module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.Connected.LocallyPathConnected | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 83
} | {
"line": 303,
"column": 2
} | [
{
"pp": "case refine_2\nX✝ : Type u_1\nY : Type u_2\ninst✝⁴ : TopologicalSpace X✝\ninst✝³ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type u_3\nF : Set X✝\ninst✝² : LocallyPathConnectedSpace X✝\nX : ι → Type u_4\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : ∀ (i : ι), LocallyPathConnectedSpace (X i)\nx : (i : ... | [] | exact isOpenMap_sigmaMk _ <| (hu.preimage continuous_sigmaMk).pathComponentIn _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Combination | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 40
} | {
"line": 354,
"column": 4
} | [
{
"pp": "case refine_2\nR : Type u_1\nE : Type u_3\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\ns : Set E\nι : Type\nsx : Finset ι\nwx : ι → R\nzx : ι → E\nhwx₀ : ∀ i ∈ sx, 0 ≤ wx i\nhwx₁ : ∑ i ∈ sx, wx i = 1\nhzx : ∀ i ∈ sx, zx i ∈ s\nι... | [
"case refine_2.refine_1\nR : Type u_1\nE : Type u_3\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\ns : Set E\nι : Type\nsx : Finset ι\nwx : ι → R\nzx : ι → E\nhwx₀ : ∀ i ∈ sx, 0 ≤ wx i\nhwx₁ : ∑ i ∈ sx, wx i = 1\nhzx : ∀ i ∈ sx, zx i ∈ s\nι' :... | refine ⟨_, _, _, _, ?_, ?_, ?_, rfl⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.Combination | {
"line": 354,
"column": 6
} | {
"line": 354,
"column": 17
} | {
"line": 355,
"column": 6
} | [
{
"pp": "case refine_2.refine_1\nR : Type u_1\nE : Type u_3\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\ns : Set E\nι : Type\nsx : Finset ι\nwx : ι → R\nzx : ι → E\nhwx₀ : ∀ i ∈ sx, 0 ≤ wx i\nhwx₁ : ∑ i ∈ sx, wx i = 1\nhzx : ∀ i ∈ sx, zx... | [
"case refine_2.refine_1\nR : Type u_1\nE : Type u_3\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\ns : Set E\nι : Type\nsx : Finset ι\nwx : ι → R\nzx : ι → E\nhwx₀ : ∀ i ∈ sx, 0 ≤ wx i\nhwx₁ : ∑ i ∈ sx, wx i = 1\nhzx : ∀ i ∈ sx, zx i ∈ s\nι' :... | rintro i hi | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Convex.Combination | {
"line": 350,
"column": 2
} | {
"line": 361,
"column": 77
} | {
"line": 362,
"column": 2
} | [
{
"pp": "case refine_2\nR : Type u_1\nE : Type u_3\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\ns : Set E\n⊢ Convex R {x | ∃ ι t w z, (∀ i ∈ t, 0 ≤ w i) ∧ ∑ i ∈ t, w i = 1 ∧ (∀ i ∈ t, z i ∈ s) ∧ t.centerMass w z = x}",
"ppTerm": "?re... | [
"case refine_3\nR : Type u_1\nE : Type u_3\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\ns : Set E\n⊢ {x | ∃ ι t w z, (∀ i ∈ t, 0 ≤ w i) ∧ ∑ i ∈ t, w i = 1 ∧ (∀ i ∈ t, z i ∈ s) ∧ t.centerMass w z = x} ⊆ (convexHull R) s"
] | · rintro x ⟨ι, sx, wx, zx, hwx₀, hwx₁, hzx, rfl⟩ y ⟨ι', sy, wy, zy, hwy₀, hwy₁, hzy, rfl⟩ a b ha
hb hab
rw [Finset.centerMass_segment' _ _ _ _ _ _ hwx₁ hwy₁ _ _ hab]
refine ⟨_, _, _, _, ?_, ?_, ?_, rfl⟩
· rintro i hi
rw [Finset.mem_disjSum] at hi
rcases hi with (⟨j, hj, rfl⟩ | ⟨j, hj, rfl⟩... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Combination | {
"line": 392,
"column": 2
} | {
"line": 392,
"column": 54
} | {
"line": 393,
"column": 2
} | [
{
"pp": "R : Type u_1\nE : Type u_3\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\ns : Finset E\n⊢ (convexHull R) ↑s = {x | ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ s.centerMass w id = x}",
"ppTerm": "?m.49",
"assigned": true,
... | [
"case refine_1\nR : Type u_1\nE : Type u_3\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\ns : Finset E\n⊢ ↑s ⊆ {x | ∃ w, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ s.centerMass w id = x}",
"case refine_2\nR : Type u_1\nE : Type u_3\ninst✝⁴ : Fi... | refine Set.Subset.antisymm (convexHull_min ?_ ?_) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.Combination | {
"line": 403,
"column": 6
} | {
"line": 403,
"column": 17
} | {
"line": 404,
"column": 6
} | [
{
"pp": "case refine_2.refine_1\nR : Type u_1\nE : Type u_3\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\ns : Finset E\nwx : E → R\nhwx₀ : ∀ y ∈ s, 0 ≤ wx y\nhwx₁ : ∑ y ∈ s, wx y = 1\nwy : E → R\nhwy₀ : ∀ y ∈ s, 0 ≤ wy y\nhwy₁ : ∑ y ∈ s, ... | [
"case refine_2.refine_1\nR : Type u_1\nE : Type u_3\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\ns : Finset E\nwx : E → R\nhwx₀ : ∀ y ∈ s, 0 ≤ wx y\nhwx₁ : ∑ y ∈ s, wx y = 1\nwy : E → R\nhwy₀ : ∀ y ∈ s, 0 ≤ wy y\nhwy₁ : ∑ y ∈ s, wy y = 1\na ... | rintro i hi | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Convex.Combination | {
"line": 593,
"column": 6
} | {
"line": 593,
"column": 15
} | {
"line": 594,
"column": 4
} | [
{
"pp": "case a.a\n𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : Finite ι\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\ns : Set ι\nt✝ : (i : ι) → Set (E i)\nx : (i : ι) → E i\nval✝ : Fintype ι\nt ... | [] | exact g.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 200,
"column": 12
} | {
"line": 200,
"column": 56
} | {
"line": 200,
"column": 57
} | [
{
"pp": "case refine_2\n𝕜 : Type u_2\nF : Type u_7\nι : Type u_9\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\np : SeminormFamily 𝕜 F ι\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : (s.sup p).ball 0 r ∈ p.basisSets\n⊢ (s.sup p).ball 0 r ∈ ⨅ i, comap (⇑(p i)) (𝓝 0)",
"ppTerm": "?... | [
"case refine_2\n𝕜 : Type u_2\nF : Type u_7\nι : Type u_9\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\np : SeminormFamily 𝕜 F ι\ns : Finset ι\nr : ℝ\nhr : 0 < r\nhU : (s.sup p).ball 0 r ∈ p.basisSets\n⊢ ⋂ i ∈ s, (p i).ball 0 r ∈ ⨅ i, comap (⇑(p i)) (𝓝 0)"
] | Seminorm.ball_finset_sup_eq_iInter _ _ _ hr, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 607,
"column": 4
} | {
"line": 607,
"column": 76
} | {
"line": 608,
"column": 4
} | [
{
"pp": "case refine_3.refine_1\n𝕜 : Type u_2\nE : Type u_6\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nh : p.ball 0 1 ∈ 𝓝 0\nh' : IsVonNBounded 𝕜 (p.ba... | [
"case refine_3.refine_1\n𝕜 : Type u_2\nE : Type u_6\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nh : p.ball 0 1 ∈ 𝓝 0\nh' : IsVonNBounded 𝕜 (p.ball 0 1)\ns :... | change p.ball 0 (‖c⁻¹‖) ∈ SeminormFamily.basisSets (fun (i : Fin 1) ↦ p) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 630,
"column": 4
} | {
"line": 632,
"column": 44
} | {
"line": 633,
"column": 4
} | [
{
"pp": "case refine_3.refine_2.inr\n𝕜 : Type u_2\nE : Type u_6\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nh : p.ball 0 1 ∈ 𝓝 0\nh' : IsVonNBounded 𝕜 (... | [
"case refine_3.refine_2.inr\n𝕜 : Type u_2\nE : Type u_6\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : TopologicalSpace E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousConstSMul 𝕜 E\np : Seminorm 𝕜 E\nh : p.ball 0 1 ∈ 𝓝 0\nh' : IsVonNBounded 𝕜 (p.ball 0 1)\... | have : c • p.ball 0 1 ⊆ p.ball 0 r := by
rw [smul_ball_zero c_ne]
exact ball_mono (by simpa using hc.le) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 77
} | {
"line": 233,
"column": 4
} | [
{
"pp": "case inr.refine_2\n𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SeminormedAddCommGroup F\ninst✝⁵ : DenselyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHom... | [
"case inr.refine_2\n𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SeminormedAddCommGroup F\ninst✝⁵ : DenselyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric σ₁... | obtain ⟨x, hx, hxf⟩ := f.exists_nnnorm_eq_one_lt_apply_of_lt_opNNNorm hub | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 1075,
"column": 6
} | {
"line": 1075,
"column": 17
} | {
"line": 1075,
"column": 17
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_6\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nt₁ t₂ : TopologicalSpace E\nht₁ : PolynormableSpace 𝕜 E\nht₂ : PolynormableSpace 𝕜 E\n⊢ PolynormableSpace 𝕜 E",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"𝕜 : Type u_2\nE : Type u_6\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nt₁ t₂ : TopologicalSpace E\nht₁ : PolynormableSpace 𝕜 E\nht₂ : PolynormableSpace 𝕜 E\n⊢ PolynormableSpace 𝕜 E"
] | ← sInf_pair | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Operator.Bilinear | {
"line": 153,
"column": 25
} | {
"line": 153,
"column": 61
} | {
"line": 153,
"column": 61
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁸ : SeminormedAddCommGroup E\ninst✝¹⁷ : SeminormedAddCommGroup Eₗ\ninst✝¹⁶ : SeminormedAddCommGroup F\ninst✝¹⁵ : SeminormedAddCommGroup Fₗ\ninst✝¹⁴... | [] | simp only [f.map_smulₛₗ, smul_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Operator.Bilinear | {
"line": 153,
"column": 25
} | {
"line": 153,
"column": 61
} | {
"line": 153,
"column": 61
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁸ : SeminormedAddCommGroup E\ninst✝¹⁷ : SeminormedAddCommGroup Eₗ\ninst✝¹⁶ : SeminormedAddCommGroup F\ninst✝¹⁵ : SeminormedAddCommGroup Fₗ\ninst✝¹⁴... | [] | simp only [f.map_smulₛₗ, smul_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.Bilinear | {
"line": 153,
"column": 25
} | {
"line": 153,
"column": 61
} | {
"line": 153,
"column": 61
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\n𝕜₃ : Type u_3\nE : Type u_4\nEₗ : Type u_5\nF : Type u_6\nFₗ : Type u_7\nG : Type u_8\nGₗ : Type u_9\n𝓕 : Type u_10\ninst✝¹⁸ : SeminormedAddCommGroup E\ninst✝¹⁷ : SeminormedAddCommGroup Eₗ\ninst✝¹⁶ : SeminormedAddCommGroup F\ninst✝¹⁵ : SeminormedAddCommGroup Fₗ\ninst✝¹⁴... | [] | simp only [f.map_smulₛₗ, smul_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.ENNReal | {
"line": 255,
"column": 52
} | {
"line": 255,
"column": 54
} | {
"line": 256,
"column": 4
} | [
{
"pp": "α : Type u_1\nf : Filter α\nu : α → ℝ\nh₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nr : ℝ≥0\nh : ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y\nx : ℝ≥0\nhx : some x > ↑r\na : α\n⊢ u a < ↑x → ENNReal.ofReal (u a) < some x",
"ppTerm": "?m.105",
"assigned": ... | [
"α : Type u_1\nf : Filter α\nu : α → ℝ\nh₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nr : ℝ≥0\nh : ∀ y > ↑r, ∀ᶠ (a : α) in f, u a < y\nx : ℝ≥0\nhx : some x > ↑r\na : α\nha : u a < ↑x\n⊢ ENNReal.ofReal (u a) < some x"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.Filter.ENNReal | {
"line": 261,
"column": 69
} | {
"line": 261,
"column": 71
} | {
"line": 262,
"column": 4
} | [
{
"pp": "α : Type u_1\nf : Filter α\nu : α → ℝ\nh₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nr : ℝ≥0\nh : ∀ y > ↑r, ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < y\nx : ℝ\nhx : x > ↑r\nthis : 0 < x\na : α\n⊢ ENNReal.ofReal (u a) < ENNReal.ofReal x → u a < x",
"pp... | [
"α : Type u_1\nf : Filter α\nu : α → ℝ\nh₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nh₂ : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\nr : ℝ≥0\nh : ∀ y > ↑r, ∀ᶠ (a : α) in f, ENNReal.ofReal (u a) < y\nx : ℝ\nhx : x > ↑r\nthis : 0 < x\na : α\nha : ENNReal.ofReal (u a) < ENNReal.ofReal x\n⊢ u a < x"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Order.Filter.ENNReal | {
"line": 270,
"column": 31
} | {
"line": 270,
"column": 33
} | {
"line": 271,
"column": 4
} | [
{
"pp": "α : Type u_1\nf : Filter α\ninst✝ : f.NeBot\nu : α → ℝ≥0∞\nC : ℝ≥0\nhf : ∀ᶠ (a : α) in f, u a ≤ ↑C\nh₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal\na : α\n⊢ u a ≤ ↑C → (u a).toReal ≤ ↑C",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"ENNReal.ofNNReal",
... | [
"α : Type u_1\nf : Filter α\ninst✝ : f.NeBot\nu : α → ℝ≥0∞\nC : ℝ≥0\nhf : ∀ᶠ (a : α) in f, u a ≤ ↑C\nh₁ : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ (u a).toReal\na : α\nha : u a ≤ ↑C\n⊢ (u a).toReal ≤ ↑C"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.UniformOn | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 35
} | {
"line": 92,
"column": 2
} | [
{
"pp": "case hcs\nΩ : Type u_1\ninst✝ : MeasurableSpace Ω\ns : Set Ω\nhs_fin : s.Finite\nhs_nonempty : s.Nonempty\nhs_meas : MeasurableSet s\n⊢ Measure.count s ≠ 0",
"ppTerm": "?hcs",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
... | [] | rwa [Measure.count_ne_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Probability.UniformOn | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 35
} | {
"line": 92,
"column": 2
} | [
{
"pp": "case hcs\nΩ : Type u_1\ninst✝ : MeasurableSpace Ω\ns : Set Ω\nhs_fin : s.Finite\nhs_nonempty : s.Nonempty\nhs_meas : MeasurableSet s\n⊢ Measure.count s ≠ 0",
"ppTerm": "?hcs",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
... | [] | rwa [Measure.count_ne_zero_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.UniformOn | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 35
} | {
"line": 92,
"column": 2
} | [
{
"pp": "case hcs\nΩ : Type u_1\ninst✝ : MeasurableSpace Ω\ns : Set Ω\nhs_fin : s.Finite\nhs_nonempty : s.Nonempty\nhs_meas : MeasurableSet s\n⊢ Measure.count s ≠ 0",
"ppTerm": "?hcs",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
... | [] | rwa [Measure.count_ne_zero_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.UniformOn | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 35
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case hcs\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns : Set Ω\nhs : s.Finite\nhs' : s.Nonempty\n⊢ Measure.count s ≠ 0",
"ppTerm": "?hcs",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
... | [] | rwa [Measure.count_ne_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Probability.UniformOn | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 35
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case hcs\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns : Set Ω\nhs : s.Finite\nhs' : s.Nonempty\n⊢ Measure.count s ≠ 0",
"ppTerm": "?hcs",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
... | [] | rwa [Measure.count_ne_zero_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.UniformOn | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 35
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case hcs\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns : Set Ω\nhs : s.Finite\nhs' : s.Nonempty\n⊢ Measure.count s ≠ 0",
"ppTerm": "?hcs",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
... | [] | rwa [Measure.count_ne_zero_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.UniformOn | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 12
} | {
"line": 126,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : MeasurableSingletonClass Ω\nω : Ω\nt : Set Ω\ninst✝ : Decidable (ω ∈ t)\n⊢ (uniformOn {ω}) t = if ω ∈ t then 1 else 0",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"MeasureTheory.Measu... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : MeasurableSingletonClass Ω\nω : Ω\nt : Set Ω\ninst✝ : Decidable (ω ∈ t)\n⊢ Measure.count ({ω} ∩ t) = if ω ∈ t then 1 else 0"
] | rw [uniformOn, cond_apply (measurableSet_singleton ω), Measure.count_singleton, inv_one,
one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.UniformOn | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 35
} | {
"line": 138,
"column": 2
} | [
{
"pp": "case h0\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns : Set Ω\nhs : s.Finite\nhs' : s.Nonempty\n⊢ Measure.count s ≠ 0",
"ppTerm": "?h0",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
... | [] | rwa [Measure.count_ne_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Probability.UniformOn | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 35
} | {
"line": 138,
"column": 2
} | [
{
"pp": "case h0\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns : Set Ω\nhs : s.Finite\nhs' : s.Nonempty\n⊢ Measure.count s ≠ 0",
"ppTerm": "?h0",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
... | [] | rwa [Measure.count_ne_zero_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.UniformOn | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 35
} | {
"line": 138,
"column": 2
} | [
{
"pp": "case h0\nΩ : Type u_1\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSingletonClass Ω\ns : Set Ω\nhs : s.Finite\nhs' : s.Nonempty\n⊢ Measure.count s ≠ 0",
"ppTerm": "?h0",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"id",
... | [] | rwa [Measure.count_ne_zero_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 580,
"column": 40
} | {
"line": 580,
"column": 42
} | {
"line": 581,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nf g : α →ₘ[μ] β\na✝ : α\n⊢ ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ → ↑f a✝ ≤ ↑(f ⊔ g) a✝",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝\n⊢ ↑f a✝ ≤ ↑(f ⊔ g) a✝"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 581,
"column": 6
} | {
"line": 581,
"column": 8
} | {
"line": 581,
"column": 8
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝\n⊢ ↑f a✝ ≤ ↑(f ⊔ g) a✝",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝\n⊢ ↑f a✝ ≤ ↑f a✝ ⊔ ↑g a✝"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 586,
"column": 40
} | {
"line": 586,
"column": 42
} | {
"line": 587,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nf g : α →ₘ[μ] β\na✝ : α\n⊢ ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝ → ↑g a✝ ≤ ↑(f ⊔ g) a✝",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝\n⊢ ↑g a✝ ≤ ↑(f ⊔ g) a✝"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 587,
"column": 6
} | {
"line": 587,
"column": 8
} | {
"line": 587,
"column": 8
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝\n⊢ ↑g a✝ ≤ ↑(f ⊔ g) a✝",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeSup β\ninst✝ : ContinuousSup β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊔ g) a✝ = ↑f a✝ ⊔ ↑g a✝\n⊢ ↑g a✝ ≤ ↑f a✝ ⊔ ↑g a✝"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 609,
"column": 40
} | {
"line": 609,
"column": 42
} | {
"line": 610,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\n⊢ ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ → ↑(f ⊓ g) a✝ ≤ ↑f a✝",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑(f ⊓ g) a✝ ≤ ↑f a✝"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 610,
"column": 6
} | {
"line": 610,
"column": 8
} | {
"line": 610,
"column": 8
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑(f ⊓ g) a✝ ≤ ↑f a✝",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑f a✝"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 615,
"column": 40
} | {
"line": 615,
"column": 42
} | {
"line": 616,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\n⊢ ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝ → ↑(f ⊓ g) a✝ ≤ ↑g a✝",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑(f ⊓ g) a✝ ≤ ↑g a✝"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 616,
"column": 6
} | {
"line": 616,
"column": 8
} | {
"line": 616,
"column": 8
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑(f ⊓ g) a✝ ≤ ↑g a✝",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑g a✝"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 155,
"column": 23
} | {
"line": 155,
"column": 33
} | {
"line": 155,
"column": 33
} | [
{
"pp": "case cons\nι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝³ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝² : (i : ι) → MeasurableSpace (α i)\nμ✝ : (i : ι) → Measure (α i)\nδ : Type u_4\nX : δ → Type u_5\ninst✝¹ : (i : δ) → MeasurableSpace (X i)\nμ : (i : δ) → Measure (X i)\ninst✝ : ∀ (i : δ... | [
"case cons\nι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝³ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝² : (i : ι) → MeasurableSpace (α i)\nμ✝ : (i : ι) → Measure (α i)\nδ : Type u_4\nX : δ → Type u_5\ninst✝¹ : (i : δ) → MeasurableSpace (X i)\nμ : (i : δ) → Measure (X i)\ninst✝ : ∀ (i : δ), SigmaFini... | tprod_cons | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Indicator | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 31
} | {
"line": 47,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\ns : Set α\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ∞\n⊢ (∫⁻ (x : α), ‖s.indicator f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) =\n (∫⁻... | [
"case neg\nα : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\ns : Set α\nhs : MeasurableSet s\nhp_zero : ¬p = 0\nhp_top : ¬p = ∞\n⊢ (∫⁻ (x : α), ‖s.indicator f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) =\n (∫⁻ (a : α), s.... | rw [← lintegral_indicator hs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 459,
"column": 4
} | {
"line": 459,
"column": 78
} | {
"line": 460,
"column": 4
} | [
{
"pp": "ι : Type u_1\ninst✝² : Fintype ι\nβ : Option ι → Type u_4\ninst✝¹ : (i : Option ι) → MeasurableSpace (β i)\nμ : (i : Option ι) → Measure (β i)\ninst✝ : ∀ (i : Option ι), SigmaFinite (μ i)\ns : (i : Option ι) → Set (β i)\nx✝ : ∀ (i : Option ι), MeasurableSet (s i)\ne_meas : ((i : ι) → β (some i)) × β no... | [
"ι : Type u_1\ninst✝² : Fintype ι\nβ : Option ι → Type u_4\ninst✝¹ : (i : Option ι) → MeasurableSpace (β i)\nμ : (i : Option ι) → Measure (β i)\ninst✝ : ∀ (i : Option ι), SigmaFinite (μ i)\ns : (i : Option ι) → Set (β i)\nx✝ : ∀ (i : Option ι), MeasurableSet (s i)\ne_meas : ((i : ι) → β (some i)) × β none ≃ᵐ ((i : ... | simp only [mem_preimage, Set.mem_pi, mem_univ, forall_true_left, mem_prod] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 47
} | {
"line": 513,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_3\ninst✝⁴ : Fintype ι\ninst✝³ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝² : ∀ (i : ι), SigmaFinite (μ i)\ninst✝¹ : (i : ι) → PartialOrder (α i)\ninst✝ : ∀ (i : ι), NullSingletonClass (μ i)\nf g : (i : ι) → α i\n⊢ (univ.pi fun i ↦ Ico (f i) (g i)) ... | [] | rw [← pi_univ_Icc]; exact pi_Ico_ae_eq_pi_Icc | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 47
} | {
"line": 513,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_3\ninst✝⁴ : Fintype ι\ninst✝³ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝² : ∀ (i : ι), SigmaFinite (μ i)\ninst✝¹ : (i : ι) → PartialOrder (α i)\ninst✝ : ∀ (i : ι), NullSingletonClass (μ i)\nf g : (i : ι) → α i\n⊢ (univ.pi fun i ↦ Ico (f i) (g i)) ... | [] | rw [← pi_univ_Icc]; exact pi_Ico_ae_eq_pi_Icc | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.Indicator | {
"line": 144,
"column": 2
} | {
"line": 145,
"column": 21
} | {
"line": 147,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nE : Type u_4\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup E\ns : Set α\np : ℝ≥0∞\nhs : MeasurableSet s\nc : E\nhμ : μ s ≠ ∞\n⊢ MemLp (fun x ↦ c) p (μ.restrict s)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measu... | [] | · have := Fact.mk hμ.lt_top
apply memLp_const | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.LpSeminorm.Indicator | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 75
} | {
"line": 182,
"column": 0
} | [
{
"pp": "case neg.inr.right\nα : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝² : TopologicalSpace ε\ninst✝¹ : ESeminormedAddMonoid ε\ns : Set α\nf : α → ε\ninst✝ : DecidablePred fun x ↦ x ∈ s\ng : α → ε\nhs : MeasurableSet s\nhf : MemLp f p (μ.restrict s)\nhg : MemLp g p (μ.res... | [] | exact lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top hp_zero hp_top hg.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSeminorm.SMul | {
"line": 71,
"column": 35
} | {
"line": 71,
"column": 55
} | {
"line": 71,
"column": 55
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_3\ninst✝⁴ : NormedRing 𝕜\nε : Type u_4\ninst✝³ : TopologicalSpace ε\ninst✝² : ESeminormedAddMonoid ε\ninst✝¹ : SMul 𝕜 ε\ninst✝ : ENormSMulClass 𝕜 ε\nc : 𝕜\nf : α → ε\nx✝ : α\n⊢ ‖(c • f) x✝‖ₑ ≤ ‖c‖ₑ * ‖f x✝‖ₑ",
"ppTerm": "?m.42",
... | [] | by simp [enorm_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 686,
"column": 47
} | {
"line": 687,
"column": 46
} | {
"line": 689,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nf : α → ε\nc : ℝ≥0∞\n⊢ eLpNorm f 1 (c • μ) = c * eLpNorm f 1 μ",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real",
... | [] | by
rw [eLpNorm_smul_measure_of_ne_top] <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Real.ConjExponents | {
"line": 541,
"column": 10
} | {
"line": 541,
"column": 32
} | {
"line": 541,
"column": 33
} | [
{
"pp": "p : ℝ≥0∞\nhp✝ : 1 ≤ p\nthis : p ≠ 0\nhp₁ : 1 < p\nhp : p ≠ ∞\n⊢ (p - 1 + 1) / (p - 1) = (1⁻¹ - p⁻¹)⁻¹",
"ppTerm": "?m.167",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"ENNReal.instAdd",
"instHDiv",
"ENNReal.instOrderedSu... | [
"p : ℝ≥0∞\nhp✝ : 1 ≤ p\nthis : p ≠ 0\nhp₁ : 1 < p\nhp : p ≠ ∞\n⊢ p / (p - 1) = (1⁻¹ - p⁻¹)⁻¹",
"p : ℝ≥0∞\nhp✝ : 1 ≤ p\nthis : p ≠ 0\nhp₁ : 1 < p\nhp : p ≠ ∞\n⊢ 1 ≤ p"
] | tsub_add_cancel_of_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 219,
"column": 46
} | {
"line": 219,
"column": 48
} | {
"line": 219,
"column": 49
} | [
{
"pp": "x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\n⊢ 0 < a → 0 < b → a + b = 1 → a • log x + b • log y < log (a • x + b • y)",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"Preorder.toLT",
"PartialOrder.toPreorder",
... | [
"x : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x ≠ y\na b : ℝ\nha : 0 < a\n⊢ 0 < b → a + b = 1 → a • log x + b • log y < log (a • x + b • y)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.Order.Monovary | {
"line": 338,
"column": 31
} | {
"line": 338,
"column": 55
} | {
"line": 338,
"column": 55
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : Semifield α\ninst✝⁴ : LinearOrder α\ninst✝³ : IsStrictOrderedRing α\ninst✝² : Semifield β\ninst✝¹ : LinearOrder β\ninst✝ : IsStrictOrderedRing β\ns : Set ι\nf : ι → α\ng : ι → β\nhf : ∀ (i : ι), i ∈ s → 0 < f i\nhg : ∀ (i : ι), i ∈ s → 0 < g i\n⊢ Monov... | [
"ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : Semifield α\ninst✝⁴ : LinearOrder α\ninst✝³ : IsStrictOrderedRing α\ninst✝² : Semifield β\ninst✝¹ : LinearOrder β\ninst✝ : IsStrictOrderedRing β\ns : Set ι\nf : ι → α\ng : ι → β\nhf : ∀ (i : ι), i ∈ s → 0 < f i\nhg : ∀ (i : ι), i ∈ s → 0 < g i\n⊢ AntivaryOn f g s ... | monovaryOn_inv_right₀ hg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.MeanInequalities | {
"line": 147,
"column": 8
} | {
"line": 147,
"column": 34
} | {
"line": 148,
"column": 4
} | [
{
"pp": "case e'_3.inr\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nhw : ∀ i ∈ s, 0 ≤ w i\nhw' : ∑ i ∈ s, w i = 1\nhz✝ : ∀ i ∈ s, 0 ≤ z i\nA : ∀ i ∈ s, z i = 0 → w i = 0\nthis : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))\ni : ι\nhi : i ∈ s\nhz : 0 < z i\n⊢ z i ^ w i = rexp (log (z i) * w i)",
... | [] | exact rpow_def_of_pos hz _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.MeanInequalities | {
"line": 147,
"column": 8
} | {
"line": 147,
"column": 34
} | {
"line": 148,
"column": 4
} | [
{
"pp": "case e'_3.inr\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nhw : ∀ i ∈ s, 0 ≤ w i\nhw' : ∑ i ∈ s, w i = 1\nhz✝ : ∀ i ∈ s, 0 ≤ z i\nA : ∀ i ∈ s, z i = 0 → w i = 0\nthis : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))\ni : ι\nhi : i ∈ s\nhz : 0 < z i\n⊢ z i ^ w i = rexp (log (z i) * w i)",
... | [] | exact rpow_def_of_pos hz _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalities | {
"line": 147,
"column": 8
} | {
"line": 147,
"column": 34
} | {
"line": 148,
"column": 4
} | [
{
"pp": "case e'_3.inr\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nhw : ∀ i ∈ s, 0 ≤ w i\nhw' : ∑ i ∈ s, w i = 1\nhz✝ : ∀ i ∈ s, 0 ≤ z i\nA : ∀ i ∈ s, z i = 0 → w i = 0\nthis : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))\ni : ι\nhi : i ∈ s\nhz : 0 < z i\n⊢ z i ^ w i = rexp (log (z i) * w i)",
... | [] | exact rpow_def_of_pos hz _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 173,
"column": 76
} | {
"line": 186,
"column": 55
} | {
"line": 188,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\np q : ℝ\nhp : 0 ≤ p\nhq : 0 ≤ q\nhpq : p + q = 1\n⊢ ∫⁻ (a : α), f a ^ p * g a ^ q ∂μ ≤ (∫⁻ (a : α), f a ∂μ) ^ p * (∫⁻ (a : α), g a ∂μ) ^ q",
"ppTerm": "?m.62",
"assigned": true,... | [] | by
rcases hp.eq_or_lt with rfl | hp
· rw [zero_add] at hpq
simp [hpq]
rcases hq.eq_or_lt with rfl | hq
· rw [add_zero] at hpq
simp [hpq]
have h2p : 1 < 1 / p := by
rw [one_div, one_lt_inv₀ hp]
linarith
have h2pq : (1 / p)⁻¹ + (1 / q)⁻¹ = 1 := by simp [hpq]
have := ENNReal.lintegral_mul_le_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 353,
"column": 8
} | {
"line": 355,
"column": 34
} | {
"line": 355,
"column": 34
} | [
{
"pp": "case pos\nz₁ z₂ : ℝ≥0∞\np : ℝ\nhp : 0 ≤ p\nh : 1 < p\nhmem : (ENNReal.ofReal p)⁻¹ ∈ Set.Ioo 0 1\n⊢ (z₁ + z₂) ^ p ≤ (ENNReal.ofReal p)⁻¹.LpAddConst * (z₁ ^ p + z₂ ^ p)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Set.decidableMemIoo",
"Eq.mpr",
"ENNReal.instAd... | [
"case pos\nz₁ z₂ : ℝ≥0∞\np : ℝ\nhp : 0 ≤ p\nh : 1 < p\nhmem : (ENNReal.ofReal p)⁻¹ ∈ Set.Ioo 0 1\n⊢ (z₁ + z₂) ^ p ≤ 2 ^ (1 / (ENNReal.ofReal p)⁻¹.toReal - 1) * (z₁ ^ p + z₂ ^ p)"
] | show LpAddConst (ENNReal.ofReal p)⁻¹ =
(2 : ℝ≥0∞) ^ (1 / ((ENNReal.ofReal p)⁻¹).toReal - 1) by
rw [LpAddConst, if_pos hmem] | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 82
} | {
"line": 60,
"column": 2
} | [
{
"pp": "α : Type u_1\nε : Type u_3\nm : MeasurableSpace α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\np : ℝ≥0∞\nμ : Measure α\nf g : α → ε\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nhp1 : 1 ≤ p\nhp0 : ¬p = 0\nhp_top : ¬p = ∞\n⊢ 1 ≤ p.toReal",
"ppTerm": "?m.69",
"as... | [] | rwa [← ENNReal.toReal_one, ENNReal.toReal_le_toReal ENNReal.one_ne_top hp_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 328,
"column": 19
} | {
"line": 328,
"column": 59
} | {
"line": 328,
"column": 60
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\na : α\nh_zero : ¬(f + g) a = 0\nh_top : (f + g) a = ∞\n⊢ ∞ ^ p ≤ ∞ * ∞ ^ (p - 1)",
"ppTerm": "?pos✝",
"assigned": true,
"usedCo... | [
"case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\na : α\nh_zero : ¬(f + g) a = 0\nh_top : (f + g) a = ∞\n⊢ ∞ ^ p ≤ ∞ * ∞"
] | ENNReal.top_rpow_of_pos hpq.sub_one_pos, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 328,
"column": 60
} | {
"line": 328,
"column": 79
} | {
"line": 328,
"column": 79
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\na : α\nh_zero : ¬(f + g) a = 0\nh_top : (f + g) a = ∞\n⊢ ∞ ^ p ≤ ∞ * ∞",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": ... | [
"case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\na : α\nh_zero : ¬(f + g) a = 0\nh_top : (f + g) a = ∞\n⊢ ∞ ^ p ≤ ∞"
] | ENNReal.top_mul_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSpace.Complete | {
"line": 223,
"column": 6
} | {
"line": 223,
"column": 32
} | {
"line": 224,
"column": 6
} | [
{
"pp": "case e_a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_3\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhp_pos : 0 < p\nhn : (∫⁻ (a : α), ‖∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ₑ ^ p ∂μ) ^ p⁻¹ ≤ ∑' (i : ℕ), B i\na : α\n⊢ ENNReal.ofRe... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_3\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\np : ℝ\nhp1 : 1 ≤ p\nB : ℕ → ℝ≥0∞\nn : ℕ\nhp_pos : 0 < p\nhn : (∫⁻ (a : α), ‖∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ₑ ^ p ∂μ) ^ p⁻¹ ≤ ∑' (i : ℕ), B i\na : α\n⊢ 0 ≤ ∑ i ∈ Finset.range (n + 1), ‖f... | rw [Real.norm_of_nonneg _] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 705,
"column": 46
} | {
"line": 705,
"column": 48
} | {
"line": 706,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\ng : E → F\nc : ℝ≥0\nhg : LipschitzWith c g\ng0 : g 0 = 0\na✝ : α\n⊢ ↑↑0 a✝ = 0 a✝ → (g ∘ ↑↑0) a✝ = 0 a✝",
"ppTerm": "?m.74",
"assigned": true,
... | [
"α : Type u_1\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\ng : E → F\nc : ℝ≥0\nhg : LipschitzWith c g\ng0 : g 0 = 0\na✝ : α\nha : ↑↑0 a✝ = 0 a✝\n⊢ (g ∘ ↑↑0) a✝ = 0 a✝"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.MeanInequalities | {
"line": 1160,
"column": 34
} | {
"line": 1160,
"column": 46
} | {
"line": 1160,
"column": 46
} | [
{
"pp": "case e'_3.a.inr\nι : Type u\ns : Finset ι\np : ℝ\nhp✝ : 1 ≤ p\nw f : ι → ℝ≥0∞\nhp : 1 < p\nhp₀ : 0 < p\nhp₁ : p⁻¹ < 1\nH : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0\nH' : (∀ i ∈ s, w i ≠ ∞) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ∞\nthis :\n ∑ x ∈ s, ↑(w x).toNNReal * ↑(f x).toNNReal ≤ (∑... | [
"case e'_3.a.inr\nι : Type u\ns : Finset ι\np : ℝ\nhp✝ : 1 ≤ p\nw f : ι → ℝ≥0∞\nhp : 1 < p\nhp₀ : 0 < p\nhp₁ : p⁻¹ < 1\nH : (∑ i ∈ s, w i) ^ (1 - p⁻¹) ≠ 0 ∧ (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹ ≠ 0\nH' : (∀ i ∈ s, w i ≠ ∞) ∧ ∀ i ∈ s, w i * f i ^ p ≠ ∞\nthis :\n ∑ x ∈ s, ↑(w x).toNNReal * ↑(f x).toNNReal ≤ (∑ i ∈ s, ↑(w ... | coe_toNNReal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Operator.Mul | {
"line": 268,
"column": 4
} | {
"line": 268,
"column": 31
} | {
"line": 269,
"column": 2
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nR : Type u_3\ninst✝⁵ : NormedDivisionRing R\ninst✝⁴ : NormedAlgebra 𝕜 R\ninst✝³ : Module R E\ninst✝² : NormSMulClass R E\ninst✝¹ : IsScalarTower 𝕜 R E\ninst✝ : No... | [] | apply opNorm_lsmul_apply_le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Operator.BoundedLinearMaps | {
"line": 123,
"column": 2
} | {
"line": 125,
"column": 24
} | {
"line": 127,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : Semiring 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : Module 𝕜 F\n⊢ IsBoundedLinearMap 𝕜 fun x ↦ x.2",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"le_max_right... | [] | refine (LinearMap.snd 𝕜 E F).isLinear.with_bound 1 fun x => ?_
rw [one_mul]
exact le_max_right _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.BoundedLinearMaps | {
"line": 123,
"column": 2
} | {
"line": 125,
"column": 24
} | {
"line": 127,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : Semiring 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : Module 𝕜 F\n⊢ IsBoundedLinearMap 𝕜 fun x ↦ x.2",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"le_max_right... | [] | refine (LinearMap.snd 𝕜 E F).isLinear.with_bound 1 fun x => ?_
rw [one_mul]
exact le_max_right _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.BoundedLinearMaps | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 16
} | {
"line": 221,
"column": 4
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : Module 𝕜 G\nf : E × F → G\nh : IsBoundedBilinearMap 𝕜 f... | [
"case inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : Module 𝕜 G\nf : E × F → G\nh : IsBoundedBilinearMap 𝕜 f\nx : E\nC :... | · grw [hx.1] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Operator.BoundedLinearMaps | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 51
} | {
"line": 263,
"column": 2
} | [
{
"pp": "case hf.refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : Module 𝕜 G\nf : E × F → G\nh : IsBoundedBilinear... | [] | exact (continuous_snd.tendsto _).isBigO_one ℝ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Operator.BoundedLinearMaps | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 51
} | {
"line": 263,
"column": 2
} | [
{
"pp": "case hf.refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : Module 𝕜 G\nf : E × F → G\nh : IsBoundedBilinear... | [] | exact (continuous_snd.tendsto _).isBigO_one ℝ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.BoundedLinearMaps | {
"line": 262,
"column": 6
} | {
"line": 262,
"column": 51
} | {
"line": 263,
"column": 2
} | [
{
"pp": "case hf.refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : Module 𝕜 G\nf : E × F → G\nh : IsBoundedBilinear... | [] | exact (continuous_snd.tendsto _).isBigO_one ℝ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent | {
"line": 62,
"column": 67
} | {
"line": 64,
"column": 82
} | {
"line": 66,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf T : M →L[R] M\nhf : IsIdempotentElem f\n⊢ (↑f).range ∈ Module.End.invtSubmodule ↑T ↔ f ∘SL T ∘SL f = T ∘SL f",
"ppTerm": "?m.136",
"assigned": true,
"usedConstants": [
... | [] | by
simpa [← toLinearMap_comp] using
LinearMap.IsIdempotentElem.range_mem_invtSubmodule_iff (T := T) hf.toLinearMap | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Homeomorph.Quotient | {
"line": 38,
"column": 28
} | {
"line": 38,
"column": 58
} | {
"line": 40,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nr : X → X → Prop\ne : X ≃ₜ Y\nx✝¹ x✝ : X\n⊢ r x✝¹ x✝ ↔ r (e.symm (e x✝¹)) (e.symm (e x✝))",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"congrArg",
"iff_self",
"Homeomorph.inst... | [] | simp only [e.symm_apply_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Homeomorph.Quotient | {
"line": 38,
"column": 28
} | {
"line": 38,
"column": 58
} | {
"line": 40,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nr : X → X → Prop\ne : X ≃ₜ Y\nx✝¹ x✝ : X\n⊢ r x✝¹ x✝ ↔ r (e.symm (e x✝¹)) (e.symm (e x✝))",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"congrArg",
"iff_self",
"Homeomorph.inst... | [] | simp only [e.symm_apply_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Homeomorph.Quotient | {
"line": 38,
"column": 28
} | {
"line": 38,
"column": 58
} | {
"line": 40,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nr : X → X → Prop\ne : X ≃ₜ Y\nx✝¹ x✝ : X\n⊢ r x✝¹ x✝ ↔ r (e.symm (e x✝¹)) (e.symm (e x✝))",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"congrArg",
"iff_self",
"Homeomorph.inst... | [] | simp only [e.symm_apply_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas | {
"line": 97,
"column": 34
} | {
"line": 97,
"column": 36
} | {
"line": 97,
"column": 37
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : StronglyMeasurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x\nA : MeasurableSet {x | f x ≠ 0}\na : α\n⊢ (↑(f a) ≠ 0 → g... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → ℝ≥0\nhf : Measurable f\ng g' : α → E\ng'meas : StronglyMeasurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x\nA : MeasurableSet {x | f x ≠ 0}\na : α\nha : ↑(f a) ≠ 0 → g a = g' a\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.Real | {
"line": 481,
"column": 2
} | {
"line": 481,
"column": 42
} | {
"line": 483,
"column": 0
} | [
{
"pp": "α : Type u_1\nx✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : IsProbabilityMeasure μ\nh : NullMeasurableSet s μ\n⊢ μ.real sᶜ = 1 - μ.real s",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Compl.compl",
"Real.in... | [] | rw [measureReal_compl₀ h, probReal_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Real | {
"line": 481,
"column": 2
} | {
"line": 481,
"column": 42
} | {
"line": 483,
"column": 0
} | [
{
"pp": "α : Type u_1\nx✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : IsProbabilityMeasure μ\nh : NullMeasurableSet s μ\n⊢ μ.real sᶜ = 1 - μ.real s",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Compl.compl",
"Real.in... | [] | rw [measureReal_compl₀ h, probReal_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Real | {
"line": 481,
"column": 2
} | {
"line": 481,
"column": 42
} | {
"line": 483,
"column": 0
} | [
{
"pp": "α : Type u_1\nx✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : IsProbabilityMeasure μ\nh : NullMeasurableSet s μ\n⊢ μ.real sᶜ = 1 - μ.real s",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Compl.compl",
"Real.in... | [] | rw [measureReal_compl₀ h, probReal_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 28
} | {
"line": 191,
"column": 2
} | [
{
"pp": "α : Type u_1\nε' : Type u_4\nmα : MeasurableSpace α\ns : Set α\nμ : Measure α\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ESeminormedAddMonoid ε'\nf g : α → ε'\ninst✝ : DecidablePred fun x ↦ x ∈ s\nhs : MeasurableSet s\nhf : IntegrableOn f s μ\nhg : IntegrableOn g sᶜ μ\n⊢ Integrable (s.piecewise f g) μ",
... | [
"α : Type u_1\nε' : Type u_4\nmα : MeasurableSpace α\ns : Set α\nμ : Measure α\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ESeminormedAddMonoid ε'\nf g : α → ε'\ninst✝ : DecidablePred fun x ↦ x ∈ s\nhs : MeasurableSet s\nhf : Integrable f (μ.restrict s)\nhg : Integrable g (μ.restrict sᶜ)\n⊢ Integrable (s.piecewise f g)... | rw [IntegrableOn] at hf hg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 215,
"column": 72
} | {
"line": 215,
"column": 74
} | {
"line": 216,
"column": 4
} | [
{
"pp": "α : Type u_1\nε' : Type u_4\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ESeminormedAddMonoid ε'\nf : α → ε'\nx : α\ninst✝ : MeasurableSingletonClass α\nhfx : ‖f x‖ₑ ≠ ∞\na✝ : α\n⊢ a✝ ∈ {x} → f a✝ = f x",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants"... | [
"α : Type u_1\nε' : Type u_4\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ESeminormedAddMonoid ε'\nf : α → ε'\nx : α\ninst✝ : MeasurableSingletonClass α\nhfx : ‖f x‖ₑ ≠ ∞\na✝ : α\nha : a✝ ∈ {x}\n⊢ f a✝ = f x"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 542,
"column": 31
} | {
"line": 542,
"column": 78
} | {
"line": 544,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : α → β\nhf : Integrable f μ\nhg : Integrable g μ\n⊢ Integrable (f - g) μ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"MeasureT... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 542,
"column": 31
} | {
"line": 542,
"column": 78
} | {
"line": 544,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : α → β\nhf : Integrable f μ\nhg : Integrable g μ\n⊢ Integrable (f - g) μ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"MeasureT... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 542,
"column": 31
} | {
"line": 542,
"column": 78
} | {
"line": 544,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : α → β\nhf : Integrable f μ\nhg : Integrable g μ\n⊢ Integrable (f - g) μ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"MeasureT... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 387,
"column": 20
} | {
"line": 387,
"column": 78
} | {
"line": 388,
"column": 6
} | [
{
"pp": "α : Type u_1\nε' : Type u_4\nmα : MeasurableSpace α\ns : Set α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε'\ninst✝ : ESeminormedAddMonoid ε'\nf : α → ε'\nhf : IntegrableOn f s μ\nh's : ∀ x ∈ s, ‖f x‖ₑ ≠ 0\nu : ℕ → ℝ≥0∞\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), u n ∈ Ioo 0 ∞\nu_lim : Tendsto u atTop (𝓝... | [
"α : Type u_1\nε' : Type u_4\nmα : MeasurableSpace α\ns : Set α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε'\ninst✝ : ESeminormedAddMonoid ε'\nf : α → ε'\nhf : IntegrableOn f s μ\nh's : ∀ x ∈ s, ‖f x‖ₑ ≠ 0\nu : ℕ → ℝ≥0∞\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), u n ∈ Ioo 0 ∞\nu_lim : Tendsto u atTop (𝓝 0)\nv : ℕ →... | ← Measure.restrict_apply (measurableSet_toMeasurable _ _), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 385,
"column": 2
} | {
"line": 389,
"column": 68
} | {
"line": 390,
"column": 2
} | [
{
"pp": "α : Type u_1\nε' : Type u_4\nmα : MeasurableSpace α\ns : Set α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε'\ninst✝ : ESeminormedAddMonoid ε'\nf : α → ε'\nhf : IntegrableOn f s μ\nh's : ∀ x ∈ s, ‖f x‖ₑ ≠ 0\nu : ℕ → ℝ≥0∞\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), u n ∈ Ioo 0 ∞\nu_lim : Tendsto u atTop (𝓝... | [
"α : Type u_1\nε' : Type u_4\nmα : MeasurableSpace α\ns : Set α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε'\ninst✝ : ESeminormedAddMonoid ε'\nf : α → ε'\nhf : IntegrableOn f s μ\nh's : ∀ x ∈ s, ‖f x‖ₑ ≠ 0\nu : ℕ → ℝ≥0∞\nleft✝ : StrictAnti u\nu_pos : ∀ (n : ℕ), u n ∈ Ioo 0 ∞\nu_lim : Tendsto u atTop (𝓝 0)\nv : ℕ →... | have A : ∀ n, μ (s ∩ v n) ≠ ∞ := by
intro n
rw [inter_comm, ← Measure.restrict_apply (measurableSet_toMeasurable _ _),
measure_toMeasurable]
exact (hf.measure_enorm_ge_lt_top (u_pos n).1 (u_pos n).2.ne).ne | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 627,
"column": 12
} | {
"line": 627,
"column": 14
} | {
"line": 628,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf₀ f₁ : α → β\ng : α → ℝ\nhf₁_m : AEStronglyMeasurable f₁ μ\nhf₀_i : Integrable f₀ μ\nhg_i : Integrable g μ\nh : ∀ᵐ (a : α) ∂μ, ‖f₀ a - f₁ a‖ ≤ g a\na : α\n⊢ ‖f₀ a - f₁ a‖ ≤ g a → ‖f₁ a‖ ≤ ‖f₀ a‖ + g a",
... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf₀ f₁ : α → β\ng : α → ℝ\nhf₁_m : AEStronglyMeasurable f₁ μ\nhf₀_i : Integrable f₀ μ\nhg_i : Integrable g μ\nh : ∀ᵐ (a : α) ∂μ, ‖f₀ a - f₁ a‖ ≤ g a\na : α\nha : ‖f₀ a - f₁ a‖ ≤ g a\n⊢ ‖f₁ a‖ ≤ ‖f₀ a‖ + g a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 421,
"column": 6
} | {
"line": 421,
"column": 59
} | {
"line": 422,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nmα : MeasurableSpace α\ns t : Set α\nμ : Measure α\nε' : Type u_7\ninst✝² : TopologicalSpace ε'\ninst✝¹ : ENormedAddMonoid ε'\ninst✝ : PseudoMetrizableSpace ε'\nf : α → ε'\nhf : IntegrableOn f s μ\nht : NullMeasurableSet t μ\nh't : ∀ᵐ (x : α) ∂μ, x ∈ t \\ s → f x = 0\nu : Set α ... | [] | exact hx.2 (subset_toMeasurable μ u ⟨h'x, Ne.symm H⟩) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 587,
"column": 2
} | {
"line": 587,
"column": 26
} | {
"line": 590,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_5\nmα : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\nμ : Measure α\nl : Filter α\n𝕜 : Type u_7\ninst✝² : NormedAddCommGroup 𝕜\ninst✝¹ : SMulZeroClass 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nf : α → E\nc : 𝕜\ns : Set α\nsl : s ∈ l\nhs : IntegrableOn f s μ\n⊢ IntegrableAtFilter... | [] | exact ⟨s, sl, hs.smul c⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 785,
"column": 47
} | {
"line": 785,
"column": 49
} | {
"line": 785,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝³ : TopologicalSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns t : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : IsCompact s\nht : MeasurableSet t\nhts : t ⊆ s\nthis✝¹ : Mea... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝³ : TopologicalSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : PseudoMetrizableSpace β\nf : α → β\ns t : Set α\nμ : Measure α\nhf : ContinuousOn f s\nhs : IsCompact s\nht : MeasurableSet t\nhts : t ⊆ s\nthis✝¹ : MeasurableSpace... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.L1Space.AEEqFun | {
"line": 145,
"column": 43
} | {
"line": 145,
"column": 45
} | {
"line": 146,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ↥(Lp β 1 μ)\na✝ : α\n⊢ ↑↑(f - g) a✝ = (↑↑f - ↑↑g) a✝ → ‖↑↑(f - g) a✝‖ₑ = ‖↑↑f a✝ - ↑↑g a✝‖ₑ",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"MeasureTheory.AEEqFun.cast",
... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ↥(Lp β 1 μ)\na✝ : α\nha : ↑↑(f - g) a✝ = (↑↑f - ↑↑g) a✝\n⊢ ‖↑↑(f - g) a✝‖ₑ = ‖↑↑f a✝ - ↑↑g a✝‖ₑ"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.L1Space.AEEqFun | {
"line": 159,
"column": 43
} | {
"line": 159,
"column": 45
} | {
"line": 160,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ↥(Lp β 1 μ)\na✝ : α\n⊢ ↑↑(f - g) a✝ = (↑↑f - ↑↑g) a✝ → edist (↑↑(f - g) a✝) 0 = edist (↑↑f a✝ - ↑↑g a✝) 0",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"MeasureTheory.AEEq... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ↥(Lp β 1 μ)\na✝ : α\nha : ↑↑(f - g) a✝ = (↑↑f - ↑↑g) a✝\n⊢ edist (↑↑(f - g) a✝) 0 = edist (↑↑f a✝ - ↑↑g a✝) 0"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1224,
"column": 4
} | {
"line": 1224,
"column": 19
} | {
"line": 1225,
"column": 4
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nE₁ : ι → Type wE₁\nG : Type wG\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E₁ i)\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝² : SeminormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E₁ G... | [
"case h\n𝕜 : Type u\nι : Type v\nE₁ : ι → Type wE₁\nG : Type wG\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E₁ i)\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝² : SeminormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E₁ G\nk ... | gcongr with e _ | Mathlib.Tactic.GCongr._aux_Mathlib_Tactic_GCongr_Core___elabRules_Mathlib_Tactic_GCongr_gcongr_1 | Mathlib.Tactic.GCongr.gcongr |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 524,
"column": 2
} | {
"line": 524,
"column": 95
} | {
"line": 525,
"column": 2
} | [
{
"pp": "X : Type u_1\nε : Type u_3\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace ε\ninst✝³ : ContinuousENorm ε\nf : X → ε\nμ : Measure X\ninst✝² : PseudoMetrizableSpace ε\na : X\ninst✝¹ : LinearOrder X\ninst✝ : CompactIccSpace X\nx✝ : IntegrableAtFilter f atBot μ ∧ Locally... | [
"X : Type u_1\nε : Type u_3\ninst✝⁶ : MeasurableSpace X\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace ε\ninst✝³ : ContinuousENorm ε\nf : X → ε\nμ : Measure X\ninst✝² : PseudoMetrizableSpace ε\na : X\ninst✝¹ : LinearOrder X\ninst✝ : CompactIccSpace X\nx✝ : IntegrableAtFilter f atBot μ ∧ LocallyIntegrableOn... | refine (integrableOn_union.mpr ⟨hs.mono ha' le_rfl, ?_⟩).mono Iic_subset_Iic_union_Icc le_rfl | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 581,
"column": 45
} | {
"line": 581,
"column": 47
} | {
"line": 581,
"column": 48
} | [
{
"pp": "X : Type u_1\nE : Type u_6\ninst✝³ : MeasurableSpace X\ninst✝² : TopologicalSpace X\ninst✝¹ : NormedAddCommGroup E\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nK : Set X\nf : X → E\nhf : ContinuousOn f K\nhK : IsCompact K\nhs : MeasurableSet s\nh's : s ⊆ K\nmus : μ s ≠ ∞\nthis : Fact (μ s... | [
"X : Type u_1\nE : Type u_6\ninst✝³ : MeasurableSpace X\ninst✝² : TopologicalSpace X\ninst✝¹ : NormedAddCommGroup E\nμ : Measure X\ns : Set X\ninst✝ : OpensMeasurableSpace X\nK : Set X\nf : X → E\nhf : ContinuousOn f K\nhK : IsCompact K\nhs : MeasurableSet s\nh's : s ⊆ K\nmus : μ s ≠ ∞\nthis : Fact (μ s < ∞)\nC : ℝ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 118,
"column": 31
} | {
"line": 118,
"column": 33
} | {
"line": 119,
"column": 4
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nhp_ne_top : p ≠ ∞\nμ : Measure β\nhμ : ∀ᵐ (x : β) ∂μ, f ... | [
"β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nhf : Measurable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nhp_ne_top : p ≠ ∞\nμ : Measure β\nhμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure[... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.FinMeasAdditive | {
"line": 139,
"column": 18
} | {
"line": 139,
"column": 21
} | {
"line": 139,
"column": 22
} | [
{
"pp": "case refine_2\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝ : AddCommMonoid β\nT : Set α → β\nT_empty : T ∅ = 0\nh_add : FinMeasAdditive μ T\nι : Type u_8\nS : ι → Set α\nsι : Finset ι\nhS_meas : ∀ (i : ι), MeasurableSet (S i)\na : ι\ns : Finset ι\nhas : a ∉ s\nh : (∀ i ∈ s, ... | [
"case refine_2\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_7\ninst✝ : AddCommMonoid β\nT : Set α → β\nT_empty : T ∅ = 0\nh_add : FinMeasAdditive μ T\nι : Type u_8\nS : ι → Set α\nsι : Finset ι\nhS_meas : ∀ (i : ι), MeasurableSet (S i)\na : ι\ns : Finset ι\nhas : a ∉ s\nh : (∀ i ∈ s, μ (S i) ≠ ∞)... | hps | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 39
} | {
"line": 229,
"column": 2
} | [
{
"pp": "β : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nμ : Measure β\ninst✝ : SeparableSpace ↑(Set.range f ∪ {0})\nfmeas : Measurable f\nhf : Integrable f μ\n⊢ Tendsto (fun n ↦ ∫⁻ (x : β), ‖(approxOn... | [
"case hμ\nβ : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : OpensMeasurableSpace E\nf : β → E\nμ : Measure β\ninst✝ : SeparableSpace ↑(Set.range f ∪ {0})\nfmeas : Measurable f\nhf : Integrable f μ\n⊢ ∀ᵐ (x : β) ∂μ, f x ∈ closure[PseudoMetricS... | apply tendsto_approxOn_L1_enorm fmeas | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Operator.Extend | {
"line": 196,
"column": 2
} | {
"line": 197,
"column": 78
} | {
"line": 198,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nEₗ : Type u_4\nF : Type u_5\ninst✝¹⁰ : NormedDivisionRing 𝕜\ninst✝⁹ : NormedDivisionRing 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : SeminormedAddCommGroup Eₗ\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ :... | [
"𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nEₗ : Type u_4\nF : Type u_5\ninst✝¹⁰ : NormedDivisionRing 𝕜\ninst✝⁹ : NormedDivisionRing 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : SeminormedAddCommGroup Eₗ\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : Module 𝕜₂ F\ninst✝³ : IsBoundedSM... | have := (f.compLeftInverse e).extend_eq (e := (LinearMap.range e).subtypeL)
(by simpa using! h_dense) isUniformEmbedding_subtype_val.isUniformInducing | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 218,
"column": 29
} | {
"line": 218,
"column": 67
} | {
"line": 220,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : DecidablePred fun x ↦ x ≠ 0\nf : α →ₛ F\ns : Finset F\nhs : {x ∈ f.range | x ≠ 0} ⊆ s\nx : F\nhx✝ : x ∉ f.range ∨ x = 0\nhx : x ∉ Set.range ⇑f\n⊢ μ.real ∅ • x = 0... | [] | simp [Set.disjoint_singleton_left, hx] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 218,
"column": 29
} | {
"line": 218,
"column": 67
} | {
"line": 220,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : DecidablePred fun x ↦ x ≠ 0\nf : α →ₛ F\ns : Finset F\nhs : {x ∈ f.range | x ≠ 0} ⊆ s\nx : F\nhx✝ : x ∉ f.range ∨ x = 0\nhx : x ∉ Set.range ⇑f\n⊢ Disjoint {x} (Se... | [] | simp [Set.disjoint_singleton_left, hx] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 307,
"column": 28
} | {
"line": 307,
"column": 30
} | {
"line": 308,
"column": 2
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_6\nx✝ : MeasurableSpace β\nν : Measure β\nf g : α → β → G\nh : ∀ᵐ (a : α) ∂μ, f a =ᵐ[ν] g a\na✝ : α\n⊢ f a✝ =ᵐ[ν] g a✝ → ∫ (b : β), f a✝ b ∂ν = ∫ (b : β), g a✝ b ∂ν",
... | [
"α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_6\nx✝ : MeasurableSpace β\nν : Measure β\nf g : α → β → G\nh : ∀ᵐ (a : α) ∂μ, f a =ᵐ[ν] g a\na✝ : α\nha : f a✝ =ᵐ[ν] g a✝\n⊢ ∫ (b : β), f a✝ b ∂ν = ∫ (b : β), g a✝ b ∂ν"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 401,
"column": 4
} | {
"line": 401,
"column": 52
} | {
"line": 402,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nι : Type u_6\nf : α → G\nhfi : AEStronglyMeasurable f μ\nF : ι → α → G\nl : Filter ι\nhFi : ∀ᶠ (i : ι) in l, Integrable (F i) μ\nhF : Tendsto (fun i ↦ ∫⁻ (x : α), ‖F ... | [] | filter_upwards [hFi] with i hi using hi.restrict | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 401,
"column": 4
} | {
"line": 401,
"column": 52
} | {
"line": 402,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nι : Type u_6\nf : α → G\nhfi : AEStronglyMeasurable f μ\nF : ι → α → G\nl : Filter ι\nhFi : ∀ᶠ (i : ι) in l, Integrable (F i) μ\nhF : Tendsto (fun i ↦ ∫⁻ (x : α), ‖F ... | [] | filter_upwards [hFi] with i hi using hi.restrict | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 401,
"column": 4
} | {
"line": 401,
"column": 52
} | {
"line": 402,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nm : MeasurableSpace α\nμ : Measure α\nι : Type u_6\nf : α → G\nhfi : AEStronglyMeasurable f μ\nF : ι → α → G\nl : Filter ι\nhFi : ∀ᶠ (i : ι) in l, Integrable (F i) μ\nhF : Tendsto (fun i ↦ ∫⁻ (x : α), ‖F ... | [] | filter_upwards [hFi] with i hi using hi.restrict | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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