module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift | {
"line": 222,
"column": 2
} | {
"line": 231,
"column": 34
} | {
"line": 233,
"column": 0
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\ninst✝¹ : LinearOrder k\ninst✝ : IsOrderedRing k\nn : ℕ\ni : Fin n\nx : k\nhxpos : 0 < x\nhx1 : x ≤ 1\nw : Fin n → k\nhw : ∑ i, w i = 1\n⊢ (∀ (j : Fin n), w j ∈ Set.Icc 0 1) ∧ w i = 1 - x ↔\n (∀ (j : Fin n), j ≠ i → x⁻¹ * w j ∈ Set.Icc 0 1) ∧ x⁻¹ * (w i - 1) + 1 = 0",
... | [] | rw [show x⁻¹ * (w i - 1) + 1 = 0 ↔ w i = 1 - x by grind]
refine and_congr_left fun hi ↦ ⟨fun hj j hji ↦ ⟨?_, ?_⟩, fun hj ↦ ?_⟩
· exact mul_nonneg (by simpa using hxpos.le) (hj j).1
· rw [eq_sub_iff_add_eq, add_comm, ← eq_sub_iff_add_eq] at hi
rw [inv_mul_le_one₀ hxpos, hi, le_sub_iff_add_le, ← hw]
exact a... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Center | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 16
} | {
"line": 79,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (r : R), r • f (b j) = (b.coord i) (f (b i)) • r • b j\ni j : ι\nhij : j = i\n⊢ (b.c... | [
"case neg\nR : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nf : V →ₗ[R] V\nhV : Nontrivial V\nhcomm : ∀ (i j : ι), i ≠ j → ∀ (r : R), r • f (b j) = (b.coord i) (f (b i)) • r • b j\ni j : ι\nhij : ¬j = i\n⊢ (b.coord i) (f ... | · simp [hij] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Center | {
"line": 153,
"column": 50
} | {
"line": 153,
"column": 73
} | {
"line": 153,
"column": 74
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nfeq : ∀ (i : ι), f (b i) = (b.coord i) (f (b i)) • b i\ni j : ι\nhij : i ≠ j\nr : R\nx : V := b.repr.symm ((Finsupp.single ... | [
"R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nfeq : ∀ (i : ι), f (b i) = (b.coord i) (f (b i)) • b i\ni j : ι\nhij : i ≠ j\nr : R\nx : V := b.repr.symm ((Finsupp.single i 1).update ... | add_eq_zero_iff_eq_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Center | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 8
} | {
"line": 161,
"column": 4
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nfeq : ∀ (i : ι), f (b i) = (b.coord i) (f (b i)) • b i\ni j : ι\nhij : i ≠ j\nr : R\nx : V := b.repr.symm ((Finsupp.single ... | [
"R : Type u_1\nV : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\nf : V →ₗ[R] V\nι : Type u_3\ninst✝ : Nontrivial ι\nb : Basis ι R V\nfeq : ∀ (i : ι), f (b i) = (b.coord i) (f (b i)) • b i\ni j : ι\nhij : i ≠ j\nr : R\nx : V := ⋯\ns t : R\nhst : s = 0 → ¬t = 0\nh : ∀ (... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.LinearAlgebra.CliffordAlgebra.BaseChange | {
"line": 55,
"column": 48
} | {
"line": 55,
"column": 81
} | {
"line": 56,
"column": 6
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nV : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : AddCommGroup V\ninst✝² : Algebra R A\ninst✝¹ : Module R V\ninst✝ : Invertible 2\nQ : QuadraticForm R V\nv : V\n⊢ (algebraMap A (CliffordAlgebra (QuadraticForm.baseChange A Q))) (Q v • 1) =\n (algebraMap R (C... | [
"R : Type u_1\nA : Type u_2\nV : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : AddCommGroup V\ninst✝² : Algebra R A\ninst✝¹ : Module R V\ninst✝ : Invertible 2\nQ : QuadraticForm R V\nv : V\n⊢ (algebraMap A (CliffordAlgebra (QuadraticForm.baseChange A Q))) ((algebraMap R A) (Q v)) =\n (algebraMap R... | ← Algebra.algebraMap_eq_smul_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 471,
"column": 47
} | {
"line": 471,
"column": 58
} | {
"line": 471,
"column": 59
} | [
{
"pp": "case h\nV : Type u_2\ninst✝³ : AddCommGroup V\nK : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nx✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1\nhe : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1\nhe' : e.fixedReduce = 1\nhe_one : ¬e = 1\nhefixed_ne_top... | [
"case h\nV : Type u_2\ninst✝³ : AddCommGroup V\nK : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\ne : V ≃ₗ[K] V\nx✝ : e ∈ dilatransvections K V ∧ e.fixedReduce = 1\nhe : finrank K (V ⧸ (↑e).fixedSubmodule) ≤ 1\nhe' : e.fixedReduce = 1\nhe_one : ¬e = 1\nhefixed_ne_top : (↑e).fixe... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Equivs | {
"line": 355,
"column": 8
} | {
"line": 355,
"column": 21
} | {
"line": 355,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ ((CliffordAlgebra.lift 0) ⟨inrHom R R, ⋯⟩).comp (DualNumber.lift ⟨(Algebra.ofId R (CliffordAlgebra 0), (ι 0) 1), ⋯⟩) =\n AlgHom.id R R[ε]",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.CliffordAlgebra.Equivs | {
"line": 355,
"column": 8
} | {
"line": 355,
"column": 21
} | {
"line": 355,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ ((CliffordAlgebra.lift 0) ⟨inrHom R R, ⋯⟩).comp (DualNumber.lift ⟨(Algebra.ofId R (CliffordAlgebra 0), (ι 0) 1), ⋯⟩) =\n AlgHom.id R R[ε]",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"... | [] | ext : 1; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.ExteriorAlgebra.Grading | {
"line": 64,
"column": 29
} | {
"line": 64,
"column": 67
} | {
"line": 64,
"column": 67
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ni : ℕ\nx x✝ y✝ : ExteriorAlgebra R M\ni✝ : ℕ\nhx✝ : x✝ ∈ (ι R).range ^ i✝\nhy✝ : y✝ ∈ (ι R).range ^ i✝\nihx : (liftι R M) x✝ = (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝) ⟨x✝, hx✝⟩\nihy : (liftι R M) y✝ ... | [
"case add\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ni : ℕ\nx x✝ y✝ : ExteriorAlgebra R M\ni✝ : ℕ\nhx✝ : x✝ ∈ (ι R).range ^ i✝\nhy✝ : y✝ ∈ (ι R).range ^ i✝\nihx : (liftι R M) x✝ = (DirectSum.of (fun i ↦ ↥(⋀[R]^i M)) i✝) ⟨x✝, hx✝⟩\nihy : (liftι R M) y✝ = (DirectSum... | simp_rw [map_add, ihx, ihy, ← map_add] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.Matrix.Charpoly.FiniteField | {
"line": 41,
"column": 4
} | {
"line": 41,
"column": 36
} | {
"line": 42,
"column": 4
} | [
{
"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⊢ (expand K (Fintype.card K)).... | [
"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⊢ matPolyEquiv ((expand K (Fintype.card K)... | refine matPolyEquiv.injective ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Matrix.Determinant.Misc | {
"line": 36,
"column": 45
} | {
"line": 36,
"column": 64
} | {
"line": 36,
"column": 65
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nM : Matrix (Fin (n + 1)) (Fin n) R\nhv : ∑ j, M j = 0\nj₁ j₂ : Fin (n + 1)\n⊢ (M.submatrix (Fin.succAbove 0) id).det = Int.negOnePow 0 • (M.submatrix (Fin.succAbove 0) id).det",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
... | [
"case zero\nR : Type u_1\ninst✝ : CommRing R\nn : ℕ\nM : Matrix (Fin (n + 1)) (Fin n) R\nhv : ∑ j, M j = 0\nj₁ j₂ : Fin (n + 1)\n⊢ (M.submatrix (Fin.succAbove 0) id).det = 1 • (M.submatrix (Fin.succAbove 0) id).det"
] | Int.negOnePow_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 111,
"column": 19
} | {
"line": 111,
"column": 71
} | {
"line": 112,
"column": 2
} | [
{
"pp": "case h\nm : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝¹ : CommRing R\ninst✝ : IsEmpty m\nA : Matrix m n R\nf : Fin 0 → m\ng✝ : Fin 0 → n\na✝¹ : Function.Injective f\na✝ : Function.Injective g✝\n⊢ ↑1 = (A.submatrix f g✝).det",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Sign... | [] | rw [submatrix_empty, det_fin_zero, SignType.coe_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 6
} | {
"line": 127,
"column": 2
} | [
{
"pp": "m : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nhc : ↑A 1 0 ≠ 0\n⊢ reduce (reduceStep A) = reduce A",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"instDecidableEqFin",
"instOfNatNat",
"Int",
"Fin.fintype",
"FixedDetMatrices.reduce",
"Int.instCommRing",... | [
"m : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\nhc : ↑A 1 0 ≠ 0\n⊢ reduce A = reduce (reduceStep A)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.LinearAlgebra.Matrix.FixedDetMatrices | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 48
} | {
"line": 145,
"column": 2
} | [
{
"pp": "m : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\ni j : Fin 2\nh10 : ↑A 1 0 = 0\nh00 : 0 < ↑A 0 0\nh01 : 0 ≤ ↑A 0 1\nh11 : |↑A 0 1| < |↑A 1 1|\nh1 : 0 < |↑A 1 1|\n⊢ |↑A i j| ≤ |m|",
"ppTerm": "?m.104",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"P... | [
"m : ℤ\nA : FixedDetMatrix (Fin 2) ℤ m\ni j : Fin 2\nh10 : ↑A 1 0 = 0\nh00 : 0 < ↑A 0 0\nh01 : 0 ≤ ↑A 0 1\nh11 : |↑A 0 1| < |↑A 1 1|\nh1 : 0 < |↑A 1 1|\nh2 : 0 < |↑A 0 0|\n⊢ |↑A i j| ≤ |m|"
] | have h2 : 0 < |A.1 0 0| := abs_pos.mpr h00.ne' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.IsDiag | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 19
} | {
"line": 89,
"column": 0
} | [
{
"pp": "α : Type u_1\nn : Type u_4\ninst✝ : AddZeroClass α\nA B : Matrix n n α\nha : A.IsDiag\nhb : B.IsDiag\ni j : n\nh : i ≠ j\n⊢ (A + B) i j = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Matrix.add",
"congrArg",
"Matrix",
"AddZeroClass.toAddZero",
"... | [] | simp [ha h, hb h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.IsDiag | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 19
} | {
"line": 94,
"column": 0
} | [
{
"pp": "α : Type u_1\nn : Type u_4\ninst✝ : SubtractionMonoid α\nA B : Matrix n n α\nha : A.IsDiag\nhb : B.IsDiag\ni j : n\nh : i ≠ j\n⊢ (A - B) i j = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"congrArg",
"sub_zero",
"Matrix",
"HSub.hSub",
"Subtractio... | [] | simp [ha h, hb h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 52
} | {
"line": 252,
"column": 2
} | [
{
"pp": "n : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n ℤ\nhA : A.IsHadamard\nhcard : 2 < Fintype.card n\nhpm : ∀ (i j : n), A i j = 1 ∨ A i j = -1\nr s t : n\nhrs : r ≠ s\nhrt : r ≠ t\nhst : s ≠ t\nhorth : ∀ ⦃i k : n⦄, i ≠ k → ∑ j, A i j * A k j = 0\nj : n\n⊢ (1 + A s j * A r j) * (1 +... | [] | obtain hr | hr := hpm r j <;> simp [hr] <;> ring | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 52
} | {
"line": 252,
"column": 2
} | [
{
"pp": "n : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n ℤ\nhA : A.IsHadamard\nhcard : 2 < Fintype.card n\nhpm : ∀ (i j : n), A i j = 1 ∨ A i j = -1\nr s t : n\nhrs : r ≠ s\nhrt : r ≠ t\nhst : s ≠ t\nhorth : ∀ ⦃i k : n⦄, i ≠ k → ∑ j, A i j * A k j = 0\nj : n\n⊢ (1 + A s j * A r j) * (1 +... | [] | obtain hr | hr := hpm r j <;> simp [hr] <;> ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.HadamardMatrix | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 52
} | {
"line": 252,
"column": 2
} | [
{
"pp": "n : Type u_2\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n ℤ\nhA : A.IsHadamard\nhcard : 2 < Fintype.card n\nhpm : ∀ (i j : n), A i j = 1 ∨ A i j = -1\nr s t : n\nhrs : r ≠ s\nhrt : r ≠ t\nhst : s ≠ t\nhorth : ∀ ⦃i k : n⦄, i ≠ k → ∑ j, A i j * A k j = 0\nj : n\n⊢ (1 + A s j * A r j) * (1 +... | [] | obtain hr | hr := hpm r j <;> simp [hr] <;> ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Multilinear.Pi | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 34
} | {
"line": 97,
"column": 2
} | [
{
"pp": "ι : Type uι\nκ : ι → Type uκ\nR : Type uR\nM : (i : ι) → κ i → Type uM\nN : ((i : ι) → κ i) → Type uN\ninst✝⁶ : Semiring R\ninst✝⁵ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)\ninst✝⁴ : (p : (i : ι) → κ i) → AddCommMonoid (N p)\ninst✝³ : (i : ι) → (k : κ i) → Module R (M i k)\ninst✝² : (p : (i : ι) → ... | [
"case inl\nι : Type uι\nκ : ι → Type uκ\nR : Type uR\nM : (i : ι) → κ i → Type uM\nN : ((i : ι) → κ i) → Type uN\ninst✝⁶ : Semiring R\ninst✝⁵ : (i : ι) → (k : κ i) → AddCommMonoid (M i k)\ninst✝⁴ : (p : (i : ι) → κ i) → AddCommMonoid (N p)\ninst✝³ : (i : ι) → (k : κ i) → Module R (M i k)\ninst✝² : (p : (i : ι) → κ ... | obtain rfl | hpq := eq_or_ne p q | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.PiTensorProduct | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 36
} | {
"line": 172,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR' : Type u_2\nR : Type u_3\nA : ι → Type u_4\ninst✝⁶ : CommSemiring R'\ninst✝⁵ : CommSemiring R\ninst✝⁴ : (i : ι) → Semiring (A i)\ninst✝³ : Algebra R' R\ninst✝² : (i : ι) → Algebra R (A i)\ninst✝¹ : (i : ι) → Algebra R' (A i)\ninst✝ : ∀ (i : ι), IsScalarTower R' R (A i)\nr : R'\nx : ⨂[R... | [
"ι : Type u_1\nR' : Type u_2\nR : Type u_3\nA : ι → Type u_4\ninst✝⁶ : CommSemiring R'\ninst✝⁵ : CommSemiring R\ninst✝⁴ : (i : ι) → Semiring (A i)\ninst✝³ : Algebra R' R\ninst✝² : (i : ι) → Algebra R (A i)\ninst✝¹ : (i : ι) → Algebra R' (A i)\ninst✝ : ∀ (i : ι), IsScalarTower R' R (A i)\nr : R'\nx : ⨂[R] (i : ι), A... | change r • (1 * x) = r • (x * 1) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.LinearAlgebra.Projectivization.Subspace | {
"line": 142,
"column": 6
} | {
"line": 142,
"column": 17
} | {
"line": 142,
"column": 18
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\n⊢ span Set.univ = ⊤",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"eq_top_iff",
"CompleteLattice.toLattice",
"con... | [
"K : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\n⊢ ⊤ ≤ span Set.univ"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Projectivization.Subspace | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 63
} | {
"line": 188,
"column": 0
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nS : Set (ℙ K V)\nu : ℙ K V\n⊢ u ∈ span S ↔ ∀ (W : Subspace K V), span S ≤ W → u ∈ W",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"le_refl",
"PartialOrder.toPreorder",
... | [] | exact ⟨fun hu W hW => hW hu, fun W => W (span S) (le_refl _)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Projectivization.Subspace | {
"line": 278,
"column": 2
} | {
"line": 280,
"column": 40
} | {
"line": 282,
"column": 0
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Submodule K V\nx : ℙ K V\n⊢ x ∈ projectivization s ↔ x.submodule ≤ s",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Projectivization.mk",
"Eq.mpr",
"Submodule"... | [] | cases x
rw [mk_mem_projectivization_iff, Projectivization.submodule_mk,
Submodule.span_singleton_le_iff_mem] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Projectivization.Subspace | {
"line": 278,
"column": 2
} | {
"line": 280,
"column": 40
} | {
"line": 282,
"column": 0
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Submodule K V\nx : ℙ K V\n⊢ x ∈ projectivization s ↔ x.submodule ≤ s",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Projectivization.mk",
"Eq.mpr",
"Submodule"... | [] | cases x
rw [mk_mem_projectivization_iff, Projectivization.submodule_mk,
Submodule.span_singleton_le_iff_mem] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Projectivization.PSL.PSL2 | {
"line": 28,
"column": 51
} | {
"line": 29,
"column": 92
} | {
"line": 29,
"column": 92
} | [
{
"pp": "ι : Type u_1\nF : Type u_2\ninst✝² : Field F\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\ni j : ι\nhij : i ≠ j\nb : F\nw : ι → F\n⊢ (b * w j) • Pi.single i 1 = SpecialLinearGroup.transvection hij b • w - w",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Pi.Function.module",
... | [] | by simp [mul_smul,
Matrix.SpecialLinearGroup.smul_def, transvection_coe, add_smul, Matrix.single_mulVec_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.QuadraticForm.Basis | {
"line": 50,
"column": 68
} | {
"line": 51,
"column": 63
} | {
"line": 53,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R M\ninst✝ : Module R N\nQ : QuadraticMap R M N\nbm : Basis ι R M\nx : M\n⊢ (linearCombination R (polarSym2 ⇑Q ∘ Sym2.map ⇑bm)) (bm.repr x).sym2Mul -\n (li... | [] | by
rw [← apply_linearCombination', Basis.linearCombination_repr] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 256,
"column": 38
} | {
"line": 256,
"column": 45
} | {
"line": 256,
"column": 46
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\nS : Type u_3\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Free R V\ninst✝ : Module.Finite R V\ng : SpecialLinearGroup R V\n⊢ (Units.map ↑(algebraMap R S)) (LinearEquiv.det ↑g) = 1",
"ppTerm... | [
"R : Type u_1\nV : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\nS : Type u_3\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : Module.Free R V\ninst✝ : Module.Finite R V\ng : SpecialLinearGroup R V\n⊢ (Units.map ↑(algebraMap R S)) 1 = 1"
] | g.prop, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 458,
"column": 6
} | {
"line": 458,
"column": 10
} | {
"line": 459,
"column": 6
} | [
{
"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 : Subsingleton V\nhR : Nontrivial R\n⊢ 1 = r",
"ppTerm": "?pos✝",
"assigned": true... | [
"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 : Subsingleton V\nhR : Nontrivial R\n⊢ r = 1"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 505,
"column": 4
} | {
"line": 505,
"column": 51
} | {
"line": 507,
"column": 0
} | [
{
"pp": "case neg\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\ng : ↥(Subgroup.center (SpecialLinearGroup R V))\nhR : ¬Subsingleton R\nhV : Nontrivial V\n⊢ ↑↑↑g = ⋯.choose • LinearMap.id",
"ppTerm": "?neg✝... | [] | rw [← (mem_center_iff.mp g.prop).choose_spec.2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 67
} | {
"line": 125,
"column": 4
} | [
{
"pp": "case e'_2\nR : Type u_1\nM : Type u_2\nM' : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : CommRing R\ninst✝² : LinearOrder R\ninst✝¹ : Module R M\nQ : QuadraticForm R M\ninst✝ : Module R M'\nQ' : QuadraticForm R M'\ne : IsometryEquiv Q Q'\nj : ℕ\n⊢ (∃ V, Module.finrank R ↥V = j ... | [
"case e'_2\nR : Type u_1\nM : Type u_2\nM' : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : CommRing R\ninst✝² : LinearOrder R\ninst✝¹ : Module R M\nQ : QuadraticForm R M\ninst✝ : Module R M'\nQ' : QuadraticForm R M'\ne : IsometryEquiv Q Q'\nj : ℕ\n⊢ ∀ (a : Submodule R M),\n Module.finrank... | apply (Submodule.orderIsoMapComap e.toLinearEquiv).exists_congr | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 33
} | {
"line": 249,
"column": 0
} | [
{
"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\nι : Type u_5\ninst✝¹ : Fintype ι\nw : ι → 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nhQ : Equivalent Q (weightedSumSquares 𝕜 w)\n⊢ sigPos (weightedSumSquares 𝕜 w) =... | [] | exact sigPos_weightedSumSquares | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 85,
"column": 2
} | {
"line": 96,
"column": 36
} | {
"line": 98,
"column": 0
} | [
{
"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✝... | [] | classical
have : Matrix.diagLinearMap (b.support ⊕ ι) R R ∘ h =
Sum.elimZeroLeft ∘ fun i : b.support ↦ algebraMap ℤ R ∘ (P.pairingIn ℤ · i) := by
ext; rw [comp_apply, h_def]; aesop
apply LinearIndependent.of_comp (Matrix.diagLinearMap _ _ _)
rw [this, LinearMap.linearIndependent_iff_of_injOn _ Sum.elim_... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 85,
"column": 2
} | {
"line": 96,
"column": 36
} | {
"line": 98,
"column": 0
} | [
{
"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✝... | [] | classical
have : Matrix.diagLinearMap (b.support ⊕ ι) R R ∘ h =
Sum.elimZeroLeft ∘ fun i : b.support ↦ algebraMap ℤ R ∘ (P.pairingIn ℤ · i) := by
ext; rw [comp_apply, h_def]; aesop
apply LinearIndependent.of_comp (Matrix.diagLinearMap _ _ _)
rw [this, LinearMap.linearIndependent_iff_of_injOn _ Sum.elim_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 85,
"column": 2
} | {
"line": 96,
"column": 36
} | {
"line": 98,
"column": 0
} | [
{
"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✝... | [] | classical
have : Matrix.diagLinearMap (b.support ⊕ ι) R R ∘ h =
Sum.elimZeroLeft ∘ fun i : b.support ↦ algebraMap ℤ R ∘ (P.pairingIn ℤ · i) := by
ext; rw [comp_apply, h_def]; aesop
apply LinearIndependent.of_comp (Matrix.diagLinearMap _ _ _)
rw [this, LinearMap.linearIndependent_iff_of_injOn _ Sum.elim_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 44
} | {
"line": 277,
"column": 0
} | [
{
"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... | [] | simp [neg_eq_iff_eq_neg, sub_eq_add_neg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 88,
"column": 2
} | {
"line": 99,
"column": 8
} | {
"line": 101,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CharZero R\ninst✝⁶ : IsDomain R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : Finite ι\ninst✝ : P.IsCrystallographic\nb : P.Base\ni j k ... | [] | let _i := P.indexNeg
replace hk : α (-k) + α j - α i ∈ Φ := by
rw [← neg_mem_range_root_iff]
convert! hk using 1
simp only [indexNeg_neg, root_reflectionPerm, reflection_apply_self]
module
rw [← neg_mem_range_root_iff]
convert!
b.root_sub_root_mem_of_mem_of_mem j i (-k) hij.symm hj hi hk (by c... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 88,
"column": 2
} | {
"line": 99,
"column": 8
} | {
"line": 101,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : CharZero R\ninst✝⁶ : IsDomain R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\nP : RootPairing ι R M N\ninst✝¹ : Finite ι\ninst✝ : P.IsCrystallographic\nb : P.Base\ni j k ... | [] | let _i := P.indexNeg
replace hk : α (-k) + α j - α i ∈ Φ := by
rw [← neg_mem_range_root_iff]
convert! hk using 1
simp only [indexNeg_neg, root_reflectionPerm, reflection_apply_self]
module
rw [← neg_mem_range_root_iff]
convert!
b.root_sub_root_mem_of_mem_of_mem j i (-k) hij.symm hj hi hk (by c... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 370,
"column": 4
} | {
"line": 370,
"column": 15
} | {
"line": 371,
"column": 2
} | [
{
"pp": "case inl\nι : 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 : CharZe... | [] | simp [f, h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 8
} | {
"line": 325,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : P.EmbeddedG2\ninst✝² : Finite ι\ninst✝¹ : CharZero R\ninst✝ : IsDomain R\nhli : Injective ⇑(Fintype.... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 406,
"column": 4
} | {
"line": 406,
"column": 72
} | {
"line": 407,
"column": 2
} | [
{
"pp": "case mem.inr.inr\nι : 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 ... | [] | · exact LieSubalgebra.subset_lieSpan <| by simp [ω_mul_f, mul_assoc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 128,
"column": 51
} | {
"line": 128,
"column": 59
} | {
"line": 128,
"column": 59
} | [
{
"pp": "case inr.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Finite ι\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : CharZero R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsCrystallographic\nb ... | [
"case inr.inl\nι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Finite ι\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : CharZero R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : RootPairing ι R M N\ninst✝² : P.IsCrystallographic\nb : P.Base\nin... | zero_sub | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.SymmetricAlgebra.Basic | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 18
} | {
"line": 128,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ ↑(lift (ι R M)) ∘ₗ ι R M = ↑(AlgHom.id R (SymmetricAlgebra R M)) ∘ₗ ι R M",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"AlgHom... | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ ι R M = ↑(AlgHom.id R (SymmetricAlgebra R M)) ∘ₗ ι R M"
] | rw [lift_comp_ι] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 292,
"column": 6
} | {
"line": 292,
"column": 88
} | {
"line": 293,
"column": 6
} | [
{
"pp": "case h₂\nι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : CharZero K\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : Fintype ι\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module K M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module K N\nP : RootPairing ι K M N\ninst✝² : P.IsCrystallographic\nb : P.... | [
"case h₂\nι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Field K\ninst✝⁹ : CharZero K\ninst✝⁸ : DecidableEq ι\ninst✝⁷ : Fintype ι\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module K M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module K N\nP : RootPairing ι K M N\ninst✝² : P.IsCrystallographic\nb : P.Base\ninst✝¹... | have : ⁅f ⟨k, hk⟩, v b l⁆ = (P.chainTopCoeff k l + 1 : K) • v b j := f_lie_v_ne h₁ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 309,
"column": 48
} | {
"line": 309,
"column": 59
} | {
"line": 310,
"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... | eq_top_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Constructions.Cylinders | {
"line": 369,
"column": 4
} | {
"line": 370,
"column": 78
} | {
"line": 371,
"column": 4
} | [
{
"pp": "case a\nι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\ni : ι\nx : Set ((i : ι) → α i)\n⊢ (x ∈ (fun t ↦ {i}.pi t) '' univ.pi fun i ↦ {s | MeasurableSet s}) → x ∈ measurableCylinders α",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Mea... | [
"case a\nι : Type u_1\nα : ι → Type u_2\ninst✝ : (i : ι) → MeasurableSpace (α i)\ni : ι\nx : Set ((i : ι) → α i)\n⊢ ∀ (x_1 : (i : ι) → Set (α i)),\n (∀ (i : ι), MeasurableSet (x_1 i)) → eval i ⁻¹' x_1 i = x → ∃ s S, MeasurableSet S ∧ x = cylinder s S"
] | simp only [singleton_pi, mem_image, mem_pi, mem_univ, mem_setOf_eq,
forall_true_left, mem_measurableCylinders, forall_exists_index, and_imp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.SetAlgebra | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 51
} | {
"line": 132,
"column": 2
} | [
{
"pp": "α : Type u_1\n𝒜 : Set (Set α)\ns : Set α\nms : MeasurableSet s\n⊢ MeasurableSet s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"MeasurableSet",
"MeasurableSet.empty",
"MeasurableSpace.measurableSet_generateFrom",
"Membership.mem",
"MeasureTheory.g... | [] | induction s, ms using generateFrom_induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 185,
"column": 9
} | {
"line": 185,
"column": 50
} | {
"line": 185,
"column": 50
} | [
{
"pp": "case h_mem\nα : Type u_1\nC : Set (Set α)\ns t : Set α\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nhC : IsSetSemiring C\nhs : s ∈ C\nht : t ∈ C\nhst : s ⊆ t\n⊢ ⋃₀ insert s ↑(hC.disjointOfDiff ht hs) ∈ C",
"ppTerm": "?h_mem",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"case h_mem\nα : Type u_1\nC : Set (Set α)\ns t : Set α\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nhC : IsSetSemiring C\nhs : s ∈ C\nht : t ∈ C\nhst : s ⊆ t\n⊢ t ∈ C"
] | hC.sUnion_insert_disjointOfDiff ht hs hst | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 102,
"column": 6
} | {
"line": 103,
"column": 55
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nhs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUn... | [] | simpa using mem_iUnion₂.mp (cthickening_subset_iUnion_closedBall_of_lt (s (f j))
(by positivity) (lt_two_mul_self hrp') (hf₀ j)) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 102,
"column": 6
} | {
"line": 103,
"column": 55
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nhs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUn... | [] | simpa using mem_iUnion₂.mp (cthickening_subset_iUnion_closedBall_of_lt (s (f j))
(by positivity) (lt_two_mul_self hrp') (hf₀ j)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 102,
"column": 6
} | {
"line": 103,
"column": 55
} | {
"line": 104,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝⁵ : PseudoMetricSpace α\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsUnifLocDoublingMeasure μ\np : ℕ → Prop\ns : ℕ → Set α\nhs : ∀ (i : ℕ), IsClosed[PseudoMetricSpace.toUn... | [] | simpa using mem_iUnion₂.mp (cthickening_subset_iUnion_closedBall_of_lt (s (f j))
(by positivity) (lt_two_mul_self hrp') (hf₀ j)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 438,
"column": 6
} | {
"line": 438,
"column": 38
} | {
"line": 439,
"column": 6
} | [
{
"pp": "case succ.inr\nα : Type u_1\nC : Set (Set α)\ns t : Set α\nI✝ : Finset (Set α)\nG✝ : Type u_2\ninst✝² : AddCommMonoid G✝\nm m' : AddContent G✝ C\ninst✝¹ : LinearOrder α\nG : Type u_3\ninst✝ : AddCommGroup G\nf : α → G\nn : ℕ\nih :\n ∀ (I : Finset (Set α)),\n ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v... | [
"case succ.inr\nα : Type u_1\nC : Set (Set α)\ns t : Set α\nI✝ : Finset (Set α)\nG✝ : Type u_2\ninst✝² : AddCommMonoid G✝\nm m' : AddContent G✝ C\ninst✝¹ : LinearOrder α\nG : Type u_3\ninst✝ : AddCommGroup G\nf : α → G\nn : ℕ\nih :\n ∀ (I : Finset (Set α)),\n ↑I ⊆ {s | ∃ u v, u ≤ v ∧ s = Set.Ioc u v} →\n (... | let I' := I.erase (Set.Ioc u' v) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 511,
"column": 4
} | {
"line": 511,
"column": 57
} | {
"line": 512,
"column": 4
} | [
{
"pp": "case insert\nα : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nι : Type u_3\nhC : IsSetRing C\ns : ι → Set α\ni : ι\nS : Finset ι\nhiS : i ∉ S\nh : (∀ n ∈ S, s n ∈ C) → (↑S).PairwiseDisjoint s → m (⋃ i ∈ S, s i) = ∑ i ∈ S, m (s i)\nhs : ∀ n ∈ insert i S, s n ∈ C\... | [
"case insert\nα : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝ : AddCommMonoid G\nm : AddContent G C\nι : Type u_3\nhC : IsSetRing C\ns : ι → Set α\ni : ι\nS : Finset ι\nhiS : i ∉ S\nh : (∀ n ∈ S, s n ∈ C) → (↑S).PairwiseDisjoint s → m (⋃ i ∈ S, s i) = ∑ i ∈ S, m (s i)\nhS : (↑(insert i S)).PairwiseDisjoint s\nhs... | simp only [Finset.mem_insert, forall_eq_or_imp] at hs | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 64
} | {
"line": 164,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nf : α → E\ns : Set α\nm m₂ m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nhm₂ : m₂ ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ninst✝ : SigmaFinite (μ.trim hm₂)\nhs_m : MeasurableSet s\nhs : ∀... | [] | rw [this, add_zero, Measure.restrict_restrict (hm _ hs_m)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 64
} | {
"line": 164,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nf : α → E\ns : Set α\nm m₂ m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nhm₂ : m₂ ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ninst✝ : SigmaFinite (μ.trim hm₂)\nhs_m : MeasurableSet s\nhs : ∀... | [] | rw [this, add_zero, Measure.restrict_restrict (hm _ hs_m)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 64
} | {
"line": 164,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nf : α → E\ns : Set α\nm m₂ m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nhm₂ : m₂ ≤ m0\ninst✝¹ : SigmaFinite (μ.trim hm)\ninst✝ : SigmaFinite (μ.trim hm₂)\nhs_m : MeasurableSet s\nhs : ∀... | [] | rw [this, add_zero, Measure.restrict_restrict (hm _ hs_m)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 35
} | {
"line": 65,
"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... | filter_upwards [lem0] with a ha | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 570,
"column": 4
} | {
"line": 570,
"column": 57
} | {
"line": 571,
"column": 4
} | [
{
"pp": "case insert\nα : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝² : AddCommMonoid G\nm : AddContent G C\ninst✝¹ : PartialOrder G\ninst✝ : CanonicallyOrderedAdd G\nι : Type u_3\nhC : IsSetRing C\ns : ι → Set α\ni : ι\nS : Finset ι\nhiS : i ∉ S\nh : (∀ n ∈ S, s n ∈ C) → m (⋃ i ∈ S, s i) ≤ ∑ i ∈ S, m (s i)... | [
"case insert\nα : Type u_1\nC : Set (Set α)\nG : Type u_2\ninst✝² : AddCommMonoid G\nm : AddContent G C\ninst✝¹ : PartialOrder G\ninst✝ : CanonicallyOrderedAdd G\nι : Type u_3\nhC : IsSetRing C\ns : ι → Set α\ni : ι\nS : Finset ι\nhiS : i ∉ S\nh : (∀ n ∈ S, s n ∈ C) → m (⋃ i ∈ S, s i) ≤ ∑ i ∈ S, m (s i)\nhs : s i ∈... | simp only [Finset.mem_insert, forall_eq_or_imp] at hs | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 254,
"column": 13
} | {
"line": 254,
"column": 46
} | {
"line": 254,
"column": 46
} | [
{
"pp": "case neg\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX Y : Ω → ℝ≥0∞\nhm : mΩ ≤ mΩ₀\nhσ : ¬SigmaFinite (P.trim hm)\n⊢ P⁻[X | mΩ] + P⁻[Y | mΩ] ≤ᵐ[P] P⁻[X + Y | mΩ]",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"ENNReal.... | [
"case neg\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX Y : Ω → ℝ≥0∞\nhm : mΩ ≤ mΩ₀\nhσ : ¬SigmaFinite (P.trim hm)\n⊢ 0 + 0 ≤ᵐ[P] 0"
] | condLExp_of_not_sigmaFinite hm hσ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 269,
"column": 19
} | {
"line": 269,
"column": 52
} | {
"line": 269,
"column": 52
} | [
{
"pp": "case neg\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX Y : Ω → ℝ≥0∞\nhX : AEMeasurable X P\nhm : mΩ ≤ mΩ₀\nhσ : ¬SigmaFinite (P.trim hm)\n⊢ P⁻[X + Y | mΩ] =ᵐ[P] P⁻[X | mΩ] + P⁻[Y | mΩ]",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"E... | [
"case neg\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX Y : Ω → ℝ≥0∞\nhX : AEMeasurable X P\nhm : mΩ ≤ mΩ₀\nhσ : ¬SigmaFinite (P.trim hm)\n⊢ 0 =ᵐ[P] 0 + 0"
] | condLExp_of_not_sigmaFinite hm hσ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 299,
"column": 13
} | {
"line": 299,
"column": 46
} | {
"line": 299,
"column": 46
} | [
{
"pp": "case neg\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX : Ω → ℝ≥0∞\nc : ℝ≥0∞\nhm : mΩ ≤ mΩ₀\nhσ : ¬SigmaFinite (P.trim hm)\n⊢ c • P⁻[X | mΩ] ≤ᵐ[P] P⁻[c • X | mΩ]",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Eq.mpr",
"instHSMul... | [
"case neg\nΩ : Type u_1\nmΩ₀ mΩ : MeasurableSpace Ω\nP : Measure Ω\nX : Ω → ℝ≥0∞\nc : ℝ≥0∞\nhm : mΩ ≤ mΩ₀\nhσ : ¬SigmaFinite (P.trim hm)\n⊢ c • 0 ≤ᵐ[P] 0"
] | condLExp_of_not_sigmaFinite hm hσ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.ConditionalLExpectation | {
"line": 327,
"column": 13
} | {
"line": 327,
"column": 46
} | {
"line": 327,
"column": 46
} | [
{
"pp": "case neg\nΩ : Type u_1\nmΩ₀ : MeasurableSpace Ω\nP : Measure Ω\nι : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝ : Countable ι\nX : ι → Ω → ℝ≥0∞\nhX : ∀ (i : ι), AEMeasurable (X i) P\nhm : mΩ ≤ mΩ₀\nhσ : ¬SigmaFinite (P.trim hm)\n⊢ P⁻[∑' (i : ι), X i | mΩ] =ᵐ[P] ∑' (i : ι), P⁻[X i | mΩ]",
"ppTerm": "?ne... | [
"case neg\nΩ : Type u_1\nmΩ₀ : MeasurableSpace Ω\nP : Measure Ω\nι : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝ : Countable ι\nX : ι → Ω → ℝ≥0∞\nhX : ∀ (i : ι), AEMeasurable (X i) P\nhm : mΩ ≤ mΩ₀\nhσ : ¬SigmaFinite (P.trim hm)\n⊢ 0 =ᵐ[P] ∑' (i : ι), 0"
] | condLExp_of_not_sigmaFinite hm hσ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 35
} | {
"line": 134,
"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\... | filter_upwards [lem0] with a ha | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 96
} | {
"line": 94,
"column": 0
} | [
{
"pp": "𝓧 : Type u_1\nm m𝓧 : MeasurableSpace 𝓧\nμ : Measure 𝓧\nf : 𝓧 → ℝ\nhf : Integrable f μ\nh'f : 0 ≤ᵐ[μ] f\nA : μ⁻[fun x ↦ ENNReal.ofReal (f x) | m] =ᵐ[μ] μ⁻[fun x ↦ ‖f x‖ₑ | m]\n⊢ (fun x ↦ ENNReal.ofReal (μ[f | m] x)) =ᵐ[μ] fun x ↦ ‖μ[f | m] x‖ₑ",
"ppTerm": "?m.126",
"assigned": true,
"us... | [] | filter_upwards [condExp_nonneg h'f (m := m)] with x hx using by simp [Real.enorm_eq_ofReal hx] | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 196,
"column": 2
} | {
"line": 198,
"column": 52
} | {
"line": 199,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhf : MemLp f 1 μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\n⊢ ∃ M, ∫⁻ (x : α), ↑‖{x | M ≤ ↑‖f x‖₊}.indicator f x‖₊ ∂μ ≤ ENNReal.ofReal ε",
"ppTerm": "?m.34",
"assigned": true,
"... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhf : MemLp f 1 μ\nhmeas : StronglyMeasurable f\nε : ℝ\nhε : 0 < ε\nhtendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun M ↦ {x | ↑M ≤ ↑‖f x‖₊}.indicator f x) atTop (𝓝 0)\n⊢ ∃ M, ∫⁻ (x : α), ↑‖{x | M ≤ ↑‖f x‖₊}.indicator f... | have htendsto :
∀ᵐ x ∂μ, Tendsto (fun M : ℕ => { x | (M : ℝ) ≤ ‖f x‖₊ }.indicator f x) atTop (𝓝 0) :=
univ_mem' (id fun x => tendsto_indicator_ge f x) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 255,
"column": 12
} | {
"line": 255,
"column": 29
} | {
"line": 255,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhf : MemLp f ∞ μ\nhmeas : StronglyMeasurable f\nhbdd : eLpNormEssSup f μ < ∞\nx : α\nhx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}\n⊢ x ∈ {x | eLpNormEssSup f μ < ↑‖f x‖₊}",
"ppTerm": ... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhf : MemLp f ∞ μ\nhmeas : StronglyMeasurable f\nhbdd : eLpNormEssSup f μ < ∞\nx : α\nhx : x ∈ {x | (eLpNormEssSup f μ + 1).toReal ≤ ‖f x‖}\n⊢ eLpNormEssSup f μ < ↑‖f x‖₊"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 6
} | {
"line": 168,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv : VectorMeasure α M\ninst✝ : T2Space M\nA B : Set α\nhA : MeasurableSet A\nhB : MeasurableSet B\nh' : v (B \\ A) = 0\n⊢ v (A \\ B) + v B = v A",
"ppTerm": "?m.25",
"assigned": true,
"... | [
"α : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\nv : VectorMeasure α M\ninst✝ : T2Space M\nA B : Set α\nhA : MeasurableSet A\nhB : MeasurableSet B\nh' : v (B \\ A) = 0\n⊢ v A = v (A \\ B) + v B"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 145,
"column": 51
} | {
"line": 146,
"column": 71
} | {
"line": 148,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nj : JordanDecomposition α\nr : ℝ\nhr : r < 0\n⊢ (r • j).negPart = (-r).toNNReal • j.posPart",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"not_le",
"Iff.mpr",
"MeasureTheory.JordanDecomposition.posPart",
"Eq.mpr",
... | [] | by
rw [real_smul_def, ← smul_posPart, if_neg (not_le.2 hr), neg_negPart] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 155,
"column": 4
} | {
"line": 155,
"column": 21
} | {
"line": 155,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ni : Set α\nhi : MeasurableSet i\n⊢ 0 = 0 i",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"AddMonoid.toAddZeroClass",
"PseudoMetricSpace.toUniformSpace",
... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\ni : Set α\nhi : MeasurableSet i\n⊢ 0 = 0 i"
] | FunLike.coe_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 508,
"column": 4
} | {
"line": 508,
"column": 44
} | {
"line": 510,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nx✝ : ℕ → Set α\nhf₁ : ∀ (i : ℕ), MeasurableSet (x✝ i)\nhf₂ : Pairwise (Disjoint on x✝)\n⊢ (∑' (b : ℕ), if MeasurableSet (x✝ b) then μ (x✝ b) else 0) = ∑' (i : ℕ), μ (x✝ i)",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants... | [] | exact tsum_congr fun n => if_pos (hf₁ n) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 21
} | {
"line": 238,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : 0 ≤[i] s\nhi₃ : s ≤[iᶜ] 0\nhμ : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi₁ hi₂\nhν : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iᶜ ⋯ hi₃\nk : Set α\nhk : MeasurableSet k\n⊢ ... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : 0 ≤[i] s\nhi₃ : s ≤[iᶜ] 0\nhμ : s.toJordanDecomposition.posPart = s.toMeasureOfZeroLE i hi₁ hi₂\nhν : s.toJordanDecomposition.negPart = s.toMeasureOfLEZero iᶜ ⋯ hi₃\nk : Set α\nhk : MeasurableSet k\n⊢ s (k ∩ i ∪ i... | Set.inter_comm i, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 751,
"column": 2
} | {
"line": 754,
"column": 23
} | {
"line": 756,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : TopologicalSpace M\ns : Set α\nx : α\nm : M\nhx : x ∉ s\n⊢ (dirac x m).restrict s = 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"MeasurableSet",
"eq_false",
"congrArg",
... | [] | classical
by_cases hs : MeasurableSet s
· simp [restrict_dirac, hs, hx]
· simp [restrict, hs] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 751,
"column": 2
} | {
"line": 754,
"column": 23
} | {
"line": 756,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : TopologicalSpace M\ns : Set α\nx : α\nm : M\nhx : x ∉ s\n⊢ (dirac x m).restrict s = 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"MeasurableSet",
"eq_false",
"congrArg",
... | [] | classical
by_cases hs : MeasurableSet s
· simp [restrict_dirac, hs, hx]
· simp [restrict, hs] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 751,
"column": 2
} | {
"line": 754,
"column": 23
} | {
"line": 756,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : TopologicalSpace M\ns : Set α\nx : α\nm : M\nhx : x ∉ s\n⊢ (dirac x m).restrict s = 0",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"MeasurableSet",
"eq_false",
"congrArg",
... | [] | classical
by_cases hs : MeasurableSet s
· simp [restrict_dirac, hs, hx]
· simp [restrict, hs] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 146,
"column": 2
} | {
"line": 148,
"column": 88
} | {
"line": 149,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable f μ\n⊢ μ.withDensityᵥ f ≪ᵥ μ.toENNRealVectorMeasure",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"MeasureTheory.... | [
"case neg\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : ¬Integrable f μ\n⊢ μ.withDensityᵥ f ≪ᵥ μ.toENNRealVectorMeasure"
] | · refine VectorMeasure.AbsolutelyContinuous.mk fun i hi₁ hi₂ => ?_
rw [toENNRealVectorMeasure_apply_measurable hi₁] at hi₂
rw [withDensityᵥ_apply hf hi₁, Measure.restrict_zero_set hi₂, integral_zero_measure] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 149,
"column": 2
} | {
"line": 150,
"column": 51
} | {
"line": 152,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : ¬Integrable f μ\n⊢ μ.withDensityᵥ f ≪ᵥ μ.toENNRealVectorMeasure",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormed... | [] | · rw [withDensityᵥ, dif_neg hf]
exact VectorMeasure.AbsolutelyContinuous.zero _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.VectorMeasure.WithDensity | {
"line": 192,
"column": 60
} | {
"line": 200,
"column": 51
} | {
"line": 202,
"column": 0
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhfi : Integrable f μ\n⊢ μ.withDensityᵥ f =\n (μ.withDensity fun x ↦ ENNReal.ofReal (f x)).toSignedMeasure -\n (μ.withDensity fun x ↦ ENNReal.ofReal (-f x)).toSignedMeasure",
"ppTerm": "?m.47",
"assigned": true,
"usedCons... | [] | by
haveI := isFiniteMeasure_withDensity_ofReal hfi.2
haveI := isFiniteMeasure_withDensity_ofReal hfi.neg.2
ext i hi
rw [withDensityᵥ_apply hfi hi,
integral_eq_lintegral_pos_part_sub_lintegral_neg_part hfi.integrableOn,
_root_.sub_apply, toSignedMeasure_apply_measurable hi,
toSignedMeasure_apply_meas... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 50,
"column": 13
} | {
"line": 50,
"column": 26
} | {
"line": 50,
"column": 26
} | [
{
"pp": "case hfm\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhp₀ : p ≠ 0\nhp : p ≠ ∞\nhf : AEStronglyMeasurable f μ\n⊢ AEMeasurable (fun x ↦ ‖f x‖ₑ ^ p.toReal) μ",
"ppTerm": "?hfm",
"assigned": true,
"usedConstants": [
... | [
"case hfm\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nhp₀ : p ≠ 0\nhp : p ≠ ∞\nhf : AEStronglyMeasurable f μ\n⊢ AEMeasurable (fun x ↦ ENNReal.ofReal ‖f x‖ ^ p.toReal) μ"
] | ← ofReal_norm | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 239,
"column": 4
} | {
"line": 241,
"column": 8
} | {
"line": 243,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nc : ℝ≥0\nhc : c ≠ 0\nhf : ¬AEStronglyMeasurable f μ\n⊢ lpNorm f p (c • μ) = c ^ p.toReal⁻¹ • lpNorm f p μ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [] | rw [lpNorm_of_not_aestronglyMeasurable hf, lpNorm_of_not_aestronglyMeasurable fun h ↦ hf <| by
simpa [hc] using h.smul_measure c⁻¹]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 239,
"column": 4
} | {
"line": 241,
"column": 8
} | {
"line": 243,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf : α → E\nc : ℝ≥0\nhc : c ≠ 0\nhf : ¬AEStronglyMeasurable f μ\n⊢ lpNorm f p (c • μ) = c ^ p.toReal⁻¹ • lpNorm f p μ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [] | rw [lpNorm_of_not_aestronglyMeasurable hf, lpNorm_of_not_aestronglyMeasurable fun h ↦ hf <| by
simpa [hc] using h.smul_measure c⁻¹]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Real | {
"line": 278,
"column": 4
} | {
"line": 279,
"column": 81
} | {
"line": 280,
"column": 4
} | [
{
"pp": "case pos\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\nf : α → E\np : ℝ≥0∞\nhp : 1 ≤ p\nhf : MemLp f p μ\nhp' : 0 < p\nhpt : p ≠ ∞\n⊢ Integrable (fun x ↦ ‖μ[f | m] x‖ ^ p.toReal) μ",
"ppTerm": ... | [
"case pos\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\nf : α → E\np : ℝ≥0∞\nhp✝ : 1 ≤ p\nhf : MemLp f p μ\nhp' : 0 < p\nhpt : p ≠ ∞\nhp : 1 ≤ p.toReal\n⊢ Integrable (fun x ↦ ‖μ[f | m] x‖ ^ p.toReal) μ"
] | have hp : 1 ≤ p.toReal := by
rwa [← ENNReal.toReal_one, ENNReal.toReal_le_toReal ENNReal.one_ne_top hpt] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 554,
"column": 11
} | {
"line": 554,
"column": 30
} | {
"line": 554,
"column": 31
} | [
{
"pp": "Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_3\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i ↦ (μs i).mass) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), (μs i).testAgainstNN f ≤ (μs i).testAgainstNN 0 + nndist f 0 * (... | [
"Ω : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_3\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nmass_lim : Tendsto (fun i ↦ (μs i).mass) F (𝓝 0)\nf : Ω →ᵇ ℝ≥0\nobs : ∀ (i : γ), (μs i).testAgainstNN f ≤ 0 + nndist f 0 * (μs i).mass\n⊢ Tendsto (fun b ↦ di... | testAgainstNN_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 839,
"column": 10
} | {
"line": 842,
"column": 85
} | {
"line": 843,
"column": 8
} | [
{
"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... | [] | rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul]
simp_rw [ENNReal.ofReal_coe_nnreal] at hℐ
refine mul_le_mul' le_rfl
(ENNReal.rpow_le_rpow (hℐ C).le (one_div_nonneg.2 ENNReal.toReal_nonneg)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 839,
"column": 10
} | {
"line": 842,
"column": 85
} | {
"line": 843,
"column": 8
} | [
{
"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... | [] | rw [ENNReal.smul_def, ENNReal.smul_def, smul_eq_mul, smul_eq_mul]
simp_rw [ENNReal.ofReal_coe_nnreal] at hℐ
refine mul_le_mul' le_rfl
(ENNReal.rpow_le_rpow (hℐ C).le (one_div_nonneg.2 ENNReal.toReal_nonneg)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Kernel.Indep | {
"line": 282,
"column": 6
} | {
"line": 282,
"column": 23
} | {
"line": 282,
"column": 24
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm' _mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\ns t : Set Ω\na✝ : s ∈ {s | MeasurableSet s}\nht : t ∈ {s | MeasurableSet s}\n⊢ ∀ᵐ (a : α) ∂μ, (κ a) (s ∩ t) = (κ a) s * (κ a) t",
"ppTerm": "?m.21",
"a... | [
"α : Type u_1\nΩ : Type u_2\n_mα : MeasurableSpace α\nm' _mΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\ninst✝ : IsZeroOrMarkovKernel κ\ns t : Set Ω\na✝ : s ∈ {s | MeasurableSet s}\nht : MeasurableSet t\n⊢ ∀ᵐ (a : α) ∂μ, (κ a) (s ∩ t) = (κ a) s * (κ a) t"
] | Set.mem_setOf_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Basic | {
"line": 277,
"column": 84
} | {
"line": 278,
"column": 36
} | {
"line": 280,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nx✝ : MeasurableSpace Ω\nβ : Type u_7\nγ : Type u_8\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nf : Ω → β\ng : Ω → γ\nμ : Measure Ω\n⊢ f ⟂ᵢ[μ] g ↔ ∀ (t1 t2 : Set Ω), MeasurableSet t1 → MeasurableSet t2 → μ (t1 ∩ t2) = μ t1 * μ t2",
"ppTerm": "?m.29",
"assigned": true,
"use... | [] | by
rw [IndepFun_iff_Indep, Indep_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 521,
"column": 6
} | {
"line": 522,
"column": 52
} | {
"line": 522,
"column": 52
} | [
{
"pp": "Ω : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\nμs : ℕ → Measure Ω\ninst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)\nf : Ω →ᵇ ℝ\nf_nn : 0 ≤ f\nh_opens : ∀ (G : Set Ω), IsOpen[inst✝²] G → μ G ≤ liminf (fun i ↦ (μs i) G) atTop\nsame : ... | [
"Ω : Type u_1\ninst✝³ : MeasurableSpace Ω\ninst✝² : TopologicalSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nμ : Measure Ω\nμs : ℕ → Measure Ω\ninst✝ : ∀ (i : ℕ), IsProbabilityMeasure (μs i)\nf : Ω →ᵇ ℝ\nf_nn : 0 ≤ f\nh_opens : ∀ (G : Set Ω), IsOpen[inst✝²] G → μ G ≤ liminf (fun i ↦ (μs i) G) atTop\nsame : ∫⁻ (x : Ω), ... | @integral_eq_lintegral_of_nonneg_ae Ω _ μ f (Eventually.of_forall f_nn)
f.continuous.measurable.aestronglyMeasurable | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 170,
"column": 23
} | {
"line": 187,
"column": 14
} | {
"line": 189,
"column": 0
} | [
{
"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 : iIndepFun f κ μ\nh : ∀ (i : ι), ∀ᵐ (a : α) ∂μ, f i =ᵐ[κ a] g i\n⊢ iIndepFun g κ μ",
"ppTerm"... | [] | by
rw [iIndepFun_iff_measure_inter_preimage_eq_mul] at hf ⊢
intro S sets hmeas
have : ∀ᵐ a ∂μ, ∀ i ∈ S, f i =ᵐ[κ a] g i :=
(ae_ball_iff (Finset.countable_toSet S)).2 (fun i hi ↦ h i)
filter_upwards [this, hf S hmeas] with a ha h'a
have A i (hi : i ∈ S) : (κ a) (g i ⁻¹' sets i) = (κ a) (f i ⁻¹' sets i) := ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Independence.Basic | {
"line": 836,
"column": 4
} | {
"line": 836,
"column": 44
} | {
"line": 838,
"column": 0
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\n_mΩ : MeasurableSpace Ω\nμ : Measure Ω\nβ : ι → Type u_10\nm : (i : ι) → MeasurableSpace (β i)\nf✝ : (i : ι) → Ω → β i\nh : ∀ (s : Finset ι), iIndepFun (s.restrict f✝) μ\ns : Finset ι\nf : ι → Set Ω\nhs : ∀ i ∈ s, MeasurableSet (f i)\nthis : ⋂ i ∈ s, f i = ⋂ i, f ↑i\n⊢ μ (⋂ ... | [] | exact (h s).meas_iInter fun i ↦ hs i i.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 309,
"column": 73
} | {
"line": 309,
"column": 99
} | {
"line": 309,
"column": 99
} | [
{
"pp": "case pos\nα : Type u_1\nΩ : Type u_2\nβ : Type u_4\nγ : Type u_6\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteKernel κ\nf : Ω → β\ng : Ω → γ\nhf : Measurable f\nhg : Measurable... | [
"case pos\nα : Type u_1\nΩ : Type u_2\nβ : Type u_4\nγ : Type u_6\nmα : MeasurableSpace α\nmΩ : MeasurableSpace Ω\nκ : Kernel α Ω\nμ : Measure α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteKernel κ\nf : Ω → β\ng : Ω → γ\nhf : Measurable f\nhg : Measurable g\nh :\n ∀... | map_apply' _ (by fun_prop) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 727,
"column": 8
} | {
"line": 727,
"column": 49
} | {
"line": 727,
"column": 49
} | [
{
"pp": "γ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\nhne : F.NeBot\nh :\n ∀ (f : Ω → ℝ),\n (∃ C, ∀ (x y : Ω), dist (f x) (f y) ≤ C) →... | [
"γ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\nhne : F.NeBot\nh :\n ∀ (f : Ω → ℝ),\n (∃ C, ∀ (x y : Ω), dist (f x) (f y) ≤ C) →\n (∃ L... | ← integral_indicator_one hs.measurableSet | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 694,
"column": 2
} | {
"line": 736,
"column": 56
} | {
"line": 738,
"column": 0
} | [
{
"pp": "γ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\n⊢ Tendsto μs F (𝓝 μ) ↔\n ∀ (f : Ω → ℝ),\n (∃ C, ∀ (x y : Ω), dist (f x) (f ... | [] | constructor
· -- A bounded Lipschitz function is in particular a bounded continuous function, and we already
-- know that weak convergence implies convergence of their integrals
intro h f hf_bounded hf_lip
simp_rw [ProbabilityMeasure.tendsto_iff_forall_integral_tendsto] at h
let f' : BoundedContinuous... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Portmanteau | {
"line": 694,
"column": 2
} | {
"line": 736,
"column": 56
} | {
"line": 738,
"column": 0
} | [
{
"pp": "γ : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\ninst✝² : PseudoEMetricSpace Ω\ninst✝¹ : OpensMeasurableSpace Ω\nF : Filter γ\ninst✝ : F.IsCountablyGenerated\nμs : γ → ProbabilityMeasure Ω\nμ : ProbabilityMeasure Ω\n⊢ Tendsto μs F (𝓝 μ) ↔\n ∀ (f : Ω → ℝ),\n (∃ C, ∀ (x y : Ω), dist (f x) (f ... | [] | constructor
· -- A bounded Lipschitz function is in particular a bounded continuous function, and we already
-- know that weak convergence implies convergence of their integrals
intro h f hf_bounded hf_lip
simp_rw [ProbabilityMeasure.tendsto_iff_forall_integral_tendsto] at h
let f' : BoundedContinuous... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.Variance | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 33
} | {
"line": 239,
"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⊢ ¬MemLp (fun x ↦ X x + c) 2 μ",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCo... | [
"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\nh_memLp : MemLp (fun x ↦ X x + c) 2 μ\n⊢ MemLp X 2 μ"
] | refine fun h_memLp ↦ hX_Lp ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Moments.Variance | {
"line": 344,
"column": 2
} | {
"line": 345,
"column": 42
} | {
"line": 346,
"column": 2
} | [
{
"pp": "case pos\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhm : AEStronglyMeasurable X μ\nhX : MemLp X 2 μ\n⊢ Var[X; μ] ≤ ∫ (x : Ω), (X ^ 2) x ∂μ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
... | [
"case neg\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhm : AEStronglyMeasurable X μ\nhX : ¬MemLp X 2 μ\n⊢ Var[X; μ] ≤ ∫ (x : Ω), (X ^ 2) x ∂μ"
] | · rw [variance_eq_sub hX]
simp only [sq_nonneg, sub_le_self_iff] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Moments.Variance | {
"line": 371,
"column": 4
} | {
"line": 371,
"column": 8
} | {
"line": 372,
"column": 4
} | [
{
"pp": "case neg\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nhℒ : ¬MemLp X 2 μ\n⊢ eVar[X; μ] = ∫⁻ (ω : Ω), ‖X ω‖ₑ ^ 2 ∂μ - ENNReal.ofReal ((∫ (x : Ω), X x ∂μ) ^ 2)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstan... | [
"case neg\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsProbabilityMeasure μ\nX : Ω → ℝ\nhX : AEStronglyMeasurable X μ\nhℒ : ¬MemLp X 2 μ\n⊢ ∫⁻ (ω : Ω), ‖X ω‖ₑ ^ 2 ∂μ - ENNReal.ofReal ((∫ (x : Ω), X x ∂μ) ^ 2) = eVar[X; μ]"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Probability.Moments.Variance | {
"line": 430,
"column": 6
} | {
"line": 430,
"column": 10
} | {
"line": 431,
"column": 6
} | [
{
"pp": "case pos\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_3\nX : ι → Ω → ℝ\ns : Finset ι\nhs : ∀ i ∈ s, MemLp (X i) 2 μ\nh : (↑s).Pairwise fun i j ↦ X i ⟂ᵢ[μ] X j\nh'' : ∀ i ∈ s, X i =ᵐ[μ] 0\n⊢ 0 = ∑ i ∈ s, Var[X i; μ]",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": ... | [
"case pos\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_3\nX : ι → Ω → ℝ\ns : Finset ι\nhs : ∀ i ∈ s, MemLp (X i) 2 μ\nh : (↑s).Pairwise fun i j ↦ X i ⟂ᵢ[μ] X j\nh'' : ∀ i ∈ s, X i =ᵐ[μ] 0\n⊢ ∑ i ∈ s, Var[X i; μ] = 0"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Probability.IdentDistrib | {
"line": 189,
"column": 6
} | {
"line": 189,
"column": 58
} | {
"line": 190,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nf : α → γ\ng : β → γ\ninst✝² : NormedAddCommGroup γ\ninst✝¹ : NormedSpace ℝ γ\ninst✝ : BorelSpace γ\nh : IdentDistrib f g μ ν\nhf : AEStronglyMeasu... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nf : α → γ\ng : β → γ\ninst✝² : NormedAddCommGroup γ\ninst✝¹ : NormedSpace ℝ γ\ninst✝ : BorelSpace γ\nh : IdentDistrib f g μ ν\nhf : AEStronglyMeasurable f μ\n⊢... | rw [aestronglyMeasurable_iff_aemeasurable_separable] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Independence.Integration | {
"line": 448,
"column": 4
} | {
"line": 448,
"column": 38
} | {
"line": 448,
"column": 39
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nβ : Type u_3\nβ' : Type u_4\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nf : Ω → β\ng : Ω → β'\nhfm : Measurable f\nhgm : Measurable g\nA : Set β\nhA : MeasurableSet A\nB : Set β'\nh : integral μ (A.indicator 1 ∘ f * ... | [
"Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nβ : Type u_3\nβ' : Type u_4\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nf : Ω → β\ng : Ω → β'\nhfm : Measurable f\nhgm : Measurable g\nA : Set β\nhA : MeasurableSet A\nB : Set β'\nh : integral μ (A.indicator 1 ∘ f * B.indicator ... | ← integral_indicator_one (hfm hA), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConvergenceInDistribution | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 28
} | {
"line": 107,
"column": 0
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nΩ' : Type u_3\nΩ : ι → Type u_5\nm : (i : ι) → MeasurableSpace (Ω i)\nμ : (i : ι) → Measure (Ω i)\ninst✝³ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nm' : MeasurableSpace Ω'\nμ' : Measure Ω'\ninst✝² : IsProbabilityMeasure μ'\nmE : MeasurableSpace E\nX : (i : ι) → Ω i → E\nZ : Ω... | [] | exact tendsto_const_nhds | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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