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.Analysis.Normed.Affine.Isometry | {
"line": 365,
"column": 47
} | {
"line": 367,
"column": 5
} | [
{
"pp": "𝕜 : Type u_1\nV₁ : Type u_3\nV₂ : Type u_5\nP₁ : Type u_8\nP₂ : Type u_11\ninst✝⁸ : NormedField 𝕜\ninst✝⁷ : SeminormedAddCommGroup V₁\ninst✝⁶ : NormedSpace 𝕜 V₁\ninst✝⁵ : PseudoMetricSpace P₁\ninst✝⁴ : NormedAddTorsor V₁ P₁\ninst✝³ : SeminormedAddCommGroup V₂\ninst✝² : NormedSpace 𝕜 V₂\ninst✝¹ : Ps... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Affine.Isometry | {
"line": 385,
"column": 55
} | {
"line": 387,
"column": 5
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nV₂ : Type u_5\ninst✝⁴ : NormedField 𝕜\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : NormedSpace 𝕜 V₂\ne : V ≃ₗᵢ[𝕜] V₂\n⊢ e.toAffineIsometryEquiv.linearIsometryEquiv = e",
"usedConstants": [
"LinearIso... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Affine.Isometry | {
"line": 725,
"column": 81
} | {
"line": 727,
"column": 5
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_10\ninst✝⁴ : NormedField 𝕜\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : NormedSpace 𝕜 V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\np : P\n⊢ (constVSub 𝕜 p).symm = (LinearIsometryEquiv.neg 𝕜).toAffineIsometryEquiv.trans (vaddConst 𝕜 p)",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 138,
"column": 4
} | {
"line": 147,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝² : PseudoMetricSpace R\ninst✝¹ : Monoid R\ninst✝ : LipschitzMul R\ns t : Set R\ns_bdd : Bornology.IsBounded s\nt_bdd : Bornology.IsBounded t\n⊢ Bornology.IsBounded (s * t)",
"usedConstants": [
"Set.instSProd",
"Set.ext",
"Eq.mpr",
"Prod.pseudoEMetricSpace... | have bdd : Bornology.IsBounded (s ×ˢ t) := Bornology.IsBounded.prod s_bdd t_bdd
obtain ⟨C, mul_lip⟩ := ‹LipschitzMul R›.lipschitz_mul
convert! mul_lip.isBounded_image bdd
ext p
simp only [Set.mem_image, Set.mem_prod, Prod.exists]
constructor
· intro ⟨a, a_in_s, b, b_in_t, eq_p⟩
exact ⟨a, b... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 138,
"column": 4
} | {
"line": 147,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝² : PseudoMetricSpace R\ninst✝¹ : Monoid R\ninst✝ : LipschitzMul R\ns t : Set R\ns_bdd : Bornology.IsBounded s\nt_bdd : Bornology.IsBounded t\n⊢ Bornology.IsBounded (s * t)",
"usedConstants": [
"Set.instSProd",
"Set.ext",
"Eq.mpr",
"Prod.pseudoEMetricSpace... | have bdd : Bornology.IsBounded (s ×ˢ t) := Bornology.IsBounded.prod s_bdd t_bdd
obtain ⟨C, mul_lip⟩ := ‹LipschitzMul R›.lipschitz_mul
convert! mul_lip.isBounded_image bdd
ext p
simp only [Set.mem_image, Set.mem_prod, Prod.exists]
constructor
· intro ⟨a, a_in_s, b, b_in_t, eq_p⟩
exact ⟨a, b... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.SumMeasure | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 26
} | [
{
"pp": "case neg\nι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝⁴ : Countable ι\nmX : MeasurableSpace X\ninst✝³ : NormedAddCommGroup E\nf : X → E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : MeasurableSingletonClass X\nx : ι → X\nc : ι → ℝ≥0∞\ninst✝ : FiniteDimensional ℝ E\nhc : ∀ (i : ι), c i ≠ ∞\nhf : ¬Integrable f ... | apply mt Summable.norm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.Bochner.SumMeasure | {
"line": 154,
"column": 2
} | {
"line": 157,
"column": 20
} | [
{
"pp": "case neg\nι : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝⁴ : Countable ι\nmX : MeasurableSpace X\ninst✝³ : NormedAddCommGroup E\nf : X → E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : MeasurableSingletonClass X\nx : ι → X\nc : ι → ℝ≥0∞\ninst✝ : FiniteDimensional ℝ E\nhc : ∀ (i : ι), c i ≠ ∞\nhf : ¬Integrable f ... | · rw [integral_undef hf, tsum_eq_zero_of_not_summable]
apply mt Summable.norm
convert! mt (integrable_sum_dirac hc) hf
simp [norm_smul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 146,
"column": 8
} | {
"line": 146,
"column": 28
} | [
{
"pp": "case refine_1.h\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf g : α →ᵇ β\nC : ℝ\nhC : ∀ ⦃x : β⦄, x ∈ range ⇑f ∪ range ⇑g → ∀ ⦃y : β⦄, y ∈ range ⇑f ∪ range ⇑g → dist x y ≤ C\nx : α\n⊢ f x ∈ range ⇑f",
"usedConstants": [
"Set.mem_range_self",
"Bounde... | apply mem_range_self | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 146,
"column": 8
} | {
"line": 146,
"column": 28
} | [
{
"pp": "case refine_2.h\nα : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf g : α →ᵇ β\nC : ℝ\nhC : ∀ ⦃x : β⦄, x ∈ range ⇑f ∪ range ⇑g → ∀ ⦃y : β⦄, y ∈ range ⇑f ∪ range ⇑g → dist x y ≤ C\nx : α\n⊢ g x ∈ range ⇑g",
"usedConstants": [
"Set.mem_range_self",
"Bounde... | apply mem_range_self | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 431,
"column": 26
} | {
"line": 438,
"column": 5
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nA : ↥(Lp E 1 μ) →... | by
suffices setToL1 hT = A by rw [this]
apply ContinuousLinearMap.extend_unique
· exact (simpleFunc.denseRange one_ne_top)
· exact simpleFunc.isUniformInducing
ext f
rw [hA f]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 516,
"column": 9
} | {
"line": 516,
"column": 46
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝⁴ : TopologicalSpace α\ninst✝³ : PseudoMetricSpace β\ninst✝² : AddMonoid β\ninst✝¹ : BoundedAdd β\ninst✝ : ContinuousAdd β\nf : α →ᵇ β\n⊢ ⇑(nsmulRec 0 f) = 0 • ⇑f",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"instHSMul",
"AddMonoid.toAd... | by rw [nsmulRec, zero_smul, coe_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.ContinuousMap.Bounded.Star | {
"line": 90,
"column": 14
} | {
"line": 90,
"column": 61
} | [
{
"pp": "F : Type u_1\nα : Type u\nβ : Type v\nγ : Type w\ninst✝³ : TopologicalSpace α\ninst✝² : NonUnitalNormedRing β\ninst✝¹ : StarRing β\ninst✝ : CStarRing β\nf : α →ᵇ β\n⊢ ‖f‖ ^ 2 ≤ ‖star f * f‖",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NonUnitalNo... | ← Real.le_sqrt (norm_nonneg _) (norm_nonneg _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.ContinuousMap.Compact | {
"line": 500,
"column": 14
} | {
"line": 500,
"column": 61
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : TopologicalSpace α\ninst✝³ : CompactSpace α\ninst✝² : NonUnitalNormedRing β\ninst✝¹ : StarRing β\ninst✝ : CStarRing β\nf : C(α, β)\n⊢ ‖f‖ ^ 2 ≤ ‖star f * f‖",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NonUnit... | ← Real.le_sqrt (norm_nonneg _) (norm_nonneg _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 794,
"column": 55
} | {
"line": 794,
"column": 82
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_6\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : Module 𝕜 E\n... | Integrable.toL1_smul' f hf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.Basic | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 46
} | [
{
"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 ⊥\n⊢ chaar K₀ ⊥ = 0",
"usedConstants": [
"Real",
"Continuous",
"Pi.topologicalSpace",
"PseudoMetricSpace.toUniformSpac... | have : Continuous eval := continuous_apply ⊥ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : TopologicalSpace R\nf : StieltjesFunction R\na b : R\n⊢ f.length (Ioc a b) = ofReal (↑f b - ↑f a)",
"usedConstants": [
"Nonempty.intro",
"Nonempty"
]
}
] | have : Nonempty R := ⟨a⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 294,
"column": 74
} | {
"line": 305,
"column": 70
} | [
{
"pp": "R : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : TopologicalSpace R\nf : StieltjesFunction R\na b : R\n⊢ f.length (Ioc a b) = ofReal (↑f b - ↑f a)",
"usedConstants": [
"Iff.mpr",
"zero_le",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Set.Ioc",
"LinearOrderedCommGroup... | by
have : Nonempty R := ⟨a⟩
rw [length_eq]
refine
le_antisymm (iInf_le_of_le a <| iInf₂_le b diff_subset)
(le_iInf fun a' => le_iInf fun b' => le_iInf fun h => ENNReal.coe_le_coe.2 ?_)
rcases le_or_gt b a with ab | ab
· rw [Real.toNNReal_of_nonpos (sub_nonpos.2 (f.mono ab))]
apply zero_le
simp... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 64
} | [
{
"pp": "R : 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\n⊢ ∀ (s : Finset ℕ) (b : R),\n Icc a b ⊆ ⋃ i ∈ ↑s, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ s, ofReal (↑f (d i) - ↑f (c i))",
... | refine fun s => Finset.strongInductionOn s fun s IH b cv => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 341,
"column": 61
} | {
"line": 371,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝³ : LinearOrder R\ninst✝² : TopologicalSpace R\nf : StieltjesFunction R\ninst✝¹ : OrderTopology R\ninst✝ : CompactIccSpace R\na b : R\nc d : ℕ → R\nss : Icc a b ⊆ ⋃ i, Iotop (c i) (d i)\n⊢ ofReal (↑f b - ↑f a) ≤ ∑' (i : ℕ), ofReal (↑f (d i) - ↑f (c i))",
"usedConstants": [
... | by
suffices
∀ (s : Finset ℕ) (b), Icc a b ⊆ (⋃ i ∈ (s : Set ℕ), Iotop (c i) (d i)) →
(ofReal (f b - f a) : ℝ≥0∞) ≤ ∑ i ∈ s, ofReal (f (d i) - f (c i)) by
rcases isCompact_Icc.elim_finite_subcover_image
(fun (i : ℕ) (_ : i ∈ univ) => @isOpen_Iotop _ _ _ _ (c i) (d i)) (by simpa using ss) with
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 26
} | [
{
"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 :=... | have : Nonempty R := ⟨a⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 518,
"column": 6
} | {
"line": 518,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝³ : LinearOrder R\ninst✝² : TopologicalSpace R\nf : StieltjesFunction R\ninst✝¹ : OrderTopology R\ninst✝ : SecondCountableTopology R\n⊢ borel R ≤ f.outer.caratheodory",
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"MeasurableSpace.instLE",
"MeasureTheory.OuterM... | borel_eq_generateFrom_Ioi | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 128,
"column": 8
} | {
"line": 128,
"column": 41
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace α\ns : Set α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : Countable G\nh_meas : NullMeasurableSet s μ\nh_ae_disjoint : ∀ (g : G), g ≠ 1 → AEDisjoint μ (g • s) s\nh_qmp : ∀ (g : G), QuasiMeasurePreservi... | rw [← inv_inv g, ← preimage_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 297,
"column": 11
} | {
"line": 297,
"column": 32
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : MulAction G α\ninst✝⁴ : MeasurableSpace α\ns : Set α\nμ : Measure α\ninst✝³ : MeasurableConstSMul G α\ninst✝² : SMulInvariantMeasure G α μ\ninst✝¹ : Countable G\ninst✝ : Finite G\nh : IsFundamentalDomain G s μ\nt : Set α\nthis : Fintype G\nht : ∀ (... | measure_congr (ht _), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 541,
"column": 67
} | {
"line": 542,
"column": 97
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nα : Type u_3\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ns : Set α\ninst✝² : Group H\ninst✝¹ : MulAction H α\ninst✝ : SMulCommClass H G α\ng : H\n⊢ fundamentalFrontier G (g • s) = g • fundamentalFrontier G s",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"I... | by
simp_rw [fundamentalFrontier, smul_set_inter, smul_set_iUnion, smul_comm g (_ : G) (_ : Set α)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 61,
"column": 2
} | {
"line": 62,
"column": 7
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nn : ℕ\nv : E\nvs : Fin n → E\n⊢ Orthonormal 𝕜 (Matrix.vecCons v vs) ↔ ‖v‖ = 1 ∧ (∀ (i : Fin n), ⟪v, vs i⟫ = 0) ∧ Orthonormal 𝕜 vs",
"usedConstants": [
"Norm.norm",
"Eq.m... | simp_rw [Orthonormal, pairwise_fin_succ_iff_of_isSymm, Fin.forall_fin_succ]
tauto | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 61,
"column": 2
} | {
"line": 62,
"column": 7
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nn : ℕ\nv : E\nvs : Fin n → E\n⊢ Orthonormal 𝕜 (Matrix.vecCons v vs) ↔ ‖v‖ = 1 ∧ (∀ (i : Fin n), ⟪v, vs i⟫ = 0) ∧ Orthonormal 𝕜 vs",
"usedConstants": [
"Norm.norm",
"Eq.m... | simp_rw [Orthonormal, pairwise_fin_succ_iff_of_isSymm, Fin.forall_fin_succ]
tauto | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 182,
"column": 29
} | {
"line": 187,
"column": 65
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv : ι → E\nhv : Orthonormal 𝕜 v\n⊢ LinearIndependent 𝕜 v",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Finsupp.instFunLike",
"Eq.mpr",
"... | by
rw [linearIndependent_iff]
intro l hl
ext i
have key : ⟪v i, Finsupp.linearCombination 𝕜 v l⟫ = ⟪v i, 0⟫ := by rw [hl]
simpa only [hv.inner_right_finsupp, inner_zero_right] using key | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 16
} | [
{
"pp": "case mpr\nV : Type u_4\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℂ V\nT : V →ₗ[ℂ] V\n⊢ T = 0 → ∀ (x : V), ⟪T x, x⟫_ℂ = 0",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"AddCommGroup.toAddCommMonoid",
"NormedSpace.toModule",
"Complex.instNormedField",... | rintro rfl x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx : E\n⊢ im ⟪x, T x⟫ = 0",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"AddMonoid.toAddSemigroup",
"Inner... | simp [← hT x x, hT] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx : E\n⊢ im ⟪x, T x⟫ = 0",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"AddMonoid.toAddSemigroup",
"Inner... | simp [← hT x x, hT] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx : E\n⊢ im ⟪x, T x⟫ = 0",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"AddMonoid.toAddSemigroup",
"Inner... | simp [← hT x x, hT] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx : E\n⊢ ↑(re ⟪x, T x⟫) = ⟪x, T x⟫",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"AddMonoid.toAddSemigroup",
... | simp [← hT x x, hT] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx : E\n⊢ ↑(re ⟪x, T x⟫) = ⟪x, T x⟫",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"AddMonoid.toAddSemigroup",
... | simp [← hT x x, hT] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx : E\n⊢ ↑(re ⟪x, T x⟫) = ⟪x, T x⟫",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"AddMonoid.toAddSemigroup",
... | simp [← hT x x, hT] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\nx : E\n⊢ (∀ y ∈ K, ⟪y, x⟫ = 0) ↔ ∀ y ∈ K.topologicalClosure, ⟪y, x⟫ = 0",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedComm... | simp_rw [← mem_orthogonal, orthogonal_closure] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\nx : E\n⊢ (∀ y ∈ K, ⟪y, x⟫ = 0) ↔ ∀ y ∈ K.topologicalClosure, ⟪y, x⟫ = 0",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedComm... | simp_rw [← mem_orthogonal, orthogonal_closure] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\nx : E\n⊢ (∀ y ∈ K, ⟪y, x⟫ = 0) ↔ ∀ y ∈ K.topologicalClosure, ⟪y, x⟫ = 0",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedComm... | simp_rw [← mem_orthogonal, orthogonal_closure] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 285,
"column": 41
} | {
"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⊢ ∃ y, T y = S (S v)",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
... | by simpa using @h (S (S v)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 66
} | [
{
"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\n⊢ Kᗮ = ⊥ ↔ K = ⊤",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Submodule",
"congrArg",... | refine ⟨?_, fun h => by rw [h, Submodule.top_orthogonal_eq_bot]⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 71,
"column": 25
} | {
"line": 74,
"column": 42
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : K.HasOrthogonalProjection\nE' : Type u_4\ninst✝¹ : NormedAddCommGroup E'\ninst✝ : In... | by
rcases HasOrthogonalProjection.exists_orthogonal (K := K) (f.symm v) with ⟨w, hwK, hw⟩
refine ⟨f w, Submodule.mem_map_of_mem hwK, Set.forall_mem_image.2 fun u hu ↦ ?_⟩
simp [← f.symm.inner_map_map, hw u hu] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK₁ K₂ : Submodule 𝕜 E\ninst✝ : FiniteDimensional 𝕜 ↥K₂\nh : K₁ ≤ K₂\nthis : FiniteDimensional 𝕜 ↥K₁\nhd : finrank 𝕜 ↥K₂ + 0 = finrank 𝕜 ↥K₁ + finrank 𝕜 ↥(K₁ᗮ ⊓ K₂)\n⊢ finrank 𝕜 ↥K₁ + ... | rw [add_zero] at hd | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 59
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝ : DecidableEq ι\nV : ι → Submodule 𝕜 E\nhV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ\nhc : IsComplete ↑(iSup V)\n⊢ DirectSum.IsInternal V ↔ (iSup V)... | haveI : CompleteSpace (↥(iSup V)) := hc.completeSpace_coe | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 265,
"column": 69
} | {
"line": 278,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝¹ : Fintype ι\nV : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace ↥(V i)\nhV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ\nx : E\nhx : x ∈ iSup V\n... | by
induction hx using iSup_induction' with
| mem i x hx =>
refine
(Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hij => ?_).trans
(starProjection_eq_self_iff.mpr hx)
rw [starProjection_apply, orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero,
Submodule.coe_zero]
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 290,
"column": 4
} | {
"line": 297,
"column": 38
} | [
{
"pp": "case of\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝¹ : DecidableEq ι\nV : ι → Submodule 𝕜 E\nhV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ\ni : ι\ninst✝ : CompleteSpace ↥(V i)\nj : ι\nx : ↥(V ... | simp_rw [DirectSum.coeAddMonoidHom_of, DirectSum.of,
-- Need to unfold `DirectSum` to see through the defeq abuse.
DirectSum, DFinsupp.singleAddHom_apply]
obtain rfl | hij := Decidable.eq_or_ne i j
· rw [orthogonalProjection_mem_subspace_eq_self, DFinsupp.single_eq_same]
· rw [orthogonalProjecti... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 290,
"column": 4
} | {
"line": 297,
"column": 38
} | [
{
"pp": "case of\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝¹ : DecidableEq ι\nV : ι → Submodule 𝕜 E\nhV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ\ni : ι\ninst✝ : CompleteSpace ↥(V i)\nj : ι\nx : ↥(V ... | simp_rw [DirectSum.coeAddMonoidHom_of, DirectSum.of,
-- Need to unfold `DirectSum` to see through the defeq abuse.
DirectSum, DFinsupp.singleAddHom_apply]
obtain rfl | hij := Decidable.eq_or_ne i j
· rw [orthogonalProjection_mem_subspace_eq_self, DFinsupp.single_eq_same]
· rw [orthogonalProjecti... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 445,
"column": 39
} | {
"line": 447,
"column": 5
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜' F\nσ' : 𝕜' →+* 𝕜\ninst✝⁵ : RingHomI... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 577,
"column": 53
} | {
"line": 579,
"column": 5
} | [
{
"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 : IsClosed[instTopologicalSpaceProd] ... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 597,
"column": 54
} | {
"line": 599,
"column": 5
} | [
{
"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... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 807,
"column": 15
} | {
"line": 809,
"column": 30
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp 2 (α × β)\n⊢ ↑‖x‖₊ = ↑(NNReal.sqrt (‖x.fst‖₊ ^ 2 + ‖x.snd‖₊ ^ 2))",
"usedConstants": [
"WithLp",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"Real",
... | by
push_cast
exact prod_norm_eq_of_L2 x | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 906,
"column": 8
} | {
"line": 906,
"column": 26
} | [
{
"pp": "case inr\np : 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 ... | ← NNReal.mul_rpow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 931,
"column": 8
} | {
"line": 931,
"column": 26
} | [
{
"pp": "case inr\np : 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 : NormSMulClass 𝕜 α\ninst✝ : NormSMulClass 𝕜 β\nc : 𝕜\nf : WithLp p ... | ← NNReal.mul_rpow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 931,
"column": 27
} | {
"line": 931,
"column": 45
} | [
{
"pp": "case inr\np : 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 : NormSMulClass 𝕜 α\ninst✝ : NormSMulClass 𝕜 β\nc : 𝕜\nf : WithLp p ... | ← NNReal.mul_rpow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 491,
"column": 4
} | {
"line": 491,
"column": 53
} | [
{
"pp": "case inr\np : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\ninst✝² : Fact (1 ≤ p)\ninst✝¹ : (i : ι) → PseudoEMetricSpace (β i)\ninst✝ : Fintype ι\nx y : WithLp p ((i : ι) → β i)\nh : 1 ≤ p.toReal\n⊢ edist x y ≤ ↑(↑(Fintype.card ι) ^ (1 / p).toReal) * edist x.ofLp y.ofLp",
"usedConstants": [
"Real.par... | have pos : 0 < p.toReal := zero_lt_one.trans_le h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 822,
"column": 15
} | {
"line": 822,
"column": 33
} | [
{
"pp": "case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\nhp✝ : Fact (1 ≤ p)\ninst✝⁴ : Fintype ι\ninst✝³ : SeminormedRing 𝕜\ninst✝² : (i : ι) → SeminormedAddCommGroup (β i)\ninst✝¹ : (i : ι) → Module 𝕜 (β i)\ninst✝ : ∀ (i : ι), IsBoundedSMul 𝕜 (β i)\nc : 𝕜\nf : PiLp p β\n... | ← NNReal.mul_rpow, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 327,
"column": 54
} | {
"line": 327,
"column": 90
} | [
{
"pp": "a : ℝ\nh : a ≠ 0\n| (ofReal |a⁻¹| * ofReal |a|) • Measure.map (fun x ↦ a * x) volume",
"usedConstants": [
"Real",
"instHSMul",
"MeasureTheory.Measure",
"HMul.hMul",
"Real.lattice",
"IsScalarTower.right",
"ENNReal.ofReal",
"abs",
"congrArg",
... | ← ENNReal.ofReal_mul (abs_nonneg _), | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 837,
"column": 15
} | {
"line": 837,
"column": 33
} | [
{
"pp": "case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\nhp✝ : Fact (1 ≤ p)\ninst✝⁴ : Fintype ι\ninst✝³ : SeminormedRing 𝕜\ninst✝² : (i : ι) → SeminormedAddCommGroup (β i)\ninst✝¹ : (i : ι) → Module 𝕜 (β i)\ninst✝ : ∀ (i : ι), NormSMulClass 𝕜 (β i)\nc : 𝕜\nf : PiLp p β\n... | ← NNReal.mul_rpow, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 382,
"column": 49
} | {
"line": 382,
"column": 85
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nD : ι → ℝ\nh : (Matrix.diagonal D).det ≠ 0\ns : ι → Set ℝ\nhs : ∀ (i : ι), MeasurableSet (s i)\nthis : (⇑(toLin' (Matrix.diagonal D)) ⁻¹' univ.pi fun i ↦ s i) = univ.pi fun i ↦ (fun x ↦ D i * x) ⁻¹' s i\ni : ι\nA : D i ≠ 0\n⊢ ofReal |D i| * ofRea... | ← ENNReal.ofReal_mul (abs_nonneg _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 418,
"column": 19
} | {
"line": 418,
"column": 55
} | [
{
"pp": "case hdiag\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : M.det ≠ 0\nD : ι → ℝ\nhD : (Matrix.diagonal D).det ≠ 0\n⊢ Measure.map (⇑(toLin' (Matrix.diagonal D))) volume =\n (ofReal |(Matrix.diagonal D).det⁻¹| * ofReal |(Matrix.diagonal D).det|) •\n Measure.map (⇑... | ← ENNReal.ofReal_mul (abs_nonneg _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 23
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁷ : Fintype ι\ninst✝⁶ : Fintype ι'\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nb : Basis ι ℝ E\nthis : FiniteDimensional... | Basis.addHaar_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 269,
"column": 6
} | {
"line": 269,
"column": 23
} | [
{
"pp": "ι : Type u_1\nE : Type u_3\ninst✝⁷ : Fintype ι\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\nb : Basis ι ℝ E\nμ : Measure E\ninst✝¹ : SigmaFinite μ\ninst✝ : μ.IsAddLeftInvariant\n⊢ b.addHaar = μ ↔ μ ↑b.pa... | Basis.addHaar_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 425,
"column": 17
} | {
"line": 425,
"column": 53
} | [
{
"pp": "case hmul\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nM : Matrix ι ι ℝ\nhM : M.det ≠ 0\nA B : Matrix ι ι ℝ\na✝¹ : A.det ≠ 0\na✝ : B.det ≠ 0\nIHA : Measure.map (⇑(toLin' A)) volume = ofReal |A.det⁻¹| • volume\nIHB : Measure.map (⇑(toLin' B)) volume = ofReal |B.det⁻¹| • volume\n⊢ (ofReal |B... | ← ENNReal.ofReal_mul (abs_nonneg _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Box.Basic | {
"line": 332,
"column": 34
} | {
"line": 334,
"column": 30
} | [
{
"pp": "ι : Type u_1\nI✝ J : Box ι\nx y : ι → ℝ\nI J₁ J₂ : WithBot (Box ι)\nh₁ : I ≤ J₁\nh₂ : I ≤ J₂\n⊢ I ≤ J₁ ⊓ J₂",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Real",
"WithBot",
"congrArg",
"BoxIntegral.Box.WithBot.inf",
"PartialOrder.toPreorder",
... | by
simp only [← withBotCoe_subset_iff, coe_inf] at *
exact subset_inter h₁ h₂ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.BoxIntegral.Box.SubboxInduction | {
"line": 132,
"column": 56
} | {
"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 ... | simpa only [J_succ] using hs (J m) (hJle m) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.BoxIntegral.Box.SubboxInduction | {
"line": 132,
"column": 56
} | {
"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 ... | simpa only [J_succ] using hs (J m) (hJle m) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Box.SubboxInduction | {
"line": 132,
"column": 56
} | {
"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 ... | simpa only [J_succ] using hs (J m) (hJle m) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 615,
"column": 2
} | {
"line": 615,
"column": 27
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ₁ π₂ : Prepartition I\ninst✝ : Fintype ι\nh : Disjoint π₁.iUnion π₂.iUnion\n⊢ (π₁.disjUnion π₂ h).distortion = max π₁.distortion π₂.distortion",
"usedConstants": [
"Classical.propDecidable",
"BoxIntegral.Box.distortion",
"NNReal.instSemilatticeSup",
... | classical exact sup_union | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 615,
"column": 2
} | {
"line": 615,
"column": 27
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ₁ π₂ : Prepartition I\ninst✝ : Fintype ι\nh : Disjoint π₁.iUnion π₂.iUnion\n⊢ (π₁.disjUnion π₂ h).distortion = max π₁.distortion π₂.distortion",
"usedConstants": [
"Classical.propDecidable",
"BoxIntegral.Box.distortion",
"NNReal.instSemilatticeSup",
... | classical exact sup_union | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 615,
"column": 2
} | {
"line": 615,
"column": 27
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ₁ π₂ : Prepartition I\ninst✝ : Fintype ι\nh : Disjoint π₁.iUnion π₂.iUnion\n⊢ (π₁.disjUnion π₂ h).distortion = max π₁.distortion π₂.distortion",
"usedConstants": [
"Classical.propDecidable",
"BoxIntegral.Box.distortion",
"NNReal.instSemilatticeSup",
... | classical exact sup_union | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 34
} | [
{
"pp": "case refine_1\nι : Type u_1\ninst✝ : Fintype ι\nI : Box ι\nr : (ι → ℝ) → ↑(Ioi 0)\nJ : Box ι\nx✝ : J ≤ I\nπi : (J' : Box ι) → TaggedPrepartition J'\nhP✝ : ∀ J' ∈ splitCenter J, (πi J').IsPartition\nhHen : ∀ J' ∈ splitCenter J, (πi J').IsHenstock\nhr : ∀ J' ∈ splitCenter J, (πi J').IsSubordinate r\na✝ :... | rcases hsub J' h' with ⟨n, hn⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.BoxIntegral.Partition.Tagged | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 27
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ₁ π₂ : TaggedPrepartition I\ninst✝ : Fintype ι\nh : Disjoint π₁.iUnion π₂.iUnion\n⊢ (π₁.disjUnion π₂ h).distortion = max π₁.distortion π₂.distortion",
"usedConstants": [
"Classical.propDecidable",
"BoxIntegral.Box.distortion",
"NNReal.instSemilatticeSup",... | classical exact sup_union | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Analysis.BoxIntegral.Partition.Tagged | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 27
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ₁ π₂ : TaggedPrepartition I\ninst✝ : Fintype ι\nh : Disjoint π₁.iUnion π₂.iUnion\n⊢ (π₁.disjUnion π₂ h).distortion = max π₁.distortion π₂.distortion",
"usedConstants": [
"Classical.propDecidable",
"BoxIntegral.Box.distortion",
"NNReal.instSemilatticeSup",... | classical exact sup_union | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Partition.Tagged | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 27
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ₁ π₂ : TaggedPrepartition I\ninst✝ : Fintype ι\nh : Disjoint π₁.iUnion π₂.iUnion\n⊢ (π₁.disjUnion π₂ h).distortion = max π₁.distortion π₂.distortion",
"usedConstants": [
"Classical.propDecidable",
"BoxIntegral.Box.distortion",
"NNReal.instSemilatticeSup",... | classical exact sup_union | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 81
} | [
{
"pp": "ι : Type u_1\nI : Box ι\ni : ι\nx : ℝ\n⊢ I.splitLower i x = ⊥ ↔ x ≤ I.lower i",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"WithBot",
"Function.update",
"congrArg",
"Classical.propDecidable",
"Exists",
"BoxIntegral.Box.upper",
"id... | rw [splitLower, mk'_eq_bot, exists_update_iff I.upper fun j y => y ≤ I.lower j] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 29
} | [
{
"pp": "ι : Type u_1\nI J : Box ι\nh : I ≤ J\ni : ι\nx : ℝ\nthis : ∀ (s : Set (ι → ℝ)), ↑I ∩ s ⊆ ↑J\n⊢ ∀ ⦃x_1 : Box ι⦄,\n (∃ J', (↑J' = ↑J ∩ {y | y i ≤ x} ∨ ↑J' = ↑J ∩ {y | x < y i}) ∧ ↑x_1 = ↑I ∩ ↑J') →\n ↑x_1 = ↑I ∩ {y | y i ≤ x} ∨ ↑x_1 = ↑I ∩ {y | x < y i}",
"usedConstants": [
"BoxIntegral... | rintro J₁ ⟨J₂, H₂ | H₂, H₁⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.BoxIntegral.Partition.Filter | {
"line": 491,
"column": 2
} | {
"line": 491,
"column": 65
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nc : ℝ≥0\nl : IntegrationParams\nI : Box ι\nhc : ⊤.distortion ≤ c\nr : (ι → ℝ) → ↑(Set.Ioi 0)\n⊢ ∃ π, l.MemBaseSet I c r π ∧ π.IsPartition",
"usedConstants": [
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"LinearOrderedCommMonoidWit... | have hc' : (⊤ : Prepartition I).compl.distortion ≤ c := by simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.BoxIntegral.Partition.Additive | {
"line": 107,
"column": 59
} | {
"line": 107,
"column": 74
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\ninst✝ : AddCommMonoid M\nI₀ : WithTop (Box ι)\nI : Box ι\nf : ι →ᵇᵃ[I₀] M\nhI : ↑I ≤ I₀\ni : ι\nx : ℝ\n⊢ Option.elim' 0 (⇑f) (I.splitLower i x) + Option.elim' 0 (⇑f) (I.splitUpper i x) = ∑ J ∈ (split I i x).boxes, f J",
"usedConstants": [
"Eq.mpr",
"Option.el... | sum_split_boxes | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Partition.Additive | {
"line": 130,
"column": 12
} | {
"line": 130,
"column": 27
} | [
{
"pp": "case pos\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.lower i... | sum_split_boxes | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 545,
"column": 45
} | {
"line": 551,
"column": 82
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝³ : RCLike 𝕜\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx : E\n⊢ √(∑ x_1, (⨆ j, ‖⟪b j, x⟫‖) ^ 2) = √↑(Fintype.card ι) * ⨆ i, ‖⟪b i, x⟫‖",
"usedConstants": [
"Real.instIsOrder... | by
simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, Nat.cast_nonneg, Real.sqrt_mul]
congr
rw [Real.sqrt_sq]
cases isEmpty_or_nonempty ι
· simp
· exact le_ciSup_of_le (by simp) (Nonempty.some inferInstance) (by positivity) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1023,
"column": 2
} | {
"line": 1023,
"column": 41
} | [
{
"pp": "𝕜 : Type u_3\ninst✝³ : RCLike 𝕜\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nv : Set E\ninst✝ : FiniteDimensional 𝕜 E\nhv : Orthonormal 𝕜 Subtype.val\nu₀ : Set E\nhu₀s : u₀ ⊇ v\nhu₀ : Orthonormal 𝕜 Subtype.val\nhu₀_max : (span 𝕜 u₀)ᗮ = ⊥\nhu₀_finite : u₀.Finite\n... | let u : Finset E := hu₀_finite.toFinset | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1043,
"column": 4
} | {
"line": 1043,
"column": 32
} | [
{
"pp": "𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\nE : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_7\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (s.restrict v)\nhsv : Injective (s.restric... | rw [Set.injOn_iff_injective] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 84
} | [
{
"pp": "case refine_1\nι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Fintype ι\nB : Box ι\nx : ι → ℝ\nhx : x ∈ B\nl u : ι → ℤ\nhl : ∀ (i : ι), B.lower i = ↑(l i)\nhu : ∀ (i : ι), B.upper i = ↑(u i)\ni : ι\n⊢ ↑(l i) ≤ (↑⌈↑n * x i⌉ - 1) / ↑n",
"usedConstants": [
"Int.cast",
"Real.instLE",
... | exact (mem_admissibleIndex_of_mem_box_aux₁ n (x i) (l i)).mp ((hl i) ▸ (hx i).1) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 84
} | [
{
"pp": "case refine_1\nι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Fintype ι\nB : Box ι\nx : ι → ℝ\nhx : x ∈ B\nl u : ι → ℤ\nhl : ∀ (i : ι), B.lower i = ↑(l i)\nhu : ∀ (i : ι), B.upper i = ↑(u i)\ni : ι\n⊢ ↑(l i) ≤ (↑⌈↑n * x i⌉ - 1) / ↑n",
"usedConstants": [
"Int.cast",
"Real.instLE",
... | exact (mem_admissibleIndex_of_mem_box_aux₁ n (x i) (l i)).mp ((hl i) ▸ (hx i).1) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 84
} | [
{
"pp": "case refine_1\nι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Fintype ι\nB : Box ι\nx : ι → ℝ\nhx : x ∈ B\nl u : ι → ℤ\nhl : ∀ (i : ι), B.lower i = ↑(l i)\nhu : ∀ (i : ι), B.upper i = ↑(u i)\ni : ι\n⊢ ↑(l i) ≤ (↑⌈↑n * x i⌉ - 1) / ↑n",
"usedConstants": [
"Int.cast",
"Real.instLE",
... | exact (mem_admissibleIndex_of_mem_box_aux₁ n (x i) (l i)).mp ((hl i) ▸ (hx i).1) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 17
} | [
{
"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\nf : ι → E\nb : ι\n⊢ ∀ (x : ι),\n (∀ y < x, ∀ a < y, ⟪gramSchmidt 𝕜 f a, gramSchmidt 𝕜 f ... | intro b ih a h₀ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 346,
"column": 2
} | {
"line": 346,
"column": 28
} | [
{
"pp": "case pos\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.car... | · rw [hk, inner_zero_left] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 375,
"column": 2
} | {
"line": 375,
"column": 73
} | [
{
"pp": "case h.e'_3.a\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : LocallyFiniteOrderBot ι\ninst✝³ : WellFoundedLT ι\ninst✝² : Fintype ι\ninst✝¹ : FiniteDimensional 𝕜 E\nh : finrank 𝕜 E = Finty... | exact ((gramSchmidtOrthonormalBasis h f).repr_apply_apply (f _) _).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Orientation | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 76
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\ninst✝⁷ : PartialOrder R\ninst✝⁶ : IsStrictOrderedRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nι : Type u_4\nι' : Type u_5\ninst✝³ : Fintype ι\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι'\ninst✝ : DecidableEq ι'\ne : Basis ι R M\neι : ι ≃ ι'\n⊢ (... | simp_rw [Basis.orientation, Orientation.reindex_apply, Basis.det_reindex'] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.Orientation | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 76
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\ninst✝⁷ : PartialOrder R\ninst✝⁶ : IsStrictOrderedRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nι : Type u_4\nι' : Type u_5\ninst✝³ : Fintype ι\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι'\ninst✝ : DecidableEq ι'\ne : Basis ι R M\neι : ι ≃ ι'\n⊢ (... | simp_rw [Basis.orientation, Orientation.reindex_apply, Basis.det_reindex'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Orientation | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 76
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\ninst✝⁷ : PartialOrder R\ninst✝⁶ : IsStrictOrderedRing R\nM : Type u_2\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nι : Type u_4\nι' : Type u_5\ninst✝³ : Fintype ι\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι'\ninst✝ : DecidableEq ι'\ne : Basis ι R M\neι : ι ≃ ι'\n⊢ (... | simp_rw [Basis.orientation, Orientation.reindex_apply, Basis.det_reindex'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Orientation | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 28
} | [
{
"pp": "case h.e'_4.h\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : IsEmpty ι\nx : M [⋀^ι]→ₗ[R] R\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constOfIsEmpty R M ι 1]\nf :... | fin_cases i <;> simp [f] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 36
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\nι : Type u_2\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ne : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\ninst✝ : Nonempty ι\n⊢ (e.adjustToOrientation x).toBasis.orientation = x",
"usedConstants": [
"Eq.mpr",
... | rw [e.toBasis_adjustToOrientation] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.BoxIntegral.Integrability | {
"line": 334,
"column": 4
} | {
"line": 335,
"column": 82
} | [
{
"pp": "case h.e'_13.right\nι : Type u\nE : Type v\ninst✝⁴ : Fintype ι\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nI : Box ι\nhb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nhc : ∀ᵐ (x : ι → ℝ) ∂μ, ContinuousA... | have : IsFiniteMeasure (μ.restrict (Box.Icc I)) :=
{ measure_univ_lt_top := by simp [I.isCompact_Icc.measure_lt_top (μ := μ)] } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 473,
"column": 64
} | {
"line": 477,
"column": 73
} | [
{
"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⊢ Cauchy (... | by
refine ⟨inferInstance, ?_⟩
rw [prod_map_map_eq, ← toFilter_inf_iUnion_eq, ← prod_inf_prod, prod_principal_principal]
exact h.tendsto_integralSum_toFilter_prod_self_inf_iUnion_eq_uniformity.mono_left
(inf_le_inf_left _ <| principal_mono.2 fun π h => h.1.trans h.2.symm) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SumOverResidueClass | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 73
} | [
{
"pp": "case neg\nm : ℕ\ninst✝ : NeZero m\nf : ℕ → ℝ\nhf : Antitone f\nk l : ZMod m\nhs : Summable ({n | ↑n = k}.indicator f)\nn : ℕ\nhn : f n < 0\n⊢ Summable ({n | ↑n = l}.indicator f)",
"usedConstants": [
"Real",
"ZMod.commRing",
"Real.instZero",
"Set.indicator",
"False.elim... | exact (not_summable_indicator_mod_of_antitone_of_neg hf hn k hs).elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 46
} | [
{
"pp": "E : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\ninst✝ : FiniteDimensional ℝ E\nh : finrank ℝ E = 1\nv : E\nhv : v ≠ 0\n⊢ ∀ (v_1 : E), ∃ r, r • ‖v‖⁻¹ • v = v_1",
"usedConstants": [
"Norm.norm",
"InnerProductS... | apply exists_smul_eq_of_finrank_eq_one h | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.PSeries | {
"line": 371,
"column": 63
} | {
"line": 372,
"column": 30
} | [
{
"pp": "p : ℝ\n⊢ (Summable fun n ↦ (↑n ^ p)⁻¹) ↔ 1 < p",
"usedConstants": [
"NNReal.instTopologicalSpace",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instPow",
"Real",
"Real.summable_nat_rpow_inv._simp_1",
"congrArg",
"Real.instInv",
"SummationFilter",
... | by
simp [← NNReal.summable_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.PSeries | {
"line": 375,
"column": 88
} | {
"line": 376,
"column": 30
} | [
{
"pp": "p : ℝ\n⊢ (Summable fun n ↦ ↑n ^ p) ↔ p < -1",
"usedConstants": [
"NNReal.instTopologicalSpace",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instPow",
"Real",
"congrArg",
"Real.summable_nat_rpow._simp_1",
"SummationFilter",
"PseudoMetricSpace.toUn... | by
simp [← NNReal.summable_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.ZLattice.Covolume | {
"line": 174,
"column": 2
} | {
"line": 175,
"column": 63
} | [
{
"pp": "E : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\nL₁ L₂ : Submodule ℤ E\ninst✝³ : DiscreteTopology ↥L₁\ninst✝² : IsZLattice ℝ L₁\ninst✝¹ : DiscreteTopology ↥L₂\ninst✝ : IsZLattice ℝ L₂\nh : L₁ ... | let f := (EuclideanSpace.equiv _ ℝ).symm.trans
(stdOrthonormalBasis ℝ E).repr.toContinuousLinearEquiv.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Module.ZLattice.Covolume | {
"line": 178,
"column": 6
} | {
"line": 178,
"column": 43
} | [
{
"pp": "E : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : InnerProductSpace ℝ E\ninst✝⁶ : FiniteDimensional ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\nL₁ L₂ : Submodule ℤ E\ninst✝³ : DiscreteTopology ↥L₁\ninst✝² : IsZLattice ℝ L₁\ninst✝¹ : DiscreteTopology ↥L₂\ninst✝ : IsZLattice ℝ L₂\nh : L₁ ... | ← covolume_comap L₁ volume volume hf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MonoidAlgebra.Grading | {
"line": 175,
"column": 6
} | {
"line": 177,
"column": 70
} | [
{
"pp": "M : Type u_1\nι : Type u_2\nR : Type u_3\ninst✝³ : AddMonoid M\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : CommSemiring R\nf : M →+ ι\n⊢ (coeAlgHom (gradeBy R ⇑f)).comp (decomposeAux f) = AlgHom.id R R[M]",
"usedConstants": [
"Subtype.coe_mk",
"Eq.mpr",
"NonAssocSemirin... | ext : 4
dsimp
rw [decomposeAux_single, DirectSum.coeAlgHom_of, Subtype.coe_mk] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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