module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 927,
"column": 4
} | {
"line": 927,
"column": 28
} | [
{
"pp": "case pos.h\nα : Type u_1\nE : Type u_4\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : MeasurableSpace E\ninst✝ : OpensMeasurableSpace E\nR : ℝ≥0\np : ℝ≥0∞\nf : ℕ → α → E\nhfmeas : ∀ (n : ℕ), Measurable (f n)\nhbdd : ∀ (n : ℕ), eLpNorm (f n) p μ ≤ ↑R\nhp0 : p.toReal = 0\... | exact ENNReal.one_lt_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.Module.StrongTopology | {
"line": 298,
"column": 2
} | {
"line": 298,
"column": 87
} | [
{
"pp": "𝕜₁ : Type u_1\n𝕜₂ : Type u_2\ninst✝¹² : NormedField 𝕜₁\ninst✝¹¹ : NormedField 𝕜₂\nσ : 𝕜₁ →+* 𝕜₂\nE : Type u_3\nF : Type u_4\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module 𝕜₁ E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\nR : Type u_6\ninst✝⁵ : NormedDivisionRing R\ni... | simp_rw [isVonNBounded_iff_absorbing_le, nhds_zero_eq, le_iInf_iff, le_principal_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Algebra.Module.StrongTopology | {
"line": 709,
"column": 56
} | {
"line": 709,
"column": 66
} | [
{
"pp": "R : Type u_1\n𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : Semiring R\ninst✝¹² : NormedField 𝕜₂\ninst✝¹¹ : NormedField 𝕜₃\ninst✝¹⁰ : AddCommMonoid E\ninst✝⁹ : Module R E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : Topol... | smul_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Module.StrongTopology | {
"line": 739,
"column": 18
} | {
"line": 739,
"column": 28
} | [
{
"pp": "𝕜₂ : Type u_3\n𝕜₃ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹² : NormedField 𝕜₂\ninst✝¹¹ : NormedField 𝕜₃\ninst✝¹⁰ : AddCommMonoid E\ninst✝⁹ : Module 𝕜₃ E\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜₂ F\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommGro... | smul_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 155,
"column": 39
} | {
"line": 155,
"column": 47
} | [
{
"pp": "case neg\np : ℝ\na b : ℝ≥0\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nh_zero : ¬a + b = 0\nh_nonzero : ¬(a = 0 ∧ b = 0)\nh_add : a / (a + b) + b / (a + b) = 1\nh : a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p ≤ 1\nhab_0 : (a + b) ^ p ≠ 0\nh_mul : (a + b) ^ p * (a ^ p * ((a + b) ^ p)⁻¹ + b ^ p * ((a + b) ^ p)⁻¹) ≤ (a +... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 121,
"column": 14
} | {
"line": 121,
"column": 95
} | [
{
"pp": "α : 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\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ∞\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\nnpf : ℝ≥0∞... | simp [npf, nqg, hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 121,
"column": 14
} | {
"line": 121,
"column": 95
} | [
{
"pp": "α : 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\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ∞\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\nnpf : ℝ≥0∞... | simp [npf, nqg, hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 121,
"column": 14
} | {
"line": 121,
"column": 95
} | [
{
"pp": "α : 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\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ∞\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\nnpf : ℝ≥0∞... | simp [npf, nqg, hf_nontop, hg_nontop, hf_nonzero, hg_nonzero, ENNReal.mul_eq_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 43,
"column": 14
} | {
"line": 43,
"column": 41
} | [
{
"pp": "case neg\nα : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\np q : ℝ\nhp0_lt : 0 < p\nhpq✝ : p ≤ q\nhf : AEStronglyMeasurable f μ\nhq0_lt : 0 < q\nhpq_eq : ¬p = q\nhpq : p < q\ng : α → ℝ≥0∞ := fun x ↦ 1\nh_rw : ∫⁻ (a : α)... | eLpNorm'_eq_lintegral_enorm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 43,
"column": 14
} | {
"line": 43,
"column": 41
} | [
{
"pp": "case neg\nα : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\np q : ℝ\nhp0_lt : 0 < p\nhpq✝ : p ≤ q\nhf : AEStronglyMeasurable f μ\nhq0_lt : 0 < q\nhpq_eq : ¬p = q\nhpq : p < q\ng : α → ℝ≥0∞ := fun x ↦ 1\nh_rw : ∫⁻ (a : α)... | eLpNorm'_eq_lintegral_enorm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 43,
"column": 14
} | {
"line": 43,
"column": 41
} | [
{
"pp": "case neg\nα : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\np q : ℝ\nhp0_lt : 0 < p\nhpq✝ : p ≤ q\nhf : AEStronglyMeasurable f μ\nhq0_lt : 0 < q\nhpq_eq : ¬p = q\nhpq : p < q\ng : α → ℝ≥0∞ := fun x ↦ 1\nh_rw : ∫⁻ (a : α)... | eLpNorm'_eq_lintegral_enorm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 155,
"column": 14
} | {
"line": 155,
"column": 75
} | [
{
"pp": "α : Type u_1\nε' : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε'\ninst✝ : ESeminormedAddMonoid ε'\np q : ℝ≥0∞\nf : α → ε'\ns : Set α\nhfq : MemLp ((toMeasurable μ s).indicator f) q μ\nhf : ∀ x ∉ s, f x = 0\nhs : μ s ≠ ∞\nhpq : p ≤ q\nthis : (toMeasurable μ s).indicator f ... | memLp_indicator_iff_restrict (measurableSet_toMeasurable μ s) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 282,
"column": 12
} | {
"line": 282,
"column": 20
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞\nhg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞\nhp1 : 1 ≤ p\nhp0_lt : 0 < p\nhp0 : 0 ≤ p\na : α\nh_zero_lt_half_rpow : 0 < (1 / 2) ^ p\nh_rw : (1 / 2) ^ p * 2 ^ (p - 1) = 1 / ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 28
} | [
{
"pp": "case neg\np : ℝ≥0∞\nh : p ∉ Set.Ioo 0 1\n⊢ 1 < ∞",
"usedConstants": [
"ENNReal.one_lt_top"
]
}
] | exact ENNReal.one_lt_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 28
} | [
{
"pp": "case neg\np : ℝ≥0∞\nh : p ∉ Set.Ioo 0 1\n⊢ 1 < ∞",
"usedConstants": [
"ENNReal.one_lt_top"
]
}
] | exact ENNReal.one_lt_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 28
} | [
{
"pp": "case neg\np : ℝ≥0∞\nh : p ∉ Set.Ioo 0 1\n⊢ 1 < ∞",
"usedConstants": [
"ENNReal.one_lt_top"
]
}
] | exact ENNReal.one_lt_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 382,
"column": 2
} | {
"line": 386,
"column": 70
} | [
{
"pp": "α : 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\nh_add_zero : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ 0\nh_add_top : ∫⁻ (a : α), (f + g) a ^ p ∂μ ≠ ∞\nh0_rpow : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≠ 0\n... | have h :
(∫⁻ a : α, (f + g) a ^ p ∂μ) ≤
((∫⁻ a : α, f a ^ p ∂μ) ^ (1 / p) + (∫⁻ a : α, g a ^ p ∂μ) ^ (1 / p)) *
(∫⁻ a : α, (f + g) a ^ p ∂μ) ^ (1 / q) :=
lintegral_rpow_add_le_add_eLpNorm_mul_lintegral_rpow_add hpq hf hg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.MeanInequalities | {
"line": 835,
"column": 2
} | {
"line": 840,
"column": 72
} | [
{
"pp": "ι : Type u\nf g : ι → ℝ\np q r : ℝ\nhpqr : p.HolderTriple q r\nhf : ∀ (i : ι), 0 ≤ f i\nhg : ∀ (i : ι), 0 ≤ g i\nhf_sum : Summable fun i ↦ f i ^ p\nhg_sum : Summable fun i ↦ g i ^ q\n⊢ (Summable fun i ↦ (f i * g i) ^ r) ∧\n ∑' (i : ι), (f i * g i) ^ r ≤ (∑' (i : ι), f i ^ p) ^ (r / p) * (∑' (i : ι),... | lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After https://github.com/leanprover/lean4/pull/2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.summable_and_Lr_rpow_le_Lp_mul_Lq_tsum hpqr hf_sum hg_sum | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.MeanInequalities | {
"line": 835,
"column": 2
} | {
"line": 840,
"column": 72
} | [
{
"pp": "ι : Type u\nf g : ι → ℝ\np q r : ℝ\nhpqr : p.HolderTriple q r\nhf : ∀ (i : ι), 0 ≤ f i\nhg : ∀ (i : ι), 0 ≤ g i\nhf_sum : Summable fun i ↦ f i ^ p\nhg_sum : Summable fun i ↦ g i ^ q\n⊢ (Summable fun i ↦ (f i * g i) ^ r) ∧\n ∑' (i : ι), (f i * g i) ^ r ≤ (∑' (i : ι), f i ^ p) ^ (r / p) * (∑' (i : ι),... | lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After https://github.com/leanprover/lean4/pull/2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.summable_and_Lr_rpow_le_Lp_mul_Lq_tsum hpqr hf_sum hg_sum | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConvergenceInMeasure | {
"line": 148,
"column": 4
} | {
"line": 148,
"column": 68
} | [
{
"pp": "case refine_2\nα : Type u_1\nι : Type u_2\nE : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : EDist E\ninst✝ : IsFiniteMeasure μ\nf : ι → α → E\nl : Filter ι\ng : α → E\nhfin : ∀ (ε : ℝ≥0∞) (i : ι), μ {x | ε ≤ edist (f i x) (g x)} ≠ ∞\nh : ∀ (ε : ℝ≥0∞), 0 < ε → Tendsto (fun i ↦ (μ {x | ε ≤ ed... | rw [← ENNReal.tendsto_toNNReal_iff ENNReal.zero_ne_top (hfin ε)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 1002,
"column": 2
} | {
"line": 1004,
"column": 52
} | [
{
"pp": "case neg\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nH : (∑ i ∈ s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i ∈ s, g i ^ q) ^ (1 / q) ≠ 0\nH' : (∀ i ∈ s, f i ≠ ∞) ∧ ∀ i ∈ s, g i ≠ ∞\nthis :\n ∑ x ∈ s, ↑(f x).toNNReal * ↑(g x).toNNReal ≤\n (∑ x ∈ s, ↑(f x).toNNReal ^ p) ^ p⁻¹ *... | convert this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Logic.Equiv.Embedding | {
"line": 45,
"column": 8
} | {
"line": 45,
"column": 49
} | [
{
"pp": "case inr.inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nx✝ : { f // Disjoint (Set.range ⇑f.1) (Set.range ⇑f.2) }\nf : α ↪ γ\ng : β ↪ γ\ndisj : Disjoint (Set.range ⇑(f, g).1) (Set.range ⇑(f, g).2)\nb₁ : β\na₂ : α\nf_eq : g b₁ = f a₂\n⊢ False",
"usedConstants": [
"Disjoint.le_bot",
"Membe... | exact disj.le_bot ⟨⟨a₂, rfl⟩, ⟨b₁, f_eq⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSpace.Complete | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 23
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nE : Type u_3\ninst✝ : NormedAddCommGroup E\nf : ℕ → α → E\nhf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ\nf_lim : α → E\nB : ℕ → ℝ≥0∞\nhB : ∑' (i : ℕ), B i ≠ ∞\nh_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm (f n - f m_1) p μ < B N\nh_lim... | refine h_sub.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 701,
"column": 79
} | {
"line": 701,
"column": 82
} | [
{
"pp": "case h\nα : 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\nf f' : ↥(Lp E p μ)\na : α\n⊢ ↑↑(hg.compLp g0 f) a = (g ∘ ↑↑f) a →\n ↑↑(hg.compLp g0... | ha1 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 791,
"column": 12
} | {
"line": 791,
"column": 15
} | [
{
"pp": "case h\nα : Type u_1\n𝕜✝ : Type u_2\n𝕜'✝ : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ng✝ : E → F\nc : ℝ≥0\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : Nontrivially... | ha1 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 796,
"column": 31
} | {
"line": 796,
"column": 34
} | [
{
"pp": "case h\nα : Type u_1\n𝕜✝ : Type u_2\n𝕜'✝ : Type u_3\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ng : E → F\nc✝ : ℝ≥0\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : Nontrivially... | ha1 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Normed.Operator.NormedSpace | {
"line": 314,
"column": 13
} | {
"line": 314,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\nι : Type u_8\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁴ : RingHomIsometric σ₁₂\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : N... | bddAbove_def, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Real | {
"line": 440,
"column": 28
} | {
"line": 440,
"column": 41
} | [
{
"pp": "α : Type u_1\nx✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : IsProbabilityMeasure μ\nh : NullMeasurableSet s μ\n⊢ μ.real univ - μ.real s = 1 - μ.real s",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instSub",
"Set.univ",
"MeasureTheory.Measur... | probReal_univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 215,
"column": 4
} | {
"line": 215,
"column": 74
} | [
{
"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 ≠ ∞\n⊢ f =ᶠ[ae (μ.restrict {x})] fun x_1 ↦ f x",
"usedConstants": [
"MeasureTheory.ae",
... | filter_upwards [ae_restrict_mem (measurableSet_singleton x)] with _ ha | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 313,
"column": 6
} | {
"line": 313,
"column": 74
} | [
{
"pp": "α : Type u_1\nε : Type u_3\nmα : MeasurableSpace α\nf : α → ε\ns : Set α\nμ : Measure α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nhs : MeasurableSet s\n⊢ IntegrableOn f s μ ↔ Integrable (f ∘ Subtype.val) (Measure.comap Subtype.val μ)",
"usedConstants": [
"Eq.mpr",
"congrA... | ← (MeasurableEmbedding.subtype_coe hs).integrableOn_range_iff_comap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 499,
"column": 4
} | {
"line": 499,
"column": 17
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nl : Filter α\ninst✝ : MeasurableSpace β\ne : α → β\nhe : MeasurableEmbedding e\nf : β → ε\ns : Set β\nhs : s ∈ map e l ∧ IntegrableOn (f ∘ e) (e ⁻¹' s) μ\n⊢... | exact ⟨_, hs⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 499,
"column": 4
} | {
"line": 499,
"column": 17
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nl : Filter α\ninst✝ : MeasurableSpace β\ne : α → β\nhe : MeasurableEmbedding e\nf : β → ε\ns : Set β\nhs : s ∈ map e l ∧ IntegrableOn (f ∘ e) (e ⁻¹' s) μ\n⊢... | exact ⟨_, hs⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntegrableOn | {
"line": 499,
"column": 4
} | {
"line": 499,
"column": 17
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nl : Filter α\ninst✝ : MeasurableSpace β\ne : α → β\nhe : MeasurableEmbedding e\nf : β → ε\ns : Set β\nhs : s ∈ map e l ∧ IntegrableOn (f ∘ e) (e ⁻¹' s) μ\n⊢... | exact ⟨_, hs⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 611,
"column": 41
} | {
"line": 611,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhf : AEStronglyMeasurable f μ\n⊢ AEStronglyMeasurable (fun a ↦ ‖f a‖) μ ∧ HasFiniteIntegral (fun a ↦ ‖f a‖) μ ↔ HasFiniteIntegral f μ",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
... | and_iff_right hf.norm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 868,
"column": 25
} | {
"line": 868,
"column": 28
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nε' : Type u_6\nε'' : Type u_7\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁹ : MeasurableSpace δ\ninst✝⁸ : NormedAddCommGroup β\ninst✝⁷ : NormedAddCommGroup γ\ninst✝⁶ : TopologicalSpace ε\ninst✝⁵ : ContinuousENorm ε\ninst✝⁴ ... | h'' | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 141,
"column": 2
} | {
"line": 144,
"column": 53
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nE : ι → Type wE\nG : Type wG\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : MultilinearMap 𝕜 E G\nhf : Continuous ⇑f\nm : (i : ι) →... | rw [← inseparable_zero_iff_norm] at hi ⊢
have : Inseparable (update m i 0) m := inseparable_pi.2 <|
(forall_update_iff m fun i a ↦ Inseparable a (m i)).2 ⟨hi.symm, fun _ _ ↦ rfl⟩
simpa only [map_update_zero] using this.symm.map hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 141,
"column": 2
} | {
"line": 144,
"column": 53
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nE : ι → Type wE\nG : Type wG\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : SeminormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : MultilinearMap 𝕜 E G\nhf : Continuous ⇑f\nm : (i : ι) →... | rw [← inseparable_zero_iff_norm] at hi ⊢
have : Inseparable (update m i 0) m := inseparable_pi.2 <|
(forall_update_iff m fun i a ↦ Inseparable a (m i)).2 ⟨hi.symm, fun _ _ ↦ rfl⟩
simpa only [map_update_zero] using this.symm.map hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 373,
"column": 2
} | {
"line": 373,
"column": 39
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nε'' : Type u_5\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : TopologicalSpace ε''\ninst✝² : ESeminormedAddMonoid ε''\ninst✝¹ : BorelSpace X\ninst✝ : BorelSpace Y\ne : X ≃ₜ Y\nf : Y → ε''\nμ : Measu... | refine ⟨fun h x => ?_, fun h x => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.L1Space.Integrable | {
"line": 881,
"column": 17
} | {
"line": 881,
"column": 20
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nε : Type u_5\nε' : Type u_6\nε'' : Type u_7\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝⁹ : MeasurableSpace δ\ninst✝⁸ : NormedAddCommGroup β\ninst✝⁷ : NormedAddCommGroup γ\ninst✝⁶ : TopologicalSpace ε\ninst✝⁵ : ContinuousENorm ε\ninst✝⁴ ... | h'' | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.LocallyIntegrable | {
"line": 773,
"column": 2
} | {
"line": 773,
"column": 87
} | [
{
"pp": "X : Type u_1\nE : Type u_6\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : NormedAddCommGroup E\nμ : Measure X\ninst✝⁶ : OpensMeasurableSpace X\ninst✝⁵ : LocallyCompactSpace X\ninst✝⁴ : T2Space X\n𝕜 : Type u_9\ninst✝³ : NormedRing 𝕜\ninst✝² : SecondCountableTopologyEither X E\ninst... | exact fun k hk_sub hk_c => (hf k hk_sub hk_c).smul_continuousOn (hg.mono hk_sub) hk_c | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 470,
"column": 4
} | {
"line": 474,
"column": 44
} | [
{
"pp": "case h\n𝕜 : Type u\nι : Type v\nE : ι → Type wE\nG : Type wG\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝² : SeminormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\ninst✝ : Fintype ι\nA : ∀ (f : ContinuousMultilinear... | calc
‖f x‖ ≤ 1 := hf _ <| (pi_norm_le_iff_of_nonneg (norm_nonneg c)).2 fun i ↦ (hx i).le
_ = ∏ i : ι, 1 := by simp
_ ≤ ∏ i, ‖x i‖ := by gcongr with i; simpa only [div_self hc₀.ne'] using hcx i
_ = 1 * ∏ i, ‖x i‖ := (one_mul _).symm | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 539,
"column": 30
} | {
"line": 540,
"column": 84
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nE : ι → Type wE\nG : Type wG\nG' : Type wG'\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁴ : SeminormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\ninst✝² : SeminormedAddCommGroup G'\ninst✝¹ :... | by
simp only [opNNNorm_le_iff, prod_apply, Prod.nnnorm_def, max_le_iff, forall_and] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 720,
"column": 2
} | {
"line": 720,
"column": 34
} | [
{
"pp": "case a\n𝕜 : Type u\nι : Type v\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : Fintype ι\nA : Type u_1\ninst✝² : NormedCommRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : IsEmpty ι\n⊢ ‖ContinuousMultilinearMap.mkPiAlgebra 𝕜 ι A‖ ≤ ‖1‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"N... | · apply opNorm_le_bound <;> simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Module.Multilinear.Basic | {
"line": 1196,
"column": 60
} | {
"line": 1196,
"column": 71
} | [
{
"pp": "𝕜 : Type u\nι : Type v\nE₁ : ι → Type wE₁\nG : Type wG\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)\ninst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁴ : SeminormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\ninst✝² : Fintype ι\nα : Type u_1\ninst✝¹ : Fintype α\nf... | prod_const, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 18
} | [
{
"pp": "case refine_2\nβ : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : BorelSpace E\nf : β → E\nhp_ne_top : p ≠ ∞\nμ : Measure β\nfmeas : Measurable f\ninst✝ : SeparableSpace ↑(Set.range f ∪ {0})\nhf : eLpNorm f p μ < ∞\n⊢ eL... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 18
} | [
{
"pp": "case refine_2\nβ : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : BorelSpace E\nf : β → E\nhp_ne_top : p ≠ ∞\nμ : Measure β\nfmeas : Measurable f\ninst✝ : SeparableSpace ↑(Set.range f ∪ {0})\nhf : eLpNorm f p μ < ∞\n⊢ eL... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 18
} | [
{
"pp": "case refine_2\nβ : Type u_2\nE : Type u_4\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\np : ℝ≥0∞\ninst✝¹ : BorelSpace E\nf : β → E\nhp_ne_top : p ≠ ∞\nμ : Measure β\nfmeas : Measurable f\ninst✝ : SeparableSpace ↑(Set.range f ∪ {0})\nhf : eLpNorm f p μ < ∞\n⊢ eL... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 564,
"column": 96
} | {
"line": 568,
"column": 24
} | [
{
"pp": "α : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nf : ↥(simpleFunc E p μ)\n⊢ ⇑(toSimpleFunc (-f)) =ᶠ[ae μ] -⇑(toSimpleFunc f)",
"usedConstants": [
"MeasureTheory.ae",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZ... | by
filter_upwards [toSimpleFunc_eq_toFun (-f), toSimpleFunc_eq_toFun f,
Lp.coeFn_neg (f : Lp E p μ)] with _
simp only [Pi.neg_apply, AddSubgroup.coe_neg]
repeat intro h; rw [h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.Extend | {
"line": 262,
"column": 14
} | {
"line": 267,
"column": 59
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_3\nEₗ : Type u_4\nF : Type u_5\nFₗ : Type u_6\ninst✝¹⁵ : NormedDivisionRing 𝕜\ninst✝¹⁴ : NormedDivisionRing 𝕜₂\ninst✝¹³ : AddCommGroup E\ninst✝¹² : NormedAddCommGroup Eₗ\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : NormedAddCommGroup Fₗ\ninst✝⁹ : Module 𝕜 E\ninst✝⁸ ... | by
refine h_dense₁.induction ?_ ?_
· rintro _ ⟨_, rfl⟩
simp [LinearMap.extendOfNorm_eq, h_dense₁, h_norm₁, h_dense₂, h_norm₂]
· exact isClosed_eq (by simp only [AddHom.toFun_eq_coe, LinearMap.coe_toAddHom,
ContinuousLinearMap.coe_coe]; fun_prop) continuous_id | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.L1 | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 23
} | [
{
"pp": "case h\nα : Type u_1\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : PartialOrder F\ninst✝¹ : IsOrderedAddMonoid F\ninst✝ : IsOrderedModule ℝ F\nf : α →ₛ F\nhf : 0 ≤ᶠ[ae μ] ⇑f\ny : α\n⊢ 0 ≤ μ.real (⇑f ⁻¹' {f y}) • f y",
"usedCon... | by_cases hy : 0 ≤ f y | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.MeasureTheory.Function.SimpleFuncDenseLp | {
"line": 875,
"column": 4
} | {
"line": 875,
"column": 50
} | [
{
"pp": "α : Type u_1\nE : Type u_4\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\nμ : Measure α\nhp_ne_top : p ≠ ∞\nP : (α → E) → Prop\nh0P :\n ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g\nh1P : ∀ (f... | rcases H f' η ηpos.ne' f'_mem with ⟨g, hg, Pg⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 79,
"column": 2
} | {
"line": 80,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v : α → β\nl : Filter α\nh : u ~[l] v\n⊢ v =O[l] u",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"HEq.refl",
"Asymptotics.IsBigO",
"HSub.hSub",
"Norm",
"Eq.casesO... | convert h.isLittleO.right_isBigO_add
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 79,
"column": 2
} | {
"line": 80,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu v : α → β\nl : Filter α\nh : u ~[l] v\n⊢ v =O[l] u",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"HEq.refl",
"Asymptotics.IsBigO",
"HSub.hSub",
"Norm",
"Eq.casesO... | convert h.isLittleO.right_isBigO_add
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 115,
"column": 2
} | {
"line": 117,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu : α → β\nl : Filter α\n⊢ u ~[l] 0 ↔ u =O[l] 0",
"usedConstants": [
"Filter.instMembership",
"Asymptotics.isBigO_zero_right_iff",
"Eq.mpr",
"congrArg",
"Asymptotics.IsBigO",
"setOf",
"Asymptotics.is... | refine ⟨IsEquivalent.isBigO, fun h ↦ ?_⟩
rw [isEquivalent_zero_iff_eventually_zero, eventuallyEq_iff_exists_mem]
exact ⟨{ x : α | u x = 0 }, isBigO_zero_right_iff.mp h, fun x hx ↦ hx⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.AsymptoticEquivalent | {
"line": 115,
"column": 2
} | {
"line": 117,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : NormedAddCommGroup β\nu : α → β\nl : Filter α\n⊢ u ~[l] 0 ↔ u =O[l] 0",
"usedConstants": [
"Filter.instMembership",
"Asymptotics.isBigO_zero_right_iff",
"Eq.mpr",
"congrArg",
"Asymptotics.IsBigO",
"setOf",
"Asymptotics.is... | refine ⟨IsEquivalent.isBigO, fun h ↦ ?_⟩
rw [isEquivalent_zero_iff_eventually_zero, eventuallyEq_iff_exists_mem]
exact ⟨{ x : α | u x = 0 }, isBigO_zero_right_iff.mp h, fun x hx ↦ hx⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 500,
"column": 4
} | {
"line": 500,
"column": 58
} | [
{
"pp": "case h.e_a.a\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable f μ\nf₁ : ↥(Lp ℝ 1 μ) := ⋯\neq₁ : (∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ).toReal = ‖Lp.posPart f₁‖\na✝ : α\nh₁ : ↑↑(Lp.negPart f₁) a✝ = -min (↑↑f₁ a✝) 0\nh₂ : ↑↑(Integrable.toL1 f hf) a✝ = f a✝\n⊢ ↑(-f a✝).toNNRe... | simp only [Real.coe_toNNReal', coe_nnnorm, nnnorm_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 674,
"column": 2
} | {
"line": 674,
"column": 17
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : PartialOrder E\ninst✝⁴ : IsOrderedAddMonoid E\ninst✝³ : IsOrderedModule ℝ E\ninst✝² : ClosedIciTopology E\nβ : Type u_6\ninst✝¹ : AddCommMonoid β\ninst✝ : Module ℝ β\nf : ... | refine ⟨hs, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.ContinuousAffineMap | {
"line": 229,
"column": 2
} | {
"line": 239,
"column": 55
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nW : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝¹² : Ring R\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module R V\ninst✝⁹ : TopologicalSpace P\ninst✝⁸ : AddTorsor V P\ninst✝⁷ : AddCommGroup W\ninst✝⁶ : Module R W\ninst✝⁵ : TopologicalSpace Q\ninst✝⁴ : AddTorsor W Q\ninst✝³ : Topolo... | have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine ⟨fun h => ?_, fun h => ?_⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine ⟨fun h =... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.ContinuousAffineMap | {
"line": 229,
"column": 2
} | {
"line": 239,
"column": 55
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nW : Type u_3\nP : Type u_4\nQ : Type u_5\ninst✝¹² : Ring R\ninst✝¹¹ : AddCommGroup V\ninst✝¹⁰ : Module R V\ninst✝⁹ : TopologicalSpace P\ninst✝⁸ : AddTorsor V P\ninst✝⁷ : AddCommGroup W\ninst✝⁶ : Module R W\ninst✝⁵ : TopologicalSpace Q\ninst✝⁴ : AddTorsor W Q\ninst✝³ : Topolo... | have h₁ : f.contLinear = 0 ↔ (f : P →ᵃ[R] Q).linear = 0 := by
refine ⟨fun h => ?_, fun h => ?_⟩ <;> ext
· rw [← coe_contLinear_eq_linear, h]; rfl
· rw [← coe_linear_eq_coe_contLinear, h]; rfl
have h₂ : ∀ q : Q, f = const R P q ↔ (f : P →ᵃ[R] Q) = AffineMap.const R P q := by
intro q
refine ⟨fun h =... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 824,
"column": 4
} | {
"line": 824,
"column": 53
} | [
{
"pp": "case refine_2\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n ↦ f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ -f... | filter_upwards [h_tendsto] with x hx using hx.neg | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 824,
"column": 4
} | {
"line": 824,
"column": 53
} | [
{
"pp": "case refine_2\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n ↦ f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ -f... | filter_upwards [h_tendsto] with x hx using hx.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 824,
"column": 4
} | {
"line": 824,
"column": 53
} | [
{
"pp": "case refine_2\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf : ∀ (n : ℕ), Integrable (f n) μ\nhF : Integrable F μ\nh_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n ↦ f n x\nh_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))\n⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ -f... | filter_upwards [h_tendsto] with x hx using hx.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 859,
"column": 2
} | {
"line": 862,
"column": 61
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : ℕ → α → ℝ\nF : α → ℝ\nhf_int : ∀ (n : ℕ), Integrable (f n) μ\nhF_int : Integrable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a\nhf_bound : ∀ᵐ (a : α) ∂μ, ∀ ... | have h_bound : ∀ᵐ a ∂μ, ∀ i, f' i a ≤ F' a := by
filter_upwards [hf_bound] with a ha_bound i
refine ENNReal.ofReal_le_ofReal ?_
simp only [tsub_le_iff_right, sub_add_cancel, ha_bound i] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nhF : ∃ x, x ∉ F\nRpos : 0 < R\nthis : ‖c‖ / R < 1\nx : E\nxF : x ∉ F\nhx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖\n⊢ ... | have x0 : x ≠ 0 := fun H => by simp [H] at xF | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 113,
"column": 53
} | {
"line": 113,
"column": 89
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nhF : ∃ x, x ∉ F\nRpos : 0 < R\nthis : ‖c‖ / R < 1\nx : E\nxF : x ∉ F\nhx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖\nx0... | simp [y', Submodule.smul_mem _ _ hy] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 113,
"column": 53
} | {
"line": 113,
"column": 89
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nhF : ∃ x, x ∉ F\nRpos : 0 < R\nthis : ‖c‖ / R < 1\nx : E\nxF : x ∉ F\nhx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖\nx0... | simp [y', Submodule.smul_mem _ _ hy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.RieszLemma | {
"line": 113,
"column": 53
} | {
"line": 113,
"column": 89
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nF : Subspace 𝕜 E\nhFc : IsClosed ↑F\nhF : ∃ x, x ∉ F\nRpos : 0 < R\nthis : ‖c‖ / R < 1\nx : E\nxF : x ∉ F\nhx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖\nx0... | simp [y', Submodule.smul_mem _ _ hy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Indicator | {
"line": 30,
"column": 37
} | {
"line": 31,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\nf : α → β\ns : Set α\ninst✝ : One β\nhs : ∀ a ∈ frontier s, f a = 1\nhf : Continuous f\n⊢ Continuous (s.mulIndicator f)",
"usedConstants": [
"continuous_const",
"Classical.propDecidable",
"Member... | by
classical exact hf.piecewise hs continuous_const | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 127,
"column": 2
} | {
"line": 132,
"column": 92
} | [
{
"pp": "R : Type u_1\nX : Type u_2\ninst✝² : PseudoMetricSpace R\ninst✝¹ : Mul R\ninst✝ : BoundedMul R\nf g : X → R\nf_bdd : ∃ C, ∀ (x y : X), dist (f x) (f y) ≤ C\ng_bdd : ∃ C, ∀ (x y : X), dist (g x) (g y) ≤ C\n⊢ ∃ C, ∀ (x y : X), dist ((f * g) x) ((f * g) y) ≤ C",
"usedConstants": [
"Set.mem_range... | obtain ⟨C, hC⟩ := Metric.isBounded_iff.mp <|
isBounded_mul (Metric.isBounded_range_iff.mpr f_bdd) (Metric.isBounded_range_iff.mpr g_bdd)
use C
intro x y
exact hC (Set.mul_mem_mul (Set.mem_range_self (f := f) x) (Set.mem_range_self (f := g) x))
(Set.mul_mem_mul (Set.mem_range_self (f := f) y) (Set.m... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 127,
"column": 2
} | {
"line": 132,
"column": 92
} | [
{
"pp": "R : Type u_1\nX : Type u_2\ninst✝² : PseudoMetricSpace R\ninst✝¹ : Mul R\ninst✝ : BoundedMul R\nf g : X → R\nf_bdd : ∃ C, ∀ (x y : X), dist (f x) (f y) ≤ C\ng_bdd : ∃ C, ∀ (x y : X), dist (g x) (g y) ≤ C\n⊢ ∃ C, ∀ (x y : X), dist ((f * g) x) ((f * g) y) ≤ C",
"usedConstants": [
"Set.mem_range... | obtain ⟨C, hC⟩ := Metric.isBounded_iff.mp <|
isBounded_mul (Metric.isBounded_range_iff.mpr f_bdd) (Metric.isBounded_range_iff.mpr g_bdd)
use C
intro x y
exact hC (Set.mul_mem_mul (Set.mem_range_self (f := f) x) (Set.mem_range_self (f := g) x))
(Set.mul_mem_mul (Set.mem_range_self (f := f) y) (Set.m... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1210,
"column": 8
} | {
"line": 1211,
"column": 21
} | [
{
"pp": "case h.e'_2.hp_ne_zero\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_6\ninst✝ : NormedAddCommGroup E\nf g : α → E\np q : ℝ\nhpq : p.HolderConjugate q\nhf : MemLp f (ENNReal.ofReal p) μ\nhg : MemLp g (ENNReal.ofReal q) μ\nh_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a :... | rw [Ne, ENNReal.ofReal_eq_zero, not_le]
exact hpq.pos | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1210,
"column": 8
} | {
"line": 1211,
"column": 21
} | [
{
"pp": "case h.e'_2.hp_ne_zero\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_6\ninst✝ : NormedAddCommGroup E\nf g : α → E\np q : ℝ\nhpq : p.HolderConjugate q\nhf : MemLp f (ENNReal.ofReal p) μ\nhg : MemLp g (ENNReal.ofReal q) μ\nh_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a :... | rw [Ne, ENNReal.ofReal_eq_zero, not_le]
exact hpq.pos | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 743,
"column": 4
} | {
"line": 744,
"column": 36
} | [
{
"pp": "case neg\nα : 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\nf : α →... | rw [setToFun_undef hT hf, setToFun_undef hT, neg_zero]
rwa [← integrable_neg_iff] at hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 743,
"column": 4
} | {
"line": 744,
"column": 36
} | [
{
"pp": "case neg\nα : 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\nf : α →... | rw [setToFun_undef hT hf, setToFun_undef hT, neg_zero]
rwa [← integrable_neg_iff] at hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 143,
"column": 2
} | {
"line": 146,
"column": 28
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf g : α →ᵇ β\n⊢ ∃ C, 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C",
"usedConstants": [
"le_max_right",
"Set.mem_range_self",
"Real.instLE",
"Real",
"PseudoMetricSpace.toBornology",
"Real... | rcases isBounded_iff.1 (f.isBounded_range.union g.isBounded_range) with ⟨C, hC⟩
refine ⟨max 0 C, le_max_left _ _, fun x => (hC ?_ ?_).trans (le_max_right _ _)⟩
<;> [left; right]
<;> apply mem_range_self | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 143,
"column": 2
} | {
"line": 146,
"column": 28
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf g : α →ᵇ β\n⊢ ∃ C, 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C",
"usedConstants": [
"le_max_right",
"Set.mem_range_self",
"Real.instLE",
"Real",
"PseudoMetricSpace.toBornology",
"Real... | rcases isBounded_iff.1 (f.isBounded_range.union g.isBounded_range) with ⟨C, hC⟩
refine ⟨max 0 C, le_max_left _ _, fun x => (hC ?_ ?_).trans (le_max_right _ _)⟩
<;> [left; right]
<;> apply mem_range_self | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1075,
"column": 2
} | {
"line": 1078,
"column": 30
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : CompleteSpace F\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_7\nl ... | have h :
{ x : ι | (fun n => AEStronglyMeasurable (fs n) μ) x } ∩
{ x : ι | (fun n => ∀ᵐ a ∂μ, ‖fs n a‖ ≤ bound a) x } ∈ l :=
inter_mem hfs_meas h_bound | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 138,
"column": 4
} | {
"line": 141,
"column": 28
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : PseudoEMetricSpace α\nδseq : ℕ → ℝ\nδseq_lim : Tendsto δseq atTop (𝓝 0)\nE : Set α\nx : α\nx_mem_closure : x ∈ closure E\n⊢ Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop (𝓝 ((closure E).indicator (fun x ↦ 1) x))",
"usedConstants": [
"Eq.mpr",
... | simp_rw [thickenedIndicatorAux_one_of_mem_closure _ E x_mem_closure]
rw [show (indicator (closure E) fun _ => (1 : ℝ≥0∞)) x = 1 by
simp only [x_mem_closure, indicator_of_mem]]
exact tendsto_const_nhds | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 138,
"column": 4
} | {
"line": 141,
"column": 28
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : PseudoEMetricSpace α\nδseq : ℕ → ℝ\nδseq_lim : Tendsto δseq atTop (𝓝 0)\nE : Set α\nx : α\nx_mem_closure : x ∈ closure E\n⊢ Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop (𝓝 ((closure E).indicator (fun x ↦ 1) x))",
"usedConstants": [
"Eq.mpr",
... | simp_rw [thickenedIndicatorAux_one_of_mem_closure _ E x_mem_closure]
rw [show (indicator (closure E) fun _ => (1 : ℝ≥0∞)) x = 1 by
simp only [x_mem_closure, indicator_of_mem]]
exact tendsto_const_nhds | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 66
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ₁ δ₂ : ℝ\nδ₁_pos : 0 < δ₁\nδ₂_pos : 0 < δ₂\nhle : δ₁ ≤ δ₂\nE : Set α\nx : α\n⊢ (thickenedIndicator δ₁_pos E) x ≤ (thickenedIndicator δ₂_pos E) x",
"usedConstants": [
"Iff.mpr",
"thickenedIndicatorAux_lt_top",
"ENNReal.toNNReal_le_toNNRe... | apply (toNNReal_le_toNNReal (by finiteness) (by finiteness)).mpr | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.Marginal | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 71
} | [
{
"pp": "case pos\nδ : Type u_1\nX : δ → Type u_3\ninst✝² : (i : δ) → MeasurableSpace (X i)\nμ : (i : δ) → Measure (X i)\ninst✝¹ : DecidableEq δ\ns : Finset δ\nf : ((i : δ) → X i) → ℝ≥0∞\ninst✝ : ∀ (i : δ), SigmaFinite (μ i)\nhf : Measurable f\ni : δ\nhi : i ∈ s\n⊢ Measurable fun x ↦ updateFinset x.1 s x.2 i",
... | simpa [hi, updateFinset] using measurable_pi_iff.1 measurable_snd _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Integral.Marginal | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 71
} | [
{
"pp": "case pos\nδ : Type u_1\nX : δ → Type u_3\ninst✝² : (i : δ) → MeasurableSpace (X i)\nμ : (i : δ) → Measure (X i)\ninst✝¹ : DecidableEq δ\ns : Finset δ\nf : ((i : δ) → X i) → ℝ≥0∞\ninst✝ : ∀ (i : δ), SigmaFinite (μ i)\nhf : Measurable f\ni : δ\nhi : i ∈ s\n⊢ Measurable fun x ↦ updateFinset x.1 s x.2 i",
... | simpa [hi, updateFinset] using measurable_pi_iff.1 measurable_snd _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Marginal | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 71
} | [
{
"pp": "case pos\nδ : Type u_1\nX : δ → Type u_3\ninst✝² : (i : δ) → MeasurableSpace (X i)\nμ : (i : δ) → Measure (X i)\ninst✝¹ : DecidableEq δ\ns : Finset δ\nf : ((i : δ) → X i) → ℝ≥0∞\ninst✝ : ∀ (i : δ), SigmaFinite (μ i)\nhf : Measurable f\ni : δ\nhi : i ∈ s\n⊢ Measurable fun x ↦ updateFinset x.1 s x.2 i",
... | simpa [hi, updateFinset] using measurable_pi_iff.1 measurable_snd _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 56
} | [
{
"pp": "f : StieltjesFunction ℝ\nx : ℝ\n⊢ ⨅ r, ↑f ↑↑r = ⨅ r, ↑f ↑r",
"usedConstants": [
"Real.iInf_Ioi_eq_iInf_rat_gt",
"Real",
"Set.Ioi",
"iInf",
"Real.instRatCast",
"Rat",
"PseudoMetricSpace.toUniformSpace",
"Real.instLT",
"Membership.mem",
"Sti... | refine (Real.iInf_Ioi_eq_iInf_rat_gt _ ?_ f.mono).symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 404,
"column": 4
} | {
"line": 404,
"column": 63
} | [
{
"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 : (𝓝[>] a).NeBot := nhdsGT_neBot_of_exists_gt ⟨b, hab⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\n⊢ ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Inner.i... | simp only [inner_sub_left, inner_sub_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 694,
"column": 2
} | {
"line": 694,
"column": 61
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : LinearOrder R\ninst✝⁷ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁶ : OrderTopology R\ninst✝⁵ : CompactIccSpace R\ninst✝⁴ : MeasurableSpace R\ninst✝³ : BorelSpace R\ninst✝² : SecondCountableTopology R\ninst✝¹ : DenselyOrdered R\ninst✝ : Nonempty R\nl u : ℝ\nhfl : Tendsto ... | refine tendsto_nhds_unique (tendsto_measure_Iic_atTop _) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 750,
"column": 4
} | {
"line": 750,
"column": 18
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\nl : ℝ\nhfg ... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 750,
"column": 4
} | {
"line": 750,
"column": 18
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\nl : ℝ\nhfg ... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 750,
"column": 4
} | {
"line": 750,
"column": 18
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\nl : ℝ\nhfg ... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 764,
"column": 6
} | {
"line": 764,
"column": 20
} | [
{
"pp": "case h.inl\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\n... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 764,
"column": 6
} | {
"line": 764,
"column": 20
} | [
{
"pp": "case h.inl\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\n... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 764,
"column": 6
} | {
"line": 764,
"column": 20
} | [
{
"pp": "case h.inl\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\n... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 772,
"column": 6
} | {
"line": 772,
"column": 20
} | [
{
"pp": "case h.inr\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\n... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 772,
"column": 6
} | {
"line": 772,
"column": 20
} | [
{
"pp": "case h.inr\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\n... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 772,
"column": 6
} | {
"line": 772,
"column": 20
} | [
{
"pp": "case h.inr\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\n... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 100,
"column": 47
} | {
"line": 100,
"column": 62
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : SeminormedAddCommGroup E\ninst✝⁶ : InnerProductSpace 𝕜 E\n𝕝 : Type u_4\ninst✝⁵ : CommSemiring 𝕝\ninst✝⁴ : StarRing 𝕝\ninst✝³ : Algebra 𝕝 𝕜\ninst✝² : Module 𝕝 E\ninst✝¹ : IsScalarTower 𝕝 𝕜 E\ninst✝ : StarModule 𝕝 𝕜\nx y : E\nr : 𝕝\n⊢ ... | inner_conj_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 266,
"column": 2
} | {
"line": 266,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\n⊢ ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"SeminormedAddGroup.toA... | simp only [inner_sub_left, inner_sub_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 567,
"column": 49
} | {
"line": 570,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nh : ⟪x, y⟫ = 0\n⊢ ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"Real.partialOr... | by
rw [@norm_add_mul_self 𝕜, add_right_cancel_iff, add_eq_left, mul_eq_zero]
apply Or.inr
simp only [h, zero_re] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 638,
"column": 4
} | {
"line": 638,
"column": 13
} | [
{
"pp": "F : Type u_3\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nι₁ : Type u_4\ns₁ : Finset ι₁\nw₁ : ι₁ → ℝ\nv₁ : ι₁ → F\nh₁ : ∑ i ∈ s₁, w₁ i = 0\nι₂ : Type u_5\ns₂ : Finset ι₂\nw₂ : ι₂ → ℝ\nv₂ : ι₂ → F\nh₂ : ∑ i ∈ s₂, w₂ i = 0\n⊢ ∑ x ∈ s₁, w₁ x * 0 + 0 - ∑ x ∈ s₁, w₁ x * ∑ i ∈ s₂, w₂ i ... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
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