module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.ENNReal.Action | {
"line": 98,
"column": 2
} | {
"line": 99,
"column": 5
} | {
"line": 101,
"column": 0
} | [
{
"pp": "r : ℝ≥0\ns : ℝ≥0∞\n⊢ (r • s).toReal = r • s.toReal",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofNNReal",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"ENNReal.smul_def",
"NNReal.instSMulOfReal",
"c... | [] | rw [ENNReal.smul_def, smul_eq_mul, toReal_mul, coe_toReal]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Action | {
"line": 98,
"column": 2
} | {
"line": 99,
"column": 5
} | {
"line": 101,
"column": 0
} | [
{
"pp": "r : ℝ≥0\ns : ℝ≥0∞\n⊢ (r • s).toReal = r • s.toReal",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofNNReal",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"ENNReal.smul_def",
"NNReal.instSMulOfReal",
"c... | [] | rw [ENNReal.smul_def, smul_eq_mul, toReal_mul, coe_toReal]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.MulAction | {
"line": 193,
"column": 11
} | {
"line": 193,
"column": 25
} | {
"line": 193,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\np : β\n⊢ p ∈ (fun x ↦ s • x) '' ball x ε ↔ p ∈ ball (s • x) (‖s‖ * ε)",
"ppTerm": "?m.38",
"assigned": true,
"usedCo... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\np : β\n⊢ (∃ x_1 ∈ ball x ε, s • x_1 = p) ↔ p ∈ ball (s • x) (‖s‖ * ε)"
] | Set.mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Normed.MulAction | {
"line": 205,
"column": 11
} | {
"line": 205,
"column": 25
} | {
"line": 205,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\np : β\n⊢ p ∈ (fun x ↦ s • x) '' closedBall x ε ↔ p ∈ closedBall (s • x) (‖s‖ * ε)",
"ppTerm": "?m.38",
"assigned": true,... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : NormSMulClass α β\ns : α\nhs : s ≠ 0\nx : β\nε : ℝ\np : β\n⊢ (∃ x_1 ∈ closedBall x ε, s • x_1 = p) ↔ p ∈ closedBall (s • x) (‖s‖ * ε)"
] | Set.mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.MeasurableSpace.Basic | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 7
} | {
"line": 160,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ng : β → α\nhg : Function.Surjective g\nS : Set α\nhS : ∃ s', MeasurableSet s' ∧ g ⁻¹' s' = g ⁻¹' S\n⊢ MeasurableSet S",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"MeasurableSet",
"congrArg",
"Exists",
"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 287,
"column": 18
} | {
"line": 287,
"column": 60
} | {
"line": 289,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\na b : ℚ\n⊢ ‖a * b‖ = ‖a‖ * ‖b‖",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Real.partialOrder",
"Semigroup.toMul",
"Real",
"Rat.cast_mul",
"HMul.hMul",
... | [] | by simp only [norm, Rat.cast_mul, abs_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 143,
"column": 22
} | {
"line": 143,
"column": 35
} | {
"line": 143,
"column": 35
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nf : β → Set α\ns : Set β\nhs : s.Countable\nh : ∀ b ∈ s, MeasurableSet (f b)\n⊢ MeasurableSet (⋂ b ∈ s, f b)ᶜ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Set.compl_iInter₂",
"Eq.mpr",
"MeasurableSet",
... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nf : β → Set α\ns : Set β\nhs : s.Countable\nh : ∀ b ∈ s, MeasurableSet (f b)\n⊢ MeasurableSet (⋃ i ∈ s, (f i)ᶜ)"
] | compl_iInter₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 298,
"column": 29
} | {
"line": 298,
"column": 73
} | {
"line": 300,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort u_6\ns t u : Set α\nm : MeasurableSpace α\np : Set α → Prop\nh : ∀ (s : Set α), p s ↔ MeasurableSet s\n⊢ ∀ (f : ℕ → Set α), (∀ (i : ℕ), p (f i)) → p (⋃ i, f i)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstan... | [] | simpa only [h] using! m.measurableSet_iUnion | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 298,
"column": 29
} | {
"line": 298,
"column": 73
} | {
"line": 300,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort u_6\ns t u : Set α\nm : MeasurableSpace α\np : Set α → Prop\nh : ∀ (s : Set α), p s ↔ MeasurableSet s\n⊢ ∀ (f : ℕ → Set α), (∀ (i : ℕ), p (f i)) → p (⋃ i, f i)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstan... | [] | simpa only [h] using! m.measurableSet_iUnion | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 298,
"column": 29
} | {
"line": 298,
"column": 73
} | {
"line": 300,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort u_6\ns t u : Set α\nm : MeasurableSpace α\np : Set α → Prop\nh : ∀ (s : Set α), p s ↔ MeasurableSet s\n⊢ ∀ (f : ℕ → Set α), (∀ (i : ℕ), p (f i)) → p (⋃ i, f i)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstan... | [] | simpa only [h] using! m.measurableSet_iUnion | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 556,
"column": 2
} | {
"line": 560,
"column": 56
} | {
"line": 562,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nι : Type u_6\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j ↦ EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h :... | [] | classical
refine ⟨fun x => if hx : x ∈ ⋃ i, t i then f ⟨x, hx⟩ else g default x,
hfm.dite ((hg default).comp measurable_subtype_coe) (.iUnion t_meas), fun i x hx => ?_⟩
simp only [dif_pos (mem_iUnion.2 ⟨i, hx⟩)]
exact iUnionLift_of_mem ⟨x, mem_iUnion.2 ⟨i, hx⟩⟩ hx | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 352,
"column": 35
} | {
"line": 352,
"column": 67
} | {
"line": 352,
"column": 68
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ns : Set β\n⊢ ⊤ (f ⁻¹' s) = ⊤ (s ∩ range f)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",
"PartialOrder.toPreorder",
"Preorder... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\ns : Set β\n⊢ ⊤ (f ⁻¹' s) = ⊤ (f '' f ⁻¹' s)"
] | ← image_preimage_eq_inter_range, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Sign.Defs | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 18
} | {
"line": 64,
"column": 0
} | [
{
"pp": "a : SignType\n⊢ a = -1 ∨ a = 0 ∨ a = 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SignType.ctorIdx",
"False",
"SignType.instOne",
"congrArg",
"False.elim",
"noConfusion_of_Nat",
"SignType.pos",
"SignType.instNeg",
"SignTy... | [] | cases a <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Sign.Defs | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 18
} | {
"line": 64,
"column": 0
} | [
{
"pp": "a : SignType\n⊢ a = -1 ∨ a = 0 ∨ a = 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SignType.ctorIdx",
"False",
"SignType.instOne",
"congrArg",
"False.elim",
"noConfusion_of_Nat",
"SignType.pos",
"SignType.instNeg",
"SignTy... | [] | cases a <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sign.Defs | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 18
} | {
"line": 64,
"column": 0
} | [
{
"pp": "a : SignType\n⊢ a = -1 ∨ a = 0 ∨ a = 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"SignType.ctorIdx",
"False",
"SignType.instOne",
"congrArg",
"False.elim",
"noConfusion_of_Nat",
"SignType.pos",
"SignType.instNeg",
"SignTy... | [] | cases a <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Sign.Basic | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 18
} | {
"line": 38,
"column": 0
} | [
{
"pp": "s : SignType\nk : ℕ\n⊢ (s * s) ^ k * s = s",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"one_pow",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"SignType.instHasDistribNeg",
"HMul.hMul",
"SignType.instCommGroupWithZero",
"Monoid.to... | [] | cases s <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Sign.Basic | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 18
} | {
"line": 47,
"column": 0
} | [
{
"pp": "case inr\ns : SignType\nhs : s ≠ 0\nk : ℤ\n⊢ (s * s) ^ k * s = s",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"SignType.instHasDistribNeg",
"InvOneClass.toOne",
"HMul.hMul",
"GroupWithZero.toDivInvMonoid",
"Di... | [] | cases s <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Sign.Defs | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 18
} | {
"line": 252,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : One α\ninst✝ : SubtractionMonoid α\ns : SignType\n⊢ ↑(-s) = -↑s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"SignType.cast",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNeg",
"SignTy... | [] | cases s <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Sign.Defs | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 18
} | {
"line": 252,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : One α\ninst✝ : SubtractionMonoid α\ns : SignType\n⊢ ↑(-s) = -↑s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"SignType.cast",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNeg",
"SignTy... | [] | cases s <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sign.Defs | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 18
} | {
"line": 252,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : One α\ninst✝ : SubtractionMonoid α\ns : SignType\n⊢ ↑(-s) = -↑s",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"SignType.cast",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNeg",
"SignTy... | [] | cases s <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Inv | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 18
} | {
"line": 249,
"column": 2
} | [
{
"pp": "a : EReal\n⊢ ↑(sign a) * (↑a.abs)⁻¹ = a⁻¹",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"SignType.cast",
"Eq.mpr",
"EReal.abs",
"NegZeroClass.toNeg",
"MulOne.toOne",
"SignType.coe_one",
"Real",
... | [] | induction a with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Data.EReal.Inv | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 18
} | {
"line": 262,
"column": 2
} | [
{
"pp": "a : EReal\n⊢ ↑(sign a) * ↑a.abs⁻¹ = a⁻¹",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"SignType.cast",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"EReal.abs",
"NegZeroClass.toNeg",
"MulOn... | [] | induction a with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 41
} | {
"line": 147,
"column": 4
} | [
{
"pp": "case neg\nx : EReal\nhx : x < ⊤\nhx_bot : ¬x = ⊥\n⊢ ∃ i', Ico ↑i' ⊤ ⊆ Ici x ∩ {⊤}ᶜ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"Set.Ici",
"Compl.compl",
"LT.lt.ne_top",
"EReal.canLift",
"PartialOrder.toPreorder... | [
"case neg\nx : ℝ\nhx : ↑x < ⊤\nhx_bot : ¬↑x = ⊥\n⊢ ∃ i', Ico ↑i' ⊤ ⊆ Ici ↑x ∩ {⊤}ᶜ"
] | lift x to ℝ using ⟨hx.ne_top, hx_bot⟩ | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 310,
"column": 34
} | {
"line": 310,
"column": 39
} | {
"line": 310,
"column": 39
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ 0 < ?m.77",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h₃"
],
"usedGoals": []
}
] | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 310,
"column": 34
} | {
"line": 310,
"column": 39
} | {
"line": 310,
"column": 39
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ 0 < ?m.77",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h₃"
],
"usedGoals": []
}
] | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 310,
"column": 34
} | {
"line": 310,
"column": 39
} | {
"line": 310,
"column": 39
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ 0 < ?m.77",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h₃"
],
"usedGoals": []
}
] | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 310,
"column": 45
} | {
"line": 310,
"column": 50
} | {
"line": 310,
"column": 50
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ c ≠ ⊤",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal",
"instTopEReal",
"id",
"True",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 310,
"column": 45
} | {
"line": 310,
"column": 50
} | {
"line": 310,
"column": 50
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ c ≠ ⊤",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal",
"instTopEReal",
"id",
"True",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 310,
"column": 45
} | {
"line": 310,
"column": 50
} | {
"line": 310,
"column": 50
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ c ≠ ⊤",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal",
"instTopEReal",
"id",
"True",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 313,
"column": 34
} | {
"line": 313,
"column": 39
} | {
"line": 313,
"column": 39
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ 0 < ?m.121",
"ppTerm": "?m.122",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h₃"
],
"usedGoals": []
}
] | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 313,
"column": 34
} | {
"line": 313,
"column": 39
} | {
"line": 313,
"column": 39
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ 0 < ?m.121",
"ppTerm": "?m.122",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h₃"
],
"usedGoals": []
}
] | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 313,
"column": 34
} | {
"line": 313,
"column": 39
} | {
"line": 313,
"column": 39
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ 0 < ?m.121",
"ppTerm": "?m.122",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h₃"
],
"usedGoals": []
}
] | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 313,
"column": 45
} | {
"line": 313,
"column": 50
} | {
"line": 313,
"column": 50
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ c ≠ ⊤",
"ppTerm": "?m.123",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal",
"instTopEReal",
"id",
"True",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 313,
"column": 45
} | {
"line": 313,
"column": 50
} | {
"line": 313,
"column": 50
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ c ≠ ⊤",
"ppTerm": "?m.123",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal",
"instTopEReal",
"id",
"True",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 313,
"column": 45
} | {
"line": 313,
"column": 50
} | {
"line": 313,
"column": 50
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\nh₃ : 0 < c\n⊢ c ≠ ⊤",
"ppTerm": "?m.123",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal",
"instTopEReal",
"id",
"True",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 319,
"column": 65
} | {
"line": 319,
"column": 70
} | {
"line": 319,
"column": 70
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ 0 ≤ -c",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"Preorder.toLE",
"instZeroEReal",
"LE.le... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 319,
"column": 65
} | {
"line": 319,
"column": 70
} | {
"line": 319,
"column": 70
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ 0 ≤ -c",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"Preorder.toLE",
"instZeroEReal",
"LE.le... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 319,
"column": 65
} | {
"line": 319,
"column": 70
} | {
"line": 319,
"column": 70
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ 0 ≤ -c",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"Preorder.toLE",
"instZeroEReal",
"LE.le... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 319,
"column": 76
} | {
"line": 319,
"column": 81
} | {
"line": 319,
"column": 81
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ -c ≠ ⊤",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal.instNeg",
"EReal",
"instTopEReal",
"id",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 319,
"column": 76
} | {
"line": 319,
"column": 81
} | {
"line": 319,
"column": 81
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ -c ≠ ⊤",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal.instNeg",
"EReal",
"instTopEReal",
"id",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 319,
"column": 76
} | {
"line": 319,
"column": 81
} | {
"line": 319,
"column": 81
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ -c ≠ ⊤",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal.instNeg",
"EReal",
"instTopEReal",
"id",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 324,
"column": 55
} | {
"line": 324,
"column": 60
} | {
"line": 324,
"column": 60
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\n⊢ 0 ≤ c",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h₁"
],
"usedGoals": []
}
] | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 324,
"column": 55
} | {
"line": 324,
"column": 60
} | {
"line": 324,
"column": 60
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\n⊢ 0 ≤ c",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h₁"
],
"usedGoals": []
}
] | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 324,
"column": 55
} | {
"line": 324,
"column": 60
} | {
"line": 324,
"column": 60
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\n⊢ 0 ≤ c",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h₁"
],
"usedGoals": []
}
] | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 324,
"column": 66
} | {
"line": 324,
"column": 71
} | {
"line": 324,
"column": 71
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\n⊢ c ≠ ⊤",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal",
"instTopEReal",
"id",
"True",
"of_eq_tru... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 324,
"column": 66
} | {
"line": 324,
"column": 71
} | {
"line": 324,
"column": 71
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\n⊢ c ≠ ⊤",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal",
"instTopEReal",
"id",
"True",
"of_eq_tru... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 324,
"column": 66
} | {
"line": 324,
"column": 71
} | {
"line": 324,
"column": 71
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\n⊢ c ≠ ⊤",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal",
"instTopEReal",
"id",
"True",
"of_eq_tru... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 329,
"column": 55
} | {
"line": 329,
"column": 60
} | {
"line": 329,
"column": 60
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ 0 ≤ -c",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"Preorder.toLE",
"instZeroEReal",
"LE.le... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 329,
"column": 55
} | {
"line": 329,
"column": 60
} | {
"line": 329,
"column": 60
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ 0 ≤ -c",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"Preorder.toLE",
"instZeroEReal",
"LE.le... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 329,
"column": 55
} | {
"line": 329,
"column": 60
} | {
"line": 329,
"column": 60
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ 0 ≤ -c",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"Preorder.toLE",
"instZeroEReal",
"LE.le... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 329,
"column": 66
} | {
"line": 329,
"column": 71
} | {
"line": 329,
"column": 71
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ -c ≠ ⊤",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal.instNeg",
"EReal",
"instTopEReal",
"id",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 329,
"column": 66
} | {
"line": 329,
"column": 71
} | {
"line": 329,
"column": 71
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ -c ≠ ⊤",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal.instNeg",
"EReal",
"instTopEReal",
"id",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 329,
"column": 66
} | {
"line": 329,
"column": 71
} | {
"line": 329,
"column": 71
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : c ≤ 0\nh₂ : c ≠ ⊥\n⊢ -c ≠ ⊤",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"EReal.instNeg",
"EReal",
"instTopEReal",
"id",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 342,
"column": 6
} | {
"line": 344,
"column": 51
} | {
"line": 345,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : TopologicalSpace α\nγ : Type u_4\ninst✝² : LinearOrder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : ClosedIciTopology γ\nf : α → γ\ns : Set α\nhs : IsClosed[inst✝³] s\nhf : ∀ x ∈ s, ∀ y < f x, ∀ᶠ (x : α) in 𝓝 x, x ∈ s → y < f x\nx : α\ny : γ\nh : (x, y).1 ∈ s → (x, y).2 < f... | [] | have ⟨y', hy', z, hz, h₁⟩ := (h hx).exists_disjoint_Iio_Ioi
filter_upwards [(hf x hx z hz).prodMk_nhds (eventually_lt_nhds hy')]
with _ ⟨h₂, h₃⟩ h₄ using h₁ _ h₃ _ <| h₂ h₄ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 342,
"column": 6
} | {
"line": 344,
"column": 51
} | {
"line": 345,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : TopologicalSpace α\nγ : Type u_4\ninst✝² : LinearOrder γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : ClosedIciTopology γ\nf : α → γ\ns : Set α\nhs : IsClosed[inst✝³] s\nhf : ∀ x ∈ s, ∀ y < f x, ∀ᶠ (x : α) in 𝓝 x, x ∈ s → y < f x\nx : α\ny : γ\nh : (x, y).1 ∈ s → (x, y).2 < f... | [] | have ⟨y', hy', z, hz, h₁⟩ := (h hx).exists_disjoint_Iio_Ioi
filter_upwards [(hf x hx z hz).prodMk_nhds (eventually_lt_nhds hy')]
with _ ⟨h₂, h₃⟩ h₄ using h₁ _ h₃ _ <| h₂ h₄ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 116,
"column": 76
} | {
"line": 116,
"column": 81
} | {
"line": 117,
"column": 2
} | [
{
"pp": "m : ℕ\nhm : 0 < m\n⊢ ∀ᶠ (n : ℕ) in atTop, 0 ≤ (fun n ↦ ↑(n % m)) n",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real.instLE",
"Real",
"Real.instZero",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 166,
"column": 6
} | {
"line": 168,
"column": 44
} | {
"line": 169,
"column": 2
} | [] | [] | μ s + μ t ≤ ∞ := le_top
_ = m (f i) := (h (f i) hs ht).symm
_ ≤ ∑' i, m (f i) := ENNReal.le_tsum i | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 62
} | {
"line": 192,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nβ : Type u_2\nf : β → α\nh : Monotone m ∨ Surjective f\n⊢ (comap f) (OuterMeasure.ofFunction m m_empty) = OuterMeasure.ofFunction (fun s ↦ m (f '' s)) ⋯",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Semir... | [
"case refine_1\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nβ : Type u_2\nf : β → α\nh : Monotone m ∨ Surjective f\ns : Set β\n⊢ ((comap f) (OuterMeasure.ofFunction m m_empty)) s ≤ m (f '' s)",
"case refine_2\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nβ : Type u_2\nf : β → α\nh : Monotone m ∨ Surj... | refine le_antisymm (le_ofFunction.2 fun s => ?_) fun s => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 216,
"column": 40
} | {
"line": 216,
"column": 72
} | {
"line": 216,
"column": 73
} | [
{
"pp": "case refine_1\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nβ : Type u_2\nf : α → β\nhf : Injective f\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\n⊢ s ∩ range f ⊆ ⋃ i, f '' t i",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"case refine_1\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nβ : Type u_2\nf : α → β\nhf : Injective f\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\n⊢ f '' f ⁻¹' s ⊆ ⋃ i, f '' t i"
] | ← image_preimage_eq_inter_range, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 262,
"column": 26
} | {
"line": 264,
"column": 67
} | {
"line": 266,
"column": 0
} | [
{
"pp": "r : ℝ≥0\nhr : r < 1\n⊢ Tendsto (fun a ↦ ↑(r ^ a)) atTop (𝓝 ↑0)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Real",
"Real.instArchimedean",
"PartialOrder.toPreorder",
"PseudoMetricSpace.toUniformSpace",
"Preorder.toLE",
"SemilatticeInf.toPar... | [] | by
simp only [NNReal.coe_pow, NNReal.coe_zero,
_root_.tendsto_pow_atTop_nhds_zero_of_lt_one r.coe_nonneg hr] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 284,
"column": 2
} | {
"line": 285,
"column": 98
} | {
"line": 286,
"column": 2
} | [
{
"pp": "r : ℝ≥0∞\nh : Tendsto (fun n ↦ r ^ n) atTop (𝓝 0)\n⊢ r ≠ ∞",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"CommSemiring.toSemiring",
"Nat.pos_iff_ne_zero",
"Filter.Eventually",
"instArchimedeanNat",
"Filter.atTop_neBot",
"Preo... | [
"r : ℝ≥0\nh : Tendsto (fun n ↦ ↑r ^ n) atTop (𝓝 0)\n⊢ ↑r < 1"
] | · refine fun hr ↦ top_ne_zero (tendsto_nhds_unique (EventuallyEq.tendsto ?_) (hr ▸ h))
exact eventually_atTop.mpr ⟨1, fun _ hn ↦ pow_eq_top_iff.mpr ⟨rfl, Nat.pos_iff_ne_zero.mp hn⟩⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 526,
"column": 54
} | {
"line": 526,
"column": 67
} | {
"line": 526,
"column": 67
} | [
{
"pp": "a : ℝ\nh : 0 < a\nx : ℝ\np : EReal × EReal\np1_gt : ↑(2 * max (x + 1) 0 / a) < p.1\np2_gt : ↑(a / 2) < p.2\np1_pos : 0 < p.1\n⊢ 0 < ↑(a / 2)",
"ppTerm": "?m.141",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"instHDiv",
"Real.instZero... | [
"a : ℝ\nh : 0 < a\nx : ℝ\np : EReal × EReal\np1_gt : ↑(2 * max (x + 1) 0 / a) < p.1\np2_gt : ↑(a / 2) < p.2\np1_pos : 0 < p.1\n⊢ 0 < a / 2"
] | EReal.coe_pos | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 543,
"column": 2
} | {
"line": 543,
"column": 65
} | {
"line": 544,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nf : ℕ → α\nhr : r < 1\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nthis : HasSum (fun n ↦ C * r ^ n) (C / (1 - r))\n⊢ dist (f n) a ≤ C * r ^ n / (1 - r)",
"ppTerm": "?m.59",
"assigned": true,
... | [
"α : Type u_1\ninst✝ : PseudoMetricSpace α\nr C : ℝ\nf : ℕ → α\nhr : r < 1\nhu : ∀ (n : ℕ), dist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\nthis : HasSum (fun n ↦ C * r ^ n) (C / (1 - r))\n⊢ C * r ^ n / (1 - r) = ∑' (m : ℕ), C * r ^ (n + m)"
] | convert! dist_le_tsum_of_dist_le_of_tendsto _ hu ⟨_, this⟩ ha n | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 498,
"column": 52
} | {
"line": 500,
"column": 24
} | {
"line": 502,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\nx : α\nγ : Type u_5\ninst✝⁴ : AddCommMonoid γ\ninst✝³ : LinearOrder γ\ninst✝² : IsOrderedAddMonoid γ\ninst✝¹ : TopologicalSpace γ\ninst✝ : OrderTopology γ\nf g : α → γ\nhf : LowerSemicontinuousAt f x\nhg : LowerSemicontinuousAt g x\nhcont : ContinuousAt (fun p... | [] | by
simp_rw [← lowerSemicontinuousWithinAt_univ_iff] at *
exact hf.add' hg hcont | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.PiSystem | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 21
} | {
"line": 90,
"column": 2
} | [
{
"pp": "α : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\n⊢ IsPiSystem (insert ∅ S)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Insert.insert",
"Set.instInter",
"Inter.inter",
"Set.Nonempty",
"Set.instInsert",
"Set.instEmpt... | [
"α : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns : Set α\nhs : s ∈ insert ∅ S\nt : Set α\nht : t ∈ insert ∅ S\nhst : (s ∩ t).Nonempty\n⊢ s ∩ t ∈ insert ∅ S"
] | intro s hs t ht hst | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.PiSystem | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 21
} | {
"line": 99,
"column": 2
} | [
{
"pp": "α : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\n⊢ IsPiSystem (insert univ S)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Set.univ",
"Membership.mem",
"Insert.insert",
"Set.instInter",
"Inter.inter",
"Set.Nonempty",
"Set.instInsert"... | [
"α : Type u_1\nS : Set (Set α)\nh_pi : IsPiSystem S\ns : Set α\nhs : s ∈ insert univ S\nt : Set α\nht : t ∈ insert univ S\nhst : (s ∩ t).Nonempty\n⊢ s ∩ t ∈ insert univ S"
] | intro s hs t ht hst | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.OuterMeasure.Induced | {
"line": 119,
"column": 4
} | {
"line": 119,
"column": 67
} | {
"line": 121,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ i, f i)\nmsU : ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)), m (⋃ i, f i) ⋯ ≤ ∑' (i : ℕ), m (f i) ⋯\ns : ℕ → Set α\ni : ℕ\nhi : ¬P (s i)\n⊢ extend m (⋃ i, s i) ≤ ∑' (i : ℕ), extend... | [] | exact le_trans (le_iInf fun h => hi.elim h) (ENNReal.le_tsum i) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.PiSystem | {
"line": 369,
"column": 6
} | {
"line": 374,
"column": 29
} | {
"line": 375,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_3\nι : Type u_4\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ x ∈ t, f x ∈ π x\nhti : ∀ y ∈ t, y = i\nhi : i ∈ t\n⊢ ⋂ x ∈ t, f x ∈ π i ∨ ⋂ x ∈ t, f x ∈ {univ}",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.mem... | [] | have ht_eq_i : t = {i} := by
ext1 x
rw [Finset.mem_singleton]
exact ⟨fun h => hti x h, fun h => h.symm ▸ hi⟩
simp only [ht_eq_i, Finset.mem_singleton, iInter_iInter_eq_left]
exact Or.inl (hfπ i hi) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.PiSystem | {
"line": 369,
"column": 6
} | {
"line": 374,
"column": 29
} | {
"line": 375,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_3\nι : Type u_4\nπ : ι → Set (Set α)\ni : ι\nt : Finset ι\nf : ι → Set α\nhfπ : ∀ x ∈ t, f x ∈ π x\nhti : ∀ y ∈ t, y = i\nhi : i ∈ t\n⊢ ⋂ x ∈ t, f x ∈ π i ∨ ⋂ x ∈ t, f x ∈ {univ}",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.mem... | [] | have ht_eq_i : t = {i} := by
ext1 x
rw [Finset.mem_singleton]
exact ⟨fun h => hti x h, fun h => h.symm ▸ hi⟩
simp only [ht_eq_i, Finset.mem_singleton, iInter_iInter_eq_left]
exact Or.inl (hfπ i hi) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.OuterMeasure.AE | {
"line": 255,
"column": 59
} | {
"line": 256,
"column": 59
} | {
"line": 258,
"column": 0
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝² : FunLike F (Set α) ℝ≥0∞\ninst✝¹ : OuterMeasureClass F α\nμ : F\nM : Type u_4\ninst✝ : One M\nf : α → M\ns : Set α\n⊢ s.mulIndicator f =ᵐ[μ] 1 ↔ μ (s ∩ Function.mulSupport f) = 0",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Filter.instM... | [] | by
simp [EventuallyEq, eventually_iff, ae, compl_setOf]; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.MeasureSpaceDef | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 18
} | {
"line": 148,
"column": 18
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ₁ μ₂ : Measure α\n⊢ μ₁ = μ₂ → ∀ (s : Set α), μ₁ s = μ₂ s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"ENNReal",
"Eq.ndrec",
"Eq",
"DFunLike.coe",
"MeasureTheory.Measure.instF... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ₁ : Measure α\ns : Set α\n⊢ μ₁ s = μ₁ s"
] | rintro rfl s | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.MeasureTheory.Function.AEMeasurableSequence | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 13
} | {
"line": 54,
"column": 2
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nβ : Type u_3\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : ι → α → β\nμ : Measure α\np : α → (ι → β) → Prop\nhf : ∀ (i : ι), AEMeasurable (f i) μ\nx✝ : α\nhx : x✝ ∈ aeSeqSet hf p\ni : ι\nx : α\nh : x ∈ {x | (∀ (i : ι), f i x = AEMeasurable.mk (f i) ⋯ x) ∧ p x f... | [] | exact h.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated | {
"line": 75,
"column": 17
} | {
"line": 75,
"column": 28
} | {
"line": 76,
"column": 4
} | [
{
"pp": "case mpr.inr.inr.inl\nα : Type u_1\ns : Set α\n⊢ ∃ s', (fun x ↦ x ∈ s) ⁻¹' s' = sᶜ",
"ppTerm": "?mpr.inr.inr.inl",
"assigned": true,
"usedConstants": [
"False",
"Compl.compl",
"Membership.mem",
"Set.instSingletonSet",
"Set.instCompl",
"Set.preimage",
... | [
"case h\nα : Type u_1\ns : Set α\n⊢ (fun x ↦ x ∈ s) ⁻¹' {False} = sᶜ"
] | (Mathlib.Tactic.useSyntax
"use"
[]
(group)
[(choice
(«term{_}» "{" [`False] "}")
(Term.structInst
"{"
[]
(Term.structInstFields [(Term.structInstField (Term.structInstLVal `False []) [])])
(Term.optEllipsis [])
[]
"}"))]) | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 38
} | {
"line": 98,
"column": 39
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : MeasurableEmbedding f\ns : Set β\n⊢ MeasurableSet (f ⁻¹' s) ↔ MeasurableSet (s ∩ range f)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"co... | [
"α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nhf : MeasurableEmbedding f\ns : Set β\n⊢ MeasurableSet (f ⁻¹' s) ↔ MeasurableSet (f '' f ⁻¹' s)"
] | ← image_preimage_eq_inter_range, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 208,
"column": 2
} | {
"line": 209,
"column": 60
} | {
"line": 211,
"column": 0
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nh : NullMeasurableSet s μ\n⊢ toMeasurable μ s =ᵐ[μ] s",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"Exists.choose_spec",
"MeasureTheory.Measure",
"Measura... | [] | rw [toMeasurable_def, dif_pos]
exact (exists_measurable_superset_ae_eq h).choose_spec.2.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.NullMeasurable | {
"line": 208,
"column": 2
} | {
"line": 209,
"column": 60
} | {
"line": 211,
"column": 0
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nh : NullMeasurableSet s μ\n⊢ toMeasurable μ s =ᵐ[μ] s",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"Exists.choose_spec",
"MeasureTheory.Measure",
"Measura... | [] | rw [toMeasurable_def, dif_pos]
exact (exists_measurable_superset_ae_eq h).choose_spec.2.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 742,
"column": 2
} | {
"line": 742,
"column": 17
} | {
"line": 743,
"column": 2
} | [
{
"pp": "case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf✝ : α → β\ng✝ : β → α\nf : α → β\ng : β → α\nhf : MeasurableEmbedding f\nhg : MeasurableEmbedding g\nF : Set α →... | [
"case a\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nδ' : Type u_5\nι : Sort uι\ns t u : Set α\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf✝ : α → β\ng✝ : β → α\nf : α → β\ng : β → α\nhf : MeasurableEmbedding f\nhg : MeasurableEmbedding g\nF : Set α → Set α := fu... | rw [mem_iInter] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving | {
"line": 153,
"column": 8
} | {
"line": 153,
"column": 17
} | {
"line": 153,
"column": 18
} | [
{
"pp": "case inr\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns : Set α\ne : α ≃ α\nhe : QuasiMeasurePreserving (⇑e) μ μ\nhe' : QuasiMeasurePreserving (⇑e.symm) μ μ\nhs : ⇑e '' s =ᵐ[μ] s\nk : ℕ\n⊢ ⇑(e ^ (-↑k)).symm ⁻¹' s =ᵐ[μ] s",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [
"case inr\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ns : Set α\ne : α ≃ α\nhe : QuasiMeasurePreserving (⇑e) μ μ\nhe' : QuasiMeasurePreserving (⇑e.symm) μ μ\nhs : ⇑e '' s =ᵐ[μ] s\nk : ℕ\n⊢ ⇑(e ^ ↑k)⁻¹.symm ⁻¹' s =ᵐ[μ] s"
] | zpow_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 26
} | {
"line": 250,
"column": 2
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nγ : Type u_5\nf g : α → γ\ns : Set α\ninst✝ : DecidablePred fun x ↦ x ∈ s\nhs_zero : s ∈ ae μ\n⊢ (fun x ↦ if x ∈ s then f x else g x) =ᵐ[μ] f",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Measu... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nγ : Type u_5\nf g : α → γ\ns : Set α\ninst✝ : DecidablePred fun x ↦ x ∈ s\nhs_zero : s ∈ ae μ\n⊢ ∀ a ∈ s, (if a ∈ s then f a else g a) = f a"
] | filter_upwards [hs_zero] | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Topology.Order.AtTopBotIxx | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 78
} | {
"line": 102,
"column": 2
} | [
{
"pp": "case inl\nX : Type u_1\ninst✝² : LinearOrder X\ninst✝¹ : TopologicalSpace X\ninst✝ : OrderTopology X\ns : Set X\nb : X\nhsb : s ⊆ Iio b\nhs : ∀ a' < b, ∃ a < b, Ioo a b ⊆ s\nhb : IsSuccPrelimit b\nhb' : Iio b = ∅\nthis : IsEmpty ↑s\n⊢ map Subtype.val atTop = 𝓝[<] b",
"ppTerm": "?inl",
"assigne... | [] | rw [filter_eq_bot_of_isEmpty atTop, Filter.map_bot, hb', nhdsWithin_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 605,
"column": 2
} | {
"line": 605,
"column": 51
} | {
"line": 606,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : SecondCountableTopology α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\nh✝ : Nonempty α\ninhabited_h : Inhabited α\n⊢ Nonempty (μ.FiniteSpanningSetsIn {K | IsOpen[inst✝²] K... | [
"case inr\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : SecondCountableTopology α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\nh✝ : Nonempty α\ninhabited_h : Inhabited α\nS : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}\n⊢ Nonempty (μ.F... | let S : Set (Set α) := { s | IsOpen s ∧ μ s < ∞ } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 606,
"column": 2
} | {
"line": 607,
"column": 46
} | {
"line": 608,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : SecondCountableTopology α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\nh✝ : Nonempty α\ninhabited_h : Inhabited α\nS : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}\n⊢ N... | [
"case inr\nα : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : SecondCountableTopology α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\nh✝ : Nonempty α\ninhabited_h : Inhabited α\nS : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}\nT : Set (Set α)... | obtain ⟨T, T_count, TS, hT⟩ : ∃ T : Set (Set α), T.Countable ∧ T ⊆ S ∧ ⋃₀ T = ⋃₀ S :=
isOpen_sUnion_countable S fun s hs => hs.1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 611,
"column": 41
} | {
"line": 611,
"column": 53
} | {
"line": 611,
"column": 53
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : SecondCountableTopology α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\nh✝ : Nonempty α\ninhabited_h : Inhabited α\nS : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}\nT : Set (Set ... | [
"α : Type u_1\nβ : Type u_2\nδ : Type u_3\nι : Type u_4\ninst✝² : TopologicalSpace α\ninst✝¹ : SecondCountableTopology α\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\nh✝ : Nonempty α\ninhabited_h : Inhabited α\nS : Set (Set α) := {s | IsOpen[inst✝²] s ∧ μ s < ∞}\nT : Set (Set α)\nT_count ... | sUnion_empty | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Trim | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 39
} | {
"line": 71,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\n⊢ Measurable f",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"le_rfl",
"MeasurableSpace.comap",
"Measurable... | [] | exact Measurable.of_comap_le le_rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Trim | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 39
} | {
"line": 71,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\n⊢ Measurable f",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"le_rfl",
"MeasurableSpace.comap",
"Measurable... | [] | exact Measurable.of_comap_le le_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Trim | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 39
} | {
"line": 71,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nf : α → β\nhf : Measurable f\ns : Set β\nhs : MeasurableSet s\n⊢ Measurable f",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"le_rfl",
"MeasurableSpace.comap",
"Measurable... | [] | exact Measurable.of_comap_le le_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 427,
"column": 52
} | {
"line": 427,
"column": 97
} | {
"line": 428,
"column": 4
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\ns : Set α\nν : Measure β\nf : α → β\nhf : QuasiMeasurePreserving f μ ν\nt : Set β\nhmaps : MapsTo f s t\nu : Set β\nhum : MeasurableSet u\nthis : ν (u ∩ t) = 0 → μ (f ⁻¹' u ∩ s) = 0\n⊢ (ν.restrict t) u = 0 → (... | [] | simpa [hum, hf.measurable, hf.measurable hum] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 427,
"column": 52
} | {
"line": 427,
"column": 97
} | {
"line": 428,
"column": 4
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\ns : Set α\nν : Measure β\nf : α → β\nhf : QuasiMeasurePreserving f μ ν\nt : Set β\nhmaps : MapsTo f s t\nu : Set β\nhum : MeasurableSet u\nthis : ν (u ∩ t) = 0 → μ (f ⁻¹' u ∩ s) = 0\n⊢ (ν.restrict t) u = 0 → (... | [] | simpa [hum, hf.measurable, hf.measurable hum] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 427,
"column": 52
} | {
"line": 427,
"column": 97
} | {
"line": 428,
"column": 4
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\ns : Set α\nν : Measure β\nf : α → β\nhf : QuasiMeasurePreserving f μ ν\nt : Set β\nhmaps : MapsTo f s t\nu : Set β\nhum : MeasurableSet u\nthis : ν (u ∩ t) = 0 → μ (f ⁻¹' u ∩ s) = 0\n⊢ (ν.restrict t) u = 0 → (... | [] | simpa [hum, hf.measurable, hf.measurable hum] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 28
} | {
"line": 190,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nmα : MeasurableSpace α\ninst✝ : CountablyGenerated α\ns : Set α\nhs : MeasurableSet s\nn : ℕ\nx✝ : MeasurableSet (natGeneratingSequence α n)\nx : α\n⊢ x ∈ natGeneratingSequence α n ↔ ∃ i, i n ∧ x ∈ countablyGeneratedAtom α i",
"ppTerm": "?refine_1",
"assigned": true... | [
"case refine_1.refine_1\nα : Type u_1\nmα : MeasurableSpace α\ninst✝ : CountablyGenerated α\ns : Set α\nhs : MeasurableSet s\nn : ℕ\nx✝ : MeasurableSet (natGeneratingSequence α n)\nx : α\nhx : x ∈ natGeneratingSequence α n\n⊢ ∃ i, i n ∧ x ∈ countablyGeneratedAtom α i",
"case refine_1.refine_2\nα : Type u_1\nmα : ... | refine ⟨fun hx ↦ ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 949,
"column": 72
} | {
"line": 950,
"column": 72
} | {
"line": 952,
"column": 0
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\nμ : Measure α\n⊢ Measure.map Subtype.val (Measure.comap Subtype.val μ) = μ.restrict s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"Memb... | [] | by
rw [(MeasurableEmbedding.subtype_coe hs).map_comap, Subtype.range_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 327,
"column": 2
} | {
"line": 329,
"column": 48
} | {
"line": 331,
"column": 0
} | [
{
"pp": "α : Type u_1\nm m' : MeasurableSpace α\nhsep : CountablySeparated α\nh : m ≤ m'\n⊢ CountablySeparated α",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HasCountableSeparatingOn.casesOn",
"MeasurableSet",
"Set.univ",
"Membership.mem",
"... | [] | simp_rw [countablySeparated_def] at *
rcases hsep with ⟨S, Sct, Smeas, hS⟩
use S, Sct, (fun s hs ↦ h _ <| Smeas _ hs), hS | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 327,
"column": 2
} | {
"line": 329,
"column": 48
} | {
"line": 331,
"column": 0
} | [
{
"pp": "α : Type u_1\nm m' : MeasurableSpace α\nhsep : CountablySeparated α\nh : m ≤ m'\n⊢ CountablySeparated α",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HasCountableSeparatingOn.casesOn",
"MeasurableSet",
"Set.univ",
"Membership.mem",
"... | [] | simp_rw [countablySeparated_def] at *
rcases hsep with ⟨S, Sct, Smeas, hS⟩
use S, Sct, (fun s hs ↦ h _ <| Smeas _ hs), hS | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated | {
"line": 475,
"column": 66
} | {
"line": 508,
"column": 53
} | {
"line": 510,
"column": 0
} | [
{
"pp": "α : Type u_1\nt : ℕ → Set α\nn : ℕ\ns : Set α\n⊢ MeasurableSet s ↔ ∃ S, ↑S ⊆ memPartition t n ∧ s = ⋃₀ ↑S",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Codisjoint",
"Lattice.toSemilatticeSup",
"Finset.coe_singleton",
"Fin... | [] | by
refine ⟨fun h ↦ ?_, fun ⟨S, hS_subset, hS_eq⟩ ↦ ?_⟩
· induction s, h using generateFrom_induction with
| hC u hu _ => exact ⟨{u}, by simp [hu], by simp⟩
| empty => exact ⟨∅, by simp, by simp⟩
| compl u _ hu =>
obtain ⟨S, hS_subset, rfl⟩ := hu
classical
refine ⟨(memPartition t n).toF... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 104,
"column": 4
} | {
"line": 105,
"column": 22
} | {
"line": 106,
"column": 4
} | [
{
"pp": "case refine_2.mp\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf : α → β\ninst✝ : Countable ι\nμ : ι → Measure α\nh : ∀ (i : ι), AEMeasurable f (μ i)\na✝ : Nontrivial β\ninhabited_h : Inhabited β\ns : ι → Set α := ⋯\nhsμ : ∀ (i : ι), (μ i) (s i) = 0\nhsm... | [
"case refine_2.mpr\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nf : α → β\ninst✝ : Countable ι\nμ : ι → Measure α\nh : ∀ (i : ι), AEMeasurable f (μ i)\na✝ : Nontrivial β\ninhabited_h : Inhabited β\ns : ι → Set α := ⋯\nhsμ : ∀ (i : ι), (μ i) (s i) = 0\nhsm : Measurab... | · rintro ⟨i, hxt, hxs⟩
rwa [hs _ _ hxs] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.AEMeasurable | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 35
} | {
"line": 246,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nh : ∀ᵐ (x : α) (y : α) ∂0, f x = f y\n⊢ AEMeasurable f 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"aemeasurable_zero_measure"
],
"usedFVars": [
"α",
... | [
"case inr\nα : Type u_2\nβ : Type u_3\nm0 : MeasurableSpace α\ninst✝ : MeasurableSpace β\nf : α → β\nμ : Measure α\nh : ∀ᵐ (x : α) (y : α) ∂μ, f x = f y\nhμ : μ ≠ 0\n⊢ AEMeasurable f μ"
] | · exact aemeasurable_zero_measure | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 413,
"column": 8
} | {
"line": 414,
"column": 39
} | {
"line": 415,
"column": 6
} | [
{
"pp": "case ofNat\nα : Type u_1\nG : Type u\ninst✝³ : DivInvMonoid G\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableMul₂ G\ninst✝ : MeasurableInv G\nn : ℕ\n⊢ Measurable fun x ↦ (x, Int.ofNat n).1 ^ (x, Int.ofNat n).2",
"ppTerm": "?ofNat",
"assigned": true,
"usedConstants": [
"zpow_natCast",... | [] | simp_rw [Int.ofNat_eq_natCast, zpow_natCast]
exact measurable_id.pow_const _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.Arithmetic | {
"line": 413,
"column": 8
} | {
"line": 414,
"column": 39
} | {
"line": 415,
"column": 6
} | [
{
"pp": "case ofNat\nα : Type u_1\nG : Type u\ninst✝³ : DivInvMonoid G\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableMul₂ G\ninst✝ : MeasurableInv G\nn : ℕ\n⊢ Measurable fun x ↦ (x, Int.ofNat n).1 ^ (x, Int.ofNat n).2",
"ppTerm": "?ofNat",
"assigned": true,
"usedConstants": [
"zpow_natCast",... | [] | simp_rw [Int.ofNat_eq_natCast, zpow_natCast]
exact measurable_id.pow_const _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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 | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
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.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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