module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Projectivization.Independence | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 45
} | {
"line": 57,
"column": 4
} | [
{
"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\nh : LinearIndependent K (Projectivization.rep ∘ f)\nx✝ : ι\n⊢ f x✝ = mk K ((Projectivization.rep ∘ f) x✝) ⋯",
"ppTerm": "?m.227",
"assigned": true,
"usedConstants":... | [
"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 : LinearIndependent K (Projectivization.rep ∘ f)\n⊢ ∀ (i : ι), (Projectivization.rep ∘ f) i ≠ 0"
] | · simp only [mk_rep, Function.comp_apply] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Projectivization.Independence | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 45
} | {
"line": 90,
"column": 4
} | [
{
"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\nh : ¬LinearIndependent K (Projectivization.rep ∘ f)\nx✝ : ι\n⊢ f x✝ = mk K ((Projectivization.rep ∘ f) x✝) ⋯",
"ppTerm": "?m.230",
"assigned": true,
"usedConstants"... | [
"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 : ¬LinearIndependent K (Projectivization.rep ∘ f)\n⊢ ∀ (i : ι), (Projectivization.rep ∘ f) i ≠ 0"
] | · simp only [mk_rep, Function.comp_apply] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Projectivization.Action | {
"line": 86,
"column": 41
} | {
"line": 86,
"column": 53
} | {
"line": 86,
"column": 53
} | [
{
"pp": "case h.left\nG : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : DivisionRing K\ninst✝³ : Module K V\ninst✝² : Group G\ninst✝¹ : DistribMulAction G V\ninst✝ : SMulCommClass G K V\nD D' E E' : ℙ K V\nhD : LinearIndependent K ![D.rep, D'.rep]\nhE : LinearIndependent K ![E.rep, E'.... | [
"case h.left\nG : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : DivisionRing K\ninst✝³ : Module K V\ninst✝² : Group G\ninst✝¹ : DistribMulAction G V\ninst✝ : SMulCommClass G K V\nD D' E E' : ℙ K V\nhD : LinearIndependent K ![D.rep, D'.rep]\nhE : LinearIndependent K ![E.rep, E'.rep]\nqD :\n... | mk_eq_mk_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Projectivization.Action | {
"line": 90,
"column": 43
} | {
"line": 90,
"column": 55
} | {
"line": 90,
"column": 55
} | [
{
"pp": "case h.right\nG : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : DivisionRing K\ninst✝³ : Module K V\ninst✝² : Group G\ninst✝¹ : DistribMulAction G V\ninst✝ : SMulCommClass G K V\nD D' E E' : ℙ K V\nhD : LinearIndependent K ![D.rep, D'.rep]\nhE : LinearIndependent K ![E.rep, E'... | [
"case h.right\nG : Type u_1\nK : Type u_2\nV : Type u_3\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : DivisionRing K\ninst✝³ : Module K V\ninst✝² : Group G\ninst✝¹ : DistribMulAction G V\ninst✝ : SMulCommClass G K V\nD D' E E' : ℙ K V\nhD : LinearIndependent K ![D.rep, D'.rep]\nhE : LinearIndependent K ![E.rep, E'.rep]\nqD :\... | mk_eq_mk_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Projectivization.PSL.Stabilizer | {
"line": 76,
"column": 2
} | {
"line": 77,
"column": 45
} | {
"line": 78,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ng : SpecialLinearGroup ι F\nL : Submodule F (ι → F)\nA : SpecialLinearGroup ι F\n⊢ A ∈ lineStab (g • L) ↔ (MulAut.conj g)⁻¹ • A ∈ lineStab L",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
... | [
"F : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ng : SpecialLinearGroup ι F\nL : Submodule F (ι → F)\nA : SpecialLinearGroup ι F\n⊢ (∀ (w : ι → F), ∃ b ∈ L, g • b = A • w - w) ↔ ∀ (w : ι → F), (g⁻¹ * A * g) • w - w ∈ L"
] | simp only [mem_lineStab_iff, Submodule.mem_smul_pointwise_iff_exists, MulAut.smul_def,
MulAut.inv_apply, MulAut.conj_symm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Projectivization.PSL.Stabilizer | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 58
} | {
"line": 134,
"column": 2
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nv : ι → F\nhv : v ≠ 0\nA B : SpecialLinearGroup ι F\nhA : A ∈ lineStab (F ∙ v)\nhB : B ∈ lineStab (F ∙ v)\nw : ι → F\n⊢ ↑(A * B) • w = ↑(B * A) • w",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants"... | [
"F : Type u_1\ninst✝² : Field F\nι : Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nv : ι → F\nhv : v ≠ 0\nA B : SpecialLinearGroup ι F\nhA : A ∈ lineStab (F ∙ v)\nhB : B ∈ lineStab (F ∙ v)\nw : ι → F\nα : F\nhα : α • v = A • w - w\n⊢ ↑(A * B) • w = ↑(B * A) • w"
] | obtain ⟨α, hα⟩ := Submodule.mem_span_singleton.mp (hA w) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 460,
"column": 6
} | {
"line": 460,
"column": 55
} | {
"line": 461,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\nV : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : Module.Free R V\ninst✝ : Module.Finite R V\nr : ↥(rootsOfUnity (max (Module.finrank R V) 1) R)\nhV : Module.finrank R V = 0\nhR : Nontrivial R\n⊢ r = 1",
"ppTerm": "?pos✝",
"assigne... | [] | simpa [hV] using (mem_rootsOfUnity _ _).mp r.prop | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 490,
"column": 6
} | {
"line": 492,
"column": 37
} | {
"line": 493,
"column": 4
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : Module.Free R V\ninst✝ : Module.Finite R V\ng h : ↥(Subgroup.center (SpecialLinearGroup R V))\nhR : ¬Subsingleton R\nhV✝ : ¬Subsingleton V\nhV : Nontrivial V\nthis✝ : FaithfulSMul R V\nHg : ∃ r, r ^ ... | [] | simp [mul_smul,
← mem_center_iff_spec (g * h).prop, ← mem_center_iff_spec h.prop,
← mem_center_iff_spec g.prop] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 191,
"column": 2
} | {
"line": 203,
"column": 88
} | {
"line": 205,
"column": 0
} | [
{
"pp": "𝕜 : 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\n⊢ sigPos (weightedSumSquares 𝕜 w) = {i | 0 < w i}.ncard",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
... | [] | classical
let p : Set ι := {i | 0 < w i}
let m : Set ι := {i | w i ≤ 0}
convert_to sigPos _ = p.ncard
have : p.ncard + m.ncard = Nat.card ι := by
convert! Set.ncard_add_ncard_compl p
ext
grind
have : p.ncard ≤ sigPos (weightedSumSquares 𝕜 w) :=
(sigPos_isGreatest _).2 ⟨Pi.spanSubset 𝕜 p, Pi.... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 191,
"column": 2
} | {
"line": 203,
"column": 88
} | {
"line": 205,
"column": 0
} | [
{
"pp": "𝕜 : 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\n⊢ sigPos (weightedSumSquares 𝕜 w) = {i | 0 < w i}.ncard",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
... | [] | classical
let p : Set ι := {i | 0 < w i}
let m : Set ι := {i | w i ≤ 0}
convert_to sigPos _ = p.ncard
have : p.ncard + m.ncard = Nat.card ι := by
convert! Set.ncard_add_ncard_compl p
ext
grind
have : p.ncard ≤ sigPos (weightedSumSquares 𝕜 w) :=
(sigPos_isGreatest _).2 ⟨Pi.spanSubset 𝕜 p, Pi.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 191,
"column": 2
} | {
"line": 203,
"column": 88
} | {
"line": 205,
"column": 0
} | [
{
"pp": "𝕜 : 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\n⊢ sigPos (weightedSumSquares 𝕜 w) = {i | 0 < w i}.ncard",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
... | [] | classical
let p : Set ι := {i | 0 < w i}
let m : Set ι := {i | w i ≤ 0}
convert_to sigPos _ = p.ncard
have : p.ncard + m.ncard = Nat.card ι := by
convert! Set.ncard_add_ncard_compl p
ext
grind
have : p.ncard ≤ sigPos (weightedSumSquares 𝕜 w) :=
(sigPos_isGreatest _).2 ⟨Pi.spanSubset 𝕜 p, Pi.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 67
} | {
"line": 96,
"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.IsCrystallographic\nb : P.Base\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝... | [
"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.IsCrystallographic\nb : P.Base\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.IsRootS... | rw [Matrix.det_transpose, ← Matrix.nondegenerate_iff_det_ne_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 50
} | {
"line": 141,
"column": 2
} | [
{
"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 ... | [] | exact Submodule.subset_span (mem_range_self k) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 429,
"column": 80
} | {
"line": 434,
"column": 6
} | {
"line": 436,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN✝ : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ninst✝⁷ : AddCommGroup N✝\ninst✝⁶ : Module R N✝\nP : RootPairing ι R M N✝\ninst✝⁵ : P.IsCrystallographic\nb : P.Base\ninst✝⁴ : Finite ι\ninst✝³ : IsDomain R\ninst✝² : CharZero R\n... | [] | by
rw [← LieSubmodule.toSubmodule_eq_top]
let e : Submodule R (b.support ⊕ ι → R) ≃o Submodule R (b.support ⊕ ι → R) :=
Submodule.orderIsoMapComapOfBijective (ω b).toLin' (Involutive.bijective fun x ↦ by simp)
change e.symm N = ⊤ ↔ _
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 562,
"column": 4
} | {
"line": 566,
"column": 22
} | {
"line": 567,
"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... | [] | induction hx using Submodule.span_induction with
| zero => simp
| mem => grind
| add => simp_all
| smul => simp_all | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basis | {
"line": 69,
"column": 4
} | {
"line": 87,
"column": 51
} | {
"line": 89,
"column": 0
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹¹ : Fintype ι\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Field K\ninst✝⁸ : CharZero K\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module K M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module K N\nP : RootPairing ι K M N\ninst✝³ : P.IsReduced\ninst✝² : P.IsCrystallog... | [] | let h₀ (i : b.support) : lieAlgebra b := ⟨h i, h_mem_lieAlgebra i⟩
let e₀ (i : b.support) : lieAlgebra b := ⟨e i, e_mem_lieAlgebra i⟩
let f₀ (i : b.support) : lieAlgebra b := ⟨f i, f_mem_lieAlgebra i⟩
change LieSubalgebra.lieSpan K (lieAlgebra b) (range e₀ ∪ range f₀) = ⊤
suffices LieSubalgebra.lieSpan ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basis | {
"line": 69,
"column": 4
} | {
"line": 87,
"column": 51
} | {
"line": 89,
"column": 0
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹¹ : Fintype ι\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Field K\ninst✝⁸ : CharZero K\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module K M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module K N\nP : RootPairing ι K M N\ninst✝³ : P.IsReduced\ninst✝² : P.IsCrystallog... | [] | let h₀ (i : b.support) : lieAlgebra b := ⟨h i, h_mem_lieAlgebra i⟩
let e₀ (i : b.support) : lieAlgebra b := ⟨e i, e_mem_lieAlgebra i⟩
let f₀ (i : b.support) : lieAlgebra b := ⟨f i, f_mem_lieAlgebra i⟩
change LieSubalgebra.lieSpan K (lieAlgebra b) (range e₀ ∪ range f₀) = ⊤
suffices LieSubalgebra.lieSpan ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 261,
"column": 2
} | {
"line": 261,
"column": 82
} | {
"line": 262,
"column": 2
} | [
{
"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... | have hik_mem : P.root i + P.root k ∈ range P.root := ⟨l, by rw [← h₁, add_comm]⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.SesquilinearForm.Star | {
"line": 67,
"column": 4
} | {
"line": 68,
"column": 13
} | {
"line": 69,
"column": 0
} | [
{
"pp": "case refine_2\nM : Type u_2\nn : Type u_3\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq n\nR : Type u_4\ninst✝³ : CommRing R\ninst✝² : StarRing R\ninst✝¹ : PartialOrder R\ninst✝ : Module R M\nB : M →ₗ⋆[R] M →ₗ[R] R\nb : Basis n R M\nh : ∀ (x : n → R), 0 ≤ star x ⬝ᵥ ((toMatrix₂ b b... | [] | rw [apply_eq_star_dotProduct_toMatrix₂_mulVec b]
exact h _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SesquilinearForm.Star | {
"line": 67,
"column": 4
} | {
"line": 68,
"column": 13
} | {
"line": 69,
"column": 0
} | [
{
"pp": "case refine_2\nM : Type u_2\nn : Type u_3\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq n\nR : Type u_4\ninst✝³ : CommRing R\ninst✝² : StarRing R\ninst✝¹ : PartialOrder R\ninst✝ : Module R M\nB : M →ₗ⋆[R] M →ₗ[R] R\nb : Basis n R M\nh : ∀ (x : n → R), 0 ≤ star x ⬝ᵥ ((toMatrix₂ b b... | [] | rw [apply_eq_star_dotProduct_toMatrix₂_mulVec b]
exact h _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 131,
"column": 8
} | {
"line": 131,
"column": 79
} | {
"line": 132,
"column": 8
} | [
{
"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... | [
"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\ninst✝¹ : Fint... | apply P.nsmul_notMem_range_root (n := P.chainTopCoeff i i + 2) (i := i) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 79,
"column": 4
} | {
"line": 89,
"column": 44
} | {
"line": 91,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\na b : SymmetricAlgebra R M\n⊢ Commute a b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"TensorAlgebra.SymRel.mul_comm",
"Commute.mul_right",
"Ten... | [] | induction b using SymmetricAlgebra.induction with
| algebraMap r => exact Algebra.commute_algebraMap_right _ _
| ι x => induction a using SymmetricAlgebra.induction with
| algebraMap r => exact Algebra.commute_algebraMap_left _ _
| ι y =>
have := RingCon.le_ringConGen (r := SymRel R M) _ _ <... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 364,
"column": 7
} | {
"line": 364,
"column": 30
} | {
"line": 364,
"column": 30
} | [
{
"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 ... | [] | by simpa only [neg_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 364,
"column": 33
} | {
"line": 364,
"column": 56
} | {
"line": 364,
"column": 56
} | [
{
"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 ... | [] | by simpa only [neg_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 371,
"column": 15
} | {
"line": 371,
"column": 38
} | {
"line": 371,
"column": 38
} | [
{
"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 ... | [] | by simpa only [neg_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 371,
"column": 41
} | {
"line": 371,
"column": 64
} | {
"line": 371,
"column": 64
} | [
{
"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 ... | [] | by simpa only [neg_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.SetSemiring | {
"line": 101,
"column": 22
} | {
"line": 101,
"column": 39
} | {
"line": 101,
"column": 39
} | [
{
"pp": "case insert\nα : Type u_1\nC : Set (Set α)\nhC : IsSetSemiring C\ns : Set α\nS : Finset (Set α)\na✝ : s ∉ S\nih : ↑S ⊆ C → ∃ P, ↑P.parts ⊆ C\nhSC : insert s ↑S ⊆ C\n⊢ ∃ P, ↑P.parts ⊆ C",
"ppTerm": "?insert",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"Membe... | [
"case insert\nα : Type u_1\nC : Set (Set α)\nhC : IsSetSemiring C\ns : Set α\nS : Finset (Set α)\na✝ : s ∉ S\nih : ↑S ⊆ C → ∃ P, ↑P.parts ⊆ C\nhSC : s ∈ C ∧ ↑S ⊆ C\n⊢ ∃ P, ↑P.parts ⊆ C"
] | insert_subset_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.SetSemiring | {
"line": 141,
"column": 22
} | {
"line": 141,
"column": 39
} | {
"line": 141,
"column": 39
} | [
{
"pp": "case insert\nα : Type u_1\nC : Set (Set α)\nhC : IsSetSemiring C\nt : Set α\nT : Finset (Set α)\na✝ : t ∉ T\nih : ∀ ⦃s : Set α⦄, s ∈ supClosure C → ↑T ⊆ C → s \\ T.sup id ∈ supClosure C\ns : Set α\nhs : s ∈ supClosure C\nhTC : insert t ↑T ⊆ C\n⊢ (s \\ t) \\ T.sup id ∈ supClosure C",
"ppTerm": "?ins... | [
"case insert\nα : Type u_1\nC : Set (Set α)\nhC : IsSetSemiring C\nt : Set α\nT : Finset (Set α)\na✝ : t ∉ T\nih : ∀ ⦃s : Set α⦄, s ∈ supClosure C → ↑T ⊆ C → s \\ T.sup id ∈ supClosure C\ns : Set α\nhs : s ∈ supClosure C\nhTC : t ∈ C ∧ ↑T ⊆ C\n⊢ (s \\ t) \\ T.sup id ∈ supClosure C"
] | insert_subset_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 360,
"column": 2
} | {
"line": 388,
"column": 15
} | {
"line": 390,
"column": 0
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrysta... | [] | refine LieModule.IsIrreducible.mk fun U hU ↦ ?_
suffices ∃ i, v b i ∈ U by obtain ⟨i, hi⟩ := this; exact instIsIrreducible_aux₂ hi
let U' : LieSubmodule K H (b.support ⊕ ι → K) := { U with lie_mem := U.lie_mem }
apply instIsIrreducible_aux₁ U'
contrapose hU
replace hU : U ≤ span K (range (u (b := b))) := by r... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 360,
"column": 2
} | {
"line": 388,
"column": 15
} | {
"line": 390,
"column": 0
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹² : Field K\ninst✝¹¹ : CharZero K\ninst✝¹⁰ : DecidableEq ι\ninst✝⁹ : Fintype ι\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module K M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module K N\nP : RootPairing ι K M N\ninst✝⁴ : P.IsRootSystem\ninst✝³ : P.IsCrysta... | [] | refine LieModule.IsIrreducible.mk fun U hU ↦ ?_
suffices ∃ i, v b i ∈ U by obtain ⟨i, hi⟩ := this; exact instIsIrreducible_aux₂ hi
let U' : LieSubmodule K H (b.support ⊕ ι → K) := { U with lie_mem := U.lie_mem }
apply instIsIrreducible_aux₁ U'
contrapose hU
replace hU : U ≤ span K (range (u (b := b))) := by r... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.WithTop | {
"line": 63,
"column": 6
} | {
"line": 63,
"column": 26
} | {
"line": 64,
"column": 4
} | [
{
"pp": "ι : 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, Ioi x = ... | [] | grind (splits := 12) | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 210,
"column": 4
} | {
"line": 219,
"column": 18
} | {
"line": 221,
"column": 0
} | [
{
"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'\n⊢ ∑ t ∈ J', ∑ s ∈ J, m (s ∩ t) = ∑ t ∈ J', m t",... | [] | apply Finset.sum_congr rfl (fun t ht ↦ ?_)
have : t = ⋃ s ∈ J, s ∩ t := by
simp_rw [← Finset.set_biUnion_coe, ← iUnion_inter, right_eq_inter, ← sUnion_eq_biUnion, h]
exact subset_sUnion_of_mem ht
nth_rewrite 2 [this]
apply (addContent_biUnion _ _ _).symm
· exact fun s hs ↦ hC.inter_mem _ (hJ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 210,
"column": 4
} | {
"line": 219,
"column": 18
} | {
"line": 221,
"column": 0
} | [
{
"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'\n⊢ ∑ t ∈ J', ∑ s ∈ J, m (s ∩ t) = ∑ t ∈ J', m t",... | [] | apply Finset.sum_congr rfl (fun t ht ↦ ?_)
have : t = ⋃ s ∈ J, s ∩ t := by
simp_rw [← Finset.set_biUnion_coe, ← iUnion_inter, right_eq_inter, ← sUnion_eq_biUnion, h]
exact subset_sUnion_of_mem ht
nth_rewrite 2 [this]
apply (addContent_biUnion _ _ _).symm
· exact fun s hs ↦ hC.inter_mem _ (hJ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 243,
"column": 2
} | {
"line": 245,
"column": 13
} | {
"line": 247,
"column": 0
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nhC : IsSetSemiring C\nm : AddContent G C\ns : Set α\nhs : s ∈ C\n⊢ m.supClosureFun s = m s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Finset.coe_singleton",
"ChainCompletePartialOrder.instOfCom... | [] | have : m.supClosureFun s = ∑ t ∈ {s}, m t :=
m.supClosureFun_apply hC (by simp [hs]) (by simp) (by simp)
simp [this] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 243,
"column": 2
} | {
"line": 245,
"column": 13
} | {
"line": 247,
"column": 0
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nhC : IsSetSemiring C\nm : AddContent G C\ns : Set α\nhs : s ∈ C\n⊢ m.supClosureFun s = m s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Finset.coe_singleton",
"ChainCompletePartialOrder.instOfCom... | [] | have : m.supClosureFun s = ∑ t ∈ {s}, m t :=
m.supClosureFun_apply hC (by simp [hs]) (by simp) (by simp)
simp [this] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 316,
"column": 32
} | {
"line": 316,
"column": 47
} | {
"line": 316,
"column": 48
} | [
{
"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\nf : Fin #J → Set α := disjointed fun j ↦ ↑(J.equivFin.symm j)\nhtJ : t ⊆ ⋃... | [
"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\nf : Fin #J → Set α := disjointed fun j ↦ ↑(J.equivFin.symm j)\nhtJ : t ⊆ ⋃ i, f i\nh1 ... | ← J.sum_attach, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.MeasureTheory.SetSemiring | {
"line": 555,
"column": 6
} | {
"line": 555,
"column": 30
} | {
"line": 555,
"column": 30
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\ns t : Set α\nhC : IsSetRing C\nhs : s ∈ C\nht : t ∈ C\n⊢ s ∩ t ∈ C",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.mem",
"id",
"Set.instInter",
"Inter.inter",
"SDiff.sdiff",
... | [
"α : Type u_1\nC : Set (Set α)\ns t : Set α\nhC : IsSetRing C\nhs : s ∈ C\nht : t ∈ C\n⊢ s \\ (s \\ t) ∈ C"
] | ← sdiff_sdiff_right_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 53
} | {
"line": 71,
"column": 4
} | [
{
"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\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\nt... | [
"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\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhf_int : Integrable f μ\nhs : IsClosed s\nhc : Convex ℝ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\nthis✝² : Meas... | refine (hf_int.congr lem1).mono ?_ (by simp [fX]) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 45
} | {
"line": 230,
"column": 2
} | [
{
"pp": "case inl\nΩ : Type u_1\nmΩ₀ : MeasurableSpace Ω\nX : Ω → ℝ≥0∞\n⊢ 0⁻[X | ⊥] =ᵐ[0] fun x ↦ (0 Set.univ)⁻¹ • ∫⁻ (ω : Ω), X ω ∂0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"instHSMul",
"MeasureTheory.Measure",
"Lattice.to... | [] | rw [ae_zero]; exact Filter.eventually_bot | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 45
} | {
"line": 230,
"column": 2
} | [
{
"pp": "case inl\nΩ : Type u_1\nmΩ₀ : MeasurableSpace Ω\nX : Ω → ℝ≥0∞\n⊢ 0⁻[X | ⊥] =ᵐ[0] fun x ↦ (0 Set.univ)⁻¹ • ∫⁻ (ω : Ω), X ω ∂0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"instHSMul",
"MeasureTheory.Measure",
"Lattice.to... | [] | rw [ae_zero]; exact Filter.eventually_bot | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 50
} | {
"line": 246,
"column": 2
} | [
{
"pp": "case pos.h12\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX Y : Ω → ℝ≥0∞\nhXY : X ≤ᵐ[P] Y\nhm : mΩ ≤ mΩ₀\nhσ : SigmaFinite (P.trim hm)\ns : Set Ω\nhs : MeasurableSet s\na✝ : (P.trim hm) s < ∞\n⊢ ∫⁻ (x : Ω) in s, P⁻[X | mΩ] x ∂P.trim hm ≤ ∫⁻ (x : Ω) in s, P⁻[Y | mΩ] x ∂P.trim hm",
"ppTe... | [
"case pos.h12\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX Y : Ω → ℝ≥0∞\nhXY : X ≤ᵐ[P] Y\nhm : mΩ ≤ mΩ₀\nhσ : SigmaFinite (P.trim hm)\ns : Set Ω\nhs : MeasurableSet s\na✝ : (P.trim hm) s < ∞\n⊢ ∫⁻ (ω : Ω) in s, X ω ∂P ≤ ∫⁻ (ω : Ω) in s, Y ω ∂P"
] | repeat rw [setLIntegral_condLExp_trim hm _ _ hs] | Lean.Elab.Tactic.evalRepeat | Lean.Parser.Tactic.tacticRepeat_ |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 260,
"column": 2
} | {
"line": 260,
"column": 50
} | {
"line": 261,
"column": 2
} | [
{
"pp": "case pos.h12\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX Y : Ω → ℝ≥0∞\nhm : mΩ ≤ mΩ₀\nhσ : SigmaFinite (P.trim hm)\ns : Set Ω\nhs : MeasurableSet s\na✝ : (P.trim hm) s < ∞\n⊢ ∫⁻ (a : Ω) in s, P⁻[X | mΩ] a ∂P.trim hm + ∫⁻ (a : Ω) in s, P⁻[Y | mΩ] a ∂P.trim hm ≤\n ∫⁻ (x : Ω) in s, P⁻[X... | [
"case pos.h12\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX Y : Ω → ℝ≥0∞\nhm : mΩ ≤ mΩ₀\nhσ : SigmaFinite (P.trim hm)\ns : Set Ω\nhs : MeasurableSet s\na✝ : (P.trim hm) s < ∞\n⊢ ∫⁻ (ω : Ω) in s, X ω ∂P + ∫⁻ (ω : Ω) in s, Y ω ∂P ≤ ∫⁻ (ω : Ω) in s, (X + Y) ω ∂P"
] | repeat rw [setLIntegral_condLExp_trim hm _ _ hs] | Lean.Elab.Tactic.evalRepeat | Lean.Parser.Tactic.tacticRepeat_ |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 53
} | {
"line": 137,
"column": 4
} | [
{
"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\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ... | [
"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\ninst✝ : IsFiniteMeasure μ\nhm : m ≤ mα\nhφ_cvx : ConvexOn ℝ s φ\nhφ_cont : LowerSemicontinuousOn φ s\nhf : ∀ᵐ (a : α) ∂μ, f a ∈ s\... | refine (hf_int.congr lem1).mono ?_ (by simp [fX]) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 140,
"column": 4
} | {
"line": 141,
"column": 59
} | {
"line": 142,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\ninst✝² : CompleteSpace G\nB : F →L[... | [] | apply condExp_congr_ae
filter_upwards [hf.ae_eq_mk] with a ha using by rw [ha] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 140,
"column": 4
} | {
"line": 141,
"column": 59
} | {
"line": 142,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\ninst✝² : CompleteSpace G\nB : F →L[... | [] | apply condExp_congr_ae
filter_upwards [hf.ae_eq_mk] with a ha using by rw [ha] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 57
} | {
"line": 146,
"column": 4
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\ninst✝² : CompleteSpace G\nB : F →L[... | [
"Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace ℝ G\ninst✝² : CompleteSpace G\nB : F →L[ℝ] E →L[ℝ] G... | filter_upwards [hf_bound, hf.ae_eq_mk] with ω hω1 hω2 | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn | {
"line": 252,
"column": 6
} | {
"line": 252,
"column": 21
} | {
"line": 253,
"column": 6
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : s i < 0\nh✝ : ¬s ≤[i] 0\nhn : ∃ n, s ≤[i \\ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0\nk : ℕ := Nat.find hn\nhk₂ : s ≤[i \\ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0\nhmeas : Measur... | [
"case e'_4\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : s i < 0\nh✝ : ¬s ≤[i] 0\nhn : ∃ n, s ≤[i \\ ⋃ l, ⋃ (_ : l < n), s.restrictNonposSeq i l] 0\nk : ℕ := Nat.find hn\nhk₂ : s ≤[i \\ ⋃ l, ⋃ (_ : l < k), s.restrictNonposSeq i l] 0\nhmeas : MeasurableSet (⋃ ... | convert! h₁ _ h | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 205,
"column": 4
} | {
"line": 206,
"column": 59
} | {
"line": 207,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\ninst✝¹ : CompleteSpace G\nB : F →L[... | [] | apply condExp_congr_ae
filter_upwards [hf.ae_eq_mk] with a ha using by rw [ha] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 205,
"column": 4
} | {
"line": 206,
"column": 59
} | {
"line": 207,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm mΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace ℝ G\ninst✝¹ : CompleteSpace G\nB : F →L[... | [] | apply condExp_congr_ae
filter_upwards [hf.ae_eq_mk] with a ha using by rw [ha] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 179,
"column": 6
} | {
"line": 179,
"column": 78
} | {
"line": 179,
"column": 79
} | [
{
"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⊢ (j.posPart A).toReal - j.negPart.real A ≤ 0 A",
"ppTerm": "?refine_1",
"assigned": ... | [
"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⊢ ENNReal.toReal 0 - j.negPart.real A ≤ 0 A"
] | show j.posPart A = 0 from nonpos_iff_eq_zero.1 (hS₂ ▸ measure_mono hA₁), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 250,
"column": 8
} | {
"line": 250,
"column": 14
} | {
"line": 250,
"column": 15
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : 0 ≤[u] s\nhw₁ : s w = 0\nhw₂ : w ⊆ u\nhwt : v ⊆ w\n⊢ s v + s (w \\ v) = 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nu v w : Set α\nhu : MeasurableSet u\nhv : MeasurableSet v\nhw : MeasurableSet w\nhsu : 0 ≤[u] s\nhw₁ : s w = 0\nhw₂ : w ⊆ u\nhwt : v ⊆ w\n⊢ s v + s (w \\ v) = s w"
] | ← hw₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn | {
"line": 396,
"column": 2
} | {
"line": 396,
"column": 70
} | {
"line": 397,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nf : ℕ → ℝ\nleft✝ : Antitone f\nhf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))\nB : ℕ → Set α\nhB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ s (B n) = f n\nhB₁ : ∀ (n : ℕ), MeasurableSet (B n)\nhB₂ : ∀ (n : ℕ), s ≤[B n] 0... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nf : ℕ → ℝ\nleft✝ : Antitone f\nhf₂ : Tendsto f atTop (nhds (sInf s.measureOfNegatives))\nB : ℕ → Set α\nhB : ∀ (n : ℕ), B n ∈ {B | MeasurableSet B ∧ s ≤[B] 0} ∧ s (B n) = f n\nhB₁ : ∀ (n : ℕ), MeasurableSet (B n)\nhB₂ : ∀ (n : ℕ), s ≤[B n] 0\nA : Set α ... | rcases exists_subset_restrict_nonpos hC₂ with ⟨D, hD₁, hD, hD₂, hD₃⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 516,
"column": 8
} | {
"line": 516,
"column": 50
} | {
"line": 517,
"column": 8
} | [
{
"pp": "case mp.refine_2\nα : Type u_1\ninst✝ : MeasurableSpace α\ns t : SignedMeasure α\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ∀ t ⊆ u, s t = 0\nhu₂ : ∀ t_1 ⊆ uᶜ, t t_1 = 0\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : 0 ≤[i] s\nhi₃ : s ≤[iᶜ] 0\nhipos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i h... | [
"case mp.refine_2\nα : Type u_1\ninst✝ : MeasurableSpace α\ns t : SignedMeasure α\nu : Set α\nhmeas : MeasurableSet u\nhu₁ : ∀ t ⊆ u, s t = 0\nhu₂ : ∀ t_1 ⊆ uᶜ, t t_1 = 0\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : 0 ≤[i] s\nhi₃ : s ≤[iᶜ] 0\nhipos : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi₁ hi₂\nhine... | toMeasureOfZeroLE_apply _ _ _ hmeas.compl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 50,
"column": 36
} | {
"line": 53,
"column": 48
} | {
"line": 53,
"column": 49
} | [
{
"pp": "α : Type u_1\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nm : MeasurableSpace α\nμ : Measure α\nf : α → E\nhf : Integrable f μ\ns : ℕ → Set α\nhs₁ : ∀ (i : ℕ), MeasurableSet (s i)\nhs₂ : Pairwise (Function.onFun Disjoint s)\n⊢ HasSum (fu... | [] | by
convert! hasSum_integral_iUnion hs₁ hs₂ hf.integrableOn with n
· rw [if_pos (hs₁ n)]
· rw [if_pos (MeasurableSet.iUnion hs₁)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1099,
"column": 2
} | {
"line": 1099,
"column": 71
} | {
"line": 1100,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : LinearOrder M\nv : VectorMeasure α M\ni : Set α\nhi : ¬v ≤[i] 0\n⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ 0 < v j",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Measur... | [
"α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : LinearOrder M\nv : VectorMeasure α M\ni : Set α\nhi : ¬v ≤[i] 0\nhi₁ : MeasurableSet i\n⊢ ∃ j, MeasurableSet j ∧ j ⊆ i ∧ 0 < v j"
] | have hi₁ : MeasurableSet i := measurable_of_not_restrict_le_zero _ hi | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 211,
"column": 59
} | {
"line": 211,
"column": 87
} | {
"line": 211,
"column": 87
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → E\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nhfi : Integrable f μ\nj : Set α\nhj₁ : MeasurableSet j\nhj₂ : (μ.trim hm) j = 0\n⊢ ((μ.withDensityᵥ f).trim hm) j = 0",
"ppTerm": "?m.59",
"assign... | [
"α : Type u_1\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : α → E\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nhfi : Integrable f μ\nj : Set α\nhj₁ : MeasurableSet j\nhj₂ : μ j = 0\n⊢ ((μ.withDensityᵥ f).trim hm) j = 0"
] | trim_measurableSet_eq hm hj₁ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 322,
"column": 53
} | {
"line": 338,
"column": 75
} | {
"line": 340,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\nr : ℝ\n⊢ (r • s).singularPart μ = r • s.singularPart μ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"MeasureTheory.JordanDecomposition.posPart",
"AddGroup.toSubtractionMonoid",... | [] | by
cases le_or_gt 0 r with
| inl hr =>
lift r to ℝ≥0 using hr
exact singularPart_smul_nnreal s μ r
| inr hr =>
rw [singularPart, singularPart]
conv_lhs =>
congr
· congr
· rw [toJordanDecomposition_smul_real,
JordanDecomposition.real_smul_posPart_neg _ _ hr, singular... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 84,
"column": 2
} | {
"line": 91,
"column": 43
} | {
"line": 93,
"column": 0
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\n⊢ |μ[f | m]| ≤ᵐ[μ] μ[|f| | m]",
... | [] | by_cases! hfint : ¬Integrable f μ
· simp only [condExp_of_not_integrable hfint, abs_zero]
apply condExp_nonneg
filter_upwards with a using abs_nonneg (f a)
have h1 := condExp_mono (m := m) hfint hfint.abs (.of_forall (fun x => le_abs_self f x))
have h2 := condExp_mono (m := m) hfint.neg hfint.abs (.of_for... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 84,
"column": 2
} | {
"line": 91,
"column": 43
} | {
"line": 93,
"column": 0
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : Lattice E\ninst✝² : HasSolidNorm E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nf : α → E\n⊢ |μ[f | m]| ≤ᵐ[μ] μ[|f| | m]",
... | [] | by_cases! hfint : ¬Integrable f μ
· simp only [condExp_of_not_integrable hfint, abs_zero]
apply condExp_nonneg
filter_upwards with a using abs_nonneg (f a)
have h1 := condExp_mono (m := m) hfint hfint.abs (.of_forall (fun x => le_abs_self f x))
have h2 := condExp_mono (m := m) hfint.neg hfint.abs (.of_for... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 634,
"column": 6
} | {
"line": 637,
"column": 92
} | {
"line": 638,
"column": 6
} | [
{
"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... | [
"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 ≤ ENNReal.ofReal ... | have : ∀ᵐ x ∂μ.restrict s, ‖{ x : α | ‖f i x‖₊ < C }.indicator (f i) x‖ ≤ C := by
filter_upwards
simp_rw [norm_indicator_eq_indicator_norm]
exact Set.indicator_le' (fun x (hx : _ < _) => hx.le) fun _ _ => NNReal.coe_nonneg _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 48
} | {
"line": 182,
"column": 2
} | [
{
"pp": "case pos.refine_2.refine_2\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nR : ℝ≥0\nf : α → ℝ\nhbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R\nhnm : m ≤ m0\nhfint : Integrable f μ\nh : 0 < μ {x | ↑R < |μ[f | m] x|}\n⊢ ∫ (x : α) in {x | ↑R < |μ[f | m] x|}, |f x| ∂μ ≤ μ.real {x | ↑R < |μ[f | m] x|} * ↑R",
... | [
"case pos.refine_2.refine_2\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nR : ℝ≥0\nf : α → ℝ\nhbdd : ∀ᵐ (x : α) ∂μ, |f x| ≤ ↑R\nhnm : m ≤ m0\nhfint : Integrable f μ\nh : 0 < μ {x | ↑R < |μ[f | m] x|}\n⊢ ∫ (x : α) in {x | ↑R < |μ[f | m] x|}, |f x| ∂μ ≤ ∫ (x : α) in {x | ↑R < |μ[f | m] x|}, ↑R ∂μ"
] | simp only [← smul_eq_mul, ← setIntegral_const] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 47
} | {
"line": 223,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ\nhp : 1 ≤ p\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : p ≠ 0\nhm : m ≤ m0\nhsig : Si... | [
"case refine_2\nα : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ\nhp : 1 ≤ p\nf : α → E\ns : Set α\nhs : MeasurableSet s\nhf : Integrable (fun x ↦ ‖f x‖ ^ p) μ\nhp' : p ≠ 0\nhm : m ≤ m0\nhsig : SigmaFinite (μ... | · filter_upwards with a using by positivity | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 242,
"column": 86
} | {
"line": 243,
"column": 41
} | {
"line": 245,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝ : MeasurableSpace Ω\nμ : ProbabilityMeasure Ω\n⊢ μ.toFiniteMeasure ≠ 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"MeasureTheory.ProbabilityMeasure.mass_toFiniteMeasure",
"MeasureTheory.FiniteMeasure.mass",
"MeasureTheory.FiniteMeasure",
... | [] | by
simp [← FiniteMeasure.mass_nonzero_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 516,
"column": 82
} | {
"line": 517,
"column": 92
} | {
"line": 519,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\ninst✝ : TopologicalSpace Ω\nnonzero : μ ≠ 0\nf : Ω →ᵇ ℝ≥0\n⊢ μ.normalize.toFiniteMeasure.testAgainstNN f = μ.mass⁻¹ * μ.testAgainstNN f",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"MeasureThe... | [] | by
simp [μ.testAgainstNN_eq_mass_mul, inv_mul_cancel_left₀ <| μ.mass_nonzero_iff.mpr nonzero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Tight | {
"line": 66,
"column": 4
} | {
"line": 67,
"column": 27
} | {
"line": 68,
"column": 2
} | [
{
"pp": "case refine_1\n𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\nS : Set (Measure 𝓧)\ninst✝ : TopologicalSpace 𝓧\nh : ∀ (ε : ENNReal), 0 < ε → ∃ s, (∃ t, IsCompact t ∧ tᶜ ⊆ s) ∧ ∀ t ⊆ s, ∀ i ∈ S, i t ≤ ε\nε : ENNReal\nhε : 0 < ε\n⊢ ∃ K, IsCompact K ∧ ∀ μ ∈ S, μ Kᶜ ≤ ε",
"ppTerm": "?refine_1",
"assigne... | [] | obtain ⟨A, ⟨K, h1, h2⟩, hA⟩ := h ε hε
exact ⟨K, h1, hA Kᶜ h2⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Tight | {
"line": 66,
"column": 4
} | {
"line": 67,
"column": 27
} | {
"line": 68,
"column": 2
} | [
{
"pp": "case refine_1\n𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\nS : Set (Measure 𝓧)\ninst✝ : TopologicalSpace 𝓧\nh : ∀ (ε : ENNReal), 0 < ε → ∃ s, (∃ t, IsCompact t ∧ tᶜ ⊆ s) ∧ ∀ t ⊆ s, ∀ i ∈ S, i t ≤ ε\nε : ENNReal\nhε : 0 < ε\n⊢ ∃ K, IsCompact K ∧ ∀ μ ∈ S, μ Kᶜ ≤ ε",
"ppTerm": "?refine_1",
"assigne... | [] | obtain ⟨A, ⟨K, h1, h2⟩, hA⟩ := h ε hε
exact ⟨K, h1, hA Kᶜ h2⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 863,
"column": 2
} | {
"line": 864,
"column": 33
} | {
"line": 865,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSpace Ω'\nf : Ω → Ω'\nμ : FiniteMeasure Ω'\n⊢ (comap f μ).mass ≤ μ.mass",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"MeasureTheory.FiniteMeasure.mass",
"Eq.mpr",
"False",
"Measur... | [
"Ω : Type u_1\nΩ' : Type u_2\ninst✝¹ : MeasurableSpace Ω\ninst✝ : MeasurableSpace Ω'\nf : Ω → Ω'\nμ : FiniteMeasure Ω'\n⊢ (Measure.comap f ↑μ) univ ≤ ↑μ univ"
] | simp only [mass, comap, mk_apply, coeFn_def, ne_eq, measure_ne_top, not_false_eq_true,
ENNReal.toNNReal_le_toNNReal] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 180,
"column": 2
} | {
"line": 186,
"column": 67
} | {
"line": 188,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\n⊢ (∀ (F : Set Ω), IsClosed[inst✝³] F → limsup (fun i ↦ ... | [] | constructor
· intro h G G_open
exact le_measure_liminf_of_limsup_measure_compl_le
G_open.measurableSet (h Gᶜ (isClosed_compl_iff.mpr G_open))
· intro h F F_closed
exact limsup_measure_le_of_le_liminf_measure_compl
F_closed.measurableSet (h Fᶜ (isOpen_compl_iff.mpr F_closed)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 180,
"column": 2
} | {
"line": 186,
"column": 67
} | {
"line": 188,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : TopologicalSpace Ω\ninst✝² : OpensMeasurableSpace Ω\nι : Type u_2\nL : Filter ι\nμ : Measure Ω\nμs : ι → Measure Ω\ninst✝¹ : IsProbabilityMeasure μ\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μs i)\n⊢ (∀ (F : Set Ω), IsClosed[inst✝³] F → limsup (fun i ↦ ... | [] | constructor
· intro h G G_open
exact le_measure_liminf_of_limsup_measure_compl_le
G_open.measurableSet (h Gᶜ (isClosed_compl_iff.mpr G_open))
· intro h F F_closed
exact limsup_measure_le_of_le_liminf_measure_compl
F_closed.measurableSet (h Fᶜ (isOpen_compl_iff.mpr F_closed)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 120,
"column": 50
} | {
"line": 120,
"column": 66
} | {
"line": 122,
"column": 0
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\nm : ι → MeasurableSpace Ω\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\n⊢ iIndep m κ 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"MeasurableSet",
"setOf",
"Measure... | [] | by simp [iIndep] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 921,
"column": 2
} | {
"line": 921,
"column": 23
} | {
"line": 923,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\np : ℝ≥0∞\nhp : 1 ≤ p\nf : ℕ → α → ℝ\nhf : UniformIntegrable f p μ\nn : ℕ\nx : α\n⊢ ((∑ i ∈ Finset.range n, f i) / ↑n) x = ((↑n)⁻¹ • ∑ i ∈ Finset.range n, f i) x",
"ppTerm": "?m.108",
"assigned": true,
"usedConstants": [
"Real",
... | [] | simp [div_eq_inv_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 210,
"column": 41
} | {
"line": 210,
"column": 57
} | {
"line": 212,
"column": 0
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nμ : Measure α\ninst✝¹ : Subsingleton ι\nm : ι → MeasurableSpace Ω\nκ : Kernel α Ω\ninst✝ : IsMarkovKernel κ\n⊢ iIndep m κ μ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Measurable... | [] | by simp [iIndep] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 74,
"column": 50
} | {
"line": 74,
"column": 68
} | {
"line": 76,
"column": 0
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nγ : Type u_6\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nβ : Type u_10\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : Ω → β\ng : Ω → γ\n⊢ IndepFun f g κ 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Measure... | [] | by simp [IndepFun] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 77,
"column": 65
} | {
"line": 77,
"column": 83
} | {
"line": 79,
"column": 0
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nγ : Type u_6\n_mα : MeasurableSpace α\n_mΩ : MeasurableSpace Ω\nμ : Measure α\nβ : Type u_10\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nf : Ω → β\ng : Ω → γ\n⊢ IndepFun f g 0 μ",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Measurab... | [] | by simp [IndepFun] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 213,
"column": 2
} | {
"line": 214,
"column": 75
} | {
"line": 215,
"column": 2
} | [
{
"pp": "α : 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\nγ : ι → Type u_9\nmβ : (i : ι) → MeasurableSpace (β i)\nmγ : (i : ι) → MeasurableSpace (γ i)\nf : (i : ι) → Ω → β i\nh : iIndepFun f κ μ\ng : (i : ι) → β i → γ i\nh... | [
"α : 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\nγ : ι → Type u_9\nmβ : (i : ι) → MeasurableSpace (β i)\nmγ : (i : ι) → MeasurableSpace (γ i)\nf : (i : ι) → Ω → β i\nh✝ : iIndepFun f κ μ\ng : (i : ι) → β i → γ i\nhf : ∀ (i : ... | have h : iIndepFun (fun i ↦ ((hg i).mk (g i)) ∘ f i) κ μ :=
iIndepFun.comp h (fun i ↦ (hg i).mk (g i)) fun i ↦ (hg i).measurable_mk | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 561,
"column": 32
} | {
"line": 561,
"column": 43
} | {
"line": 561,
"column": 44
} | [
{
"pp": "case inr\nα : 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 ... | [
"case inr\nα : 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 : ... | if_pos hi2, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Independence.Integration | {
"line": 203,
"column": 2
} | {
"line": 220,
"column": 15
} | {
"line": 222,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : ContinuousENorm E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : OpensMeasurableSpace E\ninst✝⁴ : NormedAddGroup F\ninst✝³ : MeasurableSpace F\ninst✝² : OpensMeasurableSpace F\nin... | [] | refine ⟨hX, ?_⟩
have I : (∫⁻ ω, ‖Y ω‖ₑ ∂μ) ≠ 0 := fun H ↦ by
have I : (fun ω => ‖Y ω‖ₑ : Ω → ℝ≥0∞) =ᵐ[μ] 0 := (lintegral_eq_zero_iff' hY.enorm).1 H
apply h'Y
filter_upwards [I] with ω hω
simpa using hω
refine hasFiniteIntegral_iff_enorm.mpr <| lt_top_iff_ne_top.2 fun H => ?_
have J : (‖X ·‖ₑ) ⟂ᵢ[μ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Integration | {
"line": 203,
"column": 2
} | {
"line": 220,
"column": 15
} | {
"line": 222,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : ContinuousENorm E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : OpensMeasurableSpace E\ninst✝⁴ : NormedAddGroup F\ninst✝³ : MeasurableSpace F\ninst✝² : OpensMeasurableSpace F\nin... | [] | refine ⟨hX, ?_⟩
have I : (∫⁻ ω, ‖Y ω‖ₑ ∂μ) ≠ 0 := fun H ↦ by
have I : (fun ω => ‖Y ω‖ₑ : Ω → ℝ≥0∞) =ᵐ[μ] 0 := (lintegral_eq_zero_iff' hY.enorm).1 H
apply h'Y
filter_upwards [I] with ω hω
simpa using hω
refine hasFiniteIntegral_iff_enorm.mpr <| lt_top_iff_ne_top.2 fun H => ?_
have J : (‖X ·‖ₑ) ⟂ᵢ[μ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.Covariance | {
"line": 58,
"column": 34
} | {
"line": 58,
"column": 53
} | {
"line": 58,
"column": 54
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX Y : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nhX : MemLp X 2 μ\nhY : MemLp Y 2 μ\n⊢ ∫ (a : Ω), X a * Y a ∂μ - (∫ (a : Ω), X a ∂μ) * ∫ (x : Ω), Y x ∂μ -\n (∫ (a : Ω), (∫ (a : Ω), X a ∂μ) * Y a ∂μ - (∫ (a : Ω), ∫ (a : Ω), X a ∂μ ∂μ) * ∫ (x : Ω), Y ... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX Y : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nhX : MemLp X 2 μ\nhY : MemLp Y 2 μ\n⊢ ∫ (a : Ω), X a * Y a ∂μ - (∫ (a : Ω), X a ∂μ) * ∫ (x : Ω), Y x ∂μ -\n ((∫ (a : Ω), X a ∂μ) * ∫ (x : Ω), Y x ∂μ - (∫ (a : Ω), ∫ (a : Ω), X a ∂μ ∂μ) * ∫ (x : Ω), Y x ∂μ) =\n ... | integral_const_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Moments.Variance | {
"line": 188,
"column": 6
} | {
"line": 188,
"column": 25
} | {
"line": 188,
"column": 26
} | [
{
"pp": "case e_f\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nc : ℝ\nX : Ω → ℝ\nμ : Measure Ω\nω : Ω\n⊢ ‖c * X ω - ∫ (x : Ω), c * X x ∂μ‖ₑ ^ 2 = ENNReal.ofReal (c ^ 2) * ‖X ω - ∫ (x : Ω), X x ∂μ‖ₑ ^ 2",
"ppTerm": "?e_f",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNorm... | [
"case e_f\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nc : ℝ\nX : Ω → ℝ\nμ : Measure Ω\nω : Ω\n⊢ ‖c * X ω - c * ∫ (a : Ω), X a ∂μ‖ₑ ^ 2 = ENNReal.ofReal (c ^ 2) * ‖X ω - ∫ (x : Ω), X x ∂μ‖ₑ ^ 2"
] | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Integration | {
"line": 252,
"column": 4
} | {
"line": 254,
"column": 15
} | {
"line": 255,
"column": 4
} | [
{
"pp": "case pos\nΩ : Type u_1\n𝕜 : Type u_2\ninst✝¹⁴ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\n𝓧 : Type u_3\n𝓨 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : MeasurableSpace 𝓧\ninst✝¹² : MeasurableSpace 𝓨\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : Norm... | [
"case pos\nΩ : Type u_1\n𝕜 : Type u_2\ninst✝¹⁴ : RCLike 𝕜\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\n𝓧 : Type u_3\n𝓨 : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\ninst✝¹³ : MeasurableSpace 𝓧\ninst✝¹² : MeasurableSpace 𝓨\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedSpace 𝕜 E... | have h1 : ∀ᵐ ω ∂μ, B (f (X ω)) (g (Y ω)) = 0 := by
filter_upwards [h] with ω hω
simp [hω] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Moments.Variance | {
"line": 237,
"column": 4
} | {
"line": 237,
"column": 79
} | {
"line": 238,
"column": 4
} | [
{
"pp": "case neg\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nhX : AEStronglyMeasurable X μ\nc : ℝ\nhX_Lp : ¬MemLp X 2 μ\n⊢ Var[fun ω ↦ X ω + c; μ] = Var[X; μ]",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",... | [
"case neg\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nhX : AEStronglyMeasurable X μ\nc : ℝ\nhX_Lp : ¬MemLp X 2 μ\n⊢ ¬MemLp (fun x ↦ X x + c) 2 μ"
] | rw [variance_of_not_memLp (hX.add_const _), variance_of_not_memLp hX hX_Lp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.IdentDistrib | {
"line": 137,
"column": 2
} | {
"line": 139,
"column": 50
} | {
"line": 141,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nf : α → γ\ng : β → γ\nh : IdentDistrib f g μ ν\np : γ → Prop\npmeas : MeasurableSet {x | p x}\nhp : ∀ᵐ (x : α) ∂μ, p (f x)\n⊢ ∀ᵐ (x : β) ∂ν, p (g x)... | [] | apply (ae_map_iff h.aemeasurable_snd pmeas).1
rw [← h.map_eq]
exact (ae_map_iff h.aemeasurable_fst pmeas).2 hp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.IdentDistrib | {
"line": 137,
"column": 2
} | {
"line": 139,
"column": 50
} | {
"line": 141,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nf : α → γ\ng : β → γ\nh : IdentDistrib f g μ ν\np : γ → Prop\npmeas : MeasurableSet {x | p x}\nhp : ∀ᵐ (x : α) ∂μ, p (f x)\n⊢ ∀ᵐ (x : β) ∂ν, p (g x)... | [] | apply (ae_map_iff h.aemeasurable_snd pmeas).1
rw [← h.map_eq]
exact (ae_map_iff h.aemeasurable_fst pmeas).2 hp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.Filtration | {
"line": 186,
"column": 6
} | {
"line": 187,
"column": 72
} | {
"line": 188,
"column": 4
} | [
{
"pp": "case pos\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : Preorder ι\nx✝ : Set (Filtration ι m)\nhn : x✝.Nonempty\n⊢ IsGLB x✝ { seq := fun i ↦ sInf ((fun f ↦ ↑f i) '' x✝), mono' := ⋯, le' := ⋯ }",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Set.image_image",
... | [] | refine .of_image (f := seq) .rfl ?_
simpa only [isGLB_pi, Set.image_image] using! fun _ ↦ isGLB_sInf _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Process.Filtration | {
"line": 186,
"column": 6
} | {
"line": 187,
"column": 72
} | {
"line": 188,
"column": 4
} | [
{
"pp": "case pos\nΩ : Type u_1\nι : Type u_2\nm : MeasurableSpace Ω\ninst✝ : Preorder ι\nx✝ : Set (Filtration ι m)\nhn : x✝.Nonempty\n⊢ IsGLB x✝ { seq := fun i ↦ sInf ((fun f ↦ ↑f i) '' x✝), mono' := ⋯, le' := ⋯ }",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Set.image_image",
... | [] | refine .of_image (f := seq) .rfl ?_
simpa only [isGLB_pi, Set.image_image] using! fun _ ↦ isGLB_sInf _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 497,
"column": 2
} | {
"line": 501,
"column": 89
} | {
"line": 502,
"column": 2
} | [
{
"pp": "α : 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\nhf_indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ni j k l : ι\nhik : i ≠ k\... | [
"α : 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\nhf_indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ni j k l : ι\nhik : i ≠ k\nhil : i ≠ l... | have h : IndepFun (fun a ↦ ((hf_meas i).mk (f i) a, (hf_meas j).mk (f j) a))
(fun a ↦ ((hf_meas k).mk (f k) a, (hf_meas l).mk (f l) a)) κ μ := by
refine iIndepFun.indepFun_prodMk_prodMk ?_ (fun i ↦ (hf_meas i).measurable_mk) _ _ _ _ hik hil
hjk hjl
exact iIndepFun.congr' hf_indep fun i ↦ Measure.ae_... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 507,
"column": 4
} | {
"line": 508,
"column": 65
} | {
"line": 509,
"column": 4
} | [
{
"pp": "case refine_2\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\nhf_indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ni j k l : ... | [
"α : 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\nhf_indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ni j k l : ι\nhik : i ≠ k\nhil : i ≠ l... | filter_upwards [Measure.ae_ae_of_ae_comp (hf_meas k).ae_eq_mk,
Measure.ae_ae_of_ae_comp (hf_meas l).ae_eq_mk] with a hk hl | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 648,
"column": 4
} | {
"line": 648,
"column": 92
} | {
"line": 649,
"column": 4
} | [
{
"pp": "α : 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 : Type u_8\nm : MeasurableSpace β\ninst✝¹ : CommMonoid β\ninst✝ : MeasurableMul₂ β\nf : ι → Ω → β\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ns ... | [
"α : 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 : Type u_8\nm : MeasurableSpace β\ninst✝¹ : CommMonoid β\ninst✝ : MeasurableMul₂ β\nf : ι → Ω → β\nhf_Indep : iIndepFun f κ μ\nhf_meas : ∀ (i : ι), AEMeasurable (f i) (⇑κ ∘ₘ μ)\ns : Finset ι\n... | refine iIndepFun.indepFun_finsetProd_of_notMem ?_ (fun i ↦ (hf_meas i).measurable_mk) hi | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 42
} | {
"line": 115,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n... | [
"case refine_1\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u, s n).indicator ... | refine Eventually.of_forall fun n ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 118,
"column": 6
} | {
"line": 118,
"column": 34
} | {
"line": 119,
"column": 2
} | [
{
"pp": "case refine_1.inr\nα : Type u_2\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nr : ℝ≥0∞\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhr₀ : r ≠ 0\nhr : ∀ (n : ℕ), r ≤ μ (s n)\nM : (α → ℝ) → Set α := fun f ↦ {x | eLpNormEssSup f μ < ↑‖f x‖₊}\nN : Set α := ⋃ u, M ((⋂ n ∈ u,... | [] | exact lintegral_mono (hf₁ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 9
} | {
"line": 56,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nz w : ℂ\nf : ℂ → E\nx✝ : ℝ\n⊢ ((2 * ↑π * I)⁻¹ * deriv (circleMap z R) x✝ * ((circleMap z R x✝ - w) ^ 2)⁻¹) • f (circleMap z R x✝) =\n ((circleMap z R x✝ - w)⁻¹ * (2 * ↑π * I)⁻¹ * deriv (circleMap z R) x✝ * (circleMap z R x✝... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nz w : ℂ\nf : ℂ → E\nx✝ : ℝ\n⊢ ((↑π)⁻¹ * I⁻¹ * deriv (circleMap z R) x✝ * (-(circleMap z R x✝ * w * 2) + circleMap z R x✝ ^ 2 + w ^ 2)⁻¹ * (1 / 2)) •\n f (circleMap z R x✝) =\n ((↑π)⁻¹ * I⁻¹ * deriv (circleMap z R) x✝ * (circleMap ... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.MeasureTheory.Integral.CircleTransform | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 53
} | {
"line": 96,
"column": 0
} | [
{
"pp": "case h₀\nR r : ℝ\nhr : r < R\nz a : ℂ\nb : ℝ\nha : (a, b).1 ∈ closedBall z r\nha2 : a ∈ ball z R\n⊢ circleMap z R (a, b).2 - (a, b).1 ≠ 0",
"ppTerm": "?h₀",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real",
"AddGroupWithOne.toAdd... | [] | exact sub_ne_zero.2 (circleMap_ne_mem_ball ha2 b) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 314,
"column": 13
} | {
"line": 314,
"column": 46
} | {
"line": 316,
"column": 0
} | [
{
"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 simp only [ENNReal.add_thirds] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 32,
"column": 40
} | {
"line": 32,
"column": 73
} | {
"line": 32,
"column": 74
} | [
{
"pp": "p q : ℝ\nhp : 0 < p\nhq : -1 < q\nx✝ : ℝ\nhx : x✝ ∈ Ioi 0\n⊢ (1 / p * x✝ ^ (1 / p - 1)) • ((x✝ ^ (1 / p)) ^ q * rexp (-x✝ ^ (1 / p * p))) =\n (1 / p * x✝ ^ (1 / p - 1)) • ((x✝ ^ (1 / p)) ^ q * rexp (-x✝))",
"ppTerm": "?m.270",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"G... | [
"p q : ℝ\nhp : 0 < p\nhq : -1 < q\nx✝ : ℝ\nhx : x✝ ∈ Ioi 0\n⊢ (1 / p * x✝ ^ (1 / p - 1)) • ((x✝ ^ (1 / p)) ^ q * rexp (-x✝ ^ 1)) =\n (1 / p * x✝ ^ (1 / p - 1)) • ((x✝ ^ (1 / p)) ^ q * rexp (-x✝))"
] | one_div_mul_cancel (ne_of_gt hp), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 37,
"column": 6
} | {
"line": 37,
"column": 13
} | {
"line": 38,
"column": 4
} | [
{
"pp": "p q : ℝ\nhp : 0 < p\nhq : -1 < q\nx✝ : ℝ\nhx : x✝ ∈ Ioi 0\n⊢ x✝ ^ (1 / p - 1) * (x✝ ^ (1 / p * q) * rexp (-x✝)) / p = 1 / p * (rexp (-x✝) * (x✝ ^ (1 / p - 1) * x✝ ^ (q / p)))",
"ppTerm": "?m.324",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Ma... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 51,
"column": 28
} | {
"line": 51,
"column": 35
} | {
"line": 51,
"column": 35
} | [
{
"pp": "p q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\nx✝ : ℝ\nhx : x✝ ∈ Ioi 0\n⊢ x✝ ^ q * rexp (-b * x✝ ^ p) = x✝ ^ q * rexp (-(b * x✝ ^ p))",
"ppTerm": "?m.351",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Semigroup.toMul",
"Real",
"NonUnitalComm... | [
"p q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\nx✝ : ℝ\nhx : x✝ ∈ Ioi 0\n⊢ x✝ ^ q * rexp (-(b * x✝ ^ p)) = x✝ ^ q * rexp (-(b * x✝ ^ p))",
"case hx\np q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\nx✝ : ℝ\nhx : x✝ ∈ Ioi 0\n⊢ 0 < b",
"p q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\nx✝ : ℝ\nhx : x✝ ∈ Ioi 0\n⊢ p ≠... | neg_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 58,
"column": 10
} | {
"line": 58,
"column": 29
} | {
"line": 58,
"column": 30
} | [
{
"pp": "p q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\n⊢ (b ^ p⁻¹)⁻¹ * ∫ (x : ℝ) in Ioi 0, b ^ (-p⁻¹ * q) * (x ^ q * rexp (-x ^ p)) =\n b ^ (-(q + 1) / p) * (1 / p) * Gamma ((q + 1) / p)",
"ppTerm": "?m.261",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormed... | [
"p q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\n⊢ (b ^ p⁻¹)⁻¹ * (b ^ (-p⁻¹ * q) * ∫ (a : ℝ) in Ioi 0, a ^ q * rexp (-a ^ p)) =\n b ^ (-(q + 1) / p) * (1 / p) * Gamma ((q + 1) / p)"
] | integral_const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Basic | {
"line": 298,
"column": 23
} | {
"line": 298,
"column": 47
} | {
"line": 298,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\ninst✝ : NormedSpace ℝ E\n⊢ IntervalIntegrable (curveIntegralFun ω (Path.segment a b)) volume 0 ... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\na b : E\nω : E → E →L[𝕜] F\ninst✝ : NormedSpace ℝ E\n⊢ IntervalIntegrable ?m.85 volume 0 1 ↔ IntervalIntegrable (fun t ↦ (ω ((lineMap a... | intervalIntegrable_congr | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 47
} | {
"line": 73,
"column": 4
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na b c d : E\nγ₁ : Path a b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable\n... | [
"E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na b c d : E\nγ₁ : Path a b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable\nhφt : ∀ a_1 ... | rw [hU, ← interior_Icc, ← interior_prod_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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