module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 48
} | [
{
"pp": "case neg\nR : Type u_2\ninst✝² : Bornology R\ninst✝¹ : Monoid R\ninst✝ : BoundedMul R\ns : Set R\ns_bdd : Bornology.IsBounded s\ns_empty : ¬s = ∅\n⊢ Bornology.IsBounded ((fun x ↦ x ^ 0) '' s)",
"usedConstants": [
"Eq.mp",
"Set.Nonempty",
"Set.instEmptyCollection",
"EmptyColl... | simp_rw [← nonempty_iff_ne_empty] at s_empty | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 185,
"column": 62
} | {
"line": 185,
"column": 71
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : SeminormedAddCommGroup β\nf : α →ᵇ β\nn : ℕ\n⊢ -⇑(n.succ • f) = -((n + 1) • ⇑f)",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instHSMul",
"Pi.instNeg",
"congrArg",
"AddMonoid.toNSMul",
"Ps... | coe_nsmul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.Bounded.Normed | {
"line": 311,
"column": 42
} | {
"line": 311,
"column": 87
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\nR : Type u_1\ninst✝¹ : NonUnitalSeminormedRing R\ninst✝ : IsCancelMulZero R\nf g : α →ᵇ R\nh : f * g = 0\n⊢ ∀ (x : α), f x = 0 ∨ g x = 0",
"usedConstants": [
"HMul.hMul",
"IsTopologicalRing.toIsTopologicalSemiring",
"MulZeroClass.toMul",
... | simpa [DFunLike.ext_iff, mul_eq_zero] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.Content | {
"line": 285,
"column": 2
} | {
"line": 289,
"column": 23
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\nμ : Content G\ninst✝¹ : R1Space G\ninst✝ : WeaklyLocallyCompactSpace G\nK : Set G\nhK : IsCompact K\nF : Set G\nh1F : IsCompact F\nh2F : K ⊆ interior F\n⊢ μ.outerMeasure K < ∞",
"usedConstants": [
"Trans.trans",
"Preorder.toLT",
"interior_s... | calc
μ.outerMeasure K ≤ μ.outerMeasure (interior F) := measure_mono h2F
_ ≤ μ ⟨F, h1F⟩ := by
apply μ.outerMeasure_le ⟨interior F, isOpen_interior⟩ ⟨F, h1F⟩ interior_subset
_ < ⊤ := μ.lt_top _ | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1136,
"column": 2
} | {
"line": 1136,
"column": 52
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive (∞ • μ) T C\nf : α → E\n⊢ setToFun (∞ • μ) T hT f ... | refine setToFun_measure_zero' hT fun s _ hμs => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 179,
"column": 27
} | {
"line": 179,
"column": 40
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK : PositiveCompacts G\nV : Set G\nhV : (interior V).Nonempty\n⊢ 0 < Nat.find ?m.21",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"TopologicalSpace.PositiveCompacts.instSetLike",
"Monoid... | Nat.find_pos, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 21
} | [
{
"pp": "case h\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₁ K₂ : Compacts G\nV : Set G\nhV : (interior V).Nonempty\nh : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)\ns : Finset G\nh1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V\nh2s : s.card = ... | simp [hg, h2g₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Marginal | {
"line": 119,
"column": 83
} | {
"line": 122,
"column": 41
} | [
{
"pp": "δ : Type u_1\nX : δ → Type u_3\ninst✝¹ : (i : δ) → MeasurableSpace (X i)\nμ : (i : δ) → Measure (X i)\ninst✝ : DecidableEq δ\nf : ((i : δ) → X i) → ℝ≥0∞\ni : δ\nα : Type u_1 := ↥{i}\ne : X ↑default ≃ᵐ ((i_1 : α) → X ↑i_1) := (MeasurableEquiv.piUnique fun j ↦ X ↑j).symm\nx : (i : δ) → X i\n⊢ (∫⋯∫⁻_{i}, ... | by
simp_rw [lmarginal,
measurePreserving_piUnique (fun j : ({i} : Finset δ) ↦ μ j) |>.symm _
|>.lintegral_map_equiv, e, α] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Marginal | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 46
} | [
{
"pp": "case h\nδ : Type u_1\nX : δ → Type u_3\ninst✝² : (i : δ) → MeasurableSpace (X i)\nμ : (i : δ) → Measure (X i)\ninst✝¹ : DecidableEq δ\ns t : Finset δ\ninst✝ : ∀ (i : δ), SigmaFinite (μ i)\nf : ((i : δ) → X i) → ℝ≥0∞\nhf : Measurable f\nhst : Disjoint s t\nx : (i : δ) → X i\n⊢ (∫⋯∫⁻_s ∪ t, f ∂μ) x = (∫⋯... | let e := MeasurableEquiv.piFinsetUnion X hst | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 247,
"column": 2
} | {
"line": 248,
"column": 68
} | [
{
"pp": "case hm\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK : Set G\nhK : IsCompact K\nV : Set G\nhV : (interior V).Nonempty\ng : G\ns : Finset G\nh1s : K ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V\nh2s : s.card = index K V\ng₁ g₂ : G\nhg₂ : g₂ ∈ s\nhg₁ : g₂ * g₁ ∈ V\n⊢ ... | simp only [exists_prop, mem_iUnion, Finset.mem_map, Equiv.coe_mulRight,
exists_exists_and_eq_and, mem_preimage, Equiv.toEmbedding_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Marginal | {
"line": 217,
"column": 30
} | {
"line": 217,
"column": 70
} | [
{
"pp": "δ : Type u_1\nX : δ → Type u_3\ninst✝² : (i : δ) → MeasurableSpace (X i)\nμ : (i : δ) → Measure (X i)\ninst✝¹ : DecidableEq δ\ns✝ : Finset δ\ninst✝ : ∀ (i : δ), SigmaFinite (μ i)\ni : δ\nf : ((i : δ) → X i) → ℝ≥0∞\nhf : Measurable f\ny : X i\ni' : δ\ns : Finset δ\nhi' : i' ∉ s\nih : i ∉ s → ∀ (x : (i :... | by rintro rfl; exact mem_insert_self i s | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.LeftRightLim | {
"line": 126,
"column": 2
} | {
"line": 130,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace α\nh'α : OrderTopology α\nf : α → β\na : α\nh : ∃ y, Tendsto f (𝓝[<] a) (𝓝 y)\n⊢ Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))",
"usedConstants": [
"Preorder.topology",
"Eq.mpr",
"... | rcases eq_or_neBot (𝓝[<] a) with h' | h'
· simp [h']
rw [h'α.topology_eq_generate_intervals] at h h' ⊢
simp only [leftLim, neBot_iff.1 h', h, not_true_eq_false, or_self, ↓reduceIte]
exact tendsto_nhds_limUnder h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.LeftRightLim | {
"line": 126,
"column": 2
} | {
"line": 130,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace α\nh'α : OrderTopology α\nf : α → β\na : α\nh : ∃ y, Tendsto f (𝓝[<] a) (𝓝 y)\n⊢ Tendsto f (𝓝[<] a) (𝓝 (leftLim f a))",
"usedConstants": [
"Preorder.topology",
"Eq.mpr",
"... | rcases eq_or_neBot (𝓝[<] a) with h' | h'
· simp [h']
rw [h'α.topology_eq_generate_intervals] at h h' ⊢
simp only [leftLim, neBot_iff.1 h', h, not_true_eq_false, or_self, ↓reduceIte]
exact tendsto_nhds_limUnder h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 374,
"column": 26
} | {
"line": 374,
"column": 45
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nK₀ : PositiveCompacts G\neval : (Compacts G → ℝ) → ℝ := fun f ↦ f ⊥\nthis : Continuous eval\nU : Set G\nleft✝ : U ∈ {U | U ⊆ ↑⊤.toOpens ∧ IsOpen[inst✝¹] U ∧ 1 ∈ U}\n⊢ prehaar (↑K₀) U ∈ eval ⁻¹' {0}",
"usedCon... | apply prehaar_empty | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Order.LeftRightLim | {
"line": 387,
"column": 6
} | {
"line": 388,
"column": 58
} | [
{
"pp": "case refine_2.refine_2\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nh : leftLim f x = rightLim f x\nh' : l... | rw [h] at h'
exact hf.continuousWithinAt_Ioi_iff_rightLim_eq.2 h' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.LeftRightLim | {
"line": 387,
"column": 6
} | {
"line": 388,
"column": 58
} | [
{
"pp": "case refine_2.refine_2\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : LinearOrder α\ninst✝⁴ : ConditionallyCompleteLinearOrder β\ninst✝³ : TopologicalSpace β\ninst✝² : OrderTopology β\nf : α → β\nhf : Monotone f\nx : α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nh : leftLim f x = rightLim f x\nh' : l... | rw [h] at h'
exact hf.continuousWithinAt_Ioi_iff_rightLim_eq.2 h' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 374,
"column": 6
} | {
"line": 374,
"column": 23
} | [
{
"pp": "case hst\nX : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → E\ns t : Set X\nμ : Measure X\nht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0\nhaux : StronglyMeasurable f\nH : IntegrableOn f (s ∪ t) μ\nk : Set X := ⋯\nhk : MeasurableSet k\nh's : ... | union_ae_eq_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 357,
"column": 4
} | {
"line": 358,
"column": 17
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝³ : LinearOrder R\ninst✝² : TopologicalSpace R\nf : StieltjesFunction R\ninst✝¹ : OrderTopology R\ninst✝ : CompactIccSpace R\na : R\nc d : ℕ → R\ns✝ s : Finset ℕ\nIH :\n ∀ t ⊂ s,\n ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal... | rw [ENNReal.ofReal_eq_zero.2 (sub_nonpos.2 (f.mono ab))]
exact zero_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 357,
"column": 4
} | {
"line": 358,
"column": 17
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝³ : LinearOrder R\ninst✝² : TopologicalSpace R\nf : StieltjesFunction R\ninst✝¹ : OrderTopology R\ninst✝ : CompactIccSpace R\na : R\nc d : ℕ → R\ns✝ s : Finset ℕ\nIH :\n ∀ t ⊂ s,\n ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal... | rw [ENNReal.ofReal_eq_zero.2 (sub_nonpos.2 (f.mono ab))]
exact zero_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 602,
"column": 4
} | {
"line": 604,
"column": 57
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : Group G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : μ.InnerRegularCompactLTTop\ninst✝ : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEapprox : ∃ K ⊆ E... | obtain ⟨K, hKE, hK_comp, hK_meas⟩ := hEapprox
exact ⟨closure K, hK_comp.closure_subset_measurableSet hE hKE, hK_comp.closure,
isClosed_closure, by rwa [hK_comp.measure_closure]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 602,
"column": 4
} | {
"line": 604,
"column": 57
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : Group G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : μ.InnerRegularCompactLTTop\ninst✝ : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEapprox : ∃ K ⊆ E... | obtain ⟨K, hKE, hK_comp, hK_meas⟩ := hEapprox
exact ⟨closure K, hK_comp.closure_subset_measurableSet hE hKE, hK_comp.closure,
isClosed_closure, by rwa [hK_comp.measure_closure]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 605,
"column": 2
} | {
"line": 605,
"column": 100
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : Group G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : μ.InnerRegularCompactLTTop\ninst✝ : LocallyCompactSpace G\nE : Set G\nhE : MeasurableSet E\nhEapprox : ∃ K ⊆ E... | filter_upwards [eventually_nhds_one_measure_smul_diff_lt hK K_closed hKpos.ne' (μ := μ)] with g hg | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 367,
"column": 8
} | {
"line": 367,
"column": 26
} | [
{
"pp": "case inr.h\nR : Type u_1\ninst✝³ : LinearOrder R\ninst✝² : TopologicalSpace R\nf : StieltjesFunction R\ninst✝¹ : OrderTopology R\ninst✝ : CompactIccSpace R\na : R\nc d : ℕ → R\ns✝ s : Finset ℕ\nIH :\n ∀ t ⊂ s,\n ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofRe... | sub_add_sub_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 441,
"column": 70
} | {
"line": 441,
"column": 88
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : LinearOrder R\ninst✝³ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝² : OrderTopology R\ninst✝¹ : CompactIccSpace R\ninst✝ : DenselyOrdered R\na b : R\nhab : a < b\ns : ℕ → Set R\nhs : Ioc a b ⊆ ⋃ i, s i\nε : ℝ≥0\nεpos : 0 < ε\nh : ∑' (i : ℕ), f.length (s i) < ∞\nδ : ℝ≥0 :=... | sub_add_sub_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 493,
"column": 6
} | {
"line": 495,
"column": 22
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁶ : LinearOrder R\ninst✝⁵ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁴ : OrderTopology R\ninst✝³ : CompactIccSpace R\ninst✝² : MeasurableSpace R\ninst✝¹ : BorelSpace R\ninst✝ : DenselyOrdered R\ns : Set R\nt : ℕ → Set R\nht : s ⊆ ⋃ i, t i\nε : ℝ≥0\nε0 : 0 < ε\nh : ... | conv at hl =>
lhs
rw [length_eq] | Lean.Elab.Tactic.Conv.evalConv | Lean.Parser.Tactic.Conv.conv |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 323,
"column": 77
} | {
"line": 324,
"column": 85
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\nt : ℝ\n⊢ re ⟪↑t • x, ↑t • x⟫ + re ⟪↑t • x, y⟫ + re ⟪y, ↑t • x⟫ + re ⟪y, y⟫ =\n normSq x * t * t + re (⟪x, y⟫ * ↑t) + re (⟪y, x⟫ * ↑t) + re ⟪y, y⟫",
"usedCon... | rw
[re_inner_smul_ofReal_smul_self, inner_smul_ofReal_left, inner_smul_ofReal_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 323,
"column": 77
} | {
"line": 324,
"column": 85
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\nt : ℝ\n⊢ re ⟪↑t • x, ↑t • x⟫ + re ⟪↑t • x, y⟫ + re ⟪y, ↑t • x⟫ + re ⟪y, y⟫ =\n normSq x * t * t + re (⟪x, y⟫ * ↑t) + re (⟪y, x⟫ * ↑t) + re ⟪y, y⟫",
"usedCon... | rw
[re_inner_smul_ofReal_smul_self, inner_smul_ofReal_left, inner_smul_ofReal_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 323,
"column": 77
} | {
"line": 324,
"column": 85
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\nt : ℝ\n⊢ re ⟪↑t • x, ↑t • x⟫ + re ⟪↑t • x, y⟫ + re ⟪y, ↑t • x⟫ + re ⟪y, y⟫ =\n normSq x * t * t + re (⟪x, y⟫ * ↑t) + re (⟪y, x⟫ * ↑t) + re ⟪y, y⟫",
"usedCon... | rw
[re_inner_smul_ofReal_smul_self, inner_smul_ofReal_left, inner_smul_ofReal_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 354,
"column": 4
} | {
"line": 354,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\nthis : discrim (normSq x) (2 * ‖⟪x, y⟫‖) (normSq y) ≤ 0\n⊢ ‖⟪x, y⟫‖ * ‖⟪x, y⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫",
"usedConstants": [
"Norm.norm",
"Real.inst... | rw [discrim, normSq, normSq, sq] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 650,
"column": 2
} | {
"line": 650,
"column": 23
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\nl : ℝ\nhf : Tendsto (↑f) atTop (𝓝... | simp_rw [measure_Ico] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 939,
"column": 30
} | {
"line": 939,
"column": 34
} | [
{
"pp": "case h\nX : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure X\nf g : ↥(Lp E p μ)\ns : Set X\nx : X\nhx1 : ↑↑(MemLp.toLp ↑↑(f + g) ⋯) x = ↑↑(f + g) x\nhx2 : ↑↑(MemLp.toLp ↑↑g ⋯) x = ↑↑g x\nhx3 : ↑↑(MemLp.toLp ↑↑f ⋯) x = ↑↑f x\nhx4 : ↑↑(MemLp.toLp ↑↑f ⋯... | hx2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 480,
"column": 6
} | {
"line": 480,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nh : ‖y‖ = 0\n⊢ ⟪x, y⟫ = 0",
"usedConstants": [
"Eq.mpr",
"Inner.inner",
"congrArg",
"NormedField.toField",
"id",
"Field.toSemifield",
... | inner_eq_zero_symm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 637,
"column": 68
} | {
"line": 637,
"column": 71
} | [
{
"pp": "F : Type u_3\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nι₁ : Type u_4\ns₁ : Finset ι₁\nw₁ : ι₁ → ℝ\nv₁ : ι₁ → F\nh₁ : ∑ i ∈ s₁, w₁ i = 0\nι₂ : Type u_5\ns₂ : Finset ι₂\nw₂ : ι₂ → ℝ\nv₂ : ι₂ → F\nh₂ : ∑ i ∈ s₂, w₂ i = 0\n⊢ ∑ x ∈ s₁, w₁ x * ((∑ i ∈ s₂, w₂ i) * (‖v₁ x‖ * ‖v₁ x‖ / 2... | h₂, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 711,
"column": 4
} | {
"line": 711,
"column": 92
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nh : ‖⟪x, y⟫‖ = ‖x‖ * ‖y‖\nhx₀ : ¬x = 0\nthis : ‖x‖ ^ 2 ≠ 0\n⊢ y = (⟪x, y⟫ / ⟪x, x⟫) • x",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Real.partia... | rw [← sq_eq_sq₀, mul_pow, ← mul_right_inj' this, eq_comm, ← sub_eq_zero, ← mul_sub] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 797,
"column": 2
} | {
"line": 804,
"column": 73
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : F\n⊢ ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = 1 ↔ x ≠ 0 ∧ ∃ r, 0 < r ∧ y = r • x",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"GroupWithZero.toMonoidWithZero",
"InnerProdu... | constructor
· intro h
have hx₀ : x ≠ 0 := fun h₀ => by simp [h₀] at h
have hy₀ : y ≠ 0 := fun h₀ => by simp [h₀] at h
refine ⟨hx₀, ‖y‖ / ‖x‖, div_pos (norm_pos_iff.2 hy₀) (norm_pos_iff.2 hx₀), ?_⟩
exact ((inner_eq_norm_mul_iff_div hx₀).1 (eq_of_div_eq_one h)).symm
· rintro ⟨hx, ⟨r, ⟨hr, rfl⟩⟩⟩
e... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 797,
"column": 2
} | {
"line": 804,
"column": 73
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : F\n⊢ ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = 1 ↔ x ≠ 0 ∧ ∃ r, 0 < r ∧ y = r • x",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"GroupWithZero.toMonoidWithZero",
"InnerProdu... | constructor
· intro h
have hx₀ : x ≠ 0 := fun h₀ => by simp [h₀] at h
have hy₀ : y ≠ 0 := fun h₀ => by simp [h₀] at h
refine ⟨hx₀, ‖y‖ / ‖x‖, div_pos (norm_pos_iff.2 hy₀) (norm_pos_iff.2 hx₀), ?_⟩
exact ((inner_eq_norm_mul_iff_div hx₀).1 (eq_of_div_eq_one h)).symm
· rintro ⟨hx, ⟨r, ⟨hr, rfl⟩⟩⟩
e... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 818,
"column": 2
} | {
"line": 822,
"column": 52
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nν : Measure α\ninst✝² : SMulInvariantMeasure G α ν\ninst✝¹ : Countable G\ninst✝ : MeasurableConstSMul G α\ns : Set α\nfund_dom_s : IsFundamentalDomain G s ν\nvol_s : ν s = 0\n⊢ QuotientMeasureEqMeasurePrei... | apply fund_dom_s.quotientMeasureEqMeasurePreimage
ext U meas_U
simp only [Measure.coe_zero, Pi.zero_apply]
convert! (measure_inter_null_of_null_right (h := vol_s) (Quotient.mk α_mod_G ⁻¹' U)).symm
rw [measure_map_restrict_apply (meas_U := meas_U)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 818,
"column": 2
} | {
"line": 822,
"column": 52
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nν : Measure α\ninst✝² : SMulInvariantMeasure G α ν\ninst✝¹ : Countable G\ninst✝ : MeasurableConstSMul G α\ns : Set α\nfund_dom_s : IsFundamentalDomain G s ν\nvol_s : ν s = 0\n⊢ QuotientMeasureEqMeasurePrei... | apply fund_dom_s.quotientMeasureEqMeasurePreimage
ext U meas_U
simp only [Measure.coe_zero, Pi.zero_apply]
convert! (measure_inter_null_of_null_right (h := vol_s) (Quotient.mk α_mod_G ⁻¹' U)).symm
rw [measure_map_restrict_apply (meas_U := meas_U)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 266,
"column": 12
} | {
"line": 266,
"column": 15
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nK : Submodule ℝ F\nu v : F\nhv : v ∈ K\nh✝ : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\nh : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0\nw : F\nhw : w ∈ K\nle : ⟪u - v, w⟫_ℝ ≤ 0\nw'' : F := -w + v\nthis : w'' ∈ K\nh₁ : ⟪u - v, w'' - v⟫_ℝ ≤ 0\nh₂ : w'' - v = -... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 90,
"column": 6
} | {
"line": 91,
"column": 48
} | [
{
"pp": "case mpr.left\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝ : DecidableEq ι\nv : ι → E\nh : ∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0\ni : ι\n⊢ ‖v i‖ = 1",
"usedConstants": [
"RCLike.one_re",
... | have h' : ‖v i‖ ^ 2 = 1 ^ 2 := by
rw [@norm_sq_eq_re_inner 𝕜, h i i]; simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Baire.CompleteMetrizable | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 16
} | [
{
"pp": "case refine_4\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : IsCompletelyPseudoMetrizableSpace X\nx✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X\nf : ℕ → Set X\nho : ∀ (n : ℕ), IsOpen (f n)\nhd : ∀ (n : ℕ), Dense (f n)\nB : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n\n... | show z ∈ f n | Lean.Elab.Tactic.evalShow | Lean.Parser.Tactic.show |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 269,
"column": 4
} | {
"line": 271,
"column": 45
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\n⊢ ∀ c ⊆ {b | Orthonormal 𝕜 Subtype.val},\n IsChain (fun x1 x2 ↦ x1 ⊆ x2) c → c.Nonempty → ∃ ub ∈ {b | Orthonormal 𝕜 Subtype... | refine fun c hc cc _c0 => ⟨⋃₀ c, ?_, ?_⟩
· exact orthonormal_sUnion_of_directed cc.directedOn fun x xc => hc xc
· exact fun _ => Set.subset_sUnion_of_mem | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 269,
"column": 4
} | {
"line": 271,
"column": 45
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\ns : Set E\nhs : Orthonormal 𝕜 Subtype.val\n⊢ ∀ c ⊆ {b | Orthonormal 𝕜 Subtype.val},\n IsChain (fun x1 x2 ↦ x1 ⊆ x2) c → c.Nonempty → ∃ ub ∈ {b | Orthonormal 𝕜 Subtype... | refine fun c hc cc _c0 => ⟨⋃₀ c, ?_, ?_⟩
· exact orthonormal_sUnion_of_directed cc.directedOn fun x xc => hc xc
· exact fun _ => Set.subset_sUnion_of_mem | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Subspace | {
"line": 203,
"column": 6
} | {
"line": 203,
"column": 14
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_4\nG : ι → Type u_5\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 ... | intro s₁ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 373,
"column": 6
} | {
"line": 375,
"column": 12
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : SeminormedAddCommGroup E\ninst✝⁶ : InnerProductSpace 𝕜 E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : InnerProductSpace ℝ F\nι : Type u_4\nι' : Type u_5\nι'' : Type u_6\nE' : Type u_7\ninst✝³ : SeminormedAddCommGroup E'\ninst✝² : ... | have h : v.equiv v' e ∘ v = v' ∘ e := by
ext i
simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Subspace | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 14
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_4\nG : ι → Type u_5\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 ... | intro s₁ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 383,
"column": 40
} | {
"line": 383,
"column": 48
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nF : Type u_8\nH : Type u_9\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace 𝕜 H\na c : F\nb d : H\nha : a ≠ 0\nhb : b ≠ 0\nh : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c\nh₂ : ∀ (x : F), ⟪a, x⟫_𝕜... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 383,
"column": 40
} | {
"line": 383,
"column": 48
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nF : Type u_8\nH : Type u_9\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace 𝕜 H\na c : F\nb d : H\nha : a ≠ 0\nhb : b ≠ 0\nh : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c\nh₂ : ∀ (x : F), ⟪a, x⟫_𝕜... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 383,
"column": 40
} | {
"line": 383,
"column": 48
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nF : Type u_8\nH : Type u_9\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace 𝕜 H\na c : F\nb d : H\nha : a ≠ 0\nhb : b ≠ 0\nh : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c\nh₂ : ∀ (x : F), ⟪a, x⟫_𝕜... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 387,
"column": 40
} | {
"line": 387,
"column": 48
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nF : Type u_8\nH : Type u_9\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace 𝕜 H\na c : F\nb d : H\nha : a ≠ 0\nhb : b ≠ 0\nh : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c\nh₂ : ∀ (x : F), ⟪a, x⟫_𝕜... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 387,
"column": 40
} | {
"line": 387,
"column": 48
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nF : Type u_8\nH : Type u_9\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace 𝕜 H\na c : F\nb d : H\nha : a ≠ 0\nhb : b ≠ 0\nh : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c\nh₂ : ∀ (x : F), ⟪a, x⟫_𝕜... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 387,
"column": 40
} | {
"line": 387,
"column": 48
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nF : Type u_8\nH : Type u_9\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace 𝕜 H\na c : F\nb d : H\nha : a ≠ 0\nhb : b ≠ 0\nh : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c\nh₂ : ∀ (x : F), ⟪a, x⟫_𝕜... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 390,
"column": 15
} | {
"line": 390,
"column": 18
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nF : Type u_8\nH : Type u_9\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace 𝕜 H\na c : F\nb d : H\nha : a ≠ 0\nhb : b ≠ 0\nh : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c\nh₂ : ∀ (x : F), ⟪a, x⟫_𝕜... | h₂, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 90,
"column": 2
} | {
"line": 91,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT S : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nhS : S.IsSymmetric\n⊢ (T + S).IsSymmetric",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Inner.inner",
... | intro x y
rw [add_apply, inner_add_left, hT x y, hS x y, ← inner_add_right, add_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 90,
"column": 2
} | {
"line": 91,
"column": 78
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT S : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nhS : S.IsSymmetric\n⊢ (T + S).IsSymmetric",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Inner.inner",
... | intro x y
rw [add_apply, inner_add_left, hT x y, hS x y, ← inner_add_right, add_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 68
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nS T : E →ₗ[𝕜] E\nhS : S.IsSymmetric\nhT : T.IsSymmetric\nh : S.range ≤ T.range\nv : E\nhv : T v = 0\n⊢ S v = 0",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Submod... | obtain ⟨y, hy⟩ : ∃ y, T y = S (S v) := by simpa using @h (S (S v)) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.InnerProductSpace.Projection.Reflection | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 58
} | [
{
"pp": "case hx\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nv w : F\nh : ‖v‖ = ‖w‖\nR : F ≃ₗᵢ[ℝ] F := ⋯\nh₁ : R (v - w) = -(v - w)\n⊢ v + w ∈ (ℝ ∙ (v - w))ᗮ",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Submodule",
"Real",
... | rw [Submodule.mem_orthogonal_singleton_iff_inner_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 40
} | [
{
"pp": "case intro\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝² : Preorder ι\nU : ι → Submodule 𝕜 E\ninst✝¹ : ∀ (i : ι), (U i).HasOrthogonalProjection\ninst✝ : (⨆ i, U i).topologicalClosure.HasOrthogonalProjection\nhU : ... | change _ ≤ ‖y - (⟨a, hU hi hI⟩ : U i)‖ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 55
} | [
{
"pp": "case inr\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →... | rcases (mem_image _ _ _).1 z₂im with ⟨x₂, hx₂, xz₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 92
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nv : E\np : E := K.orthogonalProjectionFn v\nh' : ⟪v - p, p⟫ = 0\n⊢ ‖v‖ * ‖v‖ = ‖v - p‖ * ‖v - p‖ + ‖p‖ * ‖p‖",
"usedConstants": [
... | convert! norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (v - p) p h' using 2 <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 136,
"column": 71
} | {
"line": 140,
"column": 92
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nv : E\n⊢ ‖v‖ * ‖v‖ =\n ‖v - K.orthogonalProjectionFn v‖ * ‖v - K.orthogonalProjectionFn v‖ +\n ‖K.orthogonalProjectionFn v‖ * ‖... | by
set p := K.orthogonalProjectionFn v
have h' : ⟪v - p, p⟫ = 0 :=
orthogonalProjectionFn_inner_eq_zero _ _ (orthogonalProjectionFn_mem v)
convert! norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (v - p) p h' using 2 <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 87
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : e ∉ v\n⊢ ∃ u ⊇ v, Orthonorm... | refine ⟨insert e v, v.subset_insert e, ⟨?_, ?_⟩, (ne_insert_of_notMem v he'').symm⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 555,
"column": 2
} | {
"line": 555,
"column": 70
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CompleteSpace E\nF : Type u_5\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\ng : E →ₗ[𝕜] F\nhg : ∀ (u : ℕ → E) (x : E) (y : F), Tend... | refine hg (Prod.fst ∘ φ) x y ((continuous_fst.tendsto _).comp hφ) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 898,
"column": 32
} | {
"line": 907,
"column": 41
} | [
{
"pp": "p : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝⁶ : SeminormedAddCommGroup α\ninst✝⁵ : SeminormedAddCommGroup β\ninst✝⁴ : SeminormedRing 𝕜\ninst✝³ : Module 𝕜 α\ninst✝² : Module 𝕜 β\ninst✝¹ : IsBoundedSMul 𝕜 α\ninst✝ : IsBoundedSMul 𝕜 β\nc : 𝕜\nf : WithLp p (α × β)\n⊢ ... | by
rcases p.dichotomy with (rfl | hp)
· simp only [← prod_nnnorm_ofLp, ofLp_smul]
exact norm_smul_le _ _
· have hp0 : 0 < p.toReal := zero_lt_one.trans_le hp
have hpt : p ≠ ⊤ := p.toReal_pos_iff_ne_top.mp hp0
rw [prod_nnnorm_eq_add hpt, prod_nnnorm_eq_add hpt, one_div, NNReal.rpow_inv_le_i... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 226,
"column": 26
} | {
"line": 226,
"column": 38
} | [
{
"pp": "case pos\ns : Set ℝ\nhs : Bornology.IsBounded s\n⊢ volume s ≤ ofReal (sSup s - sInf s)",
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"ENNReal.ofReal",
"congrArg",
"Real.instSub",
"MeasureTheory.MeasureSpace.toMeasurableSpace",
"HSub.... | ← volume_Icc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 547,
"column": 6
} | {
"line": 547,
"column": 69
} | [
{
"pp": "case hb\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nr : ℝ\nhr : r ≠ 0\nx y : E\ns t : Set E\n⊢ ENNReal.ofReal (|r| ^ finrank ℝ E) ≠ ∞ ∨ μ t ≠ 0",
... | · simp only [ENNReal.ofReal_ne_top, true_or, Ne, not_false_iff] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 681,
"column": 6
} | {
"line": 681,
"column": 84
} | [
{
"pp": "μ : Measure ℝ\ninst✝ : NoAtoms μ\ns : Set ℝ\np : ℝ → Prop\nh : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x\nT : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2\nu : Set ℝ := ⋃ i, T i\nhfinite : (s \\ u).Finite\nA : Set (↑s × ↑s)\nA_count : A.Countable\nhA : ⋃ i ∈ A, T i = ⋃ i, T ... | simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using Hx | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 681,
"column": 6
} | {
"line": 681,
"column": 84
} | [
{
"pp": "μ : Measure ℝ\ninst✝ : NoAtoms μ\ns : Set ℝ\np : ℝ → Prop\nh : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x\nT : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2\nu : Set ℝ := ⋃ i, T i\nhfinite : (s \\ u).Finite\nA : Set (↑s × ↑s)\nA_count : A.Countable\nhA : ⋃ i ∈ A, T i = ⋃ i, T ... | simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using Hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 681,
"column": 6
} | {
"line": 681,
"column": 84
} | [
{
"pp": "μ : Measure ℝ\ninst✝ : NoAtoms μ\ns : Set ℝ\np : ℝ → Prop\nh : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x\nT : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2\nu : Set ℝ := ⋃ i, T i\nhfinite : (s \\ u).Finite\nA : Set (↑s × ↑s)\nA_count : A.Countable\nhA : ⋃ i ∈ A, T i = ⋃ i, T ... | simpa only [mem_iUnion, exists_prop, SetCoe.exists, exists_and_right] using Hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Box.Basic | {
"line": 163,
"column": 65
} | {
"line": 167,
"column": 59
} | [
{
"pp": "ι : Type u_1\n⊢ Injective toSet",
"usedConstants": [
"Real.partialOrder",
"Real.instLE",
"Real",
"_private.Mathlib.Analysis.BoxIntegral.Box.Basic.0.BoxIntegral.Box.injective_coe._simp_1_1",
"congrArg",
"BoxIntegral.Box.toSet",
"Real.instLT",
"_private... | by
rintro ⟨l₁, u₁, h₁⟩ ⟨l₂, u₂, h₂⟩ h
simp only [Subset.antisymm_iff, coe_subset_coe, le_iff_bounds] at h
congr
exacts [le_antisymm h.2.1 h.1.1, le_antisymm h.1.2 h.2.2] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.BoxIntegral.Box.SubboxInduction | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 99
} | [
{
"pp": "ι : Type u_1\np : Box ι → Prop\nI : Box ι\nH_nhds :\n ∀ z ∈ Box.Icc I,\n ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J\nhpI : ¬p I\ns : Box ι → Set ι\nhs : ∀ J ≤ I, ¬p ... | have hJp (m) : ¬p (J m) := Nat.recOn m hpI fun m ↦ by simpa only [J_succ] using hs (J m) (hJle m) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 679,
"column": 33
} | {
"line": 679,
"column": 58
} | [
{
"pp": "ι : Type u_1\nI J : Box ι\nπ : Prepartition I\nh : π.IsPartition\nhJ : J ≤ I\n⊢ (π.restrict J).iUnion = ↑J",
"usedConstants": [
"Real",
"congrArg",
"BoxIntegral.Box.toSet",
"BoxIntegral.Prepartition.IsPartition.iUnion_eq",
"HasSubset.Subset",
"LE.le",
"Set.... | by simp [h.iUnion_eq, hJ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 583,
"column": 28
} | {
"line": 583,
"column": 48
} | [
{
"pp": "K : Type u_1\ninst✝⁹ : NormedField K\ninst✝⁸ : LinearOrder K\ninst✝⁷ : IsStrictOrderedRing K\ninst✝⁶ : HasSolidNorm K\ninst✝⁵ : FloorRing K\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\ninst✝² : FiniteDimensional K E\ninst✝¹ : ProperSpace E\nL : Submodule ℤ E\ninst✝ : Discrete... | Set.mapsTo_univ_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 185,
"column": 13
} | {
"line": 185,
"column": 27
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\ninst✝ : AddCommMonoid M\nI : Box ι\ni : ι\nx : ℝ\nf : Box ι → M\n⊢ ∑ J ∈ (ofWithBot {I.splitLower i x, I.splitUpper i x} ⋯ ⋯).boxes, f J =\n Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x)",
"usedConstants": [
"Eq.mpr",
"Option.el... | sum_ofWithBot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 86
} | [
{
"pp": "case refine_1\nι : Type u_1\nI J Js : Box ι\ns : Finset (ι × ℝ)\nHJs : Js ∈ splitMany I s\nHn : ¬Disjoint ↑J ↑Js\nH : ∀ (i : ι), (i, J.lower i) ∈ s ∧ (i, J.upper i) ∈ s\nx : ι → ℝ\nhx : x ∈ ↑J\nhxs : x ∈ ↑Js\ny : ι → ℝ\nhy : y ∈ Js\ni : ι\nJl : Box ι\nHmem : Jl ∈ split I i (J.lower i)\nHle : Js ≤ Jl\nt... | rw [← Box.coe_subset_coe, coe_eq_of_mem_split_of_lt_mem Hmem this (hx i).1] at Hle | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.BoxIntegral.Partition.Additive | {
"line": 127,
"column": 6
} | {
"line": 132,
"column": 67
} | [
{
"pp": "case insert\nι : Type u_1\nM : Type u_2\nn : ℕ\nN : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\nI₀✝ : WithTop (Box ι)\nI✝ : Box ι\ni : ι\ninst✝ : Finite ι\nf : Box ι → M\nI₀ : WithTop (Box ι)\nhf :\n ∀ (I : Box ι),\n ↑I ≤ I₀ →\n ∀ {i : ι} {x : ℝ},\n x ∈ Set.Ioo (I.lowe... | rw [splitMany_insert, inf_split, ← ihs, biUnion_boxes, sum_biUnion_boxes]
refine Finset.sum_congr rfl fun J' hJ' => ?_
by_cases h : a.2 ∈ Ioo (J'.lower a.1) (J'.upper a.1)
· rw [sum_split_boxes]
exact hf _ ((WithTop.coe_le_coe.2 <| le_of_mem _ hJ').trans hI) h
· rw [split_of_notMem_Ioo h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Partition.Additive | {
"line": 127,
"column": 6
} | {
"line": 132,
"column": 67
} | [
{
"pp": "case insert\nι : Type u_1\nM : Type u_2\nn : ℕ\nN : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\nI₀✝ : WithTop (Box ι)\nI✝ : Box ι\ni : ι\ninst✝ : Finite ι\nf : Box ι → M\nI₀ : WithTop (Box ι)\nhf :\n ∀ (I : Box ι),\n ↑I ≤ I₀ →\n ∀ {i : ι} {x : ℝ},\n x ∈ Set.Ioo (I.lowe... | rw [splitMany_insert, inf_split, ← ihs, biUnion_boxes, sum_biUnion_boxes]
refine Finset.sum_congr rfl fun J' hJ' => ?_
by_cases h : a.2 ∈ Ioo (J'.lower a.1) (J'.upper a.1)
· rw [sum_split_boxes]
exact hf _ ((WithTop.coe_le_coe.2 <| le_of_mem _ hJ').trans hI) h
· rw [split_of_notMem_Ioo h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Partition.Additive | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 65
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nn : ℕ\nN : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\nI₀✝ : WithTop (Box ι)\nI✝ : Box ι\ni : ι\ninst✝ : Finite ι\nf : Box ι → M\nI₀ : WithTop (Box ι)\nhf : ∀ (I : Box ι), ↑I ≤ I₀ → ∀ (s : Finset (ι × ℝ)), ∑ J ∈ (splitMany I s).boxes, f J = f I\nI : Box ι\n... | exact Finset.sum_congr rfl fun J hJ => (hf _ (Hle _ hJ) _).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1305,
"column": 31
} | {
"line": 1305,
"column": 54
} | [
{
"pp": "𝕜 : Type u_7\nE : Type u_8\nF : Type u_9\nι : Type u_10\nι' : Type u_11\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : SeminormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 F\ninst✝² : Finite ι\ninst✝¹ : Fintype ι'\ninst✝ : DecidableEq ι'\nx : E\ny : F\nb : B... | toMatrix_innerₛₗ_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 16
} | [
{
"pp": "ι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Fintype ι\nν : ι → ℤ\nB : Box ι\nhν : ν ∈ admissibleIndex n B\nh : ∃ ν' ∈ admissibleIndex n B, box n ν = box n ν'\n⊢ (if hI : ∃ ν_1 ∈ admissibleIndex n B, box n ν = box n ν_1 then tag n hI.choose else ⋯.choose) = tag n ν",
"usedConstants": [
"Eq... | dif_pos h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 93
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\ni j : ι\nhij : i < j\nb : Basis ι 𝕜 E\nthis : gramSchmidt 𝕜 (⇑b) i ∈ span 𝕜 (gramSchmidt �... | have : gramSchmidt 𝕜 b i ∈ span 𝕜 (b '' Set.Iio j) := by rwa [← span_gramSchmidt_Iio 𝕜 b j] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 330,
"column": 63
} | {
"line": 334,
"column": 20
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrderBot ι\ninst✝² : WellFoundedLT ι\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional 𝕜 E\nh : finrank 𝕜 E = Fintype.card ι\nf : ι... | by
have H : gramSchmidtNormed 𝕜 f i = (‖f i‖⁻¹ : 𝕜) • f i := by
rw [gramSchmidtNormed, gramSchmidt_of_orthogonal 𝕜 hf]
rw [gramSchmidtOrthonormalBasis_apply h, H]
simpa [H] using hi | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 341,
"column": 4
} | {
"line": 341,
"column": 31
} | [
{
"pp": "case a\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrderBot ι\ninst✝² : WellFoundedLT ι\ninst✝¹ : Fintype ι\ninst✝ : FiniteDimensional 𝕜 E\nh : finrank 𝕜 E = Fintype.card ... | rw [span_gramSchmidtNormed] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 475,
"column": 66
} | {
"line": 475,
"column": 90
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nh : Integrable I l f vol\nπ₀ : Prepartition I\n⊢ Filter.m... | prod_principal_principal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SumOverResidueClass | {
"line": 28,
"column": 2
} | {
"line": 29,
"column": 32
} | [
{
"pp": "case h\nR : Type u_1\ninst✝¹ : AddCommMonoid R\nm : ℕ\ninst✝ : NeZero m\nf : ℕ → R\nn : ℕ\n⊢ f n = (∑ a, {n | ↑n = a}.indicator f) n",
"usedConstants": [
"Set.indicator_apply",
"Pi.addCommMonoid",
"Finset.univ",
"ZMod.commRing",
"congrArg",
"Set.indicator",
... | simp only [Finset.sum_apply, Set.indicator_apply, Set.mem_setOf_eq, Finset.sum_ite_eq,
Finset.mem_univ, ↓reduceIte] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.PSeries | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 12
} | [
{
"pp": "case convert_1\nf : ℕ → ℝ\nn : ℕ\nhn : ∀ b ≥ n, 0 b ≤ f b\nm : ℕ\nhm : ∀ b ≥ m, f (b + 1) ≤ f b\n⊢ ∀ (n_1 : ℕ), 0 ≤ (fun k ↦ f (max k (n + m))) n_1",
"usedConstants": [
"Real.instLE",
"Real",
"Lattice.toSemilatticeSup",
"Nat.instIsOrderedAddMonoid",
"LinearOrderedCommM... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.PSeries | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 12
} | [
{
"pp": "case convert_1\nf : ℕ → ℝ\nn : ℕ\nhn : ∀ b ≥ n, 0 b ≤ f b\nm : ℕ\nhm : ∀ b ≥ m, f (b + 1) ≤ f b\n⊢ ∀ (n_1 : ℕ), 0 ≤ (fun k ↦ f (max k (n + m))) n_1",
"usedConstants": [
"Real.instLE",
"Real",
"Lattice.toSemilatticeSup",
"Nat.instIsOrderedAddMonoid",
"LinearOrderedCommM... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.PSeries | {
"line": 256,
"column": 4
} | {
"line": 256,
"column": 12
} | [
{
"pp": "case convert_1\nf : ℕ → ℝ\nn : ℕ\nhn : ∀ b ≥ n, 0 b ≤ f b\nm : ℕ\nhm : ∀ b ≥ m, f (b + 1) ≤ f b\n⊢ ∀ (n_1 : ℕ), 0 ≤ (fun k ↦ f (max k (n + m))) n_1",
"usedConstants": [
"Real.instLE",
"Real",
"Lattice.toSemilatticeSup",
"Nat.instIsOrderedAddMonoid",
"LinearOrderedCommM... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.PSeries | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 39
} | [
{
"pp": "p : ℝ\n⊢ (Summable fun n ↦ ↑n ^ p) ↔ p < -1",
"usedConstants": [
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring",
"NormedCommRing.toNonUnitalNormedCommRing",
"NonUnitalNonAssocRing.toHasDistribNeg",
"ne... | rcases neg_surjective p with ⟨p, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.PSeries | {
"line": 346,
"column": 28
} | {
"line": 346,
"column": 42
} | [
{
"pp": "case hf₁\nb : ℝ\nhb : 1 < b\n⊢ -b < -1",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"neg_lt_neg_iff",
"Real.partialOrder",
"Real",
"AddLeftCancelSemigroup.t... | neg_lt_neg_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.PSeries | {
"line": 346,
"column": 28
} | {
"line": 346,
"column": 42
} | [
{
"pp": "case hf₂\nb : ℝ\nhb : 1 < b\n⊢ -b < -1",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"neg_lt_neg_iff",
"Real.partialOrder",
"Real",
"AddLeftCancelSemigroup.t... | neg_lt_neg_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.ZLattice.Covolume | {
"line": 92,
"column": 2
} | {
"line": 95,
"column": 76
} | [
{
"pp": "E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\ninst✝¹ : IsZLattice ℝ L\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\n⊢ covolume L μ ≠ 0",
"usedCo... | rw [covolume_eq_measure_fundamentalDomain L μ (isAddFundamentalDomain (Free.chooseBasis ℤ L) μ),
measureReal_ne_zero_iff (ne_of_lt _)]
· exact measure_fundamentalDomain_ne_zero _
· exact Bornology.IsBounded.measure_lt_top (fundamentalDomain_isBounded _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.ZLattice.Covolume | {
"line": 92,
"column": 2
} | {
"line": 95,
"column": 76
} | [
{
"pp": "E : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\ninst✝¹ : IsZLattice ℝ L\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\n⊢ covolume L μ ≠ 0",
"usedCo... | rw [covolume_eq_measure_fundamentalDomain L μ (isAddFundamentalDomain (Free.chooseBasis ℤ L) μ),
measureReal_ne_zero_iff (ne_of_lt _)]
· exact measure_fundamentalDomain_ne_zero _
· exact Bornology.IsBounded.measure_lt_top (fundamentalDomain_isBounded _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.PSeries | {
"line": 419,
"column": 6
} | {
"line": 421,
"column": 79
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nk n : ℕ\n⊢ ((↑k + 1) ^ 2)⁻¹ + ∑ i ∈ Ioc k.succ (max (k + 1) n), (↑i ^ 2)⁻¹ ≤ ((↑k + 1) ^ 2)⁻¹ + (↑k + 1)⁻¹",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"GroupWithZero.toMonoidWithZero",
... | refine add_le_add le_rfl ((sum_Ioc_inv_sq_le_sub ?_ (le_max_left _ _)).trans ?_)
· simp only [Ne, Nat.succ_ne_zero, not_false_iff]
· simp only [Nat.cast_succ, sub_le_self_iff, inv_nonneg, Nat.cast_nonneg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.PSeries | {
"line": 419,
"column": 6
} | {
"line": 421,
"column": 79
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nk n : ℕ\n⊢ ((↑k + 1) ^ 2)⁻¹ + ∑ i ∈ Ioc k.succ (max (k + 1) n), (↑i ^ 2)⁻¹ ≤ ((↑k + 1) ^ 2)⁻¹ + (↑k + 1)⁻¹",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"GroupWithZero.toMonoidWithZero",
... | refine add_le_add le_rfl ((sum_Ioc_inv_sq_le_sub ?_ (le_max_left _ _)).trans ?_)
· simp only [Ne, Nat.succ_ne_zero, not_false_iff]
· simp only [Nat.cast_succ, sub_le_self_iff, inv_nonneg, Nat.cast_nonneg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.PSeries | {
"line": 437,
"column": 2
} | {
"line": 439,
"column": 65
} | [
{
"pp": "m : ℕ\nhm : m ≠ 0\nk : ZMod m\nthis : NeZero m\n⊢ ¬Summable fun n ↦ {n | ↑n = k}.indicator (fun n ↦ 1 / ↑n) (n + 1)",
"usedConstants": [
"Real",
"Set.indicator_apply",
"instHDiv",
"ZMod.commRing",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"AddGroupWithOn... | have h (n : ℕ) : {n : ℕ | (n : ZMod m) = k - 1}.indicator (fun n : ℕ ↦ (1 / (n + 1 :) : ℝ)) n =
if (n : ZMod m) = k - 1 then (1 / (n + 1) : ℝ) else (0 : ℝ) := by
simp only [indicator_apply, mem_setOf_eq, cast_add, cast_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.PSeries | {
"line": 481,
"column": 76
} | {
"line": 481,
"column": 84
} | [
{
"pp": "a s : ℝ\n⊢ 1 < s ∧ 1 < s ↔ 1 < s",
"usedConstants": [
"Real",
"congrArg",
"and_self",
"Real.instLT",
"iff_self",
"Real.instOne",
"And",
"Iff",
"LT.lt",
"True",
"of_eq_true",
"One.toOfNat1",
"congrFun'",
"OfNat.ofNat... | and_self | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 139,
"column": 92
} | {
"line": 144,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\nι : Type u_3\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Basis ι ℤ ↥L\nn : ℕ\nr : ℝ\ns : ℕ → Finset (ι → ℤ) := fun n ↦ Fintype.piFinset fun i ↦ Icc... | by
simp_rw [Finset.mul_sum]
congr with k
push_cast
rw [Real.mul_rpow (by positivity) (by positivity), mul_pow]
group | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.ZLattice.Covolume | {
"line": 383,
"column": 4
} | {
"line": 383,
"column": 86
} | [
{
"pp": "case convert_3\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : InnerProductSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\ninst✝¹ : IsZLattice ℝ L\ninst✝ : Nontrivial E\nX : Set E\nF : E → ℝ\nhX : ∀ ⦃x... | rw [frontier_equivFun, volume_image_eq_volume_div_covolume', h₄, ENNReal.zero_div] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.List.ToFinsupp | {
"line": 98,
"column": 2
} | {
"line": 108,
"column": 54
} | [
{
"pp": "R : Type u_2\ninst✝³ : AddZeroClass R\nl₁ l₂ : List R\ninst✝² : DecidablePred fun x ↦ (l₁ ++ l₂).getD x 0 ≠ 0\ninst✝¹ : DecidablePred fun x ↦ l₁.getD x 0 ≠ 0\ninst✝ : DecidablePred fun x ↦ l₂.getD x 0 ≠ 0\n⊢ (l₁ ++ l₂).toFinsupp = l₁.toFinsupp + Finsupp.embDomain (addLeftEmbedding l₁.length) l₂.toFinsu... | ext n
simp only [toFinsupp_apply, Finsupp.add_apply]
cases lt_or_ge n l₁.length with
| inl h =>
rw [getD_append _ _ _ _ h, Finsupp.embDomain_notin_range, add_zero]
rintro ⟨k, rfl : length l₁ + k = n⟩
lia
| inr h =>
rcases Nat.exists_eq_add_of_le h with ⟨k, rfl⟩
rw [getD_append_right _ _ _ _ ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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