module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 9
} | {
"line": 90,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\nN : ℕ\na h : ℝ\nN_nonzero : 0 < N\n⊢ h / 2 * (f (a + ↑0 * h) + f (a + (↑N - 1 + 1) * h) + 2 * ∑ i ∈ Finset.range (N - 1), f (a + (↑i + 1) * h)) =\n h * ((f a + f (a + ↑N * h)) / 2 + ∑ x ∈ Finset.range (N - 1), f (a + (↑x + 1) * h))",
"ppTerm": "?m.98",
"assigned": true,
"usedC... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 172,
"column": 38
} | {
"line": 172,
"column": 45
} | {
"line": 172,
"column": 45
} | [
{
"pp": "f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 /... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 172,
"column": 38
} | {
"line": 172,
"column": 45
} | {
"line": 172,
"column": 45
} | [
{
"pp": "f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 /... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 172,
"column": 38
} | {
"line": 172,
"column": 45
} | {
"line": 172,
"column": 45
} | [
{
"pp": "f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 /... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic | {
"line": 290,
"column": 2
} | {
"line": 292,
"column": 74
} | {
"line": 293,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nΛ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nK : Compacts X\nb : ℝ≥0∞\nhb : ∀ (i : Compacts X), b ≤ ⨅ (_ : ↑K ⊆ interior ↑i), ↑(rieszContentAux Λ i)\n⊢ ∃ p, b = ↑p ∧ p ≤ sInf (⇑Λ '' {f | ∀ x ∈ K, 1 ≤ f x})",
"ppTerm": "?... | [
"X : Type u_1\ninst✝² : TopologicalSpace X\nΛ : (X →C_c ℝ≥0) →ₗ[ℝ≥0] ℝ≥0\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nK : Compacts X\nb : ℝ≥0∞\nhb : ∀ (i : Compacts X), b ≤ ⨅ (_ : ↑K ⊆ interior ↑i), ↑(rieszContentAux Λ i)\nthis : b < ∞\n⊢ ∃ p, b = ↑p ∧ p ≤ sInf (⇑Λ '' {f | ∀ x ∈ K, 1 ≤ f x})"
] | have : b < ⊤ := by
obtain ⟨F, hF⟩ := exists_compact_superset K.2
exact (le_iInf_iff.mp (hb ⟨F, hF.1⟩) hF.2).trans_lt ENNReal.coe_lt_top | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.MeasurableSpace.Card | {
"line": 156,
"column": 78
} | {
"line": 159,
"column": 40
} | {
"line": 161,
"column": 0
} | [
{
"pp": "α : Type u\ns : Set (Set α)\ni : Ordinal.{v}\nhi : ω_ 1 ≤ i\n⊢ generateMeasurableRec s i = generateMeasurableRec s (ω_ 1)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Ordinal.partialOrder",
"M... | [] | by
apply (generateMeasurableRec_mono s hi).antisymm'
rw [← generateMeasurable_eq_rec]
exact generateMeasurableRec_subset s i | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.TaylorExpansion | {
"line": 93,
"column": 34
} | {
"line": 93,
"column": 53
} | {
"line": 93,
"column": 54
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nn : ℕ\nt : E\nhint : MemLp id (↑n) μ\nx : Fin n → E\nh : innerₗ E = (innerSL ℝ).toLinearMap₁₂\nhi... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nn : ℕ\nt : E\nhint : MemLp id (↑n) μ\nx : Fin n → E\nh : innerₗ E = (innerSL ℝ).toLinearMap₁₂\nhint' : ∀ (k :... | integral_const_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 261,
"column": 2
} | {
"line": 268,
"column": 48
} | {
"line": 269,
"column": 2
} | [
{
"pp": "case calc_2\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 ... | [
"case calc_3\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 ⇑f ⊆ Ioo a b... | · -- Use that `f ≤ y n + ε'` on `V n`
gcongr with n hn
intro x
by_cases hx : x ∈ tsupport (g n)
· rw [smul_eq_mul, mul_comm]
apply mul_le_mul_of_nonneg_right ?_ (hg.2.2.1 n x).1
exact le_of_lt <| (hV n).2.1 x <| mem_of_subset_of_mem (hg.1 n) hx
· simp [image_eq_zero_of_notMem_tsupport hx... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 349,
"column": 51
} | {
"line": 357,
"column": 6
} | {
"line": 359,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝ : OpensMeasurableSpace E\nL : StrongDual ℝ E\nu : ℝ\n⊢ charFun (Measure.map (⇑L) μ) u = charFunDual μ (u • L)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"... | [] | by
rw [charFunDual_apply]
have : ∫ x, cexp ((u • L) x * I) ∂μ = ∫ x, cexp (u * x * I) ∂(μ.map L) := by
rw [integral_map]
· simp
· fun_prop
· exact Measurable.aestronglyMeasurable <| by fun_prop
rw [this, charFun_apply]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 386,
"column": 2
} | {
"line": 389,
"column": 70
} | {
"line": 390,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : T2Space X\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : BorelSpace X\nμ ν : Measure X\ninst✝³ : LocallyCompactSpace X\ninst✝² : ν.OuterRegular\ninst✝¹ : IsFiniteMeasureOnCompacts ν\ninst✝ : IsFiniteMeasureOnCompacts μ\nhμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂... | [
"X : Type u_1\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : T2Space X\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : BorelSpace X\nμ ν : Measure X\ninst✝³ : LocallyCompactSpace X\ninst✝² : ν.OuterRegular\ninst✝¹ : IsFiniteMeasureOnCompacts ν\ninst✝ : IsFiniteMeasureOnCompacts μ\nhμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂μ ≤ ∫ (x : X... | have hfV (x : X) : f x ≤ V.indicator 1 x := by
by_cases hx : x ∈ tsupport f
· simp [(pf2 hx), (pf3 x).2]
· simp [image_eq_zero_of_notMem_tsupport hx, Set.indicator_nonneg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 402,
"column": 6
} | {
"line": 402,
"column": 82
} | {
"line": 403,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : T2Space X\ninst✝⁵ : MeasurableSpace X\ninst✝⁴ : BorelSpace X\nμ ν : Measure X\ninst✝³ : LocallyCompactSpace X\ninst✝² : ν.OuterRegular\ninst✝¹ : IsFiniteMeasureOnCompacts ν\ninst✝ : IsFiniteMeasureOnCompacts μ\nhμν : ∀ (f : X →C_c ℝ), ∫ (x : X), f x ∂... | [] | exact IntegrableOn.integrable_indicator integrableOn_const pV2.measurableSet | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 477,
"column": 4
} | {
"line": 477,
"column": 61
} | {
"line": 478,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nμ : Measure X\ninst✝ : IsFiniteMeasure μ\nK : Set X\nhK : IsCompact K\nh : μ Kᶜ = 0\nμ' : Measure ↑K := Measure.comap Subtype.val μ\n⊢ ∃ ν, ν.Regular ∧ IsFiniteMeasure ν ∧ ∀ (g : ↑K →ᵇ ℝ), ... | [
"X : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nμ : Measure X\ninst✝ : IsFiniteMeasure μ\nK : Set X\nhK : IsCompact K\nh : μ Kᶜ = 0\nμ' : Measure ↑K := Measure.comap Subtype.val μ\nthis : CompactSpace ↑K\n⊢ ∃ ν, ν.Regular ∧ IsFiniteMeasure ν ∧ ∀ (g ... | have : CompactSpace K := isCompact_iff_compactSpace.mp hK | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 485,
"column": 4
} | {
"line": 485,
"column": 70
} | {
"line": 486,
"column": 4
} | [
{
"pp": "case e'_2\nX : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nμ : Measure X\ninst✝ : IsFiniteMeasure μ\nK : Set X\nhK : IsCompact K\nh : μ Kᶜ = 0\nμ' : Measure ↑K := Measure.comap Subtype.val μ\nν' : Measure ↑K\nν'_reg : ν'.Regular\nν'_fin ... | [
"case e'_2\nX : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nμ : Measure X\ninst✝ : IsFiniteMeasure μ\nK : Set X\nhK : IsCompact K\nh : μ Kᶜ = 0\nμ' : Measure ↑K := Measure.comap Subtype.val μ\nν' : Measure ↑K\nν'_reg : ν'.Regular\nν'_fin : IsFiniteMe... | rw [← integral_map (φ := Subtype.val) (by fun_prop) (by fun_prop)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Haar.Extension | {
"line": 247,
"column": 2
} | {
"line": 249,
"column": 30
} | {
"line": 250,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝¹⁷ : Group A\ninst✝¹⁶ : Group B\ninst✝¹⁵ : Group C\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝¹¹ : IsTopologicalGroup A\ninst✝¹⁰ : IsTopologicalGroup B\ninst✝⁹ : Me... | [
"A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝¹⁷ : Group A\ninst✝¹⁶ : Group B\ninst✝¹⁵ : Group C\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : TopologicalSpace B\ninst✝¹² : TopologicalSpace C\nφ : A →* B\nψ : B →* C\nH : IsSES φ ψ\ninst✝¹¹ : IsTopologicalGroup A\ninst✝¹⁰ : IsTopologicalGroup B\ninst✝⁹ : MeasurableSpac... | replace h : μC Set.univ * μA {1} < ENNReal.ofReal (∫ c : C, pushforward H μA ⟨f, hf2⟩ c ∂μC) :=
lt_of_lt_of_le h ((RealRMK.rieszMeasure_le_of_eq_one (f := ⟨f, hf2⟩) _ (fun x ↦ (hf4 x).1)
hK (fun x hx ↦ hf1 hx))) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 108,
"column": 8
} | {
"line": 108,
"column": 39
} | {
"line": 108,
"column": 39
} | [
{
"pp": "case e_f\nμ : Measure ℝ\nr : ℝ\ninst✝ : IsProbabilityMeasure μ\nhr : 0 < r\nintegrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ\nx✝ : ℝ\n⊢ 1 = 2 * 2⁻¹",
"ppTerm": "?e_f",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Group... | [
"case e_f\nμ : Measure ℝ\nr : ℝ\ninst✝ : IsProbabilityMeasure μ\nhr : 0 < r\nintegrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ\nx✝ : ℝ\n⊢ 1 = 1"
] | mul_inv_cancel₀ (by positivity) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 137,
"column": 6
} | {
"line": 137,
"column": 13
} | {
"line": 138,
"column": 6
} | [
{
"pp": "case e_a\nμ : Measure ℝ\nr : ℝ\ninst✝ : IsProbabilityMeasure μ\nhr : 0 < r\nintegrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ\n⊢ 2 = 2⁻¹ * r * 2 * (2 * r⁻¹)",
"ppTerm": "?e_a✝",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
... | [
"case e_a\nμ : Measure ℝ\nr : ℝ\ninst✝ : IsProbabilityMeasure μ\nhr : 0 < r\nintegrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ\n⊢ 2 = r * r⁻¹ * 2"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 63,
"column": 69
} | {
"line": 63,
"column": 94
} | {
"line": 64,
"column": 8
} | [
{
"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 ... | [
"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 E) - ↑(Na... | Nat.cast_sub finrank_pos, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 54,
"column": 90
} | {
"line": 70,
"column": 38
} | {
"line": 72,
"column": 0
} | [
{
"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\n⊢ μ (Metric.ball 0 1) = ENNReal.ofReal ((∫ (x : E), Real.exp (-‖x‖ ^ p) ∂μ) / Real.Gamm... | [] | by
obtain hE | hE := subsingleton_or_nontrivial E
· rw [(Metric.nonempty_ball.mpr zero_lt_one).eq_zero, ← setIntegral_univ,
Set.univ_nonempty.eq_zero, integral_singleton, finrank_zero_of_subsingleton, Nat.cast_zero,
zero_div, zero_add, Real.Gamma_one, div_one, norm_zero, Real.zero_rpow hp.ne', neg_zero,... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosedProd | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 41
} | {
"line": 97,
"column": 4
} | [
{
"pp": "case refine_2\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) → ... | [] | exact fun j ↦ (ht₁ j).inter (ht₂ j) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 59
} | {
"line": 206,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nC : ℝ≥0\nK : Set E\nhK : IsCompact K\nf : ↑K → E := ⋯\nhf : IsClosedEmbedding f\nrf : range f = K\nF : FiniteMeasure ↑K → FiniteMeasure E := ⋯\nT : Set (FiniteMeasure ↑K) := ⋯\nthis : {μ | μ... | [
"E : Type u_1\ninst✝³ : MeasurableSpace E\ninst✝² : TopologicalSpace E\ninst✝¹ : T2Space E\ninst✝ : BorelSpace E\nC : ℝ≥0\nK : Set E\nhK : IsCompact K\nf : ↑K → E := Subtype.val\nhf : IsClosedEmbedding f\nrf : range f = K\nF : FiniteMeasure ↑K → FiniteMeasure E := fun μ ↦ μ.map f\nT : Set (FiniteMeasure ↑K) := {μ |... | have : CompactSpace K := isCompact_iff_compactSpace.mp hK | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 373,
"column": 8
} | {
"line": 373,
"column": 22
} | {
"line": 374,
"column": 4
} | [
{
"pp": "case h₂\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 ∈... | [] | simpa using hρ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 373,
"column": 8
} | {
"line": 373,
"column": 22
} | {
"line": 374,
"column": 4
} | [
{
"pp": "case h₂\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 ∈... | [] | simpa using hρ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 373,
"column": 8
} | {
"line": 373,
"column": 22
} | {
"line": 374,
"column": 4
} | [
{
"pp": "case h₂\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 ∈... | [] | simpa using hρ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 379,
"column": 2
} | {
"line": 381,
"column": 43
} | {
"line": 384,
"column": 2
} | [
{
"pp": "case refine_1\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 ... | [
"case refine_2\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 ∈ Finse... | · simp only [mass, mk_apply, μ]
rw [show C = (C : ℝ≥0∞).toNNReal by simp]
exact ENNReal.toNNReal_mono (by simp) B | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 399,
"column": 10
} | {
"line": 399,
"column": 76
} | {
"line": 400,
"column": 10
} | [
{
"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)\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.range (n + 1), μ.restrict (disjoi... | [
"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)\nI :\n ∀ (μ : FiniteMeasure E) (n : ℕ),\n ∑ i ∈ Finset.range (n + 1), μ.restrict (disjointed K i) = ... | simp only [dist, abs_le, neg_le_sub_iff_le_add, tsub_le_iff_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.LevyConvergence | {
"line": 142,
"column": 6
} | {
"line": 142,
"column": 62
} | {
"line": 143,
"column": 6
} | [
{
"pp": "case hbc.refine_1\nE : 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 (... | [
"case hbc.refine_1.refine_1\nE : 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... | refine Integrable.mono' (integrable_const (ε / 4)) ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.PreVariation | {
"line": 216,
"column": 9
} | {
"line": 233,
"column": 58
} | {
"line": 234,
"column": 2
} | [] | [] | ∑ q ∈ Q.parts, f q
_ ≤ ∑ q ∈ Q.parts, ∑' i, f (q ⊓ s' i) := by
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... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Measure.Support | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 56
} | {
"line": 161,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : MeasurableSpace X\nμ : Measure X\ninst✝ : HereditarilyLindelofSpace X\ns : Set X\nhμ : Disjoint s μ.support\n⊢ μ s ≤ 0",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"instReflLe",
"Disjoint.subset_compl_right",
... | [] | exact μ.mono hμ.subset_compl_right |>.trans <| by simp | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.ResolventTransform | {
"line": 170,
"column": 2
} | {
"line": 171,
"column": 99
} | {
"line": 173,
"column": 0
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nA : Type u_2\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : HereditarilyLindelofSpace 𝕜\ninst✝⁵ : CompleteSpace 𝕜\ninst✝⁴ : MeasurableSpace 𝕜\ninst✝³ : BorelSpace 𝕜\ninst✝² : RCLike A\ninst✝¹ : NormedAlgebra 𝕜 A\nμ : Measure 𝕜\ninst✝ : IsFiniteMeasure μ\na : A\nha : a ∉ ⇑... | [] | exact hasDerivAt_integral_of_dominated_loc_of_deriv_le hs_z resolvent_meas
(integrable_resolvent (by simp [ha])) (by fun_prop) resolvent'_bound (by fun_prop) h_deriv |>.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 404,
"column": 51
} | {
"line": 404,
"column": 63
} | {
"line": 404,
"column": 64
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nm : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\nμ : Measure X\np : ℝ≥0∞\none_le_p : Fact (1 ≤ p)\np_ne_top : Fact (p ≠ ∞)\n𝒜✝ : Set (Set X)\ninst✝¹ : CountablyGenerated X\ninst✝ : SFinite μ\nthis✝ : IsSeparable (μ.restrict μ.sigmaFiniteSet)\n𝒜 : Set (Set X)\ncount_𝒜... | [
"X : Type u_1\nE : Type u_2\nm : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\nμ : Measure X\np : ℝ≥0∞\none_le_p : Fact (1 ≤ p)\np_ne_top : Fact (p ≠ ∞)\n𝒜✝ : Set (Set X)\ninst✝¹ : CountablyGenerated X\ninst✝ : SFinite μ\nthis✝ : IsSeparable (μ.restrict μ.sigmaFiniteSet)\n𝒜 : Set (Set X)\ncount_𝒜 : 𝒜.Counta... | empty_inter, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.SeparableMeasure | {
"line": 490,
"column": 6
} | {
"line": 498,
"column": 35
} | {
"line": 499,
"column": 4
} | [
{
"pp": "case refine_2.refine_1\nX : Type u_1\nE : Type u_2\nm : MeasurableSpace X\ninst✝² : NormedAddCommGroup E\nμ : Measure X\np : ℝ≥0∞\none_le_p : Fact (1 ≤ p)\np_ne_top : Fact (p ≠ ∞)\n𝒜✝ : Set (Set X)\ninst✝¹ : IsSeparable μ\ninst✝ : SeparableSpace E\n𝒜 : Set (Set X)\ncount_𝒜 : 𝒜.Countable\nh𝒜 : μ.Me... | [] | calc
‖a - b‖ * μ.real s ^ (1 / p.toReal)
≤ (ε / (3 * (1 + μ.real s ^ (1 / p.toReal)))) * μ.real s ^ (1 / p.toReal) :=
mul_le_mul_of_nonneg_right (le_of_lt hb) μs_pow_nonneg
_ ≤ ε / 3 := by
rw [← mul_one (ε / 3), div_mul_eq_div_mul_one_div, mul_assoc, one_div_mul_eq_di... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.MeasureTheory.VectorMeasure.BoundedVariation | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 48
} | {
"line": 163,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf ... | [
"α : Type u_1\ninst✝⁸ : LinearOrder α\ninst✝⁷ : DenselyOrdered α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : OrderTopology α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : CompactIccSpace α\nhα : MeasurableSpace α\ninst✝² : BorelSpace α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : CompleteSpace E\nf : α → E\na :... | simp_rw [hf.vectorMeasure_Ioc (u_lt_a _).le] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Measure.Prokhorov | {
"line": 657,
"column": 6
} | {
"line": 657,
"column": 13
} | {
"line": 658,
"column": 6
} | [
{
"pp": "case e_a\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 : Nonempt... | [
"case e_a\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 ... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.MeasureTheory.VectorMeasure.AddContent | {
"line": 311,
"column": 4
} | {
"line": 315,
"column": 61
} | {
"line": 316,
"column": 2
} | [
{
"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 : IsSetSemiring C\nhm : ∀ s ∈ C, ‖m s‖ₑ ≤ μ s\nh'C : hα = generateFrom C\nM : ∀ s ∈ C, MeasurableSet s\nD : Set ... | [] | · apply (hm s hs).trans
simp only [Measure.restrict_apply' MD, μ']
apply measure_mono_ae
nth_rewrite 1 [← Set.inter_self s]
exact ae_le_set_inter Filter.EventuallyLE.rfl (hD s hs) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 14
} | {
"line": 119,
"column": 4
} | [
{
"pp": "X : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nmX : MeasurableSpace X\nμ : VectorMeasure X E\ns : Set X\nhs : MeasurableSet s\nh's : μ.semivariation s = ∞\nt : Set X\nts : t ⊆ s\nt_meas : MeasurableSet t\nht : 2 * ‖μ s‖ₑ + 2 < 2 * ‖μ t‖ₑ\nh't : 1 + ‖μ s‖ₑ ≤ ‖μ t‖ₑ\n... | [
"X : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nmX : MeasurableSpace X\nμ : VectorMeasure X E\ns : Set X\nhs : MeasurableSet s\nh's : μ.semivariation s = ∞\nt : Set X\nts : t ⊆ s\nt_meas : MeasurableSet t\nht : 2 * ‖μ s‖ₑ + 2 < 2 * ‖μ t‖ₑ\nh't : 1 + ‖μ s‖ₑ ≤ ‖μ t‖ₑ\n⊢ μ.semivari... | rw [← h's] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.Variation.Semivariation | {
"line": 128,
"column": 9
} | {
"line": 128,
"column": 47
} | {
"line": 128,
"column": 47
} | [
{
"pp": "case inl\nX : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nmX : MeasurableSpace X\nμ : VectorMeasure X E\ns : Set X\nhs : MeasurableSet s\nh's : μ.semivariation s = ∞\nt : Set X\nts : t ⊆ s\nt_meas : MeasurableSet t\nht : 2 * ‖μ s‖ₑ + 2 < 2 * ‖μ t‖ₑ\nh't : 1 + ‖μ s‖ₑ ... | [
"case inl\nX : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nmX : MeasurableSpace X\nμ : VectorMeasure X E\ns : Set X\nhs : MeasurableSet s\nh's : μ.semivariation s = ∞\nt : Set X\nts : t ⊆ s\nt_meas : MeasurableSet t\nht : 2 * ‖μ s‖ₑ + 2 < 2 * ‖μ t‖ₑ\nh't : 1 + ‖μ s‖ₑ ≤ ‖μ t‖ₑ\nhI... | ENNReal.add_le_add_iff_right (by simp) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 35
} | {
"line": 70,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoid M\ns : Set M\n⊢ IsLinearSet s ↔ ∃ a t, s = a +ᵥ ↑(closure ↑t)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instVAddOfAdd",
"Iff.of_eq",
"congrArg",
"Finset",
"AddMonoid.toAddZeroClass",
"Set.Finite",
... | [] | simp [IsLinearSet, Finset.exists] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 35
} | {
"line": 70,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoid M\ns : Set M\n⊢ IsLinearSet s ↔ ∃ a t, s = a +ᵥ ↑(closure ↑t)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instVAddOfAdd",
"Iff.of_eq",
"congrArg",
"Finset",
"AddMonoid.toAddZeroClass",
"Set.Finite",
... | [] | simp [IsLinearSet, Finset.exists] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 35
} | {
"line": 70,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoid M\ns : Set M\n⊢ IsLinearSet s ↔ ∃ a t, s = a +ᵥ ↑(closure ↑t)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instVAddOfAdd",
"Iff.of_eq",
"congrArg",
"Finset",
"AddMonoid.toAddZeroClass",
"Set.Finite",
... | [] | simp [IsLinearSet, Finset.exists] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 48
} | {
"line": 157,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoid M\nS : Set (Set M)\nhS : S.Finite\nhS' : ∀ s ∈ S, IsSemilinearSet s\n⊢ IsSemilinearSet (⋃₀ S)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.sUnion_insert",
"Set.sUnion",
"Set.Finite",
... | [] | induction S, hS using Finite.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 48
} | {
"line": 262,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoid M\nS : Set (Set M)\nhS : S.Finite\nhS' : ∀ t ∈ S, IsLinearSet t\n⊢ IsSemilinearSet ↑(closure (⋃₀ S))",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"AddS... | [] | induction S, hS using Finite.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Defs | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 48
} | {
"line": 321,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoid M\nS : Set (Set M)\nhS : S.Finite\nhS' : ∀ s ∈ S, IsProperSemilinearSet s\n⊢ IsProperSemilinearSet (⋃₀ S)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsProperSemilinearSet.union",
"IsProperSemilinearSet.empty._si... | [] | induction S, hS using Finite.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 158,
"column": 76
} | {
"line": 160,
"column": 86
} | {
"line": 162,
"column": 0
} | [
{
"pp": "X : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\ns : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G\nB : E →L... | [] | by
rw [← setIntegral_union disjoint_compl_right
hs hs.compl hfi.integrableOn hfi.integrableOn, union_compl_self, setIntegral_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 28
} | {
"line": 134,
"column": 4
} | [
{
"pp": "case neg.hu₁\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 := ⋯\nhf : ∀ x ∈ s, ∀ (y : M), y ∈ f x ↔ x + y ∈ s\ng :... | [
"case neg.hu₂\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 := ⋯\nhf : ∀ x ∈ s, ∀ (y : M), y ∈ f x ↔ x + y ∈ s\ng : M → AddSemi... | · exact add_mem hu'₁ hu₁ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.WithDensityVec | {
"line": 134,
"column": 4
} | {
"line": 137,
"column": 33
} | {
"line": 138,
"column": 4
} | [
{
"pp": "X : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nf : X → E\nB : E →L[ℝ] F →L[ℝ... | [
"X : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nf : X → E\nB : E →L[ℝ] F →L[ℝ] G\ninst✝ :... | obtain ⟨g, h'g, gmem⟩ : ∃ (g : X →ₛ E), eLpNorm (f - ⇑g) 1 μ.variation < ρ
∧ MemLp (⇑g) 1 μ.variation :=
(memLp_one_iff_integrable.2 hf).exists_simpleFunc_eLpNorm_sub_lt (by simp)
(by simpa using ρpos.ne') | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 307,
"column": 2
} | {
"line": 307,
"column": 48
} | {
"line": 308,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : AddMonoid.FG M\nS : Set (Set M)\nhS : S.Finite\nhS' : ∀ s ∈ S, IsSemilinearSet s\n⊢ IsSemilinearSet (⋂₀ S)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.sInter_empty",
"congrArg",
"Set.univ",
... | [] | induction S, hS using Finite.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 378,
"column": 16
} | {
"line": 378,
"column": 21
} | {
"line": 378,
"column": 22
} | [
{
"pp": "case h_add\nX : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\nmX : MeasurableSpace X\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup H\nμ : VectorMeasure X F\nf✝ : X → E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : Nor... | [
"case h_add\nX : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\nmX : MeasurableSpace X\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedAddCommGroup H\nμ : VectorMeasure X F\nf✝ : X → E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedSpace ℝ F... | f_int | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 557,
"column": 2
} | {
"line": 558,
"column": 67
} | {
"line": 560,
"column": 0
} | [
{
"pp": "case right\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\ni : ↑hs.basisSet\n⊢ (hs.basis.repr (toRatVec (hs.fract x) - toRatVec hs.base)) i < 1",
"ppTerm": "?right",
"assigned": true,
"usedConstants": [
"Rat.addCommMonoid",
"Finsupp.instFun... | [] | · rw [hs.toRatVec_fract_eq, add_sub_cancel_left]
simp [← hs.basis_apply, Finsupp.single_apply, Int.fract_lt_one] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 401,
"column": 16
} | {
"line": 401,
"column": 21
} | {
"line": 401,
"column": 22
} | [
{
"pp": "case pos.h_add\nX : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\nmX : MeasurableSpace X\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedAddCommGroup H\nμ : VectorMeasure X F\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace... | [
"case pos.h_add\nX : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\nmX : MeasurableSpace X\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedAddCommGroup H\nμ : VectorMeasure X F\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace ℝ F\ninst✝¹... | f_int | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 593,
"column": 4
} | {
"line": 593,
"column": 66
} | {
"line": 595,
"column": 0
} | [
{
"pp": "case e'_3.a\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx : ι → ℕ\nhx₁ : hs.fract x = hs.base\nhx₂ : ∀ (i : ↑hs.basisSet), 0 ≤ hs.floor x i ∧ (↑i ∉ hs.periods → hs.floor x i = 0)\ni : ↑hs.basisSet\na✝ : i ∈ Finset.univ\n⊢ 0 = (-hs.floor x i).toNat • ↑i",
"ppTerm": "?... | [] | · simp [fun i => Int.toNat_eq_zero.2 (neg_nonpos.2 (hx₂ i).1)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 613,
"column": 6
} | {
"line": 613,
"column": 15
} | {
"line": 614,
"column": 2
} | [
{
"pp": "case mpr\nι : Type u_3\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\ninst✝ : Finite ι\nx u : ι → ℕ\nhu : u ∈ hs.fundamentalDomain\nhu' : u ∉ {hs.base}\ny : ι → ℕ\nhy : y ∈ closure hs.basisSet\ny' : ι → ℕ\nhy' : y' ∈ closure hs.basisSet\nheq : hs.fract x = u\n⊢ hs.fract x ≠ hs.base",
"ppTerm": "?mpr",... | [] | rwa [heq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.VectorMeasure.SetIntegral | {
"line": 459,
"column": 4
} | {
"line": 459,
"column": 54
} | {
"line": 460,
"column": 4
} | [
{
"pp": "case neg\nX : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\ns : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G... | [
"case neg\nX : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedAddCommGroup G\nμ : VectorMeasure X F\nf : X → E\ns : Set X\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ G\nB : E →L[ℝ... | have : 0 ≤ C := le_trans (norm_nonneg _) (hC x hx) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basic | {
"line": 761,
"column": 2
} | {
"line": 762,
"column": 94
} | {
"line": 763,
"column": 2
} | [
{
"pp": "ι : Type u_3\ninst✝ : Finite ι\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\nx✝ : ι → ℕ\n⊢ x✝ ∈ sᶜ ↔ x✝ ∈ hs.setOfFractNe ∪ (hs.setOfFloorNeg ∪ hs.setOfFloorPos)",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"_private.Mathlib.ModelTheory.Arithmetic.Presburger.Semilinear.Basi... | [
"ι : Type u_3\ninst✝ : Finite ι\ns : Set (ι → ℕ)\nhs : IsProperLinearSet s\nx✝ : ι → ℕ\n⊢ ¬(hs.fract x✝ = hs.base ∧ ∀ (i : ↑hs.basisSet), 0 ≤ hs.floor x✝ i ∧ (↑i ∉ hs.periods → hs.floor x✝ i = 0)) ↔\n hs.fract x✝ ≠ hs.base ∨\n (hs.fract x✝ = hs.base ∧ ∃ i, hs.floor x✝ i < 0) ∨\n hs.fract x✝ = hs.base... | simp only [mem_compl_iff, hs.mem_iff_fract_eq_and_floor_nonneg, IsProperLinearSet.setOfFractNe,
IsProperLinearSet.setOfFloorNeg, IsProperLinearSet.setOfFloorPos, mem_union, mem_setOf_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.ModelTheory.Equivalence | {
"line": 89,
"column": 2
} | {
"line": 90,
"column": 16
} | {
"line": 92,
"column": 0
} | [
{
"pp": "L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ ψ : L.BoundedFormula α n\nM : T.ModelType\nv : α → ↑M\nxs : Fin n → ↑M\n⊢ (φ ⊓ ψ ⟹ φ).Realize v xs",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"FirstOrder.Language.BoundedFormula.imp",
"Eq.mpr",
"FirstOrder.Lan... | [] | simp only [BoundedFormula.realize_imp, BoundedFormula.realize_inf]
exact And.left | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.ModelTheory.Equivalence | {
"line": 89,
"column": 2
} | {
"line": 90,
"column": 16
} | {
"line": 92,
"column": 0
} | [
{
"pp": "L : Language\nT : L.Theory\nα : Type w\nn : ℕ\nφ ψ : L.BoundedFormula α n\nM : T.ModelType\nv : α → ↑M\nxs : Fin n → ↑M\n⊢ (φ ⊓ ψ ⟹ φ).Realize v xs",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"FirstOrder.Language.BoundedFormula.imp",
"Eq.mpr",
"FirstOrder.Lan... | [] | simp only [BoundedFormula.realize_imp, BoundedFormula.realize_inf]
exact And.left | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.DirectLimit | {
"line": 175,
"column": 6
} | {
"line": 175,
"column": 29
} | {
"line": 175,
"column": 30
} | [
{
"pp": "case mk.mk\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirectedOrder ι\ninst✝ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ :... | [
"case mk.mk\nL : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirectedOrder ι\ninst✝ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\ni fst✝¹ : ι\nsnd✝¹ : G fst✝¹\nhx : ⟨fst✝¹, snd✝¹⟩.fst ≤ i\nfst✝ : ι\nsnd✝ : G fst✝\nhy ... | DirectedSystem.map_map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.PartialEquiv | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 15
} | {
"line": 140,
"column": 4
} | [
{
"pp": "case mpr\nL : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nf g : M ≃ₚ[L] N\ndom_le_dom : f.dom ≤ g.dom\nle_cod : f.cod ≤ g.cod\nh_eq : ∀ (x : ↥f.dom), (inclusion le_cod) (f.toEquiv x) = g.toEquiv ((inclusion dom_le_dom) x)\n⊢ f ≤ g",
"ppTerm": "?mpr",
"assig... | [
"case mpr\nL : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nf g : M ≃ₚ[L] N\ndom_le_dom : f.dom ≤ g.dom\nle_cod : f.cod ≤ g.cod\nh_eq : ∀ (x : ↥f.dom), (inclusion le_cod) (f.toEquiv x) = g.toEquiv ((inclusion dom_le_dom) x)\n⊢ ∃ (h : f.dom ≤ g.dom),\n g.cod.subtype.comp (g.to... | rw [le_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.DirectLimit | {
"line": 189,
"column": 40
} | {
"line": 189,
"column": 61
} | {
"line": 189,
"column": 62
} | [
{
"pp": "L : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirectedOrder ι\ninst✝ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\nn : ℕ\nR : L.Relations n\nx : Fin n → Σˣ f\ni j : ι\nhi : i ∈ upperBounds (range (Sigma... | [
"L : Language\nι : Type v\ninst✝³ : Preorder ι\nG : ι → Type w\ninst✝² : (i : ι) → L.Structure (G i)\nf : (i j : ι) → i ≤ j → G i ↪[L] G j\ninst✝¹ : IsDirectedOrder ι\ninst✝ : DirectedSystem G fun i j h ↦ ⇑(f i j h)\nn : ℕ\nR : L.Relations n\nx : Fin n → Σˣ f\ni j : ι\nhi : i ∈ upperBounds (range (Sigma.fst ∘ x))\n... | ← (f j k jk).map_rel, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.ModelTheory.PartialEquiv | {
"line": 185,
"column": 25
} | {
"line": 185,
"column": 36
} | {
"line": 185,
"column": 36
} | [
{
"pp": "case refl.le_fg\nL : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\ng : M ≃ₚ[L] N\ncod_f : L.Substructure N\nequiv_f : ↥g.1 ≃[L] ↥cod_f\nh :\n ∀ (x : M) (h : x ∈ { dom := g.1, cod := cod_f, toEquiv := equiv_f }.dom),\n { dom := g.1, cod := cod_f, toEquiv := equiv_... | [
"case refl.le_fg\nL : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\ng : M ≃ₚ[L] N\ncod_f : L.Substructure N\nequiv_f : ↥g.1 ≃[L] ↥cod_f\nh :\n ∀ (x : M) (h : x ∈ { dom := g.1, cod := cod_f, toEquiv := equiv_f }.dom),\n { dom := g.1, cod := cod_f, toEquiv := equiv_f }.cod.subt... | rw [le_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.PartialEquiv | {
"line": 185,
"column": 25
} | {
"line": 185,
"column": 36
} | {
"line": 185,
"column": 36
} | [
{
"pp": "case refl.le_gf\nL : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\ng : M ≃ₚ[L] N\ncod_f : L.Substructure N\nequiv_f : ↥g.1 ≃[L] ↥cod_f\nh :\n ∀ (x : M) (h : x ∈ { dom := g.1, cod := cod_f, toEquiv := equiv_f }.dom),\n { dom := g.1, cod := cod_f, toEquiv := equiv_... | [
"case refl.le_gf\nL : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\ng : M ≃ₚ[L] N\ncod_f : L.Substructure N\nequiv_f : ↥g.1 ≃[L] ↥cod_f\nh :\n ∀ (x : M) (h : x ∈ { dom := g.1, cod := cod_f, toEquiv := equiv_f }.dom),\n { dom := g.1, cod := cod_f, toEquiv := equiv_f }.cod.subt... | rw [le_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.PartialEquiv | {
"line": 437,
"column": 2
} | {
"line": 444,
"column": 75
} | {
"line": 445,
"column": 2
} | [
{
"pp": "case refine_1\nL : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nh : L.IsExtensionPair M N\nS : L.Substructure M\nS_FG : S.FG\nf : ↥S ↪[L] N\nm : M\n⊢ ∃ g, f = g.comp (Substructure.inclusion ⋯)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [... | [
"case refine_2\nL : Language\nM : Type w\nN : Type w'\ninst✝¹ : L.Structure M\ninst✝ : L.Structure N\nh : ∀ (S : L.Substructure M), S.FG → ∀ (f : ↥S ↪[L] N) (m : M), ∃ g, f = g.comp (Substructure.inclusion ⋯)\nx✝ : L.FGEquiv M N\nm : M\nf : M ≃ₚ[L] N\nf_FG : f.dom.FG\n⊢ ∃ g, m ∈ (↑g).dom ∧ ⟨f, f_FG⟩ ≤ g"
] | · obtain ⟨⟨f', hf'⟩, mf', ff'1, ff'2⟩ := h ⟨⟨S, _, f.equivRange⟩, S_FG⟩ m
refine ⟨f'.toEmbedding.comp (Substructure.inclusion ?_), ?_⟩
· simp only [sup_le_iff, ff'1, closure_le, singleton_subset_iff, SetLike.mem_coe, mf',
and_self]
· ext ⟨x, hx⟩
rw [Embedding.subtype_equivRange] at ff'2
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 633,
"column": 2
} | {
"line": 633,
"column": 45
} | {
"line": 634,
"column": 2
} | [
{
"pp": "X : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : X → E\nμ : VectorMeasure X F\nB : E →L[ℝ] F →L[ℝ]... | [
"X : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nf : X → E\nμ : VectorMeasure X F\nB : E →L[ℝ] F →L[ℝ] G\n⊢ ↑(NNRe... | apply (enorm_setToFun_le _ (by simp)).trans | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Order.CountableDenseLinearOrder | {
"line": 159,
"column": 10
} | {
"line": 159,
"column": 85
} | {
"line": 160,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : Finset (α × β)\nhf : ∀ p ∈ f, ∀ q ∈ f, cmp p.1 q.1 = cmp p.2 q.2\np : β × α\nhp : p ∈ Finset.image (⇑(Equiv.prodComm α β)) f\nq : β × α\nhq : q ∈ Finset.image (⇑(Equiv.prodComm α β)) f\n⊢ (Equiv.prodComm α β).symm p ∈ f",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : Finset (α × β)\nhf : ∀ p ∈ f, ∀ q ∈ f, cmp p.1 q.1 = cmp p.2 q.2\np : β × α\nhp : p ∈ ⇑(Equiv.prodComm α β).symm ⁻¹' ↑f\nq : β × α\nhq : q ∈ Finset.image (⇑(Equiv.prodComm α β)) f\n⊢ (Equiv.prodComm α β).symm p ∈ f"
] | rw [← Finset.mem_coe, Finset.coe_image, Equiv.image_eq_preimage_symm] at hp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.ModelTheory.Order | {
"line": 382,
"column": 36
} | {
"line": 382,
"column": 60
} | {
"line": 382,
"column": 60
} | [
{
"pp": "L : Language\nα : Type w\nM : Type w'\nn : ℕ\ninst✝¹ : L.IsOrdered\ninst✝ : L.Structure M\nh : M ⊨ L.preorderTheory\n⊢ leSymb.reflexive ∈ L.preorderTheory",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"congrArg",
"true_or",
"Membership.mem",
"Set.instSing... | [] | by simp [preorderTheory] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.ModelTheory.Order | {
"line": 384,
"column": 36
} | {
"line": 384,
"column": 60
} | {
"line": 384,
"column": 60
} | [
{
"pp": "L : Language\nα : Type w\nM : Type w'\nn : ℕ\ninst✝¹ : L.IsOrdered\ninst✝ : L.Structure M\nh : M ⊨ L.preorderTheory\n⊢ leSymb.transitive ∈ L.preorderTheory",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"Set.instSingletonSet",
... | [] | by simp [preorderTheory] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Baire.LocallyCompactRegular | {
"line": 23,
"column": 87
} | {
"line": 59,
"column": 34
} | {
"line": 61,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ns : Set X\ninst✝¹ : R1Space X\ninst✝ : LocallyCompactSpace X\n⊢ BaireSpace X",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"interior_subset",
"TopologicalSpace.PositiveCompacts.instSetLike",
"_private.Ma... | [] | by
constructor
intro f ho hd
/- To prove that an intersection of open dense subsets is dense, prove that its intersection
with any open neighbourhood `U` is dense. Define recursively a decreasing sequence `K` of
compact neighbourhoods: start with some compact neighbourhood inside `U`, then at each step,
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 844,
"column": 6
} | {
"line": 844,
"column": 47
} | {
"line": 845,
"column": 4
} | [
{
"pp": "case pos\nX : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nB : E →L[ℝ] F →L[ℝ]... | [] | apply integral_map hφ.measurable hfm h'fm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 844,
"column": 6
} | {
"line": 844,
"column": 47
} | {
"line": 845,
"column": 4
} | [
{
"pp": "case pos\nX : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nB : E →L[ℝ] F →L[ℝ]... | [] | apply integral_map hφ.measurable hfm h'fm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Integral | {
"line": 844,
"column": 6
} | {
"line": 844,
"column": 47
} | {
"line": 845,
"column": 4
} | [
{
"pp": "case pos\nX : Type u_2\nE : Type u_4\nF : Type u_5\nG : Type u_6\nmX : MeasurableSpace X\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\nμ : VectorMeasure X F\nB : E →L[ℝ] F →L[ℝ]... | [] | apply integral_map hφ.measurable hfm h'fm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.ModelTheory.Topology.Types | {
"line": 80,
"column": 8
} | {
"line": 80,
"column": 25
} | {
"line": 80,
"column": 26
} | [
{
"pp": "case refine_2\nL : Language\nT : L.Theory\nα : Type u_1\nF : Ultrafilter (T.CompleteType α)\na✝ : ↑F ≤ Filter.principal univ\n⊢ IsMaximal {φ | T.typesWith φ ∈ F}",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FirstOrder.Language.Theory.IsSatisfiable",
... | [
"case refine_2\nL : Language\nT : L.Theory\nα : Type u_1\nF : Ultrafilter (T.CompleteType α)\na✝ : ↑F ≤ Filter.principal univ\n⊢ IsSatisfiable {φ | T.typesWith φ ∈ F} ∧\n ∀ (φ : L[[α]].Sentence), φ ∈ {φ | T.typesWith φ ∈ F} ∨ Formula.not φ ∈ {φ | T.typesWith φ ∈ F}"
] | Theory.IsMaximal, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ADEInequality | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 46
} | {
"line": 221,
"column": 2
} | [
{
"pp": "p q r : ℕ+\nS : List ℕ+ := {p, q, r}.sort fun a b ↦ a ≤ b\nH : 1 < sumInv ↑S\nhS : S.SortedLE\nhpqr : {p, q, r} = ↑S\n⊢ (∃ q r, A' q r = ↑S) ∨ (∃ r, D' r = ↑S) ∨ E' 3 = ↑S ∨ E' 4 = ↑S ∨ E' 5 = ↑S",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"ADEInequality.admissible_of_on... | [
"p q r : ℕ+\nS : List ℕ+ := ⋯\nH : 1 < sumInv ↑S\nhS : S.SortedLE\nhpqr : {p, q, r} = ↑S\n⊢ S.length = 3"
] | apply admissible_of_one_lt_sumInv_aux hS _ H | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 61
} | {
"line": 73,
"column": 2
} | [
{
"pp": "⊢ Nat.card (ZMod 4)ˣ = 2",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ZMod.commRing",
"congrArg",
"CommSemiring.toSemiring",
"ZMod.fintype",
"ZMod.decidableEq",
"ZMod.card_units_eq_totient",
"Units",
"instFintypeU... | [
"⊢ φ 4 = 2"
] | simp only [Nat.card_eq_fintype_card, card_units_eq_totient] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ArithmeticFunction.VonMangoldt | {
"line": 106,
"column": 35
} | {
"line": 106,
"column": 44
} | {
"line": 106,
"column": 45
} | [
{
"pp": "case refine_2\nn p k : ℕ\nhp : Nat.Prime p\n⊢ ∑ x ∈ range (k + 1), Λ (p ^ x) = Real.log ↑(p ^ k)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.vonMangoldt",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"Arithm... | [
"case refine_2\nn p k : ℕ\nhp : Nat.Prime p\n⊢ ∑ x ∈ range (k + 1), Λ (p ^ x) = Real.log (↑p ^ k)"
] | cast_pow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 120,
"column": 63
} | {
"line": 120,
"column": 70
} | {
"line": 121,
"column": 6
} | [
{
"pp": "R : 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\ni : ℕ\nhi : ¬i = p ∧ ¬i = 1 ∧ ¬i = 0 ∧ i < p + 1\nhi' : 2 ≤ i\n⊢ (u * x) ^ (2 + (i - 2)) * ↑(p.choose i) = u ^ 2 * x ^ 2 * (u * x) ^ (i - 2) * ↑(p.choose i)",
"ppTerm... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 120,
"column": 63
} | {
"line": 120,
"column": 70
} | {
"line": 121,
"column": 6
} | [
{
"pp": "R : 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\ni : ℕ\nhi : ¬i = p ∧ ¬i = 1 ∧ ¬i = 0 ∧ i < p + 1\nhi' : 2 ≤ i\n⊢ (u * x) ^ (2 + (i - 2)) * ↑(p.choose i) = u ^ 2 * x ^ 2 * (u * x) ^ (i - 2) * ↑(p.choose i)",
"ppTerm... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 120,
"column": 63
} | {
"line": 120,
"column": 70
} | {
"line": 121,
"column": 6
} | [
{
"pp": "R : 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\ni : ℕ\nhi : ¬i = p ∧ ¬i = 1 ∧ ¬i = 0 ∧ i < p + 1\nhi' : 2 ≤ i\n⊢ (u * x) ^ (2 + (i - 2)) * ↑(p.choose i) = u ^ 2 * x ^ 2 * (u * x) ^ (i - 2) * ↑(p.choose i)",
"ppTerm... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 124,
"column": 34
} | {
"line": 124,
"column": 41
} | {
"line": 125,
"column": 6
} | [
{
"pp": "R : 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\ni : ℕ\nhi : ¬i = p ∧ ¬i = 1 ∧ ¬i = 0 ∧ i < p + 1\nhi' : 2 ≤ i\n⊢ u ^ 2 * x ^ 2 * (u * x) ^ (i - 2) * (↑p * ↑(p.choose i / p)) =\n u ^ 2 * x ^ 2 * (u * x) ^ (i - 2) * ↑... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 130,
"column": 40
} | {
"line": 130,
"column": 47
} | {
"line": 132,
"column": 0
} | [
{
"pp": "R : 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⊢ ↑p * u * v * b * x ^ p = ↑p * u * (v * (b * x ^ p))",
"ppTerm": "?m.656",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 130,
"column": 40
} | {
"line": 130,
"column": 47
} | {
"line": 132,
"column": 0
} | [
{
"pp": "R : 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⊢ ↑p * u * v * b * x ^ p = ↑p * u * (v * (b * x ^ p))",
"ppTerm": "?m.656",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.ZMod.UnitsCyclic | {
"line": 130,
"column": 40
} | {
"line": 130,
"column": 47
} | {
"line": 132,
"column": 0
} | [
{
"pp": "R : 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⊢ ↑p * u * v * b * x ^ p = ↑p * u * (v * (b * x ^ p))",
"ppTerm": "?m.656",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 137,
"column": 6
} | {
"line": 140,
"column": 62
} | {
"line": 141,
"column": 2
} | [
{
"pp": "n x✝ : ℕ\na✝ : x✝ ∈ range (n + 1 - 0)\n| ∑ x ∈ range (n + 1 - x✝), (monomial x✝) (↑((n + 1).choose (x✝ + x)) * ↑((x✝ + x).choose x✝) * _root_.bernoulli x)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"NonA... | [] | apply_congr
· skip
· rw [← Nat.cast_mul, choose_mul (le_add_right _ _), Nat.cast_mul, add_tsub_cancel_left,
mul_assoc, mul_comm, ← smul_eq_mul, ← smul_monomial] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.NumberTheory.PrimeCounting | {
"line": 216,
"column": 86
} | {
"line": 218,
"column": 63
} | {
"line": 220,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ n.primesLE = filter Prime (Ioc 1 n)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Finset.mem_range._simp_1",
"Finset.mem_filter._simp_1",
"Nat.Prime",
"Preorder.toLT",
"Nat.instOne",
"congrArg",
"Finset",
"PartialOrder.toP... | [] | by
ext p
simp +contextual [primesLE_eq_filter_range, Nat.Prime.one_lt] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.BernoulliPolynomials | {
"line": 224,
"column": 6
} | {
"line": 224,
"column": 20
} | {
"line": 224,
"column": 21
} | [
{
"pp": "case succ\nn i : ℕ\n⊢ ((bernoulli (n + 1)).comp (-X)).coeff i = ((-1) ^ (n + 1) • (bernoulli (n + 1) + (n + 1) • X ^ (n + 1 - 1))).coeff i",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Polynomial.instNSMul",
"instHSMul",
"... | [
"case succ\nn i : ℕ\n⊢ ((bernoulli (n + 1)).comp (-1 * X)).coeff i =\n ((-1) ^ (n + 1) • (bernoulli (n + 1) + (n + 1) • X ^ (n + 1 - 1))).coeff i"
] | ← neg_one_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Bernoulli | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 40
} | {
"line": 137,
"column": 2
} | [
{
"pp": "⊢ bernoulli' 4 = -1 / 30",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Nat.choose",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"Bool",
"Eq.refl",
"instDecidableEqNat",
"OfNat.ofNat",
"Decidabl... | [
"this : Nat.choose 4 2 = 6\n⊢ bernoulli' 4 = -1 / 30"
] | have : Nat.choose 4 2 = 6 := by decide | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.Bernoulli | {
"line": 168,
"column": 70
} | {
"line": 169,
"column": 72
} | {
"line": 170,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : Algebra ℚ A\nn : ℕ\nthis : ∑ p ∈ antidiagonal n, bernoulli' p.1 / ↑p.1! * ((↑p.2 + 1) * ↑p.2!)⁻¹ = (↑n !)⁻¹\n⊢ (coeff (n + 1, 0).1) (PowerSeries.mk fun n ↦ (algebraMap ℚ A) (bernoulli' n / ↑n !)) *\n (coeff (n + 1, 0).2) (exp A - 1) +\n ∑ p ∈ antid... | [] | by
simpa [map_sum, Nat.factorial] using congr_arg (algebraMap ℚ A) this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Bernoulli | {
"line": 210,
"column": 51
} | {
"line": 210,
"column": 71
} | {
"line": 212,
"column": 0
} | [
{
"pp": "⊢ bernoulli 1 = -1 / 2",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Mathlib.Meta.NormNum.isInt_pow",
"Rat.instOfNat",
"NegZeroClass.toNeg",
"Rat.instMul",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Mathlib.Meta.NormN... | [] | norm_num [bernoulli] | Mathlib.Tactic._aux_Mathlib_Tactic_NormNum_Core___elabRules_Mathlib_Tactic_normNum_1 | Mathlib.Tactic.normNum |
Mathlib.NumberTheory.Bernoulli | {
"line": 210,
"column": 51
} | {
"line": 210,
"column": 71
} | {
"line": 212,
"column": 0
} | [
{
"pp": "⊢ bernoulli 1 = -1 / 2",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Mathlib.Meta.NormNum.isInt_pow",
"Rat.instOfNat",
"NegZeroClass.toNeg",
"Rat.instMul",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Mathlib.Meta.NormN... | [] | norm_num [bernoulli] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Bernoulli | {
"line": 210,
"column": 51
} | {
"line": 210,
"column": 71
} | {
"line": 212,
"column": 0
} | [
{
"pp": "⊢ bernoulli 1 = -1 / 2",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Mathlib.Meta.NormNum.isInt_pow",
"Rat.instOfNat",
"NegZeroClass.toNeg",
"Rat.instMul",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Mathlib.Meta.NormN... | [] | norm_num [bernoulli] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Bertrand | {
"line": 112,
"column": 17
} | {
"line": 112,
"column": 63
} | {
"line": 113,
"column": 4
} | [
{
"pp": "x : ℝ\nx_large : 512 ≤ x\nf : ℝ → ℝ := fun x ↦ log x + √(2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)\nhf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn ℝ (Set.Ioi 0.5) f\nthis : √(2 * 512) = 32... | [] | equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_Conv_equals_1 | Batteries.Tactic.Conv.equals |
Mathlib.NumberTheory.Bertrand | {
"line": 112,
"column": 17
} | {
"line": 112,
"column": 63
} | {
"line": 113,
"column": 4
} | [
{
"pp": "x : ℝ\nx_large : 512 ≤ x\nf : ℝ → ℝ := fun x ↦ log x + √(2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)\nhf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn ℝ (Set.Ioi 0.5) f\nthis : √(2 * 512) = 32... | [] | equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.NumberTheory.Bertrand | {
"line": 112,
"column": 17
} | {
"line": 112,
"column": 63
} | {
"line": 113,
"column": 4
} | [
{
"pp": "x : ℝ\nx_large : 512 ≤ x\nf : ℝ → ℝ := fun x ↦ log x + √(2 * x) * log (2 * x) - log 4 / 3 * x\nhf' : ∀ (x : ℝ), 0 < x → 0 < x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3)\nhf : ∀ (x : ℝ), 0 < x → f x = log (x * (2 * x) ^ √(2 * x) / 4 ^ (x / 3))\nh5 : 0 < x\nh : ConcaveOn ℝ (Set.Ioi 0.5) f\nthis : √(2 * 512) = 32... | [] | equals 2 ^ (2 : ℝ) => rw [rpow_two]; norm_num1 | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.NumberTheory.AbelSummation | {
"line": 359,
"column": 8
} | {
"line": 360,
"column": 47
} | {
"line": 361,
"column": 8
} | [
{
"pp": "case succ.calc_4\n𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nc : ℕ → 𝕜\nf : ℝ → 𝕜\nm : ℕ\nh_bdd : (fun n ↦ ‖f ↑n‖ * ∑ k ∈ Icc 0 n, ‖c k‖) =O[atTop] fun x ↦ 1\nhf_int : LocallyIntegrableOn (deriv fun t ↦ ‖f t‖) (Set.Ici ↑m) volume\nhf :\n ∀ (n : ℕ),\n ∑ k ∈ Icc 0 n, ‖f ↑k‖ * ‖c k‖ =\n ‖f ↑n‖ * ∑ k ∈ I... | [
"𝕜 : 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‖ - ∫ (t : ℝ) in S... | grw [setIntegral_mono_set ?_ (.of_forall fun _ ↦ norm_nonneg _)
Set.Ioc_subset_Ioi_self.eventuallyLE] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.NumberTheory.Chebyshev | {
"line": 555,
"column": 4
} | {
"line": 555,
"column": 11
} | {
"line": 557,
"column": 0
} | [
{
"pp": "case succ\nx : ℝ\nN : ℕ\nih :\n ∑ n ∈ Icc 1 (1 + 6 * N), b x n =\n b x 1 + ∑ n ∈ Icc 1 (3 * N), b x (2 * n) + ∑ n ∈ Icc 1 (2 * N), b x (3 * n) + ∑ n ∈ Icc 1 N, c x n\n⊢ b x 1 + ∑ n ∈ Icc 1 (3 * N), b x (2 * n) + ∑ n ∈ Icc 1 (2 * N), b x (3 * n) +\n ∑ x_1 ∈ Icc 1 N, (b x (6 * x_1 - ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.ClassNumber.Finite | {
"line": 254,
"column": 40
} | {
"line": 254,
"column": 54
} | {
"line": 254,
"column": 55
} | [
{
"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... | [
"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✝ : Algebra.IsAlgebra... | mul_comm b a', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Chebyshev | {
"line": 724,
"column": 8
} | {
"line": 724,
"column": 31
} | {
"line": 724,
"column": 32
} | [
{
"pp": "x : ℝ\nhx : 4 ≤ x\ntwo_le_sqrt : 2 ≤ √x\nsqrt_le_x : √x ≤ x\n⊢ (√x - 2) / log 2 ^ 2 + (x - √x) / log √x ^ 2 ≤ 4 * x / log x ^ 2 + √x / log 2 ^ 2",
"ppTerm": "?m.217",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.... | [
"x : ℝ\nhx : 4 ≤ x\ntwo_le_sqrt : 2 ≤ √x\nsqrt_le_x : √x ≤ x\n⊢ (√x - 2) / log 2 ^ 2 + (x - √x) / (log x / 2) ^ 2 ≤ 4 * x / log x ^ 2 + √x / log 2 ^ 2"
] | log_sqrt (by linarith), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 152,
"column": 38
} | {
"line": 152,
"column": 45
} | {
"line": 153,
"column": 6
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nr : ℚ\nhr : ‖↑r - ↑x‖ < 1\n⊢ ‖↑r‖ = ‖↑r - ↑x + ↑x‖",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.t... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 152,
"column": 38
} | {
"line": 152,
"column": 45
} | {
"line": 153,
"column": 6
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nr : ℚ\nhr : ‖↑r - ↑x‖ < 1\n⊢ ‖↑r‖ = ‖↑r - ↑x + ↑x‖",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.t... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Padics.RingHoms | {
"line": 152,
"column": 38
} | {
"line": 152,
"column": 45
} | {
"line": 153,
"column": 6
} | [
{
"pp": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nx : ℤ_[p]\nr : ℚ\nhr : ‖↑r - ↑x‖ < 1\n⊢ ‖↑r‖ = ‖↑r - ↑x + ↑x‖",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.t... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Discriminant | {
"line": 209,
"column": 8
} | {
"line": 209,
"column": 25
} | {
"line": 209,
"column": 26
} | [
{
"pp": "K : Type u\nL : Type v\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Module.Finite K L\npb : PowerBasis K L\ninst✝ : Algebra.IsSeparable K L\nE : Type v := AlgebraicClosure L\nthis : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)\n⊢ Fintype.card (Fin pb... | [
"K : Type u\nL : Type v\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Module.Finite K L\npb : PowerBasis K L\ninst✝ : Algebra.IsSeparable K L\nE : Type v := AlgebraicClosure L\nthis : (a b : E) → Decidable (a = b) := fun a b ↦ Classical.propDecidable (a = b)\n⊢ pb.dim = Fintype.card (L →ₐ[K] E... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.PellMatiyasevic | {
"line": 70,
"column": 58
} | {
"line": 70,
"column": 65
} | {
"line": 72,
"column": 0
} | [
{
"pp": "d x y : ℤ\n⊢ x * x - y * (d * y) = 1 ↔ x * x + -(y * (d * y)) = 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Mathlib.Tactic.Ring.Common.neg_zero",
"... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
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