module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 224,
"column": 4
} | {
"line": 226,
"column": 74
} | [
{
"pp": "case refine_1\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nh : ∀ᵐ (a : α) ∂μ, f a ≤ g a\nt : Set α\nhts : {x | (fun a ↦ f a ≤ g a) x}ᶜ ⊆ t\nht : MeasurableSet t\nht0 : μ t = 0\nthis : ∀ᵐ (x : α) ∂μ, x ∉ t\ns : α →ₛ ℝ≥0∞\nhfs : ⇑s ≤ fun a ↦ f a\na : α\n⊢ (s.restrict tᶜ) a ≤ (fun ... | by_cases h : a ∈ t <;>
simp only [restrict_apply s ht.compl, mem_compl_iff, h, not_true, not_false_eq_true,
indicator_of_notMem, zero_le, not_false_eq_true, indicator_of_mem] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 722,
"column": 42
} | {
"line": 722,
"column": 53
} | [
{
"pp": "α : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\ng g' : δ → α\nhf : ∀ (i : ι), Measurabl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 740,
"column": 6
} | {
"line": 740,
"column": 21
} | [
{
"pp": "case pos\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\ng g' : δ → α\nhf : ∀ (i : ι),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 840,
"column": 79
} | {
"line": 840,
"column": 90
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝³ : BorelSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nh : ∀ x ∈ s, s ∈ 𝓝[>] x\nx₀ : α\nx₀s : x₀ ∈ s\nh₀ : IsTop x₀\nthis : s = {x₀} ∪ s \\ {x₀}\nx : α\nhx : x ∈ s \\ {x₀}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 865,
"column": 4
} | {
"line": 865,
"column": 25
} | [
{
"pp": "α : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (i : ι), Measurable (f i)\nhα : ... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 331,
"column": 2
} | {
"line": 331,
"column": 29
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhg : AEMeasurable g μ\n⊢ ∫⁻ (a : α), f a + g a ∂μ = ∫⁻ (a : α), f a ∂μ + ∫⁻ (a : α), g a ∂μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 84
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nc : R\nf : α → ℝ≥0∞\n⊢ ∫⁻ (a : α), f a ∂c • μ = c • ∫⁻ (a : α), f a ∂μ",
"usedConstants": [
"MeasureTheory.lintegral_def",
"MeasureTheory.SimpleFunc.lintegral",
... | simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_smul, ENNReal.smul_iSup] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 84
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nc : R\nf : α → ℝ≥0∞\n⊢ ∫⁻ (a : α), f a ∂c • μ = c • ∫⁻ (a : α), f a ∂μ",
"usedConstants": [
"MeasureTheory.lintegral_def",
"MeasureTheory.SimpleFunc.lintegral",
... | simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_smul, ENNReal.smul_iSup] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 84
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nR : Type u_4\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nc : R\nf : α → ℝ≥0∞\n⊢ ∫⁻ (a : α), f a ∂c • μ = c • ∫⁻ (a : α), f a ∂μ",
"usedConstants": [
"MeasureTheory.lintegral_def",
"MeasureTheory.SimpleFunc.lintegral",
... | simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_smul, ENNReal.smul_iSup] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 912,
"column": 4
} | {
"line": 912,
"column": 25
} | [
{
"pp": "case inl\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (... | simp [iSup_of_empty'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 912,
"column": 4
} | {
"line": 912,
"column": 25
} | [
{
"pp": "case inl\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (... | simp [iSup_of_empty'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 912,
"column": 4
} | {
"line": 912,
"column": 25
} | [
{
"pp": "case inl\nα : Type u_1\nδ : Type u_4\ninst✝⁵ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝⁴ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : SecondCountableTopology α\nι : Sort u_5\ninst✝ : Countable ι\nf : ι → δ → α\nhf : ∀ (... | simp [iSup_of_empty'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 493,
"column": 2
} | {
"line": 493,
"column": 17
} | [
{
"pp": "case neg.e_a.h.e_6.h.h\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\ng : α →ₛ ℝ≥0∞\nhg : ⇑g ≤ fun a ↦ s.indicator f a\nthis : ⇑g ≤ f\nx : α\nH : x ∉ s\n⊢ g x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 429,
"column": 2
} | {
"line": 429,
"column": 44
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhr : r ≠ ∞\nh : ¬r = 0\nrinv : r * r⁻¹ = 1\nrinv' : r⁻¹ * r = 1\nthis : r⁻¹ * ∫⁻ (a : α), r * f a ∂μ ≤ ∫⁻ (a : α), 1 * f a ∂μ\n⊢ ∫⁻ (a : α), r * f a ∂μ ≤ r * ∫⁻ (a : α), f a ∂μ",
"usedConstants": [
"Eq.mpr",
"HM... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 981,
"column": 4
} | {
"line": 981,
"column": 23
} | [
{
"pp": "α : Type u_1\nδ : Type u_4\ninst✝⁴ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝³ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nι : Type u_5\nμ : Measure δ\ns : Set ι\nf : ι → δ → α\nhs : s.Countab... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 984,
"column": 18
} | {
"line": 984,
"column": 37
} | [
{
"pp": "α : Type u_1\nδ : Type u_4\ninst✝⁴ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝³ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nι : Type u_5\nμ : Measure δ\ns : Set ι\nf : ι → δ → α\nhs : s.Countab... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 472,
"column": 8
} | {
"line": 472,
"column": 74
} | [
{
"pp": "case refine_3\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf✝ : α → ℝ≥0∞\nhf✝ : Measurable f✝\nf : ℕ → α → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nhf_mono : Monotone f\nhf_prop : ∀ (n : ℕ), (fun f ↦ ∫⁻ (a : α), f a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ) (f n)\n⊢ ⨆ n, ∫⁻ (a : α), f n a... | lintegral_iSup (fun n => Measurable.mono (hf n) hm le_rfl) hf_mono | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 486,
"column": 2
} | {
"line": 486,
"column": 19
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf : α → ℝ≥0∞\nhf : AEMeasurable f (μ.trim hm)\ns : Set α\nhs : MeasurableSet s\n⊢ AEMeasurable f ((μ.trim hm).restrict s)",
"usedConstants": [
"MeasureTheory.Measure.trim",
"ENNReal.measurableSpace",
"AEMeasurable... | exact hf.restrict | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 1027,
"column": 2
} | {
"line": 1027,
"column": 79
} | [
{
"pp": "case neg\nα : Type u_1\nδ : Type u_4\ninst✝⁴ : TopologicalSpace α\nmα : MeasurableSpace α\ninst✝³ : BorelSpace α\nmδ : MeasurableSpace δ\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\nι : Type u_5\nι' : Type u_6\nf : ι → δ → α\nv : Filter ι\nh... | let g : ℕ → Subtype p := Classical.choose (exists_surjective_nat (Subtype p)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 43
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : MeasurableSpace α\nf : α →ₛ β\np : β → Prop\n⊢ (∃ y ∈ f.range, p y) ↔ ∃ x, p (f x)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"MeasureTheory.SimpleFunc",
"Membership.mem",
"Exists",
"id",
"MeasureTheory.Si... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 13
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝ : DecidableEq α\ns : Finset ι\nS : ι → Finset α\n⊢ ↑(#(s.biUnion S)) = ∑ t, (-1) ^ (#↑t + 1) * ↑(#((↑t).inf' ⋯ S))",
"usedConstants": [
"Int.instAddCommMonoid",
"HMul.hMul",
"Finset.univ",
"Finset.inf'",
"Finset",
"Membership.mem... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 13
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ns : Finset ι\nS : ι → Finset α\n⊢ ↑(#(s.inf fun i ↦ (S i)ᶜ)) = ∑ t ∈ s.powerset, (-1) ^ #t * ↑(#(t.inf S))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 723,
"column": 2
} | {
"line": 726,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : SemilatticeSup β\ninst✝ : OrderBot β\nf : γ → α →ₛ β\ns : Finset γ\na : α\n⊢ (s.sup f) a = s.sup fun c ↦ (f c) a",
"usedConstants": [
"Eq.mpr",
"Finset.sup_insert",
"congrArg",
"MeasureTheory.Simp... | classical
refine Finset.induction_on s rfl ?_
intro a s _ ih
rw [Finset.sup_insert, Finset.sup_insert, sup_apply, ih] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 723,
"column": 2
} | {
"line": 726,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : SemilatticeSup β\ninst✝ : OrderBot β\nf : γ → α →ₛ β\ns : Finset γ\na : α\n⊢ (s.sup f) a = s.sup fun c ↦ (f c) a",
"usedConstants": [
"Eq.mpr",
"Finset.sup_insert",
"congrArg",
"MeasureTheory.Simp... | classical
refine Finset.induction_on s rfl ?_
intro a s _ ih
rw [Finset.sup_insert, Finset.sup_insert, sup_apply, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 723,
"column": 2
} | {
"line": 726,
"column": 58
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : SemilatticeSup β\ninst✝ : OrderBot β\nf : γ → α →ₛ β\ns : Finset γ\na : α\n⊢ (s.sup f) a = s.sup fun c ↦ (f c) a",
"usedConstants": [
"Eq.mpr",
"Finset.sup_insert",
"congrArg",
"MeasureTheory.Simp... | classical
refine Finset.induction_on s rfl ?_
intro a s _ ih
rw [Finset.sup_insert, Finset.sup_insert, sup_apply, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Defs | {
"line": 325,
"column": 6
} | {
"line": 325,
"column": 59
} | [
{
"pp": "case a\n𝓕 : Type u_1\nα : Type u_2\nι : Type u_3\nκ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝² : Norm E\ninst✝¹ : Group E\ninst✝ : PseudoMetricSpace E\nh₁ : ∀ (x : E), ‖x‖ = dist 1 x\nh₂ : ∀ (x y z : E), dist x y ≤ dist (z * x) (z * y)\nx y : E\n⊢ dist x y ≤ dist 1 (x⁻¹ * y)",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 787,
"column": 34
} | {
"line": 787,
"column": 85
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : Zero β\nr : β\ns : Set α\nf : α →ₛ β\nhr : r ∈ (f.restrict s).range\nh0 : r ≠ 0\nhs : MeasurableSet s\n⊢ r ∈ ⇑f '' s",
"usedConstants": [
"Eq.mpr",
"Set.mem_image._simp_1",
"MeasureTheory.SimpleFunc",
"Membershi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Defs | {
"line": 326,
"column": 6
} | {
"line": 326,
"column": 70
} | [
{
"pp": "case a\n𝓕 : Type u_1\nα : Type u_2\nι : Type u_3\nκ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝² : Norm E\ninst✝¹ : Group E\ninst✝ : PseudoMetricSpace E\nh₁ : ∀ (x : E), ‖x‖ = dist 1 x\nh₂ : ∀ (x y z : E), dist x y ≤ dist (z * x) (z * y)\nx y : E\n⊢ dist 1 (x⁻¹ * y) ≤ dist x y",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Defs | {
"line": 337,
"column": 6
} | {
"line": 337,
"column": 70
} | [
{
"pp": "case a\n𝓕 : Type u_1\nα : Type u_2\nι : Type u_3\nκ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝² : Norm E\ninst✝¹ : Group E\ninst✝ : PseudoMetricSpace E\nh₁ : ∀ (x : E), ‖x‖ = dist 1 x\nh₂ : ∀ (x y z : E), dist (z * x) (z * y) ≤ dist x y\nx y : E\n⊢ dist x y ≤ dist 1 (x⁻¹ * y)",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Defs | {
"line": 338,
"column": 6
} | {
"line": 338,
"column": 59
} | [
{
"pp": "case a\n𝓕 : Type u_1\nα : Type u_2\nι : Type u_3\nκ : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝² : Norm E\ninst✝¹ : Group E\ninst✝ : PseudoMetricSpace E\nh₁ : ∀ (x : E), ‖x‖ = dist 1 x\nh₂ : ∀ (x y z : E), dist (z * x) (z * y) ≤ dist x y\nx y : E\n⊢ dist 1 (x⁻¹ * y) ≤ dist x y",
"u... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 888,
"column": 2
} | {
"line": 897,
"column": 46
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ ⨆ n, (eapprox f n) a = f a",
"usedConstants": [
"MeasureTheory.SimpleFunc.eapprox.eq_1",
"Rat.instOfNat",
"Eq.mpr",
"lt_of_le_of_lt",
"False",
"Real",
"ENNReal.ofNNReal",
... | rw [eapprox, iSup_approx_apply ennrealRatEmbed f a hf rfl]
refine le_antisymm (iSup_le fun i => iSup_le fun hi => hi) (le_of_not_gt ?_)
intro h
rcases ENNReal.lt_iff_exists_rat_btwn.1 h with ⟨q, _, lt_q, q_lt⟩
have :
(Real.toNNReal q : ℝ≥0∞) ≤ ⨆ (k : ℕ) (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k := by... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 888,
"column": 2
} | {
"line": 897,
"column": 46
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ ⨆ n, (eapprox f n) a = f a",
"usedConstants": [
"MeasureTheory.SimpleFunc.eapprox.eq_1",
"Rat.instOfNat",
"Eq.mpr",
"lt_of_le_of_lt",
"False",
"Real",
"ENNReal.ofNNReal",
... | rw [eapprox, iSup_approx_apply ennrealRatEmbed f a hf rfl]
refine le_antisymm (iSup_le fun i => iSup_le fun hi => hi) (le_of_not_gt ?_)
intro h
rcases ENNReal.lt_iff_exists_rat_btwn.1 h with ⟨q, _, lt_q, q_lt⟩
have :
(Real.toNNReal q : ℝ≥0∞) ≤ ⨆ (k : ℕ) (_ : ennrealRatEmbed k ≤ f a), ennrealRatEmbed k := by... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 900,
"column": 2
} | {
"line": 900,
"column": 26
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ⨆ n, ⇑(eapprox f n) = f",
"usedConstants": [
"Eq.mpr",
"congrArg",
"iSup",
"MeasureTheory.SimpleFunc",
"id",
"ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice",
"Meas... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 272,
"column": 14
} | {
"line": 272,
"column": 42
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝² : Group E\ninst✝¹ : Group F\ninst✝ : Group G\np✝ q : GroupSeminorm E\nf : F →* E\ns : Set (GroupSeminorm E)\nh : BddAbove s\nx y : E\nhs : s.Nonempty\nthis : Nonempty ↑s\np : ↑s\n⊢ BddAbove (range ((fun x_1 ↦ x_1 x) ∘ Subtype... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 272,
"column": 14
} | {
"line": 272,
"column": 42
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝² : Group E\ninst✝¹ : Group F\ninst✝ : Group G\np✝ q : GroupSeminorm E\nf : F →* E\ns : Set (GroupSeminorm E)\nh : BddAbove s\nx y : E\nhs : s.Nonempty\nthis : Nonempty ↑s\np : ↑s\n⊢ BddAbove (range ((fun x ↦ x y) ∘ Subtype.val... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 413,
"column": 8
} | {
"line": 413,
"column": 71
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝¹ : CommGroup E\ninst✝ : CommGroup F\np✝ q✝ : GroupSeminorm E\nx✝ : E\np q : GroupSeminorm E\nx : E\n⊢ p 1 + q (x / 1) ≤ (fun f ↦ ⇑f) q x",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1015,
"column": 2
} | {
"line": 1015,
"column": 34
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nR : Type u_5\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nf : α →ₛ ℝ≥0∞\nc : R\n⊢ f.lintegral (c • μ) = c • f.lintegral μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 467,
"column": 4
} | {
"line": 467,
"column": 84
} | [
{
"pp": "R : Type u_1\nE : Type u_3\ninst✝³ : AddGroup E\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\nr : R\np q : AddGroupSeminorm E\nx y : ℝ\n⊢ r • max x y = max (r • x) (r • y)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 547,
"column": 14
} | {
"line": 547,
"column": 42
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝ : AddGroup E\np✝ q : NonarchAddGroupSeminorm E\ns : Set (NonarchAddGroupSeminorm E)\nh : BddAbove s\nx y : E\nhs : s.Nonempty\nthis : Nonempty ↑s\np : ↑s\n⊢ BddAbove (range ((fun x_1 ↦ x_1 x) ∘ Subtype.val))",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 547,
"column": 14
} | {
"line": 547,
"column": 42
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝ : AddGroup E\np✝ q : NonarchAddGroupSeminorm E\ns : Set (NonarchAddGroupSeminorm E)\nh : BddAbove s\nx y : E\nhs : s.Nonempty\nthis : Nonempty ↑s\np : ↑s\n⊢ BddAbove (range ((fun x ↦ x y) ∘ Subtype.val))",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1080,
"column": 2
} | {
"line": 1080,
"column": 76
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α →ₛ ℝ≥0∞\nh : f ≤ g\n⊢ f.lintegral μ ≤ g.lintegral μ",
"usedConstants": [
"MeasureTheory.SimpleFunc.lintegral",
"PartialOrder.toPreorder",
"MeasureTheory.SimpleFunc",
"ENNReal",
"ENNReal.instPartialOrder",
... | refine Monotone.of_left_le_map_sup (f := (lintegral · μ)) (fun f g ↦ ?_) h | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1200,
"column": 6
} | {
"line": 1200,
"column": 17
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nβ : Type u_5\ninst✝ : AddZeroClass β\nf g : α →ₛ β\nhf : f.FinMeasSupp μ\nhg : g.FinMeasSupp μ\n⊢ (f + g).FinMeasSupp μ",
"usedConstants": [
"MeasureTheory.SimpleFunc.instAdd",
"Eq.mpr",
"congrArg",
"MeasureTheory.SimpleFun... | add_eq_map₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 654,
"column": 4
} | {
"line": 654,
"column": 84
} | [
{
"pp": "R : Type u_1\nE : Type u_3\ninst✝³ : Group E\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\nr : R\np q : GroupSeminorm E\nx y : ℝ\n⊢ r • max x y = max (r • x) (r • y)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 708,
"column": 4
} | {
"line": 708,
"column": 84
} | [
{
"pp": "R : Type u_1\nE : Type u_3\ninst✝³ : AddGroup E\ninst✝² : SMul R ℝ\ninst✝¹ : SMul R ℝ≥0\ninst✝ : IsScalarTower R ℝ≥0 ℝ\nr : R\np q : NonarchAddGroupSeminorm E\nx y : ℝ\n⊢ r • max x y = max (r • x) (r • y)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Real | {
"line": 115,
"column": 61
} | {
"line": 116,
"column": 55
} | [
{
"pp": "r : ℝ\n⊢ ENNReal.ofReal r ≤ ‖r‖ₑ",
"usedConstants": [
"Eq.mpr",
"Real",
"le_abs_self",
"Real.lattice",
"ENNReal.ofReal",
"abs",
"congrArg",
"SeminormedAddGroup.toNNNorm",
"ENNReal.ofReal_le_ofReal",
"id",
"Real.normedAddCommGroup",
... | by
rw [enorm_eq_ofReal_abs]; gcongr; exact le_abs_self _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Int | {
"line": 24,
"column": 4
} | {
"line": 24,
"column": 37
} | [
{
"pp": "α : Type u_1\nm n : ℤ\n⊢ |↑m - ↑n| = |-↑m + ↑n|",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"Real.lattice",
"AddMonoid.toAddSemigroup",
"AddGroupWithOne.toAddGroup",
"abs",
"neg_... | rw [abs_sub_comm, neg_add_eq_sub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Group.Int | {
"line": 55,
"column": 49
} | {
"line": 55,
"column": 79
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : SeminormedCommGroup α\na : α\nn : ℕ\n⊢ ‖a ^ ↑n‖ ≤ ‖↑n‖ * ‖a‖",
"usedConstants": [
"zpow_natCast",
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"congrArg",
"DivInvMonoid.toZPow",
"Int.norm_natCast",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Int | {
"line": 55,
"column": 49
} | {
"line": 55,
"column": 79
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : SeminormedCommGroup α\na : α\nn : ℕ\n⊢ ‖a ^ (-↑n)‖ ≤ ‖-↑n‖ * ‖a‖",
"usedConstants": [
"zpow_natCast",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Int | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 56
} | [
{
"pp": "α : Type u_1\ninst✝ : SeminormedCommGroup α\nn : ℤ\na : α\n⊢ ‖a ^ n‖₊ ≤ ‖n‖₊ * ‖a‖₊",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"SeminormedAddGroup.toNNNorm",
"NNNorm.nnnorm",
"PartialOrder.toPreorder",
"DivInvMonoid.toZPow",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sequences | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 30
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\nh :\n ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a\ns : Set X\nx : X\nhcx : x ∈ closure[inst✝] s\nhx : x ∈ s\n⊢ x ∈ seqClosure s",
"usedConstants": [
"subset_seqClos... | exact subset_seqClosure hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Sequences | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 30
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\nh :\n ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a\ns : Set X\nx : X\nhcx : x ∈ closure[inst✝] s\nhx : x ∈ s\n⊢ x ∈ seqClosure s",
"usedConstants": [
"subset_seqClos... | exact subset_seqClosure hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sequences | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 30
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\nh :\n ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a\ns : Set X\nx : X\nhcx : x ∈ closure[inst✝] s\nhx : x ∈ s\n⊢ x ∈ seqClosure s",
"usedConstants": [
"subset_seqClos... | exact subset_seqClosure hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sequences | {
"line": 140,
"column": 6
} | {
"line": 142,
"column": 32
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nh :\n ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 (f a))) → ContinuousAt f a\ns : Set X\nx : X\nhcx : x ∈ closure[inst✝] s\nhx : x ∉ s\n⊢ ∃ u, Tendsto u atTop (𝓝 x) ∧ ∃ᶠ (x : ℕ) in atTop, u x ∈ s",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Constructions | {
"line": 360,
"column": 2
} | {
"line": 360,
"column": 37
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\ninst✝² : Fintype ι\ninst✝¹ : SeminormedGroup E\ninst✝ : Nonempty ι\na : E\n⊢ ‖fun _i ↦ a‖ = ‖a‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"_private.Mathlib.Analysis.Normed.Group.Constructions.0.pi_norm_const'._simp_1_1",
"InvOneCla... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sequences | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 13
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\nx : ℕ → X\ninst✝¹ : SequentialSpace X\ninst✝ : T1Space X\nhx : ∀ (l : X) (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 l)\n⊢ IsClosed[inst✝²] (range x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sequences | {
"line": 330,
"column": 2
} | {
"line": 330,
"column": 13
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\ninst✝ : SeqCompactSpace X\nf_cont : SeqContinuous f\n⊢ IsSeqCompact (Set.range f)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sequences | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 60
} | [
{
"pp": "X : Type u_1\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), t.Finite → ∃ a ∈ s, ∀ (x : X), ¬(x ∈ t ∧ a ∈ {x_1 | (x_1, x) ∈ V})\n⊢ ∃ u, (∀ (n : ℕ), u n ∈ s) ∧ ∀ (n m : ℕ), m < n → u m ∉ ball (u n) V",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sequences | {
"line": 381,
"column": 6
} | {
"line": 382,
"column": 52
} | [
{
"pp": "X : Type u_1\ninst✝¹ : UniformSpace X\ns : Set X\ninst✝ : (𝓤 X).IsCountablyGenerated\nhs : IsSeqCompact s\nl : Filter X\nhl : Cauchy l\nhls : s ∈ l\nthis✝ : l.NeBot\nV : ℕ → Set (X × X)\nhV : (𝓤 X).HasAntitoneBasis V\nW : ℕ → Set (X × X)\nhW : ∀ (n : ℕ), W n ∈ 𝓤 X\nhWV : ∀ (n : ℕ), SetRel.comp (W n)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.AbstractCompletion | {
"line": 152,
"column": 2
} | {
"line": 153,
"column": 76
} | [
{
"pp": "α : Type uα\ninst✝² : UniformSpace α\npkg : AbstractCompletion.{vα, uα} α\nβ : Type uβ\ninst✝¹ : UniformSpace β\nf : α → β\ninst✝ : CompleteSpace β\nh : IsUniformInducing f\n⊢ IsUniformInducing (pkg.extend f)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AbstractCompletion.space",
... | rw [extend_def _ h.uniformContinuous]
exact pkg.isDenseInducing.isUniformInducing_extend pkg.isUniformInducing h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.AbstractCompletion | {
"line": 152,
"column": 2
} | {
"line": 153,
"column": 76
} | [
{
"pp": "α : Type uα\ninst✝² : UniformSpace α\npkg : AbstractCompletion.{vα, uα} α\nβ : Type uβ\ninst✝¹ : UniformSpace β\nf : α → β\ninst✝ : CompleteSpace β\nh : IsUniformInducing f\n⊢ IsUniformInducing (pkg.extend f)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AbstractCompletion.space",
... | rw [extend_def _ h.uniformContinuous]
exact pkg.isDenseInducing.isUniformInducing_extend pkg.isUniformInducing h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.AbstractCompletion | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 38
} | [
{
"pp": "α : Type uα\ninst✝³ : UniformSpace α\npkg : AbstractCompletion.{vα, uα} α\nβ : Type uβ\ninst✝² : UniformSpace β\nf : α → β\ninst✝¹ : CompleteSpace β\ninst✝ : T0Space β\nhf : UniformContinuous f\ng : pkg.space → β\nhg : UniformContinuous g\nh : ∀ (a : α), f a = g (pkg.coe a)\n⊢ ∀ (a : α), pkg.extend f (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.UniformSpace.Completion | {
"line": 268,
"column": 6
} | {
"line": 268,
"column": 27
} | [
{
"pp": "α : Type u_1\ninst✝² : UniformSpace α\ninst✝¹ : CompleteSpace α\ninst✝ : T0Space α\nf g : CauchyFilter α\n⊢ (↑f).lim = (↑g).lim ↔ Inseparable f g",
"usedConstants": [
"Eq.mpr",
"Cauchy",
"SProd.sprod",
"congrArg",
"Filter.NeBot",
"uniformity",
"PartialOrder... | ← inseparable_iff_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.UniformMulAction | {
"line": 102,
"column": 4
} | {
"line": 103,
"column": 11
} | [
{
"pp": "M : Type v\nX : Type x\nY : Type y\ninst✝⁴ : UniformSpace X\ninst✝³ : UniformSpace Y\ninst✝² : SMul M X\ninst✝¹ : SMul M Y\ninst✝ : UniformContinuousConstSMul M Y\nf : X → Y\nhf : IsUniformInducing f\nhsmul : ∀ (c : M) (x : X), f (c • x) = c • f x\nc : M\n⊢ UniformContinuous fun x ↦ c • x",
"usedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.UniformMulAction | {
"line": 111,
"column": 31
} | {
"line": 111,
"column": 65
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type w\nX : Type x\nY : Type y\ninst✝⁵ : UniformSpace X\ninst✝⁴ : UniformSpace Y\ninst✝³ : SMul M X\ninst✝² : SMul Mᵐᵒᵖ X\ninst✝¹ : IsCentralScalar M X\ninst✝ : UniformContinuousConstSMul M X\nc : M\n⊢ UniformContinuous fun x ↦ MulOpposite.op c • x",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.UniformMulAction | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 30
} | [
{
"pp": "R : Type u_3\nβ : Type u_4\ninst✝³ : DivisionRing R\ninst✝² : UniformSpace R\ninst✝¹ : UniformContinuousConstSMul Rᵐᵒᵖ R\ninst✝ : UniformSpace β\nf : β → R\nhf : UniformContinuous f\na : R\n⊢ UniformContinuous fun x ↦ f x / a",
"usedConstants": [
"UniformContinuous",
"Eq.mpr",
"Di... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.UniformMulAction | {
"line": 171,
"column": 23
} | {
"line": 171,
"column": 51
} | [
{
"pp": "M : Type v\nX : Type x\ninst✝³ : UniformSpace X\ninst✝² : Monoid M\ninst✝¹ : MulAction M X\ninst✝ : UniformContinuousConstSMul M X\nc : M\nhc : IsUnit c\nd : M\nhcd : c * d = 1\ncU : c • 𝓤 X ≤ 𝓤 X\ndU : d • 𝓤 X ≤ 𝓤 X\n⊢ 𝓤 X ≤ c • 𝓤 X",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 35
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedGroup E\n⊢ comap norm (𝓝 0) = 𝓝 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 36
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedGroup E\nx : E\n⊢ Tendsto (fun a ↦ ‖a⁻¹ * x‖) (𝓝 x) (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 13
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedGroup E\nx : E\n⊢ Tendsto (fun a ↦ ‖a‖) (𝓝 x) (𝓝 ‖x‖)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 13
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedGroup E\n⊢ Tendsto (fun a ↦ ‖a‖) (𝓝 1) (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 13
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedGroup E\n⊢ Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] fun a ↦ ‖a‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 278,
"column": 4
} | {
"line": 278,
"column": 52
} | [
{
"pp": "case refine_1\nι : Type u_2\nκ : Type u_3\nG : Type u_6\ninst✝ : SeminormedGroup G\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : UniformCauchySeqOnFilter f l l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\nε : ℝ\nhε : 0 < ε\nH : ∀ (a b : G), (a, b) ∈ {p | dist p.1 p.2 < ε} → (a, b) ∈ u\nx : (ι × ι) × κ\nhx : (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 283,
"column": 4
} | {
"line": 283,
"column": 52
} | [
{
"pp": "case refine_2\nι : Type u_2\nκ : Type u_3\nG : Type u_6\ninst✝ : SeminormedGroup G\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : TendstoUniformlyOnFilter (fun n z ↦ (f n.1 z)⁻¹ * f n.2 z) 1 (l ×ˢ l) l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\nε : ℝ\nhε : 0 < ε\nH : ∀ (a b : G), (a, b) ∈ {p | dist p.1 p.2 <... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 32
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedCommGroup E\nx : E\n⊢ Tendsto (fun a ↦ ‖a / x‖) (𝓝 x) (𝓝 0)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 337,
"column": 4
} | {
"line": 337,
"column": 40
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedCommGroup E\na : E\ns : Subgroup E\nhg : a ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑s\nb : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < b n\nu : ℕ → E\nu_in : ∀ (n : ℕ), u n ∈ s\nlim_u : Tendsto u atTop (𝓝 a)\nn₀ : ℕ\nhn₀ : ∀ n ≥ n₀, ‖(u n)⁻¹ * a‖ < b 0\nz : ℕ →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 49,
"column": 19
} | {
"line": 49,
"column": 47
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nr : ℝ\nhpos : 0 < r\nhr : ∀ (x : E), x ≠ 1 → r ≤ ‖x‖\nx y : E\nhne : x ≠ y\n⊢ x⁻¹ * y ≠ 1",
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 70,
"column": 46
} | {
"line": 70,
"column": 57
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na : E\n⊢ ‖a⁻¹‖ = ‖a‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 36
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a * b‖ ≤ ‖a‖ + ‖b‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 13
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b c : E\n⊢ ‖a / c‖ ≤ ‖a / b‖ + ‖b / c‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 28
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a‖ ≤ ‖a / b‖ + ‖b‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 30
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a / b‖ ≤ ‖a‖ + ‖b‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"SeminormedGroup.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 36
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ dist a b ≤ ‖a‖ + ‖b‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"SeminormedGroup.toGrou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 36
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ |‖a‖ - ‖b‖| ≤ ‖a⁻¹ * b‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 13
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a‖ - ‖b‖ ≤ ‖a * b‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"Monoid.toMulOneClass",
"Real.instSub",
"covariant_swap_add_of_covariant_add",
"SeminormedGro... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 19
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖x‖ = 0\n⊢ ‖x * y‖ ≤ ‖y‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 19
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖x‖ = 0\n⊢ ‖y‖ ≤ ‖x * y‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 224,
"column": 4
} | {
"line": 224,
"column": 19
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖y‖ = 0\n⊢ ‖x * y‖ ≤ ‖x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 19
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖y‖ = 0\n⊢ ‖x‖ ≤ ‖x * y‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 44
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖y‖ = 0\n⊢ ‖x‖ ≤ ‖x * y‖",
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"Real.instAddMonoid",
"Monoid.toMulOneClass",
"congrArg",
"AddMonoid.toAddZeroC... | simpa [h] using norm_le_mul_norm_add x y | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 44
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖y‖ = 0\n⊢ ‖x‖ ≤ ‖x * y‖",
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"Real.instAddMonoid",
"Monoid.toMulOneClass",
"congrArg",
"AddMonoid.toAddZeroC... | simpa [h] using norm_le_mul_norm_add x y | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 44
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nx y : E\nh : ‖y‖ = 0\n⊢ ‖x‖ ≤ ‖x * y‖",
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"Real.instAddMonoid",
"Monoid.toMulOneClass",
"congrArg",
"AddMonoid.toAddZeroC... | simpa [h] using norm_le_mul_norm_add x y | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 371,
"column": 11
} | {
"line": 371,
"column": 29
} | [
{
"pp": "E : Type u_4\nF : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nj : E →* F\nb : F\nhb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range\nf : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < f n\nv : ℕ → F\nsum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 371,
"column": 40
} | {
"line": 371,
"column": 58
} | [
{
"pp": "E : Type u_4\nF : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nj : E →* F\nb : F\nhb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range\nf : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < f n\nv : ℕ → F\nsum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 372,
"column": 21
} | {
"line": 372,
"column": 37
} | [
{
"pp": "E : Type u_4\nF : Type u_5\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nj : E →* F\nb : F\nhb : b ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑j.range\nf : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < f n\nv : ℕ → F\nsum_v : Tendsto (fun n ↦ ∏ i ∈ range (n + 1), v i) atTop (𝓝 b)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 30
} | [
{
"pp": "E : Type u_4\ninst✝ : NormedGroup E\na _x : E\nhx : _x ∈ {a}ᶜ\n⊢ _x⁻¹ * a ≠ 1",
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"SeminormedGroup.toGroup",
"Group.toDiv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 13
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nu v w : E\n⊢ ‖u / w‖ - ‖v / w‖ ≤ ‖u / v‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"Real.instSub",
"covariant_swap_add_of_covariant_add",
"SeminormedGroup.toGroup",
"A... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 286,
"column": 2
} | {
"line": 287,
"column": 9
} | [
{
"pp": "E : Type u_8\ninst✝ : SeminormedGroup E\nu v : E\n⊢ ‖u * v‖ - ‖u / v‖ ≤ 2 * ‖v‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Real.instSub",
"covariant_swap... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 446,
"column": 16
} | {
"line": 446,
"column": 47
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na : E\nn : ℕ\n⊢ ‖a ^ (n + 1)‖ ≤ ↑(n + 1) * ‖a‖",
"usedConstants": [
"add_mul",
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"Monoid.toMulOneClass",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 450,
"column": 2
} | {
"line": 450,
"column": 76
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na : E\nn : ℕ\n⊢ ‖a ^ n‖₊ ≤ ↑n * ‖a‖₊",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instLE",
"Real",
"HMul.hMul",
"congrArg",
"NNNorm.nnnorm",
"SeminormedGroup.toGroup",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 487,
"column": 2
} | {
"line": 487,
"column": 40
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\n⊢ ‖a / b‖ₑ ≤ ‖a‖ₑ + ‖b‖ₑ",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"instHDiv",
"NNNorm.nnnorm",
"SeminormedGroup.toGroup",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"Distrib.toAdd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.