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.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 179,
"column": 6
} | {
"line": 179,
"column": 49
} | {
"line": 179,
"column": 50
} | [
{
"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\nm : Measure X\nhs : MeasurableSet s\nh : ∀ (E : Set X), MeasurableSet E → E ⊆ s → ‖μ E‖ₑ ≤ m E\ni : Finpartition ⟨s, ⋯⟩\na : Subtype Mea... | [
"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\nm : Measure X\nhs : MeasurableSet s\nh : ∀ (E : Set X), MeasurableSet E → E ⊆ s → ‖μ E‖ₑ ≤ m E\ni : Finpartition ⟨s, ⋯⟩\na : Subtype MeasurableSet\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 39
} | {
"line": 202,
"column": 6
} | [
{
"pp": "case insert\nX : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝³ : TopologicalSpace V\ninst✝² : ENormedAddCommMonoid V\ninst✝¹ : T2Space V\ninst✝ : ContinuousAdd V\nι : Type u_3\nμ : ι → VectorMeasure X V\ni : ι\ns : Finset ι\nhis : i ∉ s\nih : (∑ i ∈ s, μ i).variation ≤ ∑ i ∈ s, (μ i).variation... | [
"case insert\nX : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝³ : TopologicalSpace V\ninst✝² : ENormedAddCommMonoid V\ninst✝¹ : T2Space V\ninst✝ : ContinuousAdd V\nι : Type u_3\nμ : ι → VectorMeasure X V\ni : ι\ns : Finset ι\nhis : i ∉ s\nih : (∑ i ∈ s, μ i).variation ≤ ∑ i ∈ s, (μ i).variation\n⊢ (μ i + ∑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 690,
"column": 4
} | {
"line": 690,
"column": 56
} | {
"line": 691,
"column": 4
} | [
{
"pp": "case inr.inr.refine_1\n𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\ninst✝³ : PseudoMetricSpace 𝓧\ninst✝² : OpensMeasurableSpace 𝓧\ninst✝¹ : SecondCountableTopology 𝓧\nS : Set (ProbabilityMeasure 𝓧)\ninst✝ : CompleteSpace 𝓧\nhcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)\nhnonem... | [
"case inr.inr.refine_1\n𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\ninst✝³ : PseudoMetricSpace 𝓧\ninst✝² : OpensMeasurableSpace 𝓧\ninst✝¹ : SecondCountableTopology 𝓧\nS : Set (ProbabilityMeasure 𝓧)\ninst✝ : CompleteSpace 𝓧\nhcomp : IsCompact (closure[ProbabilityMeasure.instTopologicalSpace] S)\nhnonempty : Nonemp... | refine Metric.totallyBounded_iff.mpr fun δ δpos ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.VectorMeasure.AddContent | {
"line": 41,
"column": 15
} | {
"line": 41,
"column": 26
} | {
"line": 41,
"column": 27
} | [
{
"pp": "α : Type u_1\nhα : MeasurableSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nm : Set α → E\nhm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s\ninst✝ : IsFiniteMeasure μ\nh'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t\nh''m :... | [
"α : Type u_1\nhα : MeasurableSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nm : Set α → E\nhm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s\ninst✝ : IsFiniteMeasure μ\nh'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t\nh''m : ∀ (s : Set ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.AddContent | {
"line": 45,
"column": 6
} | {
"line": 45,
"column": 32
} | {
"line": 46,
"column": 6
} | [
{
"pp": "α : Type u_1\nhα : MeasurableSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nm : Set α → E\nhm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s\ninst✝ : IsFiniteMeasure μ\nh'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t\nh''m :... | [
"α : Type u_1\nhα : MeasurableSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nm : Set α → E\nhm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s\ninst✝ : IsFiniteMeasure μ\nh'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t\nh''m : ∀ (s : Set ... | simp only [← toReal_enorm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.VectorMeasure.AddContent | {
"line": 57,
"column": 16
} | {
"line": 57,
"column": 27
} | {
"line": 57,
"column": 28
} | [
{
"pp": "case zero\nα : Type u_1\nhα : MeasurableSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nm : Set α → E\nhm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s\ninst✝ : IsFiniteMeasure μ\nh'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + ... | [
"case zero\nα : Type u_1\nhα : MeasurableSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nm : Set α → E\nhm : ∀ (s : Set α), ‖m s‖ₑ ≤ μ s\ninst✝ : IsFiniteMeasure μ\nh'm : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → Disjoint s t → m (s ∪ t) = m s + m t\nh''m : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 13
} | {
"line": 71,
"column": 14
} | [
{
"pp": "case intro\nι : Type u_1\nX : Type u_2\nE : Type u_3\nF : Type u_4\nmX : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\nμ : VectorMeasure X F\nf : X → E\ninst✝ : Finite ι\nt : ι → Set X\nht : ∀ (i : ι), MeasurableSet (t i)\nh't : ∀ (i : ι), μ.IntegrableOn f (t i)\nval✝... | [
"case intro\nι : Type u_1\nX : Type u_2\nE : Type u_3\nF : Type u_4\nmX : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\nμ : VectorMeasure X F\nf : X → E\ninst✝ : Finite ι\nt : ι → Set X\nht : ∀ (i : ι), MeasurableSet (t i)\nh't : ∀ (i : ι), μ.IntegrableOn f (t i)\nval✝ : Fintype ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 89
} | {
"line": 124,
"column": 89
} | [
{
"pp": "X : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\ns t : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G\nB : E ... | [
"X : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\ns t : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G\nB : E →L[ℝ] F →L[ℝ... | integral_add_vectorMeasure (hfs.mono hs inter_subset_left) (hfs.mono hs sdiff_subset) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 59
} | {
"line": 174,
"column": 60
} | [
{
"pp": "case neg\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 : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\ns : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G... | [
"case neg\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 : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\ns : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G\nB : E →L[ℝ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | {
"line": 144,
"column": 6
} | {
"line": 144,
"column": 44
} | {
"line": 144,
"column": 45
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteS... | [
"case pos\nα : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | {
"line": 146,
"column": 36
} | {
"line": 146,
"column": 80
} | {
"line": 146,
"column": 81
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf ... | [
"α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf : α → E\na :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | {
"line": 153,
"column": 50
} | {
"line": 153,
"column": 61
} | {
"line": 153,
"column": 62
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf ... | [
"α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf : α → E\na :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | {
"line": 206,
"column": 6
} | {
"line": 206,
"column": 22
} | {
"line": 206,
"column": 23
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf ... | [
"α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf : α → E\nhf ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 22
} | {
"line": 241,
"column": 23
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf ... | [
"α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf : α → E\nhf ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 342,
"column": 63
} | {
"line": 343,
"column": 53
} | {
"line": 345,
"column": 0
} | [
{
"pp": "X : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\nμ : VectorMeasure X F\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedSpace ℝ G\nB : E →L[ℝ] F →L[ℝ] G\ninst✝¹... | [] | by
rw [integral_indicator s_meas, ← setIntegral_const] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 45
} | {
"line": 161,
"column": 4
} | [
{
"pp": "X : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nmX : MeasurableSpace X\nμ : VectorMeasure X E\nh : μ.semivariation univ = ∞\nt : Set X → Set X\nt_meas : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → MeasurableSet (t s)\nt_subs : ∀ (s : Set X), MeasurableSe... | [
"X : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nmX : MeasurableSpace X\nμ : VectorMeasure X E\nh : μ.semivariation univ = ∞\nt : Set X → Set X\nt_meas : ∀ (s : Set X), MeasurableSet s → μ.semivariation s = ∞ → MeasurableSet (t s)\nt_subs : ∀ (s : Set X), MeasurableSet s → μ.semi... | simp only [sdiff_le_iff, sup_eq_union, u] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 28
} | {
"line": 186,
"column": 29
} | [
{
"pp": "X : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nmX : MeasurableSpace X\nμ : VectorMeasure X E\ns : Set X\n⊢ ‖μ s‖ ≤ ↑μ.bound",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
... | [
"X : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nmX : MeasurableSpace X\nμ : VectorMeasure X E\ns : Set X\n⊢ ‖μ s‖₊ ≤ μ.bound"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Prod | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 23
} | {
"line": 70,
"column": 24
} | [
{
"pp": "case pos\nX : Type u_2\nY : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\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\nμ : V... | [
"case pos\nX : Type u_2\nY : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\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\nμ : VectorMeasure... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Prod | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 47
} | {
"line": 76,
"column": 4
} | [
{
"pp": "X : Type u_2\nY : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeas... | [
"X : Type u_2\nY : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X E\nν :... | rw [map_apply _ (by fun_prop) (hs.prod ht)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.Prod | {
"line": 79,
"column": 14
} | {
"line": 79,
"column": 25
} | {
"line": 79,
"column": 26
} | [
{
"pp": "X : Type u_2\nY : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\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\nμ : VectorMeasu... | [
"X : Type u_2\nY : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\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\nμ : VectorMeasure X E\nν : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Prod | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 53
} | {
"line": 92,
"column": 54
} | [
{
"pp": "case basic\nX : Type u_2\nY : Type u_3\nF : Type u_5\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\ninst✝ : NormedAddCommGroup F\nν : VectorMeasure Y F\ns✝ : Set (X × Y)\ns : Set X\nhs : s ∈ {s | MeasurableSet s}\nt : Set Y\n⊢ StronglyMeasurable fun x ↦ ν (Prod.mk x ⁻¹' (fun x1 x2 ↦ x1 ×ˢ x2) s t)",
... | [
"case basic\nX : Type u_2\nY : Type u_3\nF : Type u_5\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\ninst✝ : NormedAddCommGroup F\nν : VectorMeasure Y F\ns✝ : Set (X × Y)\ns : Set X\nhs : s ∈ {s | MeasurableSet s}\nt : Set Y\n⊢ StronglyMeasurable fun x ↦ s.indicator (fun x ↦ ν t) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Prod | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 70
} | {
"line": 162,
"column": 2
} | [
{
"pp": "X : Type u_2\nY : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\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\nμ : VectorMeasu... | [
"X : Type u_2\nY : Type u_3\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\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\nμ : VectorMeasure X E\nν : ... | apply ext_of_generateFrom _ _ generateFrom_prod.symm isPiSystem_prod | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 442,
"column": 55
} | {
"line": 442,
"column": 66
} | {
"line": 442,
"column": 67
} | [
{
"pp": "X : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\ns : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G\nB : E →L... | [
"X : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\ns : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G\nB : E →L[ℝ] F →L[ℝ] ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Variation.SignedMeasure | {
"line": 77,
"column": 27
} | {
"line": 77,
"column": 42
} | {
"line": 78,
"column": 2
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ : SignedMeasure X\nr : Set X\nhr : MeasurableSet r\ns : Set X\nhs : MeasurableSet s\nhpos : 0 ≤[s] μ\nhneg : μ ≤[sᶜ] 0\nhposPart : μ.toJordanDecomposition.posPart = μ.toMeasureOfZeroLE s hs hpos\nhnegPart : μ.toJordanDecomposition.negPart = μ.toMeasureOfLEZero sᶜ... | [] | by congr; grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 15
} | {
"line": 160,
"column": 16
} | [
{
"pp": "case insert\nM : Type u_1\ninst✝ : AddCommMonoid M\nS : Set (Set M)\na✝¹ : Set M\ns✝ : Set (Set M)\na✝ : a✝¹ ∉ s✝\nhs✝ : s✝.Finite\nih : (∀ s ∈ s✝, IsSemilinearSet s) → IsSemilinearSet (⋃₀ s✝)\nhS' : IsSemilinearSet a✝¹ ∧ ∀ a ∈ s✝, IsSemilinearSet a\n⊢ IsSemilinearSet (⋃₀ insert a✝¹ s✝)",
"ppTerm":... | [
"case insert\nM : Type u_1\ninst✝ : AddCommMonoid M\nS : Set (Set M)\na✝¹ : Set M\ns✝ : Set (Set M)\na✝ : a✝¹ ∉ s✝\nhs✝ : s✝.Finite\nih : (∀ s ∈ s✝, IsSemilinearSet s) → IsSemilinearSet (⋃₀ s✝)\nhS' : IsSemilinearSet a✝¹ ∧ ∀ a ∈ s✝, IsSemilinearSet a\n⊢ IsSemilinearSet (a✝¹ ∪ ⋃₀ s✝)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 13
} | {
"line": 223,
"column": 14
} | [
{
"pp": "X : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nB : E →L[ℝ] F →L[ℝ] G\ninst✝ ... | [
"X : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nB : E →L[ℝ] F →L[ℝ] G\ninst✝ : Nontrivial... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 266,
"column": 4
} | {
"line": 266,
"column": 40
} | {
"line": 266,
"column": 41
} | [
{
"pp": "case insert\nM : Type u_1\ninst✝ : AddCommMonoid M\nS : Set (Set M)\na✝¹ : Set M\ns✝ : Set (Set M)\na✝ : a✝¹ ∉ s✝\nhs✝ : s✝.Finite\nih : (∀ t ∈ s✝, IsLinearSet t) → IsSemilinearSet ↑(closure (⋃₀ s✝))\nhS' : IsLinearSet a✝¹ ∧ ∀ a ∈ s✝, IsLinearSet a\n⊢ IsSemilinearSet ↑(closure (⋃₀ insert a✝¹ s✝))",
... | [
"case insert\nM : Type u_1\ninst✝ : AddCommMonoid M\nS : Set (Set M)\na✝¹ : Set M\ns✝ : Set (Set M)\na✝ : a✝¹ ∉ s✝\nhs✝ : s✝.Finite\nih : (∀ t ∈ s✝, IsLinearSet t) → IsSemilinearSet ↑(closure (⋃₀ s✝))\nhS' : IsLinearSet a✝¹ ∧ ∀ a ∈ s✝, IsLinearSet a\n⊢ IsSemilinearSet (↑(closure a✝¹) + ↑(closure (⋃₀ s✝)))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 265,
"column": 4
} | {
"line": 266,
"column": 69
} | {
"line": 268,
"column": 0
} | [
{
"pp": "case insert\nM : Type u_1\ninst✝ : AddCommMonoid M\nS : Set (Set M)\na✝¹ : Set M\ns✝ : Set (Set M)\na✝ : a✝¹ ∉ s✝\nhs✝ : s✝.Finite\nih : (∀ t ∈ s✝, IsLinearSet t) → IsSemilinearSet ↑(closure (⋃₀ s✝))\nhS' : ∀ t ∈ insert a✝¹ s✝, IsLinearSet t\n⊢ IsSemilinearSet ↑(closure (⋃₀ insert a✝¹ s✝))",
"ppTer... | [] | simp_rw [mem_insert_iff, forall_eq_or_imp] at hS'
simpa [closure_union, coe_sup] using hS'.1.closure.add (ih hS'.2) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 265,
"column": 4
} | {
"line": 266,
"column": 69
} | {
"line": 268,
"column": 0
} | [
{
"pp": "case insert\nM : Type u_1\ninst✝ : AddCommMonoid M\nS : Set (Set M)\na✝¹ : Set M\ns✝ : Set (Set M)\na✝ : a✝¹ ∉ s✝\nhs✝ : s✝.Finite\nih : (∀ t ∈ s✝, IsLinearSet t) → IsSemilinearSet ↑(closure (⋃₀ s✝))\nhS' : ∀ t ∈ insert a✝¹ s✝, IsLinearSet t\n⊢ IsSemilinearSet ↑(closure (⋃₀ insert a✝¹ s✝))",
"ppTer... | [] | simp_rw [mem_insert_iff, forall_eq_or_imp] at hS'
simpa [closure_union, coe_sup] using hS'.1.closure.add (ih hS'.2) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.WithDensityVec | {
"line": 77,
"column": 35
} | {
"line": 77,
"column": 69
} | {
"line": 77,
"column": 70
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\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\nμ : VectorMeasure X F\nf g : X → E\nB : E →L[ℝ] F →L[... | [
"X : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\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\nμ : VectorMeasure X F\nf g : X → E\nB : E →L[ℝ] F →L[ℝ] G\nh : f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 325,
"column": 4
} | {
"line": 325,
"column": 15
} | {
"line": 325,
"column": 16
} | [
{
"pp": "case insert\nM : Type u_1\ninst✝ : AddCommMonoid M\nS : Set (Set M)\na✝¹ : Set M\ns✝ : Set (Set M)\na✝ : a✝¹ ∉ s✝\nhs✝ : s✝.Finite\nih : (∀ s ∈ s✝, IsProperSemilinearSet s) → IsProperSemilinearSet (⋃₀ s✝)\nhS' : IsProperSemilinearSet a✝¹ ∧ ∀ a ∈ s✝, IsProperSemilinearSet a\n⊢ IsProperSemilinearSet (⋃₀ ... | [
"case insert\nM : Type u_1\ninst✝ : AddCommMonoid M\nS : Set (Set M)\na✝¹ : Set M\ns✝ : Set (Set M)\na✝ : a✝¹ ∉ s✝\nhs✝ : s✝.Finite\nih : (∀ s ∈ s✝, IsProperSemilinearSet s) → IsProperSemilinearSet (⋃₀ s✝)\nhS' : IsProperSemilinearSet a✝¹ ∧ ∀ a ∈ s✝, IsProperSemilinearSet a\n⊢ IsProperSemilinearSet (a✝¹ ∪ ⋃₀ s✝)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 517,
"column": 4
} | {
"line": 517,
"column": 85
} | {
"line": 518,
"column": 6
} | [
{
"pp": "case pos\nX : Type u_2\nE : Type u_4\nF : Type u_5\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nf : X → E\nμ : VectorMeasure X F\nhf : μ.Integrable f\ns : Set X\nhs : MeasurableSet s\n⊢ (μ.restrict s).Integrable f",
"ppTerm": "?pos✝",
"assigned": true,
... | [
"case pos\nX : Type u_2\nE : Type u_4\nF : Type u_5\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nf : X → E\nμ : VectorMeasure X F\nhf : μ.Integrable f\ns : Set X\nhs : MeasurableSet s\n⊢ Integrable f (μ.variation.restrict s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.WithDensityVec | {
"line": 134,
"column": 8
} | {
"line": 134,
"column": 66
} | {
"line": 135,
"column": 4
} | [
{
"pp": "case inr\nX : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nf : X → E\nB : E →L... | [] | apply mul_le_of_le_one_left (by positivity) mul_inv_le_one | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.VectorMeasure.WithDensityVec | {
"line": 138,
"column": 12
} | {
"line": 138,
"column": 23
} | {
"line": 138,
"column": 24
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nf : X → E\nB : E →L[ℝ] F →L[ℝ... | [
"X : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nf : X → E\nB : E →L[ℝ] F →L[ℝ] G\ninst✝ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 369,
"column": 24
} | {
"line": 369,
"column": 56
} | {
"line": 369,
"column": 56
} | [
{
"pp": "case a\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 ∈ t ... | [
"case a\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 ∈ t \\ t', f x •... | tsub_add_cancel_of_le (hfg j hj) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 408,
"column": 36
} | {
"line": 408,
"column": 72
} | {
"line": 408,
"column": 73
} | [
{
"pp": "S : Finset (Set ℕ)\nhS : ∀ t ∈ S, IsProperLinearSet t\na : ℕ\nt : Finset ℕ\nht : LinearIndepOn ℕ id ↑t\n⊢ t.card ≤ 1",
"ppTerm": "?m.165",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"S : Finset (Set ℕ)\nhS : ∀ t ∈ S, IsProperLinearSet t\na : ℕ\nt : Finset ℕ\nht : LinearIndepOn ℕ id ↑t\n⊢ t.card ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 413,
"column": 30
} | {
"line": 413,
"column": 51
} | {
"line": 413,
"column": 52
} | [
{
"pp": "S : Finset (Set ℕ)\nhS : ∀ t ∈ S, IsProperLinearSet t\na b : ℕ\nht : LinearIndepOn ℕ id ↑{b}\n⊢ b ≠ 0",
"ppTerm": "?m.255",
"assigned": true,
"usedConstants": [
"id",
"Ne",
"instOfNatNat",
"Nat",
"OfNat.ofNat"
],
"usedFVars": [
"b"
],
"use... | [
"S : Finset (Set ℕ)\nhS : ∀ t ∈ S, IsProperLinearSet t\na b : ℕ\nht : LinearIndepOn ℕ id ↑{b}\n⊢ ¬b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Definability | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 31
} | {
"line": 137,
"column": 32
} | [
{
"pp": "case e'_8\nα : Type u_1\nA : Set ℕ\ninst✝ : Finite α\nn✝ : ℕ\nthis : Fintype α\nn : ℕ\nφ : presburger[[↑A]].BoundedFormula α (n + 1)\ne : (α ⊕ Fin n) ⊕ Fin 1 ≃ α ⊕ Fin (n + 1) :=\n (Equiv.sumAssoc α (Fin n) (Fin 1)).trans ((_root_.Equiv.refl α).sumCongr finSumFinEquiv)\nih : IsSemilinearSet (⇑(LinearE... | [
"case e'_8.last\nα : Type u_1\nA : Set ℕ\ninst✝ : Finite α\nn✝ : ℕ\nthis : Fintype α\nn : ℕ\nφ : presburger[[↑A]].BoundedFormula α (n + 1)\ne : (α ⊕ Fin n) ⊕ Fin 1 ≃ α ⊕ Fin (n + 1) :=\n (Equiv.sumAssoc α (Fin n) (Fin 1)).trans ((_root_.Equiv.refl α).sumCongr finSumFinEquiv)\nih : IsSemilinearSet (⇑(LinearEquiv.fu... | cases i using Fin.lastCases | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.ModelTheory.Arithmetic.Presburger.Definability | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 52
} | {
"line": 169,
"column": 53
} | [
{
"pp": "A : Set ℕ\nhmul : A.Definable presburger {v | v 0 = v 1 * v 2}\nx✝ : Fin 1 → ℕ\n⊢ x✝ ∈ {x | x 0 ∈ {x | ∃ x_1, x_1 * x_1 = x}} ↔\n x✝ ∈ (fun g ↦ g ∘ ![0]) '' (fun g ↦ g ∘ ![0, 1, 1]) ⁻¹' {v | v 0 = v 1 * v 2}",
"ppTerm": "?m.137",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"A : Set ℕ\nhmul : A.Definable presburger {v | v 0 = v 1 * v 2}\nx✝ : Fin 1 → ℕ\n⊢ (∃ x, x * x = x✝ 0) ↔ ∃ a, x✝ 0 = a * a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Definability | {
"line": 163,
"column": 90
} | {
"line": 180,
"column": 9
} | {
"line": 182,
"column": 0
} | [
{
"pp": "A : Set ℕ\n⊢ ¬A.Definable presburger {v | v 0 = v 1 * v 2}",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"False",
"Nat.instMulZeroClass",
"Set.Definable₁._proof_1",
"Preorder.toLT",
"HMul.hMul",
"Nat.instOne",
... | [] | by
intro hmul
have hsqr : A.Definable₁ presburger {x * x | x : ℕ} := by
rw [Definable₁]
convert! (hmul.preimage_comp (β := Fin 2) ![0, 1, 1]).image_comp ![0]
ext
simpa [funext_iff, Fin.exists_fin_succ_pi] using exists_congr fun _ => Eq.comm
rw [definable₁_iff_ultimately_periodic] at hsqr
rcases ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 13
} | {
"line": 255,
"column": 14
} | [
{
"pp": "case e_a\nM : Type u_1\ninst✝ : AddCommMonoid M\na : M\nt : Set M\nht : t.Finite\nx : M\n⊢ x ∈ t ↔ x ∈ ⇑(closure (insert a t)).subtype '' ⇑(closure (insert a t)).subtype ⁻¹' t",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"AddSubmonoid.subtype",
"Eq.mpr",
"Iff.... | [
"case e_a\nM : Type u_1\ninst✝ : AddCommMonoid M\na : M\nt : Set M\nht : t.Finite\nx : M\n⊢ x ∈ t → x ∈ closure (insert a t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.DirectLimit | {
"line": 67,
"column": 4
} | {
"line": 68,
"column": 31
} | {
"line": 70,
"column": 0
} | [
{
"pp": "case succ\nL : Language\nG' : ℕ → Type w\ninst✝ : (i : ℕ) → L.Structure (G' i)\nf' : (n : ℕ) → G' n ↪[L] G' (n + 1)\nm : ℕ\nx : G' m\nk : ℕ\nih : ∀ (h : m ≤ m + k), (natLERec f' m (m + k) h) x = Nat.leRecOn h (fun k ↦ ⇑(f' k)) x\nh : m ≤ m + (k + 1)\n⊢ (natLERec f' m (m + (k + 1)) h) x = Nat.leRecOn h ... | [] | rw [Nat.leRecOn_succ le_self_add, natLERec, Nat.leRecOn_succ le_self_add, ← natLERec,
Embedding.comp_apply, ih] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.DirectLimit | {
"line": 67,
"column": 4
} | {
"line": 68,
"column": 31
} | {
"line": 70,
"column": 0
} | [
{
"pp": "case succ\nL : Language\nG' : ℕ → Type w\ninst✝ : (i : ℕ) → L.Structure (G' i)\nf' : (n : ℕ) → G' n ↪[L] G' (n + 1)\nm : ℕ\nx : G' m\nk : ℕ\nih : ∀ (h : m ≤ m + k), (natLERec f' m (m + k) h) x = Nat.leRecOn h (fun k ↦ ⇑(f' k)) x\nh : m ≤ m + (k + 1)\n⊢ (natLERec f' m (m + (k + 1)) h) x = Nat.leRecOn h ... | [] | rw [Nat.leRecOn_succ le_self_add, natLERec, Nat.leRecOn_succ le_self_add, ← natLERec,
Embedding.comp_apply, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.DirectLimit | {
"line": 67,
"column": 4
} | {
"line": 68,
"column": 31
} | {
"line": 70,
"column": 0
} | [
{
"pp": "case succ\nL : Language\nG' : ℕ → Type w\ninst✝ : (i : ℕ) → L.Structure (G' i)\nf' : (n : ℕ) → G' n ↪[L] G' (n + 1)\nm : ℕ\nx : G' m\nk : ℕ\nih : ∀ (h : m ≤ m + k), (natLERec f' m (m + k) h) x = Nat.leRecOn h (fun k ↦ ⇑(f' k)) x\nh : m ≤ m + (k + 1)\n⊢ (natLERec f' m (m + (k + 1)) h) x = Nat.leRecOn h ... | [] | rw [Nat.leRecOn_succ le_self_add, natLERec, Nat.leRecOn_succ le_self_add, ← natLERec,
Embedding.comp_apply, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 320,
"column": 4
} | {
"line": 320,
"column": 15
} | {
"line": 320,
"column": 16
} | [
{
"pp": "case insert\nM : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : AddMonoid.FG M\nS : Set (Set M)\na✝¹ : Set M\ns✝ : Set (Set M)\na✝ : a✝¹ ∉ s✝\nhs✝ : s✝.Finite\nih : (∀ s ∈ s✝, IsSemilinearSet s) → IsSemilinearSet (⋂₀ s✝)\nhS' : IsSemilinearSet a✝¹ ∧ ∀ a ∈ s✝, IsSemilinearSet a\n⊢ IsSemilinearSet (⋂₀ inser... | [
"case insert\nM : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : AddMonoid.FG M\nS : Set (Set M)\na✝¹ : Set M\ns✝ : Set (Set M)\na✝ : a✝¹ ∉ s✝\nhs✝ : s✝.Finite\nih : (∀ s ∈ s✝, IsSemilinearSet s) → IsSemilinearSet (⋂₀ s✝)\nhS' : IsSemilinearSet a✝¹ ∧ ∀ a ∈ s✝, IsSemilinearSet a\n⊢ IsSemilinearSet (a✝¹ ∩ ⋂₀ s✝)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 24
} | {
"line": 363,
"column": 25
} | [
{
"pp": "ι : Type u_3\nx y : ι → ℕ\nh : toRatVec x = toRatVec y\ni : ι\n⊢ x i = y i",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_3\nx y : ι → ℕ\nh : toRatVec x = toRatVec y\ni : ι\n⊢ x i = y i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 15
} | {
"line": 387,
"column": 16
} | [
{
"pp": "ι : Type u_3\ns : Set (ι → ℕ)\nt : Finset (ι → ℕ)\nf : (ι → ℕ) → ℤ\nht : ↑t ⊆ s\nhf : ∀ i ∉ t, f i = 0\nheq : ∑ i ∈ t, f i • toRatVec i = 0\ni : ι → ℕ\nhs : (Int.toNat ∘ f) i = (Int.toNat ∘ (fun x ↦ -x) ∘ f) i\nhi : i ∈ t\n⊢ (f i).toNat = (-f i).toNat",
"ppTerm": "?m.167",
"assigned": false,
... | [
"ι : Type u_3\ns : Set (ι → ℕ)\nt : Finset (ι → ℕ)\nf : (ι → ℕ) → ℤ\nht : ↑t ⊆ s\nhf : ∀ i ∉ t, f i = 0\nheq : ∑ i ∈ t, f i • toRatVec i = 0\ni : ι → ℕ\nhs : (Int.toNat ∘ f) i = (Int.toNat ∘ (fun x ↦ -x) ∘ f) i\nhi : i ∈ t\n⊢ (f i).toNat = (-f i).toNat"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 508,
"column": 4
} | {
"line": 508,
"column": 57
} | {
"line": 508,
"column": 58
} | [
{
"pp": "case mem\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx y✝ : ι → ℕ\ni : ↑hs.basisSet\nt : Set (ι → ℕ)\nht : t ⊆ hs.basisSet\nhi : ↑i ∉ t\ny : ι → ℕ\nhy : y ∈ t\n⊢ (hs.basis.repr (hs.basis ⟨y, ⋯⟩)) i = 0",
"ppTerm": "?mem",
"assigned": true,
"usedConstants": [
... | [
"case mem\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx y✝ : ι → ℕ\ni : ↑hs.basisSet\nt : Set (ι → ℕ)\nht : t ⊆ hs.basisSet\nhi : ↑i ∉ t\ny : ι → ℕ\nhy : y ∈ t\n⊢ ¬y = ↑i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 657,
"column": 8
} | {
"line": 657,
"column": 37
} | {
"line": 657,
"column": 38
} | [
{
"pp": "case mp.refine_2\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\nhx : hs.fract x = hs.base\ni : ↑hs.basisSet\nhi : hs.floor x i < 0\nj : ↑hs.basisSet\nhj : j ∈ Finset.univ.erase i\n⊢ ↑j ∈ hs.basisSet \\ {↑i}",
"ppTerm": "?mp.refine_2",
"assigned": true,
... | [
"case mp.refine_2\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\nhx : hs.fract x = hs.base\ni : ↑hs.basisSet\nhi : hs.floor x i < 0\nj : ↑hs.basisSet\nhj : j ∈ Finset.univ.erase i\n⊢ ¬j = i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 659,
"column": 8
} | {
"line": 659,
"column": 37
} | {
"line": 659,
"column": 38
} | [
{
"pp": "case mp.refine_3\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\nhx : hs.fract x = hs.base\ni : ↑hs.basisSet\nhi : hs.floor x i < 0\nj : ↑hs.basisSet\nhj : j ∈ Finset.univ.erase i\n⊢ ↑j ∈ hs.basisSet \\ {↑i}",
"ppTerm": "?mp.refine_3",
"assigned": true,
... | [
"case mp.refine_3\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\nhx : hs.fract x = hs.base\ni : ↑hs.basisSet\nhi : hs.floor x i < 0\nj : ↑hs.basisSet\nhj : j ∈ Finset.univ.erase i\n⊢ ¬j = i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 683,
"column": 8
} | {
"line": 683,
"column": 25
} | {
"line": 683,
"column": 26
} | [
{
"pp": "case mpr.refine_2\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\ni : ↑hs.basisSet\nz : ι → ℕ\nhz : z ∈ closure (hs.basisSet \\ {↑i})\nz' : ι → ℕ\nhz' : z' ∈ closure (hs.basisSet \\ {↑i})\nn : ℕ\nheq : hs.floor x i = -↑(n + 1)\n⊢ hs.floor x i < 0",
"ppTerm": "... | [
"case mpr.refine_2\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\ni : ↑hs.basisSet\nz : ι → ℕ\nhz : z ∈ closure (hs.basisSet \\ {↑i})\nz' : ι → ℕ\nhz' : z' ∈ closure (hs.basisSet \\ {↑i})\nn : ℕ\nheq : hs.floor x i = -↑(n + 1)\n⊢ -1 < ↑n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 720,
"column": 8
} | {
"line": 720,
"column": 37
} | {
"line": 720,
"column": 38
} | [
{
"pp": "case mp.refine_2\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\nhx : hs.fract x = hs.base\ni : ↑hs.basisSet\nhi : ↑i ∉ hs.periods\nhi' : 0 < hs.floor x i\nj : ↑hs.basisSet\nhj : j ∈ Finset.univ.erase i\n⊢ ↑j ∈ hs.basisSet \\ {↑i}",
"ppTerm": "?mp.refine_2",
... | [
"case mp.refine_2\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\nhx : hs.fract x = hs.base\ni : ↑hs.basisSet\nhi : ↑i ∉ hs.periods\nhi' : 0 < hs.floor x i\nj : ↑hs.basisSet\nhj : j ∈ Finset.univ.erase i\n⊢ ¬j = i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.CountableDenseLinearOrder | {
"line": 256,
"column": 6
} | {
"line": 256,
"column": 40
} | {
"line": 257,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹¹ : LinearOrder α\ninst✝¹⁰ : LinearOrder β\ninst✝⁹ : Countable α\ninst✝⁸ : DenselyOrdered α\ninst✝⁷ : NoMinOrder α\ninst✝⁶ : NoMaxOrder α\ninst✝⁵ : Nonempty α\ninst✝⁴ : Countable β\ninst✝³ : DenselyOrdered β\ninst✝² : NoMinOrder β\ninst✝¹ : NoMaxOrder β\ninst✝ : Nonemp... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹¹ : LinearOrder α\ninst✝¹⁰ : LinearOrder β\ninst✝⁹ : Countable α\ninst✝⁸ : DenselyOrdered α\ninst✝⁷ : NoMinOrder α\ninst✝⁶ : NoMaxOrder α\ninst✝⁵ : Nonempty α\ninst✝⁴ : Countable β\ninst✝³ : DenselyOrdered β\ninst✝² : NoMinOrder β\ninst✝¹ : NoMaxOrder β\ninst✝ : Nonempty β\nval✝¹ ... | rcases (F a).prop with ⟨f, hf, ha⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 722,
"column": 8
} | {
"line": 722,
"column": 37
} | {
"line": 722,
"column": 38
} | [
{
"pp": "case mp.refine_3\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\nhx : hs.fract x = hs.base\ni : ↑hs.basisSet\nhi : ↑i ∉ hs.periods\nhi' : 0 < hs.floor x i\nj : ↑hs.basisSet\nhj : j ∈ Finset.univ.erase i\n⊢ ↑j ∈ hs.basisSet \\ {↑i}",
"ppTerm": "?mp.refine_3",
... | [
"case mp.refine_3\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\nhx : hs.fract x = hs.base\ni : ↑hs.basisSet\nhi : ↑i ∉ hs.periods\nhi' : 0 < hs.floor x i\nj : ↑hs.basisSet\nhj : j ∈ Finset.univ.erase i\n⊢ ¬j = i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.AlmostPrime | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 13
} | {
"line": 66,
"column": 14
} | [
{
"pp": "p : ℕ\nhp : Prime p\n⊢ IsAlmostPrime 1 p",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Prime",
"Nat.isAlmostPrime_one_iff._simp_1",
"id",
"instOfNatNat",
"Nat.IsAlmostPrime",
"Nat",
"OfNat.ofNat",
"Eq"
],
... | [
"p : ℕ\nhp : Prime p\n⊢ Prime p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.AlmostPrime | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 13
} | {
"line": 85,
"column": 14
} | [
{
"pp": "p q : ℕ\nhp : Prime p\nhq : Prime q\n⊢ IsAlmostPrime 2 (p * q)",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q : ℕ\nhp : Prime p\nhq : Prime q\n⊢ IsAlmostPrime 2 (p * q)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.AlmostPrime | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 23
} | {
"line": 88,
"column": 24
} | [
{
"pp": "p : ℕ\nhp : Prime p\n⊢ IsAlmostPrime 2 (p ^ 2)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"pow_two",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"id",
"MulOne.toMul",
"instOfNatNat",
... | [
"p : ℕ\nhp : Prime p\n⊢ IsAlmostPrime 2 (p * p)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ADEInequality | {
"line": 209,
"column": 59
} | {
"line": 209,
"column": 70
} | {
"line": 209,
"column": 71
} | [
{
"pp": "p q r : ℕ+\nhs : [p, q, r].SortedLE\nx✝ : [p, q, r].length = 3\nH : 1 < sumInv ↑[p, q, r]\n⊢ (p ≤ q ∧ p ≤ r) ∧ q ≤ r",
"ppTerm": "?m.225",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p q r : ℕ+\nhs : [p, q, r].SortedLE\nx✝ : [p, q, r].length = 3\nH : 1 < sumInv ↑[p, q, r]\n⊢ (p ≤ q ∧ p ≤ r) ∧ q ≤ r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ArithmeticFunction.Carmichael | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 87
} | {
"line": 130,
"column": 0
} | [
{
"pp": "n : ℕ\nhn : n ≤ 2\n⊢ exponent (ZMod (2 ^ n))ˣ = Nat.card (ZMod (2 ^ n))ˣ",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"ZMod.commRing",
"CommSemiring.toSemiring",
"Nat.instMonoid",
"DivInvMonoid.toZPow",
"Units",
"Nat.card",
... | [] | exact IsCyclic.iff_exponent_eq_card.mp <| ZMod.isCyclic_units_two_pow_iff n |>.mpr hn | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.ArithmeticFunction.Carmichael | {
"line": 133,
"column": 40
} | {
"line": 135,
"column": 29
} | {
"line": 137,
"column": 0
} | [
{
"pp": "n : ℕ\nhn : n ≤ 2\n⊢ λ (2 ^ n) = 2 ^ (n - 1)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.zero_le",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"of_decide_eq_true",
"Nat.rawCast",
"congrArg",
"Nat.i... | [] | by
rw [carmichael_two_pow_of_le_two_eq_totient hn]
interval_cases n <;> decide | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt | {
"line": 120,
"column": 9
} | {
"line": 120,
"column": 49
} | {
"line": 120,
"column": 49
} | [
{
"pp": "n : ℕ\n⊢ (Λ * ↑ζ) n = log n",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNat",
"ArithmeticFunctio... | [
"n : ℕ\n⊢ Real.log ↑n = log n"
] | rw [coe_mul_zeta_apply, vonMangoldt_sum] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt | {
"line": 145,
"column": 19
} | {
"line": 145,
"column": 30
} | {
"line": 145,
"column": 31
} | [
{
"pp": "n : ℕ\nhn : ¬n = 0\nmn : 0 ∣ n\n⊢ n = 0",
"ppTerm": "?m.108",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nhn : ¬n = 0\nmn : 0 ∣ n\n⊢ n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 194,
"column": 4
} | {
"line": 194,
"column": 15
} | {
"line": 194,
"column": 16
} | [
{
"pp": "p : ℕ\nhp : Nat.Prime p\nhp2 : p ≠ 2\nn : ℕ\nH : ↑p ∣ 1\n⊢ p = 1",
"ppTerm": "?m.199",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"p : ℕ\nhp : Nat.Prime p\nhp2 : p ≠ 2\nn : ℕ\nH : ↑p ∣ 1\n⊢ p = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 266,
"column": 27
} | {
"line": 266,
"column": 38
} | {
"line": 266,
"column": 39
} | [
{
"pp": "n : ℕ\nhn0 : 2 * n ≠ 0\nhn1 : 2 * n ≠ 1\n⊢ n ≠ 0",
"ppTerm": "?m.112",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"id",
"Ne",
"Nat",
"Zero.toOfNat0",
"OfNat.ofNat",
"MulZeroClass.toZero"
],
"usedFVars": [
"n"
],
... | [
"n : ℕ\nhn0 : 2 * n ≠ 0\nhn1 : 2 * n ≠ 1\n⊢ ¬n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 274,
"column": 25
} | {
"line": 274,
"column": 60
} | {
"line": 274,
"column": 61
} | [
{
"pp": "n : ℕ\nhn0 : n ≠ 0\nhn1 : n ≠ 1\nh2n : ¬2 ∣ n\nthis✝ : Nat.Coprime 4 n\nh : (Nat.card (ZMod 4)ˣ).Coprime (Nat.card (ZMod n)ˣ)\nthis : NeZero n\n⊢ Odd (φ n)",
"ppTerm": "?m.229",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nhn0 : n ≠ 0\nhn1 : n ≠ 1\nh2n : ¬2 ∣ n\nthis✝ : Nat.Coprime 4 n\nh : (Nat.card (ZMod 4)ˣ).Coprime (Nat.card (ZMod n)ˣ)\nthis : NeZero n\n⊢ Odd (φ n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Derivative | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 65
} | {
"line": 148,
"column": 4
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\ni j : σ\nh : j ≠ i\nn : σ →₀ ℕ\n⊢ (coeff n) ((pderiv R i) (X j)) = (coeff n) 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Derivation",
"Finsupp.instFunLike",
"Eq.mpr",
"NonAssocSemiring.toAddCommMono... | [
"σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\ni j : σ\nh : j ≠ i\nn : σ →₀ ℕ\n⊢ (if n + single i 1 = single j 1 then ↑(n i) + 1 else 0) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 313,
"column": 6
} | {
"line": 313,
"column": 61
} | {
"line": 313,
"column": 62
} | [
{
"pp": "case neg.refine_1.refine_1\nn : ℕ\nhn : Odd n\nhn0 : n ≠ 0\nh1 : n ≠ 1\np : ℕ\nhp : Nat.Prime p\ndvd : p ∣ n\nodd : Odd p\nhnp : ¬n = p ^ n.factorization p\nthis : p ^ n.factorization p ∣ n\n⊢ p ^ n.factorization p ≠ 1",
"ppTerm": "?neg.refine_1.refine_1✝",
"assigned": true,
"usedConstants"... | [
"case neg.refine_1.refine_1\nn : ℕ\nhn : Odd n\nhn0 : n ≠ 0\nh1 : n ≠ 1\np : ℕ\nhp : Nat.Prime p\ndvd : p ∣ n\nodd : Odd p\nhnp : ¬n = p ^ n.factorization p\nthis : p ^ n.factorization p ∣ n\n⊢ ¬p = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Derivative | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 13
} | {
"line": 79,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf : R⟦X⟧\nn : ℕ\n⊢ constantCoeff ((⇑(d⁄dX R))^[n] f) = ↑n ! * (coeff n) f",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝ : CommSemiring R\nf : R⟦X⟧\nn : ℕ\n⊢ constantCoeff ((⇑(d⁄dX R))^[n] f) = ↑n ! * (coeff n) f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.WithDensityVec | {
"line": 354,
"column": 2
} | {
"line": 354,
"column": 45
} | {
"line": 355,
"column": 2
} | [
{
"pp": "case h₂\nX : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nf : X → E\nB : E →L[... | [
"case h₂\nX : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nf : X → E\nB : E →L[ℝ] F →L[ℝ] G... | apply ContinuousLinearMap.opNNNorm_le_bound | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.AbelSummation | {
"line": 153,
"column": 2
} | {
"line": 157,
"column": 71
} | {
"line": 158,
"column": 2
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\na b : ℝ\nha : 0 ≤ a\nhab : a ≤ b\nhf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t\nhf_int : IntegrableOn (deriv f) (Set.Icc a b) volume\naux1 : ↑⌊a⌋₊ ≤ a\naux2 : b ≤ ↑⌊b⌋₊ + 1\nhb : ⌊a⌋₊ < ⌊b⌋₊\naux3 : a ≤ ↑⌊a⌋₊ + 1\naux4 : ↑⌊a⌋₊ +... | [
"case inr\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\na b : ℝ\nha : 0 ≤ a\nhab : a ≤ b\nhf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t\nhf_int : IntegrableOn (deriv f) (Set.Icc a b) volume\naux1 : ↑⌊a⌋₊ ≤ a\naux2 : b ≤ ↑⌊b⌋₊ + 1\nhb : ⌊a⌋₊ < ⌊b⌋₊\naux3 : a ≤ ↑⌊a⌋₊ + 1\naux4 : ↑⌊a⌋₊ + 1 ≤ b\naux5... | rw [this, sum_integral_adjacent_intervals_Ico hb, Nat.cast_add, Nat.cast_one,
← integral_interval_sub_left (a := a) (c := ⌊a⌋₊ + 1),
← integral_add_adjacent_intervals (b := ⌊b⌋₊) (c := b),
integralmulsum c hf_diff hf_int _ _ _ aux3 aux1 le_rfl le_rfl aux4,
integralmulsum c hf_diff hf_int _ _ _ aux5 le_r... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 361,
"column": 2
} | {
"line": 362,
"column": 7
} | {
"line": 363,
"column": 2
} | [
{
"pp": "case neg.inr.inr.ha\nn : ℕ\nh0 : ¬2 * (2 * n) = 0\nh1 : ¬2 * (2 * n) = 1\nh2 : ¬2 * (2 * n) = 2\nh4 : ¬2 * (2 * n) = 4\nhn✝ : Even (2 * (2 * n))\nhn : Even (2 * n)\n⊢ ¬IsCyclic (ZMod (2 * (2 * n)))ˣ",
"ppTerm": "?neg.inr.inr.ha✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"case neg.inr.inr.hb\nn : ℕ\nh0 : ¬2 * (2 * n) = 0\nh1 : ¬2 * (2 * n) = 1\nh2 : ¬2 * (2 * n) = 2\nh4 : ¬2 * (2 * n) = 4\nhn✝ : Even (2 * (2 * n))\nhn : Even (2 * n)\n⊢ ¬((∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * (2 * n) = x ^ x_1) ∨\n ∃ x x_1, Nat.Prime x ∧ Odd x ∧ 1 ≤ x_1 ∧ 2 * (2 * n) = 2 * x ^ x_1)"
] | · rw [← mul_assoc, show 2 * 2 = 4 from rfl, isCyclic_units_four_mul_iff]
lia | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.WithDensityVec | {
"line": 386,
"column": 4
} | {
"line": 386,
"column": 15
} | {
"line": 386,
"column": 16
} | [
{
"pp": "case e_f.inr\nX : Type u_1\nE : Type u_2\nmX : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure X\nf : X → E\nhf : Integrable f μ\nhE : Nontrivial E\nthis : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)\nI : VectorMeasure.Integrable (μ.wi... | [
"case e_f.inr\nX : Type u_1\nE : Type u_2\nmX : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure X\nf : X → E\nhf : Integrable f μ\nhE : Nontrivial E\nthis : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)\nI : VectorMeasure.Integrable (μ.withDensity fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.VectorMeasure.WithDensityVec | {
"line": 396,
"column": 33
} | {
"line": 396,
"column": 44
} | {
"line": 396,
"column": 45
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nmX : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure X\nf : X → E\nhf : Integrable f μ\nhE : Nontrivial E\nthis✝ : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)\nI : VectorMeasure.Integrable (μ.withDensity fun... | [
"X : Type u_1\nE : Type u_2\nmX : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure X\nf : X → E\nhf : Integrable f μ\nhE : Nontrivial E\nthis✝ : IsFiniteMeasure (μ.withDensity fun x ↦ ‖f x‖ₑ)\nI : VectorMeasure.Integrable (μ.withDensity fun x ↦ ‖f x‖ₑ)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Exp | {
"line": 160,
"column": 4
} | {
"line": 161,
"column": 34
} | {
"line": 161,
"column": 35
} | [
{
"pp": "case succ\nA : Type u_4\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nk : ℕ\nh : exp A ^ k = (rescale ↑k) (exp A)\n⊢ exp A ^ (k + 1) = (rescale ↑(k + 1)) (exp A)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
... | [
"case succ\nA : Type u_4\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nk : ℕ\nh : exp A ^ k = (rescale ↑k) (exp A)\n⊢ exp A ^ (k + 1) = exp A ^ k * exp A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 20
} | {
"line": 92,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ (if n = 1 then 1 else 0) + _root_.bernoulli n = bernoulli' n",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"AddMonoid.toAddZeroClass",
"bernoulli",
"Rat",
"AddZeroClass.toAddZero",
"instOfNatNat",
"dite",
"... | [
"case pos\nn : ℕ\nh : n = 1\n⊢ (if n = 1 then 1 else 0) + _root_.bernoulli n = bernoulli' n",
"case neg\nn : ℕ\nh : ¬n = 1\n⊢ (if n = 1 then 1 else 0) + _root_.bernoulli n = bernoulli' n"
] | by_cases h : n = 1 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.NumberTheory.Bernoulli | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 40
} | {
"line": 169,
"column": 41
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nthis : ∑ p ∈ antidiagonal n, bernoulli' p.1 / ↑p.1! * ((↑p.2 + 1) * ↑p.2!)⁻¹ = (↑n !)⁻¹\n⊢ (coeff (n + 1, 0).1) (PowerSeries.mk fun n ↦ (algebraMap ℚ A) (bernoulli' n / ↑n !)) *\n (coeff (n + 1, 0).2) (exp A - 1) +\n ∑ p ∈ antid... | [
"A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nthis : ∑ p ∈ antidiagonal n, bernoulli' p.1 / ↑p.1! * ((↑p.2 + 1) * ↑p.2!)⁻¹ = (↑n !)⁻¹\n⊢ ∑ x ∈ antidiagonal n,\n (algebraMap ℚ A) (bernoulli' x.1 / ↑x.1!) * ((algebraMap ℚ A) (↑x.2!)⁻¹ * (algebraMap ℚ A) (↑x.2 + 1)⁻¹) =\n (algebraMap ℚ A) (↑n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 13
} | {
"line": 215,
"column": 14
} | [
{
"pp": "n : ℕ\nx : ℚ\nthis : (bernoulli n).comp (1 + X) = bernoulli n + n • X ^ (n - 1)\n⊢ eval (1 + x) (bernoulli n) = eval x (bernoulli n) + ↑n * x ^ (n - 1)",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nx : ℚ\nthis : (bernoulli n).comp (1 + X) = bernoulli n + n • X ^ (n - 1)\n⊢ eval (1 + x) (bernoulli n) = eval x (bernoulli n) + ↑n * x ^ (n - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 187,
"column": 6
} | {
"line": 187,
"column": 20
} | {
"line": 187,
"column": 21
} | [
{
"pp": "case inl\nn : ℕ\nh_odd : Odd n\nhlt : 1 < n\nB : ℚ⟦X⟧ := PowerSeries.mk fun n ↦ bernoulli' n / ↑n !\nthis : (B - evalNegHom B) * (exp ℚ - 1) = X * (exp ℚ - 1)\nh : (coeff n) (B - (rescale (-1)) B) = if n = 1 then 1 else 0\n⊢ -bernoulli' n = bernoulli' n",
"ppTerm": "?inl",
"assigned": true,
... | [
"case pos\nn : ℕ\nh_odd : Odd n\nhlt : 1 < n\nB : ℚ⟦X⟧ := PowerSeries.mk fun n ↦ bernoulli' n / ↑n !\nthis : (B - evalNegHom B) * (exp ℚ - 1) = X * (exp ℚ - 1)\nh✝ : n = 1\nh : (coeff n) (B - (rescale (-1)) B) = 1\n⊢ -bernoulli' n = bernoulli' n",
"case neg\nn : ℕ\nh_odd : Odd n\nhlt : 1 < n\nB : ℚ⟦X⟧ := PowerSer... | split_ifs at h | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.NumberTheory.Bernoulli | {
"line": 188,
"column": 6
} | {
"line": 188,
"column": 41
} | {
"line": 188,
"column": 42
} | [
{
"pp": "case inr\nn : ℕ\nh_odd : Odd n\nhlt : 1 < n\nB : ℚ⟦X⟧ := PowerSeries.mk fun n ↦ bernoulli' n / ↑n !\nthis : (B - evalNegHom B) * (exp ℚ - 1) = X * (exp ℚ - 1)\nh : ∀ (n : ℕ), (coeff n) (exp ℚ - 1) = (coeff n) 0\n⊢ bernoulli' n = 0",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
... | [
"case inr\nn : ℕ\nh_odd : Odd n\nhlt : 1 < n\nB : ℚ⟦X⟧ := PowerSeries.mk fun n ↦ bernoulli' n / ↑n !\nthis : (B - evalNegHom B) * (exp ℚ - 1) = X * (exp ℚ - 1)\nh : ∀ (n : ℕ), (coeff n) (exp ℚ - 1) = (coeff n) 0\n⊢ bernoulli' n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 190,
"column": 4
} | {
"line": 190,
"column": 39
} | {
"line": 190,
"column": 40
} | [
{
"pp": "n : ℕ\nh_odd : Odd n\nhlt : 1 < n\nB : ℚ⟦X⟧ := PowerSeries.mk fun n ↦ bernoulli' n / ↑n !\n⊢ B * (exp ℚ - 1) = X * exp ℚ",
"ppTerm": "?m.154",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nh_odd : Odd n\nhlt : 1 < n\nB : ℚ⟦X⟧ := PowerSeries.mk fun n ↦ bernoulli' n / ↑n !\n⊢ B * (exp ℚ - 1) = X * exp ℚ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 23
} | {
"line": 238,
"column": 24
} | [
{
"pp": "n : ℕ\nx : ℚ\n⊢ eval (-x) (bernoulli n) = (-1) ^ n * (eval x (bernoulli n) + ↑n * x ^ (n - 1))",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Distrib.leftDistribClass",
"Eq.mpr",
"Polynomial.eval",
"Rat.instMul",
"HMul.hMul",
... | [
"n : ℕ\nx : ℚ\n⊢ eval (-x) (bernoulli n) = (-1) ^ n * eval x (bernoulli n) + (-1) ^ n * (↑n * x ^ (n - 1))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 241,
"column": 59
} | {
"line": 248,
"column": 8
} | {
"line": 250,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ (bernoulli n).comp (1 - X) = (-1) ^ n * bernoulli n",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Mathlib.Tact... | [] | by
cases n with
| zero => simp
| succ n =>
trans ((bernoulli (n + 1)).comp (1 + X)).comp (-X)
· simp [comp_assoc, sub_eq_add_neg]
simp [bernoulli_comp_one_add_X, bernoulli_comp_neg_X, neg_pow (X : Polynomial ℚ)]
ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 13
} | {
"line": 252,
"column": 14
} | [
{
"pp": "n : ℕ\nx : ℚ\n⊢ eval (1 - x) (bernoulli n) = (-1) ^ n * eval x (bernoulli n)",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nx : ℚ\n⊢ eval (1 - x) (bernoulli n) = (-1) ^ n * eval x (bernoulli n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 312,
"column": 29
} | {
"line": 312,
"column": 40
} | {
"line": 312,
"column": 41
} | [
{
"pp": "n p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b ↦ bernoulli a / ↑a ! * (coeff (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m ∈ range q.succ\n⊢ m < q + 1",
"ppTerm": "?m.213",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOne",
"PartialOrder.toPreorder",
... | [
"n p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b ↦ bernoulli a / ↑a ! * (coeff (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh : m ∈ range q.succ\n⊢ m ≤ q"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.AbelSummation | {
"line": 278,
"column": 4
} | {
"line": 279,
"column": 28
} | {
"line": 281,
"column": 0
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\na : ℝ\nm : ℕ\nha : 0 ≤ a\ng : ℝ → 𝕜\nhg : LocallyIntegrableOn g (Set.Ici a) volume\nK : Set ℝ\nhK₁ : K ⊆ Set.Ici a\nhK₂ : IsCompact K\nhK₃ : ¬K.Nonempty\n⊢ IntegrableOn (fun t ↦ g t * ∑ k ∈ Icc m ⌊t⌋₊, c k) K volume",
"ppTerm": "?neg✝",
"... | [] | rw [Set.not_nonempty_iff_eq_empty.mp hK₃]
exact integrableOn_empty | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.AbelSummation | {
"line": 278,
"column": 4
} | {
"line": 279,
"column": 28
} | {
"line": 281,
"column": 0
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\na : ℝ\nm : ℕ\nha : 0 ≤ a\ng : ℝ → 𝕜\nhg : LocallyIntegrableOn g (Set.Ici a) volume\nK : Set ℝ\nhK₁ : K ⊆ Set.Ici a\nhK₂ : IsCompact K\nhK₃ : ¬K.Nonempty\n⊢ IntegrableOn (fun t ↦ g t * ∑ k ∈ Icc m ⌊t⌋₊, c k) K volume",
"ppTerm": "?neg✝",
"... | [] | rw [Set.not_nonempty_iff_eq_empty.mp hK₃]
exact integrableOn_empty | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.AbelSummation | {
"line": 291,
"column": 37
} | {
"line": 291,
"column": 78
} | {
"line": 291,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhf_diff : ∀ t ∈ Set.Ici 0, DifferentiableAt ℝ f t\nhf_int : LocallyIntegrableOn (deriv f) (Set.Ici 0) volume\nl : 𝕜\nh_lim : Tendsto (fun n ↦ f ↑n * ∑ k ∈ Icc 0 n, c k) atTop (𝓝 l)\ng : ℝ → 𝕜\nhg_dom : (fun t ↦ deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c... | [] | rw [← integral_of_le (Nat.cast_nonneg _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.AbelSummation | {
"line": 291,
"column": 37
} | {
"line": 291,
"column": 78
} | {
"line": 291,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhf_diff : ∀ t ∈ Set.Ici 0, DifferentiableAt ℝ f t\nhf_int : LocallyIntegrableOn (deriv f) (Set.Ici 0) volume\nl : 𝕜\nh_lim : Tendsto (fun n ↦ f ↑n * ∑ k ∈ Icc 0 n, c k) atTop (𝓝 l)\ng : ℝ → 𝕜\nhg_dom : (fun t ↦ deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c... | [] | rw [← integral_of_le (Nat.cast_nonneg _)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.AbelSummation | {
"line": 291,
"column": 37
} | {
"line": 291,
"column": 78
} | {
"line": 291,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhf_diff : ∀ t ∈ Set.Ici 0, DifferentiableAt ℝ f t\nhf_int : LocallyIntegrableOn (deriv f) (Set.Ici 0) volume\nl : 𝕜\nh_lim : Tendsto (fun n ↦ f ↑n * ∑ k ∈ Icc 0 n, c k) atTop (𝓝 l)\ng : ℝ → 𝕜\nhg_dom : (fun t ↦ deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, c... | [] | rw [← integral_of_le (Nat.cast_nonneg _)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Bernoulli | {
"line": 317,
"column": 83
} | {
"line": 317,
"column": 93
} | {
"line": 318,
"column": 6
} | [
{
"pp": "n p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b ↦ bernoulli a / ↑a ! * (coeff (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh✝ : m ∈ range q.succ\nh : m < q + 1\n⊢ bernoulli m * ↑((q + 1)! / (m ! * (q + 1 - m)!)) * ↑n ^ (q + 1 - m) =\n bernoulli m * ↑q.succ ! / ↑m ! * (↑(q - m + 1)!)⁻¹ * ↑n ^ (q ... | [
"n p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nq : ℕ\nf : ℕ → ℕ → ℚ := fun a b ↦ bernoulli a / ↑a ! * (coeff (b + 1)) (exp ℚ ^ n)\nm : ℕ\nh✝ : m ∈ range q.succ\nh : m < q + 1\n⊢ bernoulli m * ↑((q + 1)! / (m ! * (q + 1 - m)!)) * ↑n ^ (q + 1 - m) =\n bernoulli m * ↑q.succ ! / ↑m ! * (1 / ↑(q - m + 1)!) * ↑n ^ (q - m + 1)"
... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 333,
"column": 6
} | {
"line": 333,
"column": 17
} | {
"line": 333,
"column": 18
} | [
{
"pp": "n p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p ↦ bernoulli p / ↑p !) * PowerSeries.mk fun q ↦ (coeff (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p ↦ ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nthis :\n ∀ (n_1 : ℕ),\n (coeff n_... | [
"n p : ℕ\nhne : ∀ (m : ℕ), ↑m ! ≠ 0\nh_cauchy :\n ((PowerSeries.mk fun p ↦ bernoulli p / ↑p !) * PowerSeries.mk fun q ↦ (coeff (q + 1)) (exp ℚ ^ n)) =\n PowerSeries.mk fun p ↦ ∑ i ∈ range (p + 1), bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / ↑(p + 1)!\nthis :\n ∀ (n_1 : ℕ),\n (coeff n_1) (PowerSer... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Bertrand | {
"line": 151,
"column": 41
} | {
"line": 151,
"column": 80
} | {
"line": 151,
"column": 81
} | [
{
"pp": "n : ℕ\nn_large : 2 < n\nno_prime : ∀ (p : ℕ), Nat.Prime p → n < p → 2 * n < p\nx : ℕ\nhx : x ≤ 2 * n\nh2x : 2 * n < 3 * x\n⊢ ¬Nat.Prime x ∨ x ≤ n",
"ppTerm": "?m.174",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\nn_large : 2 < n\nno_prime : ∀ (p : ℕ), Nat.Prime p → n < p → 2 * n < p\nx : ℕ\nhx : x ≤ 2 * n\nh2x : 2 * n < 3 * x\n⊢ ¬Nat.Prime x ∨ x ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.AbelSummation | {
"line": 358,
"column": 8
} | {
"line": 364,
"column": 17
} | {
"line": 366,
"column": 0
} | [
{
"pp": "case succ.calc_4\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nm : ℕ\nh_bdd : (fun n ↦ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun x ↦ 1\nhf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici ↑m) volume\nhf :\n ∀ (n : ℕ),\n ∑ k ∈ Icc 0 n, ‖f ↑k‖ * ‖c k‖ =\n ‖f ↑n‖ * ∑ k ∈ I... | [] | unfold C₂
grw [setIntegral_mono_set ?_ (.of_forall fun _ ↦ norm_nonneg _)
Set.Ioc_subset_Ioi_self.eventuallyLE]
rw [← integrableOn_Ici_iff_integrableOn_Ioi, IntegrableOn,
integrable_norm_iff (by fun_prop)]
exact (locallyIntegrableOn_mul_sum_Icc _ m.cast_nonneg hf_int).integra... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.AbelSummation | {
"line": 358,
"column": 8
} | {
"line": 364,
"column": 17
} | {
"line": 366,
"column": 0
} | [
{
"pp": "case succ.calc_4\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nm : ℕ\nh_bdd : (fun n ↦ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun x ↦ 1\nhf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici ↑m) volume\nhf :\n ∀ (n : ℕ),\n ∑ k ∈ Icc 0 n, ‖f ↑k‖ * ‖c k‖ =\n ‖f ↑n‖ * ∑ k ∈ I... | [] | unfold C₂
grw [setIntegral_mono_set ?_ (.of_forall fun _ ↦ norm_nonneg _)
Set.Ioc_subset_Ioi_self.eventuallyLE]
rw [← integrableOn_Ici_iff_integrableOn_Ioi, IntegrableOn,
integrable_norm_iff (by fun_prop)]
exact (locallyIntegrableOn_mul_sum_Icc _ m.cast_nonneg hf_int).integra... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue | {
"line": 100,
"column": 12
} | {
"line": 100,
"column": 58
} | {
"line": 100,
"column": 59
} | [
{
"pp": "R : Type u_1\ninst✝ : EuclideanDomain R\nabv : AbsoluteValue R ℤ\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : ℕ\nih :\n ∀ {ε : ℝ},\n 0 < ε →\n ∀ {b : R},\n b ≠ 0 →\n ∀ (A : Fin (h.card ε ^ n).succ → Fin n → R),\n ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ (k : Fin n), ↑(abv (A i₁ k % ... | [
"R : Type u_1\ninst✝ : EuclideanDomain R\nabv : AbsoluteValue R ℤ\nh : abv.IsAdmissible\nthis : DecidableEq R\nn : ℕ\nih :\n ∀ {ε : ℝ},\n 0 < ε →\n ∀ {b : R},\n b ≠ 0 →\n ∀ (A : Fin (h.card ε ^ n).succ → Fin n → R),\n ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ (k : Fin n), ↑(abv (A i₁ k % b - A i₀ k %... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 72
} | {
"line": 105,
"column": 73
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : EuclideanDomain R\nabv : AbsoluteValue R ℤ\nh✝ : abv.IsAdmissible\nthis✝ : DecidableEq R\nn : ℕ\nih :\n ∀ {ε : ℝ},\n 0 < ε →\n ∀ {b : R},\n b ≠ 0 →\n ∀ (A : Fin (h✝.card ε ^ n).succ → Fin n → R),\n ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ (k : Fin n)... | [
"case refine_1\nR : Type u_1\ninst✝ : EuclideanDomain R\nabv : AbsoluteValue R ℤ\nh✝ : abv.IsAdmissible\nthis✝ : DecidableEq R\nn : ℕ\nih :\n ∀ {ε : ℝ},\n 0 < ε →\n ∀ {b : R},\n b ≠ 0 →\n ∀ (A : Fin (h✝.card ε ^ n).succ → Fin n → R),\n ∃ i₀ i₁, i₀ ≠ i₁ ∧ ∀ (k : Fin n), ↑(abv (A i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 15
} | {
"line": 45,
"column": 16
} | [
{
"pp": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : b.natDegree ≤ d\nA : Fin m.succ → Fq[X]\nhA : ∀ (i : Fin m.succ), (A i).degree < b.degree\nf : Fin m.succ → Fin d → Fq := fun i j ↦ (A i).coeff ↑j\n⊢ Fintype.card (Fin d → Fq) < Fintype.card ... | [
"Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Semiring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nhb : b.natDegree ≤ d\nA : Fin m.succ → Fq[X]\nhA : ∀ (i : Fin m.succ), (A i).degree < b.degree\nf : Fin m.succ → Fin d → Fq := fun i j ↦ (A i).coeff ↑j\n⊢ Fintype.card Fq ^ d ≤ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 15
} | {
"line": 75,
"column": 16
} | [
{
"pp": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin m.succ → Fq[X]\nhA : ∀ (i : Fin m.succ), (A i).degree < b.degree\nhb : b ≠ 0\nf : Fin m.succ → Fin d → Fq := fun i j ↦ (A i).coeff (b.natDegree - ↑j.succ)\n⊢ Fintype.card (Fin d → Fq) < Fintyp... | [
"Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nb : Fq[X]\nA : Fin m.succ → Fq[X]\nhA : ∀ (i : Fin m.succ), (A i).degree < b.degree\nhb : b ≠ 0\nf : Fin m.succ → Fin d → Fq := fun i j ↦ (A i).coeff (b.natDegree - ↑j.succ)\n⊢ Fintype.card Fq ^ d ≤ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Chebyshev | {
"line": 95,
"column": 4
} | {
"line": 95,
"column": 15
} | {
"line": 95,
"column": 16
} | [
{
"pp": "case refine_2\nx : ℝ\nhy : 2 ≤ x\nthis : 0 ≤ x\n⊢ 2 ∈ {p ∈ Ioc 0 ⌊x⌋₊ | Nat.Prime p}",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Real.partialOrder",
"Real",
"Finset.mem_filter._simp_1",
"Nat.Prime",
... | [
"case refine_2\nx : ℝ\nhy : 2 ≤ x\nthis : 0 ≤ x\n⊢ 2 ≤ ⌊x⌋₊ ∧ Nat.Prime 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.Chebyshev | {
"line": 326,
"column": 4
} | {
"line": 326,
"column": 68
} | {
"line": 327,
"column": 2
} | [
{
"pp": "case h\nn k : ℕ\nhk : k ∈ range (n + 1)\n⊢ n.choose k ≤ n.lcmUpto",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Nat.choose",
"Nat.lcmUpto_pos",
"Nat.lcmUpto",
"_private.Mathlib.NumberTheory.Chebyshev.0.Chebyshev.two_pow_le_mul_lcmUpto._proof_1_1",
"Ch... | [] | exact le_of_dvd (lcmUpto_pos n) (choose_dvd_lcmUpto <| by grind) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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