module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 468,
"column": 17
} | {
"line": 468,
"column": 18
} | [
{
"pp": "S : Type u_1\ninst✝⁵ : CommRing S\ninst✝⁴ : Nontrivial S\ninst✝³ : IsDedekindDomain S\ninst✝² : Free ℤ S\ninst✝¹ : Module.Finite ℤ S\nn : ℕ\ninst✝ : CharZero S\nthis✝ : Finite { I // I ∈ (Ideal S)⁰ ∧ absNorm I ≤ n }\nthis : Finite { I // I ∉ (Ideal S)⁰ ∧ absNorm I ≤ n }\ne : { I // absNorm I ≤ n } ≃ { ... | I | Lean.Elab.Tactic.Conv.evalEnter | null |
Mathlib.Algebra.Module.Torsion.PrimaryComponent | {
"line": 182,
"column": 2
} | {
"line": 185,
"column": 64
} | [
{
"pp": "A : Type u_1\nM : Type u_2\ninst✝³ : CommRing A\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : IsDedekindDomain A\ns : Finset (HeightOneSpectrum A)\np : HeightOneSpectrum A → M\nhsum : ∑ i ∈ s, p i = 0\nf : HeightOneSpectrum A → ℕ\nhmem : ∀ i ∈ s, p i ∈ torsionBySet A M ↑(i.asIdeal ^ f i)\nm : ... | have hSupIndep : iSupIndep fun i : HeightOneSpectrum A ↦ torsionBySet A M ↑(i.asIdeal ^ m) := by
rw [iSupIndep_iff_supIndep]
exact fun _ ↦ supIndep_torsionBySet_ideal
fun _ _ _ _ hPQ ↦ (isCoprime_pow_of_ne _ _ hPQ _ _).sup_eq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 59
} | [
{
"pp": "case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ 0\nv : HeightOneSpectrum R\n⊢ v ∈ {v | ¬(Associates.mk v.asIdeal).count (Associates.mk I).factors = 0} ↔ v ∈ {v | v.asIdeal ∣ I}",
"usedConstants": [
"CommSemiring.toSemiring",
"IsDedekindDomain... | exact Associates.count_ne_zero_iff_dvd hI v.irreducible | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 148,
"column": 19
} | {
"line": 148,
"column": 35
} | [
{
"pp": "case f.h\nR : Type u\ninst✝¹⁶ : CommRing R\nS : Type v\ninst✝¹⁵ : CommRing S\ninst✝¹⁴ : Algebra R S\np : Ideal R\nK : Type u_1\ninst✝¹³ : Field K\ninst✝¹² : Algebra R K\nL : Type u_2\ninst✝¹¹ : Field L\ninst✝¹⁰ : Algebra S L\ninst✝⁹ : IsFractionRing S L\ninst✝⁸ : IsDomain R\ninst✝⁷ : IsDomain S\ninst✝⁶... | smul_comm A.det, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 477,
"column": 4
} | {
"line": 477,
"column": 26
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : Countable β\nf : α × β → γ\nhf : ∀ (y : β), Measurable fun x ↦ f (x, y)\nh'f : ∀ (y y' : β) (x : α), y' ∈ measurableAtom y → f (x, y') = f (x, y)\ns : Set γ\nhs : MeasurableS... | rintro ⟨y', hy's, hy'⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 701,
"column": 2
} | {
"line": 701,
"column": 76
} | [
{
"pp": "δ : Type u_4\nX : δ → Type u_6\ninst✝ : (a : δ) → MeasurableSpace (X a)\ns : Set δ\nt : (i : δ) → Set (X i)\nhs : s.Countable\nht : ∀ i ∈ s, MeasurableSet (t i)\n⊢ MeasurableSet (⋂ a ∈ s, eval a ⁻¹' t a)",
"usedConstants": [
"Membership.mem",
"measurable_pi_apply",
"Function.eval"... | exact MeasurableSet.biInter hs fun i hi => measurable_pi_apply _ (ht i hi) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 717,
"column": 4
} | {
"line": 717,
"column": 30
} | [
{
"pp": "case h.e'_3\nδ : Type u_4\nX : δ → Type u_6\ninst✝ : (a : δ) → MeasurableSpace (X a)\ns : Set δ\nt : (i : δ) → Set (X i)\nhs : s.Countable\nf : (i : δ) → X i\nhf : f ∈ s.pi t\nhst : MeasurableSet (s.pi t)\ni : δ\nhi : i ∈ s\n⊢ t i = update f i ⁻¹' s.pi t",
"usedConstants": [
"Eq.mpr",
"... | rw [update_preimage_pi hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.MeasurableSpace.Embedding | {
"line": 782,
"column": 27
} | {
"line": 782,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm' : MeasurableSpace β\ninst✝ : BooleanAlgebra β\nh : Measurable compl\nf : α → β\n⊢ MeasurableSpace.comap (compl ∘ f) inferInstance = MeasurableSpace.comap f inferInstance",
"usedConstants": [
"Eq.mpr",
"MeasurableSpace.comap",
"MeasurableSpace.comap_c... | ← MeasurableSpace.comap_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 565,
"column": 2
} | {
"line": 565,
"column": 33
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J : FractionalIdeal R⁰ K\nhI : I ≠ 0\nh : I ≤ J\nhJ : ¬J = 0\nJ' : Ideal R\nhJ' : ↑J' = J⁻¹ * I\n⊢ 0 ≤ count K v ↑J'",
... | · exact count_coe_nonneg K v J' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.RamificationInertia.Basic | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 47
} | [
{
"pp": "case refine_1.convert_1\nR : Type u\ninst✝¹⁴ : CommRing R\nS : Type v\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\np : Ideal R\nK : Type u_1\ninst✝¹¹ : Field K\ninst✝¹⁰ : Algebra R K\nL : Type u_2\ninst✝⁹ : Field L\ninst✝⁸ : Algebra S L\ninst✝⁷ : IsFractionRing S L\nhRK : IsFractionRing R K\ninst✝⁶ : ... | rw [Quotient.algebraMap_eq, Ideal.mk_ker] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 811,
"column": 4
} | {
"line": 812,
"column": 24
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b : CauSeq α abs\nh : b ≈ a\n⊢ a ⊓ b ≈ b",
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"abs",
"congrArg",
"IsAbsoluteValue.abs_isAbsoluteValue",
"Fie... | refine Setoid.trans (inf_equiv_inf (Setoid.symm h) (Setoid.refl _)) ?_
rw [CauSeq.inf_idem] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 811,
"column": 4
} | {
"line": 812,
"column": 24
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b : CauSeq α abs\nh : b ≈ a\n⊢ a ⊓ b ≈ b",
"usedConstants": [
"Eq.mpr",
"AddGroupWithOne.toAddGroup",
"abs",
"congrArg",
"IsAbsoluteValue.abs_isAbsoluteValue",
"Fie... | refine Setoid.trans (inf_equiv_inf (Setoid.symm h) (Setoid.refl _)) ?_
rw [CauSeq.inf_idem] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Real.Basic | {
"line": 570,
"column": 8
} | {
"line": 570,
"column": 48
} | [
{
"pp": "b : ℕ\nhb : ∀ {a : ℝ}, 0 < a → a * ↑b + 1 ≤ (a + 1) ^ b\na : ℝ\nha' : 0 < a\n⊢ a * ↑(b + 1) + 1 = (0 + 1) ^ b * a + (a * ↑b + 1)",
"usedConstants": [
"one_pow",
"Distrib.leftDistribClass",
"MulOne.toOne",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"ad... | simp [mul_add, add_assoc, add_left_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Real.Basic | {
"line": 570,
"column": 8
} | {
"line": 570,
"column": 48
} | [
{
"pp": "b : ℕ\nhb : ∀ {a : ℝ}, 0 < a → a * ↑b + 1 ≤ (a + 1) ^ b\na : ℝ\nha' : 0 < a\n⊢ a * ↑(b + 1) + 1 = (0 + 1) ^ b * a + (a * ↑b + 1)",
"usedConstants": [
"one_pow",
"Distrib.leftDistribClass",
"MulOne.toOne",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"ad... | simp [mul_add, add_assoc, add_left_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Real.Basic | {
"line": 570,
"column": 8
} | {
"line": 570,
"column": 48
} | [
{
"pp": "b : ℕ\nhb : ∀ {a : ℝ}, 0 < a → a * ↑b + 1 ≤ (a + 1) ^ b\na : ℝ\nha' : 0 < a\n⊢ a * ↑(b + 1) + 1 = (0 + 1) ^ b * a + (a * ↑b + 1)",
"usedConstants": [
"one_pow",
"Distrib.leftDistribClass",
"MulOne.toOne",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"ad... | simp [mul_add, add_assoc, add_left_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Real.Archimedean | {
"line": 221,
"column": 54
} | {
"line": 222,
"column": 52
} | [
{
"pp": "s : Set ℝ\n⊢ sInf (-s) = -sSup s",
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"congrArg",
"Real.instSupSet",
"neg_neg",
"neg_eq_iff_eq_neg",
"id",
"NonUnitalNonAsso... | by
rw [← neg_eq_iff_eq_neg, ← Real.sSup_neg, neg_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Real.Archimedean | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 21
} | [
{
"pp": "a : ℝ\nha : ∀ (k : ℕ), 0 < k → a ≤ 1 / ↑k + 1\nq : ℚ\nhq : 1 < ↑q\n⊢ 1 ≤ ↑q * ↑q.den - ↑q.den",
"usedConstants": [
"Rat.instSub",
"Eq.mpr",
"Real.partialOrder",
"Rat.instMul",
"Real",
"Preorder.toLT",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"... | norm_cast at hq ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticNorm_cast___1 | Lean.Parser.Tactic.tacticNorm_cast__ |
Mathlib.Data.Real.Archimedean | {
"line": 399,
"column": 4
} | {
"line": 399,
"column": 21
} | [
{
"pp": "a : ℝ\nha : ∀ (k : ℕ), 0 < k → a ≤ 1 / ↑k + 1\nq : ℚ\nhq : ↑q.den < ↑q.num\n⊢ 1 ≤ ↑q.num - ↑q.den",
"usedConstants": [
"Int.cast",
"Rat.instSub",
"Eq.mpr",
"Int.cast_natCast",
"Rat.num",
"Preorder.toLT",
"congrArg",
"Rat",
"PartialOrder.toPreord... | norm_cast at hq ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticNorm_cast___1 | Lean.Parser.Tactic.tacticNorm_cast__ |
Mathlib.Data.NNReal.Defs | {
"line": 123,
"column": 31
} | {
"line": 123,
"column": 54
} | [
{
"pp": "p q : ℝ≥0\nh1p : 0 < p\nh2p : p ≤ q\n⊢ q⁻¹ ≤ p⁻¹",
"usedConstants": [
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"MulZeroClass.toMul",
"IsStrictOrderedRing.toMulPosStrictMono",
"DivisionSemiring.toGroupWithZero",
"MulPosReflectLE.toMulPosReflec... | exact inv_anti₀ h1p h2p | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.NNReal.Defs | {
"line": 123,
"column": 31
} | {
"line": 123,
"column": 54
} | [
{
"pp": "p q : ℝ≥0\nh1p : 0 < p\nh2p : p ≤ q\n⊢ q⁻¹ ≤ p⁻¹",
"usedConstants": [
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"MulZeroClass.toMul",
"IsStrictOrderedRing.toMulPosStrictMono",
"DivisionSemiring.toGroupWithZero",
"MulPosReflectLE.toMulPosReflec... | exact inv_anti₀ h1p h2p | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.NNReal.Defs | {
"line": 123,
"column": 31
} | {
"line": 123,
"column": 54
} | [
{
"pp": "p q : ℝ≥0\nh1p : 0 < p\nh2p : p ≤ q\n⊢ q⁻¹ ≤ p⁻¹",
"usedConstants": [
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"MulZeroClass.toMul",
"IsStrictOrderedRing.toMulPosStrictMono",
"DivisionSemiring.toGroupWithZero",
"MulPosReflectLE.toMulPosReflec... | exact inv_anti₀ h1p h2p | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.NNReal.Defs | {
"line": 670,
"column": 96
} | {
"line": 671,
"column": 50
} | [
{
"pp": "r : ℝ≥0\np : ℝ\nhp : 0 ≤ p\n⊢ r ≤ p.toNNReal ↔ ↑r ≤ p",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"NNReal",
"LE.le",
"Real.coe_toNNReal",
"NNRea... | by
rw [← NNReal.coe_le_coe, Real.coe_toNNReal p hp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.NNReal.Defs | {
"line": 974,
"column": 46
} | {
"line": 974,
"column": 66
} | [
{
"pp": "case neg\nΓ₀ : Type u_1\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nh : Nontrivial Γ₀ˣ\nf : Γ₀ →*₀ ℝ≥0\nhf : StrictMono ⇑f\nr : ℝ≥0\nhr : 0 < r\ng : Γ₀ˣ\nhg1 : g ≠ 1\nu : Γ₀ˣ := if g < 1 then g else g⁻¹\nhu : u = if g < 1 then g else g⁻¹\nhu1 : ¬g < 1\nhfg0 : f ↑g ≠ 0\nhg1' : 1 < g\n⊢ (f ↑g)⁻¹ < 1",
... | inv_lt_one_iff hfg0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.Operations | {
"line": 425,
"column": 4
} | {
"line": 426,
"column": 25
} | [
{
"pp": "case coe\nb x✝ : ℝ≥0\n⊢ (↑x✝).toReal - (↑b).toReal ≤ (↑x✝ - ↑b).toReal",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"ENNReal.ofNNReal",
"Real.instZero",
"congrArg",
"Real.instSub",
"HSub.hSub",
"id",
"ENNReal.coe_sub._simp_1",
... | simp only [← coe_sub, NNReal.sub_def, Real.coe_toNNReal', coe_toReal]
exact le_max_left _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Operations | {
"line": 425,
"column": 4
} | {
"line": 426,
"column": 25
} | [
{
"pp": "case coe\nb x✝ : ℝ≥0\n⊢ (↑x✝).toReal - (↑b).toReal ≤ (↑x✝ - ↑b).toReal",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"ENNReal.ofNNReal",
"Real.instZero",
"congrArg",
"Real.instSub",
"HSub.hSub",
"id",
"ENNReal.coe_sub._simp_1",
... | simp only [← coe_sub, NNReal.sub_def, Real.coe_toNNReal', coe_toReal]
exact le_max_left _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Operations | {
"line": 498,
"column": 79
} | {
"line": 498,
"column": 90
} | [
{
"pp": "x y : ℝ≥0\n⊢ ofNNReal '' uIoo x y = uIoo ↑x ↑y",
"usedConstants": [
"ENNReal.ofNNReal",
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"SemilatticeSup.toMax",
"DistribLattice.toLattice",
"NNReal",
... | simp [uIoo] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.ENNReal.Operations | {
"line": 498,
"column": 79
} | {
"line": 498,
"column": 90
} | [
{
"pp": "x y : ℝ≥0\n⊢ ofNNReal '' uIoo x y = uIoo ↑x ↑y",
"usedConstants": [
"ENNReal.ofNNReal",
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"SemilatticeSup.toMax",
"DistribLattice.toLattice",
"NNReal",
... | simp [uIoo] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Operations | {
"line": 498,
"column": 79
} | {
"line": 498,
"column": 90
} | [
{
"pp": "x y : ℝ≥0\n⊢ ofNNReal '' uIoo x y = uIoo ↑x ↑y",
"usedConstants": [
"ENNReal.ofNNReal",
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"SemilatticeSup.toMax",
"DistribLattice.toLattice",
"NNReal",
... | simp [uIoo] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Operations | {
"line": 586,
"column": 2
} | {
"line": 586,
"column": 46
} | [
{
"pp": "ι : Sort u_1\na : ℝ≥0∞\nκ : Sort u_2\nq₁ : ι → Sort u_3\nq₂ : κ → Sort u_4\nf : (i : ι) → q₁ i → ℝ≥0∞\ng : (k : κ) → q₂ k → ℝ≥0∞\nh : ∀ (i : ι) (pi : q₁ i) (k : κ) (qk : q₂ k), a ≤ f i pi + g k qk\n⊢ a ≤ ⨅ i, ⨅ j, ⨅ i_1, ⨅ j_1, f i j + g i_1 j_1",
"usedConstants": [
"ENNReal.instAdd",
"... | exact le_iInf₂ fun i hi => le_iInf₂ (h i hi) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.EReal.Basic | {
"line": 769,
"column": 4
} | {
"line": 769,
"column": 27
} | [
{
"pp": "case coe\ny : EReal\na✝ : ℝ\nh : ↑a✝ ≤ y\n⊢ (↑a✝).toENNReal ≤ y.toENNReal",
"usedConstants": [
"LinearOrder.toDecidableEq",
"EReal.toENNReal",
"EReal",
"instTopEReal",
"LE.le",
"dite",
"instLinearOrderEReal",
"ENNReal.instLE",
"ENNReal",
"... | by_cases hy_top : y = ⊤ | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Data.NNReal.Basic | {
"line": 97,
"column": 2
} | {
"line": 98,
"column": 81
} | [
{
"pp": "ι : Type u_1\ns : Finset ι\nf : ι → ℝ\nhf : ∀ a ∈ s, 0 ≤ f a\n⊢ (∏ a ∈ s, f a).toNNReal = ∏ a ∈ s, (f a).toNNReal",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"NNReal.instCommSemiring",
"Real.partialOrder",
"Real",
... | rw [← coe_inj, NNReal.coe_prod, Real.coe_toNNReal _ (Finset.prod_nonneg hf)]
exact Finset.prod_congr rfl fun x hxs => by rw [Real.coe_toNNReal _ (hf x hxs)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.NNReal.Basic | {
"line": 97,
"column": 2
} | {
"line": 98,
"column": 81
} | [
{
"pp": "ι : Type u_1\ns : Finset ι\nf : ι → ℝ\nhf : ∀ a ∈ s, 0 ≤ f a\n⊢ (∏ a ∈ s, f a).toNNReal = ∏ a ∈ s, (f a).toNNReal",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"NNReal.instCommSemiring",
"Real.partialOrder",
"Real",
... | rw [← coe_inj, NNReal.coe_prod, Real.coe_toNNReal _ (Finset.prod_nonneg hf)]
exact Finset.prod_congr rfl fun x hxs => by rw [Real.coe_toNNReal _ (hf x hxs)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.NNReal.Basic | {
"line": 119,
"column": 20
} | {
"line": 119,
"column": 59
} | [
{
"pp": "s : Set ℕ\n⊢ ∀ a ∈ upperBounds (Nat.cast '' s), ⌊a⌋₊ ∈ upperBounds s",
"usedConstants": [
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"congrArg",
"zer... | by simp [upperBounds, Nat.le_floor_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.Monotone | {
"line": 123,
"column": 2
} | {
"line": 124,
"column": 50
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝⁶ : LinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : LinearOrder β\ns : Set α\nf : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : SecondCountableTopology β\nhf : MonotoneOn f s\nx : α\nxs : x ∈ s\nhx : ... | · filter_upwards [@self_mem_nhdsWithin _ _ x (s ∩ Ioi x)] with y hy
exact hm.trans_le (hf xs hy.1 (le_of_lt hy.2)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UniformSpace.OfFun | {
"line": 42,
"column": 20
} | {
"line": 42,
"column": 44
} | [
{
"pp": "X : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nd : X → X → M\nrefl : ∀ (x : X), d x x = 0\nsymm : ∀ (x y : X), d x y = d y x\ntriangle : ∀ (x y z : X), d x z ≤ d x y + d y z\nhalf : ∀ ε > 0, ∃ δ > 0, ∀ x < δ, ∀ y < δ, x + y < ε\nr : M\nx✝ : r > 0\nx : X × X\nhx : x ∈ {x |... | by rwa [mem_setOf, symm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.LiminfLimsup | {
"line": 83,
"column": 71
} | {
"line": 87,
"column": 29
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nR : Type u_4\nS : Type u_5\nπ : ι → Type u_6\ninst✝⁹ : Preorder α\ninst✝⁸ : Preorder β\ninst✝⁷ : TopologicalSpace α\ninst✝⁶ : TopologicalSpace β\ninst✝⁵ : BoundedLENhdsClass α\ninst✝⁴ : BoundedLENhdsClass β\nf : Filter ι\nu : ι → α\na : α\ninst✝³ : Finite ι\nin... | by
refine ⟨fun x ↦ ?_⟩
rw [nhds_pi]
choose f hf using fun i ↦ isBounded_le_nhds (x i)
exact ⟨f, eventually_pi hf⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Inv | {
"line": 679,
"column": 94
} | {
"line": 680,
"column": 70
} | [
{
"pp": "n : ℤ\n⊢ 0 ^ n = if n = 0 then 1 else if 0 < n then 0 else ∞",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"zpow_natCast",
"Nat.instCanonicallyOrderedAdd",
"MulOne.toOne",
"False",
"Nat.instMulZeroClass",
"DivInvMonoid.toInv",
... | by
obtain ((_ | n) | n) := n <;> simp [-Nat.cast_add, -Int.natCast_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Inv | {
"line": 682,
"column": 93
} | {
"line": 683,
"column": 70
} | [
{
"pp": "n : ℤ\n⊢ ∞ ^ n = if n = 0 then 1 else if 0 < n then ∞ else 0",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"zpow_natCast",
"Nat.instCanonicallyOrderedAdd",
"MulOne.toOne",
"False",
"Nat.instMulZeroClass",
"DivInvMonoid.toInv",
... | by
obtain ((_ | n) | n) := n <;> simp [-Nat.cast_add, -Int.natCast_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Inv | {
"line": 747,
"column": 2
} | {
"line": 749,
"column": 87
} | [
{
"pp": "case negSucc.negSucc\nx : ℝ≥0∞\nhx : 1 ≤ x\na b : ℕ\nh : Int.negSucc a ≤ Int.negSucc b\n⊢ x ^ Int.negSucc a ≤ x ^ Int.negSucc b",
"usedConstants": [
"Eq.mpr",
"ENNReal.instIsOrderedRing",
"DivInvMonoid.toInv",
"IsOrderedRing.toPosMulMono",
"IsOrderedRing.toZeroLEOneCla... | · simp only [zpow_negSucc, ENNReal.inv_le_inv]
apply pow_right_mono₀ hx
simpa only [← Int.ofNat_le, neg_le_neg_iff, Int.natCast_add, Int.ofNat_one] using h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.EMetricSpace.Pi | {
"line": 71,
"column": 8
} | {
"line": 71,
"column": 17
} | [
{
"pp": "α : Type u\nβ : Type v\nX✝ : Type u_1\ninst✝² : PseudoEMetricSpace α\nX : β → Type u_2\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoEMetricSpace (X b)\n⊢ ⨅ i, ⨅ i_1, ⨅ (_ : 0 < i_1), 𝓟 {a | edist (a.1 i) (a.2 i) < i_1} =\n ⨅ ε, ⨅ (_ : 0 < ε), 𝓟 {p | (Finset.univ.sup fun b ↦ edist (p.1 b) (p.2 b)) ... | iInf_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.EMetricSpace.Pi | {
"line": 72,
"column": 8
} | {
"line": 72,
"column": 17
} | [
{
"pp": "case e_s.h\nα : Type u\nβ : Type v\nX✝ : Type u_1\ninst✝² : PseudoEMetricSpace α\nX : β → Type u_2\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoEMetricSpace (X b)\nε : ℝ≥0∞\n⊢ ⨅ i, ⨅ (_ : 0 < ε), 𝓟 {a | edist (a.1 i) (a.2 i) < ε} =\n ⨅ (_ : 0 < ε), 𝓟 {p | (Finset.univ.sup fun b ↦ edist (p.1 b) (p.... | iInf_comm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.EMetricSpace.Defs | {
"line": 533,
"column": 4
} | {
"line": 533,
"column": 44
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\nu : β → α\na : α\n⊢ (∀ (ib : ℝ≥0∞), 0 < ib → ∃ ia, True ∧ ∀ x ∈ Ici ia, u x ∈ eball a ib) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, edist (u n) a < ε",
"usedConstants": [
"Preorder.toLT",
"Set.Ici",
... | simp only [true_and, mem_Ici, mem_eball] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.EMetricSpace.Defs | {
"line": 533,
"column": 4
} | {
"line": 533,
"column": 44
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\nu : β → α\na : α\n⊢ (∀ (ib : ℝ≥0∞), 0 < ib → ∃ ia, True ∧ ∀ x ∈ Ici ia, u x ∈ eball a ib) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, edist (u n) a < ε",
"usedConstants": [
"Preorder.toLT",
"Set.Ici",
... | simp only [true_and, mem_Ici, mem_eball] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.Defs | {
"line": 533,
"column": 4
} | {
"line": 533,
"column": 44
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\nu : β → α\na : α\n⊢ (∀ (ib : ℝ≥0∞), 0 < ib → ∃ ia, True ∧ ∀ x ∈ Ici ia, u x ∈ eball a ib) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, edist (u n) a < ε",
"usedConstants": [
"Preorder.toLT",
"Set.Ici",
... | simp only [true_and, mem_Ici, mem_eball] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 164,
"column": 2
} | {
"line": 165,
"column": 43
} | [
{
"pp": "α : Type u_2\ninst✝ : EMetricSpace α\n⊢ Nontrivial α ↔ NontrivialTopology α",
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"congrArg",
"Exists",
"id",
"EMetricSpace.toPseudoEMetricSpace",
"_private.Mathlib.... | simp_rw [nontrivial_iff, TopologicalSpace.nontrivial_iff_exists_not_inseparable,
EMetric.inseparable_iff, edist_eq_zero] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 164,
"column": 2
} | {
"line": 165,
"column": 43
} | [
{
"pp": "α : Type u_2\ninst✝ : EMetricSpace α\n⊢ Nontrivial α ↔ NontrivialTopology α",
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"congrArg",
"Exists",
"id",
"EMetricSpace.toPseudoEMetricSpace",
"_private.Mathlib.... | simp_rw [nontrivial_iff, TopologicalSpace.nontrivial_iff_exists_not_inseparable,
EMetric.inseparable_iff, edist_eq_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 164,
"column": 2
} | {
"line": 165,
"column": 43
} | [
{
"pp": "α : Type u_2\ninst✝ : EMetricSpace α\n⊢ Nontrivial α ↔ NontrivialTopology α",
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"congrArg",
"Exists",
"id",
"EMetricSpace.toPseudoEMetricSpace",
"_private.Mathlib.... | simp_rw [nontrivial_iff, TopologicalSpace.nontrivial_iff_exists_not_inseparable,
EMetric.inseparable_iff, edist_eq_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 334,
"column": 91
} | {
"line": 337,
"column": 40
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\ns : Set α\nf : α → β\ng : α → γ\nKf Kg : ℝ≥0\nhf : LipschitzOnWith Kf f s\nhg : LipschitzOnWith Kg g s\n⊢ LipschitzOnWith (max Kf Kg) (fun x ↦ (f x, g x)) s",
"usedConstan... | by
intro _ hx _ hy
rw [ENNReal.coe_mono.map_max, Prod.edist_eq, max_mul]
exact max_le_max (hf hx hy) (hg hx hy) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 224,
"column": 4
} | {
"line": 224,
"column": 17
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : CommGroup α\ninst✝ : IsUniformGroup α\nf : β → α\n⊢ (∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e) →\n ∀ s ∈ 𝓝 1, ∃ a, ∀ b ≥ a, (∏ b ∈ b.2, f b) / ∏ b ∈ b.1, f b ∈ s",
"usedConstants": [
"Filter.... | rintro h e he | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 30
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : CommGroup α\ninst✝ : IsUniformGroup α\nf : β → α\nh✝ : ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e\ne : Set α\nhe : e ∈ 𝓝 1\nd : Set α\nhd : d ∈ 𝓝 1\nhde : ∀ v ∈ d, ∀ w ∈ d, v / w ∈ e\ns : Finset β\nh : ∀ (t ... | rintro ⟨t₁, t₂⟩ ⟨ht₁, ht₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.Order.Compact | {
"line": 196,
"column": 41
} | {
"line": 196,
"column": 56
} | [
{
"pp": "case refine_1\nα : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : NoMinOrder α\ninst✝ : ClosedIicTopology α\ns : Set α\nhs : s ∈ cocompact α\nt : Set α\nht : IsCompact t\nhts : univ ⊆ s\nh_empty : t = ∅\n⊢ s ∈ atBot",
"usedConstants": [
"Set.univ_subset_iff",
"c... | univ_subset_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Compact | {
"line": 317,
"column": 2
} | {
"line": 317,
"column": 24
} | [
{
"pp": "case inl\nα : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : ClosedIicTopology α\ninst✝ : Nonempty α\nhs : IsCompact ∅\n⊢ BddBelow ∅",
"usedConstants": [
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattice",
"bddBelo... | · exact bddBelow_empty | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.MetricSpace.Pseudo.Constructions | {
"line": 118,
"column": 4
} | {
"line": 119,
"column": 40
} | [
{
"pp": "a b x✝ : ℝ≥0\n⊢ nndist a b ≤ x✝ ↔ a ≤ x✝ + b ∧ b ≤ x✝ + a",
"usedConstants": [
"NNDist.nndist",
"Real.instLE",
"Real",
"abs",
"congrArg",
"covariant_swap_add_of_covariant_add",
"PartialOrder.toPreorder",
"AddGroup.toOrderedSub",
"HSub.hSub",
... | simp only [← NNReal.coe_le_coe, coe_nndist, dist_eq, abs_sub_le_iff,
tsub_le_iff_right, NNReal.coe_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 61,
"column": 2
} | {
"line": 65,
"column": 35
} | [
{
"pp": "β : Type u_2\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\nf g : (b : β) → X b\nr : ℝ≥0\nhr : 0 < r\n⊢ nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r",
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"nndist_pi_... | rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, not_forall, and_comm]
simp_rw [not_lt, and_congr_left_iff, le_antisymm_iff]
intro h
refine exists_congr fun b => ?_
apply (and_iff_right <| h _).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Pseudo.Pi | {
"line": 61,
"column": 2
} | {
"line": 65,
"column": 35
} | [
{
"pp": "β : Type u_2\nX : β → Type u_3\ninst✝¹ : Fintype β\ninst✝ : (b : β) → PseudoMetricSpace (X b)\nf g : (b : β) → X b\nr : ℝ≥0\nhr : 0 < r\n⊢ nndist f g = r ↔ (∃ i, nndist (f i) (g i) = r) ∧ ∀ (b : β), nndist (f b) (g b) ≤ r",
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"nndist_pi_... | rw [eq_iff_le_not_lt, nndist_pi_lt_iff hr, nndist_pi_le_iff, not_forall, and_comm]
simp_rw [not_lt, and_congr_left_iff, le_antisymm_iff]
intro h
refine exists_congr fun b => ?_
apply (and_iff_right <| h _).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Cauchy | {
"line": 80,
"column": 8
} | {
"line": 80,
"column": 27
} | [
{
"pp": "case mp\nα : Type u\nβ : Type v\ninst✝² : PseudoMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\nγ : Type u_3\nF : β → γ → α\ns : Set γ\nε : ℝ\nhε : ε > 0\nu : Set (α × α) := {a | dist a.1 a.2 < ε}\nhu : u ∈ 𝓤 α\nh : ∀ᶠ (m : β × β) in atTop ×ˢ atTop, ∀ x ∈ s, (F m.1 x, F m.2 x) ∈ u\n⊢ ∀ᶠ ... | prod_atTop_atTop_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Cauchy | {
"line": 156,
"column": 2
} | {
"line": 160,
"column": 55
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nu : ℕ → α\nhu : ∀ ε > 0, ∃ N, ∀ m ≥ N, ∀ n ≥ N, dist (u m) (u n) < ε\nb : ℕ → ℝ\nhb : ∀ (n : ℕ), 0 < b n\n⊢ ∃ f, StrictMono f ∧ ∀ (n m : ℕ), m ≥ f n → dist (u m) (u (f n)) < b n",
"usedConstants": [
"Eq.mpr",
"Nat.instLattice",
"Real",
... | have hu' : ∀ k, ∀ᶠ (n : ℕ) in atTop, ∀ m ≥ n, dist (u m) (u n) < b k := by
intro k
rw [eventually_atTop]
obtain ⟨N, hN⟩ := hu (b k) (hb k)
exact ⟨N, fun m hm r hr => hN r (hm.trans hr) m hm⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 202,
"column": 63
} | {
"line": 206,
"column": 24
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : Set α\nc : α\n⊢ Bornology.IsCobounded s ↔ ∃ r, (closedBall c r)ᶜ ⊆ s",
"usedConstants": [
"Eq.mpr",
"Real",
"PseudoMetricSpace.toBornology",
"congrArg",
"Compl.compl",
"Iff.rfl",
"Exists",
"id",
"Set.... | by
rw [← isBounded_compl_iff, isBounded_iff_subset_closedBall c]
apply exists_congr
intro r
rw [compl_subset_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Basic | {
"line": 226,
"column": 51
} | {
"line": 226,
"column": 71
} | [
{
"pp": "X : Type u_2\nm : PseudoEMetricSpace X\nd : X → X → ℝ≥0∞\nhd : d = edist\n⊢ m.replaceEDist d hd = m",
"usedConstants": [
"PseudoEMetricSpace.ext",
"EDist.ext",
"PseudoEMetricSpace.toEDist",
"PseudoEMetricSpace.replaceEDist"
]
}
] | by ext : 2; exact hd | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Basic | {
"line": 252,
"column": 49
} | {
"line": 252,
"column": 69
} | [
{
"pp": "X : Type u_2\nm : PseudoMetricSpace X\nd : X → X → ℝ\nhd : d = dist\n⊢ m.replaceDist d hd = m",
"usedConstants": [
"PseudoMetricSpace.ext",
"Dist.ext",
"PseudoMetricSpace.replaceDist",
"PseudoMetricSpace.toDist"
]
}
] | by ext : 2; exact hd | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Basic | {
"line": 282,
"column": 51
} | {
"line": 282,
"column": 71
} | [
{
"pp": "X : Type u_2\nm : EMetricSpace X\nd : X → X → ℝ≥0∞\nhd : d = edist\n⊢ m.replaceEDist d hd = m",
"usedConstants": [
"EMetricSpace.toPseudoEMetricSpace",
"EMetricSpace.ext",
"EDist.ext",
"PseudoEMetricSpace.toEDist",
"EMetricSpace.replaceEDist"
]
}
] | by ext : 2; exact hd | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Basic | {
"line": 305,
"column": 49
} | {
"line": 305,
"column": 69
} | [
{
"pp": "X : Type u_2\nm : MetricSpace X\nd : X → X → ℝ\nhd : d = dist\n⊢ m.replaceDist d hd = m",
"usedConstants": [
"MetricSpace.ext",
"MetricSpace.replaceDist",
"Dist.ext",
"MetricSpace.toPseudoMetricSpace",
"PseudoMetricSpace.toDist"
]
}
] | by ext : 2; exact hd | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 559,
"column": 2
} | {
"line": 559,
"column": 38
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\ns : ℕ → Set α\nh0 : IsComplete (s 0)\nhs : ∀ (n : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s n)\nh's : ∀ (n : ℕ), Bornology.IsBounded (s n)\nh : ∀ (N : ℕ), (⋂ n, ⋂ (_ : n ≤ N), s n).Nonempty\nh' : Tendsto (fun n ↦ diam (s n)) atTop (𝓝 ... | refine ⟨x, mem_iInter.2 fun n => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Instances.NNReal.Lemmas | {
"line": 67,
"column": 68
} | {
"line": 68,
"column": 55
} | [
{
"pp": "x : ℝ≥0\n⊢ Filter.map toReal (𝓝[>] x) = 𝓝[>] ↑x",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"Real",
"Set.Ioi",
"congrArg",
"Filter.map",
"nhdsWithin",
"PartialOrder.toPreorder",
"NNReal.isEmbedding_coe",
"PseudoMetricSp... | by
rw [isEmbedding_coe.map_nhdsWithin_eq, image_coe_Ioi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Metrizable.Uniformity | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 52
} | [
{
"pp": "case h.inr\nX : Type u_1\nd : X → X → ℝ≥0\ndist_self : ∀ (x : X), d x x = 0\ndist_comm : ∀ (x y : X), d x y = d y x\nhd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))\nthis : IsTrans X fun x y ↦ d x y = 0\nx y : X\nl : List X\nihn : ∀ m < l.length, ∀ (x y : X) (l : List X)... | · grw [hd x z z' y, max_le hxz (max_le hzz' hz'y)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Metrizable.Uniformity | {
"line": 160,
"column": 24
} | {
"line": 160,
"column": 29
} | [
{
"pp": "case h.e'_3.h.e'_6.h.e'_4\nX : Type u_1\nd : X → X → ℝ≥0\ndist_self : ∀ (x : X), d x x = 0\ndist_comm : ∀ (x y : X), d x y = d y x\nhd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))\nthis : IsTrans X fun x y ↦ d x y = 0\nx y : X\nl : List X\nihn : ∀ m < l.length, ∀ (x y : ... | take, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Semicontinuity.Defs | {
"line": 699,
"column": 39
} | {
"line": 699,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nx : α\ns : Set α\n⊢ (∀ (u : Set β), IsOpen[inst✝] u → (f x ∩ u).Nonempty → ∀ᶠ (x' : α) in 𝓝[s] x, (f x' ∩ u).Nonempty) ↔\n ∀ (t : Set β), IsClosed[inst✝] t → (∃ᶠ (x' : α) in 𝓝[s] x, f x' ⊆ t) → f x ... | compl_surjective.forall | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Semicontinuity.Defs | {
"line": 726,
"column": 36
} | {
"line": 726,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\n⊢ (∀ (x : α), LowerHemicontinuousAt f x) ↔\n ∀ (x : α) (t : Set β), IsClosed[inst✝] t → (∃ᶠ (x' : α) in 𝓝 x, f x' ⊆ t) → f x ⊆ t",
"usedConstants": [
"congrArg",
"nhds",
"HasSub... | lowerHemicontinuousAt_iff_frequently | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Semicontinuity.Defs | {
"line": 828,
"column": 74
} | {
"line": 828,
"column": 97
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nx : α\ns : Set α\n⊢ (∀ (y : Set β), (∃ᶠ (x' : α) in 𝓝[s] x, y ∉ 𝓝ˢ (f x')) → y ∉ 𝓝ˢ (f x)) ↔\n ∀ (t : Set β), IsClosed[inst✝] t → (∃ᶠ (x' : α) in 𝓝[s] x, (f x' ∩ t).Nonempty) → (f x ∩ t).Nonempty"... | compl_surjective.forall | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Semicontinuity.Defs | {
"line": 828,
"column": 2
} | {
"line": 828,
"column": 98
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nx : α\ns : Set α\n⊢ UpperHemicontinuousWithinAt f s x ↔\n ∀ (t : Set β), IsClosed[inst✝] t → (∃ᶠ (x' : α) in 𝓝[s] x, (f x' ∩ t).Nonempty) → (f x ∩ t).Nonempty",
"usedConstants": [
"Filter.i... | rw [UpperHemicontinuousWithinAt, semicontinuousWithinAt_iff_frequently, compl_surjective.forall] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Archimedean | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 57
} | [
{
"pp": "G : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : MulArchimedean G\nH : Subgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 1 < a\nhd : Disjoint (↑H) (Ioo 1 a)\ng : G\nhg : g > 1\nm : ℕ\nhm : g ≤ a ^ (↑m + 1)\nhm' : a ^ ↑m < g\n⊢ ∃ n, g ∈ Ioc (a ^ n) (a ^ (n + 1))",
... | simp only [← Nat.cast_succ, zpow_natCast] at hm hm' | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Archimedean | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 92
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝³ : CommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedMonoid G\ninst✝ : MulArchimedean G\nH : Subgroup G\nhbot : H ≠ ⊥\na : G\nh₀ : 1 < a\nhd : Disjoint (↑H) (Ioo 1 a)\nhex : ∀ g > 1, ∃ n, g ∈ Ioc (a ^ n) (a ^ (n + 1))\nthis : ∃ n, (↑H ∩ Ioc (a ^ n) (a ^ (n + 1))).None... | refine disjoint_left.1 hd (div_mem hxH hyH) ⟨one_lt_div'.2 hxy, div_lt_iff_lt_mul'.2 ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 86
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu v : α → EReal\ninst✝ : f.NeBot\nhu : 0 ≤ᶠ[f] u\nhv : 0 ≤ᶠ[f] v\nh₁ : 0 < limsup u f ∨ liminf v f ≠ ⊤\nh₂ : limsup u f ≠ ⊤ ∨ 0 < liminf v f\n⊢ liminf (u * v) f ≤ limsup u f * liminf v f",
"usedConstants": [
"Iff.mpr",
"Preorder.toLT",
"HMul.hMul",
... | refine le_mul_of_forall_lt h₁ h₂ fun a a_u b b_v ↦ (liminf_le_iff).2 fun c c_ab ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 97,
"column": 29
} | {
"line": 97,
"column": 56
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\nf : α → β\ninst✝ : LinearOrder β\ns : Set α\nne_s : s.Nonempty\nhs : IsCompact s\nhf : LowerSemicontinuousOn f s\nx✝¹ : Nonempty α\nx✝ : Nonempty ↑s\nφ : β → Filter α := fun b ↦ 𝓟 (s ∩ f ⁻¹' Iic b)\nℱ : Filter α := ⨅ a, φ (f ↑a)\nthis : ℱ.NeBot\... | by apply inter_subset_right | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 323,
"column": 2
} | {
"line": 326,
"column": 91
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝¹ : TopologicalSpace α\ns : Set α\nγ : Type u_4\ninst✝ : LinearOrder γ\nι : Type u_5\nf : ι → α → γ\nks : IsCompact s\nI : Set ι\nc : γ\nhfi : ∀ i ∈ I, LowerSemicontinuousOn (f i) s\nH : s ∩ ⋂ i ∈ I, f i ⁻¹' Iic c = ∅\n⊢ ∃ u, ∀ x ∈ s, ∃ i ∈ u, c < f (↑i) x",
"usedC... | · suffices ∀ i ∈ I, IsClosed (s ↓∩ (fun i ↦ f i ⁻¹' Iic c) i) by
simpa [Set.eq_empty_iff_forall_notMem] using
ks.elim_finite_subfamily_isClosed_subtype _ this H
exact fun i hi ↦ lowerSemicontinuous_restrict_iff.mpr (hfi i hi) |>.isClosed_preimage c | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Set.Dissipate | {
"line": 93,
"column": 4
} | {
"line": 95,
"column": 29
} | [
{
"pp": "case succ\nα : Type u_1\ns : ℕ → Set α\nhd : Directed (fun x1 x2 ↦ x1 ⊇ x2) s\nn : ℕ\nhn : ∃ m, s m ⊆ dissipate s n\n⊢ ∃ m, s m ⊆ dissipate s (n + 1)",
"usedConstants": [
"Set.dissipate",
"Eq.mpr",
"congrArg",
"Exists",
"id",
"_private.Mathlib.Data.Set.Dissipate.... | obtain ⟨m, hm⟩ := hn
obtain ⟨k, hk⟩ := hd m (n + 1)
exact ⟨k, by simp; grind⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Dissipate | {
"line": 93,
"column": 4
} | {
"line": 95,
"column": 29
} | [
{
"pp": "case succ\nα : Type u_1\ns : ℕ → Set α\nhd : Directed (fun x1 x2 ↦ x1 ⊇ x2) s\nn : ℕ\nhn : ∃ m, s m ⊆ dissipate s n\n⊢ ∃ m, s m ⊆ dissipate s (n + 1)",
"usedConstants": [
"Set.dissipate",
"Eq.mpr",
"congrArg",
"Exists",
"id",
"_private.Mathlib.Data.Set.Dissipate.... | obtain ⟨m, hm⟩ := hn
obtain ⟨k, hk⟩ := hd m (n + 1)
exact ⟨k, by simp; grind⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 39
} | [
{
"pp": "case refine_1\nα : Type u_1\nm : Set α → ℝ≥0∞\nβ : Type u_2\nf : β → α\nh : (Monotone fun s ↦ m ↑s) ∨ Surjective f\nH : Monotone fun s ↦ m ↑s\ns t : Set α\nhst : s ≤ t\nhs : s.Nonempty\n⊢ m s ≤ (fun s ↦ ⨆ (_ : s.Nonempty), m s) t",
"usedConstants": [
"Set.Nonempty.mono",
"Set.Nonempty"
... | have ht : t.Nonempty := hs.mono hst | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 347,
"column": 2
} | {
"line": 349,
"column": 51
} | [
{
"pp": "case refine_2\nα : Type u_1\nm : Set (OuterMeasure α)\n⊢ boundedBy (sInfGen m) ≤ sInf m",
"usedConstants": [
"MeasureTheory.OuterMeasure.boundedBy",
"CompletelyDistribLattice.toCompleteLattice",
"iInf₂_le",
"PartialOrder.toPreorder",
"MeasureTheory.OuterMeasure.sInfGen... | · refine le_sInf ?_
intro μ hμ t
exact le_trans (boundedBy_le t) (iInf₂_le μ hμ) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.PiSystem | {
"line": 333,
"column": 23
} | {
"line": 333,
"column": 31
} | [
{
"pp": "case refine_1.h.mp\nα : Type u_3\nβ : Type u_4\ng : β → Set (Set α)\ns : Set β\nh_pi : ∀ b ∈ s, IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s → Set α\nh_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b\nh_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)\nthis : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b, ... | h_b_in_T | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.PiSystem | {
"line": 333,
"column": 23
} | {
"line": 333,
"column": 31
} | [
{
"pp": "case refine_1.h.mpr\nα : Type u_3\nβ : Type u_4\ng : β → Set (Set α)\ns : Set β\nh_pi : ∀ b ∈ s, IsPiSystem (g b)\nT : Finset (Subtype s)\nf : Subtype s → Set α\nh_t' : ∀ b ∈ T, f b ∈ (g ∘ Subtype.val) b\nh_t : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b ∈ s, g b)\nthis : ⋂ b ∈ T, f b ∈ generatePiSystem (⋃ b,... | h_b_in_T | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.PiSystem | {
"line": 374,
"column": 6
} | {
"line": 374,
"column": 29
} | [
{
"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\nht_eq_i : t = {i}\n⊢ f i ∈ π i ∨ f i ∈ {univ}",
"usedConstants": [
"Set.univ",
"Membership.mem",
"Set.instSingletonSet",
... | exact Or.inl (hfπ i hi) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.OuterMeasure.Caratheodory | {
"line": 233,
"column": 27
} | {
"line": 233,
"column": 72
} | [
{
"pp": "α : Type u_1\nm₁ m₂ : OuterMeasure α\ns : Set α\nx✝ : MeasurableSet s\nt : Set α\nhs₁ : s ∈ (fun m ↦ {t | MeasurableSet t}) m₁.caratheodory\nhs₂ : s ∈ (fun m ↦ {t | MeasurableSet t}) m₂.caratheodory\n⊢ (m₁ + m₂) t = (m₁ + m₂) (t ∩ s) + (m₁ + m₂) (t \\ s)",
"usedConstants": [
"ENNReal.instAdd"... | simp [hs₁ t, hs₂ t, add_left_comm, add_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.OuterMeasure.Caratheodory | {
"line": 233,
"column": 27
} | {
"line": 233,
"column": 72
} | [
{
"pp": "α : Type u_1\nm₁ m₂ : OuterMeasure α\ns : Set α\nx✝ : MeasurableSet s\nt : Set α\nhs₁ : s ∈ (fun m ↦ {t | MeasurableSet t}) m₁.caratheodory\nhs₂ : s ∈ (fun m ↦ {t | MeasurableSet t}) m₂.caratheodory\n⊢ (m₁ + m₂) t = (m₁ + m₂) (t ∩ s) + (m₁ + m₂) (t \\ s)",
"usedConstants": [
"ENNReal.instAdd"... | simp [hs₁ t, hs₂ t, add_left_comm, add_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.OuterMeasure.Caratheodory | {
"line": 233,
"column": 27
} | {
"line": 233,
"column": 72
} | [
{
"pp": "α : Type u_1\nm₁ m₂ : OuterMeasure α\ns : Set α\nx✝ : MeasurableSet s\nt : Set α\nhs₁ : s ∈ (fun m ↦ {t | MeasurableSet t}) m₁.caratheodory\nhs₂ : s ∈ (fun m ↦ {t | MeasurableSet t}) m₂.caratheodory\n⊢ (m₁ + m₂) t = (m₁ + m₂) (t ∩ s) + (m₁ + m₂) (t \\ s)",
"usedConstants": [
"ENNReal.instAdd"... | simp [hs₁ t, hs₂ t, add_left_comm, add_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.Basic | {
"line": 662,
"column": 2
} | {
"line": 662,
"column": 22
} | [
{
"pp": "ε : ℝ≥0∞\nhε : ε ≠ 0\nι : Type u_4\ninst✝ : Countable ι\nw δ' : ι → ℝ≥0\nHpos : ∀ (i : ι), 0 < δ' i\nHsum : ∑' (i : ι), ↑(δ' i) < ε\nthis : ∀ (i : ι), 0 < max 1 (w i)\ni : ι\n⊢ ↑(δ' i) / ↑(max 1 (w i)) ≤ ↑(δ' i) / (fun i ↦ ↑(w i)) i",
"usedConstants": [
"le_max_right",
"le_refl",
... | grw [← le_max_right] | Mathlib.Tactic._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_grwSeq_1 | Mathlib.Tactic.grwSeq |
Mathlib.MeasureTheory.OuterMeasure.AE | {
"line": 151,
"column": 28
} | {
"line": 151,
"column": 46
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\ns t : Set α\n⊢ (∀ᵐ (x : α) ∂μ, x ∈ s → x ∈ t) ↔ μ (s \\ t) = 0",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"congrArg",
"Filter.Eventually",
"Iff.rfl",
"setOf... | simp [ae_iff]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.OuterMeasure.AE | {
"line": 151,
"column": 28
} | {
"line": 151,
"column": 46
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : FunLike F (Set α) ℝ≥0∞\ninst✝ : OuterMeasureClass F α\nμ : F\ns t : Set α\n⊢ (∀ᵐ (x : α) ∂μ, x ∈ s → x ∈ t) ↔ μ (s \\ t) = 0",
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"congrArg",
"Filter.Eventually",
"Iff.rfl",
"setOf... | simp [ae_iff]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Set.Monotone | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 45
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : OrderBot α\nn : α\nφ : α → α\nk : α\nih : StrictMonoOn φ (Iic k) → ∀ m ≤ k, m ≤ φ m\nhφ : StrictMonoOn φ (Iic (succ k))\nm : α\nhm : m ≤ succ k\nhk : ¬IsMax k\n⊢ m ≤ φ m",
"usedConstants": [
... | obtain rfl | h := le_succ_iff_eq_or_le.1 hm | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.AbsolutelyContinuous | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 77
} | [
{
"pp": "α : Type u_1\nR : Type u_5\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nh : μ ≪ ν\nc : R\ns : Set α\nhνs : ν s = 0\n⊢ (c • μ) s = 0",
"usedConstants": [
"MulOne.toOne",
"instHSMul",
"MeasureTheory.Measure",
"Monoid.toMulO... | simp only [h hνs, smul_apply, smul_zero, ← smul_one_smul ℝ≥0∞ c (0 : ℝ≥0∞)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.AbsolutelyContinuous | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 77
} | [
{
"pp": "α : Type u_1\nR : Type u_5\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nh : μ ≪ ν\nc : R\ns : Set α\nhνs : ν s = 0\n⊢ (c • μ) s = 0",
"usedConstants": [
"MulOne.toOne",
"instHSMul",
"MeasureTheory.Measure",
"Monoid.toMulO... | simp only [h hνs, smul_apply, smul_zero, ← smul_one_smul ℝ≥0∞ c (0 : ℝ≥0∞)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.AbsolutelyContinuous | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 77
} | [
{
"pp": "α : Type u_1\nR : Type u_5\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SMul R ℝ≥0∞\ninst✝ : IsScalarTower R ℝ≥0∞ ℝ≥0∞\nh : μ ≪ ν\nc : R\ns : Set α\nhνs : ν s = 0\n⊢ (c • μ) s = 0",
"usedConstants": [
"MulOne.toOne",
"instHSMul",
"MeasureTheory.Measure",
"Monoid.toMulO... | simp only [h hνs, smul_apply, smul_zero, ← smul_one_smul ℝ≥0∞ c (0 : ℝ≥0∞)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Map | {
"line": 170,
"column": 79
} | {
"line": 171,
"column": 85
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure α\nf : α → β\nhf : AEMeasurable f μ\n⊢ map f μ = 0 ↔ μ = 0",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"_private.Mathlib.MeasureTheory.Measure.Map.0.MeasureTheory.Measure.map_eq_zer... | by
simp_rw [← measure_univ_eq_zero, map_apply_of_aemeasurable hf .univ, preimage_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 206,
"column": 40
} | {
"line": 206,
"column": 90
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ns : Set β\nhs : s.Countable\nf : α → β\nhf : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y})\n⊢ ∑' (b : ↑s), μ (f ⁻¹' {↑b}) = μ (⋃ y ∈ s, f ⁻¹' {y})",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.measure_biUnion",
"MeasureTheory.... | measure_biUnion hs (pairwiseDisjoint_fiber f s) hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Typeclasses.Finite | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 37
} | [
{
"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 : μ s = 0\nh_ss : sᶜ ⊆ {a | (if a ∈ s then f a else g a) = g a}\n⊢ (fun x ↦ if x ∈ s then f x else g x) =ᶠ[ae μ] g",
"usedConstants": [
"MeasureTheory.Measur... | refine measure_mono_null ?_ hs_zero | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.Filter.CountableSeparatingOn | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 27
} | [
{
"pp": "case inr\nα : Type u_1\nl : Filter α\ninst✝¹ : CountableInterFilter l\np : Set α → Prop\ns : Set α\ninst✝ : HasCountableSeparatingOn α p s\nhs : s ∈ l\nhne : s.Nonempty\nhl : ∀ (U : Set α), p U → U ∈ l ∨ Uᶜ ∈ l\nx : α\nhts : {x} ⊆ s\nht : {x}.Subsingleton\nhtl : {x} ∈ l\n⊢ ∃ a ∈ s, {a} ∈ l",
"usedC... | · exact ⟨x, hts rfl, htl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.MeasureSpace | {
"line": 676,
"column": 4
} | {
"line": 676,
"column": 69
} | [
{
"pp": "α : Type u_8\nι : Type u_9\nx✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝¹ : SemilatticeSup ι\ninst✝ : Countable ι\nf : ι → Set α\nhm : ∀ (i : ι), NullMeasurableSet (f i) μ\nε : ℝ≥0∞\nhε : 0 < ε\nhfin : ∃ i, μ (f i) ≠ ∞\nhfem : ⋂ n, f n = ∅\nF : ι → ℝ≥0∞ := fun m ↦ μ (⋂ n, ⋂ (_ : n ≤ m), f n)\nhFAnti :... | rw [ENNReal.tendsto_atTop_zero_iff_lt_of_antitone hFAnti] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 38
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\n⊢ ae (μ.restrict (s ∪ t)) = ae (μ.restrict s) ⊔ ae (μ.restrict t)",
"usedConstants": [
"cond",
"MeasureTheory.ae",
"MeasureTheory.ae.congr_simp",
"Filter.instSupSet",
"iSup_bool_eq",
"MeasureTheory... | simp [union_eq_iUnion, iSup_bool_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 38
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\n⊢ ae (μ.restrict (s ∪ t)) = ae (μ.restrict s) ⊔ ae (μ.restrict t)",
"usedConstants": [
"cond",
"MeasureTheory.ae",
"MeasureTheory.ae.congr_simp",
"Filter.instSupSet",
"iSup_bool_eq",
"MeasureTheory... | simp [union_eq_iUnion, iSup_bool_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Restrict | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 38
} | [
{
"pp": "α : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\n⊢ ae (μ.restrict (s ∪ t)) = ae (μ.restrict s) ⊔ ae (μ.restrict t)",
"usedConstants": [
"cond",
"MeasureTheory.ae",
"MeasureTheory.ae.congr_simp",
"Filter.instSupSet",
"iSup_bool_eq",
"MeasureTheory... | simp [union_eq_iUnion, iSup_bool_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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