module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Multilinear.Pi | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 34
} | {
"line": 112,
"column": 2
} | [
{
"pp": "ι : Type uι\nκ : ι → Type uκ\nR : Type uR\nM : (i : ι) → κ i → Type uM\nN : ((i : ι) → κ i) → Type uN\ninst✝⁶ : Semiring R\ninst✝⁵ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)\ninst✝⁴ : (p : (i : ι) → κ i) → AddCommMonoid (N p)\ninst✝³ : (i : ι) → (k : κ i) → Module R (M i k)\ninst✝² : (p : (i : ι) → ... | [
"case inl\nι : Type uι\nκ : ι → Type uκ\nR : Type uR\nM : (i : ι) → κ i → Type uM\nN : ((i : ι) → κ i) → Type uN\ninst✝⁶ : Semiring R\ninst✝⁵ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)\ninst✝⁴ : (p : (i : ι) → κ i) → AddCommMonoid (N p)\ninst✝³ : (i : ι) → (k : κ i) → Module R (M i k)\ninst✝² : (p : (i : ι) → κ ... | obtain rfl | hp := eq_or_ne p p' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.PiTensorProduct | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 19
} | {
"line": 136,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nA : ι → Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → NonUnitalSemiring (A i)\ninst✝² : (i : ι) → Module R (A i)\ninst✝¹ : ∀ (i : ι), SMulCommClass R (A i) (A i)\ninst✝ : ∀ (i : ι), IsScalarTower R (A i) (A i)\nx✝ y✝ z✝ : ⨂[R] (i : ι), A i\nx y z : (i : ι) → A i\n⊢ (... | [
"ι : Type u_1\nR : Type u_3\nA : ι → Type u_4\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → NonUnitalSemiring (A i)\ninst✝² : (i : ι) → Module R (A i)\ninst✝¹ : ∀ (i : ι), SMulCommClass R (A i) (A i)\ninst✝ : ∀ (i : ι), IsScalarTower R (A i) (A i)\nx✝ y✝ z✝ : ⨂[R] (i : ι), A i\nx y z : (i : ι) → A i\n⊢ (tprod R) x *... | dsimp [← mul_def] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.LinearAlgebra.Projectivization.Independence | {
"line": 64,
"column": 2
} | {
"line": 66,
"column": 73
} | {
"line": 67,
"column": 2
} | [
{
"pp": "case refine_1\nι : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : ι → ℙ K V\n⊢ Independent f → iSupIndep fun i ↦ (f i).submodule",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Projectivization.mk",
... | [
"case refine_2\nι : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : ι → ℙ K V\nh : iSupIndep fun i ↦ (f i).submodule\n⊢ Independent f"
] | · rintro ⟨f, hf, hi⟩
simp only [submodule_mk]
exact (iSupIndep_iff_linearIndependent_of_ne_zero (R := K) hf).mpr hi | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Projectivization.Independence | {
"line": 67,
"column": 4
} | {
"line": 70,
"column": 29
} | {
"line": 72,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : ι → ℙ K V\nh : iSupIndep fun i ↦ (f i).submodule\n⊢ Independent f",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule... | [] | rw [independent_iff]
refine h.linearIndependent (Projectivization.submodule ∘ f) (fun i => ?_) fun i => ?_
· simpa only [Function.comp_apply, submodule_eq] using Submodule.mem_span_singleton_self _
· exact rep_nonzero (f i) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Projectivization.Independence | {
"line": 67,
"column": 4
} | {
"line": 70,
"column": 29
} | {
"line": 72,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : ι → ℙ K V\nh : iSupIndep fun i ↦ (f i).submodule\n⊢ Independent f",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule... | [] | rw [independent_iff]
refine h.linearIndependent (Projectivization.submodule ∘ f) (fun i => ?_) fun i => ?_
· simpa only [Function.comp_apply, submodule_eq] using Submodule.mem_span_singleton_self _
· exact rep_nonzero (f i) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Projectivization.Independence | {
"line": 80,
"column": 89
} | {
"line": 90,
"column": 38
} | {
"line": 92,
"column": 0
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nf : ι → ℙ K V\n⊢ Dependent f ↔ ¬LinearIndependent K (Projectivization.rep ∘ f)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Projectivization.mk",
"Eq.mpr... | [] | by
refine ⟨?_, fun h => ?_⟩
· rintro ⟨ff, hff, hh1⟩
contrapose hh1
choose a ha using fun i : ι => exists_smul_eq_mk_rep K (ff i) (hff i)
convert! hh1.units_smul a⁻¹
ext i
simp only [← ha, inv_smul_smul, Pi.smul_apply', Pi.inv_apply, Function.comp_apply]
· convert! Dependent.mk _ _ h
· simp... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.QuadraticForm.Radical | {
"line": 70,
"column": 27
} | {
"line": 70,
"column": 38
} | {
"line": 70,
"column": 39
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nP : Type u_4\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup P\ninst✝² : CommRing R\ninst✝¹ : Module R M\ninst✝ : Module R P\nQ : QuadraticMap R M P\nN : Submodule R M\nhN : N ≤ Q.radical\nm m' : M\nhmm' : m - m' ∈ N\n⊢ Q m' + polar (⇑Q) m' (m - m') = Q m'",
"ppTerm": "?... | [
"R : Type u_1\nM : Type u_2\nP : Type u_4\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup P\ninst✝² : CommRing R\ninst✝¹ : Module R M\ninst✝ : Module R P\nQ : QuadraticMap R M P\nN : Submodule R M\nhN : N ≤ Q.radical\nm m' : M\nhmm' : m - m' ∈ N\n⊢ Q m' + polar (⇑Q) (m - m') m' = Q m'"
] | polar_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.Radical | {
"line": 82,
"column": 30
} | {
"line": 82,
"column": 41
} | {
"line": 82,
"column": 42
} | [
{
"pp": "case w\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nP : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup P\ninst✝³ : CommRing R\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R P\nQ : QuadraticMap R M P\nN : Submodule R M\nhN : N ≤ Q.radical\nn : M\nhn : n ∈... | [
"case w\nR : Type u_1\nM : Type u_2\nM' : Type u_3\nP : Type u_4\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup M'\ninst✝⁴ : AddCommGroup P\ninst✝³ : CommRing R\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R P\nQ : QuadraticMap R M P\nN : Submodule R M\nhN : N ≤ Q.radical\nn : M\nhn : n ∈ N\n⊢ ∀ (x :... | polar_comm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 148,
"column": 2
} | {
"line": 157,
"column": 24
} | {
"line": 159,
"column": 0
} | [
{
"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\n⊢ sigPos Q + Module.finrank 𝕜 ↥V ≤ Module.finrank 𝕜 M",
"ppTerm": "?m.41",
... | [] | obtain ⟨Vp, hr, hVp⟩ := exists_finrank_eq_sigPos_and_posDef Q
rw [← hr]
apply Submodule.finrank_add_finrank_le_of_disjoint
intro W hWp hWm
rw [le_bot_iff, Submodule.eq_bot_iff]
intro x hx
by_contra hx'
have := hVp ⟨x, hWp hx⟩ (by simpa using hx')
have := hV x (hWm hx)
grind [restrict_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 148,
"column": 2
} | {
"line": 157,
"column": 24
} | {
"line": 159,
"column": 0
} | [
{
"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\n⊢ sigPos Q + Module.finrank 𝕜 ↥V ≤ Module.finrank 𝕜 M",
"ppTerm": "?m.41",
... | [] | obtain ⟨Vp, hr, hVp⟩ := exists_finrank_eq_sigPos_and_posDef Q
rw [← hr]
apply Submodule.finrank_add_finrank_le_of_disjoint
intro W hWp hWm
rw [le_bot_iff, Submodule.eq_bot_iff]
intro x hx
by_contra hx'
have := hVp ⟨x, hWp hx⟩ (by simpa using hx')
have := hV x (hWm hx)
grind [restrict_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 168,
"column": 6
} | {
"line": 168,
"column": 56
} | {
"line": 169,
"column": 4
} | [
{
"pp": "case pos\n𝕜 : Type u_4\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\nι : Type u_5\ninst✝¹ : Fintype ι\nw : ι → 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set ι\nhs : ∀ i ∈ s, 0 < w i\nv : ι → 𝕜\nhv : v ∈ Pi.spanSubset 𝕜 s\nhv' : ⟨v, hv⟩ ≠ 0\ni : ι\na✝ : i ∈ univ\nhi : i ∈ s\n⊢ 0 ≤ w i • (↑⟨v, hv⟩ i * ↑⟨... | [] | exact smul_nonneg (hs i hi).le (mul_self_nonneg _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 168,
"column": 6
} | {
"line": 168,
"column": 56
} | {
"line": 169,
"column": 4
} | [
{
"pp": "case pos\n𝕜 : Type u_4\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\nι : Type u_5\ninst✝¹ : Fintype ι\nw : ι → 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set ι\nhs : ∀ i ∈ s, 0 < w i\nv : ι → 𝕜\nhv : v ∈ Pi.spanSubset 𝕜 s\nhv' : ⟨v, hv⟩ ≠ 0\ni : ι\na✝ : i ∈ univ\nhi : i ∈ s\n⊢ 0 ≤ w i • (↑⟨v, hv⟩ i * ↑⟨... | [] | exact smul_nonneg (hs i hi).le (mul_self_nonneg _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 168,
"column": 6
} | {
"line": 168,
"column": 56
} | {
"line": 169,
"column": 4
} | [
{
"pp": "case pos\n𝕜 : Type u_4\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\nι : Type u_5\ninst✝¹ : Fintype ι\nw : ι → 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set ι\nhs : ∀ i ∈ s, 0 < w i\nv : ι → 𝕜\nhv : v ∈ Pi.spanSubset 𝕜 s\nhv' : ⟨v, hv⟩ ≠ 0\ni : ι\na✝ : i ∈ univ\nhi : i ∈ s\n⊢ 0 ≤ w i • (↑⟨v, hv⟩ i * ↑⟨... | [] | exact smul_nonneg (hs i hi).le (mul_self_nonneg _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 97,
"column": 7
} | {
"line": 97,
"column": 82
} | {
"line": 97,
"column": 82
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CharZero R\ninst✝⁶ : IsDomain 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 : Finite ι\ninst✝ : P.IsCrystallographic\nb : P.Base\ni j k ... | [] | by convert! P.neg_mem_range_root_iff.mpr hk' using 1; simp [neg_add_eq_sub] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 78,
"column": 7
} | {
"line": 82,
"column": 65
} | {
"line": 83,
"column": 2
} | [] | [
"case calc_1\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 : P.Base\ninst... | ω b * ⁅h j, f i⁆ = ω b * (h j * f i - f i * h j) := by rw [Ring.lie_def]
_ = -(h j * e i - e i * h j) * ω b := ?_
_ = -⁅h j, e i⁆ * ω b := by rw [Ring.lie_def]
_ = -(b.cartanMatrix i j • e i) * ω b := by rw [lie_h_e]
_ = ω b * (-b.c... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 454,
"column": 2
} | {
"line": 454,
"column": 45
} | {
"line": 456,
"column": 0
} | [
{
"pp": "case calc_4\nι : 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\ni : ι\nthis :... | [] | · rw [threeShortAddLongRoot_longRoot]; ring | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 169,
"column": 6
} | {
"line": 174,
"column": 31
} | {
"line": 175,
"column": 6
} | [
{
"pp": "case inr.inr.inr.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Finite ι\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : CharZero R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsCrystallogra... | [
"case inr.inr.inr.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Finite ι\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : CharZero R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsCrystallographic\nb : P.... | have hx (x : ι) : ¬ (P.root x = P.root l - P.root i ∧ P.root (-i) = P.root i + P.root x) := by
rintro ⟨-, contra⟩
refine P.nsmul_notMem_range_root (n := 2) (i := -i) ⟨x, ?_⟩
replace contra : P.root x = -(P.root i + P.root i) := by
simpa [neg_eq_iff_add_eq_zero, ← add_assoc, add_eq_zero... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 70,
"column": 21
} | {
"line": 70,
"column": 33
} | {
"line": 70,
"column": 33
} | [
{
"pp": "case mul\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nmotive : SymmetricAlgebra R M → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R (SymmetricAlgebra R M)) r)\nι : ∀ (x : M), motive ((SymmetricAlgebra.ι R M) x)\nmul : ∀ (a b : Symmetr... | [
"case mul\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nmotive : SymmetricAlgebra R M → Prop\nalgebraMap : ∀ (r : R), motive ((Algebra.algebraMap R (SymmetricAlgebra R M)) r)\nι : ∀ (x : M), motive ((SymmetricAlgebra.ι R M) x)\nmul : ∀ (a b : SymmetricAlgebra R ... | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 319,
"column": 4
} | {
"line": 326,
"column": 45
} | {
"line": 327,
"column": 2
} | [
{
"pp": "case h₁\nι : 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.... | [] | intro k hk U aux
have : ⁅e ⟨k, hk⟩, u ⟨k, hk⟩⁆ = (2 : K) • v b k := by
simpa [-lie_apply] using! e_lie_u ⟨k, hk⟩ ⟨k, hk⟩
let e' : lieAlgebra b := ⟨e ⟨k, hk⟩, e_mem_lieAlgebra ⟨k, hk⟩⟩
change ⁅e', u ⟨k, hk⟩⁆ = _ at this
replace aux := U.lie_mem (x := e') <| aux ⟨k, hk⟩
rw [this] at aux
exac... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 319,
"column": 4
} | {
"line": 326,
"column": 45
} | {
"line": 327,
"column": 2
} | [
{
"pp": "case h₁\nι : 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.... | [] | intro k hk U aux
have : ⁅e ⟨k, hk⟩, u ⟨k, hk⟩⁆ = (2 : K) • v b k := by
simpa [-lie_apply] using! e_lie_u ⟨k, hk⟩ ⟨k, hk⟩
let e' : lieAlgebra b := ⟨e ⟨k, hk⟩, e_mem_lieAlgebra ⟨k, hk⟩⟩
change ⁅e', u ⟨k, hk⟩⁆ = _ at this
replace aux := U.lie_mem (x := e') <| aux ⟨k, hk⟩
rw [this] at aux
exac... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.Projective | {
"line": 93,
"column": 8
} | {
"line": 93,
"column": 19
} | {
"line": 93,
"column": 20
} | [
{
"pp": "case inr\nι : 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\nS : Set ((i : ↥I) → α ↑i)\nT : Set ((i : ↥J) → α ↑i)\nhT : MeasurableSet T\nh_eq : cylinder I S = cylinder J T\nhJI : J ⊆ ... | [
"case inr\nι : 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\nS : Set ((i : ↥I) → α ↑i)\nT : Set ((i : ↥J) → α ↑i)\nhT : MeasurableSet T\nh_eq : cylinder I S = cylinder J T\nhJI : J ⊆ I\nh : Nonem... | hP I J hJI, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 19
} | {
"line": 179,
"column": 4
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\ns t : Set α\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nhC : IsSetSemiring C\nhs : s ∈ C\nht : t ∈ C\nhst : s ⊆ t\n⊢ ∑ u ∈ insert s (hC.disjointOfDiff ht hs), m u = m s + ∑ i ∈ hC.disjointOfDiff ht hs, m i",
"ppTerm": "?m.57",
"assigned": true,
... | [
"α : Type u_1\nC : Set (Set α)\ns t : Set α\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nhC : IsSetSemiring C\nhs : s ∈ C\nht : t ∈ C\nhst : s ⊆ t\n⊢ s ∉ hC.disjointOfDiff ht hs"
] | rw [sum_insert] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 57
} | {
"line": 73,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nhs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace... | [
"α : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nhs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUniformSpace.toTopologic... | set Y₁ : ℕ → Set α := fun i => cthickening (r₁ i) (s i) | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 306,
"column": 4
} | {
"line": 306,
"column": 32
} | {
"line": 306,
"column": 33
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nt : Set α\nG : Type u_2\ninst✝² : AddCommMonoid G\ninst✝¹ : PartialOrder G\ninst✝ : CanonicallyOrderedAdd G\nm : AddContent G C\nhC : IsSetSemiring C\nJ : Finset (Set α)\nh_ss : ↑J ⊆ C\nht : t ∈ C\nhtJ : t ⊆ ⨆ a, ↑a\n⊢ m t ≤ ∑ u ∈ J, m u",
"ppTerm": "?m.34",
"assi... | [
"α : Type u_1\nC : Set (Set α)\nt : Set α\nG : Type u_2\ninst✝² : AddCommMonoid G\ninst✝¹ : PartialOrder G\ninst✝ : CanonicallyOrderedAdd G\nm : AddContent G C\nhC : IsSetSemiring C\nJ : Finset (Set α)\nh_ss : ↑J ⊆ C\nht : t ∈ C\nhtJ : t ⊆ ⨆ x, ↑(J.equivFin.symm x)\n⊢ m t ≤ ∑ u ∈ J, m u"
] | ← J.equivFin.symm.iSup_comp, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 383,
"column": 2
} | {
"line": 383,
"column": 23
} | {
"line": 384,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nG : Type u_3\ninst✝ : AddCommGroup G\nf : α → G\nu v : α\nh : u ≤ v\nh' : ∃ p, p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2\nu' : α := h'.choose.1\n⊢ f ⋯.choose.2 - f u' = f v - f u",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\ninst✝¹ : LinearOrder α\nG : Type u_3\ninst✝ : AddCommGroup G\nf : α → G\nu v : α\nh : u ≤ v\nh' : ∃ p, p.1 ≤ p.2 ∧ Set.Ioc u v = Set.Ioc p.1 p.2\nu' : α := h'.choose.1\nv' : α := h'.choose.2\n⊢ f v' - f u' = f v - f u"
] | set v' := h'.choose.2 | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 99,
"column": 8
} | {
"line": 99,
"column": 64
} | {
"line": 100,
"column": 8
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator μ[f | m] =ᵐ[μ]... | [
"case refine_1\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator μ[f | m] =ᵐ... | refine (condExp_indicator_aux hs.compl ?_).symm.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.SetSemiring | {
"line": 461,
"column": 8
} | {
"line": 461,
"column": 19
} | {
"line": 462,
"column": 6
} | [
{
"pp": "case h.refine_3.refine_2\nα : Type u_1\nC : Set (Set α)\nJ✝ : Finset (Set α)\nhC : IsSetSemiring C\ns : Set α\nJ : Finset (Set α)\nhJ : s ∉ J\nhind :\n ↑J ⊆ C →\n ∃ K,\n (↑J).PairwiseDisjoint K ∧\n (∀ i ∈ J, ↑(K i) ⊆ C) ∧\n (⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id ∧\n (... | [] | simpa [ht2] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.SetSemiring | {
"line": 461,
"column": 8
} | {
"line": 461,
"column": 19
} | {
"line": 462,
"column": 6
} | [
{
"pp": "case h.refine_3.refine_2\nα : Type u_1\nC : Set (Set α)\nJ✝ : Finset (Set α)\nhC : IsSetSemiring C\ns : Set α\nJ : Finset (Set α)\nhJ : s ∉ J\nhind :\n ↑J ⊆ C →\n ∃ K,\n (↑J).PairwiseDisjoint K ∧\n (∀ i ∈ J, ↑(K i) ⊆ C) ∧\n (⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id ∧\n (... | [] | simpa [ht2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.SetSemiring | {
"line": 461,
"column": 8
} | {
"line": 461,
"column": 19
} | {
"line": 462,
"column": 6
} | [
{
"pp": "case h.refine_3.refine_2\nα : Type u_1\nC : Set (Set α)\nJ✝ : Finset (Set α)\nhC : IsSetSemiring C\ns : Set α\nJ : Finset (Set α)\nhJ : s ∉ J\nhind :\n ↑J ⊆ C →\n ∃ K,\n (↑J).PairwiseDisjoint K ∧\n (∀ i ∈ J, ↑(K i) ⊆ C) ∧\n (⋃ x ∈ J, ↑(K x)).PairwiseDisjoint id ∧\n (... | [] | simpa [ht2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 162,
"column": 4
} | {
"line": 163,
"column": 64
} | {
"line": 164,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nf : α → E\ns : Set α\nm m₂ m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nhm₂ : m₂ ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ninst✝ : SigmaFinite (μ.trim hm₂)\nhs_m : MeasurableSet s\nhs : ∀... | [
"α : Type u_1\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nf : α → E\ns : Set α\nm m₂ m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nhm₂ : m₂ ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ninst✝ : SigmaFinite (μ.trim hm₂)\nhs_m : MeasurableSet s\nhs : ∀ (t : Set α)... | suffices ∫ x in sᶜ, (μ[s.indicator f | m]) x ∂μ.restrict t = 0 by
rw [this, add_zero, Measure.restrict_restrict (hm _ hs_m)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 167,
"column": 8
} | {
"line": 167,
"column": 36
} | {
"line": 167,
"column": 37
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr₁ r₂ : ℕ → ℝ\nhr : Tendsto r₁ atTop (... | [
"α : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nM : ℝ\nhM : 0 < M\nr₁ r₂ : ℕ → ℝ\nhr : Tendsto r₁ atTop (𝓝[>] 0)\nhM... | mul_max_of_nonneg _ _ hM.le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.SetSemiring | {
"line": 505,
"column": 2
} | {
"line": 505,
"column": 80
} | {
"line": 506,
"column": 2
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nj : Set α\nJ : Finset (Set α)\nhC : IsSetSemiring C\nhJ : ↑J ⊆ C\nhj : j ∈ J\n⊢ (↑(hC.disjointOfUnion hJ j)).PairwiseDisjoint id",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"Finset",
... | [
"α : Type u_1\nC : Set (Set α)\nj : Set α\nJ : Finset (Set α)\nhC : IsSetSemiring C\nhJ : ↑J ⊆ C\nhj : j ∈ J\n⊢ ↑(hC.disjointOfUnion hJ j) ⊆ ⋃ x ∈ J, ↑(hC.disjointOfUnion hJ x)"
] | apply PairwiseDisjoint.subset (hC.pairwiseDisjoint_biUnion_disjointOfUnion hJ) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 83
} | {
"line": 88,
"column": 84
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ... | [
"case hm\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s... | apply (isCountablySpanning_spanningSets (μ.trim hm)).null_of_forall_restrict_null | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 83
} | {
"line": 172,
"column": 84
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a... | [
"case hm\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\ns : Set E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : ... | apply (isCountablySpanning_spanningSets (μ.trim hm)).null_of_forall_restrict_null | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 30
} | {
"line": 132,
"column": 2
} | [
{
"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✝ : Countable β\nv : VectorMeasure α M\nf : β → Set α\nhm : ∀ (i : β), MeasurableSet (f i)\nhd : Pairwise (Disjoint on f)\ne : β → ℕ\nhe : Function.Injective e\n⊢ HasSum (fun i ↦ ... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : Countable β\nv : VectorMeasure α M\nf : β → Set α\nhm : ∀ (i : β), MeasurableSet (f i)\nhd : Pairwise (Disjoint on f)\ne : β → ℕ\nhe : Function.Injective e\n⊢ HasSum (Function.extend e (f... | rw [← hasSum_extend_zero he] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 23
} | {
"line": 287,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\nhp_ne_zero : ¬p = 0\nhp_ne_top : ¬p = ∞\nM : ℝ\nhM' : 0 ≤ M\nhM :\n ∫⁻ (x : α), ‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal... | [
"case neg\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\nhp_ne_zero : ¬p = 0\nhp_ne_top : ¬p = ∞\nM : ℝ\nhM' : 0 ≤ M\nhM :\n ∫⁻ (x : α), ‖{x | M ≤ ↑‖‖f x‖ ^ p.toReal‖₊}.indicato... | rw [ENNReal.rpow_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 274,
"column": 2
} | {
"line": 304,
"column": 18
} | {
"line": 306,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\n⊢ ∃ M, eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε",
"ppTerm": "?m.31",
"assigned": true,
... | [] | by_cases hp_ne_zero : p = 0
· exact ⟨1, by simp [hp_ne_zero]⟩
by_cases hp_ne_top : p = ∞
· subst hp_ne_top
obtain ⟨M, hM⟩ := hf.eLpNormEssSup_indicator_norm_ge_eq_zero hmeas
refine ⟨M, ?_⟩
simp only [eLpNorm_exponent_top, hM, zero_le]
obtain ⟨M, hM', hM⟩ := MemLp.integral_indicator_norm_ge_nonneg_le... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 274,
"column": 2
} | {
"line": 304,
"column": 18
} | {
"line": 306,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : α → β\nhf : MemLp f p μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\n⊢ ∃ M, eLpNorm ({x | M ≤ ↑‖f x‖₊}.indicator f) p μ ≤ ENNReal.ofReal ε",
"ppTerm": "?m.31",
"assigned": true,
... | [] | by_cases hp_ne_zero : p = 0
· exact ⟨1, by simp [hp_ne_zero]⟩
by_cases hp_ne_top : p = ∞
· subst hp_ne_top
obtain ⟨M, hM⟩ := hf.eLpNormEssSup_indicator_norm_ge_eq_zero hmeas
refine ⟨M, ?_⟩
simp only [eLpNorm_exponent_top, hM, zero_le]
obtain ⟨M, hM', hM⟩ := MemLp.integral_indicator_norm_ge_nonneg_le... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 406,
"column": 43
} | {
"line": 409,
"column": 33
} | {
"line": 411,
"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∞\ng : α → β\nhp : 1 ≤ p\nhp_ne_top : p ≠ ∞\nhg : MemLp g p μ\n⊢ UnifIntegrable (fun x ↦ g) p μ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Real",
... | [] | by
intro ε hε
obtain ⟨δ, hδ_pos, hgδ⟩ := hg.eLpNorm_indicator_le hp hp_ne_top hε
exact ⟨δ, hδ_pos, fun _ => hgδ⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 71,
"column": 2
} | {
"line": 77,
"column": 28
} | {
"line": 79,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → E\n⊢ μ.withDensityᵥ (-f) = -μ.withDensityᵥ f",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Pi.inst... | [] | by_cases hf : Integrable f μ
· ext1 i hi
rw [VectorMeasure.neg_apply, withDensityᵥ_apply hf hi, ← integral_neg,
withDensityᵥ_apply hf.neg hi]
simp only [Pi.neg_apply]
· rw [withDensityᵥ, withDensityᵥ, dif_neg hf, dif_neg, neg_zero]
rwa [integrable_neg_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 71,
"column": 2
} | {
"line": 77,
"column": 28
} | {
"line": 79,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → E\n⊢ μ.withDensityᵥ (-f) = -μ.withDensityᵥ f",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Pi.inst... | [] | by_cases hf : Integrable f μ
· ext1 i hi
rw [VectorMeasure.neg_apply, withDensityᵥ_apply hf hi, ← integral_neg,
withDensityᵥ_apply hf.neg hi]
simp only [Pi.neg_apply]
· rw [withDensityᵥ, withDensityᵥ, dif_neg hf, dif_neg, neg_zero]
rwa [integrable_neg_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 17
} | {
"line": 170,
"column": 17
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\n⊢ Measurable (s.rnDeriv μ)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"MeasureTheory.JordanDecomposition.posPart",
"Eq.mpr",
"MeasureTheory.SignedMeasure.rnDeriv_def",
"Real",
... | [
"α : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\n⊢ Measurable fun x ↦\n (s.toJordanDecomposition.posPart.rnDeriv μ x).toReal - (s.toJordanDecomposition.negPart.rnDeriv μ x).toReal"
] | rnDeriv_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 986,
"column": 2
} | {
"line": 986,
"column": 13
} | {
"line": 987,
"column": 2
} | [
{
"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\n⊢ -w ≤[i] -v",
"ppTerm": "?m.41",... | [
"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⊢ ↑((-w).restri... | intro j hj₁ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 144,
"column": 52
} | {
"line": 150,
"column": 51
} | {
"line": 152,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\n⊢ μ.withDensityᵥ f ≪ᵥ μ.toENNRealVectorMeasure",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
"... | [] | by
by_cases hf : Integrable f μ
· refine VectorMeasure.AbsolutelyContinuous.mk fun i hi₁ hi₂ => ?_
rw [toENNRealVectorMeasure_apply_measurable hi₁] at hi₂
rw [withDensityᵥ_apply hf hi₁, Measure.restrict_zero_set hi₂, integral_zero_measure]
· rw [withDensityᵥ, dif_neg hf]
exact VectorMeasure.Absolutely... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1062,
"column": 6
} | {
"line": 1062,
"column": 20
} | {
"line": 1062,
"column": 21
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ 0 ≤[i] v",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ 0 ≤ v.restrict i"
] | restrict_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1065,
"column": 6
} | {
"line": 1065,
"column": 20
} | {
"line": 1065,
"column": 21
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ v ≤[i] 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv : VectorMeasure α M\ni : Set α\nhi : ¬MeasurableSet i\n⊢ v.restrict i ≤ 0"
] | restrict_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1268,
"column": 2
} | {
"line": 1268,
"column": 16
} | {
"line": 1270,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nN : Type u_5\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\nM : Type u_6\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : IsTopologicalAddGroup M\nv : VectorMeasure α M\nw : VectorMeasure α N\nu : Set α\nhmu : MeasurableSet u\nhu₁ : ∀ t ⊆ u, ↑... | [] | exact hu₁ s hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1449,
"column": 2
} | {
"line": 1449,
"column": 43
} | {
"line": 1450,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\n⊢ μ.toSignedMeasure ≤ ν.toSignedMeasure ↔ μ ≤ ν",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"MeasureTheor... | [
"α : Type u_1\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\n⊢ (∀ (i : Set α), MeasurableSet i → ↑μ.toSignedMeasure i ≤ ↑ν.toSignedMeasure i) ↔\n ∀ (s : Set α), MeasurableSet s → μ s ≤ ν s"
] | rw [Measure.le_iff, VectorMeasure.le_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 623,
"column": 40
} | {
"line": 625,
"column": 32
} | {
"line": 627,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSpace Ω'\nν : ProbabilityMeasure Ω\nf : Ω → Ω'\nf_aemble : AEMeasurable f ↑ν\nA : Set Ω'\nA_mble : MeasurableSet A\n⊢ (ν.map f_aemble) A = ν (f ⁻¹' A)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"... | [] | by
exact (ENNReal.toNNReal_eq_toNNReal_iff' (measure_ne_top _ _) (measure_ne_top _ _)).mpr <|
ν.map_apply' f_aemble A_mble | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 780,
"column": 51
} | {
"line": 790,
"column": 35
} | {
"line": 792,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\n⊢ ContinuousSMul ℝ≥0 (FiniteMeasure Ω)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"MeasureTheory.FiniteMeasure.instTopologicalSpace",
"NNReal.instTopologicalSpace... | [] | by
refine ⟨continuous_iff_continuousAt.2 (fun p ↦ ?_)⟩
apply tendsto_iff_forall_integral_tendsto.2 (fun g ↦ ?_)
have A : Tendsto (fun (i : ℝ≥0 × FiniteMeasure Ω) ↦ i.1) (𝓝 p) (𝓝 (p.1)) := by
rw [nhds_prod_eq]
exact tendsto_fst
have B : Tendsto (fun (i : ℝ≥0 × FiniteMeasure Ω) ↦ ∫ x, g x ∂i.2) (𝓝 p)
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 860,
"column": 2
} | {
"line": 860,
"column": 96
} | {
"line": 862,
"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∞\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\nδ : ℝ... | [] | exact ⟨C, fun i => hδ i _ (measurableSet_le measurable_const (hf i).nnnorm.measurable) (hC i)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 306,
"column": 10
} | {
"line": 306,
"column": 22
} | {
"line": 307,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\nΩ : Type u_2\nβ : Type u_4\nγ : Type u_6\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteKernel κ\nf : Ω → β\ng : Ω → γ\nhf : Measurable f\nhg : Measurable... | [
"case pos\nα : Type u_1\nΩ : Type u_2\nβ : Type u_4\nγ : Type u_6\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteKernel κ\nf : Ω → β\ng : Ω → γ\nhf : Measurable f\nhg : Measurable g\nh :\n ∀... | · simp [hωu] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Density | {
"line": 250,
"column": 2
} | {
"line": 253,
"column": 57
} | {
"line": 255,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\ninst✝² : MeasurableSpace E\nm : MeasurableSpace Ω\nℙ : Measure Ω\nμ : Measure E\nF : Type u_3\ninst✝¹ : MeasurableSpace F\nν : Measure F\nX : Ω → E\ninst✝ : HasPDF X ℙ μ\ng : E → F\nhg : QuasiMeasurePreserving g μ ν\nhmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν\n⊢ Ha... | [] | have hgm : AEMeasurable g (map X ℙ) := hg.aemeasurable.mono_ac HasPDF.absolutelyContinuous
rw [hasPDF_iff, ← AEMeasurable.map_map_of_aemeasurable hgm (HasPDF.aemeasurable X ℙ μ)]
refine ⟨hg.measurable.comp_aemeasurable (HasPDF.aemeasurable _ _ μ), hmap, ?_⟩
exact (HasPDF.absolutelyContinuous.map hg.1).trans hg.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Density | {
"line": 250,
"column": 2
} | {
"line": 253,
"column": 57
} | {
"line": 255,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\ninst✝² : MeasurableSpace E\nm : MeasurableSpace Ω\nℙ : Measure Ω\nμ : Measure E\nF : Type u_3\ninst✝¹ : MeasurableSpace F\nν : Measure F\nX : Ω → E\ninst✝ : HasPDF X ℙ μ\ng : E → F\nhg : QuasiMeasurePreserving g μ ν\nhmap : (map g (map X ℙ)).HaveLebesgueDecomposition ν\n⊢ Ha... | [] | have hgm : AEMeasurable g (map X ℙ) := hg.aemeasurable.mono_ac HasPDF.absolutelyContinuous
rw [hasPDF_iff, ← AEMeasurable.map_map_of_aemeasurable hgm (HasPDF.aemeasurable X ℙ μ)]
refine ⟨hg.measurable.comp_aemeasurable (HasPDF.aemeasurable _ _ μ), hmap, ?_⟩
exact (HasPDF.absolutelyContinuous.map hg.1).trans hg.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.Covariance | {
"line": 256,
"column": 59
} | {
"line": 258,
"column": 6
} | {
"line": 260,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nY : Ω → ℝ\nμ : Measure Ω\nι : Type u_2\nX : ι → Ω → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : Fintype ι\nhX : ∀ (i : ι), MemLp (X i) 2 μ\nhY : MemLp Y 2 μ\n⊢ cov[fun ω ↦ ∑ i, X i ω, Y; μ] = ∑ i, cov[X i, Y; μ]",
"ppTerm": "?m.38",
"assigned": true,
"usedCo... | [] | by
convert! covariance_sum_left hX hY
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 126,
"column": 6
} | {
"line": 127,
"column": 89
} | {
"line": 128,
"column": 4
} | [
{
"pp": "case e'_3\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (Ω i)\nμ : (i : ι) → Measure (Ω i)\ninst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁵ : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nX : (i : ι) → Ω i... | [] | simp only [ProbabilityMeasure.map, ProbabilityMeasure.coe_mk, Subtype.mk.injEq]
rw [AEMeasurable.map_map_of_aemeasurable hg.aemeasurable (h.forall_aemeasurable _)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 126,
"column": 6
} | {
"line": 127,
"column": 89
} | {
"line": 128,
"column": 4
} | [
{
"pp": "case e'_3\nι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (Ω i)\nμ : (i : ι) → Measure (Ω i)\ninst✝⁶ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝⁵ : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nX : (i : ι) → Ω i... | [] | simp only [ProbabilityMeasure.map, ProbabilityMeasure.coe_mk, Subtype.mk.injEq]
rw [AEMeasurable.map_map_of_aemeasurable hg.aemeasurable (h.forall_aemeasurable _)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Integration | {
"line": 471,
"column": 2
} | {
"line": 472,
"column": 6
} | {
"line": 474,
"column": 0
} | [
{
"pp": "Ω : Type u_1\n𝕜 : Type u_2\ninst✝¹ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_3\ninst✝ : Fintype ι\n𝓧 : ι → Type u_4\nm𝓧 : (i : ι) → MeasurableSpace (𝓧 i)\nX : (i : ι) → Ω → 𝓧 i\nf : (i : ι) → 𝓧 i → 𝕜\nhX : iIndepFun X μ\nmX : ∀ (i : ι), AEMeasurable (X i) μ\nhf : ∀ (i : ι), ... | [] | convert! hX.integral_fun_prod_comp mX hf
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Integration | {
"line": 471,
"column": 2
} | {
"line": 472,
"column": 6
} | {
"line": 474,
"column": 0
} | [
{
"pp": "Ω : Type u_1\n𝕜 : Type u_2\ninst✝¹ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_3\ninst✝ : Fintype ι\n𝓧 : ι → Type u_4\nm𝓧 : (i : ι) → MeasurableSpace (𝓧 i)\nX : (i : ι) → Ω → 𝓧 i\nf : (i : ι) → 𝓧 i → 𝕜\nhX : iIndepFun X μ\nmX : ∀ (i : ι), AEMeasurable (X i) μ\nhf : ∀ (i : ι), ... | [] | convert! hX.integral_fun_prod_comp mX hf
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.Filtration | {
"line": 366,
"column": 8
} | {
"line": 366,
"column": 81
} | {
"line": 367,
"column": 8
} | [
{
"pp": "case pos\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ ... | [
"case neg\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : PartialOrder ι\n𝓕 : Filtration ι m\nthis✝ : TopologicalSpace ι := Preorder.topology ι\nthis : OrderTopology ι\ni : ι\nhne : (𝓝[>] i).NeBot\nu : ι\nhu : u > i\nhiou : Set.Ioo i u ∈ 𝓝[>] i\nv : ι\nhv : v ∈ Set.Ioo i u\nhle₁ : ⨅ j, ⨅ (_ : j > i),... | · simpa [rightCont_eq_of_neBot_nhdsGT] using iInf₂_le_of_le u hv.2 le_rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.Piecewise | {
"line": 55,
"column": 2
} | {
"line": 86,
"column": 85
} | {
"line": 88,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : MeasurableSpace α\ns : ι → Set α\nf : ι → α → β\ninst✝¹ : Countable ι\nhs : IndexedPartition s\nhm : ∀ (i : ι), MeasurableSet (s i)\ninst✝ : TopologicalSpace β\nhf : ∀ (i : ι), StronglyMeasurable (f i)\n⊢ StronglyMeasurable (hs.piecewise f)",
"ppTe... | [] | by_cases Fi : Finite ι
· refine ⟨fun n => simpleFunc_piecewise hs hm (fun i => (hf i).approx n), fun x => ?_⟩
simp [simpleFunc_piecewise, piecewise_apply, StronglyMeasurable.tendsto_approx]
simp only [not_finite_iff_infinite] at Fi
obtain ⟨e, -⟩ := exists_true_iff_nonempty.mpr (nonempty_equiv_of_countable (α ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.Piecewise | {
"line": 55,
"column": 2
} | {
"line": 86,
"column": 85
} | {
"line": 88,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : MeasurableSpace α\ns : ι → Set α\nf : ι → α → β\ninst✝¹ : Countable ι\nhs : IndexedPartition s\nhm : ∀ (i : ι), MeasurableSet (s i)\ninst✝ : TopologicalSpace β\nhf : ∀ (i : ι), StronglyMeasurable (f i)\n⊢ StronglyMeasurable (hs.piecewise f)",
"ppTe... | [] | by_cases Fi : Finite ι
· refine ⟨fun n => simpleFunc_piecewise hs hm (fun i => (hf i).approx n), fun x => ?_⟩
simp [simpleFunc_piecewise, piecewise_apply, StronglyMeasurable.tendsto_approx]
simp only [not_finite_iff_infinite] at Fi
obtain ⟨e, -⟩ := exists_true_iff_nonempty.mpr (nonempty_equiv_of_countable (α ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.SpecialFunctions.Sinc | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 25
} | {
"line": 43,
"column": 0
} | [
{
"pp": "μ : Measure ℝ\ninst✝ : IsFiniteMeasure μ\nx : ℝ\n⊢ |sinc x| ≤ 1",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Real.abs_sinc_le_one"
],
"usedFVars": [
"x"
],
"usedGoals": []
}
] | [] | exact abs_sinc_le_one x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 85
} | {
"line": 85,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : ι → α → β\np : ℝ≥0∞\nhf : UnifTight f p μ\nε : ℝ≥0∞\nhε : ε ≠ 0\nhε_top : ¬ε = ∞\n⊢ ∀ᶠ (s : Set α) in μ.cofinite.smallSets, ∀ (i : ι), eLpNorm (s.indicator (f i)) p μ ≤ ε",
"p... | [
"case neg\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : ι → α → β\np : ℝ≥0∞\nhf : UnifTight f p μ\nε : ℝ≥0∞\nhε : ε ≠ 0\nhε_top : ¬ε = ∞\ns : Set α\nhμs : μ s ≠ ∞\nhfs : ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ↑ε.toNNReal\n⊢ ∀ᶠ (s : Set α) i... | rcases hf (pos_iff_ne_zero.2 (toNNReal_ne_zero.mpr ⟨hε,hε_top⟩)) with ⟨s, hμs, hfs⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 100,
"column": 4
} | {
"line": 103,
"column": 67
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifTight f p μ\nhg : UnifTight g p μ\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ≥0\nh... | [] | replace hη := hη_top ▸ hη
refine ⟨∅, (by simp), fun i ↦ ?_⟩
simp only [compl_empty, indicator_univ, Pi.add_apply]
exact (hη (f i) (g i) (hf_meas i) (hg_meas i) le_top le_top).le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 100,
"column": 4
} | {
"line": 103,
"column": 67
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf g : ι → α → β\np : ℝ≥0∞\nhf : UnifTight f p μ\nhg : UnifTight g p μ\nhf_meas : ∀ (i : ι), AEStronglyMeasurable (f i) μ\nhg_meas : ∀ (i : ι), AEStronglyMeasurable (g i) μ\nε : ℝ≥0\nh... | [] | replace hη := hη_top ▸ hη
refine ⟨∅, (by simp), fun i ↦ ?_⟩
simp only [compl_empty, indicator_univ, Pi.add_apply]
exact (hη (f i) (g i) (hf_meas i) (hg_meas i) le_top le_top).le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.GeometryOfNumbers | {
"line": 71,
"column": 4
} | {
"line": 73,
"column": 90
} | {
"line": 74,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : MeasurableSpace E\nμ : Measure E\nF s : Set E\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : μ.IsAddHaarMeasure\nL : AddSubgroup E\ninst✝ : Countable ↥L\nfund : IsAddFundamentalDomain (↥L) F μ\nh_symm : ∀ ... | [
"E : Type u_1\ninst✝⁶ : MeasurableSpace E\nμ : Measure E\nF s : Set E\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : μ.IsAddHaarMeasure\nL : AddSubgroup E\ninst✝ : Countable ↥L\nfund : IsAddFundamentalDomain (↥L) F μ\nh_symm : ∀ x ∈ s, -x ∈ ... | rw [addHaar_smul_of_nonneg μ (by simp : 0 ≤ (2 : ℝ)⁻¹) s,
← ENNReal.mul_lt_mul_iff_left (pow_ne_zero (finrank ℝ E) (two_ne_zero' _)) (by finiteness),
mul_right_comm, ofReal_pow (by simp : 0 ≤ (2 : ℝ)⁻¹), ofReal_inv_of_pos zero_lt_two] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 102,
"column": 6
} | {
"line": 103,
"column": 27
} | {
"line": 104,
"column": 4
} | [
{
"pp": "case hg.hf\nR r : ℝ\nhr : r < R\nz : ℂ\n⊢ ContinuousOn (fun x ↦ circleMap 0 R x.2 * I) (closedBall z r ×ˢ univ)",
"ppTerm": "?hg.hf",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNo... | [] | apply_rules [ContinuousOn.mul, (continuous_circleMap 0 R).comp_continuousOn continuousOn_snd,
continuousOn_const] | Lean.Elab.Tactic.SolveByElim.evalApplyRules | Lean.Parser.Tactic.applyRules |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 305,
"column": 30
} | {
"line": 305,
"column": 43
} | {
"line": 305,
"column": 43
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup β\nμ : Measure α\np : ℝ≥0∞\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\nhut : UnifTight f p μ\nhfg : ∀... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 305,
"column": 46
} | {
"line": 305,
"column": 59
} | {
"line": 305,
"column": 59
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup β\nμ : Measure α\np : ℝ≥0∞\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\nhut : UnifTight f p μ\nhfg : ∀... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 313,
"column": 30
} | {
"line": 313,
"column": 43
} | {
"line": 313,
"column": 43
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup β\nμ : Measure α\np : ℝ≥0∞\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\nhut : UnifTight f p μ\nhfg : ∀... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 313,
"column": 46
} | {
"line": 313,
"column": 59
} | {
"line": 313,
"column": 59
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup β\nμ : Measure α\np : ℝ≥0∞\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\nhut : UnifTight f p μ\nhfg : ∀... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 52,
"column": 6
} | {
"line": 52,
"column": 26
} | {
"line": 53,
"column": 4
} | [
{
"pp": "case hx\np q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\nx✝ : ℝ\nhx : x✝ ∈ Ioi 0\n⊢ 0 < b",
"ppTerm": "?hx",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hb"
],
"usedGoals": []
},
{
"pp": "p q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\nx✝ : ℝ\nhx : x✝ ∈ ... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 56,
"column": 6
} | {
"line": 56,
"column": 26
} | {
"line": 57,
"column": 4
} | [
{
"pp": "p q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\n⊢ 0 < b ^ p⁻¹",
"ppTerm": "?m.395",
"assigned": true,
"usedConstants": [
"Real.rpow_pos_of_pos",
"Real",
"Real.instInv",
"Inv.inv"
],
"usedFVars": [
"b",
"hb",
"p"
],
"usedGoals": []
... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 61,
"column": 6
} | {
"line": 61,
"column": 26
} | {
"line": 63,
"column": 0
} | [
{
"pp": "case hx\np q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\n⊢ 0 < b",
"ppTerm": "?hx",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hb"
],
"usedGoals": []
},
{
"pp": "case hx\np q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\n⊢ 0 ≤ b",
"ppTerm": "?hx✝",
... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DistLEIntegral | {
"line": 69,
"column": 4
} | {
"line": 70,
"column": 43
} | {
"line": 71,
"column": 2
} | [
{
"pp": "E✝ : Type u_1\ninst✝³ : NormedAddCommGroup E✝\ninst✝² : NormedSpace ℝ E✝\nf✝ : ℝ → E✝\na b : ℝ\nB : ℝ → ℝ\nhab : a ≤ b\nhBi : IntervalIntegrable B volume a b\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nhfc : ContinuousOn f (Icc a b)\nhfd : DifferentiableOn ℝ f (Ioo... | [] | rwa [uIoc_of_le hab, ← Measure.restrict_congr_set Ioo_ae_eq_Ioc, EventuallyLE,
ae_restrict_iff' measurableSet_Ioo] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 22
} | {
"line": 136,
"column": 4
} | [
{
"pp": "f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 /... | [
"f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 / 2 * (f a + ... | have := Fact.mk hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Basic | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 19
} | {
"line": 173,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\ns₀ s₁ t : Set X\ns₀_compact : IsCompact s₀\ns₁_compact : IsCompact s₁\nt_compact : IsCompact t\ndisj : s₀ᶜ ∪ s₁ᶜ = univ\nhst : s₀ ∪ s₁ ⊆ t\nso : Fin 2 → Set X := fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ\nhso : so = ... | [
"case left\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\ns₀ s₁ t : Set X\ns₀_compact : IsCompact s₀\ns₁_compact : IsCompact s₁\nt_compact : IsCompact t\ndisj : s₀ᶜ ∪ s₁ᶜ = univ\nhst : s₀ ∪ s₁ ⊆ t\nso : Fin 2 → Set X := ⋯\nhso : so = fun j ↦ if j = 0 then s₀ᶜ else s₁ᶜ... | apply Or.elim h | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.IntervalIntegral.TrapezoidalRule | {
"line": 155,
"column": 59
} | {
"line": 155,
"column": 71
} | {
"line": 155,
"column": 71
} | [
{
"pp": "f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 /... | [
"f : ℝ → ℝ\nζ a b : ℝ\na_lt_b : a < b\nh_df : DifferentiableOn ℝ f (Set.Icc a b)\nh_ddf : DifferentiableOn ℝ (_root_.derivWithin f (Set.Icc a b)) (Set.Icc a b)\nfpp_bound : ∀ (x : ℝ), |iteratedDerivWithin 2 f (Set.Icc a b) x| ≤ ζ\ng : ℝ → ℝ := fun t ↦ trapezoidal_error f 1 a t\ndg : ℝ → ℝ := fun t ↦ 1 / 2 * (f a + ... | div_mul_comm | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 150,
"column": 58
} | {
"line": 150,
"column": 86
} | {
"line": 151,
"column": 4
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nt : ι → TopologicalSpace X\nht : ∀ (i : ι), CompletelyRegularSpace X\nthis : TopologicalSpace X := ⨅ i, t i\nx : X\nI' : Finset ι\nV U : ↥I' → Set X\nhUV : ∀ (i : ↥I'), U i ⊆ V i\nfs : ↥I' → X → ↑I\nhfs : ∀ (i : ↥I'), Continuous[t ↑i, _] (fs i)\nhxfs : ∀ (i : ↥I'), fs i x = ... | [] | simp [Finset.sup_eq_top_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 150,
"column": 58
} | {
"line": 150,
"column": 86
} | {
"line": 151,
"column": 4
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nt : ι → TopologicalSpace X\nht : ∀ (i : ι), CompletelyRegularSpace X\nthis : TopologicalSpace X := ⨅ i, t i\nx : X\nI' : Finset ι\nV U : ↥I' → Set X\nhUV : ∀ (i : ↥I'), U i ⊆ V i\nfs : ↥I' → X → ↑I\nhfs : ∀ (i : ↥I'), Continuous[t ↑i, _] (fs i)\nhxfs : ∀ (i : ↥I'), fs i x = ... | [] | simp [Finset.sup_eq_top_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 150,
"column": 58
} | {
"line": 150,
"column": 86
} | {
"line": 151,
"column": 4
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nt : ι → TopologicalSpace X\nht : ∀ (i : ι), CompletelyRegularSpace X\nthis : TopologicalSpace X := ⨅ i, t i\nx : X\nI' : Finset ι\nV U : ↥I' → Set X\nhUV : ∀ (i : ↥I'), U i ⊆ V i\nfs : ↥I' → X → ↑I\nhfs : ∀ (i : ↥I'), Continuous[t ↑i, _] (fs i)\nhxfs : ∀ (i : ↥I'), fs i x = ... | [] | simp [Finset.sup_eq_top_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 444,
"column": 97
} | {
"line": 448,
"column": 6
} | {
"line": 450,
"column": 0
} | [
{
"pp": "p : ℝ≥0∞\ninst✝⁵ : Fact (1 ≤ p)\nι : Type u_4\ninst✝⁴ : Fintype ι\ninst✝³ : DecidableEq ι\nE : ι → Type u_5\ninst✝² : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹ : (i : ι) → NormedSpace ℝ (E i)\nmE : (i : ι) → MeasurableSpace (E i)\nμ : (i : ι) → Measure (E i)\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\nL : ... | [] | by
simp_rw [charFunDual_apply, ← integral_fintype_prod_eq_prod, ← Complex.exp_sum, ← Finset.sum_mul,
← ofReal_sum, L.comp_apply, ← map_sum, ContinuousLinearMap.sum_comp_single]
rw [← MeasurableEquiv.coe_toLp, integral_map_equiv]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 493,
"column": 4
} | {
"line": 494,
"column": 56
} | {
"line": 495,
"column": 4
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\nmE : MeasurableSpace E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace ℝ F\nmF : MeasurableSpace F\nμ : Measure E\nν : Measure F\ninst✝⁹ : BorelSpace E\ninst✝⁸ : SecondCountableTopology E\np : ℝ≥0∞\ninst✝⁷ : F... | [
"E : Type u_2\nF : Type u_3\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\nmE : MeasurableSpace E\ninst✝¹¹ : NormedAddCommGroup F\ninst✝¹⁰ : NormedSpace ℝ F\nmF : MeasurableSpace F\nμ : Measure E\nν : Measure F\ninst✝⁹ : BorelSpace E\ninst✝⁸ : SecondCountableTopology E\np : ℝ≥0∞\ninst✝⁷ : Fact (1 ≤ p)\... | refine (MeasurableEquiv.toLp p (E × F)).map_measurableEquiv_injective
<| Measure.ext_of_charFunDual <| funext fun L ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 77
} | {
"line": 201,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ ν : Measure Ω\nhLP : levyProkhorovEDist μ ν = 0\ns : Set Ω\ns_mble : MeasurableSet s\nh_finite : ∃ δ > 0, ν (thickening δ s) ≠ ∞\nkey :\n Tendsto (fun ε ↦ ν (thickening ε.toReal s)) (𝓝[>] 0)\n ... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ ν : Measure Ω\nhLP : levyProkhorovEDist μ ν = 0\ns : Set Ω\ns_mble : MeasurableSet s\nh_finite : ∃ δ > 0, ν (thickening δ s) ≠ ∞\nkey :\n Tendsto (fun ε ↦ ν (thickening ε.toReal s)) (𝓝[>] 0)\n (𝓝 (ν (cl... | have obs := Tendsto.add key (tendsto_nhdsWithin_of_tendsto_nhds tendsto_id) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Haar.DistribChar | {
"line": 71,
"column": 6
} | {
"line": 71,
"column": 56
} | {
"line": 71,
"column": 56
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝⁹ : Group G\ninst✝⁸ : AddCommGroup A\ninst✝⁷ : DistribMulAction G A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : IsTopologicalAddGroup A\ninst✝⁴ : LocallyCompactSpace A\ninst✝³ : ContinuousConstSMul G A\ninst✝² : MeasurableSpace A\ninst✝¹ : BorelSpace A\nμ : Measure A\ninst✝ ... | [
"G : Type u_1\nA : Type u_2\ninst✝⁹ : Group G\ninst✝⁸ : AddCommGroup A\ninst✝⁷ : DistribMulAction G A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : IsTopologicalAddGroup A\ninst✝⁴ : LocallyCompactSpace A\ninst✝³ : ContinuousConstSMul G A\ninst✝² : MeasurableSpace A\ninst✝¹ : BorelSpace A\nμ : Measure A\ninst✝ : μ.IsAddHaa... | ← addHaarScalarFactor_domSMul _ _ (DomMulAct.mk g) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.DistribChar | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 33
} | {
"line": 95,
"column": 2
} | [
{
"pp": "G : Type u_1\nA : Type u_2\ninst✝¹⁰ : Group G\ninst✝⁹ : AddCommGroup A\ninst✝⁸ : DistribMulAction G A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : IsTopologicalAddGroup A\ninst✝⁵ : LocallyCompactSpace A\ninst✝⁴ : ContinuousConstSMul G A\ng : G\ninst✝³ : MeasurableSpace A\ninst✝² : BorelSpace A\nμ : Measure A... | [
"G : Type u_1\nA : Type u_2\ninst✝¹⁰ : Group G\ninst✝⁹ : AddCommGroup A\ninst✝⁸ : DistribMulAction G A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : IsTopologicalAddGroup A\ninst✝⁵ : LocallyCompactSpace A\ninst✝⁴ : ContinuousConstSMul G A\ng : G\ninst✝³ : MeasurableSpace A\ninst✝² : BorelSpace A\nμ : Measure A\ninst✝¹ : μ... | refine ENNReal.coe_injective ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Haar.MulEquivHaarChar | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 33
} | {
"line": 134,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : Group G\ninst✝⁴ : TopologicalSpace G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : LocallyCompactSpace G\nφ : G ≃ₜ* G\n⊢ mulEquivHaarChar φ * mulEquivHaarChar φ.symm = 1",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants... | [] | simp [← mulEquivHaarChar_trans] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 562,
"column": 4
} | {
"line": 564,
"column": 36
} | {
"line": 565,
"column": 4
} | [
{
"pp": "case inr.refine_4\nΩ : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nε : ℝ\nε_pos : 0 < ε\nh✝ : Nonempty Ω\nxs : ℕ → Ω\nxs_dense : DenseRange xs\nhalf_ε_pos : 0 < ε / 2\nBs : ℕ → Set Ω := fun n ↦ ball (xs n) (ε / 2)\nAs : ... | [
"case inr.refine_4\nΩ : Type u_1\ninst✝³ : PseudoMetricSpace Ω\ninst✝² : MeasurableSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\ninst✝ : SeparableSpace Ω\nε : ℝ\nε_pos : 0 < ε\nh✝ : Nonempty Ω\nxs : ℕ → Ω\nxs_dense : DenseRange xs\nhalf_ε_pos : 0 < ε / 2\nBs : ℕ → Set Ω := fun n ↦ ball (xs n) (ε / 2)\nAs : ℕ → Set Ω :=... | have aux : ⋃ n, Bs n = univ := by
convert! DenseRange.iUnion_uniformity_ball xs_dense <| Metric.dist_mem_uniformity half_ε_pos
exact (ball_eq_ball' _ _).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 66
} | {
"line": 116,
"column": 4
} | [
{
"pp": "case hbc.refine_2\nμ : Measure ℝ\nr : ℝ\ninst✝ : IsProbabilityMeasure μ\nhr : 0 < r\nintegrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ\n⊢ MeasurableSet {x | 2 < |2 * r⁻¹ * x|}",
"ppTerm": "?hbc.refine_2",
"assigned": true,
"usedConstants": [
"MeasurableSet.pre... | [] | exact MeasurableSet.preimage measurableSet_Ioi (by fun_prop) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 66
} | {
"line": 116,
"column": 4
} | [
{
"pp": "case hbc.refine_2\nμ : Measure ℝ\nr : ℝ\ninst✝ : IsProbabilityMeasure μ\nhr : 0 < r\nintegrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ\n⊢ MeasurableSet {x | 2 < |2 * r⁻¹ * x|}",
"ppTerm": "?hbc.refine_2",
"assigned": true,
"usedConstants": [
"MeasurableSet.pre... | [] | exact MeasurableSet.preimage measurableSet_Ioi (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 66
} | {
"line": 116,
"column": 4
} | [
{
"pp": "case hbc.refine_2\nμ : Measure ℝ\nr : ℝ\ninst✝ : IsProbabilityMeasure μ\nhr : 0 < r\nintegrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ\n⊢ MeasurableSet {x | 2 < |2 * r⁻¹ * x|}",
"ppTerm": "?hbc.refine_2",
"assigned": true,
"usedConstants": [
"MeasurableSet.pre... | [] | exact MeasurableSet.preimage measurableSet_Ioi (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 29
} | {
"line": 132,
"column": 2
} | [
{
"pp": "case hbc\nμ : Measure ℝ\nr : ℝ\ninst✝ : IsProbabilityMeasure μ\nhr : 0 < r\nintegrable_sinc_const_mul : ∀ (r : ℝ), Integrable (fun x ↦ sinc (r * x)) μ\n⊢ ∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ ≤ ‖∫ (x : ℝ), 1 - sinc (2 * r⁻¹ * x) ∂μ‖",
"ppTerm": "?hbc",
"assigned": true,
"usedConstants": [
... | [] | exact Real.le_norm_self _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 66
} | {
"line": 160,
"column": 2
} | [
{
"pp": "case e'_3\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nmE : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure E\ninst✝ : IsProbabilityMeasure μ\nL : StrongDual ℝ E\nr : ℝ\nhr : 0 < r\nthis : IsProbabilityMeasure (Measure.map (⇑L) μ)\n⊢ MeasurableSet {x | r < |x|... | [] | exact MeasurableSet.preimage measurableSet_Ioi (by fun_prop) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 66
} | {
"line": 160,
"column": 2
} | [
{
"pp": "case e'_3\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nmE : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure E\ninst✝ : IsProbabilityMeasure μ\nL : StrongDual ℝ E\nr : ℝ\nhr : 0 < r\nthis : IsProbabilityMeasure (Measure.map (⇑L) μ)\n⊢ MeasurableSet {x | r < |x|... | [] | exact MeasurableSet.preimage measurableSet_Ioi (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 66
} | {
"line": 160,
"column": 2
} | [
{
"pp": "case e'_3\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nmE : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure E\ninst✝ : IsProbabilityMeasure μ\nL : StrongDual ℝ E\nr : ℝ\nhr : 0 < r\nthis : IsProbabilityMeasure (Measure.map (⇑L) μ)\n⊢ MeasurableSet {x | r < |x|... | [] | exact MeasurableSet.preimage measurableSet_Ioi (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 173,
"column": 6
} | {
"line": 173,
"column": 66
} | {
"line": 174,
"column": 2
} | [
{
"pp": "case e'_3\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nmE : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure E\ninst✝ : IsProbabilityMeasure μ\na : E\nr : ℝ\nhr : 0 < r\nthis : IsProbabilityMeasure (Measure.map (fun x ↦ ⟪a, x⟫) μ)\n⊢ MeasurableSet {x ... | [] | exact MeasurableSet.preimage measurableSet_Ioi (by fun_prop) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 173,
"column": 6
} | {
"line": 173,
"column": 66
} | {
"line": 174,
"column": 2
} | [
{
"pp": "case e'_3\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nmE : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure E\ninst✝ : IsProbabilityMeasure μ\na : E\nr : ℝ\nhr : 0 < r\nthis : IsProbabilityMeasure (Measure.map (fun x ↦ ⟪a, x⟫) μ)\n⊢ MeasurableSet {x ... | [] | exact MeasurableSet.preimage measurableSet_Ioi (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.IntegralCharFun | {
"line": 173,
"column": 6
} | {
"line": 173,
"column": 66
} | {
"line": 174,
"column": 2
} | [
{
"pp": "case e'_3\nE : Type u_1\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nmE : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\nμ : Measure E\ninst✝ : IsProbabilityMeasure μ\na : E\nr : ℝ\nhr : 0 < r\nthis : IsProbabilityMeasure (Measure.map (fun x ↦ ⟪a, x⟫) μ)\n⊢ MeasurableSet {x ... | [] | exact MeasurableSet.preimage measurableSet_Ioi (by fun_prop) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 89
} | {
"line": 164,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\np : ℝ\nhp : 1 ≤ p\nh₁ : 0 < p\nthis : (ENNReal.ofReal p).toReal = p\nh₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i, |x i| ^ p\neq_norm : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)\n⊢ volume {x | ∑ i, |x i| ^ p < 1} = ENNReal.ofReal ((2 * Gamma (1 / p + 1)) ^ ... | [
"ι : Type u_1\ninst✝ : Fintype ι\np : ℝ\nhp : 1 ≤ p\nh₁ : 0 < p\nthis✝ : (ENNReal.ofReal p).toReal = p\nh₂ : ∀ (x : ι → ℝ), 0 ≤ ∑ i, |x i| ^ p\neq_norm : ∀ (x : ι → ℝ), ‖toLp (ENNReal.ofReal p) x‖ = (∑ i, |x i| ^ p) ^ (1 / p)\nthis : Fact (1 ≤ ENNReal.ofReal p)\n⊢ volume {x | ∑ i, |x i| ^ p < 1} = ENNReal.ofReal ((... | have : Fact (1 ≤ ENNReal.ofReal p) := fact_iff.mpr (ofReal_one ▸ (ofReal_le_ofReal hp)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
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