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 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 89
} | {
"line": 215,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\np : ℝ\nhp : 1 ≤ p\nr : ℝ\nh₁ : 0 < p\nthis : (ENNReal.ofReal p).toReal = p\neq_norm : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)\n⊢ volume {x | (∑ i, |x i| ^ p) ^ (1 / p) ≤ r} =\n ENNReal.ofReal r ^ card ι * ENNReal.o... | [
"ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\np : ℝ\nhp : 1 ≤ p\nr : ℝ\nh₁ : 0 < p\nthis✝ : (ENNReal.ofReal p).toReal = p\neq_norm : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)\nthis : Fact (1 ≤ ENNReal.ofReal p)\n⊢ volume {x | (∑ i, |x i| ^ p) ^ (1 / p) ≤ r} =\n ENNReal.ofRe... | have : Fact (1 ≤ ENNReal.ofReal p) := fact_iff.mpr (ofReal_one ▸ (ofReal_le_ofReal hp)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.TightNormed | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 66
} | {
"line": 224,
"column": 2
} | [
{
"pp": "E : Type u_1\nmE : MeasurableSpace E\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : BorelSpace E\nμ : ℕ → Measure E\ninst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)\nh : ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (... | [] | exact MeasurableSet.preimage measurableSet_Ioi (by fun_prop) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.TightNormed | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 66
} | {
"line": 224,
"column": 2
} | [
{
"pp": "E : Type u_1\nmE : MeasurableSpace E\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : BorelSpace E\nμ : ℕ → Measure E\ninst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)\nh : ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (... | [] | exact MeasurableSet.preimage measurableSet_Ioi (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.TightNormed | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 66
} | {
"line": 224,
"column": 2
} | [
{
"pp": "E : Type u_1\nmE : MeasurableSpace E\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : BorelSpace E\nμ : ℕ → Measure E\ninst✝ : ∀ (i : ℕ), IsFiniteMeasure (μ i)\nh : ∀ (y : E), Tendsto (fun r ↦ limsup (fun n ↦ (... | [] | exact MeasurableSet.preimage measurableSet_Ioi (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 354,
"column": 6
} | {
"line": 373,
"column": 22
} | {
"line": 374,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.r... | [] | gcongr
· have : ρ = ρ.restrict (partialSups K n)ᶜ +
∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i) := by
rw [I, ← FiniteMeasure.restrict_union disjoint_compl_left (A n).measurableSet]
simp
nth_rewrite 1 [this]
rw [toMeasure_add, integral_add_measure (g.inte... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 354,
"column": 6
} | {
"line": 373,
"column": 22
} | {
"line": 374,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nu : ℕ → ℝ≥0\nK : ℕ → Set E\nC : ℝ≥0\nhu : Tendsto u atTop (𝓝 0)\nhK : ∀ (n : ℕ), IsCompact (K n)\nh : NormalSpace E ∨ Monotone K\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.r... | [] | gcongr
· have : ρ = ρ.restrict (partialSups K n)ᶜ +
∑ i ∈ Finset.range (n + 1), ρ.restrict (disjointed K i) := by
rw [I, ← FiniteMeasure.restrict_union disjoint_compl_left (A n).measurableSet]
simp
nth_rewrite 1 [this]
rw [toMeasure_add, integral_add_measure (g.inte... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.MeasuredSets | {
"line": 104,
"column": 15
} | {
"line": 104,
"column": 18
} | {
"line": 104,
"column": 19
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nhC : IsSetRing C\nh'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0\nh : mα = generateFrom C\ns✝ : Set α\nhs✝ : MeasurableSet s✝\nε : ℝ≥0∞\nhε : 0 < ε\ns : Set α\nhs : MeasurableSet s\n⊢ (∀ (ε : ℝ≥0∞), 0 < ε →... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nhC : IsSetRing C\nh'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0\nh : mα = generateFrom C\ns✝ : Set α\nhs✝ : MeasurableSet s✝\nε : ℝ≥0∞\nhε : 0 < ε\ns : Set α\nhs : MeasurableSet s\nh's : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ t ∈ C,... | h's | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.MeasuredSets | {
"line": 121,
"column": 8
} | {
"line": 121,
"column": 45
} | {
"line": 122,
"column": 8
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nhC : IsSetRing C\nh'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0\nh : mα = generateFrom C\ns✝ : Set α\nhs✝ : MeasurableSet s✝\nε✝ : ℝ≥0∞\nhε : 0 < ε✝\ns : Set α\nhs : MeasurableSet s\nh's : ∀ (ε : ℝ≥0∞), 0 ... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nhC : IsSetRing C\nh'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0\nh : mα = generateFrom C\ns✝ : Set α\nhs✝ : MeasurableSet s✝\nε✝ : ℝ≥0∞\nhε : 0 < ε✝\ns : Set α\nhs : MeasurableSet s\nh's : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ t ∈ ... | apply MeasurableSet.nullMeasurableSet | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.MeasuredSets | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 96
} | {
"line": 188,
"column": 2
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nhC : IsSetRing C\nh'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0\nh : mα = generateFrom C\ns : MeasuredSets μ\nε : ℝ≥0∞\nεpos : ε > 0\n⊢ (Metric.eball s ε ∩ SetLike.coe ⁻¹' C).Nonempty",
"ppTerm": "?m.3... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nhC : IsSetRing C\nh'C : ∃ D, D.Countable ∧ D ⊆ C ∧ μ (⋃₀ D)ᶜ = 0\nh : mα = generateFrom C\ns : MeasuredSets μ\nε : ℝ≥0∞\nεpos : ε > 0\nt : Set α\ntC : t ∈ C\nht : μ (t ∆ ↑s) < ε\n⊢ (Metric.eball s ε ∩ SetLike.coe ⁻¹' C... | rcases exists_measure_symmDiff_lt_of_generateFrom_isSetRing hC h'C h s.2 εpos with ⟨t, tC, ht⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Measure.LevyConvergence | {
"line": 140,
"column": 6
} | {
"line": 140,
"column": 27
} | {
"line": 141,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : ℕ → Measure E\ninst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μ i)\nf : E → ℂ\nhf : ContinuousAt f 0\nh : ∀ (t : E), Tendsto (fun n ↦ charFun (μ ... | [] | rwa [tendsto_pi_nhds] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Measure.SubFinite | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 35
} | {
"line": 65,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\ninst✝ : IsFiniteMeasure (μ.withDensity g)\nhg : Measurable g\nhgf : g ≤ᵐ[μ] f\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (a : α) in s, g a ∂μ ≠ ∞",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"MeasureTheory.M... | [] | simp [← withDensity_apply _ hs] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.SubFinite | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 35
} | {
"line": 65,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\ninst✝ : IsFiniteMeasure (μ.withDensity g)\nhg : Measurable g\nhgf : g ≤ᵐ[μ] f\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (a : α) in s, g a ∂μ ≠ ∞",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"MeasureTheory.M... | [] | simp [← withDensity_apply _ hs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.SubFinite | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 35
} | {
"line": 65,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\ninst✝ : IsFiniteMeasure (μ.withDensity g)\nhg : Measurable g\nhgf : g ≤ᵐ[μ] f\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (a : α) in s, g a ∂μ ≠ ∞",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"MeasureTheory.M... | [] | simp [← withDensity_apply _ hs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.SubFinite | {
"line": 82,
"column": 13
} | {
"line": 82,
"column": 42
} | {
"line": 82,
"column": 42
} | [
{
"pp": "case refine_1\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\ninst✝ : IsFiniteMeasure (μ.withDensity g)\nhf : Measurable f\nhg : Measurable g\nt : Set α := {x | f x ≤ g x}\nht : MeasurableSet t\nh_zero : (μ.withDensity (f - g)).restrict t = 0\n⊢ (μ.withDensity (f - g)).restrict tᶜ... | [
"case refine_1\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\ninst✝ : IsFiniteMeasure (μ.withDensity g)\nhf : Measurable f\nhg : Measurable g\nt : Set α := {x | f x ≤ g x}\nht : MeasurableSet t\nh_zero : (μ.withDensity (f - g)).restrict t = 0\n⊢ (μ.restrict tᶜ).withDensity (f - g) ≤ (μ.restri... | restrict_withDensity ht.compl | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 55
} | {
"line": 354,
"column": 4
} | [
{
"pp": "X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\ninst✝¹ : CountablyGenerated X\ninst✝ : SigmaFinite μ\n⊢ ∃ 𝒜, 𝒜.Countable ∧ μ.MeasureDense 𝒜",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"MeasurableSpace.countableGeneratingSet",
"MeasurableSpace.countable_count... | [
"X : Type u_1\nm : MeasurableSpace X\nμ : Measure X\ninst✝¹ : CountablyGenerated X\ninst✝ : SigmaFinite μ\nh : (countableGeneratingSet X).Countable\n⊢ ∃ 𝒜, 𝒜.Countable ∧ μ.MeasureDense 𝒜"
] | have h := countable_countableGeneratingSet (α := X) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Order.UpperLower | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 22
} | {
"line": 86,
"column": 0
} | [
{
"pp": "case refine_1\nι : Type u_1\ninst✝ : Fintype ι\ns : Set (ι → ℝ)\nx : ι → ℝ\nf : (δ : ℝ) → 0 < δ → ι → ℝ\nhf₀ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ closedBall x δ\nhf₁ : ∀ (δ : ℝ) (a : 0 < δ), closedBall (f δ a) (δ / 4) ⊆ interior s\nH : Tendsto (fun r ↦ volume (closure s ∩ closedBall x ... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 651,
"column": 11
} | {
"line": 651,
"column": 52
} | {
"line": 651,
"column": 53
} | [
{
"pp": "case h\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 : Nonempty ... | [
"case h\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 : Nonempty 𝓧\nD : ℕ → ... | ← subset_closure (s := ball (D i) (u m)), | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Variation.Basic | {
"line": 306,
"column": 2
} | {
"line": 306,
"column": 71
} | {
"line": 307,
"column": 2
} | [
{
"pp": "X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝ : NormedAddCommGroup V\nμ ν : VectorMeasure X V\nx : X\nv : V\n⊢ IsFiniteMeasure (dirac x v).variation",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NNReal.instCommSemiring",
"MeasureTheory.Ve... | [
"X : Type u_1\nV : Type u_2\nmX : MeasurableSpace X\ninst✝ : NormedAddCommGroup V\nμ ν : VectorMeasure X V\nx : X\nv : V\n⊢ IsFiniteMeasure (‖v‖₊ • Measure.dirac x)"
] | simp only [variation_dirac, enorm_eq_nnnorm, Measure.coe_nnreal_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Arithmetic.Presburger.Definability | {
"line": 96,
"column": 8
} | {
"line": 96,
"column": 17
} | {
"line": 97,
"column": 6
} | [
{
"pp": "case func.inl.one\nα : Type u_1\nA : Set ℕ\ninst✝ : Fintype α\nts : Fin 0 → presburger[[↑A]].Term α\nk : Fin 0 → ℕ\nu : Fin 0 → α → ℕ\nih : ∀ (a : Fin 0) (v : α → ℕ), Term.realize v (ts a) = k a + u a ⬝ᵥ v\nv : α → ℕ\n⊢ (Structure.funMap presburgerFunc.one fun i ↦ Term.realize v (ts i)) = 1 + 0 ⬝ᵥ v",
... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Arithmetic.Presburger.Definability | {
"line": 101,
"column": 8
} | {
"line": 101,
"column": 17
} | {
"line": 102,
"column": 4
} | [
{
"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\nv : α → ℕ\n⊢ (Structure.funMap presburgerFunc.add fun i ↦ Term.realize v (ts i)) = k 0 + u 0 ⬝ᵥ ... | [] | simp [ih] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 42
} | {
"line": 117,
"column": 4
} | [
{
"pp": "case neg\nM : 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 ∈ ... | [
"case neg\nM : 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_mem'... | rcases hy with ⟨w, u, ⟨hu₁, hu₂⟩, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.ModelTheory.DirectLimit | {
"line": 277,
"column": 4
} | {
"line": 277,
"column": 15
} | {
"line": 278,
"column": 4
} | [
{
"pp": "L : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirectedOrder ι\ninst✝¹ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝ : Nonempty ι\ni : ι\n⊢ ∀ {n : ℕ} (r : L.Relations n) (x : Fin n → G i), RelMap r... | [
"L : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirectedOrder ι\ninst✝¹ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝ : Nonempty ι\ni : ι\nn : ℕ\nR : L.Relations n\nx : Fin n → G i\n⊢ RelMap R ((fun a ↦ ⟦Struct... | intro n R x | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.ModelTheory.DirectLimit | {
"line": 323,
"column": 4
} | {
"line": 323,
"column": 30
} | {
"line": 323,
"column": 31
} | [
{
"pp": "case h\nL : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirectedOrder ι\ninst✝¹ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝ : Nonempty ι\nS : L.Substructure (DirectLimit G f)\nS_fg : S.FG\nA : Fin... | [
"case h\nL : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : IsDirectedOrder ι\ninst✝¹ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝ : Nonempty ι\nS : L.Substructure (DirectLimit G f)\nS_fg : S.FG\nA : Finset (DirectL... | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.DirectLimit | {
"line": 466,
"column": 8
} | {
"line": 466,
"column": 29
} | {
"line": 466,
"column": 30
} | [
{
"pp": "L : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : Nonempty ι\ninst✝¹ : IsDirectedOrder ι\nM : Type u_1\ninst✝ : L.Structure M\nS : ι →o L.Substructure M\nliftInclusion_in_sup :\n ∀ (x : DirectLimit (fun i... | [
"L : Language\nι : Type v\ninst✝⁴ : Preorder ι\nG : ι → Type w\ninst✝³ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝² : Nonempty ι\ninst✝¹ : IsDirectedOrder ι\nM : Type u_1\ninst✝ : L.Structure M\nS : ι →o L.Substructure M\nliftInclusion_in_sup :\n ∀ (x : DirectLimit (fun i ↦ ↥(S i)) f... | ← rangeLiftInclusion, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.DirectLimit | {
"line": 479,
"column": 2
} | {
"line": 479,
"column": 36
} | {
"line": 480,
"column": 2
} | [
{
"pp": "L : Language\nι : Type v\ninst✝³ : Preorder ι\ninst✝² : Nonempty ι\ninst✝¹ : IsDirectedOrder ι\nM : Type u_1\ninst✝ : L.Structure M\nS : ι →o L.Substructure M\ni : ι\nx : ↥(S i)\n⊢ (Equiv_iSup S) ((Equiv_iSup S).symm ((Substructure.inclusion ⋯) x)) =\n (Equiv_iSup S) ((of L ι (fun x ↦ ↥(S x)) (fun x... | [
"L : Language\nι : Type v\ninst✝³ : Preorder ι\ninst✝² : Nonempty ι\ninst✝¹ : IsDirectedOrder ι\nM : Type u_1\ninst✝ : L.Structure M\nS : ι →o L.Substructure M\ni : ι\nx : ↥(S i)\n⊢ (Substructure.inclusion ⋯) x = (Equiv_iSup S) ((of L ι (fun x ↦ ↥(S x)) (fun x x_1 h ↦ Substructure.inclusion ⋯) i) x)"
] | simp only [Equiv.apply_symm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 663,
"column": 10
} | {
"line": 663,
"column": 63
} | {
"line": 664,
"column": 10
} | [
{
"pp": "case mpr.refine_1\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\ni : ↑hs.basisSet\ny : ι → ℕ\nhy : y ∈ closure {↑i}\nz : ι → ℕ\nhz : z ∈ closure (hs.basisSet \\ {↑i})\nz' : ι → ℕ\nhz' : z' ∈ closure (hs.basisSet \\ {↑i})\nheq : hs.fract (x + ↑i) = hs.fract (hs.ba... | [
"case mpr.refine_1\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\ni : ↑hs.basisSet\ny : ι → ℕ\nhy : y ∈ closure {↑i}\nz : ι → ℕ\nhz : z ∈ closure (hs.basisSet \\ {↑i})\nz' : ι → ℕ\nhz' : z' ∈ closure (hs.basisSet \\ {↑i})\nheq : hs.fract x = hs.fract (hs.base + z)\n⊢ hs.fract... | hs.fract_add_of_mem_closure (mem_closure_of_mem i.2), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.PartialEquiv | {
"line": 492,
"column": 48
} | {
"line": 492,
"column": 90
} | {
"line": 492,
"column": 90
} | [
{
"pp": "case e'_4\nL : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\ng : L.FGEquiv M N\nH : L.IsExtensionPair M N\nX : Set M\nleft✝ : X.Countable\nX_gen : (closure L).toFun X = ⊤\nx✝² : Countable ↑X\nx✝¹ : Encodable ↑X\nD : ↑X → Order.Cofinal (L.FGEquiv M N) := fun x ↦ H.def... | [] | apply Embedding.toPartialEquiv_toEmbedding | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.ModelTheory.Fraisse | {
"line": 417,
"column": 29
} | {
"line": 425,
"column": 29
} | {
"line": 427,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : Countable M\ninst✝¹ : Infinite M\ninst✝ : Language.empty.Structure M\nS : Language.empty.Substructure M\nhS : S.FG\nf : ↥S ↪[Language.empty] M\n⊢ ∃ g, f = g.toEmbedding.comp S.subtype",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"FirstOrder.Language... | [] | by
classical
have : Finite S := hS.finite
have : Infinite { x // x ∉ S } := ((Set.toFinite _).infinite_compl).to_subtype
have : Finite f.toHom.range := (((Substructure.fg_iff_structure_fg S).1 hS).range _).finite
have : Infinite { x // x ∉ f.toHom.range } := ((Set.toFinite _).infinite_compl).to_subt... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 726,
"column": 10
} | {
"line": 726,
"column": 63
} | {
"line": 726,
"column": 64
} | [
{
"pp": "case mpr.refine_1\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\ni : ↑hs.basisSet\nhi : ↑i ∉ hs.periods\ny : ι → ℕ\nhy : y ∈ closure {↑i}\nz : ι → ℕ\nhz : z ∈ closure (hs.basisSet \\ {↑i})\nz' : ι → ℕ\nhz' : z' ∈ closure (hs.basisSet \\ {↑i})\nheq : hs.fract x = ... | [
"case mpr.refine_1\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\ni : ↑hs.basisSet\nhi : ↑i ∉ hs.periods\ny : ι → ℕ\nhy : y ∈ closure {↑i}\nz : ι → ℕ\nhz : z ∈ closure (hs.basisSet \\ {↑i})\nz' : ι → ℕ\nhz' : z' ∈ closure (hs.basisSet \\ {↑i})\nheq : hs.fract x = hs.fract hs.... | hs.fract_add_of_mem_closure (mem_closure_of_mem i.2), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.PartialEquiv | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 44
} | {
"line": 531,
"column": 0
} | [
{
"pp": "case e'_4\nL : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\ng : L.FGEquiv M N\next_dom : L.IsExtensionPair M N\next_cod : L.IsExtensionPair N M\nX : Set M\nX_count : X.Countable\nX_gen : (closure L).toFun X = ⊤\nY : Set N\nY_count : Y.Countable\nY_gen : (closure L).... | [] | apply Embedding.toPartialEquiv_toEmbedding | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.ModelTheory.Order | {
"line": 307,
"column": 2
} | {
"line": 312,
"column": 25
} | {
"line": 314,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w'\ninst✝³ : L.IsOrdered\ninst✝² : L.Structure M\ninst✝¹ : Preorder M\ninst✝ : L.OrderedStructure M\n⊢ M ⊨ L.denselyOrderedSentence ↔ DenselyOrdered M",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"FirstOrder.Language.BoundedFormula.imp",
"First... | [] | simp only [denselyOrderedSentence, Sentence.Realize, Formula.Realize,
BoundedFormula.realize_imp, BoundedFormula.realize_all, Term.realize_lt,
BoundedFormula.realize_ex, BoundedFormula.realize_inf]
refine ⟨fun h => ⟨fun a b ab => h a b ab⟩, ?_⟩
intro h a b ab
exact exists_between ab | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Order | {
"line": 307,
"column": 2
} | {
"line": 312,
"column": 25
} | {
"line": 314,
"column": 0
} | [
{
"pp": "L : Language\nM : Type w'\ninst✝³ : L.IsOrdered\ninst✝² : L.Structure M\ninst✝¹ : Preorder M\ninst✝ : L.OrderedStructure M\n⊢ M ⊨ L.denselyOrderedSentence ↔ DenselyOrdered M",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"FirstOrder.Language.BoundedFormula.imp",
"First... | [] | simp only [denselyOrderedSentence, Sentence.Realize, Formula.Realize,
BoundedFormula.realize_imp, BoundedFormula.realize_all, Term.realize_lt,
BoundedFormula.realize_ex, BoundedFormula.realize_inf]
refine ⟨fun h => ⟨fun a b ab => h a b ab⟩, ?_⟩
intro h a b ab
exact exists_between ab | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Types | {
"line": 135,
"column": 53
} | {
"line": 135,
"column": 69
} | {
"line": 135,
"column": 70
} | [
{
"pp": "L : Language\nT : L.Theory\nα : Type w\nφ : L[[α]].Sentence\n⊢ {p | φ ∈ p}ᶜ = ∅ ↔ ¬((L.lhomWithConstants α).onTheory T ∪ {Formula.not φ}).IsSatisfiable",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"FirstOrder.Language.Theory.CompleteType.compl_setOf_mem",
"Eq.mpr",
... | [
"L : Language\nT : L.Theory\nα : Type w\nφ : L[[α]].Sentence\n⊢ {p | Formula.not φ ∈ p} = ∅ ↔ ¬((L.lhomWithConstants α).onTheory T ∪ {Formula.not φ}).IsSatisfiable"
] | compl_setOf_mem, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ADEInequality | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 80
} | {
"line": 161,
"column": 2
} | [
{
"pp": "p q r : ℕ+\nhpq : p ≤ q\nhqr : q ≤ r\nH : 3 ≤ p\nh3q : 3 ≤ q\nh3r : 3 ≤ r\nhp : (↑↑p)⁻¹ ≤ 3⁻¹\n⊢ (↑↑p)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹ ≤ 1",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"PNat.val",
"Rat.instOfNat",
"instLinearOrderPNat",
"MulZeroClass.toMul",
... | [
"p q r : ℕ+\nhpq : p ≤ q\nhqr : q ≤ r\nH : 3 ≤ p\nh3q : 3 ≤ q\nh3r : 3 ≤ r\nhp : (↑↑p)⁻¹ ≤ 3⁻¹\nhq : (↑↑q)⁻¹ ≤ 3⁻¹\n⊢ (↑↑p)⁻¹ + (↑↑q)⁻¹ + (↑↑r)⁻¹ ≤ 1"
] | have hq : (q : ℚ)⁻¹ ≤ 3⁻¹ := inv_anti₀ (by positivity) (by exact_mod_cast h3q) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.ArithmeticFunction.Carmichael | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 29
} | {
"line": 137,
"column": 0
} | [
{
"pp": "n : ℕ\nhn : n ≤ 2\n⊢ φ (2 ^ n) = 2 ^ (n - 1)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Nat.zero_le",
"of_decide_eq_true",
"Nat.rawCast",
"Nat.instMonoid",
"HSub.hSub",
"id",
"instSubNat",
"instOfNatNat",
"Mathlib.Meta.No... | [] | interval_cases n <;> decide | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 35
} | {
"line": 112,
"column": 2
} | [
{
"pp": "case h.e_a.e_a\nR : Type u_1\ninst✝ : CommSemiring R\nu v : R\np : ℕ\nhp : Nat.Prime p\nx a : R\nha : u = v * a\nb : R\nhb : u ^ p = ↑p * u * v * b\n⊢ (u * x) ^ 1 * ↑(p.choose 1) = ↑p * u * x",
"ppTerm": "?h.e_a.e_a✝",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.m... | [
"case h.e_a.e_a\nR : Type u_1\ninst✝ : CommSemiring R\nu v : R\np : ℕ\nhp : Nat.Prime p\nx a : R\nha : u = v * a\nb : R\nhb : u ^ p = ↑p * u * v * b\n⊢ ∑ x_1 ∈ (((Finset.range (p + 1)).erase 0).erase 1).erase p, (u * x) ^ x_1 * ↑(p.choose x_1) +\n (u * x) ^ p * ↑(p.choose p) =\n ↑p * u *\n (v *\n ... | · rw [Nat.choose_one_right]; ring | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.AbelSummation | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 75
} | {
"line": 151,
"column": 4
} | [
{
"pp": "𝕜 : 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\nau... | [
"𝕜 : 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 : ↑⌊b⌋₊ ≤... | rw [← Ico_add_one_add_one_eq_Ioc, Nat.sub_add_cancel (by lia), Eq.comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 235,
"column": 9
} | {
"line": 235,
"column": 35
} | {
"line": 236,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ IsCyclic (ZMod (2 ^ 2))ˣ ↔ 2 ≤ 2",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"ZMod.commRing",
"instReflLe",
"congrArg",
"CommSemiring.toSemiring",
"Nat.instMonoid",
"DivInvMonoid.toZPow",
"Std.le_refl._simp_1",
"Units",
... | [] | simp [isCyclic_units_four] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 235,
"column": 9
} | {
"line": 235,
"column": 35
} | {
"line": 236,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ IsCyclic (ZMod (2 ^ 2))ˣ ↔ 2 ≤ 2",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"ZMod.commRing",
"instReflLe",
"congrArg",
"CommSemiring.toSemiring",
"Nat.instMonoid",
"DivInvMonoid.toZPow",
"Std.le_refl._simp_1",
"Units",
... | [] | simp [isCyclic_units_four] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 235,
"column": 9
} | {
"line": 235,
"column": 35
} | {
"line": 236,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ IsCyclic (ZMod (2 ^ 2))ˣ ↔ 2 ≤ 2",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"ZMod.commRing",
"instReflLe",
"congrArg",
"CommSemiring.toSemiring",
"Nat.instMonoid",
"DivInvMonoid.toZPow",
"Std.le_refl._simp_1",
"Units",
... | [] | simp [isCyclic_units_four] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 237,
"column": 4
} | {
"line": 237,
"column": 43
} | {
"line": 238,
"column": 4
} | [
{
"pp": "n✝ n : ℕ\n⊢ IsCyclic (ZMod (2 ^ (n + 3)))ˣ ↔ n + 3 ≤ 2",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"of_decide_eq_true",
"ZMod.commRing",
"iff_false",
"congrArg",
"CommSemiring.toSemiring",
"Nat.Simproc.add_le_g... | [
"n✝ n : ℕ\n⊢ ¬IsCyclic (ZMod (2 ^ (n + 3)))ˣ"
] | simp only [Nat.reduceLeDiff, iff_false] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.Exp | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 32
} | {
"line": 97,
"column": 4
} | [
{
"pp": "case succ\nA : Type u_3\ninst✝² : CommRing A\ninst✝¹ : Algebra ℚ A\ninst✝ : IsAddTorsionFree A\nf : A⟦X⟧\nhd : (d⁄dX A) f = f\nhc : constantCoeff f = 1\nn : ℕ\nih : (coeff n) f = (coeff n) (exp A)\neq1 : (coeff n) ((d⁄dX A) f) = (coeff n) f\n⊢ (coeff (n + 1)) f = (coeff (n + 1)) (exp A)",
"ppTerm":... | [
"case succ\nA : Type u_3\ninst✝² : CommRing A\ninst✝¹ : Algebra ℚ A\ninst✝ : IsAddTorsionFree A\nf : A⟦X⟧\nhd : (d⁄dX A) f = f\nhc : constantCoeff f = 1\nn : ℕ\nih : (coeff n) f = (coeff n) (exp A)\neq1 : (coeff (n + 1)) f * (↑n + 1) = (coeff n) f\n⊢ (coeff (n + 1)) f = (coeff (n + 1)) (exp A)"
] | rw [coeff_derivative] at eq1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 30
} | {
"line": 264,
"column": 2
} | [
{
"pp": "case inr.inl\nhn0 : 1 ≠ 0\n⊢ IsCyclic (ZMod (4 * 1))ˣ ↔ 1 = 0 ∨ 1 = 1",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"HMul.hMul",
"ZMod.commRing",
"Nat.instOne",
"congrArg",
"CommSemiring.toSemiring",
... | [] | simp [isCyclic_units_four] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 30
} | {
"line": 264,
"column": 2
} | [
{
"pp": "case inr.inl\nhn0 : 1 ≠ 0\n⊢ IsCyclic (ZMod (4 * 1))ˣ ↔ 1 = 0 ∨ 1 = 1",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"HMul.hMul",
"ZMod.commRing",
"Nat.instOne",
"congrArg",
"CommSemiring.toSemiring",
... | [] | simp [isCyclic_units_four] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 30
} | {
"line": 264,
"column": 2
} | [
{
"pp": "case inr.inl\nhn0 : 1 ≠ 0\n⊢ IsCyclic (ZMod (4 * 1))ˣ ↔ 1 = 0 ∨ 1 = 1",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"HMul.hMul",
"ZMod.commRing",
"Nat.instOne",
"congrArg",
"CommSemiring.toSemiring",
... | [] | simp [isCyclic_units_four] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 337,
"column": 13
} | {
"line": 337,
"column": 39
} | {
"line": 338,
"column": 2
} | [
{
"pp": "case pos\nn : ℕ\nh0 : ¬n = 0\nh1 : ¬n = 1\nh2 : ¬n = 2\nh4 : n = 4\n⊢ IsCyclic (ZMod 4)ˣ ↔ 4 = 0 ∨ 4 = 1 ∨ 4 = 2 ∨ 4 = 4 ∨ ∃ p m, Nat.Prime p ∧ Odd p ∧ 1 ≤ m ∧ (4 = p ^ m ∨ 4 = 2 * p ^ m)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"False",
"Nat.Prime",
"HMul... | [] | simp [isCyclic_units_four] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 20
} | {
"line": 163,
"column": 21
} | [
{
"pp": "n p : ℕ\n⊢ (↑p + 1) * ∑ k ∈ range n, ↑k ^ p = eval (↑n) (bernoulli p.succ) - _root_.bernoulli p.succ",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"Rat.instOfNat",
"Rat.instSub",
"Eq.mpr",
"Polynomial.eval",
"Rat.instMul",... | [
"n p : ℕ\n⊢ (↑p + 1) * ∑ i ∈ range (p + 1), _root_.bernoulli i * ↑((p + 1).choose i) * ↑n ^ (p + 1 - i) / (↑p + 1) =\n eval (↑n) (bernoulli p.succ) - _root_.bernoulli p.succ"
] | sum_range_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 195,
"column": 42
} | {
"line": 195,
"column": 55
} | {
"line": 195,
"column": 55
} | [
{
"pp": "n d : ℕ\nhd : ∀ m < d + 1, (bernoulli m).comp (1 + X) = bernoulli m + m • X ^ (m - 1)\nx✝ : ℕ\na✝ : x✝ ∈ range (d + 1)\n⊢ x✝ ∈ range (d + 1)",
"ppTerm": "?m.147",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"a✝"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.PrimeCounting | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 31
} | {
"line": 95,
"column": 0
} | [
{
"pp": "x✝ : ℕ\n⊢ (π ∘ fun n ↦ n - 1) x✝ = π' x✝",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Nat.primeCounting_sub_one"
],
"usedFVars": [
"x✝"
],
"usedGoals": []
}
] | [] | exact primeCounting_sub_one _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Primorial | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 94
} | {
"line": 74,
"column": 0
} | [
{
"pp": "case hab\nm n : ℕ\n⊢ 0 ≤ m + 1",
"ppTerm": "?hab",
"assigned": true,
"usedConstants": [
"Nat.zero_le",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
"Nat",
"instAddNat",
"OfNat.ofNat"
],
"usedFVars": [
"m"
],
"usedGoals": []
},
... | [] | exacts [Nat.zero_le _, by lia, disjoint_filter_filter <| Ico_disjoint_Ico_consecutive _ _ _] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.NumberTheory.AbelSummation | {
"line": 314,
"column": 4
} | {
"line": 314,
"column": 100
} | {
"line": 315,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhc : c 0 = 0\nhf_diff : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ f t\nhf_int : LocallyIntegrableOn (deriv f) (Set.Ici 1) volume\nl : 𝕜\nh_lim : Tendsto (fun n ↦ f ↑n * ∑ k ∈ Icc 0 n, c k) atTop (𝓝 l)\ng : ℝ → ℝ\nhg_dom : (fun t ↦ deriv f t * ∑ k ∈ ... | [
"𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhc : c 0 = 0\nhf_diff : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ f t\nhf_int : LocallyIntegrableOn (deriv f) (Set.Ici 1) volume\nl : 𝕜\nh_lim : Tendsto (fun n ↦ f ↑n * ∑ k ∈ Icc 0 n, c k) atTop (𝓝 l)\ng : ℝ → ℝ\nhg_dom : (fun t ↦ deriv f t * ∑ k ∈ Icc 0 ⌊t⌋₊, ... | refine Tendsto.congr' h (intervalIntegral_tendsto_integral_Ioi _ ?_ tendsto_natCast_atTop_atTop) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.AbelSummation | {
"line": 356,
"column": 8
} | {
"line": 356,
"column": 70
} | {
"line": 357,
"column": 8
} | [
{
"pp": "case succ.calc_3\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... | [
"case succ.calc_3\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 ∈ Icc 0 n, ‖c k... | grw [sub_eq_add_neg, neg_le_abs, abs_integral_le_integral_abs] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.NumberTheory.Bertrand | {
"line": 185,
"column": 4
} | {
"line": 185,
"column": 69
} | {
"line": 186,
"column": 4
} | [
{
"pp": "case h₂\nn : ℕ\nn_large : 2 < n\nno_prime : ∀ (p : ℕ), Nat.Prime p → n < p → 2 * n < p\nn_pos : 0 < n\nn2_pos : 1 ≤ 2 * n\nS : Finset ℕ := {p ∈ Finset.range (2 * n / 3 + 1) | Nat.Prime p}\nf : ℕ → ℕ := fun x ↦ x ^ n.centralBinom.factorization x\nthis : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1),... | [
"case h₂.refine_1\nn : ℕ\nn_large : 2 < n\nno_prime : ∀ (p : ℕ), Nat.Prime p → n < p → 2 * n < p\nn_pos : 0 < n\nn2_pos : 1 ≤ 2 * n\nS : Finset ℕ := {p ∈ Finset.range (2 * n / 3 + 1) | Nat.Prime p}\nf : ℕ → ℕ := fun x ↦ x ^ n.centralBinom.factorization x\nthis : ∏ x ∈ S, f x = ∏ x ∈ Finset.range (2 * n / 3 + 1), f ... | refine (Finset.prod_le_prod' fun p hp => (?_ : f p ≤ p)).trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.AbelSummation | {
"line": 393,
"column": 4
} | {
"line": 395,
"column": 80
} | {
"line": 396,
"column": 2
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhf_diff : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x ↦ ‖f x‖) t\nhf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici 1) volume\ng : ℝ → ℝ\nhg₂ : IntegrableAtFilter g atTop volume\nh : ∀ (n : ℕ), ∑ k ∈ Icc 1 n, ‖c k‖ = ∑ k ... | [] | exact_mod_cast sum_mul_eq_sub_integral_mul₀' _ (by simp only [reduceIte, norm_zero]) n
(fun _ ht ↦ hf_diff _ ht.1)
(hf_int.integrableOn_compact_subset Set.Icc_subset_Ici_self isCompact_Icc) | Lean.Parser.Tactic._aux_Init_TacticsExtra___macroRules_Lean_Parser_Tactic_tacticExact_mod_cast__1 | Lean.Parser.Tactic.tacticExact_mod_cast_ |
Mathlib.NumberTheory.Bertrand | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 55
} | {
"line": 229,
"column": 2
} | [
{
"pp": "case inl\nn : ℕ\nhn0 : n ≠ 0\nh : 511 < n\n⊢ ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Nat.exists_prime_lt_and_le_two_mul_eventually"
],
"usedFVars": [
"n",
"h"
],
"usedGoals": []
},
{
"pp": "case inr... | [
"case inr\nn : ℕ\nhn0 : n ≠ 0\nh : n ≤ 511\n⊢ ∃ p, Prime p ∧ n < p ∧ p ≤ 2 * n"
] | · exact exists_prime_lt_and_le_two_mul_eventually n h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.AbelSummation | {
"line": 393,
"column": 4
} | {
"line": 395,
"column": 80
} | {
"line": 396,
"column": 2
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhf_diff : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x ↦ ‖f x‖) t\nhf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici 1) volume\ng : ℝ → ℝ\nhg₂ : IntegrableAtFilter g atTop volume\nh : ∀ (n : ℕ), ∑ k ∈ Icc 1 n, ‖c k‖ = ∑ k ... | [] | exact_mod_cast sum_mul_eq_sub_integral_mul₀' _ (by simp only [reduceIte, norm_zero]) n
(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": 393,
"column": 4
} | {
"line": 395,
"column": 80
} | {
"line": 396,
"column": 2
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhf_diff : ∀ t ∈ Set.Ici 1, DifferentiableAt ℝ (fun x ↦ ‖f x‖) t\nhf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici 1) volume\ng : ℝ → ℝ\nhg₂ : IntegrableAtFilter g atTop volume\nh : ∀ (n : ℕ), ∑ k ∈ Icc 1 n, ‖c k‖ = ∑ k ... | [] | exact_mod_cast sum_mul_eq_sub_integral_mul₀' _ (by simp only [reduceIte, norm_zero]) n
(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.Bernoulli | {
"line": 367,
"column": 30
} | {
"line": 367,
"column": 41
} | {
"line": 368,
"column": 2
} | [
{
"pp": "case succ\nn p : ℕ\n⊢ ∑ k ∈ Ico 1 (n + 1), ↑k ^ (p + 1) =\n ∑ i ∈ range (p + 1 + 1), bernoulli' i * ↑((p + 1 + 1).choose i) * ↑n ^ (p + 1 + 1 - i) / ↑(p + 1).succ",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"Mathlib.Tactic.Ring.Common.mul_pf... | [] | | succ p => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 520,
"column": 2
} | {
"line": 520,
"column": 28
} | {
"line": 521,
"column": 2
} | [
{
"pp": "k m : ℕ\nx : ℚ\nhm_lt : m < k\n⊢ ↑((2 * k + 1).choose (2 * m)) * (2 * ↑k - 2 * ↑m + 1) * x = ↑((2 * k).choose (2 * m)) * (2 * ↑k + 1) * x",
"ppTerm": "?m.108",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Rat.instSub",
"Rat.instMul",
"Nat.choose",
"HM... | [
"k m : ℕ\nx : ℚ\nhm_lt : m < k\n⊢ ↑((2 * k + 1).choose (2 * m)) * (2 * ↑k - 2 * ↑m + 1) = ↑((2 * k).choose (2 * m)) * (2 * ↑k + 1)"
] | refine congrArg (· * x) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.Bernoulli | {
"line": 558,
"column": 4
} | {
"line": 558,
"column": 26
} | {
"line": 559,
"column": 4
} | [
{
"pp": "case hx\nk m p : ℕ\nhm_lt : m < k\ninst✝ : Fact (Nat.Prime p)\nih : pIntegral p (bernoulli (2 * m) + vonStaudtIndicator (2 * m) p / ↑p)\nhp_ne : ↑p ≠ 0\nP : ℚ := ↑p ^ (2 * k - 2 * m - 1)\nhpow : ↑p ^ (2 * k - 2 * m) = P * ↑p\nhdecomp :\n bernoulli (2 * m) * ↑((2 * k + 1).choose (2 * m)) * ↑p ^ (2 * k ... | [
"case hx\nk m p : ℕ\nhm_lt : m < k\ninst✝ : Fact (Nat.Prime p)\nih : pIntegral p (bernoulli (2 * m) + vonStaudtIndicator (2 * m) p / ↑p)\nhp_ne : ↑p ≠ 0\nP : ℚ := ⋯\nhpow : ↑p ^ (2 * k - 2 * m) = P * ↑p\nhdecomp :\n bernoulli (2 * m) * ↑((2 * k + 1).choose (2 * m)) * ↑p ^ (2 * k - 2 * m) / (2 * ↑k + 1) =\n (ber... | apply pIntegral_mul ih | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.Chebyshev | {
"line": 318,
"column": 6
} | {
"line": 318,
"column": 57
} | {
"line": 319,
"column": 6
} | [
{
"pp": "case refine_2\nR : Type u_1\ninst✝ : AddCommMonoid R\nf : ℕ → R\nx : ℝ\nhx : 0 ≤ x\n⊢ ∀ (a b : ℕ),\n a ≠ 0 →\n a ≤ ⌊log x / log 2⌋₊ →\n b ≠ 0 →\n ↑b ≤ x →\n Nat.Prime b →\n ↑b ≤ x ^ (↑a)⁻¹ →\n ∀ (a_7 b_1 : ℕ),\n a_7 ≠ 0 →\n... | [
"case refine_2\nR : Type u_1\ninst✝ : AddCommMonoid R\nf : ℕ → R\nx : ℝ\nhx : 0 ≤ x\nk₁ p₁ : ℕ\nhk₁ : k₁ ≠ 0\na✝⁷ : k₁ ≤ ⌊log x / log 2⌋₊\na✝⁶ : p₁ ≠ 0\na✝⁵ : ↑p₁ ≤ x\nhp₁ : Nat.Prime p₁\na✝⁴ : ↑p₁ ≤ x ^ (↑k₁)⁻¹\nk₂ p₂ : ℕ\nhk₂ : k₂ ≠ 0\na✝³ : k₂ ≤ ⌊log x / log 2⌋₊\na✝² : p₂ ≠ 0\na✝¹ : ↑p₂ ≤ x\nhp₂ : Nat.Prime p₂\n... | intro k₁ p₁ hk₁ _ _ _ hp₁ _ k₂ p₂ hk₂ _ _ _ hp₂ _ H | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.NumberTheory.Bernoulli | {
"line": 604,
"column": 15
} | {
"line": 604,
"column": 29
} | {
"line": 604,
"column": 30
} | [
{
"pp": "k p : ℕ\nhk : 0 < k\nhfilter : ∑ v ∈ Ico 1 p, ↑v ^ (2 * k) = ∑ v ∈ range p, ↑v ^ (2 * k)\n⊢ ∑ v ∈ range p, ↑v ^ (2 * k) =\n ∑ i ∈ range (2 * k), bernoulli i * ↑((2 * k + 1).choose i) * ↑p ^ (2 * k + 1 - i) / (2 * ↑k + 1) +\n ↑p * bernoulli (2 * k)",
"ppTerm": "?m.178",
"assigned": true,... | [
"k p : ℕ\nhk : 0 < k\nhfilter : ∑ v ∈ Ico 1 p, ↑v ^ (2 * k) = ∑ v ∈ range p, ↑v ^ (2 * k)\n⊢ ∑ i ∈ range (2 * k + 1), bernoulli i * ↑((2 * k + 1).choose i) * ↑p ^ (2 * k + 1 - i) / (↑(2 * k) + 1) =\n ∑ i ∈ range (2 * k), bernoulli i * ↑((2 * k + 1).choose i) * ↑p ^ (2 * k + 1 - i) / (2 * ↑k + 1) +\n ↑p * be... | sum_range_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Chebyshev | {
"line": 439,
"column": 2
} | {
"line": 440,
"column": 58
} | {
"line": 441,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ ↑n * log 2 - log (↑n + 1) ≤ ψ ↑n",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"False",
"Real.partialOrder",
"Real.instLE",
"Real",
"HMul.hMul",
"Nat.ble",
"FloorRing.toFloorSemiri... | [
"n : ℕ\n⊢ log (2 ^ n) ≤ log ((↑n + 1) * ↑n.lcmUpto)"
] | rw [tsub_le_iff_left, psi_eq_log_lcmUpto, ← log_pow 2,
← log_mul (by positivity) (by simp [lcmUpto_ne_zero])] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 378,
"column": 12
} | {
"line": 378,
"column": 21
} | {
"line": 378,
"column": 22
} | [
{
"pp": "case neg.calc_2\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⊢ p ^ n + p ^ n * (p - 1) = p ^ (n + 1)",
"ppTerm": "?neg.calc_2✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"case neg.calc_2\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⊢ p ^ n + (p ^ n * p - p ^ n * 1) = p ^ (n + 1)"
] | mul_tsub, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 499,
"column": 66
} | {
"line": 508,
"column": 17
} | {
"line": 510,
"column": 0
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\n⊢ DenseRange Nat.cast",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"lt_of_le_of_lt",
"Real.instLE",
"Padic... | [] | by
intro x
rw [Metric.mem_closure_range_iff]
intro ε hε
obtain ⟨n, hn⟩ := exists_pow_neg_lt p hε
use x.appr n
rw [dist_eq_norm]
apply lt_of_le_of_lt _ hn
rw [norm_le_pow_iff_mem_span_pow]
apply appr_spec | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Cyclotomic.Discriminant | {
"line": 81,
"column": 6
} | {
"line": 81,
"column": 13
} | {
"line": 82,
"column": 6
} | [
{
"pp": "case e_a.inl\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)\nk : ℕ\ninst✝ : IsCyclotomicExtension {2 ^ (k.succ + 1)} K L\nhζ : IsPrimitiveRoot ζ (2 ^ (k.succ + 1))\nhirr : Irreducibl... | [
"case e_a.inl.zero\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)\ninst✝ : IsCyclotomicExtension {2 ^ (succ 0 + 1)} K L\nhζ : IsPrimitiveRoot ζ (2 ^ (succ 0 + 1))\nhirr : Irreducible (cyclotomic ... | cases k | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 234,
"column": 32
} | {
"line": 234,
"column": 62
} | {
"line": 234,
"column": 62
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn : ℕ\npn : xz a1 n * xz a1 n - ↑(d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n\nh : ↑(xn a1 n * xn a1 n) - ↑(d a1 * yn a1 n * yn a1 n) = 1\nhl : d a1 * yn a1 n * yn a1 n ≤ xn a1 n * xn a1 n\n⊢ ↑(xn a1 n * xn a1 n - d a1 * yn a1 n * yn a1 n) = ↑1",
"ppTerm": "?m.104",
"assig... | [] | rw [Int.ofNat_sub hl]; exact h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 234,
"column": 32
} | {
"line": 234,
"column": 62
} | {
"line": 234,
"column": 62
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nn : ℕ\npn : xz a1 n * xz a1 n - ↑(d a1) * yz a1 n * yz a1 n = 1 := pell_eqz a1 n\nh : ↑(xn a1 n * xn a1 n) - ↑(d a1 * yn a1 n * yn a1 n) = 1\nhl : d a1 * yn a1 n * yn a1 n ≤ xn a1 n * xn a1 n\n⊢ ↑(xn a1 n * xn a1 n - d a1 * yn a1 n * yn a1 n) = ↑1",
"ppTerm": "?m.104",
"assig... | [] | rw [Int.ofNat_sub hl]; exact h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 464,
"column": 6
} | {
"line": 465,
"column": 60
} | {
"line": 465,
"column": 60
} | [
{
"pp": "d : ℤ\nhd : d ≤ 0\nn : ℤ√d\n⊢ 0 ≤ -(d * n.im * n.im)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"mul_self_nonneg",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"mul_nonneg",
"Int.instIsStrictOrderedRing",... | [] | rw [mul_assoc, neg_mul_eq_neg_mul]
exact mul_nonneg (neg_nonneg.2 hd) (mul_self_nonneg _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 464,
"column": 6
} | {
"line": 465,
"column": 60
} | {
"line": 465,
"column": 60
} | [
{
"pp": "d : ℤ\nhd : d ≤ 0\nn : ℤ√d\n⊢ 0 ≤ -(d * n.im * n.im)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"mul_self_nonneg",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"mul_nonneg",
"Int.instIsStrictOrderedRing",... | [] | rw [mul_assoc, neg_mul_eq_neg_mul]
exact mul_nonneg (neg_nonneg.2 hd) (mul_self_nonneg _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 581,
"column": 4
} | {
"line": 584,
"column": 27
} | {
"line": 585,
"column": 2
} | [
{
"pp": "case inl.inr.inr\nd x y : ℕ\nha : { re := ↑x, im := ↑y }.Nonneg\nz w : ℕ\nhb : { re := -↑z, im := ↑w }.Nonneg\n⊢ ({ re := ↑x, im := ↑y } + { re := -↑z, im := ↑w }).Nonneg",
"ppTerm": "?inl.inr.inr",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroCla... | [] | refine nonnegg_cases_left fun i h => sqLe_of_le ?_ ?_ (nonnegg_neg_pos.1 hb)
· dsimp only at h
exact Int.ofNat_le.1 (le_of_neg_le_neg (Int.le.intro x (by simp [add_comm, *])))
· apply Nat.le_add_left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 581,
"column": 4
} | {
"line": 584,
"column": 27
} | {
"line": 585,
"column": 2
} | [
{
"pp": "case inl.inr.inr\nd x y : ℕ\nha : { re := ↑x, im := ↑y }.Nonneg\nz w : ℕ\nhb : { re := -↑z, im := ↑w }.Nonneg\n⊢ ({ re := ↑x, im := ↑y } + { re := -↑z, im := ↑w }).Nonneg",
"ppTerm": "?inl.inr.inr",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroCla... | [] | refine nonnegg_cases_left fun i h => sqLe_of_le ?_ ?_ (nonnegg_neg_pos.1 hb)
· dsimp only at h
exact Int.ofNat_le.1 (le_of_neg_le_neg (Int.le.intro x (by simp [add_comm, *])))
· apply Nat.le_add_left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 601,
"column": 4
} | {
"line": 602,
"column": 80
} | {
"line": 603,
"column": 4
} | [
{
"pp": "case inr.inr.inr.inr\nd x y : ℕ\nha : { re := -↑x, im := ↑y }.Nonneg\nz w : ℕ\nhb : { re := -↑z, im := ↑w }.Nonneg\n⊢ ({ re := -↑x, im := ↑y } + { re := -↑z, im := ↑w }).Nonneg",
"ppTerm": "?inr.inr.inr.inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Zsqrtd.Nonnegg",
... | [
"case inr.inr.inr.inr\nd x y : ℕ\nha : { re := -↑x, im := ↑y }.Nonneg\nz w : ℕ\nhb : { re := -↑z, im := ↑w }.Nonneg\nthis : { re := -↑(x + z), im := ↑(y + w) }.Nonneg\n⊢ ({ re := -↑x, im := ↑y } + { re := -↑z, im := ↑w }).Nonneg"
] | have : Nonneg ⟨_, _⟩ :=
nonnegg_neg_pos.2 (sqLe_add (nonnegg_neg_pos.1 ha) (nonnegg_neg_pos.1 hb)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 881,
"column": 2
} | {
"line": 881,
"column": 21
} | {
"line": 883,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : NonAssocRing R\nd : ℤ\nf g : ℤ√d →+* R\nh : f sqrtd = g sqrtd\nre_x im_x : ℤ\n⊢ f { re := re_x, im := im_x } = g { re := re_x, im := im_x }",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Zsqrtd.instMul",
"Int.cast",
"NonAssocSemiring.toAdd... | [] | simp [decompose, h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 897,
"column": 8
} | {
"line": 902,
"column": 12
} | {
"line": 902,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nd : ℤ\nr : { r // r * r = ↑d }\na b : ℤ√d\n⊢ ↑(a * b).re + ↑(a * b).im * ↑r = (↑a.re + ↑a.im * ↑r) * (↑b.re + ↑b.im * ↑r)",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.cast",
"Eq.m... | [] | have :
(a.re + a.im * r : R) * (b.re + b.im * r) =
a.re * b.re + (a.re * b.im + a.im * b.re) * r + a.im * b.im * (r * r) := by
ring
simp only [re_mul, Int.cast_add, Int.cast_mul, im_mul, this, r.prop]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 897,
"column": 8
} | {
"line": 902,
"column": 12
} | {
"line": 902,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nd : ℤ\nr : { r // r * r = ↑d }\na b : ℤ√d\n⊢ ↑(a * b).re + ↑(a * b).im * ↑r = (↑a.re + ↑a.im * ↑r) * (↑b.re + ↑b.im * ↑r)",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Int.cast",
"Eq.m... | [] | have :
(a.re + a.im * r : R) * (b.re + b.im * r) =
a.re * b.re + (a.re * b.im + a.im * b.re) * r + a.im * b.im * (r * r) := by
ring
simp only [re_mul, Int.cast_add, Int.cast_mul, im_mul, this, r.prop]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.DiophantineApproximation.Basic | {
"line": 284,
"column": 9
} | {
"line": 284,
"column": 86
} | {
"line": 284,
"column": 86
} | [
{
"pp": "ξ : ℝ\na b : ℤ\nx✝ : b ≠ 0\nh : ξ = ↑a / ↑b\nq : ℚ\n⊢ |↑a / ↑b - ↑q| < 1 / ↑q.den ^ 2 ↔ |↑a / ↑b - q| < 1 / ↑q.den ^ 2",
"ppTerm": "?m.177",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"Int.cast",
"Rat.instSub",
"Eq.mpr",
"Real.partialOrder",
"R... | [
"ξ : ℝ\na b : ℤ\nx✝ : b ≠ 0\nh : ξ = ↑a / ↑b\nq : ℚ\n⊢ |↑a / ↑b - ↑q| < ↑(1 / ↑q.den ^ 2) ↔ |↑a / ↑b - q| < 1 / ↑q.den ^ 2"
] | (by (push_cast; rfl) : (1 : ℝ) / (q.den : ℝ) ^ 2 = (1 / (q.den : ℚ) ^ 2 : ℚ)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 666,
"column": 30
} | {
"line": 666,
"column": 32
} | {
"line": 666,
"column": 32
} | [
{
"pp": "a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : j + 1 ≤ 2 * 1\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\no : j = n ∨ n < j\njn✝ : j > n\njn : j ≠ n\ns : xn a1 j % xn a1 n < ... | [
"a : ℕ\na1 : 1 < a\ni n : ℕ\nnpos : 0 < n\nj : ℕ\nij : i < j + 1\nj2n : 2 + 1 ≤ 2 * 1\njnn : j + 1 ≠ n\nntriv : ¬(a = 2 ∧ n = 1 ∧ i = 0 ∧ j + 1 = 2)\nlem2 : ∀ k > n, k ≤ 2 * n → ↑(xn a1 k % xn a1 n) = ↑(xn a1 n) - ↑(xn a1 (2 * n - k))\no : j = n ∨ n < j\njn✝ : j > n\njn : j ≠ n\ns : xn a1 j % xn a1 n < xn a1 (j + 1... | j2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.SmoothNumbers | {
"line": 62,
"column": 30
} | {
"line": 62,
"column": 43
} | {
"line": 62,
"column": 43
} | [
{
"pp": "s : Finset ℕ\nm k : ℕ\nh' : k ∣ m\nh₁ : m ≠ 0\nh₂ : ∀ p ∈ m.primeFactorsList, p ∈ s\nhk : k ≠ 0\np : ℕ\nhp : p ∈ k.primeFactorsList\n⊢ k ≠ 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hk"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.SmoothNumbers | {
"line": 62,
"column": 30
} | {
"line": 62,
"column": 43
} | {
"line": 62,
"column": 43
} | [
{
"pp": "s : Finset ℕ\nm k : ℕ\nh' : k ∣ m\nh₁ : m ≠ 0\nh₂ : ∀ p ∈ m.primeFactorsList, p ∈ s\nhk : k ≠ 0\np : ℕ\nhp : Prime p ∧ p ∣ k\n⊢ m ≠ 0",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h₁"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.SmoothNumbers | {
"line": 424,
"column": 4
} | {
"line": 424,
"column": 48
} | {
"line": 425,
"column": 2
} | [
{
"pp": "case refine_1\nn k : ℕ\nh : n ∈ k.smoothNumbers\nl m : ℕ\nH₁ : m ^ 2 * l = n\nH₂ : Squarefree l\nhl : l ∈ k.smoothNumbers\np : ℕ\nhp : Prime p ∧ p ∣ l ∧ l ≠ 0\n⊢ p < k",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Nat.Prime",
"Dvd.dvd",
"Nat.mem_smoothNumb... | [] | exact mem_smoothNumbers'.mp hl p hp.1 hp.2.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 105,
"column": 10
} | {
"line": 106,
"column": 96
} | {
"line": 107,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : ∀ {p : ℕ}, Nat.Prime p → Summable fun n ↦ ‖f (p ^ n)‖\np : ℕ\ns : Finset ℕ\nhp : p ∉ s\nih : (Summable fun m ↦ ‖f ↑m‖) ∧ HasSum (fun m ↦ f ↑m) (∏ p... | [
"R : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : ∀ {p : ℕ}, Nat.Prime p → Summable fun n ↦ ‖f (p ^ n)‖\np : ℕ\ns : Finset ℕ\nhp : p ∉ s\nih : (Summable fun m ↦ ‖f ↑m‖) ∧ HasSum (fun m ↦ f ↑m) (∏ p ∈ s with Na... | enter [1, x]
rw [equivProdNatFactoredNumbers_apply', factoredNumbers.map_prime_pow_mul hmul hpp hp] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.NumberTheory.EulerProduct.Basic | {
"line": 105,
"column": 10
} | {
"line": 106,
"column": 96
} | {
"line": 107,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : ∀ {p : ℕ}, Nat.Prime p → Summable fun n ↦ ‖f (p ^ n)‖\np : ℕ\ns : Finset ℕ\nhp : p ∉ s\nih : (Summable fun m ↦ ‖f ↑m‖) ∧ HasSum (fun m ↦ f ↑m) (∏ p... | [
"R : Type u_1\ninst✝¹ : NormedCommRing R\nf : ℕ → R\ninst✝ : CompleteSpace R\nhf₁ : f 1 = 1\nhmul : ∀ {m n : ℕ}, m.Coprime n → f (m * n) = f m * f n\nhsum : ∀ {p : ℕ}, Nat.Prime p → Summable fun n ↦ ‖f (p ^ n)‖\np : ℕ\ns : Finset ℕ\nhp : p ∉ s\nih : (Summable fun m ↦ ‖f ↑m‖) ∧ HasSum (fun m ↦ f ↑m) (∏ p ∈ s with Na... | enter [1, x]
rw [equivProdNatFactoredNumbers_apply', factoredNumbers.map_prime_pow_mul hmul hpp hp] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.NumberTheory.LSeries.Positivity | {
"line": 49,
"column": 58
} | {
"line": 49,
"column": 71
} | {
"line": 49,
"column": 71
} | [
{
"pp": "a : ℕ → ℂ\nhn : 0 ≤ a\nx : ℝ\nh : abscissaOfAbsConv a < ↑x\nn k : ℕ\nh✝ : ¬k = 0\n⊢ k ≠ 0",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h✝"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 86
} | {
"line": 61,
"column": 2
} | [
{
"pp": "⊢ (fun t ↦ (1 - rexp (-π * t))⁻¹) =O[atTop] fun x ↦ 1",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Filter.Tendsto.isBigO_one",
"False",
"Real.partialOrder",
"Real",
"Real.pi",
"HMul.hMul",
"... | [
"⊢ Tendsto (fun x ↦ rexp (-π * x)) atTop (𝓝 0)"
] | refine ((Tendsto.const_sub _ ?_).inv₀ (by simp)).isBigO_one ℝ (c := ((1 - 0)⁻¹ : ℝ)) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 61,
"column": 2
} | {
"line": 62,
"column": 44
} | {
"line": 64,
"column": 0
} | [
{
"pp": "⊢ Tendsto (fun x ↦ rexp (-π * x)) atTop (𝓝 0)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Filter.Tendsto.const_mul_atTop",
"Real.partialOrder",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssoc... | [] | simpa only [neg_mul, tendsto_exp_comp_nhds_zero, tendsto_neg_atBot_iff]
using! tendsto_id.const_mul_atTop pi_pos | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 21
} | {
"line": 95,
"column": 4
} | [
{
"pp": "case a\nk : ℕ\na t : ℝ\nht : 0 < t\nn : ℕ\n⊢ -π * ((↑n + a) ^ 2 * t) ≤ -π * (t * ↑n ^ 2 - 2 * (|a| * (t * ↑n))) + -π * (a ^ 2 * t)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Semigroup.toMul",
"Real",
"Real... | [
"case a\nk : ℕ\na t : ℝ\nht : 0 < t\nn : ℕ\n⊢ 0 ≤ -π * (t * ↑n ^ 2 - 2 * (|a| * (t * ↑n))) + -π * (a ^ 2 * t) - -π * ((↑n + a) ^ 2 * t)"
] | rw [← sub_nonneg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 147,
"column": 73
} | {
"line": 165,
"column": 9
} | {
"line": 167,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nr : ℝ\n⊢ (fun x ↦ P.f x - (P.ε * ↑(x ^ (-P.k))) • P.g₀) =O[𝓝[>] 0] fun x ↦ x ^ r",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr"... | [] | by
have := (P.hg_top (-(r + P.k))).comp_tendsto tendsto_inv_nhdsGT_zero
simp_rw [IsBigO, IsBigOWith, eventually_nhdsWithin_iff] at this ⊢
obtain ⟨C, hC⟩ := this
use ‖P.ε‖ * C
filter_upwards [hC] with x hC' (hx : 0 < x)
have h_nv2 : ↑(x ^ P.k) ≠ (0 : ℂ) := ofReal_ne_zero.mpr (rpow_pos_of_pos hx _).ne'
have... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.AbstractFuncEq | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 47
} | {
"line": 283,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nx : ℝ\nhx : 0 < x\n⊢ P.f_modif (1 / x) = (P.ε * ↑(x ^ P.k)) • P.g_modif x",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real",
"instHSMul",
"Preorder.... | [
"case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nx : ℝ\nhx : 0 < x\nhx' : 1 < x\n⊢ P.f_modif (1 / x) = (P.ε * ↑(x ^ P.k)) • P.g_modif x",
"case inr.inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nP : WeakFEPair E\nhx : 0 < 1\n⊢ P.f_modif... | rcases lt_trichotomy 1 x with hx' | rfl | hx' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 115,
"column": 10
} | {
"line": 115,
"column": 40
} | {
"line": 115,
"column": 40
} | [
{
"pp": "case neg\nι : 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 ... | [
"case neg\nι : 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 0... | norm_of_nonneg (by positivity) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable | {
"line": 396,
"column": 49
} | {
"line": 396,
"column": 77
} | {
"line": 396,
"column": 77
} | [
{
"pp": "z τ : ℂ\n| ∑' (x : ℤ), cexp (-↑π * I * (τ + 2 * z)) * jacobiTheta₂_term x z τ",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Equiv.addRight",
"NormedCommRing.toSeminormedCommRing",
"Real.pi",
"Equiv.instEquivLike",
"HMul.hMul",
"jacobiTheta₂_t... | [
"z τ : ℂ\n| ∑' (c : ℤ), cexp (-↑π * I * (τ + 2 * z)) * jacobiTheta₂_term ((Equiv.addRight 1) c) z τ"
] | ← (Equiv.addRight 1).tsum_eq | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.NumberTheory.LSeries.MellinEqDirichlet | {
"line": 121,
"column": 70
} | {
"line": 131,
"column": 36
} | {
"line": 133,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝ : Countable ι\na : ι → ℂ\nr : ι → ℝ\nF : ℝ → ℂ\ns : ℂ\nhs : 0 < s.re\nhF : ∀ t ∈ Ioi 0, HasSum (fun i ↦ if r i = 0 then 0 else a i * ↑(rexp (-π * r i ^ 2 * t))) (F t)\nh_sum : Summable fun i ↦ ‖a i‖ / |r i| ^ s.re\n⊢ HasSum (fun i ↦ s.Gammaℝ * a i / ↑|r i| ^ s) (mellin F (s / 2))",
... | [] | by
have hs' : 0 < (s / 2).re := by rw [div_ofNat_re]; positivity
simp_rw [← sq_eq_zero_iff (a := r _)] at hF
convert! hasSum_mellin_pi_mul₀ (fun i ↦ sq_nonneg (r i)) hs' hF ?_ using 3 with i
· rw [← neg_div, Gammaℝ_def]
· rw [← sq_abs, ofReal_pow, ← cpow_nat_mul']
· ring_nf
all_goals rw [arg_ofReal_of... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 425,
"column": 2
} | {
"line": 432,
"column": 79
} | {
"line": 434,
"column": 0
} | [
{
"pp": "a b : UnitAddCircle\n⊢ DifferentiableAt ℂ (fun s ↦ completedHurwitzZetaEven a s - completedHurwitzZetaEven b s) 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toS... | [] | have (s : _) : completedHurwitzZetaEven a s - completedHurwitzZetaEven b s =
completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -
((if a = 0 then 1 else 0) - (if b = 0 then 1 else 0)) / s := by
simp_rw [completedHurwitzZetaEven_eq, sub_div]
abel
rw [funext this]
refine .sub ?_ <| (dif... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.LSeries.HurwitzZetaEven | {
"line": 425,
"column": 2
} | {
"line": 432,
"column": 79
} | {
"line": 434,
"column": 0
} | [
{
"pp": "a b : UnitAddCircle\n⊢ DifferentiableAt ℂ (fun s ↦ completedHurwitzZetaEven a s - completedHurwitzZetaEven b s) 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toS... | [] | have (s : _) : completedHurwitzZetaEven a s - completedHurwitzZetaEven b s =
completedHurwitzZetaEven₀ a s - completedHurwitzZetaEven₀ b s -
((if a = 0 then 1 else 0) - (if b = 0 then 1 else 0)) / s := by
simp_rw [completedHurwitzZetaEven_eq, sub_div]
abel
rw [funext this]
refine .sub ?_ <| (dif... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 77
} | {
"line": 60,
"column": 2
} | [
{
"pp": "s : ℂ\nhs : 1 < s.re\n⊢ Summable fun n ↦ ‖(riemannZetaSummandHom ⋯) n‖",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Nat.instMulZeroOneClass",
"Real",
"riemannZetaSummandHom",
"PseudoMetricSpace.toUniformSpace",
"MonoidWithZeroHo... | [
"s : ℂ\nhs : 1 < s.re\n⊢ Summable fun n ↦ ‖↑n ^ (-s)‖"
] | simp only [riemannZetaSummandHom, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.EulerProduct.DirichletLSeries | {
"line": 74,
"column": 78
} | {
"line": 77,
"column": 70
} | {
"line": 79,
"column": 0
} | [
{
"pp": "s : ℂ\nN : ℕ\nχ : DirichletCharacter ℂ N\nhs : 1 < s.re\n⊢ Summable fun n ↦ ‖(dirichletSummandHom χ ⋯) n‖",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"mul_nonneg",
"NormedCommRing.toSeminormedCommRi... | [] | by
simp only [dirichletSummandHom, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, norm_mul]
exact (summable_riemannZetaSummand hs).of_nonneg_of_le (fun _ ↦ by positivity)
(fun n ↦ mul_le_of_le_one_left (norm_nonneg _) <| χ.norm_le_one n) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 22
} | {
"line": 185,
"column": 0
} | [
{
"pp": "case pos\nN : ℕ\nhN : N ≠ 0\nχ : DirichletCharacter ℂ N\nh : LSeriesSummable (fun n ↦ χ ↑n) 1\nn : ℕ\nh₁ : ¬↑n = 1\nh✝ : n = 0\n⊢ 0 ≤ 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real",
"Real.instZero",
... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.NumberTheory.LSeries.Dirichlet | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 30
} | {
"line": 282,
"column": 2
} | [
{
"pp": "s : ℂ\nhs : 1 < s.re\n⊢ ∑' (n : ℕ), term (fun n ↦ if n = 0 then 0 else 1) s n = ∑' (n : ℕ), 1 / ↑n ^ s",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"instHDiv",
"tsum_congr",
"Complex.instNormedField",
"PseudoM... | [
"s : ℂ\nhs : 1 < s.re\nn : ℕ\n⊢ term (fun n ↦ if n = 0 then 0 else 1) s n = 1 / ↑n ^ s"
] | refine tsum_congr fun n ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.FLT.Four | {
"line": 91,
"column": 2
} | {
"line": 92,
"column": 75
} | {
"line": 93,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Nat.Prime p\na1 : ℤ\nhpa : p ∣ (↑p * a1).natAbs\nb1 : ℤ\nhpb : p ∣ (↑p * b1).natAbs\nhab : ¬(↑p * a1).gcd (↑p * b1) = 1\nc1 : ℤ\nh : Minimal (↑p * a1) (↑p * b1) (↑p ^ 2 * c1)\n⊢ False",
"ppTerm": "?m.162",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"HMul.hMul",
... | [
"p : ℕ\nhp : Nat.Prime p\na1 : ℤ\nhpa : p ∣ (↑p * a1).natAbs\nb1 : ℤ\nhpb : p ∣ (↑p * b1).natAbs\nhab : ¬(↑p * a1).gcd (↑p * b1) = 1\nc1 : ℤ\nh : Minimal (↑p * a1) (↑p * b1) (↑p ^ 2 * c1)\nhf : Fermat42 a1 b1 c1\n⊢ False"
] | have hf : Fermat42 a1 b1 c1 :=
(Fermat42.mul (Int.natCast_ne_zero.mpr (Nat.Prime.ne_zero hp))).mpr h.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.PythagoreanTriples | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 13
} | {
"line": 33,
"column": 14
} | [
{
"pp": "z : Fin 4\n⊢ z * z ≠ 2",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Fintype.elems",
"HMul.hMul",
"ZMod.commRing",
"Nat.le_refl",
"CommSemiring.toSemiring",
"HEq.refl",
"Finset",
"List.Mem.tail",
"False.elim",
"Nat.ins... | [
"case «0»\n⊢ (fun i ↦ i) ⟨0, ⋯⟩ * (fun i ↦ i) ⟨0, ⋯⟩ ≠ 2",
"case «1»\n⊢ (fun i ↦ i) ⟨1, ⋯⟩ * (fun i ↦ i) ⟨1, ⋯⟩ ≠ 2",
"case «2»\n⊢ (fun i ↦ i) ⟨2, ⋯⟩ * (fun i ↦ i) ⟨2, ⋯⟩ ≠ 2",
"case «3»\n⊢ (fun i ↦ i) ⟨3, ⋯⟩ * (fun i ↦ i) ⟨3, ⋯⟩ ≠ 2"
] | fin_cases z | Lean.Elab.Tactic._aux_Mathlib_Tactic_FinCases___elabRules_Lean_Elab_Tactic_finCases_1 | Lean.Elab.Tactic.finCases |
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