module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 195,
"column": 12
} | {
"line": 195,
"column": 17
} | {
"line": 195,
"column": 17
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nγ₂ : Type u_4\nδ : Type u_5\nι : Sort y\ns t u : Set α\ninst✝⁴ : Countable α\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSingletonClass α\ninst✝¹ : TopologicalSpace α\ninst✝ : DiscreteTopology α\nthis✝ : ∀ (s : Set α), MeasurableSet s\nthis : ∀ (s : Set α),... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.BorelSpace.Basic | {
"line": 377,
"column": 4
} | {
"line": 377,
"column": 59
} | {
"line": 378,
"column": 4
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nγ₂ : Type u_4\nδ : Type u_5\nι✝ : Sort y\ns t u : Set α\ninst✝¹⁶ : TopologicalSpace α\ninst✝¹⁵ : MeasurableSpace α\ninst✝¹⁴ : OpensMeasurableSpace α\ninst✝¹³ : TopologicalSpace β\ninst✝¹² : MeasurableSpace β\ninst✝¹¹ : OpensMeasurableSpace β\ninst✝¹⁰ ... | [
"case inr\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nγ₂ : Type u_4\nδ : Type u_5\nι✝ : Sort y\ns t u : Set α\ninst✝¹⁶ : TopologicalSpace α\ninst✝¹⁵ : MeasurableSpace α\ninst✝¹⁴ : OpensMeasurableSpace α\ninst✝¹³ : TopologicalSpace β\ninst✝¹² : MeasurableSpace β\ninst✝¹¹ : OpensMeasurableSpace β\ninst✝¹⁰ : Topologica... | · exact fun s _ ↦ Subsingleton.set_cases .empty .univ s | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 47
} | {
"line": 77,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : SecondCountableTopology α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\n⊢ MeasurableSet (Iio a)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"PartialOrder.t... | [] | exact GenerateMeasurable.basic _ isOpen_Iio | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 53
} | {
"line": 177,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\nE : Set α\n⊢ 0 < infEDist x E ↔ x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"Preorder.toLT",
... | [] | rw [mem_closure_iff_infEDist_zero, pos_iff_ne_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 53
} | {
"line": 177,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\nE : Set α\n⊢ 0 < infEDist x E ↔ x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"Preorder.toLT",
... | [] | rw [mem_closure_iff_infEDist_zero, pos_iff_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 53
} | {
"line": 177,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nx : α\nE : Set α\n⊢ 0 < infEDist x E ↔ x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"Preorder.toLT",
... | [] | rw [mem_closure_iff_infEDist_zero, pos_iff_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.HausdorffDistance | {
"line": 786,
"column": 2
} | {
"line": 786,
"column": 47
} | {
"line": 787,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns t : Set α\nbs : Bornology.IsBounded s\nbt : Bornology.IsBounded t\ncs : α\nhcs : cs ∈ s\nct : α\nhct : ct ∈ t\n⊢ hausdorffEDist s t ≠ ∞",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Real",
"Exists",
"Ne",
"LE.le... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\ns t : Set α\nbs : Bornology.IsBounded s\nbt : Bornology.IsBounded t\ncs : α\nhcs : cs ∈ s\nct : α\nhct : ct ∈ t\nrs : ℝ\nhrs : s ⊆ closedBall ct rs\n⊢ hausdorffEDist s t ≠ ∞"
] | rcases bs.subset_closedBall ct with ⟨rs, hrs⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Constructions.BorelSpace.Order | {
"line": 558,
"column": 2
} | {
"line": 561,
"column": 61
} | {
"line": 562,
"column": 2
} | [
{
"pp": "case refine_2\nα : Type u_5\ninst✝⁴ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝³ : SecondCountableTopology α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : BorelSpace α\nμ ν : Measure α\nhμ : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) ≠ ∞\nh : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) = ν (Icc a b)\ns ... | [
"case refine_3\nα : Type u_5\ninst✝⁴ : TopologicalSpace α\nm : MeasurableSpace α\ninst✝³ : SecondCountableTopology α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : BorelSpace α\nμ ν : Measure α\nhμ : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) ≠ ∞\nh : ∀ ⦃a b : α⦄, a ≤ b → μ (Icc a b) = ν (Icc a b)\ns : Set α\nhsc... | · refine sUnion_eq_univ_iff.2 fun x => ?_
rcases hsd.exists_le' hsb x with ⟨l, hls, hlx⟩
rcases hsd.exists_ge' hst x with ⟨u, hus, hxu⟩
exact ⟨_, ⟨l, hls, u, hus, hlx.trans hxu, rfl⟩, hlx, hxu⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metric | {
"line": 215,
"column": 2
} | {
"line": 217,
"column": 92
} | {
"line": 220,
"column": 0
} | [
{
"pp": "α : Type u_5\nm : MeasurableSpace α\ninst✝ : CountablySeparated α\n⊢ ∃ x, SecondCountableTopology α ∧ T4Space α ∧ OpensMeasurableSpace α",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"MeasurableSpace.instLE",
"MeasurableSpace.exists_countablyGenerated_le_of_countablyS... | [] | rcases exists_countablyGenerated_le_of_countablySeparated α with ⟨m', _, _, m'le⟩
rcases exists_borelSpace_of_countablyGenerated_of_separatesPoints (m := m') with ⟨τ, _, _, τm'⟩
exact ⟨τ, ‹_›, ‹_›, @OpensMeasurableSpace.mk _ _ m (τm'.measurable_eq.symm.le.trans m'le)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.BorelSpace.Metric | {
"line": 215,
"column": 2
} | {
"line": 217,
"column": 92
} | {
"line": 220,
"column": 0
} | [
{
"pp": "α : Type u_5\nm : MeasurableSpace α\ninst✝ : CountablySeparated α\n⊢ ∃ x, SecondCountableTopology α ∧ T4Space α ∧ OpensMeasurableSpace α",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"MeasurableSpace.instLE",
"MeasurableSpace.exists_countablyGenerated_le_of_countablyS... | [] | rcases exists_countablyGenerated_le_of_countablySeparated α with ⟨m', _, _, m'le⟩
rcases exists_borelSpace_of_countablyGenerated_of_separatesPoints (m := m') with ⟨τ, _, _, τm'⟩
exact ⟨τ, ‹_›, ‹_›, @OpensMeasurableSpace.mk _ _ m (τm'.measurable_eq.symm.le.trans m'le)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 793,
"column": 6
} | {
"line": 793,
"column": 23
} | {
"line": 793,
"column": 24
} | [
{
"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\nhs : MeasurableSet s\n⊢ r ∈ (f.restrict s).range ↔ r = 0 ∧ s ≠ univ ∨ r ∈ ⇑f '' s",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : Zero β\nr : β\ns : Set α\nf : α →ₛ β\nhs : MeasurableSet s\n⊢ r ∈ ↑(f.restrict s).range ↔ r = 0 ∧ s ≠ univ ∨ r ∈ ⇑f '' s"
] | ← Finset.mem_coe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Monoid.Canonical.Basic | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 7
} | {
"line": 47,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : LinearOrder α\nP : α → Prop\nb c : α\ninst✝³ : Add α\ninst✝² : CanonicallyOrderedAdd α\ninst✝¹ : AddLeftReflectLT α\ninst✝ : IsLeftCancelAdd α\n⊢ (∀ (a : α), (a < b ∨ ∃ d < c, a = b + d) → P a) ↔ (∀ a < b, P a) ∧ ∀ d < c, P (b + d)",
"ppTerm": "?m.24",
"assigned": true,
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Monoid.Canonical.Basic | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 7
} | {
"line": 52,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : LinearOrder α\nP : α → Prop\nb c : α\ninst✝³ : Add α\ninst✝² : CanonicallyOrderedAdd α\ninst✝¹ : AddLeftReflectLT α\ninst✝ : IsLeftCancelAdd α\n⊢ (∃ a, (a < b ∨ ∃ d < c, a = b + d) ∧ P a) ↔ (∃ a < b, P a) ∨ ∃ d < c, P (b + d)",
"ppTerm": "?m.27",
"assigned": true,
"us... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Monoid.Canonical.Basic | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 7
} | {
"line": 65,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : LinearOrder α\nP : α → Prop\nb c : α\ninst✝³ : Add α\ninst✝² : CanonicallyOrderedAdd α\ninst✝¹ : AddLeftMono α\ninst✝ : IsLeftCancelAdd α\n⊢ (∀ (a : α), (a < b ∨ ∃ d ≤ c, a = b + d) → P a) ↔ (∀ a < b, P a) ∧ ∀ d ≤ c, P (b + d)",
"ppTerm": "?m.24",
"assigned": true,
"u... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Monoid.Canonical.Basic | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 7
} | {
"line": 70,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : LinearOrder α\nP : α → Prop\nb c : α\ninst✝³ : Add α\ninst✝² : CanonicallyOrderedAdd α\ninst✝¹ : AddLeftMono α\ninst✝ : IsLeftCancelAdd α\n⊢ (∃ a, (a < b ∨ ∃ d ≤ c, a = b + d) ∧ P a) ↔ (∃ a < b, P a) ∨ ∃ d ≤ c, P (b + d)",
"ppTerm": "?m.27",
"assigned": true,
"usedCon... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Monoid.Canonical.Basic | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 7
} | {
"line": 85,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : LinearOrder α\nP : α → Prop\nb c : α\ninst✝³ : AddCommMagma α\ninst✝² : CanonicallyOrderedAdd α\ninst✝¹ : AddLeftReflectLT α\ninst✝ : IsLeftCancelAdd α\n⊢ (∀ (a : α), (a < c ∨ ∃ d < b, a = d + c) → P a) ↔ (∀ a < c, P a) ∧ ∀ d < b, P (d + c)",
"ppTerm": "?m.24",
"assigned"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Monoid.Canonical.Basic | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 7
} | {
"line": 90,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : LinearOrder α\nP : α → Prop\nb c : α\ninst✝³ : AddCommMagma α\ninst✝² : CanonicallyOrderedAdd α\ninst✝¹ : AddLeftReflectLT α\ninst✝ : IsLeftCancelAdd α\n⊢ (∃ a, (a < c ∨ ∃ d < b, a = d + c) ∧ P a) ↔ (∃ a < c, P a) ∨ ∃ d < b, P (d + c)",
"ppTerm": "?m.27",
"assigned": true... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Monoid.Canonical.Basic | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 7
} | {
"line": 100,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : LinearOrder α\nP : α → Prop\nb c : α\ninst✝³ : AddCommMagma α\ninst✝² : CanonicallyOrderedAdd α\ninst✝¹ : AddLeftMono α\ninst✝ : IsLeftCancelAdd α\n⊢ (∀ (a : α), (a < c ∨ ∃ d ≤ b, a = d + c) → P a) ↔ (∀ a < c, P a) ∧ ∀ d ≤ b, P (d + c)",
"ppTerm": "?m.24",
"assigned": tru... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Algebra.Order.Monoid.Canonical.Basic | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 7
} | {
"line": 105,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : LinearOrder α\nP : α → Prop\nb c : α\ninst✝³ : AddCommMagma α\ninst✝² : CanonicallyOrderedAdd α\ninst✝¹ : AddLeftMono α\ninst✝ : IsLeftCancelAdd α\n⊢ (∃ a, (a < c ∨ ∃ d ≤ b, a = d + c) ∧ P a) ↔ (∃ a < c, P a) ∨ ∃ d ≤ b, P (d + c)",
"ppTerm": "?m.27",
"assigned": true,
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Instances.Real.Lemmas | {
"line": 114,
"column": 12
} | {
"line": 114,
"column": 17
} | {
"line": 115,
"column": 2
} | [
{
"pp": "case e'_2\nq : ℚ\n⊢ Rat.cast '' {x | q < x} = Ioi ↑q ∩ range Rat.cast",
"ppTerm": "?e'_2",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Set.Ioi",
"Preorder.toLT",
"DivisionRing.toRatCast",
"FloorRing.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.IndicatorConstPointwise | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 7
} | {
"line": 121,
"column": 0
} | [
{
"pp": "α : Type u_1\nA : Set α\nβ : Type u_2\ninst✝¹ : Zero β\ninst✝ : TopologicalSpace β\nι : Type u_3\nL : Filter ι\nAs : ι → Set α\nb : β\nnhds_b : {0}ᶜ ∈ 𝓝 b\nnhds_o : {b}ᶜ ∈ 𝓝 0\n⊢ (∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A) ↔ ∀ (i : α), ∀ᶠ (x : ι) in L, (i ∈ As x) = (i ∈ A)",
"ppTerm": "?m.71",... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.IndicatorConstPointwise | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 7
} | {
"line": 127,
"column": 0
} | [
{
"pp": "α : Type u_1\nA : Set α\nβ : Type u_2\ninst✝³ : Zero β\ninst✝² : TopologicalSpace β\nι : Type u_3\nL : Filter ι\nAs : ι → Set α\ninst✝¹ : T1Space β\nb : β\ninst✝ : NeZero b\n⊢ (∀ (x : α), ∀ᶠ (i : ι) in L, x ∈ As i ↔ x ∈ A) ↔ ∀ (i : α), ∀ᶠ (x : ι) in L, (i ∈ As x) = (i ∈ A)",
"ppTerm": "?m.51",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1289,
"column": 6
} | {
"line": 1289,
"column": 33
} | {
"line": 1290,
"column": 6
} | [
{
"pp": "α : Type u_5\nγ : Type u_6\ninst✝¹ : MeasurableSpace α\ninst✝ : AddZeroClass γ\nmotive : (α →ₛ γ) → Prop\nconst :\n ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), motive (piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))\nadd : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ⇑f) (support ⇑g) → motive f... | [
"case pos\nα : Type u_5\nγ : Type u_6\ninst✝¹ : MeasurableSpace α\ninst✝ : AddZeroClass γ\nmotive : (α →ₛ γ) → Prop\nconst :\n ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), motive (piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))\nadd : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ⇑f) (support ⇑g) → motive f →... | («tacticBy_cases_:_»
"by_cases"
[`hy ":"]
(«term_∈_»
`y
"∈"
(Set.«term_⁻¹'_»
`f
"⁻¹'"
(choice
(«term{_}» "{" [`x] "}")
(Term.structInst
"{"
[]
(Term.structInstFields [(Term.structInstField (Term.structInstLVal `x []) [])])
(Term.optEllipsis [])
[]
"}"))))) | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1294,
"column": 4
} | {
"line": 1294,
"column": 31
} | {
"line": 1294,
"column": 32
} | [
{
"pp": "case insert\nα : Type u_5\nγ : Type u_6\ninst✝¹ : MeasurableSpace α\ninst✝ : AddZeroClass γ\nmotive : (α →ₛ γ) → Prop\nconst :\n ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), motive (piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))\nadd : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ⇑f) (support ⇑... | [
"case pos\nα : Type u_5\nγ : Type u_6\ninst✝¹ : MeasurableSpace α\ninst✝ : AddZeroClass γ\nmotive : (α →ₛ γ) → Prop\nconst :\n ∀ (c : γ) {s : Set α} (hs : MeasurableSet s), motive (piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))\nadd : ∀ ⦃f g : α →ₛ γ⦄, Disjoint (support ⇑f) (support ⇑g) → motive f →... | («tacticBy_cases_:_»
"by_cases"
[`hy ":"]
(«term_∈_»
`y
"∈"
(Set.«term_⁻¹'_»
`f
"⁻¹'"
(choice
(«term{_}» "{" [`x] "}")
(Term.structInst
"{"
[]
(Term.structInstFields [(Term.structInstField (Term.structInstLVal `x []) [])])
(Term.optEllipsis [])
[]
"}"))))) | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.MeasureTheory.Function.SimpleFunc | {
"line": 1331,
"column": 6
} | {
"line": 1331,
"column": 33
} | {
"line": 1332,
"column": 6
} | [
{
"pp": "α : Type u_5\nγ : Type u_6\ninst✝¹ : MeasurableSpace α\ninst✝ : Nonempty γ\nP : (α →ₛ γ) → Prop\nconst : ∀ (c : γ), P (SimpleFunc.const α c)\npcw : ∀ ⦃f g : α →ₛ γ⦄ {s : Set α} (hs : MeasurableSet s), P f → P g → P (piecewise s hs f g)\nc : γ := Classical.ofNonempty\nx : γ\ns : Finset γ\nhxs : x ∉ s\ni... | [
"case pos\nα : Type u_5\nγ : Type u_6\ninst✝¹ : MeasurableSpace α\ninst✝ : Nonempty γ\nP : (α →ₛ γ) → Prop\nconst : ∀ (c : γ), P (SimpleFunc.const α c)\npcw : ∀ ⦃f g : α →ₛ γ⦄ {s : Set α} (hs : MeasurableSet s), P f → P g → P (piecewise s hs f g)\nc : γ := Classical.ofNonempty\nx : γ\ns : Finset γ\nhxs : x ∉ s\nih ... | («tacticBy_cases_:_»
"by_cases"
[`hy ":"]
(«term_∈_»
`y
"∈"
(Set.«term_⁻¹'_»
`f
"⁻¹'"
(choice
(«term{_}» "{" [`x] "}")
(Term.structInst
"{"
[]
(Term.structInstFields [(Term.structInstField (Term.structInstLVal `x []) [])])
(Term.optEllipsis [])
[]
"}"))))) | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.MeasureTheory.Function.SimpleFuncDense | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 21
} | {
"line": 222,
"column": 6
} | [
{
"pp": "case hnhds\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Con... | [
"case hnhds\nX : Type u_3\nY : Type u_4\nα : Type u_5\ninst✝⁷ : Zero α\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : MeasurableSpace X\ninst✝³ : MeasurableSpace Y\ninst✝² : OpensMeasurableSpace X\ninst✝¹ : OpensMeasurableSpace Y\ninst✝ : PseudoMetricSpace α\nf : X × Y → α\nhf : Continuous[inst... | rintro ⟨x, y⟩ - | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 148,
"column": 42
} | {
"line": 148,
"column": 69
} | {
"line": 148,
"column": 69
} | [
{
"pp": "α : Type u_1\nm₀ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetrizableSpace α\ninst✝ : OpensMeasurableSpace α\ns : Set α\nh1 : IsSeparable s\nh2 : μ sᶜ = 0\na✝ : Nontrivial α\na x : α\nhx : x ∈ s\n⊢ x ∈ {x | (fun x ↦ id x = (closure[inst✝²] s).piecewise id (fun x ↦ ... | [] | by simp [subset_closure hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable | {
"line": 546,
"column": 4
} | {
"line": 547,
"column": 98
} | {
"line": 548,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace β\nm₀ : MeasurableSpace α\nμ : Measure α\nf : α → β\ninst✝ : Zero β\ns : Set α\nhs : MeasurableSet s\nh : AEStronglyMeasurable f (μ.restrict s)\n⊢ s.indicator f =ᵐ[μ] s.indicator (AEStronglyMeasurable.mk f h)",
"ppTerm": "?mpr",
"a... | [
"case mpr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace β\nm₀ : MeasurableSpace α\nμ : Measure α\nf : α → β\ninst✝ : Zero β\ns : Set α\nhs : MeasurableSet s\nh : AEStronglyMeasurable f (μ.restrict s)\nA : s.indicator f =ᵐ[μ.restrict s] s.indicator (AEStronglyMeasurable.mk f h)\n⊢ s.indicator f =ᵐ[μ] s.indi... | have A : s.indicator f =ᵐ[μ.restrict s] s.indicator (h.mk f) :=
(indicator_ae_eq_restrict hs).trans (h.ae_eq_mk.trans <| (indicator_ae_eq_restrict hs).symm) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 62
} | {
"line": 221,
"column": 2
} | [
{
"pp": "α : 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\n⊢ ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), g a ∂μ",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
... | [
"α : 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\n⊢ ∫⁻ (a : α), f a ∂μ ≤ ∫⁻ (a : α), g a ∂μ"
] | have : ∀ᵐ x ∂μ, x ∉ t := measure_eq_zero_iff_ae_notMem.1 ht0 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.Lebesgue.Basic | {
"line": 647,
"column": 59
} | {
"line": 649,
"column": 58
} | {
"line": 651,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nι : Type u_4\ninst✝ : Countable ι\nf : α → ℝ≥0∞\ns : ι → Set α\nhd : Directed (fun x1 x2 ↦ x1 ⊆ x2) s\n⊢ ∫⁻ (x : α) in ⋃ i, s i, f x ∂μ = ⨆ i, ∫⁻ (x : α) in s i, f x ∂μ",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Measu... | [] | by
simp only [lintegral_def, iSup_comm (ι := ι),
SimpleFunc.lintegral_restrict_iUnion_of_directed _ hd] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion | {
"line": 178,
"column": 4
} | {
"line": 184,
"column": 41
} | {
"line": 185,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝ : IsFiniteMeasure ν\nthis : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))\nf : ℕ × ℕ → Set α :=\n fun p ↦\n (μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFinite... | [] | simp only [Nat.pairEquiv_symm_apply, measure_union_lt_top_iff, f, e]
rw [Measure.restrict_apply' measurableSet_sigmaFiniteSetWRT', Set.compl_inter_self,
Measure.restrict_apply' measurableSet_sigmaFiniteSetWRT']
simp only [measure_empty, ENNReal.zero_lt_top, true_and]
refine (measure_mono Set.inter_sub... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion | {
"line": 178,
"column": 4
} | {
"line": 184,
"column": 41
} | {
"line": 185,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝ : IsFiniteMeasure ν\nthis : ∀ (n : ℕ), SigmaFinite (μ.restrict (μ.sigmaFiniteSetGE ν n))\nf : ℕ × ℕ → Set α :=\n fun p ↦\n (μ.sigmaFiniteSetWRT' ν)ᶜ ∪ spanningSets (μ.restrict (μ.sigmaFiniteSetGE ν p.1)) p.2 ∩ μ.sigmaFinite... | [] | simp only [Nat.pairEquiv_symm_apply, measure_union_lt_top_iff, f, e]
rw [Measure.restrict_apply' measurableSet_sigmaFiniteSetWRT', Set.compl_inter_self,
Measure.restrict_apply' measurableSet_sigmaFiniteSetWRT']
simp only [measure_empty, ENNReal.zero_lt_top, true_and]
refine (measure_mono Set.inter_sub... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 294,
"column": 10
} | {
"line": 294,
"column": 24
} | {
"line": 294,
"column": 24
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ ∫⁻ (a : α), ⨆ n, (⇑(eapprox f n) + ⇑(eapprox g n)) a ∂μ = ⨆ n, (eapprox f n).lintegral μ + (eapprox g n).lintegral μ",
"ppTerm": "?m.99",
"assigned": true,
"usedConstants": [
"M... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : Measurable f\nhg : Measurable g\n⊢ ⨆ n, ∫⁻ (a : α), (⇑(eapprox f n) + ⇑(eapprox g n)) a ∂μ = ⨆ n, (eapprox f n).lintegral μ + (eapprox g n).lintegral μ",
"case hf\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : Mea... | lintegral_iSup | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Perfect | {
"line": 159,
"column": 2
} | {
"line": 163,
"column": 21
} | {
"line": 165,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nC U : Set α\nhC : Perfect C\nx : α\nxC : x ∈ C\nxU : x ∈ U\nUop : IsOpen[inst✝] U\n⊢ Perfect (closure[inst✝] (U ∩ C)) ∧ (closure[inst✝] (U ∩ C)).Nonempty",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Perfect",
"Perfect.acc",... | [] | constructor
· apply Preperfect.perfect_closure
exact hC.acc.open_inter Uop
apply Nonempty.closure
exact ⟨x, ⟨xU, xC⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Perfect | {
"line": 159,
"column": 2
} | {
"line": 163,
"column": 21
} | {
"line": 165,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nC U : Set α\nhC : Perfect C\nx : α\nxC : x ∈ C\nxU : x ∈ U\nUop : IsOpen[inst✝] U\n⊢ Perfect (closure[inst✝] (U ∩ C)) ∧ (closure[inst✝] (U ∩ C)).Nonempty",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Perfect",
"Perfect.acc",... | [] | constructor
· apply Preperfect.perfect_closure
exact hC.acc.open_inter Uop
apply Nonempty.closure
exact ⟨x, ⟨xU, xC⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 388,
"column": 10
} | {
"line": 388,
"column": 24
} | {
"line": 388,
"column": 24
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ∫⁻ (a : α), ⨆ n, (const α r * eapprox f n) a ∂μ = ⨆ n, r * (eapprox f n).lintegral μ",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"MeasureTheory.SimpleFunc.lintegral",
"... | [
"α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ⨆ n, ∫⁻ (a : α), (const α r * eapprox f n) a ∂μ = ⨆ n, r * (eapprox f n).lintegral μ",
"case hf\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ∀ (n : ℕ), Measurable ... | lintegral_iSup | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Add | {
"line": 459,
"column": 2
} | {
"line": 460,
"column": 98
} | {
"line": 461,
"column": 2
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ∫⁻ (a : α), f a ∂μ.trim hm = ∫⁻ (a : α), f a ∂μ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.trim",
"ENNReal",
"Measurable.ennreal_ind... | [
"case refine_1\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ ∀ (c : ℝ≥0∞) ⦃s : Set α⦄,\n MeasurableSet s → ∫⁻ (a : α), s.indicator (fun x ↦ c) a ∂μ.trim hm = ∫⁻ (a : α), s.indicator (fun x ↦ c) a ∂μ",
"case refine_2\nα : Type u_1\nm m0 : MeasurableSpace... | refine
@Measurable.ennreal_induction α m (fun f => ∫⁻ a, f a ∂μ.trim hm = ∫⁻ a, f a ∂μ) ?_ ?_ ?_ f hf | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Group.FunctionSeries | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 9
} | {
"line": 72,
"column": 2
} | [
{
"pp": "β : Type u_2\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nι : Type u_4\nf : ι → β → F\nu : ι → ℝ\nhu : Summable u\ns : Set β\nε : ℝ\nεpos : ε > 0\nt : Finset ι\nht : ∀ (b : Finset ι), t ⊆ b → ∑' (a : { x // x ∉ b }), u ↑a < ε\nN : Set ι\nhN : N ∈ cofinite\nHN : ∀ y ∈ N, ∀ x ∈ ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Algebra.MetricSpace.Lipschitz | {
"line": 41,
"column": 4
} | {
"line": 44,
"column": 56
} | {
"line": 45,
"column": 2
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\ns : Set α\nf : α → β\nhf : LipschitzOnWith K f s\nu : ℕ → α\nhu : CauchySeq u\nh'u : range u ⊆ s\nb : ℕ → ℝ\nb_nonneg : ∀ (n : ℕ), 0 ≤ b n\nhb : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (u n) (u m)... | [] | intro n m N hn hm
have A n : u n ∈ s := h'u (mem_range_self _)
apply (hf.dist_le_mul _ (A n) _ (A m)).trans
exact mul_le_mul_of_nonneg_left (hb n m N hn hm) K.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.MetricSpace.Lipschitz | {
"line": 41,
"column": 4
} | {
"line": 44,
"column": 56
} | {
"line": 45,
"column": 2
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\ns : Set α\nf : α → β\nhf : LipschitzOnWith K f s\nu : ℕ → α\nhu : CauchySeq u\nh'u : range u ⊆ s\nb : ℕ → ℝ\nb_nonneg : ∀ (n : ℕ), 0 ≤ b n\nhb : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (u n) (u m)... | [] | intro n m N hn hm
have A n : u n ∈ s := h'u (mem_range_self _)
apply (hf.dist_le_mul _ (A n) _ (A m)).trans
exact mul_le_mul_of_nonneg_left (hb n m N hn hm) K.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 118,
"column": 2
} | {
"line": 122,
"column": 43
} | {
"line": 124,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nx : X\ny z : Y\n⊢ glueDist Φ Ψ ε (Sum.inl x) (Sum.inr z) ≤\n glueDist Φ Ψ ε (Sum.inl x) (Sum.inr y) + glueDist Φ Ψ ε (Sum.inr y) (Sum.inr z)",
"ppTerm": "?m.34",
... | [] | simp only [glueDist]
rw [add_right_comm, add_le_add_iff_right]
refine le_ciInf_add fun p => ciInf_le_of_le ⟨0, ?_⟩ p ?_
· exact forall_mem_range.2 fun _ => by positivity
· linarith [dist_triangle_left z (Ψ p) y] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 118,
"column": 2
} | {
"line": 122,
"column": 43
} | {
"line": 124,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nx : X\ny z : Y\n⊢ glueDist Φ Ψ ε (Sum.inl x) (Sum.inr z) ≤\n glueDist Φ Ψ ε (Sum.inl x) (Sum.inr y) + glueDist Φ Ψ ε (Sum.inr y) (Sum.inr z)",
"ppTerm": "?m.34",
... | [] | simp only [glueDist]
rw [add_right_comm, add_le_add_iff_right]
refine le_ciInf_add fun p => ciInf_le_of_le ⟨0, ?_⟩ p ?_
· exact forall_mem_range.2 fun _ => by positivity
· linarith [dist_triangle_left z (Ψ p) y] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 221,
"column": 89
} | {
"line": 223,
"column": 16
} | {
"line": 225,
"column": 0
} | [
{
"pp": "X✝ : Type u_1\nY : Type u_2\nι : Type u_3\ninst✝² : Countable ι\nX : ι → Type u_4\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : ∀ (i : ι), IsCompletelyMetrizableSpace (X i)\n⊢ IsCompletelyMetrizableSpace ((i : ι) → X i)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"i... | [] | by
letI := fun i ↦ upgradeIsCompletelyMetrizable (X i)
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 7
} | {
"line": 90,
"column": 0
} | [
{
"pp": "E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhn : n < if h : x ≠ y then Nat.find ⋯ else 0\n⊢ x n = y n",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Classical.dite_not",
"not_lt_zero._simp_1",
"instDecidableNot",
"False",
"Nat.instMulZeroClass",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.MetricSpace.Polish | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 97
} | {
"line": 126,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : Countable ι\nt : ι → TopologicalSpace α\nht : ∀ (i : ι), PolishSpace α\ni₀ : ι\nhi₀ : ∀ (i : ι), t i ≤ t i₀\nu : UniformSpace α\nhcomp : CompleteSpace α\nhcount : (𝓤 α).IsCountablyGenerated\nhtop : u.toTopologicalSpace = ⨅ i, t i\nthis : SecondCountableTopology α\n⊢... | [
"α : Type u_1\nι : Type u_3\ninst✝ : Countable ι\nt : ι → TopologicalSpace α\nht : ∀ (i : ι), PolishSpace α\ni₀ : ι\nhi₀ : ∀ (i : ι), t i ≤ t i₀\nu : UniformSpace α\nhcomp : CompleteSpace α\nhcount : (𝓤 α).IsCountablyGenerated\nhtop : u.toTopologicalSpace = ⨅ i, t i\nthis✝ : SecondCountableTopology α\nthis : T1Spa... | have : @T1Space α u.toTopologicalSpace :=
htop.symm ▸ t1Space_antitone (iInf_le _ i₀) (by letI := t i₀; haveI := ht i₀; infer_instance) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.MetricSpace.Polish | {
"line": 115,
"column": 2
} | {
"line": 126,
"column": 16
} | {
"line": 128,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : Countable ι\nt : ι → TopologicalSpace α\nht₀ : ∃ i₀, ∀ (i : ι), t i ≤ t i₀\nht : ∀ (i : ι), PolishSpace α\n⊢ PolishSpace α",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"instT4SpaceOfT1SpaceOfNormalSpace",
"UniformSpace",
"Eq... | [] | rcases ht₀ with ⟨i₀, hi₀⟩
rcases CompletePseudometrizable.iInf ⟨t i₀, letI := t i₀; haveI := ht i₀; inferInstance, hi₀⟩
fun i ↦
letI := t i; haveI := ht i; letI := upgradeIsCompletelyMetrizable α
⟨inferInstance, inferInstance, inferInstance, rfl⟩
with ⟨u, hcomp, hcount, htop⟩
rw [← htop]
have ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Polish | {
"line": 115,
"column": 2
} | {
"line": 126,
"column": 16
} | {
"line": 128,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : Countable ι\nt : ι → TopologicalSpace α\nht₀ : ∃ i₀, ∀ (i : ι), t i ≤ t i₀\nht : ∀ (i : ι), PolishSpace α\n⊢ PolishSpace α",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"instT4SpaceOfT1SpaceOfNormalSpace",
"UniformSpace",
"Eq... | [] | rcases ht₀ with ⟨i₀, hi₀⟩
rcases CompletePseudometrizable.iInf ⟨t i₀, letI := t i₀; haveI := ht i₀; inferInstance, hi₀⟩
fun i ↦
letI := t i; haveI := ht i; letI := upgradeIsCompletelyMetrizable α
⟨inferInstance, inferInstance, inferInstance, rfl⟩
with ⟨u, hcomp, hcount, htop⟩
rw [← htop]
have ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Polish | {
"line": 201,
"column": 6
} | {
"line": 201,
"column": 100
} | {
"line": 202,
"column": 4
} | [
{
"pp": "case refine_4.refine_1\nα : Type u_1\nβ : Type u_2\ninst✝ : MetricSpace α\ns : Opens α\nt : Set s.CompleteCopy\nh : IsOpen t\nx : s.CompleteCopy\nhx : x ∈ t\nε : ℝ\nε0 : ε > 0\nhε : ball x ε ⊆ t\n⊢ ∃ ε > 0, ∀ (y : s.CompleteCopy), dist x y < ε → y ∈ t",
"ppTerm": "?refine_4.refine_1",
"assigned... | [] | exact ⟨ε, ε0, fun y hy ↦ hε <| (dist_comm _ _).trans_lt <| (dist_val_le_dist _ _).trans_lt hy⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 15
} | {
"line": 220,
"column": 2
} | [
{
"pp": "case mp.succ.inr\nα : Type u_2\nx y : ℕ → α\nn : ℕ\nih : res x n = res y n → ∀ ⦃m : ℕ⦄, m < n → x m = y m\nm : ℕ\nh : x n = y n ∧ res x n = res y n\nhm : m = n\n⊢ x n = y n",
"ppTerm": "?mp.succ.inr",
"assigned": true,
"usedConstants": [
"List",
"And.left",
"Eq",
"Pi... | [] | exact h.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 399,
"column": 6
} | {
"line": 402,
"column": 47
} | {
"line": 403,
"column": 2
} | [
{
"pp": "case mp.inr\nι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ns : Set ((i : ι) × E i)\nhs : IsOpen[instTopologicalSpaceSigma] s\ni : ι\nx : E i\nhx : ⟨i, x⟩ ∈ s\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ Sigma.mk i ⁻¹' s\nj : ι\ny : E j\nhy : dist ⟨i, x⟩ ⟨j, y⟩ < min ε 1\nhij : i ≠ j\n⊢ ... | [] | apply (lt_irrefl (1 : ℝ) _).elim
calc
1 ≤ Sigma.dist ⟨i, x⟩ ⟨j, y⟩ := Sigma.one_le_dist_of_ne hij _ _
_ < 1 := hy.trans_le (min_le_right _ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 399,
"column": 6
} | {
"line": 402,
"column": 47
} | {
"line": 403,
"column": 2
} | [
{
"pp": "case mp.inr\nι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ns : Set ((i : ι) × E i)\nhs : IsOpen[instTopologicalSpaceSigma] s\ni : ι\nx : E i\nhx : ⟨i, x⟩ ∈ s\nε : ℝ\nεpos : ε > 0\nhε : ball x ε ⊆ Sigma.mk i ⁻¹' s\nj : ι\ny : E j\nhy : dist ⟨i, x⟩ ⟨j, y⟩ < min ε 1\nhij : i ≠ j\n⊢ ... | [] | apply (lt_irrefl (1 : ℝ) _).elim
calc
1 ≤ Sigma.dist ⟨i, x⟩ ⟨j, y⟩ := Sigma.one_le_dist_of_ne hij _ _
_ < 1 := hy.trans_le (min_le_right _ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Polish | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 70
} | {
"line": 244,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝¹ : MetricSpace α\ns : Opens α\ninst✝ : CompleteSpace α\nu : ℕ → s.CompleteCopy\nhu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (1 / 2) ^ N\nA : CauchySeq fun n ↦ ↑(u n)\nx : α\nxlim : Tendsto (fun n ↦ ↑(u n)) atTop (𝓝 x)\nxs : x ∉ s\nC : ℝ\nhC : ∀ (n... | [] | exact absurd (Hmem.2 <| lt_of_lt_of_le (div_pos one_pos Cpos) I') xs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.MetricSpace.Polish | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 28
} | {
"line": 249,
"column": 2
} | [
{
"pp": "α : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\ns : Set α\nhs : IsOpen[inst✝¹] s\nthis : UpgradedIsCompletelyMetrizableSpace α := upgradeIsCompletelyMetrizable α\n⊢ PolishSpace ↑s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"TopologicalSpace.Opens",
... | [
"α : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\nthis : UpgradedIsCompletelyMetrizableSpace α := upgradeIsCompletelyMetrizable α\ns : Opens α\n⊢ PolishSpace ↑↑s"
] | lift s to Opens α using hs | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.Topology.MetricSpace.Polish | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 43
} | {
"line": 277,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\ns : Set α\nhs : IsClosed[inst✝¹] s\nthis✝ : PolishSpace ↑s\nt : Set α := sᶜ\nthis : PolishSpace ↑t\n⊢ IsClopenable s",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Classical.propDecidable",
"Membership... | [
"α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : PolishSpace α\ns : Set α\nhs : IsClosed[inst✝¹] s\nthis✝ : PolishSpace ↑s\nt : Set α := sᶜ\nthis : PolishSpace ↑t\nf : ↑s ⊕ ↑t ≃ α := Equiv.Set.sumCompl s\n⊢ IsClopenable s"
] | let f : s ⊕ t ≃ α := Equiv.Set.sumCompl s | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.MetricSpace.CantorScheme | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 17
} | {
"line": 178,
"column": 2
} | [
{
"pp": "case h\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\ninst✝¹ : PseudoMetricSpace α\ninst✝ : CompleteSpace α\nhdiam : VanishingDiam A\nhanti : ClosureAntitone A\nhnonempty : ∀ (l : List β), (A l).Nonempty\nx : ℕ → β\nu : ℕ → α\nhu : ∀ (n : ℕ), u n ∈ A (res x n)\numem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (re... | [
"case h\nβ : Type u_1\nα : Type u_2\nA : List β → Set α\ninst✝¹ : PseudoMetricSpace α\ninst✝ : CompleteSpace α\nhdiam : VanishingDiam A\nhanti : ClosureAntitone A\nhnonempty : ∀ (l : List β), (A l).Nonempty\nx : ℕ → β\nu : ℕ → α\nhu : ∀ (n : ℕ), u n ∈ A (res x n)\numem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)\nthis... | rw [mem_iInter] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Dynamics.Ergodic.MeasurePreserving | {
"line": 63,
"column": 54
} | {
"line": 66,
"column": 17
} | {
"line": 68,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμa : Measure α\nμb : Measure β\nf f' : α → β\nhf : MeasurePreserving f μa μb\nhf' : Measurable f'\nh : f =ᵐ[μa] f'\n⊢ MeasurePreserving f' μa μb",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
... | [] | by
refine ⟨hf', ?_⟩
rw [Measure.map_congr h.symm]
exact hf.map_eq | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 112,
"column": 56
} | {
"line": 112,
"column": 61
} | {
"line": 112,
"column": 61
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable α\nμ ν : Measure α\nh : ∀ (a : α), μ {a} = ν {a}\n⊢ ⋃₀ range singleton = univ",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Set.ext",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Set.m... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 112,
"column": 56
} | {
"line": 112,
"column": 61
} | {
"line": 112,
"column": 61
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable α\nμ ν : Measure α\nh : ∀ (a : α), μ {a} = ν {a}\n⊢ ⋃₀ range singleton = univ",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Set.ext",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Set.m... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 112,
"column": 56
} | {
"line": 112,
"column": 61
} | {
"line": 112,
"column": 61
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable α\nμ ν : Measure α\nh : ∀ (a : α), μ {a} = ν {a}\n⊢ ⋃₀ range singleton = univ",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Set.ext",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Set.m... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 640,
"column": 19
} | {
"line": 640,
"column": 22
} | {
"line": 640,
"column": 23
} | [
{
"pp": "case pos\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some\nfs : ∀ x ∈ s, f x = x\nx y : (n ... | [
"case pos\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some\nfs : ∀ x ∈ s, f x = x\nx y : (n : ℕ) → E n\n... | I2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 669,
"column": 20
} | {
"line": 669,
"column": 23
} | {
"line": 669,
"column": 24
} | [
{
"pp": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some\nfs : ∀ x ∈ s, f x = x\nx y : (n : ℕ) → E n... | [
"E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nhne : s.Nonempty\nf : ((n : ℕ) → E n) → (n : ℕ) → E n := fun x ↦ if x ∈ s then x else ⋯.some\nfs : ∀ x ∈ s, f x = x\nx y : (n : ℕ) → E n\nhxy : x ≠ ... | I2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 738,
"column": 71
} | {
"line": 740,
"column": 22
} | {
"line": 741,
"column": 4
} | [
{
"pp": "α : Type u_2\ninst✝³ : MetricSpace α\ninst✝² : CompleteSpace α\ninst✝¹ : SecondCountableTopology α\ninst✝ : Nonempty α\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : ℕ → α\nhu : DenseRange u\ns : Set (ℕ → ℕ) := {x | (⋂ n, closedBall (u (x n)) ((1 / 2) ^ n)).Nonemp... | [] | by
rcases hu.exists_dist_lt y (by simp : (0 : ℝ) < (1 / 2) ^ n) with ⟨j, hj⟩
exact ⟨j, hj.le⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 848,
"column": 6
} | {
"line": 878,
"column": 39
} | {
"line": 879,
"column": 4
} | [
{
"pp": "case left\nE : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\n⊢ ∀ (i : ℝ≥0∞), 0 < i → {p | edist p.1 p.2 < i} ∈ ⨅ i, ⨅ i_2, ⨅ (_ : 0 < i_2), 𝓟 {a | edist (a.1 i) (a.2 i) < i_2}",
"ppTerm": "?left",
"assigned": true,
"usedCons... | [] | intro ε hε
classical
obtain ⟨K, hK⟩ : ∃ K : Finset ι, ∑' i : {j // j ∉ K}, 2⁻¹ ^ encode (i : ι) < ε / 2 :=
((tendsto_order.1 <| ENNReal.tendsto_tsum_compl_atTop_zero
(tsum_geometric_encode_lt_top ENNReal.one_half_lt_one).ne).2 _
<| by simpa using hε.ne').exists
obtain ⟨δ,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 848,
"column": 6
} | {
"line": 878,
"column": 39
} | {
"line": 879,
"column": 4
} | [
{
"pp": "case left\nE : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\n⊢ ∀ (i : ℝ≥0∞), 0 < i → {p | edist p.1 p.2 < i} ∈ ⨅ i, ⨅ i_2, ⨅ (_ : 0 < i_2), 𝓟 {a | edist (a.1 i) (a.2 i) < i_2}",
"ppTerm": "?left",
"assigned": true,
"usedCons... | [] | intro ε hε
classical
obtain ⟨K, hK⟩ : ∃ K : Finset ι, ∑' i : {j // j ∉ K}, 2⁻¹ ^ encode (i : ι) < ε / 2 :=
((tendsto_order.1 <| ENNReal.tendsto_tsum_compl_atTop_zero
(tsum_geometric_encode_lt_top ENNReal.one_half_lt_one).ne).2 _
<| by simpa using hε.ne').exists
obtain ⟨δ,... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Complement | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 28
} | {
"line": 117,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS : Set G\ng : G\nh : IsComplement S {g}\nx : G\nx✝ : x ∈ ⊤\ny : ↑S × ↑{g}\nhy : (fun x ↦ ↑x.1 * ↑x.2) y = x * ↑y.2\n⊢ x ∈ S",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CancelMonoid.toRightCancelMonoid",
"Monoid.toMul... | [
"G : Type u_1\ninst✝ : Group G\nS : Set G\ng : G\nh : IsComplement S {g}\nx : G\nx✝ : x ∈ ⊤\ny : ↑S × ↑{g}\nhy : (fun x ↦ ↑x.1 * ↑x.2) y = x * ↑y.2\n⊢ ↑y.1 ∈ S"
] | rw [← mul_right_cancel hy] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Complement | {
"line": 201,
"column": 37
} | {
"line": 201,
"column": 42
} | {
"line": 201,
"column": 42
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑S\nhx : (↑x)⁻¹ * g ∈ T\nhx' : ∀ (y : ↑S), (fun s ↦ (↑s)⁻¹ * g ∈ T) y → y = x\n⊢ ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = (x, ⟨(↑x)⁻¹ * g, hx⟩)",
"ppTerm": "?m.168",
"assigned": true,
"usedConstants": [
"Subtype.mk.congr... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.GroupTheory.Complement | {
"line": 201,
"column": 37
} | {
"line": 201,
"column": 42
} | {
"line": 201,
"column": 42
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑S\nhx : (↑x)⁻¹ * g ∈ T\nhx' : ∀ (y : ↑S), (fun s ↦ (↑s)⁻¹ * g ∈ T) y → y = x\n⊢ ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = (x, ⟨(↑x)⁻¹ * g, hx⟩)",
"ppTerm": "?m.168",
"assigned": true,
"usedConstants": [
"Subtype.mk.congr... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Complement | {
"line": 201,
"column": 37
} | {
"line": 201,
"column": 42
} | {
"line": 201,
"column": 42
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑S\nhx : (↑x)⁻¹ * g ∈ T\nhx' : ∀ (y : ↑S), (fun s ↦ (↑s)⁻¹ * g ∈ T) y → y = x\n⊢ ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = (x, ⟨(↑x)⁻¹ * g, hx⟩)",
"ppTerm": "?m.168",
"assigned": true,
"usedConstants": [
"Subtype.mk.congr... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Complement | {
"line": 199,
"column": 2
} | {
"line": 202,
"column": 97
} | {
"line": 204,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\n⊢ IsComplement S T ↔ ∀ (g : G), ∃! s, (↑s)⁻¹ * g ∈ T",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"inv_mul_cancel_left",
... | [] | convert! isComplement_iff_existsUnique with g
constructor <;> rintro ⟨x, hx, hx'⟩
· exact ⟨(x, ⟨_, hx⟩), by simp, by aesop⟩
· exact ⟨x.1, by simp [← hx], fun y hy ↦ (Prod.ext_iff.1 <| by simpa using hx' (y, ⟨_, hy⟩)).1⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Complement | {
"line": 199,
"column": 2
} | {
"line": 202,
"column": 97
} | {
"line": 204,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\n⊢ IsComplement S T ↔ ∀ (g : G), ∃! s, (↑s)⁻¹ * g ∈ T",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subtype.mk.congr_simp",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"inv_mul_cancel_left",
... | [] | convert! isComplement_iff_existsUnique with g
constructor <;> rintro ⟨x, hx, hx'⟩
· exact ⟨(x, ⟨_, hx⟩), by simp, by aesop⟩
· exact ⟨x.1, by simp [← hx], fun y hy ↦ (Prod.ext_iff.1 <| by simpa using hx' (y, ⟨_, hy⟩)).1⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Complement | {
"line": 209,
"column": 37
} | {
"line": 209,
"column": 42
} | {
"line": 209,
"column": 42
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑T\nhx : g * (↑x)⁻¹ ∈ S\nhx' : ∀ (y : ↑T), (fun t ↦ g * (↑t)⁻¹ ∈ S) y → y = x\n⊢ ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = (⟨g * (↑x)⁻¹, hx⟩, x)",
"ppTerm": "?m.168",
"assigned": true,
"usedConstants": [
"mul_inv_cancel_r... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.GroupTheory.Complement | {
"line": 209,
"column": 37
} | {
"line": 209,
"column": 42
} | {
"line": 209,
"column": 42
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑T\nhx : g * (↑x)⁻¹ ∈ S\nhx' : ∀ (y : ↑T), (fun t ↦ g * (↑t)⁻¹ ∈ S) y → y = x\n⊢ ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = (⟨g * (↑x)⁻¹, hx⟩, x)",
"ppTerm": "?m.168",
"assigned": true,
"usedConstants": [
"mul_inv_cancel_r... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Complement | {
"line": 209,
"column": 37
} | {
"line": 209,
"column": 42
} | {
"line": 209,
"column": 42
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\ng : G\nx : ↑T\nhx : g * (↑x)⁻¹ ∈ S\nhx' : ∀ (y : ↑T), (fun t ↦ g * (↑t)⁻¹ ∈ S) y → y = x\n⊢ ∀ (y : ↑S × ↑T), (fun x ↦ ↑x.1 * ↑x.2 = g) y → y = (⟨g * (↑x)⁻¹, hx⟩, x)",
"ppTerm": "?m.168",
"assigned": true,
"usedConstants": [
"mul_inv_cancel_r... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Complement | {
"line": 440,
"column": 2
} | {
"line": 442,
"column": 24
} | {
"line": 443,
"column": 2
} | [
{
"pp": "case mp\nG : Type u_1\ninst✝ : Group G\nS T : Set G\nhST : IsComplement S T\ng : G\nh1 : 1 ∈ T\n⊢ ↑(hST.equiv g).1 = g → g ∈ S",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"Membership.mem",
"Set.Elem",
... | [
"case mpr\nG : Type u_1\ninst✝ : Group G\nS T : Set G\nhST : IsComplement S T\ng : G\nh1 : 1 ∈ T\n⊢ g ∈ S → ↑(hST.equiv g).1 = g"
] | · intro h
rw [← h]
exact Subtype.prop _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.Complement | {
"line": 449,
"column": 2
} | {
"line": 451,
"column": 24
} | {
"line": 452,
"column": 2
} | [
{
"pp": "case mp\nG : Type u_1\ninst✝ : Group G\nS T : Set G\nhST : IsComplement S T\ng : G\nh1 : 1 ∈ S\n⊢ ↑(hST.equiv g).2 = g → g ∈ T",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"Membership.mem",
"Set.Elem",
... | [
"case mpr\nG : Type u_1\ninst✝ : Group G\nS T : Set G\nhST : IsComplement S T\ng : G\nh1 : 1 ∈ S\n⊢ g ∈ T → ↑(hST.equiv g).2 = g"
] | · intro h
rw [← h]
exact Subtype.prop _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 453,
"column": 2
} | {
"line": 461,
"column": 67
} | {
"line": 462,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\ninst✝ : OpensMeasurableSpace α\nμ : Measure α\ns : ℕ → Set α\nh : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular\nh' : ∀ (n : ℕ), IsOpen[inst✝¹] (s n)\nh'' : univ ⊆ ⋃ n, s n\nr : ℝ≥0∞\nhm : ∀ (n : ℕ), MeasurableSet (s n)\nA : ℕ → Set α\... | [
"α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\ninst✝ : OpensMeasurableSpace α\nμ : Measure α\ns : ℕ → Set α\nh : ∀ (n : ℕ), (μ.restrict (s n)).OuterRegular\nh' : ∀ (n : ℕ), IsOpen[inst✝¹] (s n)\nh'' : univ ⊆ ⋃ n, s n\nr : ℝ≥0∞\nhm : ∀ (n : ℕ), MeasurableSet (s n)\nA : ℕ → Set α\nhAm : ∀ (n ... | have : ∀ n, ∃ U ⊇ A n, IsOpen U ∧ μ U < μ (A n) + δ n := by
intro n
have H₁ : ∀ t, μ.restrict (s n) t = μ (t ∩ s n) := fun t => restrict_apply' (hm n)
have Ht : μ.restrict (s n) (A n) ≠ ∞ := by
rw [H₁]
exact ((measure_mono (inter_subset_left.trans (subset_iUnion A n))).trans_lt HA).ne
rcases... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 525,
"column": 2
} | {
"line": 525,
"column": 75
} | {
"line": 526,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : OpensMeasurableSpace α\ns : Set α\nhs : AnalyticSet s\nhsc : AnalyticSet sᶜ\nu : Set α\nhsu : s ⊆ u\nhdu : Disjoint sᶜ u\nhmu : MeasurableSet u\n⊢ MeasurableSet s",
"ppTerm": "?m.45",
"assigned": ... | [
"α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : OpensMeasurableSpace α\ns : Set α\nhs : AnalyticSet s\nhsc : AnalyticSet sᶜ\nhsu : s ⊆ s\nhdu : Disjoint sᶜ s\nhmu : MeasurableSet s\n⊢ MeasurableSet s"
] | obtain rfl : s = u := hsu.antisymm (disjoint_compl_left_iff_subset.1 hdu) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.ContinuousMap.CocompactMap | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 65
} | {
"line": 201,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α ≃ₜ β\n⊢ Tendsto (⇑f) (cocompact α) (cocompact β)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"CocompactMap.tendsto_of_forall_preimage",
"Homeomorph.instEquivLike",
"Hom... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α ≃ₜ β\nK : Set β\nhK : IsCompact K\n⊢ IsCompact (⇑f ⁻¹' K)"
] | refine CocompactMap.tendsto_of_forall_preimage fun K hK => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 758,
"column": 2
} | {
"line": 759,
"column": 88
} | {
"line": 760,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ninst✝ : SFinite τ\nthis :\n (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ninst✝ : SFinite τ\nthis :\n (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod (sfiniteSeq... | rw [← sum_sfiniteSeq μ, ← sum_sfiniteSeq ν, ← sum_sfiniteSeq τ, prod_sum, prod_sum,
map_sum MeasurableEquiv.prodAssoc.measurable.aemeasurable, prod_sum, prod_sum, this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 58,
"column": 15
} | {
"line": 58,
"column": 58
} | {
"line": 58,
"column": 58
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ : Measure G\ninst✝ : μ.IsMulLeftInvariant\nc : ℝ≥0∞\ng : G\n⊢ Measure.map (fun x ↦ g * x) (c • μ) = c • μ",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"in... | [] | rw [Measure.map_smul, map_mul_left_eq_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 58,
"column": 15
} | {
"line": 58,
"column": 58
} | {
"line": 58,
"column": 58
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ : Measure G\ninst✝ : μ.IsMulLeftInvariant\nc : ℝ≥0∞\ng : G\n⊢ Measure.map (fun x ↦ g * x) (c • μ) = c • μ",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"in... | [] | rw [Measure.map_smul, map_mul_left_eq_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.Measure | {
"line": 58,
"column": 15
} | {
"line": 58,
"column": 58
} | {
"line": 58,
"column": 58
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ : Measure G\ninst✝ : μ.IsMulLeftInvariant\nc : ℝ≥0∞\ng : G\n⊢ Measure.map (fun x ↦ g * x) (c • μ) = c • μ",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"in... | [] | rw [Measure.map_smul, map_mul_left_eq_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 83,
"column": 82
} | {
"line": 86,
"column": 6
} | {
"line": 88,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nf : α → ℝ≥0∞\n⊢ withDensity 0 f = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withDensity",
"MeasureTheory.Measure",
"MeasurableSet",
"congrArg",
"MeasureTheory.Measure... | [] | by
ext s hs
rw [withDensity_apply _ hs]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 907,
"column": 2
} | {
"line": 907,
"column": 12
} | {
"line": 909,
"column": 2
} | [
{
"pp": "γ : Type u_3\ns : Set γ\ntγ : TopologicalSpace γ\ninst✝² : PolishSpace γ\ninst✝¹ : MeasurableSpace γ\ninst✝ : BorelSpace γ\nhs : IsClopenable s\n⊢ MeasurableSet s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MeasurableSet",
"BorelSpace.measurable_eq",
"BorelS... | [
"γ : Type u_3\ns : Set γ\ntγ : TopologicalSpace γ\ninst✝¹ : PolishSpace γ\nhs : IsClopenable s\ninst✝ : BorelSpace γ\nthis✝ : MeasurableSpace γ := borel γ\n⊢ MeasurableSet s"
] | borelize γ | Mathlib.Tactic.Borelize._aux_Mathlib_MeasureTheory_Constructions_BorelSpace_Basic___elabRules_Mathlib_Tactic_Borelize_tacticBorelize____1 | Mathlib.Tactic.Borelize.tacticBorelize___ |
Mathlib.Analysis.Real.Sqrt | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 59
} | {
"line": 165,
"column": 0
} | [
{
"pp": "case mpr.inl\ny : ℝ\nhy : 0 ≤ y\n⊢ √(y * y) = y",
"ppTerm": "?mpr.inl",
"assigned": true,
"usedConstants": [
"Real.sqrt_mul_self"
],
"usedFVars": [
"y",
"hy"
],
"usedGoals": []
},
{
"pp": "case mpr.inr\nx : ℝ\nhx : x < 0\n⊢ √x = 0",
"ppTerm": "?... | [] | exacts [sqrt_mul_self hy, sqrt_eq_zero_of_nonpos hx.le] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 644,
"column": 4
} | {
"line": 644,
"column": 81
} | {
"line": 645,
"column": 4
} | [
{
"pp": "case inr\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\nhfm : Measurable f\nthis : ∀ {μ : Measure α} [SFinite μ], IsFiniteMeasure μ → SFinite (μ.withDensity f)\nhμ : ¬IsFiniteMeasure μ\n⊢ SFinite (sum fun n ↦ (sfiniteSeq μ n).withDensity f)",
"ppTerm": "?inr"... | [
"case inr\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\nhfm : Measurable f\nthis✝ : ∀ {μ : Measure α} [SFinite μ], IsFiniteMeasure μ → SFinite (μ.withDensity f)\nhμ : ¬IsFiniteMeasure μ\nthis : ∀ (n : ℕ), SFinite ((sfiniteSeq μ n).withDensity f)\n⊢ SFinite (sum fun n ↦ (sfin... | have (n : ℕ) : SFinite ((sfiniteSeq μ n).withDensity f) := this inferInstance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 652,
"column": 10
} | {
"line": 652,
"column": 47
} | {
"line": 652,
"column": 47
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ≥0∞\nhfm : Measurable f\nμ : Measure α\ninst✝ : SFinite μ\nhμ : IsFiniteMeasure μ\ns : Set α := {x | f x = ∞}\nhs : MeasurableSet s\n⊢ μ.withDensity (sᶜ.indicator f) + μ.withDensity (s.indicator f) =\n μ.withDensity (sᶜ.indicator f) + su... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ≥0∞\nhfm : Measurable f\nμ : Measure α\ninst✝ : SFinite μ\nhμ : IsFiniteMeasure μ\ns : Set α := {x | f x = ∞}\nhs : MeasurableSet s\n⊢ μ.withDensity (sᶜ.indicator f) + μ.withDensity (s.indicator f) =\n μ.withDensity (sᶜ.indicator f) + μ.withDensity ... | ← withDensity_tsum (by measurability) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Complex.Basic | {
"line": 65,
"column": 67
} | {
"line": 65,
"column": 72
} | {
"line": 66,
"column": 0
} | [
{
"pp": "p : ℂ → Prop\n⊢ (∀ (x : ℂ), p x) ↔ ∀ (a b : ℝ), p { re := a, im := b }",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Real",
"Complex.im",
"Complex.re",
"Iff.intro",
"eq_true",
"of_eq_true",
"Complex",
"Complex.mk"
],
"usedF... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Complex.Basic | {
"line": 65,
"column": 67
} | {
"line": 65,
"column": 72
} | {
"line": 66,
"column": 0
} | [
{
"pp": "p : ℂ → Prop\n⊢ (∀ (x : ℂ), p x) ↔ ∀ (a b : ℝ), p { re := a, im := b }",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Real",
"Complex.im",
"Complex.re",
"Iff.intro",
"eq_true",
"of_eq_true",
"Complex",
"Complex.mk"
],
"usedF... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Complex.Basic | {
"line": 65,
"column": 67
} | {
"line": 65,
"column": 72
} | {
"line": 66,
"column": 0
} | [
{
"pp": "p : ℂ → Prop\n⊢ (∀ (x : ℂ), p x) ↔ ∀ (a b : ℝ), p { re := a, im := b }",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Real",
"Complex.im",
"Complex.re",
"Iff.intro",
"eq_true",
"of_eq_true",
"Complex",
"Complex.mk"
],
"usedF... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Complex.Basic | {
"line": 66,
"column": 67
} | {
"line": 66,
"column": 72
} | {
"line": 68,
"column": 0
} | [
{
"pp": "p : ℂ → Prop\n⊢ (∃ x, p x) ↔ ∃ a b, p { re := a, im := b }",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Real",
"Complex.im",
"Exists",
"Complex.re",
"Exists.casesOn",
"Iff.intro",
"Exists.intro",
"Complex",
"Complex.mk"
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Complex.Basic | {
"line": 66,
"column": 67
} | {
"line": 66,
"column": 72
} | {
"line": 68,
"column": 0
} | [
{
"pp": "p : ℂ → Prop\n⊢ (∃ x, p x) ↔ ∃ a b, p { re := a, im := b }",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Real",
"Complex.im",
"Exists",
"Complex.re",
"Exists.casesOn",
"Iff.intro",
"Exists.intro",
"Complex",
"Complex.mk"
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Complex.Basic | {
"line": 66,
"column": 67
} | {
"line": 66,
"column": 72
} | {
"line": 68,
"column": 0
} | [
{
"pp": "p : ℂ → Prop\n⊢ (∃ x, p x) ↔ ∃ a b, p { re := a, im := b }",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Real",
"Complex.im",
"Exists",
"Complex.re",
"Exists.casesOn",
"Iff.intro",
"Exists.intro",
"Complex",
"Complex.mk"
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Norm | {
"line": 39,
"column": 2
} | {
"line": 41,
"column": 23
} | {
"line": 43,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ |z.re| ≤ ‖z‖",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"Complex.norm_mul_self_eq_normSq",
"Complex.norm_nonneg",
"Real.partialOrder",
"Real.instLE",
"Real",
"HMul.h... | [] | rw [mul_self_le_mul_self_iff (abs_nonneg z.re) (Complex.norm_nonneg _), abs_mul_abs_self,
norm_mul_self_eq_normSq]
apply re_sq_le_normSq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Norm | {
"line": 39,
"column": 2
} | {
"line": 41,
"column": 23
} | {
"line": 43,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ |z.re| ≤ ‖z‖",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"Complex.norm_mul_self_eq_normSq",
"Complex.norm_nonneg",
"Real.partialOrder",
"Real.instLE",
"Real",
"HMul.h... | [] | rw [mul_self_le_mul_self_iff (abs_nonneg z.re) (Complex.norm_nonneg _), abs_mul_abs_self,
norm_mul_self_eq_normSq]
apply re_sq_le_normSq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Complex.Basic | {
"line": 757,
"column": 79
} | {
"line": 757,
"column": 90
} | {
"line": 757,
"column": 90
} | [
{
"pp": "z : ℂ\nx : ℚ\n⊢ (z / ↑x).re = z.re / ↑x",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"congrArg",
"Complex.im",
"Real.instDivInvMonoid",
"Real.instRatCast",
"Complex.instDivInvMonoid",
"id",
... | [
"z : ℂ\nx : ℚ\n⊢ { re := z.re / ↑x, im := z.im / ↑x }.re = z.re / ↑x"
] | div_ratCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Complex.Basic | {
"line": 758,
"column": 79
} | {
"line": 758,
"column": 90
} | {
"line": 758,
"column": 90
} | [
{
"pp": "z : ℂ\nx : ℚ\n⊢ (z / ↑x).im = z.im / ↑x",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"congrArg",
"Complex.im",
"Real.instDivInvMonoid",
"Real.instRatCast",
"Complex.instDivInvMonoid",
"id",
... | [
"z : ℂ\nx : ℚ\n⊢ { re := z.re / ↑x, im := z.im / ↑x }.im = z.im / ↑x"
] | div_ratCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Order | {
"line": 78,
"column": 12
} | {
"line": 78,
"column": 17
} | {
"line": 78,
"column": 17
} | [
{
"pp": "z : ℂ\n⊢ 0 ≤ z.re ^ 2 - z.im ^ 2 ∧ (z.re = 0 ∨ z.im = 0) → z.im = 0",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"False",
"Real.partialOrder",
"Real.instLE",
"Real",
"IsOrderedRing.toPosMulMono",
"Real.instZero... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Complex.Order | {
"line": 78,
"column": 12
} | {
"line": 78,
"column": 17
} | {
"line": 78,
"column": 17
} | [
{
"pp": "z : ℂ\n⊢ 0 ≤ z.re ^ 2 - z.im ^ 2 ∧ (z.re = 0 ∨ z.im = 0) → z.im = 0",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"False",
"Real.partialOrder",
"Real.instLE",
"Real",
"IsOrderedRing.toPosMulMono",
"Real.instZero... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Order | {
"line": 78,
"column": 12
} | {
"line": 78,
"column": 17
} | {
"line": 78,
"column": 17
} | [
{
"pp": "z : ℂ\n⊢ 0 ≤ z.re ^ 2 - z.im ^ 2 ∧ (z.re = 0 ∨ z.im = 0) → z.im = 0",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"False",
"Real.partialOrder",
"Real.instLE",
"Real",
"IsOrderedRing.toPosMulMono",
"Real.instZero... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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