 14
} | [
{
"pp": "case neg\nι : Type u_1\nX : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : X → E\nB : E →L[ℝ] F →L[ℝ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 324,
"column": 6
} | {
"line": 324,
"column": 14
} | [
{
"pp": "case neg\nι : Type u_1\nX : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : X → E\nB : E →L[ℝ] F →L[ℝ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 379,
"column": 6
} | {
"line": 379,
"column": 14
} | [
{
"pp": "case neg\nι : Type u_1\nX : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : X → E\nμ : VectorMeasure ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 379,
"column": 6
} | {
"line": 379,
"column": 14
} | [
{
"pp": "case neg\nι : Type u_1\nX : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : X → E\nμ : VectorMeasure ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 379,
"column": 6
} | {
"line": 379,
"column": 14
} | [
{
"pp": "case neg\nι : Type u_1\nX : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : X → E\nμ : VectorMeasure ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 145,
"column": 8
} | {
"line": 150,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\np : ℕ\nu v : R\nhp : Nat.Prime p\nhvu : v ∣ u\nhpuv : ↑p * u * v ∣ u ^ p\nx : R\nm✝ m : ℕ\ny : R\nhy : (1 + u * x) ^ p = 1 + ↑p * u * (x + v * y)\n⊢ ↑p * (↑p * u) * (↑p * v) ∣ (↑p * u) ^ p",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMo... | rw [mul_pow]
simp only [← mul_assoc]
rw [mul_assoc, mul_assoc, ← mul_assoc u, mul_comm u]
apply mul_dvd_mul _ hpuv
rw [← pow_two]
exact pow_dvd_pow _ hp.two_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 145,
"column": 8
} | {
"line": 150,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\np : ℕ\nu v : R\nhp : Nat.Prime p\nhvu : v ∣ u\nhpuv : ↑p * u * v ∣ u ^ p\nx : R\nm✝ m : ℕ\ny : R\nhy : (1 + u * x) ^ p = 1 + ↑p * u * (x + v * y)\n⊢ ↑p * (↑p * u) * (↑p * v) ∣ (↑p * u) ^ p",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMo... | rw [mul_pow]
simp only [← mul_assoc]
rw [mul_assoc, mul_assoc, ← mul_assoc u, mul_comm u]
apply mul_dvd_mul _ hpuv
rw [← pow_two]
exact pow_dvd_pow _ hp.two_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 178,
"column": 4
} | {
"line": 179,
"column": 29
} | [
{
"pp": "case refine_1\np : ℕ\nhp : Nat.Prime p\nm : ℕ\nhm0 : m ≠ 0\nhpm : m + 2 ≤ p * m\na : ℤ\nha : ¬↑p ∣ a\nn✝ n : ℕ\nthis : Fact (Nat.Prime p)\n⊢ ↑p * ↑p ^ m * ↑p ∣ (↑p ^ m) ^ p",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Dvd.dvd",
"HMul.hMul",
... | · rw [← pow_succ', ← pow_succ, ← pow_mul, mul_comm]
exact pow_dvd_pow _ hpm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.PowerSeries.Derivative | {
"line": 122,
"column": 16
} | {
"line": 122,
"column": 24
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommSemiring R\nn : ℕ\nh✝ : n = 0\n⊢ ↑n + 1 = 1",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"Nat.instAddMonoid",
"AddZeroClass.toAddZero",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.PowerSeries.Derivative | {
"line": 122,
"column": 16
} | {
"line": 122,
"column": 24
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommSemiring R\nn : ℕ\nh✝ : ¬n = 0\n⊢ 0 = 0",
"usedConstants": [
"CommSemiring.toSemiring",
"instMulZeroOneClassOfSemiring",
"eq_self",
"of_eq_true",
"Zero.toOfNat0",
"MulZeroOneClass.toMulZeroClass",
"OfNat.ofNat",
"Eq... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.PowerSeries.Exp | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 59
} | [
{
"pp": "case h.e_a\nA : Type u_4\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\na b : A\nn x : ℕ\nhx : x ∈ Finset.range n.succ\n⊢ 1 / ↑x ! * (1 / ↑(n - x)!) = ↑(n.choose x) * (1 / ↑n !)",
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.to... | rw [mul_one_div (↑(n.choose x) : ℚ), one_div_mul_one_div] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.AbelSummation | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nn m : ℕ\nh : n ≤ m\nhf_diff : ∀ t ∈ Set.Icc ↑n ↑m, DifferentiableAt ℝ f t\nhf_int : IntegrableOn (deriv f) (Set.Icc ↑n ↑m) volume\n⊢ ∑ k ∈ Ioc n m, f ↑k * c k =\n f ↑m * ∑ k ∈ Icc 0 m, c k - f ↑n * ∑ k ∈ Icc 0 n, c k -\n ∫ (t : ℝ) in Set... | convert! sum_mul_eq_sub_sub_integral_mul c n.cast_nonneg (Nat.cast_le.mpr h) hf_diff hf_int
all_goals rw [Nat.floor_natCast] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.AbelSummation | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nn m : ℕ\nh : n ≤ m\nhf_diff : ∀ t ∈ Set.Icc ↑n ↑m, DifferentiableAt ℝ f t\nhf_int : IntegrableOn (deriv f) (Set.Icc ↑n ↑m) volume\n⊢ ∑ k ∈ Ioc n m, f ↑k * c k =\n f ↑m * ∑ k ∈ Icc 0 m, c k - f ↑n * ∑ k ∈ Icc 0 n, c k -\n ∫ (t : ℝ) in Set... | convert! sum_mul_eq_sub_sub_integral_mul c n.cast_nonneg (Nat.cast_le.mpr h) hf_diff hf_int
all_goals rw [Nat.floor_natCast] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 65
} | [
{
"pp": "n : ℕ\n⊢ bernoulli n = ∑ i ∈ range (n + 1), (monomial i) (_root_.bernoulli (n - i) * ↑(n.choose i))",
"usedConstants": [
"Eq.mpr",
"Rat.instMul",
"Nat.choose",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"HMul.hMul",
"CommRing.toNo... | rw [← sum_range_reflect, add_succ_sub_one, add_zero, bernoulli] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 109,
"column": 72
} | {
"line": 109,
"column": 93
} | [
{
"pp": "k : ℕ\n⊢ ∑ x ∈ range (k + 1 + 1), (monomial (k + 1 - (x + 1))) (_root_.bernoulli x * ↑((k + 1).choose x) * ↑(k + 1 - x)) =\n (↑k + 1) * ∑ i ∈ range (k + 1), (monomial (k - i)) (_root_.bernoulli i * ↑(k.choose i))",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne... | Nat.add_sub_add_right | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 179,
"column": 2
} | {
"line": 180,
"column": 37
} | [
{
"pp": "n p : ℕ\n⊢ eval (↑n) (bernoulli p.succ) = _root_.bernoulli p.succ + (↑p + 1) * ∑ k ∈ range n, ↑k ^ p",
"usedConstants": [
"Rat.addCommMonoid",
"Rat.instOfNat",
"Rat.instSub",
"Eq.mpr",
"Polynomial.eval",
"eq_add_of_sub_eq'",
"Rat.instMul",
"HMul.hMul"... | apply eq_add_of_sub_eq'
rw [sum_range_pow_eq_bernoulli_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 179,
"column": 2
} | {
"line": 180,
"column": 37
} | [
{
"pp": "n p : ℕ\n⊢ eval (↑n) (bernoulli p.succ) = _root_.bernoulli p.succ + (↑p + 1) * ∑ k ∈ range n, ↑k ^ p",
"usedConstants": [
"Rat.addCommMonoid",
"Rat.instOfNat",
"Rat.instSub",
"Eq.mpr",
"Polynomial.eval",
"eq_add_of_sub_eq'",
"Rat.instMul",
"HMul.hMul"... | apply eq_add_of_sub_eq'
rw [sum_range_pow_eq_bernoulli_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.AbelSummation | {
"line": 373,
"column": 2
} | {
"line": 375,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhf_diff : ∀ t ∈ Set.Ici 0, DifferentiableAt ℝ (fun x ↦ ‖f x‖) t\nhf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici 0) volume\nh_bdd : (fun n ↦ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun x ↦ 1\ng : ℝ → ℝ\nhg₁ : (fun t ↦ deriv (fun t ↦... | refine summable_mul_of_bigO_atTop_aux c 0 h_bdd (by rwa [Nat.cast_zero]) (fun n ↦ ?_) hg₁ hg₂
exact_mod_cast sum_mul_eq_sub_integral_mul' _ _ (fun _ ht ↦ hf_diff _ ht.1)
(hf_int.integrableOn_compact_subset Set.Icc_subset_Ici_self isCompact_Icc) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.AbelSummation | {
"line": 373,
"column": 2
} | {
"line": 375,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhf_diff : ∀ t ∈ Set.Ici 0, DifferentiableAt ℝ (fun x ↦ ‖f x‖) t\nhf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici 0) volume\nh_bdd : (fun n ↦ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun x ↦ 1\ng : ℝ → ℝ\nhg₁ : (fun t ↦ deriv (fun t ↦... | refine summable_mul_of_bigO_atTop_aux c 0 h_bdd (by rwa [Nat.cast_zero]) (fun n ↦ ?_) hg₁ hg₂
exact_mod_cast sum_mul_eq_sub_integral_mul' _ _ (fun _ ht ↦ hf_diff _ ht.1)
(hf_int.integrableOn_compact_subset Set.Icc_subset_Ici_self isCompact_Icc) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Bertrand | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 34
} | [
{
"pp": "case refine_2\nx : ℝ\nx_large : 512 ≤ x\nf : ℝ → ℝ := fun x ↦ log x + √(2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)\nhf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn ℝ (Set.Ioi 0.5) f\nthis : ... | div_le_one (by positivity) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Bertrand | {
"line": 112,
"column": 39
} | {
"line": 112,
"column": 52
} | [
{
"pp": "x : ℝ\nx_large : 512 ≤ x\nf : ℝ → ℝ := fun x ↦ log x + √(2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)\nhf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn ℝ (Set.Ioi 0.5) f\nthis : √(2 * 512) = 32... | rw [rpow_two] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 161,
"column": 6
} | {
"line": 161,
"column": 33
} | [
{
"pp": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Field Fq\nx y z : Fq[X]\na : ℤ\nhxy : cardPowDegree (x - y) < a\nhyz : cardPowDegree (y - z) < a\nha : 0 < a\nhxy' : x - y ≠ 0\nhyz' : y - z ≠ 0\nhxz' : x - z ≠ 0\nthis : 1 ≤ ↑(Fintype.card Fq)\n⊢ (x - z).natDegree ≤ (x - y).natDegree ∨ (x - z).natD... | natDegree_le_iff_degree_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 161,
"column": 34
} | {
"line": 161,
"column": 61
} | [
{
"pp": "case neg\nFq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Field Fq\nx y z : Fq[X]\na : ℤ\nhxy : cardPowDegree (x - y) < a\nhyz : cardPowDegree (y - z) < a\nha : 0 < a\nhxy' : x - y ≠ 0\nhyz' : y - z ≠ 0\nhxz' : x - z ≠ 0\nthis : 1 ≤ ↑(Fintype.card Fq)\n⊢ (x - z).degree ≤ ↑(x - y).natDegree ∨ (x - z).natDeg... | natDegree_le_iff_degree_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Chebyshev | {
"line": 337,
"column": 24
} | {
"line": 337,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : AddCommMonoid R\nf : ℕ → R\nx : ℝ\nhx : 0 ≤ x\nk : ℕ\nx✝¹ : k ∈ Icc 1 ⌊log x / log 2⌋₊\np : ℕ\nx✝ : ((0 < p ∧ p ≤ ⌊x⌋₊) ∧ Nat.Prime p) ∧ p ≤ ⌊x ^ (↑k)⁻¹⌋₊\n⊢ (0 < p ∧ p ≤ ⌊x ^ (1 / ↑k)⌋₊) ∧ Nat.Prime p",
"usedConstants": [
"Real.instIsOrderedRing",
"Real.instPow",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.Chebyshev | {
"line": 337,
"column": 24
} | {
"line": 337,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : AddCommMonoid R\nf : ℕ → R\nx : ℝ\nhx : 0 ≤ x\nk : ℕ\nx✝¹ : k ∈ Icc 1 ⌊log x / log 2⌋₊\np : ℕ\nx✝ : ((0 < p ∧ p ≤ ⌊x⌋₊) ∧ Nat.Prime p) ∧ p ≤ ⌊x ^ (↑k)⁻¹⌋₊\n⊢ (0 < p ∧ p ≤ ⌊x ^ (1 / ↑k)⌋₊) ∧ Nat.Prime p",
"usedConstants": [
"Real.instIsOrderedRing",
"Real.instPow",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Chebyshev | {
"line": 337,
"column": 24
} | {
"line": 337,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝ : AddCommMonoid R\nf : ℕ → R\nx : ℝ\nhx : 0 ≤ x\nk : ℕ\nx✝¹ : k ∈ Icc 1 ⌊log x / log 2⌋₊\np : ℕ\nx✝ : ((0 < p ∧ p ≤ ⌊x⌋₊) ∧ Nat.Prime p) ∧ p ≤ ⌊x ^ (↑k)⁻¹⌋₊\n⊢ (0 < p ∧ p ≤ ⌊x ^ (1 / ↑k)⌋₊) ∧ Nat.Prime p",
"usedConstants": [
"Real.instIsOrderedRing",
"Real.instPow",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Chebyshev | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 10
} | [
{
"pp": "x : ℝ\nhx : 0 ≤ x\nx✝² : ℕ\nhk : x✝² ∈ Icc 1 ⌊log x / log 2⌋₊\nx✝¹ : ℕ\nx✝ : x✝¹ ∈ {p ∈ Ioc 0 ⌊x ^ (1 / ↑x✝²)⌋₊ | Nat.Prime p}\n⊢ Nat.Prime x✝¹",
"usedConstants": [
"Real.instIsOrderedRing",
"Real.instPow",
"Real.partialOrder",
"Real",
"Finset.mem_filter._simp_1",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.Chebyshev | {
"line": 346,
"column": 65
} | {
"line": 350,
"column": 10
} | [
{
"pp": "x : ℝ\nhx : 0 ≤ x\n⊢ ψ x = ∑ n ∈ Icc 1 ⌊log x / log 2⌋₊, θ (x ^ (1 / ↑n))",
"usedConstants": [
"Int.instAddCommGroup",
"ArithmeticFunction.vonMangoldt",
"Real.instIsOrderedRing",
"Not.intro",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.to... | by
simp_rw [psi, vonMangoldt_apply, ← sum_filter, sum_PrimePow_eq_sum_sum _ hx]
apply sum_congr rfl fun _ hk ↦ sum_congr rfl fun _ _ ↦ ?_
rw [Prime.pow_minFac _ (by linarith [mem_Icc.mp hk])]
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Bernoulli | {
"line": 466,
"column": 4
} | {
"line": 467,
"column": 90
} | [
{
"pp": "case pos\nk p : ℕ\nhk : k > 0\ninst✝ : Fact (Nat.Prime p)\nhdvd : p - 1 ∣ 2 * k\n⊢ ∑ q ∈ vonStaudtPrimes k, 1 / ↑q = vonStaudtIndicator (2 * k) p / ↑p + ∑ q ∈ (vonStaudtPrimes k).erase p, 1 / ↑q",
"usedConstants": [
"Iff.mpr",
"Nat.Prime",
"Dvd.dvd",
"HMul.hMul",
"Nat.... | have hp_mem : p ∈ vonStaudtPrimes k := Finset.mem_filter.mpr
⟨Finset.mem_range.mpr (by have := Nat.le_of_dvd (by lia) hdvd; lia), Fact.out, hdvd⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Chebyshev | {
"line": 381,
"column": 12
} | {
"line": 381,
"column": 20
} | [
{
"pp": "case h.hbc.hyz.hdb\nx : ℝ\nhx✝ : 1 ≤ x\nhx : 2 ≤ x\ni : ℕ\nhi : i ∈ Icc 2 ⌊log x / log 2⌋₊\n⊢ 2 ≤ ↑i",
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real",
"instHDiv",
"FloorRing.toFloorSemiring",
"Real.instRCLike",
"Real.instZeroLEOneC... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.Chebyshev | {
"line": 392,
"column": 4
} | {
"line": 392,
"column": 29
} | [
{
"pp": "case hbc.h\nx : ℝ\nhx✝ : 1 ≤ x\nhx : 2 ≤ x\n⊢ 1 ≤ log x / log 2",
"usedConstants": [
"Iff.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"instHDiv",
"GroupWithZero.toDivInvMonoid",
"MulZeroClass.toMul",
"PartialOrder.toP... | apply one_le_div _ |>.mpr | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 29
} | [
{
"pp": "case h\nR : Type u_1\nS : Type u_2\ninst✝³ : EuclideanDomain R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nabv : AbsoluteValue R ℤ\nI : ↥(Ideal S)⁰\n⊢ ∀ (z : ℤ), (fun a ↦ ∃ b ∈ ↑I, b ≠ 0 ∧ abv ((Algebra.norm R) b) = a) z → 0 ≤ z",
"usedConstants": [
"Int"
]
}
] | rintro _ ⟨b, _, _, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 236,
"column": 67
} | {
"line": 236,
"column": 74
} | [
{
"pp": "case refine_2.refine_1.hab.e_a.h.e_6.h\nR : Type u_1\nS : Type u_2\ninst✝⁷ : EuclideanDomain R\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝³ : DecidableEq ι\ninst✝² : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝¹ : Infinite ... | if_true | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 671,
"column": 4
} | {
"line": 671,
"column": 32
} | [
{
"pp": "case inl\n⊢ bernoulli (2 * 0) + ∑ p ∈ range (2 * 0 + 2) with Nat.Prime p ∧ p - 1 ∣ 2 * 0, 1 / ↑p ∈ Set.range Int.cast",
"usedConstants": [
"Rat.addCommMonoid",
"Rat.instOfNat",
"Int.cast",
"Nat.Prime",
"Dvd.dvd",
"instHDiv",
"HMul.hMul",
"Nat.decidabl... | exact ⟨1, by decide +kernel⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Bernoulli | {
"line": 671,
"column": 4
} | {
"line": 671,
"column": 32
} | [
{
"pp": "case inl\n⊢ bernoulli (2 * 0) + ∑ p ∈ range (2 * 0 + 2) with Nat.Prime p ∧ p - 1 ∣ 2 * 0, 1 / ↑p ∈ Set.range Int.cast",
"usedConstants": [
"Rat.addCommMonoid",
"Rat.instOfNat",
"Int.cast",
"Nat.Prime",
"Dvd.dvd",
"instHDiv",
"HMul.hMul",
"Nat.decidabl... | exact ⟨1, by decide +kernel⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Bernoulli | {
"line": 671,
"column": 4
} | {
"line": 671,
"column": 32
} | [
{
"pp": "case inl\n⊢ bernoulli (2 * 0) + ∑ p ∈ range (2 * 0 + 2) with Nat.Prime p ∧ p - 1 ∣ 2 * 0, 1 / ↑p ∈ Set.range Int.cast",
"usedConstants": [
"Rat.addCommMonoid",
"Rat.instOfNat",
"Int.cast",
"Nat.Prime",
"Dvd.dvd",
"instHDiv",
"HMul.hMul",
"Nat.decidabl... | exact ⟨1, by decide +kernel⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 32
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nh : ‖↑r‖ ≤ 1\nn : ℤ := modPart p r\n⊢ ↑p ∣ r.num - n * ↑r.den",
"usedConstants": [
"PadicInt.norm_sub_modPart_aux"
]
}
] | exact norm_sub_modPart_aux r h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 303,
"column": 5
} | {
"line": 305,
"column": 43
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nr : ℚ\nx : ℤ_[p]\n⊢ ∀ (x : ℤ_[p]) (a b : ℕ), x - ↑a ∈ Ideal.span {↑p} → x - ↑b ∈ Ideal.span {↑p} → ↑a = ↑b",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Semiring.toModule",
"NormedRing.toRing",
"ZMod.commRing",
... | by
rw [← maximalIdeal_eq_span_p]
exact zmod_congr_of_sub_mem_max_ideal | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 370,
"column": 6
} | {
"line": 370,
"column": 30
} | [
{
"pp": "case pos\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (x : ℤ_[p]), x.appr n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : x - ↑(x.appr n) = 0\n⊢ x.appr n < p ^ (n + 1)",
"usedConstants": [
"Nat.instMonoid",
"lt_trans",
"instOfNatNat",
"Monoid.toPow",
"instH... | apply lt_trans (ih _) hp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 370,
"column": 6
} | {
"line": 370,
"column": 30
} | [
{
"pp": "case pos\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (x : ℤ_[p]), x.appr n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : x - ↑(x.appr n) = 0\n⊢ x.appr n < p ^ (n + 1)",
"usedConstants": [
"Nat.instMonoid",
"lt_trans",
"instOfNatNat",
"Monoid.toPow",
"instH... | apply lt_trans (ih _) hp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 370,
"column": 6
} | {
"line": 370,
"column": 30
} | [
{
"pp": "case pos\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nn : ℕ\nih : ∀ (x : ℤ_[p]), x.appr n < p ^ n\nx : ℤ_[p]\nhp : p ^ n < p ^ (n + 1)\nh : x - ↑(x.appr n) = 0\n⊢ x.appr n < p ^ (n + 1)",
"usedConstants": [
"Nat.instMonoid",
"lt_trans",
"instOfNatNat",
"Monoid.toPow",
"instH... | apply lt_trans (ih _) hp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.CyclotomicCharacter | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 57
} | [
{
"pp": "L : Type u\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\nn : ℕ\ninst✝ : NeZero n\ng : L ≃+* L\n⊢ ∃ m, ∀ t ∈ rootsOfUnity n L, g ↑t = ↑(t ^ m)",
"usedConstants": [
"Units.val",
"Eq.mpr",
"rootsOfUnity.integer_power_of_ringEquiv",
"congrArg",
"CommSemiring.toSemiring",
... | simpa using rootsOfUnity.integer_power_of_ringEquiv n g | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.NumberTheory.Cyclotomic.CyclotomicCharacter | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 57
} | [
{
"pp": "L : Type u\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\nn : ℕ\ninst✝ : NeZero n\ng : L ≃+* L\n⊢ ∃ m, ∀ t ∈ rootsOfUnity n L, g ↑t = ↑(t ^ m)",
"usedConstants": [
"Units.val",
"Eq.mpr",
"rootsOfUnity.integer_power_of_ringEquiv",
"congrArg",
"CommSemiring.toSemiring",
... | simpa using rootsOfUnity.integer_power_of_ringEquiv n g | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.CyclotomicCharacter | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 57
} | [
{
"pp": "L : Type u\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\nn : ℕ\ninst✝ : NeZero n\ng : L ≃+* L\n⊢ ∃ m, ∀ t ∈ rootsOfUnity n L, g ↑t = ↑(t ^ m)",
"usedConstants": [
"Units.val",
"Eq.mpr",
"rootsOfUnity.integer_power_of_ringEquiv",
"congrArg",
"CommSemiring.toSemiring",
... | simpa using rootsOfUnity.integer_power_of_ringEquiv n g | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.Gal | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 57
} | [
{
"pp": "n : ℕ\ninst✝⁵ : NeZero n\nK : Type u_1\ninst✝⁴ : Field K\nL : Type u_2\nμ : L\ninst✝³ : CommRing L\ninst✝² : IsDomain L\nhμ✝ : IsPrimitiveRoot μ n\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension {n} K L\nh✝ : Irreducible (cyclotomic n K)\nhζ : IsPrimitiveRoot (zeta n K L) n\nh : ∃ i < n, zeta n K ... | rw [PowerBasis.equivOfMinpoly_gen, hμ.powerBasis_gen K] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Cyclotomic.Discriminant | {
"line": 83,
"column": 6
} | {
"line": 84,
"column": 59
} | [
{
"pp": "case e_a.inl.succ\nK : Type u\nL : Type v\nζ : L\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\nmf : Module.Finite K L\nse : Algebra.IsSeparable K L\nhp : Fact (Nat.Prime 2)\nn✝ : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ ((n✝ + 1).succ + 1)} K L\nhζ : IsPrimitiveRoot ζ (2 ^ ((n✝ + 1).succ + 1)... | · simp_rw [_root_.pow_succ', (even_two.mul_right _).neg_one_pow,
((even_two.mul_right _).mul_right _).neg_one_pow] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Cyclotomic.Discriminant | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 87
} | [
{
"pp": "case pos\nK : Type u\nL : Type v\nζ : L\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nhp : Fact (Nat.Prime 2)\nhcycl✝ : IsCyclotomicExtension {2 ^ (0 + 1)} K L\nhcycl : IsCyclotomicExtension {2} K L\nhζ✝ : IsPrimitiveRoot ζ (2 ^ (0 + 1))\nhζ : IsPrimitiveRoot ζ 2\nhirr : Irreducible (cyclot... | simp_rw [hζ.eq_neg_one_of_two_right, show (-1 : L) = algebraMap K L (-1) by simp] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 575,
"column": 4
} | {
"line": 575,
"column": 55
} | [
{
"pp": "case inl\nd : ℕ\nb : ℤ√↑d\nhb : b.Nonneg\nx y : ℕ\nha : { re := ↑x, im := ↑y }.Nonneg\n⊢ ({ re := ↑x, im := ↑y } + b).Nonneg",
"usedConstants": [
"Zsqrtd.nonneg_cases"
]
}
] | rcases nonneg_cases hb with ⟨z, w, rfl | rfl | rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 575,
"column": 4
} | {
"line": 575,
"column": 55
} | [
{
"pp": "case inr.inl\nd : ℕ\nb : ℤ√↑d\nhb : b.Nonneg\nx y : ℕ\nha : { re := ↑x, im := -↑y }.Nonneg\n⊢ ({ re := ↑x, im := -↑y } + b).Nonneg",
"usedConstants": [
"Zsqrtd.nonneg_cases"
]
}
] | rcases nonneg_cases hb with ⟨z, w, rfl | rfl | rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 575,
"column": 4
} | {
"line": 575,
"column": 55
} | [
{
"pp": "case inr.inr\nd : ℕ\nb : ℤ√↑d\nhb : b.Nonneg\nx y : ℕ\nha : { re := -↑x, im := ↑y }.Nonneg\n⊢ ({ re := -↑x, im := ↑y } + b).Nonneg",
"usedConstants": [
"Zsqrtd.nonneg_cases"
]
}
] | rcases nonneg_cases hb with ⟨z, w, rfl | rfl | rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 666,
"column": 37
} | {
"line": 666,
"column": 50
} | [
{
"pp": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha : a.Nonneg\nx y : ℕ\nx✝ : { re := ↑x, im := ↑y }.Nonneg\n⊢ (↑↑n * { re := ↑x, im := ↑y }).Nonneg",
"usedConstants": [
"Zsqrtd.instMul",
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Int",
"AddGroupWithOne.toIntCast",... | rw [smul_val] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 668,
"column": 6
} | {
"line": 668,
"column": 19
} | [
{
"pp": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha✝ : a.Nonneg\nx y : ℕ\nha : { re := ↑x, im := -↑y }.Nonneg\n⊢ (↑↑n * { re := ↑x, im := -↑y }).Nonneg",
"usedConstants": [
"Zsqrtd.instMul",
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Int.instNegInt",
"Int",
"... | rw [smul_val] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 670,
"column": 6
} | {
"line": 670,
"column": 19
} | [
{
"pp": "d : ℕ\na : ℤ√↑d\nn : ℕ\nha✝ : a.Nonneg\nx y : ℕ\nha : { re := -↑x, im := ↑y }.Nonneg\n⊢ (↑↑n * { re := -↑x, im := ↑y }).Nonneg",
"usedConstants": [
"Zsqrtd.instMul",
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Int.instNegInt",
"Int",
"... | rw [smul_val] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 686,
"column": 8
} | {
"line": 686,
"column": 18
} | [
{
"pp": "d x y : ℕ\na : ℤ√↑d\nha : a.Nonneg\n⊢ { re := ↑x, im := ↑y } * a = ↑x * a + sqrtd * (↑y * a)",
"usedConstants": [
"Zsqrtd.instMul",
"Int.cast",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"Zsqrtd.sqrtd",
"id",
"AddMo... | decompose, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 679,
"column": 26
} | {
"line": 679,
"column": 34
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nij : i ≤ j\nj2n : j ≤ 2 * n\nh : xn a1 i ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)\nnpos : n = 0\n⊢ i = j",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"HMul.hM... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 679,
"column": 26
} | {
"line": 679,
"column": 34
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nij : i ≤ j\nj2n : j ≤ 2 * n\nh : xn a1 i ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)\nnpos : n = 0\n⊢ i = j",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"HMul.hM... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 679,
"column": 26
} | {
"line": 679,
"column": 34
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ni j n : ℕ\nij : i ≤ j\nj2n : j ≤ 2 * n\nh : xn a1 i ≡ xn a1 j [MOD xn a1 n]\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j = 2)\nnpos : n = 0\n⊢ i = j",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"HMul.hM... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 423,
"column": 4
} | {
"line": 424,
"column": 28
} | [
{
"pp": "ξ : ℝ\nu v : ℤ\nhv : 2 ≤ v\nhcop : IsCoprime u v\nleft✝ : v = 1 → -(1 / 2) < ξ - ↑u\nhv₀ : 0 < ↑v\nhv₀' : 0 < 2 * ↑v - 1\nhv₁ : 0 < 2 * v - 1\nhu₀ : 0 ≤ u - ⌊ξ⌋ * v\nh : ↑u * (2 * ↑v - 1) < 1 + ξ * (↑v * (2 * ↑v - 1))\n⊢ (u - ⌊ξ⌋ * v) * (2 * v - 1) < v * (2 * v - 1) + 1",
"usedConstants": [
"... | rw [← sub_lt_iff_lt_add, ← mul_assoc, ← sub_mul, ← add_lt_add_iff_right (v * (2 * v - 1) : ℝ),
add_comm (1 : ℝ)] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Dioph | {
"line": 435,
"column": 26
} | {
"line": 438,
"column": 57
} | [
{
"pp": "n : ℕ\nS : Set (Vector3 ℕ n.succ)\nd : Dioph S\nv : Fin2 n → ℕ\nx : ℕ\n⊢ x ::ₒ v ∈ {v | (v ∘ none :: some) ∈ S} ↔ (x :: v) ∈ S",
"usedConstants": [
"Eq.mpr",
"Option.elim'",
"congrArg",
"HEq.refl",
"Iff.rfl",
"setOf",
"Fin2.fz",
"Fin2.casesOn",
... | by
dsimp
rw [show Option.elim' x v ∘ cons none some = x :: v from
funext fun s => by rcases s with a | b <;> rfl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 149,
"column": 31
} | {
"line": 149,
"column": 56
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nr : ℝ\nthis :\n ∃ c,\n ∀ᶠ (x : ℝ) in 𝓝[>] 0,\n ‖((fun x ↦ P.g x - P.g₀) ∘ fun x ↦ x⁻¹) x‖ ≤ c * ‖((fun x ↦ x ^ (-(r + P.k))) ∘ fun x ↦ x⁻¹) x‖\n⊢ ∃ c, ∀ᶠ (x : ℝ) in 𝓝[>] 0, ‖P.f x - (P.ε * ↑(x ^ (-P.k))) ... | eventually_nhdsWithin_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 182,
"column": 88
} | {
"line": 186,
"column": 28
} | [
{
"pp": "a : ℝ\nha : 0 ≤ a\nthis :\n ∃ p,\n 0 < p ∧\n (fun t ↦\n rexp (-π * (a ^ 2 + 1) * t) / (1 - rexp (-π * t)) ^ 2 +\n a * rexp (-π * a ^ 2 * t) / (1 - rexp (-π * t))) =O[atTop]\n fun t ↦ rexp (-p * t)\n⊢ ∃ p, 0 < p ∧ F_nat 1 a =O[atTop] fun t ↦ rexp (-p * t)",
"usedC... | by
let ⟨p, hp, hp'⟩ := this
refine ⟨p, hp, (Eventually.isBigO ?_).trans hp'⟩
filter_upwards [eventually_gt_atTop 0] with t ht
exact F_nat_one_le ha ht | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 109,
"column": 2
} | {
"line": 115,
"column": 41
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\np : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhp : ∀ (i : ι), 0 ≤ p i\nhs : 0 < s.re\nh_sum : Summable fun i ↦ ‖a i‖ / p i ^ s.re\nhs' : s ≠ 0\na' : ι → ℂ := fun i ↦ if p i = 0 then 0 else a i\nhp' : ∀ (i : ι), a' i = 0 ∨ 0 < p i\nthis : ∀ (i : ι) (t : ℝ), (if p i = 0 then ... | · refine h_sum.of_norm_bounded (fun i ↦ ?_)
simp only [a']
split_ifs
· simp only [norm_zero, zero_div]
positivity
· have := hp i
rw [norm_of_nonneg (by positivity)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 412,
"column": 2
} | {
"line": 413,
"column": 64
} | [
{
"pp": "case e_f.h\nz τ : ℂ\nn : ℤ\n⊢ (starRingEnd ℂ) (jacobiTheta₂_term n (-z) τ) = jacobiTheta₂_term n ((starRingEnd ℂ) z) (-(starRingEnd ℂ) τ)",
"usedConstants": [
"Int.cast",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instRingHomClass",
"NonUnitalCommRing.toNonUnitalNon... | simp only [jacobiTheta₂_term, mul_neg, ← exp_conj, map_add, map_neg, map_mul, map_ofNat,
conj_ofReal, conj_I, map_intCast, neg_mul, neg_neg, map_pow] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 86,
"column": 85
} | {
"line": 86,
"column": 92
} | [
{
"pp": "s : ℂ\n⊢ completedHurwitzZetaEven₀ 0 s - (if True then 1 else 0) / s - 1 / (1 - s) =\n completedHurwitzZetaEven₀ 0 s - 1 / s - 1 / (1 - s)",
"usedConstants": [
"Real",
"instHDiv",
"HurwitzZeta.completedHurwitzZetaEven₀",
"instDecidableTrue",
"if_true",
"congrA... | if_true | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 150,
"column": 63
} | {
"line": 150,
"column": 70
} | [
{
"pp": "⊢ (if True then -1 / 2 else 0) = -1 / 2",
"usedConstants": [
"instHDiv",
"instDecidableTrue",
"if_true",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instZero",
"Complex.instDivInvMonoid",
"HDiv.hDiv",
"instOfNatNat",
"Complex.instN... | if_true | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.LSeries.HurwitzZeta | {
"line": 175,
"column": 2
} | {
"line": 178,
"column": 8
} | [
{
"pp": "a : UnitAddCircle\ns : ℂ\nhs : ∀ (n : ℕ), s ≠ 1 - ↑n\n⊢ expZeta a (1 - s) =\n (2 * ↑π) ^ (-s) * Complex.Gamma s *\n (cexp (↑π * I * s / 2) * hurwitzZeta a s + cexp (-↑π * I * s / 2) * hurwitzZeta (-a) s)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
... | have hs' (n : ℕ) : s ≠ -↑n := by
convert! hs (n + 1) using 1
push_cast
ring | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 242,
"column": 74
} | {
"line": 242,
"column": 82
} | [
{
"pp": "x✝¹ : ℝ\nx✝ : x✝¹ ∈ Ioi 1\n⊢ ↑x✝¹ ∈ {s | 1 < s.re}",
"usedConstants": [
"Real",
"Set.Ioi",
"Preorder.toLT",
"Membership.mem",
"Set.mem_Ioi._simp_1",
"Eq.mp",
"id",
"Real.instOne",
"LT.lt",
"One.toOfNat1",
"OfNat.ofNat",
"Set.in... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 242,
"column": 74
} | {
"line": 242,
"column": 82
} | [
{
"pp": "x✝¹ : ℝ\nx✝ : x✝¹ ∈ Ioi 1\n⊢ ↑x✝¹ ∈ {s | 1 < s.re}",
"usedConstants": [
"Real",
"Set.Ioi",
"Preorder.toLT",
"Membership.mem",
"Set.mem_Ioi._simp_1",
"Eq.mp",
"id",
"Real.instOne",
"LT.lt",
"One.toOfNat1",
"OfNat.ofNat",
"Set.in... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.RiemannZeta | {
"line": 242,
"column": 74
} | {
"line": 242,
"column": 82
} | [
{
"pp": "x✝¹ : ℝ\nx✝ : x✝¹ ∈ Ioi 1\n⊢ ↑x✝¹ ∈ {s | 1 < s.re}",
"usedConstants": [
"Real",
"Set.Ioi",
"Preorder.toLT",
"Membership.mem",
"Set.mem_Ioi._simp_1",
"Eq.mp",
"id",
"Real.instOne",
"LT.lt",
"One.toOfNat1",
"OfNat.ofNat",
"Set.in... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 640,
"column": 20
} | {
"line": 640,
"column": 22
} | [
{
"pp": "a : UnitAddCircle\n| fun s ↦ completedHurwitzZetaEven₀ a s - ((if a = 0 then 1 else 0) / s + (1 / (1 - s) + 1 / (s - 1)))",
"usedConstants": [
"Real",
"instHDiv",
"HurwitzZeta.completedHurwitzZetaEven₀",
"AddMonoid.toAddSemigroup",
"AddGroupWithOne.toAddMonoidWithOne",... | s, | Lean.Elab.Tactic.Conv.evalEnter | null |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 656,
"column": 2
} | {
"line": 656,
"column": 93
} | [
{
"pp": "case inl\na b : UnitAddCircle\nz : ℂ\nhz : z ≠ 1\n⊢ DifferentiableAt ℂ (fun s ↦ hurwitzZetaEven a s - hurwitzZetaEven b s) z",
"usedConstants": [
"HurwitzZeta.differentiableAt_hurwitzZetaEven",
"InnerProductSpace.toNormedSpace",
"Complex.instNormedAddCommGroup",
"Complex.ins... | · exact (differentiableAt_hurwitzZetaEven a hz).sub (differentiableAt_hurwitzZetaEven b hz) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 175,
"column": 2
} | {
"line": 176,
"column": 57
} | [
{
"pp": "M N : ℕ\ninst✝ : NeZero N\nhMN : M ∣ N\nχ : DirichletCharacter ℂ M\ns : ℂ\nhs : 1 < s.re\n⊢ L (fun n ↦ ((changeLevel hMN) χ) ↑n) s = L (fun n ↦ χ ↑n) s * ∏ p ∈ N.primeFactors, (1 - χ ↑p * ↑p ^ (-s))",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DirichletCh... | rw [prod_eq_tprod_mulIndicator, ← DirichletCharacter.LSeries_eulerProduct_tprod _ hs,
← DirichletCharacter.LSeries_eulerProduct_tprod _ hs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.