module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Order.Interval.Set.Union | {
"line": 29,
"column": 17
} | {
"line": 32,
"column": 69
} | [
{
"pp": "case succ\nX : Type u_1\ninst✝ : LinearOrder X\na : ℕ → X\nN : ℕ\nih : Ioc (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ioc (a i) (a (i + 1))\n⊢ Ioc (a 0) (a (N + 1)) ⊆ ⋃ i ∈ Finset.range (N + 1), Ioc (a i) (a (i + 1))",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Finset.mem_range._simp_1",
... | calc
_ ⊆ Ioc (a 0) (a N) ∪ Ioc (a N) (a (N + 1)) := Ioc_subset_Ioc_union_Ioc
_ ⊆ _ := by simpa [Finset.range_add_one] using
union_subset_union_right (Ioc (a N) (a (N + 1))) ih | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.Union | {
"line": 29,
"column": 17
} | {
"line": 32,
"column": 69
} | [
{
"pp": "case succ\nX : Type u_1\ninst✝ : LinearOrder X\na : ℕ → X\nN : ℕ\nih : Ioc (a 0) (a N) ⊆ ⋃ i ∈ Finset.range N, Ioc (a i) (a (i + 1))\n⊢ Ioc (a 0) (a (N + 1)) ⊆ ⋃ i ∈ Finset.range (N + 1), Ioc (a i) (a (i + 1))",
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Finset.mem_range._simp_1",
... | calc
_ ⊆ Ioc (a 0) (a N) ∪ Ioc (a N) (a (N + 1)) := Ioc_subset_Ioc_union_Ioc
_ ⊆ _ := by simpa [Finset.range_add_one] using
union_subset_union_right (Ioc (a N) (a (N + 1))) ih | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 118,
"column": 31
} | {
"line": 118,
"column": 39
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : MeasurableSpace X\ninst✝ : BorelSpace X\nf : X →C_c ℝ\na ε : ℝ\nhε : 0 < ε\nN : ℕ\nhf : range ⇑f ⊆ Ioo a (a + ↑N * ε)\nb : ℝ := a + ↑N * ε\ny : Fin N → ℝ := fun n ↦ a + ε * (↑↑n + 1)\nn m : Fin N\nh : n < m\n⊢ a + ε * ↑↑m + ε = a + ε * (↑↑m + 1)",
... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 59,
"column": 53
} | {
"line": 59,
"column": 80
} | [
{
"pp": "ι : Type u\ninst✝ : Small.{v, u} ι\nf : ι → Ordinal.{v}\n⊢ Cardinal.lift.{u, v} #(⨆ i, f i).ToType ≤ Cardinal.lift.{v, max v u} #((x : ι) × (f x).ToType)",
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"iSup",
"Cardinal.lift",
"Cardinal.mk",
"id",
... | ← Cardinal.lift_umax.{v, u} | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 255,
"column": 8
} | {
"line": 255,
"column": 17
} | [
{
"pp": "case succ\na b o✝ : Ordinal.{u}\na✝ : a < ω ^ o✝ → a + ω ^ o✝ ≤ ω ^ o✝\nh : a < ω ^ succ o✝\n⊢ a + ω ^ succ o✝ ≤ ω ^ succ o✝",
"usedConstants": [
"Preorder.toLT",
"HMul.hMul",
"Order.succ",
"Ordinal.omega0",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
"co... | opow_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 294,
"column": 4
} | {
"line": 296,
"column": 78
} | [
{
"pp": "case inr.hab\no : Ordinal.{u}\nho : o ≠ 0\nH : ∀ a < o, a + o = o\nh : ω ^ log ω o < o\nthis : ω ^ log ω o + o = o\nn : ℕ\n⊢ ω ^ log ω o * ↑n + o = o",
"usedConstants": [
"Nat.cast_succ",
"Nat.recAux",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"Ordinal.omega0",
"M... | induction n with
| zero => simp [Nat.cast_zero, mul_zero, zero_add]
| succ n IH => simp only [Nat.cast_succ, mul_add_one, add_assoc, this, IH] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 415,
"column": 8
} | {
"line": 415,
"column": 17
} | [
{
"pp": "case inr.inl\na b : Ordinal.{u}\nhb : b < ω\nc : Ordinal.{u}\nc0 : 0 < succ c\nha : a < ω ^ succ c\n⊢ a * b < ω ^ succ c",
"usedConstants": [
"Preorder.toLT",
"HMul.hMul",
"Order.succ",
"Ordinal.omega0",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
"congrA... | opow_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 486,
"column": 19
} | {
"line": 486,
"column": 28
} | [
{
"pp": "case a\na b : Ordinal.{u}\nha : a ≠ 0\nhb : IsPrincipal (fun x1 x2 ↦ x1 * x2) b\nhb₂ : 2 < b\nhbl : IsSuccLimit b\nc : Ordinal.{u}\nhcb : c ∈ Set.Iio b\nhb₁ : 1 < b\nhbo₀ : b ^ log b a ≠ 0\n⊢ b ^ log b a * (succ (a / b ^ log b a) * c) ≤ b ^ succ (log b a)",
"usedConstants": [
"Eq.mpr",
... | opow_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 491,
"column": 2
} | {
"line": 491,
"column": 42
} | [
{
"pp": "case a\na b : Ordinal.{u}\nha : a ≠ 0\nhb : IsPrincipal (fun x1 x2 ↦ x1 * x2) b\nhb₂ : 2 < b\n⊢ b ^ succ (log b a) ≤ a * b",
"usedConstants": [
"Eq.mpr",
"le_refl",
"Ordinal.mulRightMono",
"HMul.hMul",
"Order.succ",
"Ordinal.partialOrder",
"MulZeroClass.toM... | · grw [opow_succ, opow_log_le_self b ha] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 300,
"column": 13
} | {
"line": 300,
"column": 21
} | [
{
"pp": "case calc_5\nX : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : LocallyCompactSpace X\nf : X →C_c ℝ\nμ : Measure X := rieszMeasure Λ\nK : Set X := tsupport ⇑f\nε : ℝ\nhε : 0 < ε\na b : ℝ\nhab : a < b ∧ range ... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 68
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝ : IsProbabilityMeasure P\nX : Ω → ℝ\nhX : AEMeasurable X P\nh0 : ∫ (x : Ω), X x ∂P = 0\nh1 : ∫ (x : Ω), (X ^ 2) x ∂P = 1\n⊢ (fun t ↦ charFun (Measure.map X P) t - (1 - ↑t ^ 2 / 2)) =o[𝓝 0] fun t ↦ t ^ 2",
"usedConstants": [
"instInne... | simp_rw [← taylorWithinEval_charFun_two_zero' (by fun_prop) h0 h1] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 94,
"column": 32
} | {
"line": 107,
"column": 25
} | [
{
"pp": "X : Type u\ninst✝² : TopologicalSpace X\ninst✝¹ : NormalSpace X\ninst✝ : R0Space X\n⊢ CompletelyRegularSpace X",
"usedConstants": [
"Iff.mpr",
"Set.mem_singleton",
"Real.instIsOrderedRing",
"Eq.mpr",
"Specializes.symm",
"Real.partialOrder",
"Real",
"C... | by
rw [completelyRegularSpace_iff]
intro x K hK hx
have cx : IsClosed (closure {x}) := isClosed_closure
have d : Disjoint (closure {x}) K := by
rw [Set.disjoint_iff]
intro a ⟨hax, haK⟩
exact hx ((specializes_iff_mem_closure.mpr hax).symm.mem_closed hK haK)
let ⟨⟨f, cf⟩, hfx, hfK, hficc⟩ := exists_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 150,
"column": 26
} | {
"line": 150,
"column": 35
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nt : ι → TopologicalSpace X\nht : ∀ (i : ι), CompletelyRegularSpace X\nthis : TopologicalSpace X := ⨅ i, t i\nx✝ : X\nI' : Finset ι\nV U : ↥I' → Set X\nhUV : ∀ (i : ↥I'), U i ⊆ V i\nfs : ↥I' → X → ↑I\nhfs : ∀ (i : ↥I'), Continuous (fs i)\nhxfs : ∀ (i : ↥I'), fs i x✝ = 0\nhfsU... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 150,
"column": 26
} | {
"line": 150,
"column": 35
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nt : ι → TopologicalSpace X\nht : ∀ (i : ι), CompletelyRegularSpace X\nthis : TopologicalSpace X := ⨅ i, t i\nx✝ : X\nI' : Finset ι\nV U : ↥I' → Set X\nhUV : ∀ (i : ↥I'), U i ⊆ V i\nfs : ↥I' → X → ↑I\nhfs : ∀ (i : ↥I'), Continuous (fs i)\nhxfs : ∀ (i : ↥I'), fs i x✝ = 0\nhfsU... | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 150,
"column": 26
} | {
"line": 150,
"column": 35
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nt : ι → TopologicalSpace X\nht : ∀ (i : ι), CompletelyRegularSpace X\nthis : TopologicalSpace X := ⨅ i, t i\nx✝ : X\nI' : Finset ι\nV U : ↥I' → Set X\nhUV : ∀ (i : ↥I'), U i ⊆ V i\nfs : ↥I' → X → ↑I\nhfs : ∀ (i : ↥I'), Continuous (fs i)\nhxfs : ∀ (i : ↥I'), fs i x✝ = 0\nhfsU... | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 214,
"column": 2
} | {
"line": 216,
"column": 33
} | [
{
"pp": "case refine_2\nX : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\nhX : #X < 𝔠\nx : X\ns : Set X\nhxs : x ∈ s\nhs : IsOpen s\nf : X → ↑I\nhfc : Continuous f\nhf₀ : f x = 0\nhf₁ : EqOn f 1 sᶜ\nR : Set ↑I := range f\nhR : #↑R < #↑I\nr : ↑I\nhr : r ∈ Rᶜ\nhr' : ∀ (x : X), f x ≠ r\nh... | · refine preimage_subset_iff.mpr (fun x ↦ ?_)
contrapose!; intro hxs
simpa [hf₁ hxs] using le_one' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 347,
"column": 6
} | {
"line": 347,
"column": 76
} | [
{
"pp": "case a.a.refine_1\nΩ : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : BorelSpace Ω\nμ ν : LevyProkhorov (ProbabilityMeasure Ω)\nh : dist μ ν = 0\n⊢ inst✝³ = MeasurableSpace.generateFrom {s | IsClosed s}",
"usedConstants": [
"Eq.mp... | rw [BorelSpace.measurable_eq (α := Ω), borel_eq_generateFrom_isClosed] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 347,
"column": 6
} | {
"line": 347,
"column": 76
} | [
{
"pp": "case a.a.refine_1\nΩ : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : BorelSpace Ω\nμ ν : LevyProkhorov (ProbabilityMeasure Ω)\nh : dist μ ν = 0\n⊢ inst✝³ = MeasurableSpace.generateFrom {s | IsClosed s}",
"usedConstants": [
"Eq.mp... | rw [BorelSpace.measurable_eq (α := Ω), borel_eq_generateFrom_isClosed] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 347,
"column": 6
} | {
"line": 347,
"column": 76
} | [
{
"pp": "case a.a.refine_1\nΩ : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : BorelSpace Ω\nμ ν : LevyProkhorov (ProbabilityMeasure Ω)\nh : dist μ ν = 0\n⊢ inst✝³ = MeasurableSpace.generateFrom {s | IsClosed s}",
"usedConstants": [
"Eq.mp... | rw [BorelSpace.measurable_eq (α := Ω), borel_eq_generateFrom_isClosed] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 26
} | [
{
"pp": "case refine_2\nE : Type u_3\ninst✝⁷ : MeasurableSpace E\nμ ν : Measure E\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : BorelSpace E\ninst✝³ : SecondCountableTopology E\ninst✝² : CompleteSpace E\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : ∀ (x : E), ∫ (v : E), ... | · exact continuous_inner | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 447,
"column": 7
} | {
"line": 450,
"column": 18
} | [] | rieszMeasure Λ univ
_ ≤ ENNReal.ofReal (Λ o) :=
rieszMeasure_le_of_eq_one _ (fun x ↦ zero_le_one) isCompact_univ (fun x hx ↦ rfl)
_ < ⊤ := by simp | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 391,
"column": 2
} | {
"line": 420,
"column": 43
} | [
{
"pp": "Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nμ ν : Measure Ω\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nε : ℝ\nε_pos : 0 < ε\nhμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε\nf : Ω →ᵇ ℝ\nf_nn : 0 ≤ᶠ[ae μ] ⇑f\n⊢ ∫ (ω : Ω), f ω ∂μ ≤... | rw [BoundedContinuousFunction.integral_eq_integral_meas_le f μ f_nn]
have key : (fun (t : ℝ) ↦ μ.real {a | t ≤ f a})
≤ (fun (t : ℝ) ↦ ν.real (thickening ε {a | t ≤ f a}) + ε) := by
intro t
simp only [measureReal_def]
convert ENNReal.toReal_mono ?_ <| left_measure_le_of_levyProkhorovEDist_lt ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 391,
"column": 2
} | {
"line": 420,
"column": 43
} | [
{
"pp": "Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nμ ν : Measure Ω\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nε : ℝ\nε_pos : 0 < ε\nhμν : levyProkhorovEDist μ ν < ENNReal.ofReal ε\nf : Ω →ᵇ ℝ\nf_nn : 0 ≤ᶠ[ae μ] ⇑f\n⊢ ∫ (ω : Ω), f ω ∂μ ≤... | rw [BoundedContinuousFunction.integral_eq_integral_meas_le f μ f_nn]
have key : (fun (t : ℝ) ↦ μ.real {a | t ≤ f a})
≤ (fun (t : ℝ) ↦ ν.real (thickening ε {a | t ≤ f a}) + ε) := by
intro t
simp only [measureReal_def]
convert ENNReal.toReal_mono ?_ <| left_measure_le_of_levyProkhorovEDist_lt ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.DistribChar | {
"line": 90,
"column": 29
} | {
"line": 90,
"column": 57
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝¹⁰ : Group G\ninst✝⁹ : AddCommGroup A\ninst✝⁸ : DistribMulAction G A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : IsTopologicalAddGroup A\ninst✝⁵ : LocallyCompactSpace A\ninst✝⁴ : ContinuousConstSMul G A\ninst✝³ : MeasurableSpace A\ninst✝² : BorelSpace A\nμ : Measure A\ninst✝... | ENNReal.mul_div_cancel_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.DistribChar | {
"line": 95,
"column": 43
} | {
"line": 95,
"column": 71
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝¹⁰ : Group G\ninst✝⁹ : AddCommGroup A\ninst✝⁸ : DistribMulAction G A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : IsTopologicalAddGroup A\ninst✝⁵ : LocallyCompactSpace A\ninst✝⁴ : ContinuousConstSMul G A\ng : G\ninst✝³ : MeasurableSpace A\ninst✝² : BorelSpace A\nμ : Measure A... | ENNReal.mul_div_cancel_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 535,
"column": 2
} | {
"line": 539,
"column": 25
} | [
{
"pp": "case neg\nΩ : Type u_1\ninst✝² : PseudoMetricSpace Ω\ninst✝¹ : MeasurableSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nP : ProbabilityMeasure Ω\nG : Set Ω\nG_open : IsOpen G\nε : ℝ≥0∞\nε_pos : 0 < ε\neasy : ε ≤ ↑P G\nε_top : ¬ε = ∞\n⊢ {Q | ↑P G < ↑Q G + ε} ∈ 𝓝 P",
"usedConstants": [
"instHasOuter... | have aux : P.toMeasure G - ε < liminf (fun Q ↦ Q.toMeasure G) (𝓝 P) := by
apply lt_of_lt_of_le (ENNReal.sub_lt_self (by finiteness) _ _)
<| ProbabilityMeasure.le_liminf_measure_open_of_tendsto tendsto_id G_open
· exact (lt_of_lt_of_le ε_pos easy).ne.symm
· exact ε_pos.ne.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 71
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : LocallyCompactSpace G\nφ ψ : G ≃ₜ* G\n⊢ mulEquivHaarChar (ψ.trans φ) = mulEquivHaarChar ψ * mulEquivHaarChar φ",
"usedConstants": [
"Eq.mpr",
... | rw [mulEquivHaarChar_eq haar ψ, mulEquivHaarChar_eq haar (ψ.trans φ)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 29
} | [
{
"pp": "μ : Measure ℝ\nr : ℝ\ninst✝ : IsFiniteMeasure μ\nhr : 0 < r\nh_int : Integrable (Function.uncurry fun x y ↦ cexp (↑x * ↑y * I)) ((volume.restrict (Set.uIoc (-r) r)).prod μ)\n⊢ ∫ (x : ℝ), 2 * r * sinc (r * x) ∂μ = 2 * r * ∫ (x : ℝ), sinc (r * x) ∂μ",
"usedConstants": [
"Eq.mpr",
"InnerPr... | rw [← integral_const_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 29
} | [
{
"pp": "μ : Measure ℝ\nr : ℝ\ninst✝ : IsProbabilityMeasure μ\nhr : 0 < r\nintegrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ\n⊢ ∫ (x : ℝ) in {x | 2 < |2 * r⁻¹ * x|}, 1 ∂μ = 2 * ∫ (x : ℝ) in {x | 2 < |2 * r⁻¹ * x|}, 2⁻¹ ∂μ",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace... | rw [← integral_const_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 63,
"column": 15
} | {
"line": 63,
"column": 55
} | [
{
"pp": "E : Type u_1\np : ℝ\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhp : 0 < p\nhE : Nontrivial E\nthis : 0 < ↑(finrank ℝ E)\n⊢ (∫ (y : ℝ) in Set.Ioi 0, y ^ (finrank ℝ... | ← Real.rpow_natCast _ (finrank ℝ E - 1), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 63
} | [
{
"pp": "E : Type u_1\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\nmE : MeasurableSpace E\ntE : TopologicalSpace E\ninst✝⁴ : IsTopologicalAddGroup E\ninst✝³ : BorelSpace E\ninst✝² : T2Space E\ninst✝¹ : ContinuousSMul ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ng : E → ℝ\nh1... | convert (measure_unitBall_eq_integral_div_gamma ν hp) using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.MeasureTheory.Measure.Haar.Extension | {
"line": 105,
"column": 10
} | {
"line": 105,
"column": 55
} | [
{
"pp": "case neg\nA : Type u_1\nB : Type u_2\nC : Type u_3\nE : Type u_4\ninst✝¹³ : Group A\ninst✝¹² : Group B\ninst✝¹¹ : Group C\ninst✝¹⁰ : TopologicalSpace A\ninst✝⁹ : TopologicalSpace B\ninst✝⁸ : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝⁷ : IsTopologicalGroup A\ninst✝⁶ : IsTopological... | refine ⟨ε / 2 / μA.real S, by positivity, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Haar.Extension | {
"line": 162,
"column": 19
} | {
"line": 164,
"column": 46
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\nE : Type u_4\ninst✝¹⁵ : Group A\ninst✝¹⁴ : Group B\ninst✝¹³ : Group C\ninst✝¹² : TopologicalSpace A\ninst✝¹¹ : TopologicalSpace B\ninst✝¹⁰ : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝⁹ : IsTopologicalGroup A\ninst✝⁸ : IsTopologicalGroup B\... | by
rw [map_smul]
exact integral_smul x (H.pushforward μA f) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Tight | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 38
} | [
{
"pp": "case inr\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 S)\nhnonempty : Nonempty 𝓧\n⊢ ∀ (ε : ℝ≥0∞), 0 < ε → ∃ K, IsComp... | obtain ⟨D, hD⟩ := exists_dense_seq 𝓧 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.TightNormed | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 63
} | [
{
"pp": "case inr\nE : Type u_1\nmE : MeasurableSpace E\nS : Set (Measure E)\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_2\nι : Type u_3\ninst✝³ : RCLike 𝕜\ninst✝² : Fintype ι\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nb : OrthonormalBasis ι 𝕜 E\nh : ∀ (i : ι), Tendsto (fun r ↦ ⨆ μ ∈ S,... | refine tendsto_finset_sum Finset.univ fun i _ ↦ (h i).comp ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 252,
"column": 6
} | {
"line": 252,
"column": 50
} | [
{
"pp": "case h\nE : 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 ∈ ... | convert measure_empty (μ := (ρ : Measure E)) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 313,
"column": 12
} | {
"line": 313,
"column": 56
} | [
{
"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... | integral_sum_measure (g.integrable (μ := μ)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.MeasuredSets | {
"line": 119,
"column": 12
} | {
"line": 119,
"column": 16
} | [
{
"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 ... | ← hD | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd | {
"line": 104,
"column": 4
} | {
"line": 106,
"column": 79
} | [
{
"pp": "case hD\nι : Type u_1\nκ : Type u_2\nX : ι → Type u_5\nY : κ → Type u_6\nmX : (i : ι) → MeasurableSpace (X i)\ninst✝⁸ : (i : ι) → TopologicalSpace (X i)\ninst✝⁷ : ∀ (i : ι), BorelSpace (X i)\ninst✝⁶ : ∀ (i : ι), HasOuterApproxClosed (X i)\nmY : (j : κ) → MeasurableSpace (Y j)\ninst✝⁵ : (j : κ) → Topolo... | · rw [← generateFrom_eq_pi (C := fun _ ↦ {t | IsClosed t})]
· simp [BorelSpace.measurable_eq, borel_eq_generateFrom_isClosed]
· exact fun _ ↦ ⟨fun _ ↦ Set.univ, fun _ ↦ isClosed_univ, iUnion_const _⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.PreVariation | {
"line": 204,
"column": 10
} | {
"line": 214,
"column": 41
} | [
{
"pp": "X : Type u_1\ninst✝ : MeasurableSpace X\nf : Set X → ℝ≥0∞\nhf : IsSigmaSubadditiveSetFun f\nhf' : f ∅ = 0\ns : ℕ → Set X\nhs : ∀ (i : ℕ), MeasurableSet (s i)\nhs' : Pairwise (Disjoint on s)\nb : ℝ≥0∞\nQ : Finpartition ⟨⋃ i, s i, ⋯⟩\nhQ : b < ∑ p ∈ Q.parts, f ↑p\ns' : ℕ → Subtype MeasurableSet := fun i ... | apply Finset.sum_le_sum fun q hq => ?_
have hq_eq : q.val = ⋃ i, q.val ∩ s i := by
rw [← Set.inter_iUnion]; exact (Set.inter_eq_left.mpr (Q.le hq)).symm
let t (i : ℕ) : Subtype MeasurableSet := ⟨q.val ∩ s i, q.2.inter (hs i)⟩
have ht_disj : Pairwise (Disjoint on (Subtype.val ∘ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.PreVariation | {
"line": 204,
"column": 10
} | {
"line": 214,
"column": 41
} | [
{
"pp": "X : Type u_1\ninst✝ : MeasurableSpace X\nf : Set X → ℝ≥0∞\nhf : IsSigmaSubadditiveSetFun f\nhf' : f ∅ = 0\ns : ℕ → Set X\nhs : ∀ (i : ℕ), MeasurableSet (s i)\nhs' : Pairwise (Disjoint on s)\nb : ℝ≥0∞\nQ : Finpartition ⟨⋃ i, s i, ⋯⟩\nhQ : b < ∑ p ∈ Q.parts, f ↑p\ns' : ℕ → Subtype MeasurableSet := fun i ... | apply Finset.sum_le_sum fun q hq => ?_
have hq_eq : q.val = ⋃ i, q.val ∩ s i := by
rw [← Set.inter_iUnion]; exact (Set.inter_eq_left.mpr (Q.le hq)).symm
let t (i : ℕ) : Subtype MeasurableSet := ⟨q.val ∩ s i, q.2.inter (hs i)⟩
have ht_disj : Pairwise (Disjoint on (Subtype.val ∘ ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 427,
"column": 44
} | {
"line": 427,
"column": 56
} | [
{
"pp": "case h\nE : 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)\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.range (n + 1), μ.restrict... | (hm : n ≤ m) | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.typeAscription |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 505,
"column": 21
} | {
"line": 505,
"column": 34
} | [
{
"pp": "E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nS : Set (ProbabilityMeasure E)\nhS : IsTightMeasureSet {x | ∃ μ ∈ S, ↑μ = x}\nu : ℕ → ℝ≥0\nu_pos : ∀ (n : ℕ), 0 < u n\nu_lim : Tendsto u atTop (𝓝 0)\nn : ℕ\n⊢ 0 < u n",
"usedConstants":... | exact u_pos n | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.JordanSub | {
"line": 89,
"column": 2
} | {
"line": 91,
"column": 31
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ ν : Measure X\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ns : Set X\nhs : IsHahnDecomposition μ ν s\nhsc : IsHahnDecomposition ν μ sᶜ\nh₁ :\n ((ν - μ).restrict s).toSignedMeasure =\n VectorMeasure.restrict ν.toSignedMeasure s - VectorMeasure.restr... | rw [← VectorMeasure.restrict_add_restrict_compl μ.toSignedMeasure hs.measurableSet,
← VectorMeasure.restrict_add_restrict_compl ν.toSignedMeasure hs.measurableSet,
← partition₁, ← partition₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.JordanSub | {
"line": 111,
"column": 45
} | {
"line": 113,
"column": 98
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ ν : Measure X\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\n⊢ (μ.jordanDecompositionOfToSignedMeasureSub ν).toSignedMeasure = μ.toSignedMeasure - ν.toSignedMeasure",
"usedConstants": [
"MeasureTheory.JordanDecomposition.posPart",
"Measur... | by
simp_rw [JordanDecomposition.toSignedMeasure, jordanDecompositionOfToSignedMeasureSub_posPart,
jordanDecompositionOfToSignedMeasureSub_negPart, ← sub_toSignedMeasure_eq_toSignedMeasure_sub] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.AddContent | {
"line": 107,
"column": 2
} | {
"line": 113,
"column": 12
} | [
{
"pp": "α : Type u_1\nhα : MeasurableSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nC : Set (Set α)\nm : AddContent E C\nhC : IsSetRing C\nhCmeas : ∀ s ∈ C, MeasurableSet s\nhm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s\nh'C : ∀ (t : Set α) (ε : ℝ≥0∞), Me... | have C'_dense : Dense C' := by
simp only [Dense, EMetric.mem_closure_iff, gt_iff_lt]
intro x ε εpos
rcases h'C x ε x.2 εpos with ⟨s, sC, hs⟩
refine ⟨⟨s, hCmeas s sC⟩, ⟨s, sC, rfl⟩, ?_⟩
rw [edist_comm]
exact hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 297,
"column": 25
} | {
"line": 299,
"column": 56
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoid M\ns : Set M\nhs : IsProperSemilinearSet s\n⊢ IsSemilinearSet s",
"usedConstants": [
"IsProperSemilinearSet",
"IsProperLinearSet",
"IsProperLinearSet.isLinearSet",
"Set.sUnion",
"Set.Finite",
"Membership.mem",
"IsLinearSe... | by
rcases hs with ⟨S, hS, hS', rfl⟩
exact ⟨S, hS, fun s hs => (hS' s hs).isLinearSet, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 412,
"column": 12
} | {
"line": 412,
"column": 35
} | [
{
"pp": "case inr\nS : Finset (Set ℕ)\nhS : ∀ t ∈ S, IsProperLinearSet t\na b : ℕ\nht : LinearIndepOn ℕ id ↑{b}\nhb : b ≠ 0\n⊢ ∃ k, ∃ p > 0, ∀ x ≥ k, x ∈ a +ᵥ ↑(closure ↑{b}) ↔ x + p ∈ a +ᵥ ↑(closure ↑{b})",
"usedConstants": [
"congrArg",
"Nat.ne_zero_iff_zero_lt",
"Eq.mp",
"Ne",
... | Nat.ne_zero_iff_zero_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.FinitelyGenerated | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 74
} | [
{
"pp": "L : Language\nM : Type u_1\ninst✝¹ : L.Structure M\nN : Type u_2\ninst✝ : L.Structure N\nf : M ↪[L] N\ns : L.Substructure M\nt : Set N\nh1 : t.Countable\nh2 : (closure L).toFun t = Substructure.map f.toHom s\nhf : Function.Injective ⇑f.toHom\n⊢ Substructure.map f.toHom ((closure L).toFun (⇑f ⁻¹' t)) = ... | rw [← h2, map_closure, Embedding.coe_toHom, image_preimage_eq_of_subset] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 200,
"column": 6
} | {
"line": 200,
"column": 45
} | [
{
"pp": "ι : Type u_3\ns : Set (ι → ℕ)\n⊢ IsLinearSet s ↔ ∃ v n A, s = {x | ∃ x_1, v + A *ᵥ x_1 = x}",
"usedConstants": [
"Eq.mpr",
"Pi.addCommMonoid",
"AddMonoid.toAddSemigroup",
"congrArg",
"Matrix",
"AddMonoid.toAddZeroClass",
"setOf",
"Nat.instAddMonoid",
... | isLinearSet_iff_exists_fin_addMonoidHom | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 207,
"column": 6
} | {
"line": 207,
"column": 45
} | [
{
"pp": "M : Type u_1\nι : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Finite ι\nF : Type u_5\ninst✝¹ : FunLike F (ι → ℕ) M\ninst✝ : AddMonoidHomClass F (ι → ℕ) M\ns : Set M\nhs : IsLinearSet s\nf : F\n⊢ IsSemilinearSet (⇑f ⁻¹' s)",
"usedConstants": [
"AddMonoid.toAddSemigroup",
"congrArg",
... | isLinearSet_iff_exists_fin_addMonoidHom | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.DirectLimit | {
"line": 61,
"column": 74
} | {
"line": 69,
"column": 31
} | [
{
"pp": "L : Language\nG' : ℕ → Type w\ninst✝ : (i : ℕ) → L.Structure (G' i)\nf' : (n : ℕ) → G' n ↪[L] G' (n + 1)\nm n : ℕ\nh : m ≤ n\n⊢ ⇑(natLERec f' m n h) = fun a ↦ Nat.leRecOn h (fun k ↦ ⇑(f' k)) a",
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Nat.recAux",
"Fir... | by
obtain ⟨k, rfl⟩ := Nat.exists_eq_add_of_le h
ext x
induction k with
| zero => simp [natLERec, Nat.leRecOn_self]
| succ k ih =>
-- This used to be `rw`, but we need `erw` after https://github.com/leanprover/lean4/pull/2644
erw [Nat.leRecOn_succ le_self_add, natLERec, Nat.leRecOn_succ le_self_add, ← ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 481,
"column": 49
} | {
"line": 481,
"column": 67
} | [
{
"pp": "case hab\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx y : ι → ℕ\ni : ↑hs.basisSet\nhy : y ∈ closure hs.basisSet\n⊢ (hs.basis.repr (toRatVec x - toRatVec hs.base)) i ≤\n (hs.basis.repr (toRatVec x - toRatVec hs.base) + hs.basis.repr (toRatVec y)) i",
"usedConstant... | Finsupp.add_apply, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.ModelTheory.PartialEquiv | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 58
} | [
{
"pp": "L : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nf g : M ≃ₚ[L] N\nh : f ≤ g\n⊢ g.toEquiv.toEmbedding.comp (inclusion ⋯) = (inclusion ⋯).comp f.toEquiv.toEmbedding",
"usedConstants": [
"FirstOrder.Language.PartialEquiv.toEquiv",
"Eq.mpr",
"First... | rw [← (subtype _).comp_inj, subtype_toEquiv_inclusion h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.PartialEquiv | {
"line": 136,
"column": 18
} | {
"line": 136,
"column": 40
} | [
{
"pp": "L : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nf g : M ≃ₚ[L] N\nh : f ≤ g\nx : ↥f.dom\n⊢ (inclusion ⋯) (f.toEquiv x) = g.toEquiv ((inclusion ⋯) x)",
"usedConstants": [
"FirstOrder.Language.PartialEquiv.toEquiv",
"FirstOrder.Language.PartialEquiv.do... | apply (subtype _).inj' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.ModelTheory.DirectLimit | {
"line": 252,
"column": 2
} | {
"line": 262,
"column": 16
} | [
{
"pp": "L : Language\nι : Type v\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝³ : IsDirectedOrder ι\ninst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝¹ : Nonempty ι\nα : Type u_1\ninst✝ : Finite α\nx : α → DirectLimit G f\n⊢ ∃ i y,... | obtain ⟨i, hi⟩ := Finite.bddAbove_range fun a => (x a).out.1
refine ⟨i, unify f (Quotient.out ∘ x) i hi, ?_⟩
ext a
rw [Quotient.eq_mk_iff_out, unify]
generalize_proofs r
change _ ≈ Structure.Sigma.mk f i (f (Quotient.out (x a)).fst i r (Quotient.out (x a)).snd)
have : (.mk f i (f (Quotient.out (x a)).fst i ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.DirectLimit | {
"line": 252,
"column": 2
} | {
"line": 262,
"column": 16
} | [
{
"pp": "L : Language\nι : Type v\ninst✝⁵ : Preorder ι\nG : ι → Type w\ninst✝⁴ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝³ : IsDirectedOrder ι\ninst✝² : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ninst✝¹ : Nonempty ι\nα : Type u_1\ninst✝ : Finite α\nx : α → DirectLimit G f\n⊢ ∃ i y,... | obtain ⟨i, hi⟩ := Finite.bddAbove_range fun a => (x a).out.1
refine ⟨i, unify f (Quotient.out ∘ x) i hi, ?_⟩
ext a
rw [Quotient.eq_mk_iff_out, unify]
generalize_proofs r
change _ ≈ Structure.Sigma.mk f i (f (Quotient.out (x a)).fst i r (Quotient.out (x a)).snd)
have : (.mk f i (f (Quotient.out (x a)).fst i ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Fraisse | {
"line": 327,
"column": 2
} | {
"line": 334,
"column": 7
} | [
{
"pp": "case mpr\nL : Language\nM : Type w\ninst✝ : L.Structure M\nM_CG : Structure.CG L M\n⊢ L.IsExtensionPair M M → L.IsUltrahomogeneous M",
"usedConstants": [
"FirstOrder.Language.PartialEquiv.toEquiv",
"Eq.mpr",
"FirstOrder.Language.Embedding.comp",
"congrArg",
"FirstOrder... | · intro h S S_FG f
let ⟨g, ⟨dom_le_dom, eq⟩⟩ :=
equiv_between_cg M_CG M_CG ⟨⟨S, f.toHom.range, f.equivRange⟩, S_FG⟩ h h
use g
simp only [Embedding.subtype_equivRange] at eq
rw [← eq]
ext
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.ModelTheory.DirectLimit | {
"line": 418,
"column": 4
} | {
"line": 418,
"column": 20
} | [
{
"pp": "case refine_2\nL : Language\nι : Type u_1\ninst✝⁵ : Countable ι\ninst✝⁴ : Preorder ι\ninst✝³ : IsDirectedOrder ι\ninst✝² : Nonempty ι\nG : ι → Type w\ninst✝¹ : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\nh : ∀ (i : ι), CG L (G i)\ninst✝ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\n... | rw [le_iSup_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.Order | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 92
} | [
{
"pp": "L : Language\nα : Type w\nM : Type w'\nn : ℕ\nthis : IsEmpty ((n : ℕ) × Language.order.Functions n)\n⊢ ∀ (a : Language.order.Symbols), a = default",
"usedConstants": [
"Eq.mpr",
"Inhabited.default",
"False",
"Sum.ctorIdx",
"congrArg",
"Sum.inr.injEq",
"Fals... | simp only [Symbols, Sum.forall, reduceCtorEq, Sum.inr.injEq, IsEmpty.forall_iff, true_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.CountableDenseLinearOrder | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 28
} | [
{
"pp": "case intro\nα : 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\nin... | cases nonempty_encodable β | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.ModelTheory.Types | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 69
} | [
{
"pp": "L : Language\nT : L.Theory\nα : Type w\n⊢ Nonempty (T.CompleteType α) ↔ T.IsSatisfiable",
"usedConstants": [
"Eq.mpr",
"FirstOrder.Language.Theory.IsSatisfiable",
"congrArg",
"id",
"FirstOrder.Language.withConstants",
"FirstOrder.Language.Theory.isSatisfiable_onT... | rw [← isSatisfiable_onTheory_iff (lhomWithConstants_injective L α)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 101,
"column": 43
} | {
"line": 104,
"column": 22
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nu v : R\np : ℕ\nhp : Nat.Prime p\nhvu : v ∣ u\nhpuv : ↑p * u * v ∣ u ^ p\nx : R\n⊢ p ∈ ((Finset.range (p + 1)).erase 0).erase 1",
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Finset.mem_range._simp_1",
"Preorder.toLT",
... | by -- aesop works but is slow
simp only [Finset.mem_erase]
rw [← and_assoc, and_comm (a := ¬ _), ← Nat.two_le_iff]
simp [hp.two_le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPowerSeries.Evaluation | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 18
} | [
{
"pp": "case h\nσ : Type u_1\nR : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : UniformSpace R\nS : Type u_3\ninst✝⁴ : CommRing S\ninst✝³ : UniformSpace S\nφ : R →+* S\na : σ → S\ninst✝² : IsUniformAddGroup R\ninst✝¹ : IsUniformAddGroup S\ninst✝ : IsLinearTopology S S\nhφ : Continuous ⇑φ\nha : HasEval a\nI : Ideal S... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 171,
"column": 40
} | {
"line": 171,
"column": 48
} | [
{
"pp": "case refine_2.hnot\np : ℕ\nhp : Nat.Prime p\nm : ℕ\nhm0 : m ≠ 0\nhpm : m + 2 ≤ p * m\na : ℤ\nha : ¬↑p ∣ a\nn✝ n : ℕ\nthis✝ : Fact (Nat.Prime p)\nthis : ∀ (m_1 : ℕ), ∃ y, (1 + ↑p ^ m * ↑a) ^ p ^ m_1 = 1 + ↑p ^ m_1 * ↑p ^ m * (↑a + ↑p * y)\ny : ZMod (p ^ (n + 1 + m))\nhy : (1 + ↑p ^ m * ↑a) ^ p ^ n = 1 +... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Evaluation | {
"line": 88,
"column": 2
} | {
"line": 89,
"column": 21
} | [
{
"pp": "S : Type u_2\ninst✝² : CommRing S\ninst✝¹ : TopologicalSpace S\ninst✝ : IsLinearTopology S S\nc x : S\nhx : HasEval x\n⊢ HasEval (c * x)",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Eq.mp"... | simp only [hasEval_iff] at hx ⊢
exact hx.mul_left _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Evaluation | {
"line": 88,
"column": 2
} | {
"line": 89,
"column": 21
} | [
{
"pp": "S : Type u_2\ninst✝² : CommRing S\ninst✝¹ : TopologicalSpace S\ninst✝ : IsLinearTopology S S\nc x : S\nhx : HasEval x\n⊢ HasEval (c * x)",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Eq.mp"... | simp only [hasEval_iff] at hx ⊢
exact hx.mul_left _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 58
} | [
{
"pp": "n✝ n : ℕ\nH : IsCyclic (ZMod (2 ^ (n + 3)))ˣ\nh : 2 ^ 3 ∣ 2 ^ (n + 3)\n⊢ IsCyclic (ZMod 8)ˣ",
"usedConstants": [
"isCyclic_of_surjective",
"ZMod.unitsMap_surjective",
"MonoidHom.instMonoidHomClass",
"ZMod.unitsMap",
"MonoidHom.instFunLike",
"ZMod.commRing",
... | exact isCyclic_of_surjective _ (unitsMap_surjective h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 39
} | [
{
"pp": "case inl\nn : ℕ\nhn : Odd n\nhn1 : n ≠ 1\nx✝¹ : NeZero n\nhm : Odd 2\nhm1 : 2 ≠ 1\nhmn : Nat.Coprime 2 n\nx✝ : NeZero 2\ne : (ZMod (2 * n))ˣ ≃* (ZMod 2)ˣ × (ZMod n)ˣ :=\n (Units.mapEquiv (chineseRemainder hmn).toMulEquiv).trans MulEquiv.prodUnits\n⊢ False",
"usedConstants": [
"False",
... | · simp [← Nat.not_even_iff_odd] at hm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 359,
"column": 6
} | {
"line": 359,
"column": 26
} | [
{
"pp": "case neg.inr.inl\nn : ℕ\nh0 : ¬2 * n = 0\nh1 : ¬2 * n = 1\nh2 : ¬2 * n = 2\nh4 : ¬2 * n = 4\nhn✝ : Even (2 * n)\nhn : Odd n\np m : ℕ\nodd : Odd p\neq : 2 * n = p ^ m\n⊢ False",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semigroup.toMul",
"HMul.hMul",
"Nat... | have := eq ▸ odd.pow | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.PowerSeries.Substitution | {
"line": 208,
"column": 82
} | {
"line": 216,
"column": 14
} | [
{
"pp": "R : Type u_2\ninst✝² : CommRing R\nτ : Type u_3\nS : Type u_4\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\na : MvPowerSeries τ S\nha : HasSubst a\nf : R⟦X⟧\ne : τ →₀ ℕ\n⊢ Function.HasFiniteSupport fun d ↦ (coeff d) f • (MvPowerSeries.coeff e) (a ^ d)",
"usedConstants": [
"Finsupp.instFunLike",
... | by
rw [Function.HasFiniteSupport]
convert (MvPowerSeries.coeff_subst_finite ha.const f e).image
(Finsupp.LinearEquiv.finsuppUnique ℕ ℕ Unit).toEquiv
rw [← Equiv.preimage_eq_iff_eq_image, ← Function.support_comp_eq_preimage]
apply congr_arg
rw [← Equiv.eq_comp_symm]
ext
simp [coeff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 14
} | [
{
"pp": "case h\nσ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\nthis✝ : UniformSpace R := ⊥\nf : MvPowerSeries σ R\nthis : ∀ (x : MvPowerSeries σ R), (aeval ⋯) x = (AlgHom.id R (MvPowerSeries σ R)) x\n⊢ (aeval ⋯) f = id f",
"usedConstants": []
}
] | exact this f | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 122,
"column": 28
} | {
"line": 122,
"column": 44
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Ring R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : constantCoeff φ = ↑u\nn : σ →₀ ℕ\nthis✝ : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)\nH : ¬n = 0\nthis : (0, n) ∈ antidiagonal n\ni j : σ →₀ ℕ\nhij : (i, j) ≠ (0, n) ∧ (i, j) ∈ antidiagonal n\n⊢ (coeff (i, j).1) φ * (c... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 114,
"column": 6
} | {
"line": 127,
"column": 16
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Ring R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : constantCoeff φ = ↑u\nn : σ →₀ ℕ\nthis : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)\nH : ¬n = 0\n⊢ (coeff n) (φ * φ.invOfUnit u) = (coeff n) 1",
"usedConstants": [
"MvPowerSeries.coeff_zero_eq_constantCoeff_a... | have : ((0 : σ →₀ ℕ), n) ∈ antidiagonal n := by rw [mem_antidiagonal, zero_add]
rw [coeff_one, if_neg H, coeff_mul, ← Finset.insert_erase this,
Finset.sum_insert (Finset.notMem_erase _ _), coeff_zero_eq_constantCoeff_apply, h,
coeff_invOfUnit, if_neg H, neg_mul, mul_neg, Units.mul_inv_cancel_left,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 114,
"column": 6
} | {
"line": 127,
"column": 16
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Ring R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : constantCoeff φ = ↑u\nn : σ →₀ ℕ\nthis : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)\nH : ¬n = 0\n⊢ (coeff n) (φ * φ.invOfUnit u) = (coeff n) 1",
"usedConstants": [
"MvPowerSeries.coeff_zero_eq_constantCoeff_a... | have : ((0 : σ →₀ ℕ), n) ∈ antidiagonal n := by rw [mem_antidiagonal, zero_add]
rw [coeff_one, if_neg H, coeff_mul, ← Finset.insert_erase this,
Finset.sum_insert (Finset.notMem_erase _ _), coeff_zero_eq_constantCoeff_apply, h,
coeff_invOfUnit, if_neg H, neg_mul, mul_neg, Units.mul_inv_cancel_left,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 452,
"column": 6
} | {
"line": 452,
"column": 42
} | [
{
"pp": "case neg\nσ : Type u_1\nR : Type u_3\ninst✝² : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\na : σ → MvPowerSeries τ S\nw : τ → ℕ\nha : HasSubst a\nf : MvPowerSeries σ R\nd : τ →₀ ℕ\nhd : ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weight... | coeff_eq_zero_of_lt_weightedOrder w, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 497,
"column": 4
} | {
"line": 497,
"column": 53
} | [
{
"pp": "case h\nσ : Type u_1\nA : Type u_2\ninst✝²¹ : CommSemiring A\nR✝ : Type u_3\ninst✝²⁰ : CommRing R✝\ninst✝¹⁹ : Algebra A R✝\nτ : Type u_4\nS : Type u_5\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra A S\ninst✝¹⁶ : Algebra R✝ S\ninst✝¹⁵ : IsScalarTower A R✝ S\na✝¹ a✝ : σ → MvPowerSeries τ S\nT✝ : Type u_6\nins... | simp only [coeff_one, mul_ite, mul_one, mul_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.AbelSummation | {
"line": 223,
"column": 35
} | {
"line": 223,
"column": 44
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhc : c 0 = 0\nb : ℝ\nhf_diff : ∀ t ∈ Set.Icc 1 b, DifferentiableAt ℝ f t\nhf_int : IntegrableOn (deriv f) (Set.Icc 1 b) volume\nhb : 1 ≤ b\nthis : 1 ≤ ⌊b⌋₊\n⊢ f ↑0 * 0 +\n (f 1 * c 1 +\n (f b * ∑ k ∈ Icc 0 ⌊b⌋₊, c k - f 1 *... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.AbelSummation | {
"line": 225,
"column": 83
} | {
"line": 225,
"column": 92
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nhc : c 0 = 0\nb : ℝ\nhf_diff : ∀ t ∈ Set.Icc 1 b, DifferentiableAt ℝ f t\nhf_int : IntegrableOn (deriv f) (Set.Icc 1 b) volume\nhb : b < 1\n⊢ f 0 * 0 = f b * 0 - ∫ (t : ℝ) in Set.Ioc 1 b, deriv f t * ∑ x ∈ Icc 0 ⌊t⌋₊, c x",
"usedCo... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Bertrand | {
"line": 97,
"column": 4
} | {
"line": 98,
"column": 95
} | [
{
"pp": "x : ℝ\nx_large : 512 ≤ x\nf : ℝ → ℝ := fun x ↦ log x + √(2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)\nhf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn ℝ (Set.Ioi 0.5) f\nthis : ∃ x1 x2, 0.5 < ... | obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Bertrand | {
"line": 97,
"column": 4
} | {
"line": 98,
"column": 95
} | [
{
"pp": "x : ℝ\nx_large : 512 ≤ x\nf : ℝ → ℝ := fun x ↦ log x + √(2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)\nhf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn ℝ (Set.Ioi 0.5) f\nthis : ∃ x1 x2, 0.5 < ... | obtain ⟨x1, x2, h1, h2, h0, h3, h4⟩ := this
exact (h.right_le_of_le_left'' h1 ((h1.trans h2).trans_le h0) h2 h0 (h4.trans h3)).trans h4 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 115,
"column": 73
} | {
"line": 115,
"column": 82
} | [
{
"pp": "k : ℕ\n⊢ (monomial (k - (k + 1))) (_root_.bernoulli (k + 1) * ↑((k + 1).choose (k + 1)) * 0) +\n ∑ x ∈ range (k + 1), (monomial (k - x)) (_root_.bernoulli x * ↑((k + 1).choose x) * ↑(k + 1 - x)) =\n (↑k + 1) * ∑ i ∈ range (k + 1), (monomial (k - i)) (_root_.bernoulli i * ↑(k.choose i))",
"u... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 18
} | [
{
"pp": "n : ℕ\n⊢ ↑(n + 1) • bernoulli n = (monomial n) ↑(n + 1) - ∑ x ∈ range n, ↑((n + 1).choose x) • bernoulli x",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.cast_succ",
"instHSMul",
"Nat.choose",
"NonUnitalCommRing.toNonUnitalNonAssoc... | Nat.cast_succ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 156,
"column": 11
} | {
"line": 156,
"column": 27
} | [
{
"pp": "case h.e'_2\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\n⊢ ∀ x ∈ antidiagonal n,\n bernoulli' x.1 / ↑x.1! * ((↑x.2 + 1) * ↑x.2!)⁻¹ * ↑n ! = ↑((x.1 + x.2).choose x.2) / (↑x.2 + 1) * bernoulli' x.1",
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"Rat.instMul",
... | mem_antidiagonal | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 29
} | [
{
"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 : ∀ (n : ℕ), (coeff n) (B - (rescale (-1)) B) = if n = 1 then 1 else 0\n⊢ bernoulli' n = 0",
"usedConstants": [
"Rat",
"Rat.l... | apply eq_zero_of_neg_eq | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.Bernoulli | {
"line": 270,
"column": 6
} | {
"line": 270,
"column": 22
} | [
{
"pp": "case h.succ.succ\nA : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nhfact : ∀ (m : ℕ), ↑m ! ≠ 0\nhite2 : (if n.succ = 0 then 1 else 0) = 0\nx : ℕ × ℕ\nh : x ∈ antidiagonal n.succ\n⊢ bernoulli x.1 / ↑x.1! * ((↑x.2 + 1) * ↑x.2!)⁻¹ * ↑n.succ ! = ↑((x.1 + x.2).choose x.2) / (↑x.2 + 1) * bernou... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ClassNumber.AdmissibleAbs | {
"line": 40,
"column": 2
} | {
"line": 41,
"column": 81
} | [
{
"pp": "n : ℕ\nε : ℝ\nhε : 0 < ε\nb : ℤ\nhb : b ≠ 0\nA : Fin n → ℤ\nhb' : 0 < ↑|b|\nhbε : 0 < |b| • ε\n⊢ ∃ t, ∀ (i₀ i₁ : Fin n), t i₀ = t i₁ → ↑|A i₁ % b - A i₀ % b| < |b| • ε",
"usedConstants": [
"Iff.mpr",
"Int.cast",
"Real.partialOrder",
"Real",
"instHSMul",
"instHDiv... | have hfloor : ∀ i, 0 ≤ floor ((A i % b : ℤ) / abs b • ε : ℝ) :=
fun _ ↦ floor_nonneg.mpr (div_nonneg (cast_nonneg (emod_nonneg _ hb)) hbε.le) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.ClassNumber.AdmissibleCardPowDegree | {
"line": 67,
"column": 8
} | {
"line": 67,
"column": 19
} | [
{
"pp": "Fq : Type u_1\ninst✝¹ : Fintype Fq\ninst✝ : Ring Fq\nd m : ℕ\nhm : Fintype.card Fq ^ d ≤ m\nA : Fin m.succ → Fq[X]\nhA : (A 0).degree < degree 0\n⊢ False",
"usedConstants": [
"WithBot.instPreorder",
"WithBot",
"Preorder.toLT",
"congrArg",
"Eq.mp",
"Fin.instOfNat"... | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.FunctionField | {
"line": 168,
"column": 14
} | {
"line": 168,
"column": 23
} | [
{
"pp": "case pos\nFq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : DecidableEq (RatFunc Fq)\nx y : RatFunc Fq\nhx : ¬x = 0\nhy : y = 0\n⊢ (if x * 0 = 0 then 0 else exp (x * 0).intDegree) =\n (if x = 0 then 0 else exp x.intDegree) * if 0 = 0 then 0 else exp (RatFunc.intDegree 0)",
"usedConstants": [
"Eq.m... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Chebyshev | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 20
} | [
{
"pp": "x : ℝ\nhx : 0 ≤ x\n⊢ log ↑(primorial ⌊x⌋₊) ≤ log (4 ^ ⌊x⌋₊)",
"usedConstants": [
"Real.partialOrder",
"Real",
"FloorRing.toFloorSemiring",
"Nat.instAtLeastTwoHAddOfNat",
"Real.semiring",
"Real.instFloorRing",
"Real.instRing",
"instOfNatNat",
"Re... | apply log_le_log | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.NumberTheory.FunctionField | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 73
} | [
{
"pp": "Fq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : DecidableEq (RatFunc Fq)\np : Fq[X]\nhp : p ≠ 0\n⊢ inftyValuationDef Fq ((algebraMap Fq[X] (RatFunc Fq)) p) = exp ↑p.natDegree",
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
... | rw [inftyValuationDef, if_neg (by simpa), RatFunc.intDegree_polynomial] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.FunctionField | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 73
} | [
{
"pp": "Fq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : DecidableEq (RatFunc Fq)\np : Fq[X]\nhp : p ≠ 0\n⊢ inftyValuationDef Fq ((algebraMap Fq[X] (RatFunc Fq)) p) = exp ↑p.natDegree",
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
... | rw [inftyValuationDef, if_neg (by simpa), RatFunc.intDegree_polynomial] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.FunctionField | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 73
} | [
{
"pp": "Fq : Type u_1\ninst✝¹ : Field Fq\ninst✝ : DecidableEq (RatFunc Fq)\np : Fq[X]\nhp : p ≠ 0\n⊢ inftyValuationDef Fq ((algebraMap Fq[X] (RatFunc Fq)) p) = exp ↑p.natDegree",
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"congrArg",
"CommSemiring.toSemiring",
... | rw [inftyValuationDef, if_neg (by simpa), RatFunc.intDegree_polynomial] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 103,
"column": 4
} | {
"line": 103,
"column": 32
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : EuclideanDomain R\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDomain S\ninst✝⁵ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nT : Type u_6\ninst✝² : Ring T\ninst✝¹ : LinearOrder T\ninst✝ : IsStrictOrderedRing... | simp only [Finset.mem_image] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 14
} | [
{
"pp": "case hf\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nn : ℕ\n⊢ x.appr n ≤ x.appr (n + 1)",
"usedConstants": [
"Preorder.toLE",
"id",
"instOfNatNat",
"LE.le",
"instHAdd",
"HAdd.hAdd",
"Nat.instPreorder",
"Nat",
"instAddNat",
"PadicInt.a... | dsimp [appr] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 395,
"column": 4
} | {
"line": 395,
"column": 16
} | [
{
"pp": "case succ\np : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nm k : ℕ\nih : p ^ m ∣ x.appr (m + k) - x.appr m\n⊢ p ^ m ∣ x.appr (m + k + 1) - x.appr m",
"usedConstants": [
"Dvd.dvd",
"Nat.instMonoid",
"HSub.hSub",
"id",
"instSubNat",
"instOfNatNat",
"Monoid.t... | dsimp [appr] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.NumberTheory.Cyclotomic.CyclotomicCharacter | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 56
} | [
{
"pp": "case inr\nL : Type u\ninst✝² : CommRing L\ninst✝¹ : IsDomain L\nn : ℕ\ninst✝ : NeZero n\nt : ↥(rootsOfUnity n L)\nthis : 1 ≤ Fintype.card ↥(rootsOfUnity n L)\nh : 1 = Fintype.card ↥(rootsOfUnity n L)\n⊢ (RingEquiv.refl L) ↑↑t = ↑(↑t ^ ZMod.val 1)",
"usedConstants": [
"Eq.ge",
"Membershi... | have := Fintype.card_le_one_iff_subsingleton.mp h.ge | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 23
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : EuclideanDomain R\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDomain S\ninst✝⁵ : Algebra R S\nabv : AbsoluteValue R ℤ\nι : Type u_5\ninst✝⁴ : DecidableEq ι\ninst✝³ : Fintype ι\nbS : Basis ι R S\nadm : abv.IsAdmissible\ninst✝² : Infinite R\ninst✝¹ : DecidableEq R\ninst✝ : Algeb... | refine ⟨q, r, hr, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 366,
"column": 22
} | {
"line": 366,
"column": 32
} | [
{
"pp": "a : ℕ\na1 : 1 < a\nm n : ℕ\nh : n ≤ m\n⊢ yz a1 (m - n) = xz a1 n * yz a1 m + -(xz a1 m * yz a1 n)",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Pell.yz",
"congrArg",
"AddMonoid.... | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Zsqrtd.Basic | {
"line": 379,
"column": 6
} | {
"line": 386,
"column": 10
} | [
{
"pp": "case refine_1\nd x y z w : ℕ\n⊢ SqLe x 1 y d → SqLe z 1 w d → SqLe (x * w + y * z) d (x * z + d * y * w) 1",
"usedConstants": [
"Mathlib.Tactic.Ring.mul_pp_pf_overlap",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.N... | intro xy zw
have :=
Int.mul_nonneg (sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le xy))
(sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le zw))
refine Int.le_of_ofNat_le_ofNat (le_of_sub_nonneg ?_)
convert this using 1
simp only [one_mul, Int.natCast_add, Int.natCast_mul]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.