module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Function.LpSeminorm.Trim | {
"line": 29,
"column": 2
} | {
"line": 29,
"column": 20
} | {
"line": 30,
"column": 2
} | [
{
"pp": "α : Type u_1\nε : Type u_3\nm m0 : MeasurableSpace α\nq : ℝ\nμ : Measure α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nhm : m ≤ m0\nf : α → ε\nhf : StronglyMeasurable f\n⊢ eLpNorm' f q (μ.trim hm) = eLpNorm' f q μ",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nε : Type u_3\nm m0 : MeasurableSpace α\nq : ℝ\nμ : Measure α\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nhm : m ≤ m0\nf : α → ε\nhf : StronglyMeasurable f\n⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ.trim hm) ^ (1 / q) = (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q)"
] | simp_rw [eLpNorm'] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable | {
"line": 367,
"column": 2
} | {
"line": 367,
"column": 40
} | {
"line": 368,
"column": 2
} | [
{
"pp": "α : Type u_1\nF : Type u_2\np : ℝ≥0∞\ninst✝² : NormedAddCommGroup F\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝¹ : Fact (1 ≤ p)\ninst✝ : NormedSpace ℝ F\nhm : m ≤ m0\nhp_ne_top : p ≠ ∞\nP : ↥(Lp F p μ) → Prop\nh_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(simpleFunc.indi... | [
"α : Type u_1\nF : Type u_2\np : ℝ≥0∞\ninst✝² : NormedAddCommGroup F\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝¹ : Fact (1 ≤ p)\ninst✝ : NormedSpace ℝ F\nhm : m ≤ m0\nhp_ne_top : p ≠ ∞\nP : ↥(Lp F p μ) → Prop\nh_ind : ∀ (c : F) {s : Set α} (hs : MeasurableSet s) (hμs : μ s < ∞), P ↑(simpleFunc.indicatorConst p... | let f' := (⟨f, hf⟩ : lpMeas F ℝ m p μ) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 457,
"column": 4
} | {
"line": 457,
"column": 31
} | {
"line": 459,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\ninst✝ : SigmaFinite (μ.trim hm)\nh : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2 hs hμs 1))) μ\nt : Set α\nht : MeasurableSet t\nhμt : (μ.trim hm) t < ... | [] | exact ENNReal.toReal_nonneg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 391,
"column": 76
} | {
"line": 397,
"column": 54
} | {
"line": 399,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_3\n𝕜 : Type u_4\ninst✝⁵ : RCLike 𝕜\nm m₀ : MeasurableSpace α\nμ : Measure α\nf : α → E\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : InnerProductSpace 𝕜 E\nhm : m ≤ m₀\nhf1 : Integrable f μ\nhf2 : MemLp f 2 μ\ninst✝ : SigmaFinit... | [] | by
refine ae_eq_condExp_of_forall_setIntegral_eq hm hf1
(fun s hs htop ↦ integrableOn_condExpL2_of_measure_ne_top hm htop.ne _) (fun s hs htop ↦ ?_)
(aestronglyMeasurable_condExpL2 hm _)
rw [integral_condExpL2_eq hm (hf2.toLp _) hs htop.ne]
refine setIntegral_congr_ae (hm _ hs) ?_
filter_upwards [hf2.co... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Matrix | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 21
} | {
"line": 83,
"column": 4
} | [
{
"pp": "ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A ... | [
"ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix... | rintro q ⟨i, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 39
} | {
"line": 122,
"column": 0
} | [
{
"pp": "case inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : AddTorsor V P\ninst✝² : Module k V\nι : Type u_4\ninst✝¹ : Fintype ι\ninst✝ : Nontrivial ι\np : ι → P\ni : ι\nw : ι → k\nhw : ∑ i, w i = 1\nh✝ : Nontrivial k\nj : ι\nhj : j ≠ i\n⊢ ∑ i_1, (-(w i • (Pi.... | [] | simp [sum_add_distrib, ← mul_sum, hw] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.FixedSubmodule | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 97
} | {
"line": 100,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\nV : Type u_3\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\nW : Submodule R V\nhe : e ∈ fixingSubgroup (V ≃ₗ[R] V) W.carrier\nv : V\n⊢ v ∈ map (↑e) W ↔ v ∈ W",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"Submodule",
... | [
"R : Type u_1\ninst✝² : Semiring R\nV : Type u_3\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\ne : V ≃ₗ[R] V\nW : Submodule R V\nv : V\nhe : ∀ y ∈ W, e y = y\n⊢ v ∈ map (↑e) W ↔ v ∈ W"
] | simp only [mem_fixingSubgroup_iff, carrier_eq_coe, SetLike.mem_coe, LinearEquiv.smul_def] at he | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.TensorAlgebra.ToTensorPower | {
"line": 43,
"column": 71
} | {
"line": 56,
"column": 83
} | {
"line": 58,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ni j : ℕ\na : ⨂[R]^i M\nb : ⨂[R]^j M\n⊢ toTensorAlgebra (GradedMonoid.GMul.mul a b) = toTensorAlgebra a * toTensorAlgebra b",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"PiTe... | [] | by
-- change `a` and `b` to `tprod R a` and `tprod R b`
rw [TensorPower.gMul_eq_coe_linearMap, ← LinearMap.compr₂_apply, ← @LinearMap.mul_apply' R, ←
LinearMap.compl₂_apply, ← LinearMap.comp_apply]
refine LinearMap.congr_fun (LinearMap.congr_fun ?_ a) b
clear! a b
ext (a b)
simp only [LinearMap.compMult... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | {
"line": 381,
"column": 13
} | {
"line": 381,
"column": 27
} | {
"line": 381,
"column": 28
} | [
{
"pp": "R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ : GradedAlgebra ℬ\... | [
"R : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ : GradedAlgebra ℬ\ninst✝ : Mod... | auxEquiv_comm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.FreeModule.Int | {
"line": 106,
"column": 10
} | {
"line": 106,
"column": 25
} | {
"line": 107,
"column": 10
} | [
{
"pp": "case e'_2.a\nι : Type u_1\nR : Type u_2\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Fintype ι\ninst✝ : Module R M\nN : Submodule R M\nbM : Basis ι R M\nbN : Basis (Fin n) R ↥N\nf : Fin n ↪ ι\na : Fin n → R\nsnf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)\nN' : Submodule R... | [
"case e'_2.a\nι : Type u_1\nR : Type u_2\nM : Type u_3\nn : ℕ\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Fintype ι\ninst✝ : Module R M\nN : Submodule R M\nbM : Basis ι R M\nbN : Basis (Fin n) R ↥N\nf : Fin n ↪ ι\na : Fin n → R\nsnf : ∀ (i : Fin n), ↑(bN i) = a i • bM (f i)\nN' : Submodule R (ι → R) := ... | specialize hj x | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.LinearAlgebra.LinearIndependent.BaseChange | {
"line": 39,
"column": 48
} | {
"line": 39,
"column": 82
} | {
"line": 39,
"column": 82
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\ninst✝³ : Finite ι'\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nv : ι → ι' → K\nhv : LinearIndependent K v\nthis : Fintype ι' := Fintype.ofFinite ι'\nI : Set (ι' → K) := ⋯.extend ⋯\nb : Module.Basis (↑I) K (ι' → K) := Module.Basis.ex... | [
"ι : Type u_1\nι' : Type u_2\ninst✝³ : Finite ι'\nK : Type u_3\nL : Type u_4\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nv : ι → ι' → K\nhv : LinearIndependent K v\nthis✝ : Fintype ι' := Fintype.ofFinite ι'\nI : Set (ι' → K) := ⋯.extend ⋯\nb : Module.Basis (↑I) K (ι' → K) := Module.Basis.extend ⋯\nb' ... | have : Injective v := hv.injective | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.Determinant.Bird | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 10
} | {
"line": 127,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nn k : ℕ\nA : Array R\nhn : n = k + 1\n⊢ birdDet n A = (-1) ^ k * BirdDet.iter n A k (BirdDet.get n A) 0 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMu... | [
"R : Type u_1\ninst✝ : CommRing R\nk : ℕ\nA : Array R\n⊢ birdDet (k + 1) A = (-1) ^ k * BirdDet.iter (k + 1) A k (BirdDet.get (k + 1) A) 0 0"
] | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 68,
"column": 9
} | {
"line": 68,
"column": 31
} | {
"line": 68,
"column": 32
} | [
{
"pp": "case mp\nm : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝² : CommRing R\nA : Matrix m n R\nι : Type w\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : ι → m\ng : ι → n\nhA : (A.submatrix (f ∘ ⇑(Fintype.equivFin ι).symm) (g ∘ ⇑(Fintype.equivFin ι).symm)).det ∈ Set.range SignType.cast\n⊢ (A.submatrix f g).... | [
"case mp\nm : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝² : CommRing R\nA : Matrix m n R\nι : Type w\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : ι → m\ng : ι → n\nhA : ((A.submatrix f g).submatrix ⇑(Fintype.equivFin ι).symm ⇑(Fintype.equivFin ι).symm).det ∈ Set.range SignType.cast\n⊢ (A.submatrix f g).det ∈ Se... | ← submatrix_submatrix, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 71,
"column": 9
} | {
"line": 71,
"column": 31
} | {
"line": 71,
"column": 32
} | [
{
"pp": "case mpr\nm : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝ : CommRing R\nA : Matrix m n R\nk : ℕ\nf : Fin k → m\ng : Fin k → n\nhA : (A.submatrix (f ∘ ⇑Equiv.ulift) (g ∘ ⇑Equiv.ulift)).det ∈ Set.range SignType.cast\n⊢ (A.submatrix f g).det ∈ Set.range SignType.cast",
"ppTerm": "?mpr",
"assigned"... | [
"case mpr\nm : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝ : CommRing R\nA : Matrix m n R\nk : ℕ\nf : Fin k → m\ng : Fin k → n\nhA : ((A.submatrix f g).submatrix ⇑Equiv.ulift ⇑Equiv.ulift).det ∈ Set.range SignType.cast\n⊢ (A.submatrix f g).det ∈ Set.range SignType.cast"
] | ← submatrix_submatrix, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 47
} | {
"line": 83,
"column": 0
} | [
{
"pp": "m : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nh : ↑A 1 0 ≠ 0\n⊢ (!![0, -1; 1, 0] * (!![1, -(↑A 0 0 / ↑A 1 0); 0, 1] * ↑A)) 1 0 = ↑A 0 0 + -(↑A 1 0 * (↑A 0 0 / ↑A 1 0))",
"ppTerm": "?m.92",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"NonUnitalNonAssocCommRing.toNonUnit... | [] | norm_num [vecMul, vecHead, vecTail, mul_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_NormNum_Core___elabRules_Mathlib_Tactic_normNum_1 | Mathlib.Tactic.normNum |
Mathlib.LinearAlgebra.Matrix.Integer | {
"line": 81,
"column": 58
} | {
"line": 81,
"column": 71
} | {
"line": 81,
"column": 72
} | [
{
"pp": "m : Type u_1\nn : Type u_2\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nA : Matrix m n ℚ\ni : m\nj : n\nk : ℕ\nhk : A.den = (A i j).den * k\n⊢ ↑(A i j * ↑((A i j).den * k)).num = A i j * ↑((A i j).den * k)",
"ppTerm": "?m.91",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing... | [
"m : Type u_1\nn : Type u_2\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nA : Matrix m n ℚ\ni : m\nj : n\nk : ℕ\nhk : A.den = (A i j).den * k\n⊢ ↑(A i j * (↑(A i j).den * ↑k)).num = A i j * (↑(A i j).den * ↑k)"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Unique | {
"line": 35,
"column": 28
} | {
"line": 35,
"column": 91
} | {
"line": 36,
"column": 2
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nA : Type u_3\nR : Type u_4\ninst✝¹ : Unique m\ninst✝ : Unique n\nM : Matrix m n A\ni : m\nj : n\n⊢ (fun a ↦ of fun x x_1 ↦ a) ((fun M ↦ M default default) M) i j = M i j",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Inhabited.default",
"E... | [] | simp [Subsingleton.elim i default, Subsingleton.elim j default] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.WithConv | {
"line": 104,
"column": 2
} | {
"line": 108,
"column": 39
} | {
"line": 110,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝² : Semiring α\ninst✝¹ : IsLeftCancelMulZero α\ninst✝ : StarRing α\nf : WithConv (Matrix m n α)\nhf : IsIdempotentElem f\n⊢ IsSelfAdjoint f",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Is... | [] | simp_rw [IsIdempotentElem, WithConv.ext_iff, ← Matrix.ext_iff, convMul_def, hadamard_apply,
← isIdempotentElem_iff, IsIdempotentElem.iff_eq_zero_or_one] at hf
rw [IsSelfAdjoint, WithConv.ext_iff]
ext i j
obtain (h | h) := hf i j <;> simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.WithConv | {
"line": 104,
"column": 2
} | {
"line": 108,
"column": 39
} | {
"line": 110,
"column": 0
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝² : Semiring α\ninst✝¹ : IsLeftCancelMulZero α\ninst✝ : StarRing α\nf : WithConv (Matrix m n α)\nhf : IsIdempotentElem f\n⊢ IsSelfAdjoint f",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Is... | [] | simp_rw [IsIdempotentElem, WithConv.ext_iff, ← Matrix.ext_iff, convMul_def, hadamard_apply,
← isIdempotentElem_iff, IsIdempotentElem.iff_eq_zero_or_one] at hf
rw [IsSelfAdjoint, WithConv.ext_iff]
ext i j
obtain (h | h) := hf i j <;> simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.PiTensorProduct.Dual | {
"line": 50,
"column": 2
} | {
"line": 51,
"column": 6
} | {
"line": 53,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : ι → Type u_3\ninst✝³ : CommSemiring R\ninst✝² : (i : ι) → AddCommMonoid (M i)\ninst✝¹ : (i : ι) → Module R (M i)\ninst✝ : Fintype ι\nf : (i : ι) → Dual R (M i)\nm : (i : ι) → M i\n⊢ (dualDistrib ((tprod R) fun i ↦ f i)) (⨂ₜ[R] (i : ι), m i) = ∏ i, (f i) (m i)",
"ppTe... | [] | rw [dualDistrib, Subsingleton.elim (Fintype.ofFinite ι) ‹_›]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.PiTensorProduct.Dual | {
"line": 50,
"column": 2
} | {
"line": 51,
"column": 6
} | {
"line": 53,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : ι → Type u_3\ninst✝³ : CommSemiring R\ninst✝² : (i : ι) → AddCommMonoid (M i)\ninst✝¹ : (i : ι) → Module R (M i)\ninst✝ : Fintype ι\nf : (i : ι) → Dual R (M i)\nm : (i : ι) → M i\n⊢ (dualDistrib ((tprod R) fun i ↦ f i)) (⨂ₜ[R] (i : ι), m i) = ∏ i, (f i) (m i)",
"ppTe... | [] | rw [dualDistrib, Subsingleton.elim (Fintype.ofFinite ι) ‹_›]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.TensorProduct.Isometries | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 32
} | {
"line": 52,
"column": 0
} | [
{
"pp": "R : Type uR\nM₁ : Type uM₁\nM₂ : Type uM₂\nM₃ : Type uM₃\nM₄ : Type uM₄\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M₁\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : AddCommGroup M₄\ninst✝⁴ : Module R M₁\ninst✝³ : Module R M₂\ninst✝² : Module R M₃\ninst✝¹ : Module R M₄\ninst✝ : Invertibl... | [] | simp [h₁, h₃, associated_tmul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.QuadraticForm.QuadraticModuleCat.Monoidal | {
"line": 145,
"column": 12
} | {
"line": 145,
"column": 29
} | {
"line": 145,
"column": 30
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Invertible 2\nX Y Z : QuadraticModuleCat R\n⊢ (α_ X.toModuleCat Y.toModuleCat Z.toModuleCat).hom =\n (𝟙 ((X.toModuleCat ⊗ Y.toModuleCat) ⊗ Z.toModuleCat) ≫ (𝟙 (X.toModuleCat ⊗ Y.toModuleCat) ⊗ₘ 𝟙 Z.toModuleCat)) ≫\n (α_ X.toModuleCat Y.toModuleCat Z.t... | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : Invertible 2\nX Y Z : QuadraticModuleCat R\n⊢ (α_ X.toModuleCat Y.toModuleCat Z.toModuleCat).hom =\n (𝟙 (X.toModuleCat ⊗ Y.toModuleCat) ⊗ₘ 𝟙 Z.toModuleCat) ≫\n (α_ X.toModuleCat Y.toModuleCat Z.toModuleCat).hom ≫\n 𝟙 (X.toModuleCat ⊗ Y.toModuleCat ⊗ Z.toMo... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.QuadraticModuleCat.Monoidal | {
"line": 145,
"column": 30
} | {
"line": 145,
"column": 47
} | {
"line": 145,
"column": 48
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : Invertible 2\nX Y Z : QuadraticModuleCat R\n⊢ (α_ X.toModuleCat Y.toModuleCat Z.toModuleCat).hom =\n (𝟙 (X.toModuleCat ⊗ Y.toModuleCat) ⊗ₘ 𝟙 Z.toModuleCat) ≫\n (α_ X.toModuleCat Y.toModuleCat Z.toModuleCat).hom ≫\n 𝟙 (X.toModuleCat ⊗ Y.toModule... | [
"R : Type u\ninst✝¹ : CommRing R\ninst✝ : Invertible 2\nX Y Z : QuadraticModuleCat R\n⊢ (α_ X.toModuleCat Y.toModuleCat Z.toModuleCat).hom =\n (𝟙 (X.toModuleCat ⊗ Y.toModuleCat) ⊗ₘ 𝟙 Z.toModuleCat) ≫\n (α_ X.toModuleCat Y.toModuleCat Z.toModuleCat).hom ≫ 𝟙 (X.toModuleCat ⊗ Y.toModuleCat ⊗ Z.toModuleCat)"... | Category.id_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 11
} | {
"line": 150,
"column": 2
} | [
{
"pp": "M : Type u_2\ninst✝⁴ : AddCommGroup M\n𝕜 : Type u_4\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : Module 𝕜 M\nQ : QuadraticForm 𝕜 M\ninst✝ : FiniteDimensional 𝕜 M\nV : Subspace 𝕜 M\nhV : ∀ x ∈ V, Q x ≤ 0\nVp : Submodule 𝕜 M\nhr : Module.finrank 𝕜 ↥Vp = sigPos Q\nhVp : (QuadraticMap.restr... | [
"M : Type u_2\ninst✝⁴ : AddCommGroup M\n𝕜 : Type u_4\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : Module 𝕜 M\nQ : QuadraticForm 𝕜 M\ninst✝ : FiniteDimensional 𝕜 M\nV : Subspace 𝕜 M\nhV : ∀ x ∈ V, Q x ≤ 0\nVp : Submodule 𝕜 M\nhr : Module.finrank 𝕜 ↥Vp = sigPos Q\nhVp : (QuadraticMap.restrict Q Vp).Po... | rw [← hr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 124,
"column": 27
} | {
"line": 129,
"column": 30
} | {
"line": 131,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nP : RootPairing ι R M N\ninst✝ : P.IsCrystallographic\nb : P.Base\ni : ↥b.support\nj : ↥b.support ⊕ ι\nx : Matrix (↥b.support ⊕ ι) (↥b... | [] | by
induction hx using span_induction with
| mem x h => obtain ⟨i, rfl⟩ := h; cases j <;> simp [h]
| zero => simp
| add u v _ _ hu hv => simp [hu, hv]
| smul t u _ hu => simp [hu] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 319,
"column": 2
} | {
"line": 323,
"column": 8
} | {
"line": 325,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.EmbeddedG2\ninst✝² : Finite ι\ninst✝¹ : CharZero R\ninst✝ : IsDomain R\n⊢ (allRoots P).Nodup",
... | [] | have hli : Injective (Fintype.linearCombination ℤ ![shortRoot P, longRoot P]) := by
rw [← linearIndependent_iff_injective_fintypeLinearCombination]
exact (linearIndependent_short_long P).restrict_scalars' ℤ
rw [allRoots_eq_map_allCoeffs, nodup_map_iff hli]
decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 319,
"column": 2
} | {
"line": 323,
"column": 8
} | {
"line": 325,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.EmbeddedG2\ninst✝² : Finite ι\ninst✝¹ : CharZero R\ninst✝ : IsDomain R\n⊢ (allRoots P).Nodup",
... | [] | have hli : Injective (Fintype.linearCombination ℤ ![shortRoot P, longRoot P]) := by
rw [← linearIndependent_iff_injective_fintypeLinearCombination]
exact (linearIndependent_short_long P).restrict_scalars' ℤ
rw [allRoots_eq_map_allCoeffs, nodup_map_iff hli]
decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 15
} | {
"line": 50,
"column": 2
} | [
{
"pp": "case inl.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : CharZero R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb :... | [] | simp [h, e] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 15
} | {
"line": 50,
"column": 2
} | [
{
"pp": "case inl.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : CharZero R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb :... | [] | simp [h, e] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 15
} | {
"line": 50,
"column": 2
} | [
{
"pp": "case inl.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : Finite ι\ninst✝⁸ : CommRing R\ninst✝⁷ : IsDomain R\ninst✝⁶ : CharZero R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : P.IsCrystallographic\nb :... | [] | simp [h, e] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 172,
"column": 8
} | {
"line": 172,
"column": 20
} | {
"line": 172,
"column": 21
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j k l m : ι\ni... | ← two_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 531,
"column": 2
} | {
"line": 531,
"column": 84
} | {
"line": 532,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.IsIrreducible\n... | [
"ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.IsIrreducible\nthis :\n (∀... | rw [show P.root '' {long P, short P} = {longRoot P, shortRoot P} by aesop] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 95,
"column": 34
} | {
"line": 95,
"column": 58
} | {
"line": 95,
"column": 58
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\ninst✝² : P.IsReduced\nb : P.Base\ni... | [] | rw [contra, hk₁]; module | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 95,
"column": 34
} | {
"line": 95,
"column": 58
} | {
"line": 95,
"column": 58
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\ninst✝² : P.IsReduced\nb : P.Base\ni... | [] | rw [contra, hk₁]; module | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 329,
"column": 74
} | {
"line": 329,
"column": 89
} | {
"line": 329,
"column": 89
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹¹ : Finite ι\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\nb : P.Base\nins... | [] | by rw [hl', hk] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 318,
"column": 23
} | {
"line": 318,
"column": 37
} | {
"line": 319,
"column": 2
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : CharZero K\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : Fintype ι\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module K M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module K N\nP : RootPairing ι K M N\ninst✝² : P.IsCrystallographic\nb : P.Base\nins... | [] | simp [hωu, hU] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 318,
"column": 23
} | {
"line": 318,
"column": 37
} | {
"line": 319,
"column": 2
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : CharZero K\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : Fintype ι\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module K M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module K N\nP : RootPairing ι K M N\ninst✝² : P.IsCrystallographic\nb : P.Base\nins... | [] | simp [hωu, hU] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 318,
"column": 23
} | {
"line": 318,
"column": 37
} | {
"line": 319,
"column": 2
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : CharZero K\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : Fintype ι\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module K M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module K N\nP : RootPairing ι K M N\ninst✝² : P.IsCrystallographic\nb : P.Base\nins... | [] | simp [hωu, hU] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.Projective | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 64
} | {
"line": 74,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\nP : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)\nI J : Finset ι\nhP : IsProjectiveMeasureFamily P\nhJI : J ⊆ I\nthis : univ = Finset.restrict₂ hJI ⁻¹' univ\n⊢ Measurable (Finset.restrict₂ hJI)",
"ppTerm": "?m.41",
"assigned... | [] | · exact measurable_pi_lambda _ (fun _ ↦ measurable_pi_apply _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 65
} | {
"line": 189,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nh_nonempty : Nonempty ((i : ι) → α i)\ns : Finset ι\nS : Set ((i : ↥s) → α ↑i)\nh : cylinder s S = ∅\nhS : S.Nonempty\nf : (i : ↥s) → α ↑i := hS.some\nhf : f ∈ S\nf' : (i : ι) → α i := fun i ↦ if hi : i ∈ s then f ⟨i, hi⟩ else h_nonempty.some i\n⊢ s.restrict f' ∈ S",
... | [] | simpa only [Finset.restrict_def, Finset.coe_mem, dif_pos, f'] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 30
} | {
"line": 337,
"column": 2
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\ns : Finset ι\nS : Set ((i : ↥s) → α ↑i)\nhS : MeasurableSet S\n⊢ ∃ s_1 S_1, MeasurableSet S_1 ∧ (cylinder s S)ᶜ = cylinder s_1 S_1",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"MeasurableSet",
... | [
"ι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\ns : Finset ι\nS : Set ((i : ↥s) → α ↑i)\nhS : MeasurableSet S\n⊢ (cylinder s S)ᶜ = cylinder s Sᶜ"
] | refine ⟨s, Sᶜ, hS.compl, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Order.WithTop | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 45
} | {
"line": 70,
"column": 4
} | [
{
"pp": "case inr.a\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι := ⋯\nd : S... | [
"case inr.a\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι := ⋯\nd : Set (WithTop ... | apply le_generateFrom_iff_subset_isOpen.2 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.SetAlgebra | {
"line": 184,
"column": 4
} | {
"line": 188,
"column": 22
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case base\nα : Type u_1\n𝒜 : Set (Set α)\ns u : Set α\nu_mem : u ∈ 𝒜\n⊢ ∃ A, A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ u = ⋃ a ∈ A, ⋂ t ∈ a, t",
"ppTerm": "?base",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.eq_of_mem_singleton",
"If... | [] | refine ⟨{{u}}, finite_singleton {u},
fun a ha ↦ eq_of_mem_singleton ha ▸ finite_singleton u,
fun a ha t ht ↦ ?_, by simp⟩
rw [eq_of_mem_singleton ha, ha, eq_of_mem_singleton ht, ht] at *
exact Or.inl u_mem | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.SetAlgebra | {
"line": 184,
"column": 4
} | {
"line": 188,
"column": 22
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case base\nα : Type u_1\n𝒜 : Set (Set α)\ns u : Set α\nu_mem : u ∈ 𝒜\n⊢ ∃ A, A.Finite ∧ (∀ a ∈ A, a.Finite) ∧ (∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜) ∧ u = ⋃ a ∈ A, ⋂ t ∈ a, t",
"ppTerm": "?base",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.eq_of_mem_singleton",
"If... | [] | refine ⟨{{u}}, finite_singleton {u},
fun a ha ↦ eq_of_mem_singleton ha ▸ finite_singleton u,
fun a ha t ht ↦ ?_, by simp⟩
rw [eq_of_mem_singleton ha, ha, eq_of_mem_singleton ht, ht] at *
exact Or.inl u_mem | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.WithTop | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 43
} | {
"line": 79,
"column": 2
} | [
{
"pp": "case a\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι := if h : ∃ x, ... | [
"case a\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι := ⋯\nd : Set (WithTop ι) :... | apply le_generateFrom_iff_subset_isOpen.2 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 13
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case h_mem\nα : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nι : Type u_3\na : Finset ι\nf : ι → Set α\nhf : ∀ i ∈ a, f i ∈ C\nh_dis : (↑a).PairwiseDisjoint f\nh_mem : ⋃ i ∈ a, f i ∈ C\nA : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)\n⊢ ⋃₀ ↑(Finset.image f a) ∈ C",
... | [] | rwa [← A] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 13
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case h_mem\nα : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nι : Type u_3\na : Finset ι\nf : ι → Set α\nhf : ∀ i ∈ a, f i ∈ C\nh_dis : (↑a).PairwiseDisjoint f\nh_mem : ⋃ i ∈ a, f i ∈ C\nA : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)\n⊢ ⋃₀ ↑(Finset.image f a) ∈ C",
... | [] | rwa [← A] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 13
} | {
"line": 117,
"column": 2
} | [
{
"pp": "case h_mem\nα : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nι : Type u_3\na : Finset ι\nf : ι → Set α\nhf : ∀ i ∈ a, f i ∈ C\nh_dis : (↑a).PairwiseDisjoint f\nh_mem : ⋃ i ∈ a, f i ∈ C\nA : ⋃ i ∈ a, f i = ⋃₀ ↑(Finset.image f a)\n⊢ ⋃₀ ↑(Finset.image f a) ∈ C",
... | [] | rwa [← A] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 13
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case convert_3\nα : Type u_1\nC : Set (Set α)\ns t : Set α\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nhs : s ∈ C\nht : t ∈ C\nhst : s ∪ t ∈ C\nh_dis : Disjoint s t\nA : s ∪ t = ⋃ i, ![s, t] i\n⊢ ⋃ i, ![s, t] i ∈ C",
"ppTerm": "?convert_3",
"assigned": true,
"usedConstants":... | [] | rwa [← A] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 13
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case convert_3\nα : Type u_1\nC : Set (Set α)\ns t : Set α\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nhs : s ∈ C\nht : t ∈ C\nhst : s ∪ t ∈ C\nh_dis : Disjoint s t\nA : s ∪ t = ⋃ i, ![s, t] i\n⊢ ⋃ i, ![s, t] i ∈ C",
"ppTerm": "?convert_3",
"assigned": true,
"usedConstants":... | [] | rwa [← A] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 13
} | {
"line": 148,
"column": 0
} | [
{
"pp": "case convert_3\nα : Type u_1\nC : Set (Set α)\ns t : Set α\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nhs : s ∈ C\nht : t ∈ C\nhst : s ∪ t ∈ C\nh_dis : Disjoint s t\nA : s ∪ t = ⋃ i, ![s, t] i\n⊢ ⋃ i, ![s, t] i ∈ C",
"ppTerm": "?convert_3",
"assigned": true,
"usedConstants":... | [] | rwa [← A] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 24
} | {
"line": 215,
"column": 4
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nhC : IsSetSemiring C\nJ J' : Finset (Set α)\nhJ : ↑J ⊆ C\nhJdisj : (↑J).PairwiseDisjoint id\nhJ' : ↑J' ⊆ C\nhJ'disj : (↑J').PairwiseDisjoint id\nh : ⋃₀ ↑J = ⋃₀ ↑J'\nt : Set α\nht : t ∈ J'\nthis : t = ⋃ s ∈ J, s ∩ ... | [
"α : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nhC : IsSetSemiring C\nJ J' : Finset (Set α)\nhJ : ↑J ⊆ C\nhJdisj : (↑J).PairwiseDisjoint id\nhJ' : ↑J' ⊆ C\nhJ'disj : (↑J').PairwiseDisjoint id\nh : ⋃₀ ↑J = ⋃₀ ↑J'\nt : Set α\nht : t ∈ J'\nthis : t = ⋃ s ∈ J, s ∩ t\n⊢ ∑ s ∈ J... | nth_rewrite 2 [this] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rewrite______1 | Mathlib.Tactic.tacticNth_rewrite_____ |
Mathlib.Topology.Order.WithTop | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 38
} | {
"line": 235,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\na : ↑{a | a ≠ ⊤}\n⊢ Tendsto (fun x ↦ (↑x).untop ⋯) (𝓝 a) (𝓝 ((↑a).untop ⋯))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"setOf",
"Membership.mem",
"Ne",
"Nonem... | [
"ι : Type u_1\ninst✝² : LinearOrder ι\ninst✝¹ : TopologicalSpace ι\ninst✝ : OrderTopology ι\na : ↑{a | a ≠ ⊤}\nthis : Nonempty ι\n⊢ Tendsto (fun x ↦ (↑x).untop ⋯) (𝓝 a) (𝓝 ((↑a).untop ⋯))"
] | have : Nonempty ι := ⟨untop a.1 a.2⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.SetSemiring | {
"line": 544,
"column": 4
} | {
"line": 544,
"column": 80
} | {
"line": 545,
"column": 4
} | [
{
"pp": "case inr.inr\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : Nonempty α\nu v : α\nhuv : u ≤ v\nu' v' : α\nhu'v' : u' ≤ v'\nhu : u < u'\nhv : v' < v\n⊢ ∃ I, ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} ∧ (↑I).PairwiseDisjoint id ∧ Set.Ioc u v \\ Set.Ioc u' v' = ⋃₀ ↑I",
"ppTerm": "?inr.inr",
"assigne... | [
"case inr.inr\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : Nonempty α\nu v : α\nhuv : u ≤ v\nu' v' : α\nhu'v' : u' ≤ v'\nhu : u < u'\nhv : v' < v\n⊢ ∃ I, ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} ∧ (↑I).PairwiseDisjoint id ∧ Set.Ioc u u' ∪ Set.Ioc v' v = ⋃₀ ↑I"
] | rw [show Set.Ioc u v \ Set.Ioc u' v' = Set.Ioc u u' ∪ Set.Ioc v' v by grind] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 59
} | {
"line": 228,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\nmE : MeasurableSpace E\ninst✝¹ : BorelSpace E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhφ_cvx : ConvexOn ℝ univ φ\nhφ_cont ... | [] | exact hφ_cvx.map_condExp_le_univ hm hφ_cont hf_int hφ_int | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 235,
"column": 6
} | {
"line": 235,
"column": 25
} | {
"line": 235,
"column": 25
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv : VectorMeasure α M\ninst✝ : T2Space M\ns : ℕ → Set α\nhm : Monotone s\nhs : ∀ (i : ℕ), MeasurableSet (s i)\nt : ℕ → Set α := disjointed s\nht : ∀ (n : ℕ), MeasurableSet (t n)\nthis : HasSum (fun... | [
"α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv : VectorMeasure α M\ninst✝ : T2Space M\ns : ℕ → Set α\nhm : Monotone s\nhs : ∀ (i : ℕ), MeasurableSet (s i)\nt : ℕ → Set α := disjointed s\nht : ∀ (n : ℕ), MeasurableSet (t n)\nthis : HasSum (fun n ↦ v (t n)... | ← of_biUnion_finset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 418,
"column": 6
} | {
"line": 418,
"column": 39
} | {
"line": 419,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : MeasurableSpace β\nx✝ : β\nv✝ : M\ns : Set β\nx : β\nv : M\nf : ℕ → Set β\nf_meas : ∀ (i : ℕ), MeasurableSet (f i)\nf_disj : Pairwise (Disjoint on f)\nhx : x ∈ ⋃ i, f i\n⊢ Mea... | [] | apply MeasurableSet.iUnion f_meas | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 31
} | {
"line": 182,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝ : MeasurableSpace α\nj : JordanDecomposition α\nS : Set α\nhS₁ : MeasurableSet S\nhS₂ : j.posPart S = 0\nhS₃ : j.negPart Sᶜ = 0\nA : Set α\nhA : MeasurableSet A\nhA₁ : A ⊆ S\n⊢ 0 ≤ j.negPart.real A",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants"... | [] | exact ENNReal.toReal_nonneg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 31
} | {
"line": 188,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : MeasurableSpace α\nj : JordanDecomposition α\nS : Set α\nhS₁ : MeasurableSet S\nhS₂ : j.posPart S = 0\nhS₃ : j.negPart Sᶜ = 0\nA : Set α\nhA : MeasurableSet A\nhA₁ : A ⊆ Sᶜ\n⊢ 0 A ≤ j.posPart.real A",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstan... | [] | exact ENNReal.toReal_nonneg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 363,
"column": 4
} | {
"line": 372,
"column": 23
} | {
"line": 373,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhMpos : 0 < M\nhM : eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal... | [] | rw [eLpNorm_indicator_eq_eLpNorm_restrict hs]
refine le_trans (eLpNorm_add_le ?_ ?_ hp_one) ?_
· exact StronglyMeasurable.aestronglyMeasurable
(hmeas.indicator (measurableSet_le measurable_const hmeas.nnnorm.measurable.subtype_coe))
· exact StronglyMeasurable.aestronglyMeasurable
(hmeas.indi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 363,
"column": 4
} | {
"line": 372,
"column": 23
} | {
"line": 373,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\nM : ℝ\nhMpos : 0 < M\nhM : eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal... | [] | rw [eLpNorm_indicator_eq_eLpNorm_restrict hs]
refine le_trans (eLpNorm_add_le ?_ ?_ hp_one) ?_
· exact StronglyMeasurable.aestronglyMeasurable
(hmeas.indicator (measurableSet_le measurable_const hmeas.nnnorm.measurable.subtype_coe))
· exact StronglyMeasurable.aestronglyMeasurable
(hmeas.indi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 998,
"column": 6
} | {
"line": 998,
"column": 32
} | {
"line": 998,
"column": 33
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝⁴ : TopologicalSpace M\ninst✝³ : AddCommGroup M\ninst✝² : PartialOrder M\ninst✝¹ : IsOrderedAddMonoid M\ninst✝ : IsTopologicalAddGroup M\nv w : VectorMeasure α M\ni : Set α\nhi : MeasurableSet i\nh : v ≤[i] w\nj : Set α\nhj₁ : MeasurableSet j\n⊢ v... | [
"α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝⁴ : TopologicalSpace M\ninst✝³ : AddCommGroup M\ninst✝² : PartialOrder M\ninst✝¹ : IsOrderedAddMonoid M\ninst✝ : IsTopologicalAddGroup M\nv w : VectorMeasure α M\ni : Set α\nhi : MeasurableSet i\nh : v ≤[i] w\nj : Set α\nhj₁ : MeasurableSet j\n⊢ (v.restrict i... | ← restrict_apply _ hi hj₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 487,
"column": 4
} | {
"line": 489,
"column": 47
} | {
"line": 491,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nh : s.totalVariation ≪ μ.ennrealToMeasure\n⊢ s ≪ᵥ μ",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Real",
"MeasureTheory.Measure",
"MeasureTheory.VectorMeasure.ennr... | [] | refine VectorMeasure.AbsolutelyContinuous.mk fun S hS₁ hS₂ => ?_
rw [← VectorMeasure.ennrealToMeasure_apply hS₁] at hS₂
exact null_of_totalVariation_zero s (h hS₂) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 487,
"column": 4
} | {
"line": 489,
"column": 47
} | {
"line": 491,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nμ : VectorMeasure α ℝ≥0∞\nh : s.totalVariation ≪ μ.ennrealToMeasure\n⊢ s ≪ᵥ μ",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Real",
"MeasureTheory.Measure",
"MeasureTheory.VectorMeasure.ennr... | [] | refine VectorMeasure.AbsolutelyContinuous.mk fun S hS₁ hS₂ => ?_
rw [← VectorMeasure.ennrealToMeasure_apply hS₁] at hS₂
exact null_of_totalVariation_zero s (h hS₂) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 464,
"column": 8
} | {
"line": 464,
"column": 18
} | {
"line": 464,
"column": 18
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nf : ℕ → α → β\ng : α → β\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : MemLp g p μ\nhui : UnifIntegrable... | [
"case pos\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nf : ℕ → α → β\ng : α → β\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : MemLp g p μ\nhui : UnifIntegrable f p μ\nhfg ... | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 495,
"column": 4
} | {
"line": 495,
"column": 85
} | {
"line": 496,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nf : ℕ → α → β\ng : α → β\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : MemLp g p μ\nhui : UnifIntegrable f p μ\nhf... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nf : ℕ → α → β\ng : α → β\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : MemLp g p μ\nhui : UnifIntegrable f p μ\nhfg : ∀ᵐ (x : ... | have : 0 ≤ ε.toReal / (3 * measureUnivNNReal μ ^ (1 / p.toReal)) := by positivity | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 182,
"column": 2
} | {
"line": 183,
"column": 8
} | {
"line": 185,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhf : MemLp f p μ\nhp : 1 ≤ p\nhg : ¬MemLp g p μ\n⊢ lpNorm (f + g) p μ ≤ lpNorm f p μ + lpNorm g p μ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [] | · rw [lpNorm_of_not_memLp fun hfg ↦ hg <| by simpa using hfg.sub hf, lpNorm_of_not_memLp hg]
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 230,
"column": 2
} | {
"line": 231,
"column": 73
} | {
"line": 232,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\ng : α → ℝ\nhg : MemLp g p μ\nh : ∀ (x : α), ‖f x‖ ≤ g x\nhf : AEStronglyMeasurable f μ\n⊢ lpNorm f p μ ≤ lpNorm g p μ",
"ppTerm": "?pos✝",
"assigned": true,
"usedCo... | [
"case neg\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\ng : α → ℝ\nhg : MemLp g p μ\nh : ∀ (x : α), ‖f x‖ ≤ g x\nhf : ¬AEStronglyMeasurable f μ\n⊢ lpNorm f p μ ≤ lpNorm g p μ"
] | · rw [← toReal_eLpNorm hf, ← toReal_eLpNorm hg.aestronglyMeasurable]
exact ENNReal.toNNReal_mono (hg.eLpNorm_ne_top) (eLpNorm_mono_real h) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1310,
"column": 12
} | {
"line": 1310,
"column": 26
} | {
"line": 1310,
"column": 26
} | [
{
"pp": "case e'_5\nα : Type u_1\nβ : Type u_2\nm✝ : MeasurableSpace α\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : AddCommMonoid L\ninst✝⁴ : TopologicalSpace L\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid N\ninst✝ : TopologicalSpace N\nm n✝ : MeasurableSpace α\nv : Vect... | [
"case e'_5\nα : Type u_1\nβ : Type u_2\nm✝ : MeasurableSpace α\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : AddCommMonoid L\ninst✝⁴ : TopologicalSpace L\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid N\ninst✝ : TopologicalSpace N\nm n✝ : MeasurableSpace α\nv : VectorMeasure α ... | if_pos (hf₁ n) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1309,
"column": 6
} | {
"line": 1310,
"column": 27
} | {
"line": 1311,
"column": 6
} | [
{
"pp": "case e'_5\nα : Type u_1\nβ : Type u_2\nm✝ : MeasurableSpace α\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : AddCommMonoid L\ninst✝⁴ : TopologicalSpace L\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid N\ninst✝ : TopologicalSpace N\nm n : MeasurableSpace α\nv : Vecto... | [
"case e'_6\nα : Type u_1\nβ : Type u_2\nm✝ : MeasurableSpace α\nL : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : AddCommMonoid L\ninst✝⁴ : TopologicalSpace L\ninst✝³ : AddCommMonoid M\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid N\ninst✝ : TopologicalSpace N\nm n : MeasurableSpace α\nv : VectorMeasure α M... | · ext n
rw [if_pos (hf₁ n)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.HasOuterApproxClosed | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 59
} | {
"line": 129,
"column": 2
} | [
{
"pp": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nδ : ℝ\nδ_pos : 0 < δ\n⊢ Integrable (fun ω ↦ ↑((thickenedIndicator δ_pos F) ω)) μ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants":... | [
"Ω : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nF : Set Ω\nδ : ℝ\nδ_pos : 0 < δ\nx : Ω\n⊢ ‖↑((thickenedIndicator δ_pos F) x)‖ ≤ 1"
] | refine .of_bound (by fun_prop) 1 (ae_of_all _ fun x ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 645,
"column": 59
} | {
"line": 648,
"column": 59
} | {
"line": 649,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nhp : 1 ≤ p\nhp' : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, 0 < C ∧ ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENN... | [] | by
gcongr
rw [← ENNReal.ofReal_coe_nnreal, ← ENNReal.ofReal_mul (NNReal.coe_nonneg _), ← div_div,
mul_div_cancel₀ _ (NNReal.coe_pos.2 hCpos).ne.symm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 656,
"column": 28
} | {
"line": 673,
"column": 21
} | {
"line": 675,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nhp : 1 ≤ p\nhp' : p ≠ ∞\nf : ι → α → β\nhf : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nh : ∀ (ε : ℝ), 0 < ε → ∃ C, ∀ (i : ι), eLpNorm ({x | C ≤ ‖f i x‖₊}.indicator (f i)) p μ ≤ ENNReal... | [] | by
set g : ι → α → β := fun i => (hf i).choose
refine
(unifIntegrable_of' hp hp' (fun i => (Exists.choose_spec <| hf i).1) fun ε hε => ?_).ae_eq
fun i => (Exists.choose_spec <| hf i).2.symm
obtain ⟨C, hC⟩ := h ε hε
have hCg : ∀ i, eLpNorm ({ x | C ≤ ‖g i x‖₊ }.indicator (g i)) p μ ≤ ENNReal.ofReal ε :... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 472,
"column": 11
} | {
"line": 472,
"column": 13
} | {
"line": 473,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nf₁ : Ω →ᵇ ℝ≥0\n⊢ ∀ (y : Ω →ᵇ ℝ≥0), dist (μ.testAgainstNN f₁) (μ.testAgainstNN y) ≤ ↑μ.mass * dist f₁ y",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nf₁ f₂ : Ω →ᵇ ℝ≥0\n⊢ dist (μ.testAgainstNN f₁) (μ.testAgainstNN f₂) ≤ ↑μ.mass * dist f₁ f₂"
] | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 235,
"column": 2
} | {
"line": 240,
"column": 6
} | {
"line": 242,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\n⊢ range toFiniteMeasure = {μ | μ.mass = 1}",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MeasureTheory.ProbabilityMeasure.mass_toFiniteMeasure",
"MeasureTheory.FiniteMeasure.mass",
"Iff.mpr",
"Set.ext",
"Mea... | [] | ext μ
simp only [mem_range, mem_setOf_eq]
refine ⟨fun ⟨ν, hν⟩ ↦ by simp [← hν], fun h ↦ ?_⟩
refine ⟨⟨μ, isProbabilityMeasure_iff_real.2 (by simpa using! h)⟩, ?_⟩
ext s hs
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 235,
"column": 2
} | {
"line": 240,
"column": 6
} | {
"line": 242,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\n⊢ range toFiniteMeasure = {μ | μ.mass = 1}",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MeasureTheory.ProbabilityMeasure.mass_toFiniteMeasure",
"MeasureTheory.FiniteMeasure.mass",
"Iff.mpr",
"Set.ext",
"Mea... | [] | ext μ
simp only [mem_range, mem_setOf_eq]
refine ⟨fun ⟨ν, hν⟩ ↦ by simp [← hν], fun h ↦ ?_⟩
refine ⟨⟨μ, isProbabilityMeasure_iff_real.2 (by simpa using! h)⟩, ?_⟩
ext s hs
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 551,
"column": 56
} | {
"line": 560,
"column": 40
} | {
"line": 562,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_3\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i ↦ (μs i).mass) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\n⊢ Tendsto (fun i ↦ (μs i).testAgainstNN f) F (𝓝 0)",
"ppTerm": "?m.31",
... | [] | by
apply tendsto_iff_dist_tendsto_zero.mpr
have obs := fun i ↦ (μs i).testAgainstNN_lipschitz_estimate f 0
simp_rw [testAgainstNN_zero, zero_add] at obs
simp_rw [show ∀ i, dist ((μs i).testAgainstNN f) 0 = (μs i).testAgainstNN f by
simp only [dist_nndist, NNReal.nndist_zero_eq_val', imp_true_iff]]
apply... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 373,
"column": 73
} | {
"line": 375,
"column": 25
} | {
"line": 377,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nX : Type u_2\ninst✝ : TopologicalSpace X\nμs : X → ProbabilityMeasure Ω\n⊢ Continuous[inst✝, instTopologicalSpace] μs ↔ ∀ (f : Ω →ᵇ ℝ), Continuous[inst✝, _] fun x ↦ ∫ (ω : Ω), f ω ∂↑(μs x)",
"ppT... | [] | by
simp [continuous_iff_continuousAt, ContinuousAt, tendsto_iff_forall_integral_tendsto,
forall_comm (α := X)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 546,
"column": 2
} | {
"line": 548,
"column": 22
} | {
"line": 549,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝² : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_2\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμs_lim : Tendsto μs F (𝓝 μ)\nnonzero : μ ≠ 0\nf : Ω →ᵇ ℝ≥0\nlim_mass : Tendsto (fun i ↦ (μs i).mass) F (𝓝 ... | [
"Ω : Type u_1\ninst✝² : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_2\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμs_lim : Tendsto μs F (𝓝 μ)\nnonzero : μ ≠ 0\nf : Ω →ᵇ ℝ≥0\nlim_mass : Tendsto (fun i ↦ (μs i).mass) F (𝓝 μ.mass)\naux... | have eventually_nonzero : ∀ᶠ i in F, μs i ≠ 0 := by
simp_rw [← mass_nonzero_iff]
exact lim_mass aux | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 568,
"column": 84
} | {
"line": 571,
"column": 94
} | {
"line": 573,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝² : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_2\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμs_lim : Tendsto (fun i ↦ (μs i).normalize) F (𝓝 μ.normalize)\nmass_lim : Tendsto (fun i ↦ (μs i).mass) F (... | [] | by
rw [tendsto_iff_forall_testAgainstNN_tendsto]
exact fun f ↦
tendsto_testAgainstNN_of_tendsto_normalize_testAgainstNN_of_tendsto_mass μs_lim mass_lim f | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 589,
"column": 2
} | {
"line": 593,
"column": 68
} | {
"line": 595,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝² : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_2\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nnonzero : μ ≠ 0\n⊢ Tendsto (fun i ↦ (μs i).normalize) F (𝓝 μ.normalize) ∧ Tendsto (fun i ↦ (μs i).mass) F (... | [] | constructor
· rintro ⟨normalized_lim, mass_lim⟩
exact tendsto_of_tendsto_normalize_testAgainstNN_of_tendsto_mass normalized_lim mass_lim
· intro μs_lim
exact ⟨tendsto_normalize_of_tendsto μs_lim nonzero, μs_lim.mass⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 589,
"column": 2
} | {
"line": 593,
"column": 68
} | {
"line": 595,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝² : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_2\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nnonzero : μ ≠ 0\n⊢ Tendsto (fun i ↦ (μs i).normalize) F (𝓝 μ.normalize) ∧ Tendsto (fun i ↦ (μs i).mass) F (... | [] | constructor
· rintro ⟨normalized_lim, mass_lim⟩
exact tendsto_of_tendsto_normalize_testAgainstNN_of_tendsto_mass normalized_lim mass_lim
· intro μs_lim
exact ⟨tendsto_normalize_of_tendsto μs_lim nonzero, μs_lim.mass⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 803,
"column": 73
} | {
"line": 805,
"column": 25
} | {
"line": 807,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nX : Type u_2\ninst✝ : TopologicalSpace X\nμs : X → FiniteMeasure Ω\n⊢ Continuous[inst✝, instTopologicalSpace] μs ↔ ∀ (f : Ω →ᵇ ℝ), Continuous[inst✝, _] fun x ↦ ∫ (ω : Ω), f ω ∂↑(μs x)",
"ppTerm":... | [] | by
simp [continuous_iff_continuousAt, ContinuousAt, tendsto_iff_forall_integral_tendsto,
forall_comm (α := X)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 857,
"column": 4
} | {
"line": 857,
"column": 84
} | {
"line": 858,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\nhp : p ≠ 0\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nε : ℝ\nhε : 0 < ε\nhfu : UnifIntegrable f p μ\nM : ℝ≥0\nhM : ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑M\nℐ : ℝ... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\nhp : p ≠ 0\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nε : ℝ\nhε : 0 < ε\nhfu : UnifIntegrable f p μ\nM : ℝ≥0\nhM : ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑M\nℐ : ℝ≥0 → ι\nδ : ... | rw [← ENNReal.coe_one, ← ENNReal.coe_max, ← ENNReal.coe_mul, ENNReal.coe_lt_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 572,
"column": 24
} | {
"line": 572,
"column": 38
} | {
"line": 572,
"column": 39
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s κ μ\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ x ∈ p1, f1 x ∈ s x\nht1_eq ... | [
"α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s κ μ\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ x ∈ p1, f1 x ∈ s x\nht1_eq : t1 = ⋂ x ∈... | h_p1_inter_p2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Density | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 96
} | {
"line": 183,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\ninst✝¹ : MeasurableSpace E\nm : MeasurableSpace Ω\nℙ : Measure Ω\nμ : Measure E\nX : Ω → E\ninst✝ : HasPDF X ℙ μ\n⊢ ∫⁻ (x : E), pdf X ℙ μ x ∂μ = ℙ Set.univ",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
... | [] | rw [← setLIntegral_univ, ← map_eq_setLIntegral_pdf X ℙ μ MeasurableSet.univ,
map_apply_of_aemeasurable (HasPDF.aemeasurable X ℙ μ) MeasurableSet.univ, Set.preimage_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Density | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 96
} | {
"line": 183,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\ninst✝¹ : MeasurableSpace E\nm : MeasurableSpace Ω\nℙ : Measure Ω\nμ : Measure E\nX : Ω → E\ninst✝ : HasPDF X ℙ μ\n⊢ ∫⁻ (x : E), pdf X ℙ μ x ∂μ = ℙ Set.univ",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
... | [] | rw [← setLIntegral_univ, ← map_eq_setLIntegral_pdf X ℙ μ MeasurableSet.univ,
map_apply_of_aemeasurable (HasPDF.aemeasurable X ℙ μ) MeasurableSet.univ, Set.preimage_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Density | {
"line": 180,
"column": 2
} | {
"line": 181,
"column": 96
} | {
"line": 183,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\ninst✝¹ : MeasurableSpace E\nm : MeasurableSpace Ω\nℙ : Measure Ω\nμ : Measure E\nX : Ω → E\ninst✝ : HasPDF X ℙ μ\n⊢ ∫⁻ (x : E), pdf X ℙ μ x ∂μ = ℙ Set.univ",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
... | [] | rw [← setLIntegral_univ, ← map_eq_setLIntegral_pdf X ℙ μ MeasurableSet.univ,
map_apply_of_aemeasurable (HasPDF.aemeasurable X ℙ μ) MeasurableSet.univ, Set.preimage_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 720,
"column": 4
} | {
"line": 723,
"column": 9
} | {
"line": 724,
"column": 2
} | [
{
"pp": "case mpr.inr.refine_1\nγ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\nhne : F.NeBot\nh :\n ∀ (f : Ω → ℝ),\n (∃ C, ∀ (x y : Ω), ... | [] | simp only [Real.dist_eq, abs_le]
have h1 x : fs M x ≤ 1 := thickenedIndicator_le_one _ _ _
have h2 x : 0 ≤ fs M x := by simp [fs]
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 720,
"column": 4
} | {
"line": 723,
"column": 9
} | {
"line": 724,
"column": 2
} | [
{
"pp": "case mpr.inr.refine_1\nγ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\nhne : F.NeBot\nh :\n ∀ (f : Ω → ℝ),\n (∃ C, ∀ (x y : Ω), ... | [] | simp only [Real.dist_eq, abs_le]
have h1 x : fs M x ≤ 1 := thickenedIndicator_le_one _ _ _
have h2 x : 0 ≤ fs M x := by simp [fs]
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 360,
"column": 2
} | {
"line": 360,
"column": 19
} | {
"line": 361,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nS T : Finset ι\nhST : Disjoint S T\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), Measurabl... | [
"case inr\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nS T : Finset ι\nhST : Disjoint S T\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), Measurable (f i)\nhμ ... | rw [← hs2, ← ht2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 730,
"column": 6
} | {
"line": 732,
"column": 20
} | {
"line": 733,
"column": 2
} | [
{
"pp": "case refine_2\nγ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\nhne : F.NeBot\nh :\n ∀ (f : Ω → ℝ),\n (∃ C, ∀ (x y : Ω), dist (f ... | [] | have h : _ ≤ fs M :=
indicator_le_thickenedIndicator (δ := (1 : ℝ) / (M + 1)) (by positivity) s
simpa using! h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 730,
"column": 6
} | {
"line": 732,
"column": 20
} | {
"line": 733,
"column": 2
} | [
{
"pp": "case refine_2\nγ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\nhne : F.NeBot\nh :\n ∀ (f : Ω → ℝ),\n (∃ C, ∀ (x y : Ω), dist (f ... | [] | have h : _ ≤ fs M :=
indicator_le_thickenedIndicator (δ := (1 : ℝ) / (M + 1)) (by positivity) s
simpa using! h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.Variance | {
"line": 126,
"column": 28
} | {
"line": 126,
"column": 38
} | {
"line": 126,
"column": 38
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhX : AEStronglyMeasurable X μ\n⊢ ¬MemLp X 2 μ → ∞ ≤ eVar[X; μ]",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhX : AEStronglyMeasurable X μ\n⊢ ¬MemLp X 2 μ → eVar[X; μ] = ∞"
] | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.Covariance | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 22
} | {
"line": 77,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX Y : Ω → ℝ\nμ : Measure Ω\n⊢ cov[X, Y; μ] = cov[Y, X; μ]",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Real",
"id",
"ProbabilityTheory.covariance",
"Eq"
],
"usedFVars": [
"Ω",
"mΩ",
"X",
... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX Y : Ω → ℝ\nμ : Measure Ω\n⊢ ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) * (Y ω - ∫ (x : Ω), Y x ∂μ) ∂μ =\n ∫ (ω : Ω), (Y ω - ∫ (x : Ω), Y x ∂μ) * (X ω - ∫ (x : Ω), X x ∂μ) ∂μ"
] | simp_rw [covariance] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Moments.Covariance | {
"line": 91,
"column": 2
} | {
"line": 91,
"column": 22
} | {
"line": 92,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX Y : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nhX : Integrable X μ\nc : ℝ\n⊢ cov[fun ω ↦ X ω + c, Y; μ] = cov[X, Y; μ]",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Real",
"id",
"Real.instAdd",
"instHAdd... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX Y : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nhX : Integrable X μ\nc : ℝ\n⊢ ∫ (ω : Ω), (X ω + c - ∫ (x : Ω), X x + c ∂μ) * (Y ω - ∫ (x : Ω), Y x ∂μ) ∂μ =\n ∫ (ω : Ω), (X ω - ∫ (x : Ω), X x ∂μ) * (Y ω - ∫ (x : Ω), Y x ∂μ) ∂μ"
] | simp_rw [covariance] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Moments.Covariance | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 39
} | {
"line": 113,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX Y : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nhY : Integrable Y μ\nc : ℝ\n⊢ cov[X, fun ω ↦ Y ω + c; μ] = cov[X, Y; μ]",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"ProbabilityTheory.covariance_add_const_right"
],
... | [] | exact covariance_add_const_right hY c | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Moments.Variance | {
"line": 325,
"column": 4
} | {
"line": 325,
"column": 57
} | {
"line": 327,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\nΩ' : Type u_3\nmΩ' : MeasurableSpace Ω'\nμ : Measure Ω'\nY : Ω' → Ω\nhX : AEMeasurable X (Measure.map Y μ)\nhY : AEMeasurable Y μ\n⊢ AEStronglyMeasurable (fun ω ↦ X ω - ∫ (x : Ω), X x ∂Measure.map Y μ) (Measure.map Y μ)",
"ppTerm": "?m.106",
"ass... | [] | exact AEMeasurable.aestronglyMeasurable (by fun_prop) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Moments.Variance | {
"line": 364,
"column": 2
} | {
"line": 379,
"column": 42
} | {
"line": 381,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\n⊢ eVar[X; μ] = ∫⁻ (ω : Ω), ‖X ω‖ₑ ^ 2 ∂μ - ENNReal.ofReal ((∫ (x : Ω), X x ∂μ) ^ 2)",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Real.instIsOrde... | [] | by_cases hℒ : MemLp X 2 μ
· rw [← ofReal_variance hℒ, variance_eq_sub hℒ, ENNReal.ofReal_sub _ (sq_nonneg _)]
congr
simp_rw [← enorm_pow, enorm]
rw [lintegral_coe_eq_integral]
· simp
· simpa using hℒ.abs.integrable_sq
· symm
rw [evariance_eq_top hX hℒ, ENNReal.sub_eq_top_iff]
refine ⟨?_,... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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