module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 304,
"column": 34
} | {
"line": 304,
"column": 37
} | {
"line": 304,
"column": 38
} | [
{
"pp": "case add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx✝ y✝ : CliffordAlgebra Q\nhx : (changeForm ⋯) x✝ = x✝\nhy : (changeForm ⋯) y✝ = y✝\n⊢ (changeForm ⋯) x✝ + (changeForm ⋯) y✝ = x✝ + y✝",
"ppTerm": "?add",
"assigned": tru... | [
"case add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx✝ y✝ : CliffordAlgebra Q\nhx : (changeForm ⋯) x✝ = x✝\nhy : (changeForm ⋯) y✝ = y✝\n⊢ x✝ + (changeForm ⋯) y✝ = x✝ + y✝"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 305,
"column": 42
} | {
"line": 305,
"column": 45
} | {
"line": 305,
"column": 46
} | [
{
"pp": "case ι_mul\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx : (changeForm ⋯) x✝ = x✝\n⊢ (ι Q) m✝ * (changeForm ⋯) x✝ - (contractLeft (0 m✝)) ((changeForm ⋯) x✝) = (ι Q) m✝ * x✝",
"ppTerm": "?ι_mul"... | [
"case ι_mul\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx : (changeForm ⋯) x✝ = x✝\n⊢ (ι Q) m✝ * x✝ - (contractLeft (0 m✝)) x✝ = (ι Q) m✝ * x✝"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 318,
"column": 52
} | {
"line": 318,
"column": 55
} | {
"line": 318,
"column": 56
} | [
{
"pp": "case add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\nh' : BilinMap.toQuadraticMap B' = Q'' - Q'\nx✝ y✝ : CliffordAlgebra Q\nhx : (changeForm h') ((changeForm h) x... | [
"case add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\nh' : BilinMap.toQuadraticMap B' = Q'' - Q'\nx✝ y✝ : CliffordAlgebra Q\nhx : (changeForm h') ((changeForm h) x✝) = (change... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 319,
"column": 87
} | {
"line": 319,
"column": 90
} | {
"line": 319,
"column": 91
} | [
{
"pp": "case ι_mul\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\nh' : BilinMap.toQuadraticMap B' = Q'' - Q'\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx : (changeForm h') ((changeFo... | [
"case ι_mul\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\nh' : BilinMap.toQuadraticMap B' = Q'' - Q'\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx : (changeForm h') ((changeForm h) x✝) = ... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 320,
"column": 82
} | {
"line": 320,
"column": 85
} | {
"line": 321,
"column": 6
} | [
{
"pp": "case ι_mul\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\nh' : BilinMap.toQuadraticMap B' = Q'' - Q'\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx : (changeForm h') ((changeFo... | [
"case ι_mul\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\nh' : BilinMap.toQuadraticMap B' = Q'' - Q'\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx : (changeForm h') ((changeForm h) x✝) = ... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | {
"line": 213,
"column": 6
} | {
"line": 213,
"column": 23
} | {
"line": 213,
"column": 24
} | [
{
"pp": "case e'_3\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ : Grade... | [
"case e'_3\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ : GradedAlgebra ℬ\n... | SetLike.coe_gOne, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 23
} | {
"line": 218,
"column": 24
} | [
{
"pp": "case e'_3\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ : Grade... | [
"case e'_3\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ : GradedAlgebra ℬ\n... | SetLike.coe_gOne, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | {
"line": 225,
"column": 8
} | {
"line": 225,
"column": 25
} | {
"line": 225,
"column": 26
} | [
{
"pp": "case e'_3.e'_16\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ :... | [
"case e'_3.e'_16\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ : GradedAlgeb... | SetLike.coe_gOne, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | {
"line": 226,
"column": 8
} | {
"line": 226,
"column": 25
} | {
"line": 226,
"column": 26
} | [
{
"pp": "case e'_3.e'_17\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ :... | [
"case e'_3.e'_17\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ : GradedAlgeb... | SetLike.coe_gOne, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Graded.Internal | {
"line": 242,
"column": 6
} | {
"line": 242,
"column": 23
} | {
"line": 242,
"column": 24
} | [
{
"pp": "case e_f\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ : Graded... | [
"case e_f\nR : Type u_1\nι : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring ι\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : CommRing R\ninst✝⁶ : Ring A\ninst✝⁵ : Ring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\n𝒜 : ι → Submodule R A\nℬ : ι → Submodule R B\ninst✝² : GradedAlgebra 𝒜\ninst✝¹ : GradedAlgebra ℬ\ni... | SetLike.coe_gOne, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FreeProduct.Basic | {
"line": 217,
"column": 6
} | {
"line": 217,
"column": 17
} | {
"line": 217,
"column": 18
} | [
{
"pp": "I : Type u\ninst✝⁵ : DecidableEq I\nR : Type v\ninst✝⁴ : CommSemiring R\nA : I → Type w\ninst✝³ : (i : I) → Semiring (A i)\ninst✝² : (i : I) → Algebra R (A i)\nB : Type w'\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nmaps : {i : I} → A i →ₐ[R] B\nr : R\n⊢ ((lift R A) fun {i} ↦ maps) ((algebraMap R (FreeP... | [
"I : Type u\ninst✝⁵ : DecidableEq I\nR : Type v\ninst✝⁴ : CommSemiring R\nA : I → Type w\ninst✝³ : (i : I) → Semiring (A i)\ninst✝² : (i : I) → Algebra R (A i)\nB : Type w'\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nmaps : {i : I} → A i →ₐ[R] B\nr : R\n⊢ ((ringCon R A).liftₐ ((TensorAlgebra.lift R) (toModule R I B ... | lift_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Charpoly.FiniteField | {
"line": 36,
"column": 12
} | {
"line": 36,
"column": 46
} | {
"line": 36,
"column": 46
} | [
{
"pp": "case inl\nn : Type u_1\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Fintype K\nM : Matrix n n K\nh✝ : Nonempty n\np : ℕ\nhp✝ : CharP K p\nk : ℕ\nkpos : 0 < k\nhp : Nat.Prime p\nhk : Fintype.card K = p ^ k\nthis : Fact (Nat.Prime p)\n⊢ (⇑(frobenius K[X] p))^[k] (M... | [
"case inl\nn : Type u_1\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Fintype K\nM : Matrix n n K\nh✝ : Nonempty n\np : ℕ\nhp✝ : CharP K p\nk : ℕ\nkpos : 0 < k\nhp : Nat.Prime p\nhk : Fintype.card K = p ^ k\nthis : Fact (Nat.Prime p)\n⊢ (M ^ p ^ k).charpoly ^ p ^ k = (⇑(froben... | rw [iterate_frobenius (R := K[X])] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Charpoly.FiniteField | {
"line": 36,
"column": 12
} | {
"line": 36,
"column": 46
} | {
"line": 36,
"column": 46
} | [
{
"pp": "case inl\nn : Type u_1\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Fintype K\nM : Matrix n n K\nh✝ : Nonempty n\np : ℕ\nhp✝ : CharP K p\nk : ℕ\nkpos : 0 < k\nhp : Nat.Prime p\nhk : Fintype.card K = p ^ k\nthis : Fact (Nat.Prime p)\n⊢ (M ^ Fintype.card K).charpol... | [
"case inl\nn : Type u_1\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Fintype K\nM : Matrix n n K\nh✝ : Nonempty n\np : ℕ\nhp✝ : CharP K p\nk : ℕ\nkpos : 0 < k\nhp : Nat.Prime p\nhk : Fintype.card K = p ^ k\nthis : Fact (Nat.Prime p)\n⊢ (M ^ Fintype.card K).charpoly ^ Fintype.... | rw [iterate_frobenius (R := K[X])] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Charpoly.FiniteField | {
"line": 36,
"column": 12
} | {
"line": 36,
"column": 46
} | {
"line": 36,
"column": 46
} | [
{
"pp": "case inl\nn : Type u_1\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Fintype K\nM : Matrix n n K\nh✝ : Nonempty n\np : ℕ\nhp✝ : CharP K p\nk : ℕ\nkpos : 0 < k\nhp : Nat.Prime p\nhk : Fintype.card K = p ^ k\nthis : Fact (Nat.Prime p)\n⊢ (M ^ Fintype.card K).charpol... | [
"case inl\nn : Type u_1\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Fintype K\nM : Matrix n n K\nh✝ : Nonempty n\np : ℕ\nhp✝ : CharP K p\nk : ℕ\nkpos : 0 < k\nhp : Nat.Prime p\nhk : Fintype.card K = p ^ k\nthis : Fact (Nat.Prime p)\n⊢ (M ^ Fintype.card K).charpoly ^ Fintype.... | rw [iterate_frobenius (R := K[X])] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Gershgorin | {
"line": 49,
"column": 49
} | {
"line": 49,
"column": 64
} | {
"line": 50,
"column": 6
} | [
{
"pp": "K : Type u_1\nn : Type u_2\ninst✝² : NormedField K\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n K\nμ : K\nhμ : Module.End.HasEigenvalue (Matrix.toLin' A) μ\nh✝ : Nonempty n\nv : n → K\nh_eg : v ∈ (Module.End.genEigenspace (Matrix.toLin' A) μ) 1\nh_nz✝ : v ≠ 0\ni : n\nh_i : (Finset.univ.su... | [] | by congr; field | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Multilinear.Pi | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 47
} | {
"line": 53,
"column": 2
} | [
{
"pp": "ι : Type uι\nκ : ι → Type uκ\nR : Type uR\nM : (i : ι) → κ i → Type uM\nN : Type uN\ninst✝⁷ : Semiring R\ninst✝⁶ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : (i : ι) → (k : κ i) → Module R (M i k)\ninst✝³ : Module R N\ninst✝² : Finite ι\ninst✝¹ : ∀ (i : ι), Finite (... | [
"ι : Type uι\nκ : ι → Type uκ\nR : Type uR\nM : (i : ι) → κ i → Type uM\nN : Type uN\ninst✝⁷ : Semiring R\ninst✝⁶ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : (i : ι) → (k : κ i) → Module R (M i k)\ninst✝³ : Module R N\ninst✝² : Finite ι\ninst✝¹ : ∀ (i : ι), Finite (κ i)\ninst✝ ... | have (i : _) := (nonempty_fintype (κ i)).some | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.PiTensorProduct | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 18
} | {
"line": 186,
"column": 19
} | [
{
"pp": "ι : Type u_1\nR' : Type u_2\nR : Type u_3\nA : ι → Type u_4\ninst✝⁷ : CommSemiring R'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : ι) → Semiring (A i)\ninst✝⁴ : Algebra R' R\ninst✝³ : (i : ι) → Algebra R (A i)\ninst✝² : (i : ι) → Algebra R' (A i)\ninst✝¹ : ∀ (i : ι), IsScalarTower R' R (A i)\nr : R'\ni : ι\... | [
"ι : Type u_1\nR' : Type u_2\nR : Type u_3\nA : ι → Type u_4\ninst✝⁷ : CommSemiring R'\ninst✝⁶ : CommSemiring R\ninst✝⁵ : (i : ι) → Semiring (A i)\ninst✝⁴ : Algebra R' R\ninst✝³ : (i : ι) → Algebra R (A i)\ninst✝² : (i : ι) → Algebra R' (A i)\ninst✝¹ : ∀ (i : ι), IsScalarTower R' R (A i)\nr : R'\ni : ι\ninst✝ : Dec... | smul_one_smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Projectivization.Cardinality | {
"line": 80,
"column": 2
} | {
"line": 85,
"column": 88
} | {
"line": 87,
"column": 0
} | [
{
"pp": "case inl.inl\nk : Type u_1\nV : Type u_2\ninst✝² : DivisionRing k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\na✝ : Nontrivial V\nh✝ : Finite k\nh : Finite V\n⊢ Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1)",
"ppTerm": "?inl.inl",
"assigned": true,
"usedConstants": [
"Fintype.... | [] | haveI : Fintype V := Fintype.ofFinite V
haveI : Fintype (ℙ k V) := Fintype.ofFinite (ℙ k V)
haveI : Fintype k := Fintype.ofFinite k
have hV : Fintype.card { v : V // v ≠ 0 } = Fintype.card V - 1 := by simp
simp_rw [← Fintype.card_eq_nat_card, ← Fintype.card_units (α := k), ← hV]
rw [Fintype.card_congr (nonZer... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Projectivization.Cardinality | {
"line": 80,
"column": 2
} | {
"line": 85,
"column": 88
} | {
"line": 87,
"column": 0
} | [
{
"pp": "case inl.inl\nk : Type u_1\nV : Type u_2\ninst✝² : DivisionRing k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\na✝ : Nontrivial V\nh✝ : Finite k\nh : Finite V\n⊢ Nat.card V - 1 = Nat.card (ℙ k V) * (Nat.card k - 1)",
"ppTerm": "?inl.inl",
"assigned": true,
"usedConstants": [
"Fintype.... | [] | haveI : Fintype V := Fintype.ofFinite V
haveI : Fintype (ℙ k V) := Fintype.ofFinite (ℙ k V)
haveI : Fintype k := Fintype.ofFinite k
have hV : Fintype.card { v : V // v ≠ 0 } = Fintype.card V - 1 := by simp
simp_rw [← Fintype.card_eq_nat_card, ← Fintype.card_units (α := k), ← hV]
rw [Fintype.card_congr (nonZer... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Projectivization.Collinear | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 84
} | {
"line": 89,
"column": 2
} | [
{
"pp": "case h.h\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nhv : v ≠ 0\nw : V\nhw : w ≠ 0\nh : ¬mk K v hv = mk K w hw\n⊢ IsCollinear {mk K v hv, mk K w hw}",
"ppTerm": "?h.h",
"assigned": true,
"usedConstants": [
"Projectivization... | [
"case h.h\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nhv : v ≠ 0\nw : V\nhw : w ≠ 0\nh : LinearIndependent K ![v, w]\n⊢ IsCollinear {mk K v hv, mk K w hw}"
] | rw [← ne_eq, ← independent_pair_iff_ne, independent_mk_iff_LinearIndependent] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 553,
"column": 4
} | {
"line": 579,
"column": 8
} | {
"line": 581,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_4\ninst✝⁶ : CommRing R\nV : Type u_5\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : Module.Free R V\ninst✝² : Module.Finite R V\nι : Type u_6\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Module.Basis ι R V\ng : ↥(center (Matrix.SpecialLinearGroup ι R))\na✝ : Nontrivial ... | [] | have hι : ¬ IsEmpty ι := fun hι ↦ hV (by
rw [← Module.finrank_eq_zero_iff_of_free (R := R),
Module.finrank_eq_card_basis b, Fintype.card_of_isEmpty])
rw [not_subsingleton_iff_nontrivial] at hV
have := Module.Free.instFaithfulSMulOfNontrivial R V
suffices (((((Subgroup.centerCongr (Matrix.Speci... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 553,
"column": 4
} | {
"line": 579,
"column": 8
} | {
"line": 581,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_4\ninst✝⁶ : CommRing R\nV : Type u_5\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : Module.Free R V\ninst✝² : Module.Finite R V\nι : Type u_6\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nb : Module.Basis ι R V\ng : ↥(center (Matrix.SpecialLinearGroup ι R))\na✝ : Nontrivial ... | [] | have hι : ¬ IsEmpty ι := fun hι ↦ hV (by
rw [← Module.finrank_eq_zero_iff_of_free (R := R),
Module.finrank_eq_card_basis b, Fintype.card_of_isEmpty])
rw [not_subsingleton_iff_nontrivial] at hV
have := Module.Free.instFaithfulSMulOfNontrivial R V
suffices (((((Subgroup.centerCongr (Matrix.Speci... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 237,
"column": 6
} | {
"line": 237,
"column": 18
} | {
"line": 237,
"column": 19
} | [
{
"pp": "M : Type u_2\ninst✝⁵ : AddCommGroup M\n𝕜 : Type u_4\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : Module 𝕜 M\nQ : QuadraticForm 𝕜 M\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : FiniteDimensional 𝕜 M\nthis : Invertible 2\nw : Fin (Module.finrank 𝕜 M) → 𝕜\ne : Equivalent Q (weightedSumSquares ... | [
"M : Type u_2\ninst✝⁵ : AddCommGroup M\n𝕜 : Type u_4\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : Module 𝕜 M\nQ : QuadraticForm 𝕜 M\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : FiniteDimensional 𝕜 M\nthis : Invertible 2\nw : Fin (Module.finrank 𝕜 M) → 𝕜\ne : Equivalent Q (weightedSumSquares 𝕜 w)\n⊢ sig... | e.sigPos_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 274,
"column": 4
} | {
"line": 274,
"column": 72
} | {
"line": 275,
"column": 4
} | [
{
"pp": "case inr.inr\nι : 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 : IsDomain R\ninst✝¹ : Cha... | [
"case inr.inr\nι : 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 : IsDomain R\ninst✝¹ : CharZero R\nins... | rw [Finset.sum_eq_single_of_mem (-k) (Finset.mem_univ _) (by aesop)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 287,
"column": 66
} | {
"line": 287,
"column": 92
} | {
"line": 288,
"column": 22
} | [
{
"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 : IsDomain R\ninst✝¹ : CharZero R\ninst✝... | [] | rw [Ring.lie_def, sub_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 287,
"column": 66
} | {
"line": 287,
"column": 92
} | {
"line": 288,
"column": 22
} | [
{
"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 : IsDomain R\ninst✝¹ : CharZero R\ninst✝... | [] | rw [Ring.lie_def, sub_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 287,
"column": 66
} | {
"line": 287,
"column": 92
} | {
"line": 288,
"column": 22
} | [
{
"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 : IsDomain R\ninst✝¹ : CharZero R\ninst✝... | [] | rw [Ring.lie_def, sub_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 173,
"column": 10
} | {
"line": 173,
"column": 91
} | {
"line": 174,
"column": 8
} | [
{
"pp": "ι : 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.Base\ninst... | [] | simpa [neg_eq_iff_add_eq_zero, ← add_assoc, add_eq_zero_iff_eq_neg'] using contra | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 304,
"column": 6
} | {
"line": 304,
"column": 71
} | {
"line": 305,
"column": 4
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹¹ : Finite ι\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\nb : P... | [] | simp [Finset.sum_ite_of_false aux₃, Finset.sum_ite_of_false aux₄] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 315,
"column": 6
} | {
"line": 315,
"column": 71
} | {
"line": 316,
"column": 4
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹¹ : Finite ι\ninst✝¹⁰ : CommRing R\ninst✝⁹ : IsDomain R\ninst✝⁸ : CharZero R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.IsCrystallographic\nb : P... | [] | simp [Finset.sum_ite_of_false aux₃, Finset.sum_ite_of_false aux₄] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Logic.Hydra | {
"line": 67,
"column": 2
} | {
"line": 75,
"column": 33
} | {
"line": 77,
"column": 0
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ninst✝¹ : DecidableEq α\ninst✝ : Std.Irrefl r\ns t u : Multiset α\na : α\nhe : s + {a} = t + u\nhr : ∀ (a' : α), ¬r a' a → a' ∉ u\n⊢ InvImage (Finsupp.Lex (rᶜ ⊓ fun x1 x2 ↦ x1 ≠ x2) fun x1 x2 ↦ x1 < x2) (⇑toFinsupp) s t",
"ppTerm": "?m.60",
"assigned": true,
"... | [] | classical
refine ⟨a, fun b h ↦ ?_, ?_⟩ <;> simp_rw [toFinsupp_apply]
· apply_fun count b at he
simpa only [count_add, count_singleton, if_neg h.2, add_zero, count_eq_zero.2 (hr b h.1)]
using he
· apply_fun count a at he
simp only [count_add, count_singleton_self, count_eq_zero.2 (hr _ (irrefl_of r a... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Logic.Hydra | {
"line": 105,
"column": 2
} | {
"line": 106,
"column": 67
} | {
"line": 108,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nr : α → α → Prop\ninst✝¹ : DecidableEq α\ninst✝ : Std.Irrefl r\ns' s t : Multiset α\na : α\n⊢ (∀ a' ∈ t, r a' a) ∧ a ∈ s ∧ s' = s.erase a + t → (∀ a' ∈ t, r a' a) ∧ a ∈ s + t ∧ s' = (s + t).erase a",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
... | [] | · rintro ⟨ht, h, rfl⟩
exact ⟨ht, mem_add.2 (Or.inl h), (erase_add_left_pos t h).symm⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Logic.Hydra | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 21
} | {
"line": 170,
"column": 22
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ninst✝ : Std.Irrefl r\na : α\nhacc : Acc r a\n⊢ Acc (CutExpand r) {a}",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Relation.CutExpand",
"Multiset",
"Multiset.instSingleton",
"Acc",
"Acc.rec",
"Singleton.singleto... | [
"case intro\nα : Type u_1\nr : α → α → Prop\ninst✝ : Std.Irrefl r\na✝ a : α\nh : ∀ (y : α), r y a → Acc r y\nih : ∀ (y : α), r y a → Acc (CutExpand r) {y}\n⊢ Acc (CutExpand r) {a}"
] | induction hacc with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 310,
"column": 68
} | {
"line": 310,
"column": 85
} | {
"line": 311,
"column": 6
} | [
{
"pp": "ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : CharZero K\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : Fintype ι\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module K M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module K N\nP : RootPairing ι K M N\ninst✝² : P.IsCrystallographic\nb : P.Base\nins... | [
"ι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : CharZero K\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : Fintype ι\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module K M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module K N\nP : RootPairing ι K M N\ninst✝² : P.IsCrystallographic\nb : P.Base\ninst✝¹ : P.IsRe... | range_subset_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 125,
"column": 6
} | {
"line": 125,
"column": 46
} | {
"line": 126,
"column": 6
} | [
{
"pp": "case pos\nι : Type u_2\nα : ι → Type u_1\nm : (i : ι) → MeasurableSpace (α i)\ni : ι\nS : Set ((x : ι) → α x)\nt : Set (α i)\nht : MeasurableSet t\nh : eval i ⁻¹' t = S\nj : ι\nhji : j = i\n⊢ MeasurableSet ((fun j ↦ if hji : j = i then ⋯.mpr t else univ) j)",
"ppTerm": "?pos✝",
"assigned": true... | [
"case pos\nι : Type u_2\nα : ι → Type u_1\nm : (i : ι) → MeasurableSpace (α i)\ni : ι\nS : Set ((x : ι) → α x)\nt : Set (α i)\nht : MeasurableSet t\nh : eval i ⁻¹' t = S\nj : ι\nhji : j = i\n⊢ MeasurableSet (cast ⋯ t)"
] | simp only [hji, eq_mpr_eq_cast, dif_pos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 144,
"column": 2
} | {
"line": 151,
"column": 22
} | {
"line": 153,
"column": 0
} | [
{
"pp": "case a\nι : Type u_2\nα : ι → Type u_1\ninst✝ : (i : ι) → MeasurableSpace (α i)\n⊢ MeasurableSpace.pi ≤ generateFrom (squareCylinders fun i ↦ {s | MeasurableSet s})",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.coe_singleton",
"MeasurableSpace.c... | [] | · refine iSup_le fun i ↦ ?_
refine (comap_eval_le_generateFrom_squareCylinders_singleton α i).trans ?_
refine MeasurableSpace.generateFrom_mono ?_
rw [← Finset.coe_singleton, squareCylinders_eq_iUnion_image]
exact subset_iUnion
(fun (s : Finset ι) ↦
(fun t : ∀ i, Set (α i) ↦ (s : Set ι).pi... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Constructions.Projective | {
"line": 136,
"column": 2
} | {
"line": 138,
"column": 26
} | {
"line": 140,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → MeasurableSpace (α i)\nP : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)\nμ : Measure ((i : ι) → α i)\ninst✝ : ∀ (i : Finset ι), IsFiniteMeasure (P i)\nhμ : IsProjectiveLimit μ P\n⊢ IsFiniteMeasure μ",
"ppTerm": "?m.16",
"assigned": true,
"use... | [] | constructor
rw [hμ.measure_univ_eq (∅ : Finset ι)]
exact measure_lt_top _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.Projective | {
"line": 136,
"column": 2
} | {
"line": 138,
"column": 26
} | {
"line": 140,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → MeasurableSpace (α i)\nP : (J : Finset ι) → Measure ((j : ↥J) → α ↑j)\nμ : Measure ((i : ι) → α i)\ninst✝ : ∀ (i : Finset ι), IsFiniteMeasure (P i)\nhμ : IsProjectiveLimit μ P\n⊢ IsFiniteMeasure μ",
"ppTerm": "?m.16",
"assigned": true,
"use... | [] | constructor
rw [hμ.measure_univ_eq (∅ : Finset ι)]
exact measure_lt_top _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 415,
"column": 88
} | {
"line": 417,
"column": 63
} | {
"line": 419,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nX : ι → Type u_3\nmα : MeasurableSpace α\nm : (i : ι) → MeasurableSpace (X i)\nΔ : Set ι\ng : α → (i : ι) → X i\n⊢ Measurable g ↔ ∀ ⦃i : ι⦄, i ∈ Δ → Measurable fun a ↦ g a i",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Measurab... | [] | by
simp_rw [measurable_iff_comap_le, cylinderEvents, MeasurableSpace.comap_iSup,
MeasurableSpace.comap_comp, Function.comp_def, iSup_le_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.WithTop | {
"line": 87,
"column": 6
} | {
"line": 87,
"column": 28
} | {
"line": 88,
"column": 4
} | [
{
"pp": "case a.inl.top\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι := ⋯\nd... | [] | exact ⟨⊤, by simp [d]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.SetSemiring | {
"line": 279,
"column": 17
} | {
"line": 279,
"column": 35
} | {
"line": 279,
"column": 36
} | [
{
"pp": "case insert.refine_2\nα : Type u_1\nC : Set (Set α)\ns : Set α\nI : Finset (Set α)\nhC : IsSetSemiring C\nhs : s ∈ C\nt : Set α\nI' : Finset (Set α)\na✝ : t ∉ I'\nh : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \\ ⋃₀ ↑I' = ⋃₀ ↑J\nhI : insert t ↑I' ⊆ C\nht : t ∈ C\nJ : Finset (Set α)\nh_ss : ↑J... | [
"case insert.refine_2\nα : Type u_1\nC : Set (Set α)\ns : Set α\nI : Finset (Set α)\nhC : IsSetSemiring C\nhs : s ∈ C\nt : Set α\nI' : Finset (Set α)\na✝ : t ∉ I'\nh : ↑I' ⊆ C → ∃ J, ↑J ⊆ C ∧ (↑J).PairwiseDisjoint id ∧ s \\ ⋃₀ ↑I' = ⋃₀ ↑J\nhI : insert t ↑I' ⊆ C\nht : t ∈ C\nJ : Finset (Set α)\nh_ss : ↑J ⊆ C\nh_dis ... | sUnion_eq_biUnion, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.SetSemiring | {
"line": 334,
"column": 6
} | {
"line": 334,
"column": 53
} | {
"line": 334,
"column": 53
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\ns : Set α\nI : Finset (Set α)\nhC : IsSetSemiring C\nhs : s ∈ C\nhI : ↑I ⊆ C\n⊢ ⋃₀ ↑(hC.disjointOfDiffUnion hs hI) ⊆ s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Set.sUnion",
"id",
... | [
"α : Type u_1\nC : Set (Set α)\ns : Set α\nI : Finset (Set α)\nhC : IsSetSemiring C\nhs : s ∈ C\nhI : ↑I ⊆ C\n⊢ s \\ ⋃₀ ↑I ⊆ s"
] | ← hC.sdiff_sUnion_eq_sUnion_disjointOfDiffUnion | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.SetSemiring | {
"line": 353,
"column": 6
} | {
"line": 353,
"column": 53
} | {
"line": 353,
"column": 53
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\ns : Set α\nI : Finset (Set α)\nhC : IsSetSemiring C\nhs : s ∈ C\nhI : ↑I ⊆ C\n⊢ Disjoint (⋃₀ ↑I) (⋃₀ ↑(hC.disjointOfDiffUnion hs hI))",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CompleteBooleanAlgebra.toCompleteDistribLatti... | [
"α : Type u_1\nC : Set (Set α)\ns : Set α\nI : Finset (Set α)\nhC : IsSetSemiring C\nhs : s ∈ C\nhI : ↑I ⊆ C\n⊢ Disjoint (⋃₀ ↑I) (s \\ ⋃₀ ↑I)"
] | ← hC.sdiff_sUnion_eq_sUnion_disjointOfDiffUnion | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.WithTop | {
"line": 145,
"column": 6
} | {
"line": 145,
"column": 28
} | {
"line": 146,
"column": 4
} | [
{
"pp": "case a.inr.top\nι : Type u_1\ninst✝¹ : Preorder ι\nts : TopologicalSpace ι\nht : OrderTopology ι\ninst✝ : SecondCountableTopology ι\nx₀ : ι\nc : Set ι\nc_count : c.Countable\nhc : ts = generateFrom {s | ∃ a ∈ c, s = Ioi a ∨ s = Iio a}\nc' : Set ι\nc'_count : c'.Countable\nhc' : Dense c'\nx₁ : ι := ⋯\nd... | [] | exact ⟨⊤, by simp [d]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.SetSemiring | {
"line": 455,
"column": 8
} | {
"line": 461,
"column": 58
} | {
"line": 462,
"column": 4
} | [
{
"pp": "case h.refine_3.refine_3\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 (... | [] | simp only [mem_coe, mem_iUnion, exists_prop, ne_eq, id_eq, forall_exists_index, and_imp]
intro i hi j x hx h3 h4
obtain ki : i ⊆ s \ ⋃₀ J := hC.subset_of_diffUnion_disjointOfDiffUnion h1.1 h1.2 _
(hK1s ▸ hi)
obtain hx2 : j ⊆ x := subset_trans (subset_sUnion_of_mem (ht1' x hx ▸ h3)) (hK... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.SetSemiring | {
"line": 455,
"column": 8
} | {
"line": 461,
"column": 58
} | {
"line": 462,
"column": 4
} | [
{
"pp": "case h.refine_3.refine_3\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 (... | [] | simp only [mem_coe, mem_iUnion, exists_prop, ne_eq, id_eq, forall_exists_index, and_imp]
intro i hi j x hx h3 h4
obtain ki : i ⊆ s \ ⋃₀ J := hC.subset_of_diffUnion_disjointOfDiffUnion h1.1 h1.2 _
(hK1s ▸ hi)
obtain hx2 : j ⊆ x := subset_trans (subset_sUnion_of_mem (ht1' x hx ▸ h3)) (hK... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 195,
"column": 2
} | {
"line": 227,
"column": 74
} | {
"line": 229,
"column": 0
} | [
{
"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\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)... | [] | have : ∀ (p : ℕ → Prop) {r : ℕ → ℝ} (_ : Tendsto r atTop (𝓝[>] 0)),
(blimsup (fun i => cthickening (M * r i) (s i)) atTop p : Set α) =ᵐ[μ]
(blimsup (fun i => cthickening (r i) (s i)) atTop p : Set α) := by
clear p hr r; intro p r hr
have hr' : Tendsto (fun i => M * r i) atTop (𝓝[>] 0) := by
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 195,
"column": 2
} | {
"line": 227,
"column": 74
} | {
"line": 229,
"column": 0
} | [
{
"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\nr : ℕ → ℝ\nhr : Tendsto r atTop (𝓝 0)... | [] | have : ∀ (p : ℕ → Prop) {r : ℕ → ℝ} (_ : Tendsto r atTop (𝓝[>] 0)),
(blimsup (fun i => cthickening (M * r i) (s i)) atTop p : Set α) =ᵐ[μ]
(blimsup (fun i => cthickening (r i) (s i)) atTop p : Set α) := by
clear p hr r; intro p r hr
have hr' : Tendsto (fun i => M * r i) atTop (𝓝[>] 0) := by
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 95
} | {
"line": 250,
"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\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr' : ∀ᶠ (i : ℕ) in 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\nr : ℕ → ℝ\nh₁ : blimsup (fun i ↦ cthickening (r i)... | replace hr' : ∀ᶠ i in atTop, p i → 0 < M * r i := hr'.mono fun i hi hip ↦ mul_pos hM (hi hip) | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 232,
"column": 2
} | {
"line": 233,
"column": 38
} | {
"line": 234,
"column": 2
} | [
{
"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\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhφ_cvx : ConcaveOn ℝ univ φ\nhφ_cont : UpperSemicontinuous φ\nhf_int : Integrable f... | [
"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\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhφ_cvx : ConcaveOn ℝ univ φ\nhφ_cont : UpperSemicontinuous φ\nhf_int : Integrable f μ\nhφ_int :... | filter_upwards [hφ_cvx.neg.map_condExp_le_univ hm hφ_cont.neg hf_int hφ_int.neg,
condExp_neg (φ ∘ f) m] with a h ha | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 240,
"column": 47
} | {
"line": 242,
"column": 59
} | {
"line": 244,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nα : Type u_2\nf : α → E\nφ : E → ℝ\nm mα : MeasurableSpace α\nμ : Measure α\nmE : MeasurableSpace E\ninst✝¹ : BorelSpace E\nhm : m ≤ mα\ninst✝ : SigmaFinite (μ.trim hm)\nhφ_cvx : ConcaveOn ℝ univ φ\nhφ_cont... | [] | by
rw [StronglyMeasurable.ae_le_trim_iff hm (by fun_prop) (by fun_prop)]
exact hφ_cvx.condExp_map_le_univ hm hφ_cont hf_int hφ_int | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 170,
"column": 55
} | {
"line": 173,
"column": 17
} | {
"line": 174,
"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[... | [] | by
refine (condExp_restrict_ae_eq_restrict hm (h_meas n) hfg).symm.trans ?_
filter_upwards [this, (condExp_restrict_ae_eq_restrict hm (h_meas n) hg)] with ω hω1 hω2
rw [hω1, hω2] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn | {
"line": 252,
"column": 6
} | {
"line": 255,
"column": 22
} | {
"line": 256,
"column": 4
} | [
{
"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... | [] | convert! h₁ _ h
ext x
rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
exact fun _ => h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Hahn | {
"line": 252,
"column": 6
} | {
"line": 255,
"column": 22
} | {
"line": 256,
"column": 4
} | [
{
"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... | [] | convert! h₁ _ h
ext x
rw [Set.mem_iUnion, exists_prop, and_iff_right_iff_imp]
exact fun _ => h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 244,
"column": 2
} | {
"line": 251,
"column": 26
} | {
"line": 253,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : T2Space M\ninst✝ : ContinuousSub M\nv : VectorMeasure α M\ns : ℕ → Set α\nhm : Antitone s\nhs : ∀ (i : ℕ), MeasurableSet (s i)\n⊢ Tendsto (fun n ↦ v (s n)) atTop (𝓝 (v (⋂ n, s n)))",
"... | [] | have I n : v (s n) = v univ - v (s n)ᶜ := by simp [of_compl (hs n)]
have J : v (⋂ n, s n) = v univ - v (⋃ n, (s n)ᶜ) := by
rw [← of_compl (MeasurableSet.iUnion (fun n ↦ (hs n).compl))]
simp
simp_rw [I, J]
apply tendsto_const_nhds.sub
exact tendsto_vectorMeasure_iUnion_atTop_nat (fun i j hij ↦ by simpa u... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 244,
"column": 2
} | {
"line": 251,
"column": 26
} | {
"line": 253,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_4\ninst✝³ : AddCommGroup M\ninst✝² : TopologicalSpace M\ninst✝¹ : T2Space M\ninst✝ : ContinuousSub M\nv : VectorMeasure α M\ns : ℕ → Set α\nhm : Antitone s\nhs : ∀ (i : ℕ), MeasurableSet (s i)\n⊢ Tendsto (fun n ↦ v (s n)) atTop (𝓝 (v (⋂ n, s n)))",
"... | [] | have I n : v (s n) = v univ - v (s n)ᶜ := by simp [of_compl (hs n)]
have J : v (⋂ n, s n) = v univ - v (⋃ n, (s n)ᶜ) := by
rw [← of_compl (MeasurableSet.iUnion (fun n ↦ (hs n).compl))]
simp
simp_rw [I, J]
apply tendsto_const_nhds.sub
exact tendsto_vectorMeasure_iUnion_atTop_nat (fun i j hij ↦ by simpa u... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 446,
"column": 10
} | {
"line": 446,
"column": 39
} | {
"line": 447,
"column": 8
} | [
{
"pp": "case neg.posPart\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : r < 0\n⊢ (r • s).toJordanDecomposition.posPart = (r • s.toJordanDecomposition).posPart",
"ppTerm": "?neg.posPart✝",
"assigned": true,
"usedConstants": [
"MeasureTheory.JordanDecomposition.posPa... | [
"case neg.posPart\nα : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nr : ℝ\nhr : r < 0\n⊢ (r • s).toJordanDecomposition.posPart = (-r).toNNReal • s.toJordanDecomposition.negPart"
] | real_smul_posPart_neg _ _ hr, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1024,
"column": 6
} | {
"line": 1026,
"column": 93
} | {
"line": 1027,
"column": 4
} | [
{
"pp": "case refine_3\nα : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝⁴ : TopologicalSpace M\ninst✝³ : AddCommMonoid M\ninst✝² : PartialOrder M\ninst✝¹ : IsOrderedAddMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), ... | [] | refine (v.m_iUnion (fun n => ?_) ?_).summable
· exact ha₁.inter (MeasurableSet.disjointed hf₁ n)
· exact (disjoint_disjointed _).mono fun i j => Disjoint.mono inf_le_right inf_le_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1024,
"column": 6
} | {
"line": 1026,
"column": 93
} | {
"line": 1027,
"column": 4
} | [
{
"pp": "case refine_3\nα : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝⁴ : TopologicalSpace M\ninst✝³ : AddCommMonoid M\ninst✝² : PartialOrder M\ninst✝¹ : IsOrderedAddMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), ... | [] | refine (v.m_iUnion (fun n => ?_) ?_).summable
· exact ha₁.inter (MeasurableSet.disjointed hf₁ n)
· exact (disjoint_disjointed _).mono fun i j => Disjoint.mono inf_le_right inf_le_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 487,
"column": 4
} | {
"line": 487,
"column": 44
} | {
"line": 488,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nf : ℕ → α → β\ng : α → β\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : MemLp g p μ\nhui : UnifIntegrable f p μ\nhf... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : IsFiniteMeasure μ\nhp : 1 ≤ p\nhp' : p ≠ ∞\nf : ℕ → α → β\ng : α → β\nhf : ∀ (n : ℕ), StronglyMeasurable (f n)\nhg : StronglyMeasurable g\nhg' : MemLp g p μ\nhui : UnifIntegrable f p μ\nhfg : ∀ᵐ (x : ... | rw [ENNReal.ofReal_le_ofReal_iff hδ₁.le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 177,
"column": 2
} | {
"line": 184,
"column": 46
} | {
"line": 186,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhfm : AEMeasurable f μ\nhf : ∫⁻ (x : α), f x ∂μ ≠ ∞\n⊢ (μ.withDensityᵥ fun x ↦ (f x).toReal) = (μ.withDensity f).toSignedMeasure",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpa... | [] | have hfi := integrable_toReal_of_lintegral_ne_top hfm hf
haveI := isFiniteMeasure_withDensity hf
ext i hi
rw [withDensityᵥ_apply hfi hi, toSignedMeasure_apply_measurable hi, measureReal_def,
withDensity_apply _ hi, integral_toReal hfm.restrict]
refine ae_lt_top' hfm.restrict (ne_top_of_le_ne_top hf ?_)
co... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 177,
"column": 2
} | {
"line": 184,
"column": 46
} | {
"line": 186,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhfm : AEMeasurable f μ\nhf : ∫⁻ (x : α), f x ∂μ ≠ ∞\n⊢ (μ.withDensityᵥ fun x ↦ (f x).toReal) = (μ.withDensity f).toSignedMeasure",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpa... | [] | have hfi := integrable_toReal_of_lintegral_ne_top hfm hf
haveI := isFiniteMeasure_withDensity hf
ext i hi
rw [withDensityᵥ_apply hfi hi, toSignedMeasure_apply_measurable hi, measureReal_def,
withDensity_apply _ hi, integral_toReal hfm.restrict]
refine ae_lt_top' hfm.restrict (ne_top_of_le_ne_top hf ?_)
co... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 59
} | {
"line": 139,
"column": 4
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\n⊢ (s.singularPart μ).toJordanDecomposition =\n { posPart := s.toJordanDecomposition.posPart.singularPart μ,\n negPart := s.toJordanDecomposition.negPart.singularPart μ, posPart_finite := ⋯, negPart_finite := ⋯,\n mutu... | [
"α : Type u_1\nm : MeasurableSpace α\ns : SignedMeasure α\nμ : Measure α\n⊢ (s.singularPart μ).toJordanDecomposition.toSignedMeasure =\n { posPart := s.toJordanDecomposition.posPart.singularPart μ,\n negPart := s.toJordanDecomposition.negPart.singularPart μ, posPart_finite := ⋯, negPart_finite := ⋯,\n ... | refine JordanDecomposition.toSignedMeasure_injective ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 92,
"column": 2
} | {
"line": 95,
"column": 76
} | {
"line": 97,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n⊢ lpNorm (-f) p μ = lpNorm f p μ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNe... | [] | by_cases hf : AEStronglyMeasurable f μ
· simp [← toReal_eLpNorm, hf, hf.neg]
· rw [lpNorm_of_not_aestronglyMeasurable hf,
lpNorm_of_not_aestronglyMeasurable fun h ↦ hf <| by simpa using h.neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 92,
"column": 2
} | {
"line": 95,
"column": 76
} | {
"line": 97,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf : α → E\np : ℝ≥0∞\nμ : Measure α\n⊢ lpNorm (-f) p μ = lpNorm f p μ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNe... | [] | by_cases hf : AEStronglyMeasurable f μ
· simp [← toReal_eLpNorm, hf, hf.neg]
· rw [lpNorm_of_not_aestronglyMeasurable hf,
lpNorm_of_not_aestronglyMeasurable fun h ↦ hf <| by simpa using h.neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 250,
"column": 2
} | {
"line": 262,
"column": 78
} | {
"line": 264,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns t : SignedMeasure α\nμ : Measure α\nf : α → ℝ\nhf : Measurable f\nhfi : Integrable f μ\nhtμ : t ⟂ᵥ μ.toENNRealVectorMeasure\nhadd : s = t + μ.withDensityᵥ f\n⊢ s.HaveLebesgueDecomposition μ",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
... | [] | have htμ' := htμ
rw [mutuallySingular_ennreal_iff] at htμ
change _ ⟂ₘ VectorMeasure.equivMeasure.toFun (VectorMeasure.equivMeasure.invFun μ) at htμ
rw [VectorMeasure.equivMeasure.right_inv, totalVariation_mutuallySingular_iff] at htμ
refine
{ posPart := by
use ⟨t.toJordanDecomposition.posPart, fun x... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 250,
"column": 2
} | {
"line": 262,
"column": 78
} | {
"line": 264,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns t : SignedMeasure α\nμ : Measure α\nf : α → ℝ\nhf : Measurable f\nhfi : Integrable f μ\nhtμ : t ⟂ᵥ μ.toENNRealVectorMeasure\nhadd : s = t + μ.withDensityᵥ f\n⊢ s.HaveLebesgueDecomposition μ",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
... | [] | have htμ' := htμ
rw [mutuallySingular_ennreal_iff] at htμ
change _ ⟂ₘ VectorMeasure.equivMeasure.toFun (VectorMeasure.equivMeasure.invFun μ) at htμ
rw [VectorMeasure.equivMeasure.right_inv, totalVariation_mutuallySingular_iff] at htμ
refine
{ posPart := by
use ⟨t.toJordanDecomposition.posPart, fun x... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 57
} | {
"line": 86,
"column": 4
} | [
{
"pp": "case pos\nα : 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\nhfint : ¬Integrable f μ\n... | [
"case pos\nα : 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\nhfint : ¬Integrable f μ\n⊢ 0 ≤ᵐ[μ] μ[... | simp only [condExp_of_not_integrable hfint, abs_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 454,
"column": 67
} | {
"line": 467,
"column": 39
} | {
"line": 469,
"column": 0
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : FiniteMeasure Ω\nf g : Ω →ᵇ ℝ≥0\n⊢ μ.testAgainstNN f ≤ μ.testAgainstNN g + nndist f g * μ.mass",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"MeasureTheory.FiniteMeas... | [] | by
simp only [← μ.testAgainstNN_const (nndist f g), ← testAgainstNN_add, ← ENNReal.coe_le_coe,
BoundedContinuousFunction.coe_add, const_apply, ENNReal.coe_add, Pi.add_apply,
coe_nnreal_ennreal_nndist, testAgainstNN_coe_eq]
apply lintegral_mono
have le_dist : ∀ ω, dist (f ω) (g ω) ≤ nndist f g := BoundedCo... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 342,
"column": 64
} | {
"line": 342,
"column": 87
} | {
"line": 343,
"column": 6
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : IsFiniteMeasure μ\ng : α → ℝ\nhint : Integrable g μ\nℱ : ι → MeasurableSpace α\nhℱ : ∀ (i : ι), ℱ i ≤ m0\nA : MeasurableSpace α := m0\nhmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ≤ ‖μ[g | ℱ n] x‖₊}\nhg : MemLp g 1 μ\nε : ℝ... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nι : Type u_2\ninst✝ : IsFiniteMeasure μ\ng : α → ℝ\nhint : Integrable g μ\nℱ : ι → MeasurableSpace α\nhℱ : ∀ (i : ι), ℱ i ≤ m0\nA : MeasurableSpace α := m0\nhmeas : ∀ (n : ι) (C : ℝ≥0), MeasurableSet {x | C ≤ ‖μ[g | ℱ n] x‖₊}\nhg : MemLp g 1 μ\nε : ℝ\nhε : 0 < ε... | mul_inv_cancel₀ hδ.ne', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 788,
"column": 4
} | {
"line": 788,
"column": 79
} | {
"line": 789,
"column": 2
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\np : ℝ≥0 × FiniteMeasure Ω\ng : Ω →ᵇ ℝ\nA : Tendsto (fun i ↦ i.1) (𝓝 p) (𝓝 p.1)\n⊢ Tendsto (fun i ↦ ∫ (x : Ω), g x ∂↑i.2) (𝓝 p.1 ×ˢ 𝓝 p.2) (𝓝 (∫ (x : Ω), g x ∂↑p.2))",
"ppTerm": "?m.100",
... | [] | exact (tendsto_iff_forall_integral_tendsto.1 tendsto_id g).comp tendsto_snd | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 854,
"column": 4
} | {
"line": 856,
"column": 40
} | {
"line": 857,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\nhp : p ≠ 0\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nε : ℝ\nhε : 0 < ε\nhfu : UnifIntegrable f p μ\nM : ℝ≥0\nhM : ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑M\nℐ : ℝ... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nf : ι → α → β\nhp : p ≠ 0\nhp' : p ≠ ∞\nhf : ∀ (i : ι), StronglyMeasurable (f i)\nε : ℝ\nhε : 0 < ε\nhfu : UnifIntegrable f p μ\nM : ℝ≥0\nhM : ∀ (i : ι), eLpNorm (f i) p μ ≤ ↑M\nℐ : ℝ≥0 → ι\nδ : ... | refine (lt_of_le_of_lt (le_trans
(hM <| ℐ <| 2 * max M 1 * δ⁻¹ ^ (1 / p.toReal)) (le_max_left (M : ℝ≥0∞) 1))
(lt_of_lt_of_le ?_ this)).ne rfl | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 184,
"column": 4
} | {
"line": 186,
"column": 67
} | {
"line": 188,
"column": 0
} | [
{
"pp": "case mpr\nΩ : 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⊢ (∀ (G : Set Ω), IsOpen[inst✝³] G → μ G ≤ li... | [] | 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": 184,
"column": 4
} | {
"line": 186,
"column": 67
} | {
"line": 188,
"column": 0
} | [
{
"pp": "case mpr\nΩ : 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⊢ (∀ (G : Set Ω), IsOpen[inst✝³] G → μ G ≤ li... | [] | 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.MeasureTheory.Measure.Portmanteau | {
"line": 228,
"column": 6
} | {
"line": 231,
"column": 46
} | {
"line": 232,
"column": 2
} | [] | [] | (L.limsup fun i ↦ μs i E) ≤ L.limsup fun i ↦ μs i E₁ :=
limsup_le_limsup (.of_forall fun _ ↦ measure_mono subset_E₁)
_ ≤ μ E₁ := h_E₁
_ = μ E := measure_congr E_ae_eq_E₁.symm | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 511,
"column": 4
} | {
"line": 512,
"column": 61
} | {
"line": 514,
"column": 0
} | [
{
"pp": "case calc_2\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : Measure Ω\nμs : ℕ → Measure Ω\nf : Ω → ℝ\nf_cont : Continuous[inst✝¹, _] f\nf_nn : 0 ≤ f\nh_opens : ∀ (G : Set Ω), IsOpen[inst✝¹] G → μ G ≤ liminf (fun i ↦ (μs i) G) atTop\n⊢ ∫⁻ (t : ... | [] | exact lintegral_liminf_le (fun n ↦ Antitone.measurable (fun s t hst ↦
measure_mono (fun ω hω ↦ lt_of_le_of_lt hst hω))) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 511,
"column": 4
} | {
"line": 512,
"column": 61
} | {
"line": 514,
"column": 0
} | [
{
"pp": "case calc_2\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : Measure Ω\nμs : ℕ → Measure Ω\nf : Ω → ℝ\nf_cont : Continuous[inst✝¹, _] f\nf_nn : 0 ≤ f\nh_opens : ∀ (G : Set Ω), IsOpen[inst✝¹] G → μ G ≤ liminf (fun i ↦ (μs i) G) atTop\n⊢ ∫⁻ (t : ... | [] | exact lintegral_liminf_le (fun n ↦ Antitone.measurable (fun s t hst ↦
measure_mono (fun ω hω ↦ lt_of_le_of_lt hst hω))) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 511,
"column": 4
} | {
"line": 512,
"column": 61
} | {
"line": 514,
"column": 0
} | [
{
"pp": "case calc_2\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : Measure Ω\nμs : ℕ → Measure Ω\nf : Ω → ℝ\nf_cont : Continuous[inst✝¹, _] f\nf_nn : 0 ≤ f\nh_opens : ∀ (G : Set Ω), IsOpen[inst✝¹] G → μ G ≤ liminf (fun i ↦ (μs i) G) atTop\n⊢ ∫⁻ (t : ... | [] | exact lintegral_liminf_le (fun n ↦ Antitone.measurable (fun s t hst ↦
measure_mono (fun ω hω ↦ lt_of_le_of_lt hst hω))) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 72
} | {
"line": 185,
"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 : ι → Type u_8\nmβ : (i : ι) → MeasurableSpace (β i)\nf g : (i : ι) → Ω → β i\nhf :\n ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},\n (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a... | [
"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\nmβ : (i : ι) → MeasurableSpace (β i)\nf g : (i : ι) → Ω → β i\nhf :\n ∀ (S : Finset ι) {sets : (i : ι) → Set (β i)},\n (∀ i ∈ S, MeasurableSet (sets i)) → ∀ᵐ (a : α) ∂μ, (κ... | change (ω ∈ ⋂ i ∈ S, g i ⁻¹' sets i) = (ω ∈ ⋂ i ∈ S, f i ⁻¹' sets i) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 522,
"column": 26
} | {
"line": 522,
"column": 75
} | {
"line": 522,
"column": 76
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm1 m2 m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : Indep... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm1 m2 m : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\np1 p2 : Set (Set Ω)\nh1 : m1 ≤ m\nh2 : m2 ≤ m\nhp1 : IsPiSystem p1\nhp2 : IsPiSystem p2\nhpm1 : m1 = generateFrom p1\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2 κ... | measure_compl (h1 _ ht) (measure_ne_top (κ a) t), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Density | {
"line": 207,
"column": 79
} | {
"line": 210,
"column": 57
} | {
"line": 212,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\ninst✝¹ : MeasurableSpace E\nm : MeasurableSpace Ω\nℙ : Measure Ω\nμ : Measure E\nX : Ω → E\ninst✝ : HasPDF X ℙ μ\nf : E → ℝ≥0∞\nhf : AEMeasurable f μ\n⊢ ∫⁻ (x : E), pdf X ℙ μ x * f x ∂μ = ∫⁻ (x : Ω), f (X x) ∂ℙ",
"ppTerm": "?m.26",
"assigned": true,
"usedConstant... | [] | by
rw [pdf_def,
← lintegral_map' (hf.mono_ac HasPDF.absolutelyContinuous) (HasPDF.aemeasurable X ℙ μ),
lintegral_rnDeriv_mul HasPDF.absolutelyContinuous hf] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Density | {
"line": 227,
"column": 2
} | {
"line": 229,
"column": 59
} | {
"line": 231,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\ninst✝⁴ : MeasurableSpace E\nm : MeasurableSpace Ω\nℙ : Measure Ω\nμ : Measure E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : IsFiniteMeasure ℙ\nX : Ω → E\ninst✝ : HasPDF X ℙ μ\nf : E → F\nhf : AEStronglyMeasurable f μ\n⊢ ∫ (x : E), (pdf X ... | [] | rw [← integral_map (HasPDF.aemeasurable X ℙ μ) (hf.mono_ac HasPDF.absolutelyContinuous),
map_eq_withDensity_pdf X ℙ μ, pdf_def, integral_rnDeriv_smul HasPDF.absolutelyContinuous,
withDensity_rnDeriv_eq _ _ HasPDF.absolutelyContinuous] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Density | {
"line": 227,
"column": 2
} | {
"line": 229,
"column": 59
} | {
"line": 231,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\ninst✝⁴ : MeasurableSpace E\nm : MeasurableSpace Ω\nℙ : Measure Ω\nμ : Measure E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : IsFiniteMeasure ℙ\nX : Ω → E\ninst✝ : HasPDF X ℙ μ\nf : E → F\nhf : AEStronglyMeasurable f μ\n⊢ ∫ (x : E), (pdf X ... | [] | rw [← integral_map (HasPDF.aemeasurable X ℙ μ) (hf.mono_ac HasPDF.absolutelyContinuous),
map_eq_withDensity_pdf X ℙ μ, pdf_def, integral_rnDeriv_smul HasPDF.absolutelyContinuous,
withDensity_rnDeriv_eq _ _ HasPDF.absolutelyContinuous] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Density | {
"line": 227,
"column": 2
} | {
"line": 229,
"column": 59
} | {
"line": 231,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\ninst✝⁴ : MeasurableSpace E\nm : MeasurableSpace Ω\nℙ : Measure Ω\nμ : Measure E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : IsFiniteMeasure ℙ\nX : Ω → E\ninst✝ : HasPDF X ℙ μ\nf : E → F\nhf : AEStronglyMeasurable f μ\n⊢ ∫ (x : E), (pdf X ... | [] | rw [← integral_map (HasPDF.aemeasurable X ℙ μ) (hf.mono_ac HasPDF.absolutelyContinuous),
map_eq_withDensity_pdf X ℙ μ, pdf_def, integral_rnDeriv_smul HasPDF.absolutelyContinuous,
withDensity_rnDeriv_eq _ _ HasPDF.absolutelyContinuous] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Density | {
"line": 284,
"column": 13
} | {
"line": 284,
"column": 27
} | {
"line": 284,
"column": 28
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsFiniteMeasure ℙ\nf : ℝ → ℝ\ng : ℝ → ℝ≥0∞\nhg : pdf X ℙ volume =ᵐ[volume] g\nhgi : ∫⁻ (x : ℝ), ‖f x‖ₑ * g x ≠ ∞\n⊢ (fun i ↦ ‖f i‖ₑ * ENNReal.ofReal (pdf X ℙ volume i).toReal) =ᵐ[volume] fun x ↦ ‖f x * (pdf X ℙ volume x).toReal‖ₑ",
... | [
"Ω : Type u_1\nm : MeasurableSpace Ω\nℙ : Measure Ω\nX : Ω → ℝ\ninst✝ : IsFiniteMeasure ℙ\nf : ℝ → ℝ\ng : ℝ → ℝ≥0∞\nhg : pdf X ℙ volume =ᵐ[volume] g\nhgi : ∫⁻ (x : ℝ), ‖f x‖ₑ * g x ≠ ∞\n⊢ (fun i ↦ ‖f i‖ₑ • ENNReal.ofReal (pdf X ℙ volume i).toReal) =ᵐ[volume] fun x ↦ ‖f x • (pdf X ℙ volume x).toReal‖ₑ"
] | ← smul_eq_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 738,
"column": 6
} | {
"line": 739,
"column": 75
} | {
"line": 740,
"column": 4
} | [
{
"pp": "case inr.refine_2.refine_2\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\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ _mΩ\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateF... | [] | filter_upwards [h_rec hf_m_S, h] with a' ha' h'
rwa [Finset.set_biInter_insert, Finset.prod_insert ha_notin_S, ← ha'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 738,
"column": 6
} | {
"line": 739,
"column": 75
} | {
"line": 740,
"column": 4
} | [
{
"pp": "case inr.refine_2.refine_2\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\nm : ι → MeasurableSpace Ω\nh_le : ∀ (i : ι), m i ≤ _mΩ\nπ : ι → Set (Set Ω)\nh_pi : ∀ (n : ι), IsPiSystem (π n)\nh_generate : ∀ (i : ι), m i = generateF... | [] | filter_upwards [h_rec hf_m_S, h] with a' ha' h'
rwa [Finset.set_biInter_insert, Finset.prod_insert ha_notin_S, ← ha'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.Variance | {
"line": 104,
"column": 11
} | {
"line": 104,
"column": 27
} | {
"line": 104,
"column": 27
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhX : MemLp X 2 μ\nthis : ∫⁻ (x : Ω), ‖X x - ∫ (x : Ω), X x ∂μ‖ₑ ^ 2 ∂μ < ∞\n⊢ eVar[X; μ] < ∞",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhX : MemLp X 2 μ\nthis : ∫⁻ (x : Ω), ‖X x - ∫ (x : Ω), X x ∂μ‖ₑ ^ 2 ∂μ < ∞\n⊢ eVar[X; μ] < ∞"
] | ENNReal.rpow_two | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Moments.Variance | {
"line": 206,
"column": 60
} | {
"line": 208,
"column": 5
} | {
"line": 210,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nc : ℝ\nX : Ω → ℝ\nμ : Measure Ω\n⊢ Var[fun ω ↦ c * X ω; μ] = c ^ 2 * Var[X; μ]",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real",
"IsOrderedRing.toPosM... | [] | by
rw [variance, evariance_mul, ENNReal.toReal_mul, ENNReal.toReal_ofReal (sq_nonneg _)]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Moments.Variance | {
"line": 392,
"column": 49
} | {
"line": 392,
"column": 65
} | {
"line": 392,
"column": 65
} | [
{
"pp": "case e'_4\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nc : ℝ≥0\nhc : c ≠ 0\nA : ↑c ≠ 0\nB : AEStronglyMeasurable (fun x ↦ ∫ (x : Ω), X x ∂μ) μ\n⊢ (↑c)⁻¹ ^ 2 * eVar[X; μ] = (↑c)⁻¹ ^ 2 * ((∫⁻ (x : Ω), ‖X x - ∫ (x : Ω), X x ∂μ‖ₑ ^ 2 ∂μ) ^ 2⁻¹) ^ 2",
"... | [
"case e'_4\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nc : ℝ≥0\nhc : c ≠ 0\nA : ↑c ≠ 0\nB : AEStronglyMeasurable (fun x ↦ ∫ (x : Ω), X x ∂μ) μ\n⊢ (↑c)⁻¹ ^ 2 * eVar[X; μ] = (↑c)⁻¹ ^ 2 * ((∫⁻ (x : Ω), ‖X x - ∫ (x : Ω), X x ∂μ‖ₑ ^ 2 ∂μ) ^ 2⁻¹) ^ 2"
] | ENNReal.rpow_two | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 404,
"column": 10
} | {
"line": 404,
"column": 32
} | {
"line": 404,
"column": 33
} | [
{
"pp": "case inr\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nS T : Finset ι\nhST : Disjoint S T\nhf_meas : ∀ (i : ι), Measurable (f i)\nhμ : μ ≠ 0\nη : Ker... | [
"case inr\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nβ : ι → Type u_8\nm : (i : ι) → MeasurableSpace (β i)\nf : (i : ι) → Ω → β i\nS T : Finset ι\nhST : Disjoint S T\nhf_meas : ∀ (i : ι), Measurable (f i)\nhμ : μ ≠ 0\nη : Kernel α Ω\nη_e... | h_sets_s'_univ hi_mem, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.Variance | {
"line": 471,
"column": 2
} | {
"line": 471,
"column": 53
} | {
"line": 472,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\na b : ℝ\nX : Ω → ℝ\nh : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nhX : AEMeasurable X μ\n⊢ Var[X; μ] ≤ (b - ∫ (x : Ω), X x ∂μ) * (∫ (x : Ω), X x ∂μ - a)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\na b : ℝ\nX : Ω → ℝ\nh : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b\nhX : AEMeasurable X μ\nha : ∀ᵐ (ω : Ω) ∂μ, a ≤ X ω\n⊢ Var[X; μ] ≤ (b - ∫ (x : Ω), X x ∂μ) * (∫ (x : Ω), X x ∂μ - a)"
] | have ha : ∀ᵐ ω ∂μ, a ≤ X ω := h.mono fun ω h => h.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.FactorsThrough | {
"line": 68,
"column": 59
} | {
"line": 68,
"column": 73
} | {
"line": 68,
"column": 73
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nmY : MeasurableSpace Y\nf : X → Y\ng : X → Z\ninst✝² : Nonempty Z\ninst✝¹ : TopologicalSpace Z\ninst✝ : IsCompletelyMetrizableSpace Z\nmX : MeasurableSpace X := MeasurableSpace.comap f mY\nt : Set Y\nht : MeasurableSet t\nh₁ : Y → Z\nmh₁ : StronglyMeasurable h₁... | [
"X : Type u_1\nY : Type u_2\nZ : Type u_3\nmY : MeasurableSpace Y\nf : X → Y\ng : X → Z\ninst✝² : Nonempty Z\ninst✝¹ : TopologicalSpace Z\ninst✝ : IsCompletelyMetrizableSpace Z\nmX : MeasurableSpace X := MeasurableSpace.comap f mY\nt : Set Y\nht : MeasurableSet t\nh₁ : Y → Z\nmh₁ : StronglyMeasurable h₁\nhg₁ : Stro... | piecewise_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.Intersectivity | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 63
} | {
"line": 50,
"column": 2
} | [
{
"pp": "α : 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)\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (⋂ n ∈ u, s n)",
"ppTerm": "?m.33",
"assigned":... | [
"α : 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‖₊}\n⊢ ∃ t, t.Infinite ∧ ∀ ⦃u : Set ℕ⦄, u ⊆ t → u.Finite → 0 < μ (... | let M (f : α → ℝ) : Set α := {x | eLpNormEssSup f μ < ‖f x‖₊} | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 651,
"column": 2
} | {
"line": 657,
"column": 60
} | {
"line": 658,
"column": 2
} | [
{
"pp": "case refine_1\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 : 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... | [
"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 : 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)\... | · have : ∀ᵐ a ∂μ, ∀ (i : s), f i =ᵐ[κ a] (hf_meas i).mk := by
rw [ae_all_iff]
exact fun i ↦ Measure.ae_ae_of_ae_comp (hf_meas i).ae_eq_mk
filter_upwards [this] with a ha
filter_upwards [ae_all_iff.2 ha] with ω hω
simp only [Finset.prod_apply]
exact Finset.prod_congr rfl fun i hi ↦ (hω ⟨i, hi... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.Piecewise | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 85
} | {
"line": 88,
"column": 0
} | [
{
"pp": "case neg.refine_2\nι : 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)\nFi : Infinite ι\ne : ℕ ≃ ι\ng ... | [] | exact (Filter.tendsto_congr' this).mp (by simp [StronglyMeasurable.tendsto_approx]) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 64,
"column": 2
} | {
"line": 67,
"column": 53
} | {
"line": 69,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : NormedAddCommGroup β\nx✝ : MeasurableSpace α\nf : ι → α → β\np : ℝ≥0∞\nμ : Measure α\n⊢ UnifTight f p μ ↔ ∀ ⦃ε : ℝ≥0∞⦄, 0 < ε → ∃ s, μ s ≠ ∞ ∧ ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ε",
"ppTerm": "?m.29",
"assigned": true,
"usedConsta... | [] | simp only [ENNReal.forall_ennreal, ENNReal.coe_pos]
refine (and_iff_left ?_).symm
simp only [zero_lt_top, le_top, implies_true, and_true, true_implies]
use ∅; simpa only [measure_empty] using zero_ne_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.UnifTight | {
"line": 64,
"column": 2
} | {
"line": 67,
"column": 53
} | {
"line": 69,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : NormedAddCommGroup β\nx✝ : MeasurableSpace α\nf : ι → α → β\np : ℝ≥0∞\nμ : Measure α\n⊢ UnifTight f p μ ↔ ∀ ⦃ε : ℝ≥0∞⦄, 0 < ε → ∃ s, μ s ≠ ∞ ∧ ∀ (i : ι), eLpNorm (sᶜ.indicator (f i)) p μ ≤ ε",
"ppTerm": "?m.29",
"assigned": true,
"usedConsta... | [] | simp only [ENNReal.forall_ennreal, ENNReal.coe_pos]
refine (and_iff_left ?_).symm
simp only [zero_lt_top, le_top, implies_true, and_true, true_implies]
use ∅; simpa only [measure_empty] using zero_ne_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 107,
"column": 32
} | {
"line": 107,
"column": 48
} | {
"line": 108,
"column": 2
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝⁴ : MeasurableSpace X\nμ : Measure X\ninst✝³ : Group G\ninst✝² : MulAction G X\nι : Type u_3\nl : Filter ι\ninst✝¹ : NeZero μ\ninst✝ : IsFiniteMeasure μ\n⊢ ∀ᶠ (i : ι) in l, μ univ ≠ 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"_private.... | [] | simp [NeZero.ne] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Group.FoelnerFilter | {
"line": 107,
"column": 32
} | {
"line": 107,
"column": 48
} | {
"line": 108,
"column": 2
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝⁴ : MeasurableSpace X\nμ : Measure X\ninst✝³ : Group G\ninst✝² : MulAction G X\nι : Type u_3\nl : Filter ι\ninst✝¹ : NeZero μ\ninst✝ : IsFiniteMeasure μ\n⊢ ∀ᶠ (i : ι) in l, μ univ ≠ 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"_private.... | [] | simp [NeZero.ne] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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