module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 46,
"column": 2
} | {
"line": 57,
"column": 19
} | [
{
"pp": "case mp\nm : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝ : CommRing R\nA : Matrix m n R\nhA : A.IsTotallyUnimodular\n⊢ ∀ (k : ℕ) (f : Fin k → m) (g : Fin k → n), (A.submatrix f g).det ∈ Set.range SignType.cast",
"usedConstants": [
"SignType.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
... | · intro k f g
by_cases hfg : f.Injective ∧ g.Injective
· exact hA k f g hfg.1 hfg.2
· use 0
rw [SignType.coe_zero, eq_comm]
simp_rw [not_and_or, Function.not_injective_iff] at hfg
obtain ⟨i, j, hfij, hij⟩ | ⟨i, j, hgij, hij⟩ := hfg
· rw [← det_transpose, transpose_submatrix]
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 63
} | [
{
"pp": "m : Type u_1\nm' : Type u_2\nn : Type u_3\nR : Type u_5\ninst✝ : CommRing R\nA : Matrix m n R\n⊢ (A.fromRows (replicateRow m' 0)).IsTotallyUnimodular ↔ A.IsTotallyUnimodular",
"usedConstants": [
"CommSemiring.toSemiring",
"Classical.propDecidable",
"Matrix.fromRows_isTotallyUnimod... | refine fromRows_isTotallyUnimodular_iff_rows <| fun _ _ => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Matrix.Gershgorin | {
"line": 46,
"column": 4
} | {
"line": 46,
"column": 38
} | [
{
"pp": "case inr\nK : 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 : (Fins... | simp_rw [mem_closedBall_iff_norm'] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.LinearAlgebra.Multilinear.FiniteDimensional | {
"line": 47,
"column": 6
} | {
"line": 49,
"column": 70
} | [
{
"pp": "case intro\nR : Type u_2\nM₂ : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : Module R M₂\ninst✝⁵ : Module.Finite R M₂\ninst✝⁴ : Module.Free R M₂\nn : ℕ\nih :\n ∀ (N : Fin n → Type u_5) [inst : (i : Fin n) → AddCommGroup (N i)] [inst_1 : (i : Fin n) → Module R (N i)]\n [∀ (i : Fi... | exact
⟨Module.Free.of_equiv (multilinearCurryLeftEquiv R N M₂).symm,
Module.Finite.equiv (multilinearCurryLeftEquiv R N M₂).symm⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.PiTensorProduct | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 26
} | [
{
"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... | change _ = r • (1 * x) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.LinearAlgebra.Projectivization.Subspace | {
"line": 100,
"column": 19
} | {
"line": 100,
"column": 47
} | [
{
"pp": "case mem_add\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nA : Set (ℙ K V)\nB : Subspace K V\nh : A ≤ ↑B\nx : ℙ K V\nv✝ w✝ : V\nhv✝ : v✝ ≠ 0\nhw✝ : w✝ ≠ 0\nhvw✝ : v✝ + w✝ ≠ 0\na✝¹ : spanCarrier A (Projectivization.mk K v✝ hv✝)\na✝ : spanCarrier A (Pr... | apply B.mem_add; assumption' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Projectivization.Subspace | {
"line": 100,
"column": 19
} | {
"line": 100,
"column": 47
} | [
{
"pp": "case mem_add\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nA : Set (ℙ K V)\nB : Subspace K V\nh : A ≤ ↑B\nx : ℙ K V\nv✝ w✝ : V\nhv✝ : v✝ ≠ 0\nhw✝ : w✝ ≠ 0\nhvw✝ : v✝ + w✝ ≠ 0\na✝¹ : spanCarrier A (Projectivization.mk K v✝ hv✝)\na✝ : spanCarrier A (Pr... | apply B.mem_add; assumption' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Projectivization.Independence | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 63
} | [
{
"pp": "K : Type u_2\nV : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nu v : ℙ K V\n⊢ Independent ![u, v] ↔ u ≠ v",
"usedConstants": [
"Projectivization.dependent_pair_iff_eq",
"Eq.mpr",
"Projectivization.Independent",
"Projectivization.Dependent",... | rw [independent_iff_not_dependent, dependent_pair_iff_eq u v] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Projectivization.Independence | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 63
} | [
{
"pp": "K : Type u_2\nV : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nu v : ℙ K V\n⊢ Independent ![u, v] ↔ u ≠ v",
"usedConstants": [
"Projectivization.dependent_pair_iff_eq",
"Eq.mpr",
"Projectivization.Independent",
"Projectivization.Dependent",... | rw [independent_iff_not_dependent, dependent_pair_iff_eq u v] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Projectivization.Independence | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 63
} | [
{
"pp": "K : Type u_2\nV : Type u_3\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nu v : ℙ K V\n⊢ Independent ![u, v] ↔ u ≠ v",
"usedConstants": [
"Projectivization.dependent_pair_iff_eq",
"Eq.mpr",
"Projectivization.Independent",
"Projectivization.Dependent",... | rw [independent_iff_not_dependent, dependent_pair_iff_eq u v] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Projectivization.Action | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 54
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝² : AddCommGroup V\ninst✝¹ : Field K\ninst✝ : Module K V\nthis : ∀ {a b c d : ℙ K V}, a ≠ b → c ≠ d → ∃ g, g • a = c ∧ g • b = d\nD D' E E' : ℙ K V\nhD : D ≠ D'\nhE : E ≠ E'\ng : V ≃ₗ[K] V\ngD : g • D = E\ngE : g • D' = E'\nhV : ¬FiniteDimensional K V\n⊢ LinearMap.det ↑... | apply LinearMap.det_eq_one_of_not_module_finite hV | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.SpecialLinearGroup | {
"line": 225,
"column": 8
} | {
"line": 225,
"column": 13
} | [
{
"pp": "case mp\nR : Type u_1\nV : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nu : LinearMap.GeneralLinearGroup R V\nv : SpecialLinearGroup R V\nhv : toGeneralLinearGroup v = u\n⊢ LinearEquiv.det u.toLinearEquiv = 1",
"usedConstants": [
"LinearEquiv.det",
"Eq.mpr... | ← hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.Signature | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 15
} | [
{
"pp": "𝕜 : Type u_4\ninst✝³ : Field 𝕜\ninst✝² : LinearOrder 𝕜\nι : Type u_5\ninst✝¹ : Fintype ι\nw : ι → 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set ι\nhs : ∀ i ∈ s, 0 < w i\n⊢ ((weightedSumSquares 𝕜 w).restrict (Pi.spanSubset 𝕜 s)).PosDef",
"usedConstants": [
"Pi.Function.module",
"Submo... | intro ⟨v, hv⟩ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Lemmas | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 31
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CharZero R\ninst✝⁸ : IsDomain R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : RootPairing ι R M N\ninst✝³ : Finite ι\ninst✝² : P.IsCrystallographic\nb : P.Base\ni j ... | rw [← P.not_isG2_iff_isNotG2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 252,
"column": 2
} | {
"line": 257,
"column": 9
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing 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✝¹ : CharZero R\ninst✝ : Fintype ι\ni : ↥b.support\n⊢ ω b * h ... | ext (k | k) (l | l)
· simp [ω, h]
· simp [ω, h]
· simp [ω, h]
· simp only [ω, h, Matrix.mul_apply, Fintype.sum_sum_type, Matrix.fromBlocks_apply₂₂]
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 252,
"column": 2
} | {
"line": 257,
"column": 9
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁷ : CommRing 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✝¹ : CharZero R\ninst✝ : Fintype ι\ni : ↥b.support\n⊢ ω b * h ... | ext (k | k) (l | l)
· simp [ω, h]
· simp [ω, h]
· simp [ω, h]
· simp only [ω, h, Matrix.mul_apply, Fintype.sum_sum_type, Matrix.fromBlocks_apply₂₂]
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 354,
"column": 4
} | {
"line": 355,
"column": 30
} | [
{
"pp": "case h.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 : Char... | replace h : i ≠ -k := by rintro rfl; exact P.ne_zero j <| by simpa using h
simp [e, h, -indexNeg_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 354,
"column": 4
} | {
"line": 355,
"column": 30
} | [
{
"pp": "case h.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 : Char... | replace h : i ≠ -k := by rintro rfl; exact P.ne_zero j <| by simpa using h
simp [e, h, -indexNeg_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 404,
"column": 6
} | {
"line": 404,
"column": 72
} | [
{
"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.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 404,
"column": 6
} | {
"line": 404,
"column": 72
} | [
{
"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.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Basic | {
"line": 404,
"column": 6
} | {
"line": 404,
"column": 72
} | [
{
"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.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 52
} | [
{
"pp": "case pos\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... | by_cases h₆ : P.root l + P.root i ∈ range P.root | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 504,
"column": 21
} | {
"line": 506,
"column": 7
} | [
{
"pp": "a b c d e f a' b' c' d' e' f' : ℤ\nS : Set (ℤ × ℤ)\nS_def : S = {(0, 0), (1, 1), (-1, -1), (1, 2), (2, 1), (-1, -2), (-2, -1), (1, 3), (3, 1), (-1, -3), (-3, -1)}\nha : (a, a') ∈ S\nhb : (b, b') ∈ S\nhc : (c, c') ∈ S\nhd : (d, d') ∈ S\nhe : (e, e') ∈ S\nhf : (f, f') ∈ S\nh₁ : c = a + 3 * b\nh₂ : c' = a... | by
simp [S_def] at ha hb hc hd he hf
omega | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 276,
"column": 4
} | {
"line": 329,
"column": 13
} | [
{
"pp": "case a.inr.inr\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\... | rcases eq_or_ne l j with rfl | h₃
· rw [← ⁅e i, f j⁆.transpose_apply, lie_e_f_ne_aux₁ hij, Pi.zero_apply, Matrix.zero_apply]
rcases eq_or_ne l (-i) with rfl | h₄
· rw [← ⁅e i, f j⁆.transpose_apply, lie_e_f_ne_aux₂ hij, Pi.zero_apply, Matrix.zero_apply]
/- Geck Case 2.
It's all just definition unfold... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Relations | {
"line": 276,
"column": 4
} | {
"line": 329,
"column": 13
} | [
{
"pp": "case a.inr.inr\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\... | rcases eq_or_ne l j with rfl | h₃
· rw [← ⁅e i, f j⁆.transpose_apply, lie_e_f_ne_aux₁ hij, Pi.zero_apply, Matrix.zero_apply]
rcases eq_or_ne l (-i) with rfl | h₄
· rw [← ⁅e i, f j⁆.transpose_apply, lie_e_f_ne_aux₂ hij, Pi.zero_apply, Matrix.zero_apply]
/- Geck Case 2.
It's all just definition unfold... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 531,
"column": 2
} | {
"line": 531,
"column": 34
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : Module R N\nP : RootPairing ι R M N\ninst✝⁴ : P.EmbeddedG2\ninst✝³ : Finite ι\ninst✝² : CharZero R\ninst✝¹ : IsDomain R\ninst✝ : P.IsIrreducible\n... | refine this fun k hk ij hij ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.RootSystem.Finite.G2 | {
"line": 621,
"column": 4
} | {
"line": 621,
"column": 27
} | [
{
"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.IsG2\nb : P.Base\ninst✝² : Finite ι\ninst✝¹ : CharZero R\ninst✝ : IsDomain R\ni j : ι\nhi : i ∈ b.... | b.span_int_root_support | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Logic.Hydra | {
"line": 87,
"column": 44
} | {
"line": 87,
"column": 58
} | [
{
"pp": "case h.e'_1.h.e'_3\nα : Type u_1\nr : α → α → Prop\ns' s t : Multiset α\n⊢ s' + t = t + s'",
"usedConstants": [
"Multiset.instAddCancelCommMonoid",
"Multiset",
"add_comm",
"AddCancelCommMonoid.toAddCommMonoid",
"AddCommSemigroup.toAddCommMagma",
"AddCommMonoid.to... | apply add_comm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Logic.Hydra | {
"line": 87,
"column": 44
} | {
"line": 87,
"column": 58
} | [
{
"pp": "case h.e'_1.h.e'_4\nα : Type u_1\nr : α → α → Prop\ns' s t : Multiset α\n⊢ s + t = t + s",
"usedConstants": [
"Multiset.instAddCancelCommMonoid",
"Multiset",
"add_comm",
"AddCancelCommMonoid.toAddCommMonoid",
"AddCommSemigroup.toAddCommMagma",
"AddCommMonoid.toAd... | apply add_comm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Logic.Hydra | {
"line": 142,
"column": 25
} | {
"line": 144,
"column": 9
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\na a₁ a₂ : α\nh₁ : r a₁ a\nh₂ : r a₂ a\n⊢ ∀ x' ∈ {a₁, a₂}, r x' a",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Multiset.instInsert",
"_private.Mathlib.Logic.Hydra.0.Relation.cutExpand_double._simp_1_3",
"Membership.mem",
"Multiset",
... | by
simp only [insert_eq_cons, mem_cons, mem_singleton, forall_eq_or_imp, forall_eq]
tauto | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.SetSemiring | {
"line": 111,
"column": 8
} | {
"line": 111,
"column": 53
} | [
{
"pp": "α : Type u_1\nC : Set (Set α)\nhC : IsSetSemiring C\ns : Set α\nS : Finset (Set α)\na✝ : s ∉ S\nih : ↑S ⊆ C → ∃ P, ↑P.parts ⊆ C\nhsC : s ∈ C\nhSC : ↑S ⊆ C\nP : Finpartition (S.sup id)\nhP : ↑P.parts ⊆ C\nhs : s ≠ ⊥\nt : Set α\nht : t ∈ P.parts\n⊢ ∃ Q, ↑Q.parts ⊆ C",
"usedConstants": [
"Measur... | exact hC.exists_finpartition_diff (hP ht) hsC | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.SetSemiring | {
"line": 144,
"column": 6
} | {
"line": 144,
"column": 48
} | [
{
"pp": "case insert\nα : Type u_1\nC : Set (Set α)\nhC : IsSetSemiring C\nt : Set α\nT : Finset (Set α)\na✝ : t ∉ T\nih : ∀ ⦃s : Set α⦄, s ∈ supClosure C → ↑T ⊆ C → s \\ T.sup id ∈ supClosure C\nhtC : t ∈ C\nhTC : ↑T ⊆ C\nS : Finset (Set α)\nhS : S.Nonempty\nhSC : ↑S ⊆ C\n⊢ S.sup' hS id \\ t ∈ supClosure C",
... | rw [sup'_eq_sup, ← Finset.sup_sdiff_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.RootSystem.GeckConstruction.Semisimple | {
"line": 380,
"column": 2
} | {
"line": 385,
"column": 14
} | [
{
"pp": "case h\nι : Type u_1\nK : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹³ : Field K\ninst✝¹² : CharZero K\ninst✝¹¹ : DecidableEq ι\ninst✝¹⁰ : Fintype ι\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module K M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : Module K N\nP : RootPairing ι K M N\ninst✝⁵ : P.IsRootSystem\ninst✝⁴ : P... | have key : v b j ∈ U := by
have : ⁅e j, x⁆ ∈ U := U.lie_mem (x := ⟨e j, e_mem_lieAlgebra j⟩) hx
have aux (k : b.support) : ⁅e j, u k⁆ = |b.cartanMatrix j k| • v b j := e_lie_u j k
simp_rw [← hc, lie_sum, lie_smul, aux, smul_comm (M := K), ← smul_assoc, ← Finset.sum_smul,
zsmul_eq_mul, mul_comm, ← LieS... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.SetAlgebra | {
"line": 197,
"column": 6
} | {
"line": 197,
"column": 31
} | [
{
"pp": "case compl.refine_1\nα : Type u_1\n𝒜 : Set (Set α)\ns u : Set α\nhs✝ : generateSetAlgebra 𝒜 u\nA : Set (Set (Set α))\nA_fin : A.Finite\nmem_A : ∀ a ∈ A, a.Finite\nhA : ∀ a ∈ A, ∀ t ∈ a, t ∈ 𝒜 ∨ tᶜ ∈ 𝒜\nu_eq : u = ⋃ a ∈ A, ⋂ t ∈ a, t\nthis✝ : Finite ↑A\nthis : ∀ (a : ↑A), Finite ↑↑a\nf : (a : ↑A) → ... | rw [compl_compl, or_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 101,
"column": 39
} | {
"line": 101,
"column": 53
} | [
{
"pp": "case refine_2\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator... | Set.inter_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 111,
"column": 37
} | {
"line": 111,
"column": 51
} | [
{
"pp": "case refine_2\nα : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\nf : α → E\ns : Set α\nhf_int : Integrable f μ\nhs : MeasurableSet s\nhm : m ≤ m0\nhμm this✝ : SigmaFinite (μ.trim hm)\nthis : s.indicator... | Set.inter_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.LiminfLimsup | {
"line": 213,
"column": 4
} | {
"line": 213,
"column": 73
} | [
{
"pp": "case neg\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\nM : ℝ\nhM : 0 < M\nr : ℕ → ℝ\nhr : Tendsto r at... | · simp only [r', hi, one_div, mem_Ioi, if_false, inv_pos]; positivity | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator | {
"line": 178,
"column": 62
} | {
"line": 178,
"column": 76
} | [
{
"pp": "case pos.refine_1\nα : 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 : Meas... | Set.inter_self | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 496,
"column": 35
} | {
"line": 496,
"column": 40
} | [
{
"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... | ← IH, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.LebesgueBochner | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 22
} | [
{
"pp": "𝓧 : Type u_1\nm m𝓧 : MeasurableSpace 𝓧\nμ : Measure 𝓧\nf : 𝓧 → ℝ≥0∞\nhf_meas : AEMeasurable f μ\nhf : ∫⁻ (x : 𝓧), f x ∂μ ≠ ∞\n⊢ (fun x ↦ (μ⁻[f|m] x).toReal) =ᶠ[ae μ] μ[fun x ↦ (f x).toReal | m]",
"usedConstants": [
"MeasureTheory.ae",
"InnerProductSpace.toNormedSpace",
"Real... | by_cases hm : m ≤ m𝓧 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.MeasureTheory.Measure.AddContent | {
"line": 496,
"column": 68
} | {
"line": 496,
"column": 86
} | [
{
"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... | onIocAux_apply uu' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.RadonNikodym | {
"line": 95,
"column": 24
} | {
"line": 95,
"column": 33
} | [
{
"pp": "case h\n𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\ninst✝ : IsFiniteMeasure μ\nhμν : μ ≪ ν\ng : 𝓧 → 𝓨\nhg : Measurable g\nhσ : SigmaFinite (Measure.map g ν)\nthis : SigmaFinite ν\nh_ne_top1 : ∀ᵐ (x : 𝓧) ∂ν, (Measure.map g μ).rnDeriv (Measure.ma... | h_ne_top1 | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 56
} | [
{
"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\n⊢ ∃ M, eLpNormEssSup ({x | M ≤ ↑‖f x‖₊}.indicator f) μ = 0",
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"P... | have hbdd : eLpNormEssSup f μ < ∞ := hf.eLpNorm_lt_top | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 252,
"column": 6
} | {
"line": 252,
"column": 76
} | [
{
"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 < ∞\n⊢ μ.restrict {x | (eLpNorm f ∞ μ + 1).toReal ≤ ↑‖f x‖₊} = 0",
"usedConstants": [
"Norm.norm",
"Seminor... | simp only [coe_nnnorm, eLpNorm_exponent_top, Measure.restrict_eq_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 23
} | [
{
"pp": "case pos\nΩ : 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\... | filter_upwards with ω | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 366,
"column": 8
} | {
"line": 366,
"column": 26
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nj₁ j₂ : JordanDecomposition α\nhj : j₁.toSignedMeasure = j₂.toSignedMeasure\nS : Set α\nhS₁ : MeasurableSet S\nhS₂ : j₁.toSignedMeasure ≤[S] 0\nhS₃ : 0 ≤[Sᶜ] j₁.toSignedMeasure\nhS₄ : j₁.posPart S = 0\nhS₅ : j₁.negPart Sᶜ = 0\nT : Set α\nhT₁ : MeasurableSet T\nh... | measureReal_union, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Jordan | {
"line": 381,
"column": 8
} | {
"line": 381,
"column": 26
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nj₁ j₂ : JordanDecomposition α\nhj : j₁.toSignedMeasure = j₂.toSignedMeasure\nS : Set α\nhS₁ : MeasurableSet S\nhS₂ : j₁.toSignedMeasure ≤[S] 0\nhS₃ : 0 ≤[Sᶜ] j₁.toSignedMeasure\nhS₄ : j₁.posPart S = 0\nhS₅ : j₁.negPart Sᶜ = 0\nT : Set α\nhT₁ : MeasurableSet T\nh... | measureReal_union, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 211,
"column": 4
} | {
"line": 212,
"column": 24
} | [
{
"pp": "case insert\nα : 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\nι : Type u_4\nf : ι → Set α\na : ι\ns : Finset ι\nhas : a ∉ s\nih : (↑s).PairwiseDisjoint f → (∀ b ∈ s, MeasurableSet (f b)) → ↑v (⋃ b ∈ s, f ... | simp only [Finset.mem_insert, iUnion_iUnion_eq_or_left, has, not_false_eq_true,
Finset.sum_insert] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 440,
"column": 6
} | {
"line": 441,
"column": 83
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nn : ℕ\nh : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifIntegrable f p μ\nf : Fin (n + 1) → α → β\nhfLp : ∀ (i : Fin (n + 1)), MemLp (f i)... | obtain rfl : i = Fin.last n := Fin.ext (le_antisymm (Fin.is_le i) hi)
exact hδ₂ _ hs (le_trans hμs <| ENNReal.ofReal_le_ofReal <| min_le_right _ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.UniformIntegrable | {
"line": 440,
"column": 6
} | {
"line": 441,
"column": 83
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\np : ℝ≥0∞\nhp_one : 1 ≤ p\nhp_top : p ≠ ∞\nn : ℕ\nh : ∀ {f : Fin n → α → β}, (∀ (i : Fin n), MemLp (f i) p μ) → UnifIntegrable f p μ\nf : Fin (n + 1) → α → β\nhfLp : ∀ (i : Fin (n + 1)), MemLp (f i)... | obtain rfl : i = Fin.last n := Fin.ext (le_antisymm (Fin.is_le i) hi)
exact hδ₂ _ hs (le_trans hμs <| ENNReal.ofReal_le_ofReal <| min_le_right _ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 308,
"column": 26
} | {
"line": 308,
"column": 98
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousAdd M\nv w : VectorMeasure α M\nx✝ : Set α\nhi : ¬MeasurableSet x✝\n⊢ (↑v + ↑w) x✝ = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddMonoid.to... | by rw [Pi.add_apply, v.not_measurable hi, w.not_measurable hi, add_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 618,
"column": 74
} | {
"line": 620,
"column": 5
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\nM : Type u_3\ninst✝¹ : AddCommMonoid M\ninst✝ : TopologicalSpace M\nv : VectorMeasure α M\n⊢ v.mapRange (AddMonoidHom.id M) ⋯ = v",
"usedConstants": [
"MeasurableSet",
"MeasureTheory.VectorMeasure.ext",
"AddMonoid.toAddZeroClass",
"M... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 882,
"column": 4
} | {
"line": 882,
"column": 47
} | [
{
"pp": "case pos\nα : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv w : VectorMeasure α M\ni : Set α\nh : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑v j ≤ ↑w j\nhi : MeasurableSet i\n⊢ v ≤[i] w",
"usedConstants": [
"Iff.... | exact (restrict_le_restrict_iff _ _ hi).2 h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 882,
"column": 4
} | {
"line": 882,
"column": 47
} | [
{
"pp": "case pos\nα : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv w : VectorMeasure α M\ni : Set α\nh : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑v j ≤ ↑w j\nhi : MeasurableSet i\n⊢ v ≤[i] w",
"usedConstants": [
"Iff.... | exact (restrict_le_restrict_iff _ _ hi).2 h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 882,
"column": 4
} | {
"line": 882,
"column": 47
} | [
{
"pp": "case pos\nα : Type u_1\nm : MeasurableSpace α\nM : Type u_3\ninst✝² : TopologicalSpace M\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nv w : VectorMeasure α M\ni : Set α\nh : ∀ ⦃j : Set α⦄, MeasurableSet j → j ⊆ i → ↑v j ≤ ↑w j\nhi : MeasurableSet i\n⊢ v ≤[i] w",
"usedConstants": [
"Iff.... | exact (restrict_le_restrict_iff _ _ hi).2 h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 254,
"column": 2
} | {
"line": 262,
"column": 78
} | [
{
"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\nhadd : s = t + μ.withDensityᵥ f\nhtμ' : t ⟂ᵥ μ.toENNRealVectorMeasure\nhtμ : t.toJordanDecomposition.posPart ⟂ₘ μ ∧ t.toJordanDecomposition.negPart ⟂ₘ μ\n⊢ s.HaveLebesgueDecomp... | refine
{ posPart := by
use ⟨t.toJordanDecomposition.posPart, fun x => ENNReal.ofReal (f x)⟩
refine ⟨hf.ennreal_ofReal, htμ.1, ?_⟩
rw [toJordanDecomposition_eq_of_eq_add_withDensity hf hfi htμ' hadd]
negPart := by
use ⟨t.toJordanDecomposition.negPart, fun x => ENNReal.ofReal (-f... | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 403,
"column": 57
} | {
"line": 406,
"column": 32
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\ns t : SignedMeasure α\nμ : Measure α\ninst✝¹ : s.HaveLebesgueDecomposition μ\ninst✝ : t.HaveLebesgueDecomposition μ\nhst : (s - t).HaveLebesgueDecomposition μ\n⊢ (s - t).rnDeriv μ =ᶠ[ae μ] s.rnDeriv μ - t.rnDeriv μ",
"usedConstants": [
"MeasureTheory.ae",
... | by
rw [sub_eq_add_neg] at hst
rw [sub_eq_add_neg, sub_eq_add_neg]
grw [rnDeriv_add, rnDeriv_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 442,
"column": 14
} | {
"line": 442,
"column": 56
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nc : ComplexMeasure α\ninst✝ : c.HaveLebesgueDecomposition μ\n| c",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"NormedSpace.t... | rw [← c.toComplexMeasure_to_signedMeasure] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 442,
"column": 14
} | {
"line": 442,
"column": 56
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nc : ComplexMeasure α\ninst✝ : c.HaveLebesgueDecomposition μ\n| c",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"NormedSpace.t... | rw [← c.toComplexMeasure_to_signedMeasure] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Decomposition.Lebesgue | {
"line": 442,
"column": 14
} | {
"line": 442,
"column": 56
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nc : ComplexMeasure α\ninst✝ : c.HaveLebesgueDecomposition μ\n| c",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"NormedSpace.t... | rw [← c.toComplexMeasure_to_signedMeasure] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1070,
"column": 17
} | {
"line": 1070,
"column": 22
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nN : Type u_5\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\nM : Type u_6\ninst✝² : AddCommGroup M\ninst✝¹ : TopologicalSpace M\ninst✝ : IsTopologicalAddGroup M\nv : VectorMeasure α M\nw : VectorMeasure α N\nh : v ≪ᵥ w\ns : Set α\nhs : ↑w s = 0\n⊢ -↑v s = 0"... | h hs, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1092,
"column": 18
} | {
"line": 1092,
"column": 23
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nM : Type u_4\nN : Type u_5\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\nR : Type u_6\ninst✝² : Semiring R\ninst✝¹ : DistribMulAction R M\ninst✝ : ContinuousConstSMul R M\nr : R\nv : VectorMeasure α M\... | h hs, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1155,
"column": 4
} | {
"line": 1156,
"column": 15
} | [
{
"pp": "case refine_1\nα : Type u_1\nm : MeasurableSpace α\nM : Type u_4\nN : Type u_5\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : TopologicalSpace M\ninst✝³ : AddCommMonoid N\ninst✝² : TopologicalSpace N\nv₁ v₂ : VectorMeasure α M\nw : VectorMeasure α N\ninst✝¹ : T2Space N\ninst✝ : ContinuousAdd M\nu : Set α\nhmu : M... | rw [add_apply, hu₁ _ (Set.subset_inter_iff.1 ht).1, hv₁ _ (Set.subset_inter_iff.1 ht).2,
zero_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1155,
"column": 4
} | {
"line": 1156,
"column": 15
} | [
{
"pp": "case refine_1\nα : Type u_1\nm : MeasurableSpace α\nM : Type u_4\nN : Type u_5\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : TopologicalSpace M\ninst✝³ : AddCommMonoid N\ninst✝² : TopologicalSpace N\nv₁ v₂ : VectorMeasure α M\nw : VectorMeasure α N\ninst✝¹ : T2Space N\ninst✝ : ContinuousAdd M\nu : Set α\nhmu : M... | rw [add_apply, hu₁ _ (Set.subset_inter_iff.1 ht).1, hv₁ _ (Set.subset_inter_iff.1 ht).2,
zero_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1155,
"column": 4
} | {
"line": 1156,
"column": 15
} | [
{
"pp": "case refine_1\nα : Type u_1\nm : MeasurableSpace α\nM : Type u_4\nN : Type u_5\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : TopologicalSpace M\ninst✝³ : AddCommMonoid N\ninst✝² : TopologicalSpace N\nv₁ v₂ : VectorMeasure α M\nw : VectorMeasure α N\ninst✝¹ : T2Space N\ninst✝ : ContinuousAdd M\nu : Set α\nhmu : M... | rw [add_apply, hu₁ _ (Set.subset_inter_iff.1 ht).1, hv₁ _ (Set.subset_inter_iff.1 ht).2,
zero_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.VectorMeasure.Basic | {
"line": 1265,
"column": 17
} | {
"line": 1282,
"column": 64
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\ns : SignedMeasure α\ni : Set α\nhi₁ : MeasurableSet i\nhi₂ : 0 ≤[i] s\n⊢ Measure α",
"usedConstants": [
"Nonneg.addCommMonoid",
"NNReal.instTopologicalSpace",
"Iff.mpr",
"Eq.mpr",
"MeasureTheory.VectorMeasure.empty",
... | by
refine Measure.ofMeasurable (s.toMeasureOfZeroLE' i hi₂) ?_ ?_
· simp_rw [toMeasureOfZeroLE', s.restrict_apply hi₁ MeasurableSet.empty, Set.empty_inter i,
s.empty]
rfl
· intro f hf₁ hf₂
have h₁ : ∀ n, MeasurableSet (i ∩ f n) := fun n => hi₁.inter (hf₁ n)
have h₂ : Pairwise (Disjoint on fun n ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.ProbabilityMeasure | {
"line": 570,
"column": 2
} | {
"line": 570,
"column": 47
} | [
{
"pp": "Ω : Type u_1\ninst✝² : Nonempty Ω\nm0 : MeasurableSpace Ω\nμ : FiniteMeasure Ω\ninst✝¹ : TopologicalSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nγ : Type u_2\nF : Filter γ\nμs : γ → FiniteMeasure Ω\nμs_lim : Tendsto (fun i ↦ (μs i).normalize) F (𝓝 μ.normalize)\nmass_lim : Tendsto (fun i ↦ (μs i).mass) F (... | rw [tendsto_iff_forall_testAgainstNN_tendsto] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.FiniteMeasure | {
"line": 566,
"column": 2
} | {
"line": 566,
"column": 47
} | [
{
"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)\n⊢ Tendsto μs F (𝓝 0)",
"usedConstants": [
"MeasureTheory.FiniteMeasure.instTopologic... | rw [tendsto_iff_forall_testAgainstNN_tendsto] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Independence.Kernel.IndepFun | {
"line": 316,
"column": 15
} | {
"line": 316,
"column": 18
} | [
{
"pp": "case refine_2.refine_3\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\nh... | hμu | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Probability.Moments.Covariance | {
"line": 160,
"column": 61
} | {
"line": 163,
"column": 57
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX Y : Ω → ℝ\nμ : Measure Ω\n⊢ cov[-X, Y; μ] = -cov[X, Y; μ]",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"NegZeroClass.toNeg",
"neg_smul",
"Real",
"instHSMul",
"Trans.trans",
"MeasureTheory.Measure",
... | by
calc cov[-X, Y; μ]
_ = cov[(-1 : ℝ) • X, Y; μ] := by simp
_ = -cov[X, Y; μ] := by rw [covariance_smul_left]; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Moments.Covariance | {
"line": 173,
"column": 23
} | {
"line": 173,
"column": 58
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX Y : Ω → ℝ\nμ : Measure Ω\n⊢ cov[X, -1 • Y; μ] = -cov[X, Y; μ]",
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul",
"HMul.hMul",
"congrArg",
"Real.... | by rw [covariance_smul_right]; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.HasLaw | {
"line": 169,
"column": 2
} | {
"line": 170,
"column": 33
} | [
{
"pp": "case hY\nΩ : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nX : Ω → 𝓧\nμ : Measure 𝓧\nP : Measure Ω\nhX : HasLaw X μ P\nf g : 𝓧 → ℝ\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\n⊢ AEStronglyMeasurable g (map X P)",
"usedConstants": [
"AEMeasurable.aestronglyMea... | · rw [hX.map_eq]
exact hg.aestronglyMeasurable | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.IdentDistrib | {
"line": 182,
"column": 2
} | {
"line": 183,
"column": 64
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nν : Measure β\nf : α → ℝ≥0∞\ng : β → ℝ≥0∞\nh : IdentDistrib f g μ ν\n⊢ ∫⁻ (x : α), id (f x) ∂μ = ∫⁻ (x : β), id (g x) ∂ν",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congr... | rw [← lintegral_map' aemeasurable_id h.aemeasurable_fst, ←
lintegral_map' aemeasurable_id h.aemeasurable_snd, h.map_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.IdentDistrib | {
"line": 198,
"column": 8
} | {
"line": 198,
"column": 44
} | [
{
"pp": "case pos\nα : 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 : AEStr... | ← integral_map h.aemeasurable_snd A, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.Variance | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 54
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nhXm : AEStronglyMeasurable X μ\nhX : ¬MemLp X 2 μ\nh : eVar[X; μ] < ∞\n⊢ (∫⁻ (x : Ω), ‖X x - ∫ (x : Ω), X x ∂μ‖ₑ ^ ENNReal.toReal 2 ∂μ) ^ (1 / ENNReal.toReal 2) < ∞",
"usedConstants": [
"Eq.mpr",
... | simp only [ENNReal.toReal_ofNat, ENNReal.rpow_two] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Moments.Variance | {
"line": 428,
"column": 2
} | {
"line": 435,
"column": 52
} | [
{
"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 =ᶠ[ae μ] 0\n⊢ Var[∑ i ∈ s, X i; μ] = ∑ i ∈ s, Var[X i; μ]",
"usedConstants": [
"MeasureTheory.... | · rw [variance_congr (Y := 0), variance_zero]
· symm
refine Finset.sum_eq_zero fun i hi ↦ ?_
simp [variance_congr (h'' i hi)]
· have := fun (i : s) ↦ h'' i.1 i.2
filter_upwards [ae_all_iff.2 this] with ω hω
simp only [sum_apply, Pi.zero_apply]
exact Finset.sum_eq_zero fun i hi ↦ hω... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.Piecewise | {
"line": 37,
"column": 2
} | {
"line": 39,
"column": 66
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : MeasurableSpace α\ns : ι → Set α\nf : ι → α → β\nμ : Measure α\ninst✝¹ : MeasurableSpace β\ninst✝ : Countable ι\nhs : IndexedPartition s\nhm : ∀ (i : ι), MeasurableSet (s i)\nhf : ∀ (i : ι), AEMeasurable (f i) μ\n⊢ AEMeasurable (hs.piecewise f) μ",
... | choose p hp hq using hf
refine ⟨hs.piecewise p, hs.measurable_piecewise hm hp, ?_⟩
filter_upwards [ae_all_iff.2 hq] with x hx using hx (hs.index x) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.Piecewise | {
"line": 37,
"column": 2
} | {
"line": 39,
"column": 66
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : MeasurableSpace α\ns : ι → Set α\nf : ι → α → β\nμ : Measure α\ninst✝¹ : MeasurableSpace β\ninst✝ : Countable ι\nhs : IndexedPartition s\nhm : ∀ (i : ι), MeasurableSet (s i)\nhf : ∀ (i : ι), AEMeasurable (f i) μ\n⊢ AEMeasurable (hs.piecewise f) μ",
... | choose p hp hq using hf
refine ⟨hs.piecewise p, hs.measurable_piecewise hm hp, ?_⟩
filter_upwards [ae_all_iff.2 hq] with x hx using hx (hs.index x) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm | {
"line": 177,
"column": 48
} | {
"line": 177,
"column": 89
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝ : NormedAddCommGroup E\nf g : α → E\nhf : MemLp f p μ\nhp : 1 ≤ p\nhg : MemLp g p μ\n⊢ (eLpNorm (f + g) p μ).toReal ≤ (eLpNorm f p μ).toReal + lpNorm g p μ",
"usedConstants": [
"Eq.mpr",
"Real.in... | ← toReal_eLpNorm hg.aestronglyMeasurable, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Group.GeometryOfNumbers | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 35
} | [
{
"pp": "case refine_2.bc\nE : Type u_1\ninst✝⁸ : MeasurableSpace E\nμ : Measure E\nF s : Set E\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : Nontrivial E\ninst✝² : μ.IsAddHaarMeasure\nL : AddSubgroup E\ninst✝¹ : Countable ↥L\ninst✝ : D... | rw [ofReal_pow (by positivity)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 46,
"column": 2
} | {
"line": 61,
"column": 26
} | [
{
"pp": "p q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\n⊢ ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-b * x ^ p) = b ^ (-(q + 1) / p) * (1 / p) * Gamma ((q + 1) / p)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"GroupWithZero... | calc
_ = ∫ x in Ioi (0 : ℝ), b ^ (-p⁻¹ * q) * ((b ^ p⁻¹ * x) ^ q * rexp (-(b ^ p⁻¹ * x) ^ p)) := by
refine setIntegral_congr_fun measurableSet_Ioi (fun _ hx => ?_)
rw [mul_rpow _ (le_of_lt hx), mul_rpow _ (le_of_lt hx), ← rpow_mul, ← rpow_mul,
inv_mul_cancel₀, rpow_one, mul_assoc, ← mul_assoc, ←... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 46,
"column": 2
} | {
"line": 61,
"column": 26
} | [
{
"pp": "p q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\n⊢ ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-b * x ^ p) = b ^ (-(q + 1) / p) * (1 / p) * Gamma ((q + 1) / p)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"GroupWithZero... | calc
_ = ∫ x in Ioi (0 : ℝ), b ^ (-p⁻¹ * q) * ((b ^ p⁻¹ * x) ^ q * rexp (-(b ^ p⁻¹ * x) ^ p)) := by
refine setIntegral_congr_fun measurableSet_Ioi (fun _ hx => ?_)
rw [mul_rpow _ (le_of_lt hx), mul_rpow _ (le_of_lt hx), ← rpow_mul, ← rpow_mul,
inv_mul_cancel₀, rpow_one, mul_assoc, ← mul_assoc, ←... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 46,
"column": 2
} | {
"line": 61,
"column": 26
} | [
{
"pp": "p q b : ℝ\nhp : 0 < p\nhq : -1 < q\nhb : 0 < b\n⊢ ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-b * x ^ p) = b ^ (-(q + 1) / p) * (1 / p) * Gamma ((q + 1) / p)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"GroupWithZero... | calc
_ = ∫ x in Ioi (0 : ℝ), b ^ (-p⁻¹ * q) * ((b ^ p⁻¹ * x) ^ q * rexp (-(b ^ p⁻¹ * x) ^ p)) := by
refine setIntegral_congr_fun measurableSet_Ioi (fun _ hx => ?_)
rw [mul_rpow _ (le_of_lt hx), mul_rpow _ (le_of_lt hx), ← rpow_mul, ← rpow_mul,
inv_mul_cancel₀, rpow_one, mul_assoc, ← mul_assoc, ←... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Gamma | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 75
} | [
{
"pp": "case h.e'_3\np b : ℝ\nhp : 0 < p\nhb : 0 < b\n⊢ b ^ (-1 / p) * Gamma (1 / p + 1) = b ^ (-(0 + 1) / p) * (1 / p) * Gamma ((0 + 1) / p)",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.instPow",
"Semigroup.toMul",
"Real",
"instHDiv",
"Gr... | · rw [zero_add, Gamma_add_one (one_div_ne_zero (ne_of_gt hp)), mul_assoc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 55
} | [
{
"pp": "case refine_3\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | rwa [← lintegral_indicator_one B_mble] at B_finmeas | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 55
} | [
{
"pp": "case refine_3\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | rwa [← lintegral_indicator_one B_mble] at B_finmeas | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Indicator | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 55
} | [
{
"pp": "case refine_3\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nA : Set α\nι : Type u_2\nL : Filter ι\ninst✝ : L.IsCountablyGenerated\nAs : ι → Set α\nμ : Measure α\nA_mble : MeasurableSet A\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nB : Set α\nB_mble : MeasurableSet B\nB_finmeas : μ B ≠ ∞\nAs_le_B : ∀ᶠ (i :... | rwa [← lintegral_indicator_one B_mble] at B_finmeas | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 45
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na b c d : E\nγ₁ : Path a b\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countable\n... | simp [Prod.mk_zero_zero, Prod.mk_one_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.TorusIntegral | {
"line": 109,
"column": 94
} | {
"line": 110,
"column": 44
} | [
{
"pp": "n : ℕ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\na : E\nc : Fin n → ℂ\nR : Fin n → ℝ\n⊢ TorusIntegrable (fun x ↦ a) c R",
"usedConstants": [
"instWeaklyLocallyCompactSpaceOfLocallyCompactSpace",
"locallyCompact_of_proper",
"False",
"ConditionallyCompleteLinearOrder.toCompa... | by
simp [TorusIntegrable, measure_Icc_lt_top] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.CurveIntegral.Poincare | {
"line": 183,
"column": 4
} | {
"line": 187,
"column": 59
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\na✝ b✝ c d : E\nγ₁ : Path a✝ b✝\nγ₂ : Path c d\ns : Set (↑I × ↑I)\nt : Set E\nω : E → E →L[ℝ] F\ndω : E → E →L[ℝ] E →L[ℝ] F\nφ : (↑γ₁).Homotopy ↑γ₂\nhs : s.Countab... | have hdψ_mem (u) : dψ (a, b) u ∈ tangentConeAt ℝ t (φ (a, b)) := by
refine tangentConeAt_mono hψUt.image_subset ?_
rw [← hψφ]
refine (hdψ _ hU).hasFDerivWithinAt.mapsTo_tangent_cone ?_
simp [tangentConeAt_of_mem_nhds (hUopen.mem_nhds hU)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Separation.CompletelyRegular | {
"line": 185,
"column": 53
} | {
"line": 185,
"column": 57
} | [
{
"pp": "case a\nX : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompletelyRegularSpace X\nx : X\nU : Set X\nhxU : x ∈ U\nhU : IsOpen[inst✝¹] U\nf : X → ↑I\nhf : Continuous[inst✝¹, _] f\nefx : f x = 0\nhfU : stoneCechUnit ⁻¹' stoneCechExtend hf ⁻¹' {1}ᶜ ⊆ U\n⊢ f x ∈ {1}ᶜ",
"usedConstants": [
"Real.in... | efx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 169,
"column": 39
} | {
"line": 169,
"column": 56
} | [
{
"pp": "E : Type u_2\nmE : MeasurableSpace E\nμ : Measure E\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nt x : E\n⊢ 2 * ↑π ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"GroupWithZero.toMonoidWithZero",
"False",
"Real",
"Real.pi",
"HMul.hM... | simp [pi_ne_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 169,
"column": 39
} | {
"line": 169,
"column": 56
} | [
{
"pp": "E : Type u_2\nmE : MeasurableSpace E\nμ : Measure E\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nt x : E\n⊢ 2 * ↑π ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"GroupWithZero.toMonoidWithZero",
"False",
"Real",
"Real.pi",
"HMul.hM... | simp [pi_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 169,
"column": 39
} | {
"line": 169,
"column": 56
} | [
{
"pp": "E : Type u_2\nmE : MeasurableSpace E\nμ : Measure E\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nt x : E\n⊢ 2 * ↑π ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"GroupWithZero.toMonoidWithZero",
"False",
"Real",
"Real.pi",
"HMul.hM... | simp [pi_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 36
} | [
{
"pp": "E : Type u_3\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : OpensMeasurableSpace E\nx t : E\n⊢ charFun (Measure.dirac x) t = cexp (↑⟪x, t⟫ * I)",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpac... | rw [charFun_apply, integral_dirac] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 36
} | [
{
"pp": "E : Type u_3\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : OpensMeasurableSpace E\nx t : E\n⊢ charFun (Measure.dirac x) t = cexp (↑⟪x, t⟫ * I)",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpac... | rw [charFun_apply, integral_dirac] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 36
} | [
{
"pp": "E : Type u_3\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : OpensMeasurableSpace E\nx t : E\n⊢ charFun (Measure.dirac x) t = cexp (↑⟪x, t⟫ * I)",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpac... | rw [charFun_apply, integral_dirac] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.LevyProkhorovMetric | {
"line": 481,
"column": 4
} | {
"line": 487,
"column": 25
} | [
{
"pp": "case h.refine_4\nΩ : Type u_1\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμs : ℕ → LevyProkhorov (ProbabilityMeasure Ω)\nν : LevyProkhorov (ProbabilityMeasure Ω)\nhμs : Tendsto μs atTop (𝓝 ν)\nP : ProbabilityMeasure Ω := ν.toMeasure\nPs : ℕ → ProbabilityM... | · grw [bound, hn, BoundedContinuousFunction.integral_eq_integral_meas_le _ _ <| .of_forall f_nn,
add_assoc, mul_comm]
gcongr
calc
δ / 2 + ‖f‖ * (dist (μs n) ν + εs n)
_ ≤ δ / 2 + ‖f‖ * (‖f‖⁻¹ * δ / 2) := by gcongr
_ = δ := by field | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 447,
"column": 2
} | {
"line": 450,
"column": 18
} | [
{
"pp": "case measure_univ_lt_top\nX : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : T2Space X\ninst✝² : MeasurableSpace X\ninst✝¹ : BorelSpace X\nΛ✝ : (X →C_c ℝ) →ₚ[ℝ] ℝ\ninst✝ : CompactSpace X\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ\no : X →C_c ℝ := { toContinuousMap := 1, hasCompactSupport' := ⋯ }\n⊢ (rieszMeasure Λ) univ... | calc rieszMeasure Λ univ
_ ≤ ENNReal.ofReal (Λ o) :=
rieszMeasure_le_of_eq_one _ (fun x ↦ zero_le_one) isCompact_univ (fun x hx ↦ rfl)
_ < ⊤ := by simp | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.MeasureTheory.Integral.RieszMarkovKakutani.Real | {
"line": 463,
"column": 2
} | {
"line": 463,
"column": 83
} | [
{
"pp": "X : Type u_1\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : T2Space X\ninst✝³ : MeasurableSpace X\ninst✝² : BorelSpace X\ninst✝¹ : CompactSpace X\nμ : Measure X\ninst✝ : IsFiniteMeasure μ\nΛ : (X →C_c ℝ) →ₚ[ℝ] ℝ := { toFun := fun g ↦ ∫ (x : X), g x ∂μ, map_add' := ⋯, map_smul' := ⋯, monotone' := ⋯ }\n⊢ ∃ ν, ν.... | refine ⟨RealRMK.rieszMeasure Λ, by infer_instance, by infer_instance, fun g ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
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