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.Probability.Process.Stopping | {
"line": 715,
"column": 27
} | {
"line": 715,
"column": 41
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : LinearOrder ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : SecondCountableTopology ι\ninst✝ : OrderTopology ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f π\ns : Set Ω\nh : MeasurableSet (s ∩ {ω | τ ω ≤ π ... | Set.inter_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Process.Stopping | {
"line": 728,
"column": 24
} | {
"line": 728,
"column": 38
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝ : LinearOrder ι\nf : Filtration ι m\nτ : Ω → WithTop ι\nhτ : IsStoppingTime f τ\ns : Set Ω\ni : ι\nh : MeasurableSet (s ∩ {ω | τ ω ≤ ↑i}) ∧ ∀ (i_1 : ι), MeasurableSet (s ∩ {ω | τ ω ≤ ↑i} ∩ {ω | τ ω ≤ ↑i_1})\nh' : MeasurableSet (s ∩ ({ω | τ ω ≤ ↑i... | Set.inter_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.MeasurableStieltjes | {
"line": 368,
"column": 52
} | {
"line": 381,
"column": 67
} | [
{
"pp": "α : Type u_1\nf : α → ℚ → ℝ\ninst✝ : MeasurableSpace α\nhf : IsMeasurableRatCDF f\na : α\n⊢ Tendsto (↑(hf.stieltjesFunction a)) atBot (𝓝 0)",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"Exists.choose_spec",
"tendsto_of_tendsto_of_tendsto_of_le_of_le",
"Rea... | by
have h_exists : ∀ x : ℝ, ∃ q : ℚ, x < q ∧ ↑q < x + 1 := fun x ↦ exists_rat_btwn (lt_add_one x)
let qs : ℝ → ℚ := fun x ↦ (h_exists x).choose
have hqs_tendsto : Tendsto qs atBot atBot := by
rw [tendsto_atBot_atBot]
refine fun q ↦ ⟨q - 1, fun y hy ↦ ?_⟩
have h_le : ↑(qs y) ≤ (q : ℝ) - 1 + 1 :=
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Process.Stopping | {
"line": 936,
"column": 6
} | {
"line": 936,
"column": 50
} | [
{
"pp": "case coe.inl\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁸ : Nonempty ι\ninst✝⁷ : LinearOrder ι\nu : ι → Ω → β\nτ : Ω → WithTop ι\ninst✝⁶ : MeasurableSpace ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : SecondCountableTopology ι\ninst✝² : BorelSpace ι\nins... | simp [(mod_cast h_it : (i : WithTop ι) ≤ t)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Process.Stopping | {
"line": 936,
"column": 6
} | {
"line": 936,
"column": 50
} | [
{
"pp": "case coe.inl\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁸ : Nonempty ι\ninst✝⁷ : LinearOrder ι\nu : ι → Ω → β\nτ : Ω → WithTop ι\ninst✝⁶ : MeasurableSpace ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : SecondCountableTopology ι\ninst✝² : BorelSpace ι\nins... | simp [(mod_cast h_it : (i : WithTop ι) ≤ t)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Process.Stopping | {
"line": 936,
"column": 6
} | {
"line": 936,
"column": 50
} | [
{
"pp": "case coe.inl\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝⁸ : Nonempty ι\ninst✝⁷ : LinearOrder ι\nu : ι → Ω → β\nτ : Ω → WithTop ι\ninst✝⁶ : MeasurableSpace ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : SecondCountableTopology ι\ninst✝² : BorelSpace ι\nins... | simp [(mod_cast h_it : (i : WithTop ι) ≤ t)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 161,
"column": 27
} | {
"line": 161,
"column": 73
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\nr : ℚ\ninst✝ : IsFiniteMeasure ρ\n⊢ ∫⁻ (x : α) in univ, preCDF ρ r x ∂ρ.fst = (ρ.IicSnd ↑r) univ",
"usedConstants": [
"MeasureTheory.Measure.IicSnd",
"ProbabilityTheory.preCDF",
"Eq.mpr",
"ProbabilityTheory.setLInteg... | setLIntegral_preCDF_fst ρ r MeasurableSet.univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 47
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\nt : β\n⊢ 0 ≤ ↑(stieltjesOfMeasurableRat f ⋯ (a, t)) x",
"usedConstants": [
"Proba... | exact stieltjesOfMeasurableRat_nonneg _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Process.Stopping | {
"line": 1130,
"column": 39
} | {
"line": 1132,
"column": 51
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝² : Nonempty ι\nμ : Measure Ω\nτ : Ω → WithTop ι\nE : Type u_4\nu : ι → Ω → E\ninst✝¹ : PartialOrder ι\nℱ : Filtration ι m\ninst✝ : NormedAddCommGroup E\nhτ : IsStoppingTime ℱ τ\nhu : ∀ (n : ι), Integrable (u n) μ\ns : Finset ι\nhbdd : ∀ (ω : Ω), ... | by
simp_rw [← memLp_one_iff_integrable] at hu ⊢
exact memLp_stoppedValue_of_mem_finset hτ hu hbdd | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Moments.IntegrableExpMul | {
"line": 227,
"column": 4
} | {
"line": 227,
"column": 16
} | [
{
"pp": "case h.e'_4.h.e'_6.h.e'_4.h.e'_1.h.e'_5\nx t p : ℝ\nhp : 0 ≤ p\nht : 0 < t\nhp_zero : ¬p = 0\nh_x_le : ∀ (c : ℝ), 0 < c → x ≤ c⁻¹ * rexp (c * x)\nh_neg_x_le : ∀ (c : ℝ), 0 < c → -x ≤ c⁻¹ * rexp (-c * x)\nh_abs_le : ∀ (c : ℝ), 0 < c → |x| ≤ c⁻¹ * max (rexp (c * x)) (rexp (-c * x))\n⊢ -t / p = -(t / p)",... | rw [neg_div] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 286,
"column": 4
} | {
"line": 290,
"column": 32
} | [
{
"pp": "case h.e'_2.h.h.e'_2.h.e'_2.h.e'_6\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\ninst✝ : IsZeroOrProbabilityMeasure μ\nht : 0 < t\nhc : ∫ (x : Ω), X x ∂μ = 0\nhs : Set.Icc 0 t ⊆ interior (integrableExpSet X μ)\nhu : UniqueDiffOn ℝ (Set.Icc 0 t)\nx✝ : ℝ\n⊢ 0 = taylorWithinEval (... | have hd : derivWithin (cgf X μ) (Set.Icc 0 t) 0 = 0 := by
convert! (analyticAt_cgf (hs ⟨le_refl 0, le_of_lt ht⟩)).differentiableAt.derivWithin _
· simpa [hc] using (deriv_cgf_zero (hs ⟨le_refl 0, le_of_lt ht⟩)).symm
· exact hu 0 ⟨le_refl 0, le_of_lt ht⟩
simp [hd, Set.uIcc_of_lt ht] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.MGFAnalytic | {
"line": 286,
"column": 4
} | {
"line": 290,
"column": 32
} | [
{
"pp": "case h.e'_2.h.h.e'_2.h.e'_2.h.e'_6\nΩ : Type u_1\nm : MeasurableSpace Ω\nX : Ω → ℝ\nμ : Measure Ω\nt : ℝ\ninst✝ : IsZeroOrProbabilityMeasure μ\nht : 0 < t\nhc : ∫ (x : Ω), X x ∂μ = 0\nhs : Set.Icc 0 t ⊆ interior (integrableExpSet X μ)\nhu : UniqueDiffOn ℝ (Set.Icc 0 t)\nx✝ : ℝ\n⊢ 0 = taylorWithinEval (... | have hd : derivWithin (cgf X μ) (Set.Icc 0 t) 0 = 0 := by
convert! (analyticAt_cgf (hs ⟨le_refl 0, le_of_lt ht⟩)).differentiableAt.derivWithin _
· simpa [hc] using (deriv_cgf_zero (hs ⟨le_refl 0, le_of_lt ht⟩)).symm
· exact hu 0 ⟨le_refl 0, le_of_lt ht⟩
simp [hd, Set.uIcc_of_lt ht] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 27
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\nc : ℝ\nhc : c ≠ 0\nx : ℝ\n⊢ (√↑v)⁻¹ * (√(2 * π))⁻¹ * rexp (-(c⁻¹ * x - μ) ^ 2 / (2 * ↑v)) =\n |c| * (√(2 * π * (c ^ 2 * ↑v)))⁻¹ * rexp (-(x - c * μ) ^ 2 / (2 * (c ^ 2 * ↑v)))",
"usedConstants": [
"Semigroup.toMul",
"Real",
"DivInvMonoid.toInv",
"instHDiv",... | refine congr_arg₂ _ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Independence.CharacteristicFunction | {
"line": 81,
"column": 42
} | {
"line": 81,
"column": 59
} | [
{
"pp": "case h\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : IsFiniteMeasure P\nE : Type u_2\nmE : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nX : Ω → E\ninst✝ : NormedSpace ℝ E\nY : Ω → E\nmX : AEMeasurable X P\nmY : AEMeasurabl... | charFunDual_conv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 509,
"column": 34
} | {
"line": 509,
"column": 58
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\n⊢ deriv (mgf (fun x ↦ x) (gaussianReal μ v)) 0 = μ",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Semiring.toModule",
"HMul.hMul",
"Real.denselyNormedField",
"Real.instZero",
"congrArg",
"deriv",
"Real.instDivInvMonoid... | mgf_fun_id_gaussianReal, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 528,
"column": 8
} | {
"line": 528,
"column": 32
} | [
{
"pp": "μ : ℝ\nv : ℝ≥0\n⊢ iteratedDeriv 2 (mgf (fun x ↦ x) (gaussianReal 0 v)) 0 = ↑v",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"instHDiv",
"HMul.hMul",
"Real.denselyNormedField",
"Real.instZero",
"Real.instRCLike",
"cong... | mgf_fun_id_gaussianReal, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 615,
"column": 2
} | {
"line": 620,
"column": 9
} | [
{
"pp": "m₁ m₂ : ℝ\nv₁ v₂ : ℝ≥0\n⊢ gaussianReal m₁ v₁ ∗ gaussianReal m₂ v₂ = gaussianReal (m₁ + m₂) (v₁ + v₂)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidW... | refine Measure.ext_of_charFun ?_
ext t
simp_rw [charFun_conv, charFun_gaussianReal]
rw [← Complex.exp_add]
simp only [Complex.ofReal_add, NNReal.coe_add]
ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Gaussian.Real | {
"line": 615,
"column": 2
} | {
"line": 620,
"column": 9
} | [
{
"pp": "m₁ m₂ : ℝ\nv₁ v₂ : ℝ≥0\n⊢ gaussianReal m₁ v₁ ∗ gaussianReal m₂ v₂ = gaussianReal (m₁ + m₂) (v₁ + v₂)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidW... | refine Measure.ext_of_charFun ?_
ext t
simp_rw [charFun_conv, charFun_gaussianReal]
rw [← Complex.exp_add]
simp only [Complex.ofReal_add, NNReal.coe_add]
ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.CharacteristicFunction | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 50
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nι : Type u_2\ns : Finset ι\nE : Type u_3\ninst✝⁴ : MeasurableSpace E\ninst✝³ : NormedAddCommGroup E\ninst✝² : BorelSpace E\ninst✝¹ : SecondCountableTopology E\nX : ι → Ω → E\ninst✝ : NormedSpace ℝ E\nmX : ∀ i ∈ s, AEMeasurable (X i) P\nhX : iIndepFun... | convert! hX.charFunDual_map_finsetSum_eq_prod mX | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Probability.CentralLimitTheorem | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 70
} | [
{
"pp": "Ω : Type u_1\nΩ' : Type u_2\nmΩ : MeasurableSpace Ω\nmΩ' : MeasurableSpace Ω'\nP : Measure Ω\nP' : Measure Ω'\nX : ℕ → Ω → ℝ\nY : Ω' → ℝ\ninst✝¹ : IsProbabilityMeasure P\ninst✝ : IsProbabilityMeasure P'\nhY : HasLaw Y (gaussianReal 0 1) P'\nh0 : ∫ (x : Ω), X 0 x ∂P = 0\nh1 : ∫ (x : Ω), (X 0 ^ 2) x ∂P =... | refine ProbabilityMeasure.tendsto_iff_tendsto_charFun.2 fun t ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Kernel.Disintegration.StandardBorel | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 21
} | [
{
"pp": "α : Type u_1\nγ : Type u_3\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\ninst✝¹ : CountablyGenerated γ\nκ : Kernel α (γ × ℝ)\ninst✝ : IsFiniteKernel κ\n⊢ κ.fst ⊗ₖ κ.condKernelReal = κ",
"usedConstants": [
"Eq.mpr",
"Real",
"ProbabilityTheory.Kernel.condKernelCDF",
"Probab... | condKernelReal, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.Unique | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 48
} | [
{
"pp": "α : Type u_1\nΩ : Type u_3\nmα : MeasurableSpace α\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nρ : Measure (α × Ω)\ninst✝¹ : IsFiniteMeasure ρ\nκ : Kernel α Ω\ninst✝ : IsSFiniteKernel κ\nhκ : ρ = ρ.fst ⊗ₘ κ\ns : Set Ω\nhs : MeasurableSet s\nt : Set α\nht : Measurabl... | exact (Measure.compProd_apply_prod ht hs).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Disintegration.Unique | {
"line": 75,
"column": 8
} | {
"line": 75,
"column": 47
} | [
{
"pp": "case h\nα : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝¹ : IsFiniteMeasure ρ\nκ : Kernel α ℝ\ninst✝ : IsFiniteKernel κ\nhκ : ρ = ρ.fst ⊗ₘ κ\nhuniv : ∀ᵐ (x : α) ∂ρ.fst, (κ x) univ = (ρ.condKernel x) univ\nx : α\nhxuniv : (κ x) univ = (ρ.condKernel x) univ\nt : Set ℝ\nht : MeasurableSet ... | measure_compl ht <| measure_ne_top _ _, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Disintegration.Integral | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 51
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nF : Type u_4\nmα : MeasurableSpace α\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\ninst✝¹ : NormedAddCommGroup F\nρ : Measure (α × Ω)\ninst✝ : IsFiniteMeasure ρ\nf : α × Ω → F\nhf_int : Integrable f ρ\n⊢ ∀ᵐ (a : α) ∂ρ.fst, Integrable (fun ω... | have hf_ae : AEStronglyMeasurable f ρ := hf_int.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.Disintegration.Integral | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 51
} | [
{
"pp": "α : Type u_1\nΩ : Type u_2\nF : Type u_4\nmα : MeasurableSpace α\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\ninst✝¹ : NormedAddCommGroup F\nρ : Measure (α × Ω)\ninst✝ : IsFiniteMeasure ρ\nf : α × Ω → F\nhf_int : Integrable f ρ\n⊢ Integrable (fun x ↦ ∫ (y : Ω), ‖f (x... | have hf_ae : AEStronglyMeasurable f ρ := hf_int.1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 174,
"column": 70
} | {
"line": 174,
"column": 83
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nY : α → Ω\nmβ : MeasurableSpace β\nX : α → β\nhY : AEMeasurable Y μ\nκ : Kernel β Ω\ninst✝ : IsFiniteKernel κ\nhκ ... | rw [haX, haY] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 174,
"column": 70
} | {
"line": 174,
"column": 83
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nY : α → Ω\nmβ : MeasurableSpace β\nX : α → β\nhY : AEMeasurable Y μ\nκ : Kernel β Ω\ninst✝ : IsFiniteKernel κ\nhκ ... | rw [haX, haY] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.CondDistrib | {
"line": 174,
"column": 70
} | {
"line": 174,
"column": 83
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nΩ : Type u_3\ninst✝⁴ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\ninst✝² : Nonempty Ω\nmα : MeasurableSpace α\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\nY : α → Ω\nmβ : MeasurableSpace β\nX : α → β\nhY : AEMeasurable Y μ\nκ : Kernel β Ω\ninst✝ : IsFiniteKernel κ\nhκ ... | rw [haX, haY] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.Density | {
"line": 702,
"column": 2
} | {
"line": 712,
"column": 21
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝ : CountablyGenerated γ\nκ : Kernel α (γ × β)\nn : ℕ\na : α\nx : γ\nseq : ℕ → Set β\nhseq : Monotone seq\nhseq_iUnion : ⋃ i, seq i = univ\nh0 : (κ.fst a) (countablePartitionS... | · rw [fst_apply' _ _ (measurableSet_countablePartitionSet _ _)] at h0 ⊢
suffices ∀ m, κ a (countablePartitionSet n x ×ˢ seq m) = 0 by
simp only [this, h0, ENNReal.zero_div, tendsto_const_nhds_iff]
suffices κ a (countablePartitionSet n x ×ˢ univ) = 0 by
simp only [this, ENNReal.zero_div]
co... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Distributions.SetBernoulli | {
"line": 76,
"column": 2
} | {
"line": 88,
"column": 46
} | [
{
"pp": "ι : Type u_1\nu : Set ι\np : ↑I\ninst✝ : Countable ι\n⊢ ∀ᵐ (s : Set ι) ∂setBer(u, p), s ⊆ u",
"usedConstants": [
"Filter.instMembership",
"MeasureTheory.ae",
"Set.ext",
"Eq.mpr",
"Inhabited.default",
"ENNReal.instAdd",
"_private.Mathlib.Probability.Distribu... | classical
simp only [Filter.Eventually, mem_ae_iff, Set.compl_setOf, Set.not_subset_iff_exists_mem_notMem,
Set.setOf_exists, Set.setOf_and, measure_iUnion_null_iff]
rintro i
by_cases hi : i ∈ u
· simp [*]
calc
setBer(u, p) ({s | i ∈ s} ∩ {s | i ∉ u})
_ = setBer(u, p) {s | i ∈ s} := by simp [hi]
... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Probability.Distributions.SetBernoulli | {
"line": 76,
"column": 2
} | {
"line": 88,
"column": 46
} | [
{
"pp": "ι : Type u_1\nu : Set ι\np : ↑I\ninst✝ : Countable ι\n⊢ ∀ᵐ (s : Set ι) ∂setBer(u, p), s ⊆ u",
"usedConstants": [
"Filter.instMembership",
"MeasureTheory.ae",
"Set.ext",
"Eq.mpr",
"Inhabited.default",
"ENNReal.instAdd",
"_private.Mathlib.Probability.Distribu... | classical
simp only [Filter.Eventually, mem_ae_iff, Set.compl_setOf, Set.not_subset_iff_exists_mem_notMem,
Set.setOf_exists, Set.setOf_and, measure_iUnion_null_iff]
rintro i
by_cases hi : i ∈ u
· simp [*]
calc
setBer(u, p) ({s | i ∈ s} ∩ {s | i ∉ u})
_ = setBer(u, p) {s | i ∈ s} := by simp [hi]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.SetBernoulli | {
"line": 76,
"column": 2
} | {
"line": 88,
"column": 46
} | [
{
"pp": "ι : Type u_1\nu : Set ι\np : ↑I\ninst✝ : Countable ι\n⊢ ∀ᵐ (s : Set ι) ∂setBer(u, p), s ⊆ u",
"usedConstants": [
"Filter.instMembership",
"MeasureTheory.ae",
"Set.ext",
"Eq.mpr",
"Inhabited.default",
"ENNReal.instAdd",
"_private.Mathlib.Probability.Distribu... | classical
simp only [Filter.Eventually, mem_ae_iff, Set.compl_setOf, Set.not_subset_iff_exists_mem_notMem,
Set.setOf_exists, Set.setOf_and, measure_iUnion_null_iff]
rintro i
by_cases hi : i ∈ u
· simp [*]
calc
setBer(u, p) ({s | i ∈ s} ∩ {s | i ∉ u})
_ = setBer(u, p) {s | i ∈ s} := by simp [hi]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs | {
"line": 80,
"column": 57
} | {
"line": 80,
"column": 88
} | [
{
"pp": "V : Type u_1\ninst✝ : Countable V\n⊢ G(V, 0) = dirac ⊥",
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"Set.Icc.instZero",
"congrArg",
"SimpleGraph.fromEdgeSet",
"Compl.compl",
"ProbabilityThe... | by simp [binomialRandom_eq_map] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Combinatorics.BinomialRandomGraph.Defs | {
"line": 83,
"column": 56
} | {
"line": 83,
"column": 87
} | [
{
"pp": "V : Type u_1\ninst✝ : Countable V\n⊢ G(V, 1) = dirac ⊤",
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"congrArg",
"SimpleGraph.fromEdgeSet",
"Compl.compl",
"ProbabilityTheory.setBernoulli_one",
... | by simp [binomialRandom_eq_map] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.CondVar | {
"line": 80,
"column": 4
} | {
"line": 81,
"column": 80
} | [
{
"pp": "case neg\nΩ : Type u_1\nm₀ m : MeasurableSpace Ω\nμ : Measure Ω\nhm : m ≤ m₀\nc : ℝ\nhc : c ≠ 0\nhμm : ¬IsFiniteMeasure μ\n⊢ Var[fun x ↦ c; μ | m] = 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"InnerProductSpace.toNormedSpace",
"False",
"Real",
"MeasureTheory... | simp [condVar, condExp_of_not_integrable, integrable_const_iff_isFiniteMeasure hc,
integrable_const_iff_isFiniteMeasure <| pow_ne_zero _ hc, hμm, Pi.pow_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.CondVar | {
"line": 80,
"column": 4
} | {
"line": 81,
"column": 80
} | [
{
"pp": "case neg\nΩ : Type u_1\nm₀ m : MeasurableSpace Ω\nμ : Measure Ω\nhm : m ≤ m₀\nc : ℝ\nhc : c ≠ 0\nhμm : ¬IsFiniteMeasure μ\n⊢ Var[fun x ↦ c; μ | m] = 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"InnerProductSpace.toNormedSpace",
"False",
"Real",
"MeasureTheory... | simp [condVar, condExp_of_not_integrable, integrable_const_iff_isFiniteMeasure hc,
integrable_const_iff_isFiniteMeasure <| pow_ne_zero _ hc, hμm, Pi.pow_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.CondVar | {
"line": 80,
"column": 4
} | {
"line": 81,
"column": 80
} | [
{
"pp": "case neg\nΩ : Type u_1\nm₀ m : MeasurableSpace Ω\nμ : Measure Ω\nhm : m ≤ m₀\nc : ℝ\nhc : c ≠ 0\nhμm : ¬IsFiniteMeasure μ\n⊢ Var[fun x ↦ c; μ | m] = 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"InnerProductSpace.toNormedSpace",
"False",
"Real",
"MeasureTheory... | simp [condVar, condExp_of_not_integrable, integrable_const_iff_isFiniteMeasure hc,
integrable_const_iff_isFiniteMeasure <| pow_ne_zero _ hc, hμm, Pi.pow_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.CondVar | {
"line": 170,
"column": 6
} | {
"line": 170,
"column": 35
} | [
{
"pp": "case h\nΩ : Type u_1\nm₀ m : MeasurableSpace Ω\nμ : Measure Ω\nc : ℝ\nX : Ω → ℝ\nω : Ω\nhω : μ[c • X | m] ω = (c • μ[X | m]) ω\n⊢ ((c • X - μ[c • X | m]) ^ 2) ω = (c ^ 2 • (X - μ[X | m]) ^ 2) ω",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"instHSMul",
"Non... | simp [hω, ← mul_sub, mul_pow] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.ProductMeasure | {
"line": 396,
"column": 2
} | {
"line": 399,
"column": 17
} | [
{
"pp": "case a\nι : Type u_1\nX : ι → Type u_2\nmX : (i : ι) → MeasurableSpace (X i)\nμ : (i : ι) → Measure (X i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\nν : Measure ((i : ι) → X i)\nhν : ∀ (s : Finset ι) (t : (i : ι) → Set (X i)), (∀ (i : ι), MeasurableSet (t i)) → ν ((↑s).pi t) = ∏ i ∈ s, (μ i) (t i)\ns... | · rintro i
split_ifs with hi
· exact ht ⟨i, hi⟩
· exact .univ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.ProductMeasure | {
"line": 531,
"column": 62
} | {
"line": 531,
"column": 79
} | [
{
"pp": "case hν\nι : Type u_3\nκ : ι → Type u_4\nX : (i : ι) → κ i → Type u_5\nmX : (i : ι) → (j : κ i) → MeasurableSpace (X i j)\nμ : (i : ι) → (j : κ i) → Measure (X i j)\nhμ : ∀ (i : ι) (j : κ i), IsProbabilityMeasure (μ i j)\ns : Finset ((i : ι) × κ i)\nt : (i : (i : ι) × κ i) → Set (X i.fst i.snd)\nht : ∀... | Finset.coe_sigma, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.Basic | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 45
} | [
{
"pp": "m : ℝ\nv : ℝ≥0\nL : StrongDual ℝ ℝ\n⊢ Measure.map (⇑L) (gaussianReal m v) =\n gaussianReal (∫ (x : ℝ), L x ∂gaussianReal m v) Var[⇑L; gaussianReal m v].toNNReal",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"MeasureTheory.Measure",
"Semir... | rw [gaussianReal_map_continuousLinearMap] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 778,
"column": 71
} | {
"line": 783,
"column": 46
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝² : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝¹ : ∀ (n : ℕ), IsMarkovKernel (κ n)\nμ₀ : Measure (X 0)\ninst✝ : IsProbabilityMeasure μ₀\na : ℕ\n⊢ Measure.map (frestrictLe a) (trajMeasure μ₀ κ) ⊗ₘ κ a =\n Measure.map (fun x ↦... | by
rw [Measure.compProd_eq_comp_prod, trajMeasure, Measure.map_comp _ _ (by fun_prop),
traj_map_frestrictLe, Measure.comp_assoc, Measure.map_comp _ _ (by fun_prop)]
congr with x₀ : 1
rw [comp_apply, ← Measure.compProd_eq_comp_prod, map_apply _ (by fun_prop),
partialTraj_compProd_eq_map_traj zero_le'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Moments.CovarianceBilin | {
"line": 145,
"column": 62
} | {
"line": 152,
"column": 97
} | [
{
"pp": "ι : Type u_2\nΩ : Type u_3\ninst✝¹ : Fintype ι\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nX : ι → Ω → ℝ\nhX : ∀ (i : ι), MemLp (X i) 2 μ\ni j : ι\n⊢ ((covarianceBilin (Measure.map (fun ω ↦ toLp 2 fun x ↦ X x ω) μ)) ((basisFun ι ℝ) i)) ((basisFun ι ℝ) j) =\n cov[X i, X j; μ]",... | by
have (i : ι) := (hX i).aemeasurable
rw [covarianceBilin_apply_eq_cov, covariance_map]
· simp [basisFun_inner]; rfl
· exact Measurable.aestronglyMeasurable (by fun_prop)
· exact Measurable.aestronglyMeasurable (by fun_prop)
· fun_prop
· exact (memLp_map_measure_iff aestronglyMeasurable_id (by fun_prop))... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Fernique | {
"line": 298,
"column": 2
} | {
"line": 299,
"column": 14
} | [
{
"pp": "case hab.hxy.hd\nc d : ℝ≥0∞\nhc : 2⁻¹ < c\nhd : d < 1\nh : c ≤ d\n⊢ 0 < (1 - d).toReal",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"ENNReal.instOrderedSu... | · simp only [ENNReal.toReal_pos_iff, tsub_pos_iff_lt, hd, true_and]
finiteness | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Distributions.Fernique | {
"line": 385,
"column": 2
} | {
"line": 385,
"column": 26
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : SecondCountableTopology E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\na : ℝ\ninst✝ : IsProbabilityMeasure μ\nh_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ\... | let t := normThreshold a | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic | {
"line": 104,
"column": 2
} | {
"line": 105,
"column": 6
} | [
{
"pp": "T : Type u_2\nΩ : Type u_3\nE : Type u_4\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nX : T → Ω → E\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SecondCountableTopology E\nhX : IsGaussianProcess X P\nI : Finset T\n⊢ HasGaussianLaw (fun ... | convert! hX.hasGaussianLaw_sum (I := I)
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic | {
"line": 104,
"column": 2
} | {
"line": 105,
"column": 6
} | [
{
"pp": "T : Type u_2\nΩ : Type u_3\nE : Type u_4\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nX : T → Ω → E\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SecondCountableTopology E\nhX : IsGaussianProcess X P\nI : Finset T\n⊢ HasGaussianLaw (fun ... | convert! hX.hasGaussianLaw_sum (I := I)
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProbabilityMassFunction.Constructions | {
"line": 277,
"column": 20
} | {
"line": 277,
"column": 77
} | [
{
"pp": "α : Type u_1\np : PMF α\ns : Set α\nh : ∃ a ∈ s, a ∈ p.support\na : α\nha : a ∉ s\n⊢ s.indicator (⇑p) a * (∑' (a' : α), s.indicator (⇑p) a')⁻¹ = 0",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"HMul.hMul",
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"CommSe... | Set.indicator_apply_eq_zero.mpr fun ha' => absurd ha' ha, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 171,
"column": 9
} | {
"line": 171,
"column": 32
} | [
{
"pp": "case h.h\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\nh_le : κ ≤ κ + η\nthis :\n (((κ + η).withDensity fun a x ↦ ↑(1 - κ.rnDerivAux (κ + η) a ... | withDensity_rnDerivAux, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 195,
"column": 73
} | {
"line": 196,
"column": 42
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\na : α\nx : γ\n⊢ x ∈ κ.mutuallySingularSetSlice η a ↔ 1 ≤ κ.rnDerivAux (κ + η) a x",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"R... | by
rw [mutuallySingularSetSlice, mem_setOf] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 33
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\na : α\nx : γ\n⊢ x ∉ κ.mutuallySingularSetSlice η a ↔ κ.rnDerivAux (κ + η) a x < 1",
"usedConstants": [
"Real",
"Preorder.toLT",
"c... | simp [mutuallySingularSetSlice] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 33
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\na : α\nx : γ\n⊢ x ∉ κ.mutuallySingularSetSlice η a ↔ κ.rnDerivAux (κ + η) a x < 1",
"usedConstants": [
"Real",
"Preorder.toLT",
"c... | simp [mutuallySingularSetSlice] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 33
} | [
{
"pp": "α : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\na : α\nx : γ\n⊢ x ∉ κ.mutuallySingularSetSlice η a ↔ κ.rnDerivAux (κ + η) a x < 1",
"usedConstants": [
"Real",
"Preorder.toLT",
"c... | simp [mutuallySingularSetSlice] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 296,
"column": 4
} | {
"line": 297,
"column": 32
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : Type u_2\nF : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : MeasurableSpace E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : SecondCountableTopology E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : MeasurableSpace F\ninst✝⁴ : Complete... | simp only [this, map_add, ofReal_add, add_mul, diagonalStrongDualProd_apply, add_div,
add_sub_add_comm, exp_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 307,
"column": 8
} | {
"line": 307,
"column": 44
} | [
{
"pp": "case e_a.h\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : Type u_2\nF : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : MeasurableSpace E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : SecondCountableTopology E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : MeasurableSpace F\ninst✝... | exact hY.isGaussian_map.memLp_two_id | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 307,
"column": 8
} | {
"line": 307,
"column": 44
} | [
{
"pp": "case e_a.h\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : Type u_2\nF : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : MeasurableSpace E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : SecondCountableTopology E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : MeasurableSpace F\ninst✝... | exact hY.isGaussian_map.memLp_two_id | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence | {
"line": 307,
"column": 8
} | {
"line": 307,
"column": 44
} | [
{
"pp": "case e_a.h\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nP : Measure Ω\nE : Type u_2\nF : Type u_3\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : MeasurableSpace E\ninst✝⁹ : CompleteSpace E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : SecondCountableTopology E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : MeasurableSpace F\ninst✝... | exact hY.isGaussian_map.memLp_two_id | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Conditional | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 57
} | [
{
"pp": "Ω : Type u_1\nι : Type u_2\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nπ : ι → Set (Set Ω)\nhπ : ∀ (i : ι), ∀ s ∈ π i, MeasurableSet s\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\nh_eq' :\n ∀ (s : Finset ι) (f : ι → Set Ω),\n (∀ i ∈ s, f i ∈ π i) →\n ∀ i ∈ s, (fun ω... | refine fun s f H ↦ condExpKernel_ae_eq_condExp hm' ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Kernel.Deterministic | {
"line": 75,
"column": 39
} | {
"line": 75,
"column": 69
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\n⊢ IsDeterministic (copy α)",
"usedConstants": [
"ProbabilityTheory.Kernel.instIsDeterministicDeterministic",
"inferInstance",
"ProbabilityTheory.IsDeterministic",
"id",
"Prod.mk",
"Probab... | by unfold copy; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Category.Stoch | {
"line": 108,
"column": 4
} | {
"line": 112,
"column": 44
} | [
{
"pp": "X Y : Stoch\nκ✝ : X ⟶ Y\nX✝ Y✝ Z✝ : Stoch\nκ : X✝ ⟶ Y✝\nη : Y✝ ⟶ Z✝\nx✝ : Deterministic (κ ≫ η)\n⊢ κ ≫ Δ ≫ (η ⊗ₘ 𝟙 Y✝) = Δ ≫ (κ ≫ η ⊗ₘ κ)",
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"Eq.mpr",
"instIsStableUnderComonoidSFinKerStochHom",
"SFinKer.instMonoidalCategor... | ext : 2
dsimp
simp only [id_parallelComp_id, id_comp, id_parallelComp_comp_parallelComp_id]
have : IsDeterministic (κ ≫ η).hom.hom := inferInstance
exact (comp_parallelComp_comp_copy).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Category.Stoch | {
"line": 108,
"column": 4
} | {
"line": 112,
"column": 44
} | [
{
"pp": "X Y : Stoch\nκ✝ : X ⟶ Y\nX✝ Y✝ Z✝ : Stoch\nκ : X✝ ⟶ Y✝\nη : Y✝ ⟶ Z✝\nx✝ : Deterministic (κ ≫ η)\n⊢ κ ≫ Δ ≫ (η ⊗ₘ 𝟙 Y✝) = Δ ≫ (κ ≫ η ⊗ₘ κ)",
"usedConstants": [
"CategoryTheory.ComonObj.comul",
"Eq.mpr",
"instIsStableUnderComonoidSFinKerStochHom",
"SFinKer.instMonoidalCategor... | ext : 2
dsimp
simp only [id_parallelComp_id, id_comp, id_parallelComp_comp_parallelComp_id]
have : IsDeterministic (κ ≫ η).hom.hom := inferInstance
exact (comp_parallelComp_comp_copy).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.ZeroOne | {
"line": 250,
"column": 2
} | {
"line": 255,
"column": 55
} | [
{
"pp": "case refine_1\nα : Type u_1\nΩ : Type u_2\nι : Type u_3\n_mα : MeasurableSpace α\ns : ι → MeasurableSpace Ω\nm0 : MeasurableSpace Ω\nκ : Kernel α Ω\nμα : Measure α\ninst✝² : SemilatticeSup ι\ninst✝¹ : NoMaxOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\nns : ι → Set ι ... | · simp only [mem_atTop_sets, Set.mem_compl_iff, BddAbove, upperBounds, Set.Nonempty]
rintro t ⟨a, ha⟩
obtain ⟨b, hb⟩ : ∃ b, a < b := exists_gt a
refine ⟨b, fun c hc hct => ?_⟩
suffices ∀ i ∈ t, i < c from lt_irrefl c (this c hct)
exact fun i hi => (ha hi).trans_lt (hb.trans_le hc) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Posterior | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 56
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteKernel κ\ninst✝¹ : StandardBorelSpace Ω\ninst✝ : Nonempty Ω\n⊢ (⇑κ ∘ₘ μ) ⊗ₘ κ†μ = ⇑(Kernel.swap Ω 𝓧) ∘ₘ μ ⊗ₘ κ",
"usedConstants": [
"Eq.... | rw [compProd_posterior_eq_map_swap, Measure.swap_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Posterior | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 56
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteKernel κ\ninst✝¹ : StandardBorelSpace Ω\ninst✝ : Nonempty Ω\n⊢ (⇑κ ∘ₘ μ) ⊗ₘ κ†μ = ⇑(Kernel.swap Ω 𝓧) ∘ₘ μ ⊗ₘ κ",
"usedConstants": [
"Eq.... | rw [compProd_posterior_eq_map_swap, Measure.swap_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Posterior | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 56
} | [
{
"pp": "Ω : Type u_1\n𝓧 : Type u_2\nmΩ : MeasurableSpace Ω\nm𝓧 : MeasurableSpace 𝓧\nκ : Kernel Ω 𝓧\nμ : Measure Ω\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteKernel κ\ninst✝¹ : StandardBorelSpace Ω\ninst✝ : Nonempty Ω\n⊢ (⇑κ ∘ₘ μ) ⊗ₘ κ†μ = ⇑(Kernel.swap Ω 𝓧) ∘ₘ μ ⊗ₘ κ",
"usedConstants": [
"Eq.... | rw [compProd_posterior_eq_map_swap, Measure.swap_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 208,
"column": 4
} | {
"line": 212,
"column": 7
} | [
{
"pp": "X : SFinKer\n⊢ { hom := Kernel.copy X.carrier, property := ⋯ } ≫ { hom := Kernel.discard X.carrier, property := ⋯ } ▷ X = (λ_ X).inv",
"usedConstants": [
"ProbabilityTheory.Kernel.deterministic_map",
"Eq.mpr",
"SFinKer.instMonoidalCategory",
"SFinKer.hom_ext",
"SFinKer... | ext : 1; dsimp
simp only [Kernel.discard, Kernel.copy, Kernel.id]
rw [Kernel.deterministic_parallelComp_deterministic,
Kernel.deterministic_comp_deterministic, Kernel.deterministic_map measurable_id (by fun_prop)]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 208,
"column": 4
} | {
"line": 212,
"column": 7
} | [
{
"pp": "X : SFinKer\n⊢ { hom := Kernel.copy X.carrier, property := ⋯ } ≫ { hom := Kernel.discard X.carrier, property := ⋯ } ▷ X = (λ_ X).inv",
"usedConstants": [
"ProbabilityTheory.Kernel.deterministic_map",
"Eq.mpr",
"SFinKer.instMonoidalCategory",
"SFinKer.hom_ext",
"SFinKer... | ext : 1; dsimp
simp only [Kernel.discard, Kernel.copy, Kernel.id]
rw [Kernel.deterministic_parallelComp_deterministic,
Kernel.deterministic_comp_deterministic, Kernel.deterministic_map measurable_id (by fun_prop)]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 214,
"column": 4
} | {
"line": 218,
"column": 7
} | [
{
"pp": "X : SFinKer\n⊢ { hom := Kernel.copy X.carrier, property := ⋯ } ≫ X ◁ { hom := Kernel.discard X.carrier, property := ⋯ } = (ρ_ X).inv",
"usedConstants": [
"ProbabilityTheory.Kernel.deterministic_map",
"Eq.mpr",
"SFinKer.instMonoidalCategory",
"SFinKer.hom_ext",
"Categor... | ext : 1; dsimp
simp only [Kernel.discard, Kernel.copy, Kernel.id]
rw [Kernel.deterministic_parallelComp_deterministic,
Kernel.deterministic_comp_deterministic, Kernel.deterministic_map measurable_id (by fun_prop)]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 214,
"column": 4
} | {
"line": 218,
"column": 7
} | [
{
"pp": "X : SFinKer\n⊢ { hom := Kernel.copy X.carrier, property := ⋯ } ≫ X ◁ { hom := Kernel.discard X.carrier, property := ⋯ } = (ρ_ X).inv",
"usedConstants": [
"ProbabilityTheory.Kernel.deterministic_map",
"Eq.mpr",
"SFinKer.instMonoidalCategory",
"SFinKer.hom_ext",
"Categor... | ext : 1; dsimp
simp only [Kernel.discard, Kernel.copy, Kernel.id]
rw [Kernel.deterministic_parallelComp_deterministic,
Kernel.deterministic_comp_deterministic, Kernel.deterministic_map measurable_id (by fun_prop)]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 93,
"column": 58
} | {
"line": 107,
"column": 50
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : CompleteSpace E\nι : Type u_3\ninst✝⁶ : LinearOrder ι\ninst✝⁵ : TopologicalSpace ι\ninst✝⁴ : OrderTopology ι\ninst✝³ : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝²... | by
have : Set.univ = ⋃ i ∈ Set.range τ, {x | τ x = i} := by
ext1 x
simp only [Set.mem_univ, Set.mem_range, Set.iUnion_exists, Set.iUnion_iUnion_eq',
Set.mem_iUnion, Set.mem_setOf_eq, exists_apply_eq_apply']
nth_rw 1 [← @Measure.restrict_univ Ω _ μ]
rw [this, ae_eq_restrict_biUnion_iff _ h_countable_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Action | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 21
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : Monoid G\ninst✝ : Semiring k\nX Y : Action (Type w) G\nf : X ⟶ Y\nx : X.V\nr : k\n⊢ (linearizeMap f) (Finsupp.single x r) = Finsupp.single ((ConcreteCategory.hom f.hom) x) r",
"usedConstants": [
"Semiring.toModule",
"Finsupp.module",
"congrArg",
... | simp [linearizeMap] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Action | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 21
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : Monoid G\ninst✝ : Semiring k\nX Y : Action (Type w) G\nf : X ⟶ Y\nx : X.V\nr : k\n⊢ (linearizeMap f) (Finsupp.single x r) = Finsupp.single ((ConcreteCategory.hom f.hom) x) r",
"usedConstants": [
"Semiring.toModule",
"Finsupp.module",
"congrArg",
... | simp [linearizeMap] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Action | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 21
} | [
{
"pp": "k : Type u\nG : Type v\ninst✝¹ : Monoid G\ninst✝ : Semiring k\nX Y : Action (Type w) G\nf : X ⟶ Y\nx : X.V\nr : k\n⊢ (linearizeMap f) (Finsupp.single x r) = Finsupp.single ((ConcreteCategory.hom f.hom) x) r",
"usedConstants": [
"Semiring.toModule",
"Finsupp.module",
"congrArg",
... | simp [linearizeMap] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.StrongLaw | {
"line": 233,
"column": 8
} | {
"line": 233,
"column": 81
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK N : ℕ\nhKN : K ≤ N\nρ : Measure ℝ := Measure.map X ℙ\nthis : IsProbabilityMeasure ρ\n⊢ ∑ x ∈ (range K).sigma fun a ↦ Ico a N, ∫ (x : ℝ) in ↑x.snd..↑(x.snd + 1), 1 ∂ρ =\n ∑ x ∈ ... | refine sum_nbij' (fun p ↦ ⟨p.2, p.1⟩) (fun p ↦ ⟨p.2, p.1⟩) ?_ ?_ ?_ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.Moments.SubGaussian | {
"line": 771,
"column": 2
} | {
"line": 775,
"column": 34
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nι : Type u_2\nX : ι → Ω → ℝ\nh_indep : iIndepFun X μ\nc : ι → ℝ≥0\ns : Finset ι\nh_subG : ∀ i ∈ s, HasSubgaussianMGF (X i) (c i) μ\n⊢ HasSubgaussianMGF (fun ω ↦ ∑ i ∈ s, X i ω) (∑ i ∈ s, c i) μ",
"usedConstants": [
"Real",
"Finset.uni... | have : HasSubgaussianMGF (fun ω ↦ ∑ (i : s), X i ω) (∑ (i : s), c i) μ := by
apply sum_of_iIndepFun_of_forall_aemeasurable
· exact h_indep.precomp Subtype.val_injective
· exact fun i ↦ (h_subG i i.2).aemeasurable
· exact fun i _ ↦ h_subG i i.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.StrongLaw | {
"line": 333,
"column": 6
} | {
"line": 333,
"column": 79
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : Ω → ℝ\nhint : Integrable X ℙ\nhnonneg : 0 ≤ X\nK : ℕ\nY : ℕ → Ω → ℝ := fun n ↦ truncation X ↑n\nρ : Measure ℝ := Measure.map X ℙ\nY2 : ∀ (n : ℕ), ∫ (a : Ω), (Y n ^ 2) a = ∫ (x : ℝ) in 0..↑n, x ^ 2 ∂ρ\n⊢ ∑ x ∈ (range K).sigma ran... | refine sum_nbij' (fun p ↦ ⟨p.2, p.1⟩) (fun p ↦ ⟨p.2, p.1⟩) ?_ ?_ ?_ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.StrongLaw | {
"line": 423,
"column": 8
} | {
"line": 423,
"column": 81
} | [
{
"pp": "Ω : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 < ε\nc_pos : 0 < ... | refine sum_nbij' (fun p ↦ ⟨p.2, p.1⟩) (fun p ↦ ⟨p.2, p.1⟩) ?_ ?_ ?_ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Probability.StrongLaw | {
"line": 455,
"column": 10
} | {
"line": 455,
"column": 27
} | [
{
"pp": "case h\nΩ : Type u_1\ninst✝¹ : MeasureSpace Ω\ninst✝ : IsProbabilityMeasure ℙ\nX : ℕ → Ω → ℝ\nhint : Integrable (X 0) ℙ\nhindep : Pairwise ((fun f g ↦ f ⟂ᵢ g) on X)\nhident : ∀ (i : ℕ), IdentDistrib (X i) (X 0) ℙ ℙ\nhnonneg : ∀ (i : ℕ) (ω : Ω), 0 ≤ X i ω\nc : ℝ\nc_one : 1 < c\nε : ℝ\nεpos : 0 < ε\nc_po... | rw [Nat.cast_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RepresentationTheory.FDRep | {
"line": 115,
"column": 51
} | {
"line": 115,
"column": 74
} | [
{
"pp": "R : Type u\nG : Type v\ninst✝¹ : CommRing R\ninst✝ : Monoid G\nV W : FDRep R G\ni : V ≅ W\ng : G\n⊢ ModuleCat.Hom.hom (ModuleCat.ofHom (W.ρ g)) =\n ModuleCat.Hom.hom\n (((Action.forget (FGModuleCat R) G).mapIso i).inv ≫ V.ρ g ≫ ((Action.forget (FGModuleCat R) G).mapIso i).hom).hom",
"usedCo... | ← ModuleCat.hom_ext_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 41
} | [
{
"pp": "case neg\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng : G\nX✝ Y✝ : Rep k G\nf : X✝ ⟶ Y✝\nj : ℕ\nhj : ¬Even (j + 1)\n⊢ f ≫ (HomologicalComplex.alternatingConst Y✝ ⋯ ⋯ ⋯).d (j + 1) j =\n (HomologicalComplex.alternatingConst X✝ ⋯ ⋯ ⋯).d (j + 1) j ≫ f",
... | simp [if_neg hj, applyAsHom_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 41
} | [
{
"pp": "case neg\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng : G\nX✝ Y✝ : Rep k G\nf : X✝ ⟶ Y✝\nj : ℕ\nhj : ¬Even (j + 1)\n⊢ f ≫ (HomologicalComplex.alternatingConst Y✝ ⋯ ⋯ ⋯).d (j + 1) j =\n (HomologicalComplex.alternatingConst X✝ ⋯ ⋯ ⋯).d (j + 1) j ≫ f",
... | simp [if_neg hj, applyAsHom_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 41
} | [
{
"pp": "case neg\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\nA : Rep k G\ng : G\nX✝ Y✝ : Rep k G\nf : X✝ ⟶ Y✝\nj : ℕ\nhj : ¬Even (j + 1)\n⊢ f ≫ (HomologicalComplex.alternatingConst Y✝ ⋯ ⋯ ⋯).d (j + 1) j =\n (HomologicalComplex.alternatingConst X✝ ⋯ ⋯ ⋯).d (j + 1) j ≫ f",
... | simp [if_neg hj, applyAsHom_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 211,
"column": 8
} | {
"line": 213,
"column": 74
} | [
{
"pp": "case zero.left.hS\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : CommGroup G\ninst✝ : Fintype G\ng : G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\n⊢ ({\n X₁ :=\n ((HomologicalComplex.shortComplexFunctor' (Rep k G) (ComplexShape.down ℕ) 1 0 0).obj\n ((chainComplexFunctor k ... | simpa [ShortComplex.moduleCat_exact_iff_range_eq_ker,
HomologicalComplex.alternatingConst, ChainComplex.toSingle₀Equiv] using
leftRegular.range_applyAsHom_sub_eq_ker_linearCombination k g hg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RepresentationTheory.Rep.Basic | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 57
} | [
{
"pp": "case hf\nk : Type u\nG : Type v\ninst✝¹ : Semiring k\ninst✝ : Monoid G\nA B C : Rep k G\nf₁ f₂ : A ⟶ B\ng : B ⟶ C\n⊢ Hom.hom ((f₁ + f₂) ≫ g) = Hom.hom (f₁ ≫ g + f₂ ≫ g)",
"usedConstants": [
"Rep.V",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"A... | simp [add_hom, Representation.IntertwiningMap.add_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Hilbert90 | {
"line": 141,
"column": 8
} | {
"line": 141,
"column": 34
} | [
{
"pp": "case h\nK L : Type\ninst✝⁵ : Field K\ninst✝⁴ : Field L\ninst✝³ : Algebra K L\ninst✝² : FiniteDimensional K L\ninst✝¹ : IsGalois K L\ninst✝ : IsCyclic Gal(L/K)\ng : Gal(L/K)\nhg : ∀ (x : Gal(L/K)), x ∈ Subgroup.zpowers g\nx : L\nhx : (Algebra.norm K) x = 1\nH : ∀ (x : L), (Algebra.norm K) x = 1 → ∃ y, g... | IsUnit.div_eq_iff y.isUnit | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Homological.GroupCohomology.Hilbert90 | {
"line": 189,
"column": 8
} | {
"line": 189,
"column": 13
} | [
{
"pp": "case refine_1\nK L : Type\ninst✝¹⁶ : Field K\ninst✝¹⁵ : Field L\ninst✝¹⁴ : Algebra K L\ninst✝¹³ : FiniteDimensional K L\ninst✝¹² : IsGalois K L\ninst✝¹¹ : IsCyclic Gal(L/K)\ng : Gal(L/K)\nA : Type u_1\nB : Type u_2\ninst✝¹⁰ : CommRing A\ninst✝⁹ : CommRing B\ninst✝⁸ : Algebra A B\ninst✝⁷ : Algebra A L\n... | ← hε, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 70,
"column": 6
} | {
"line": 70,
"column": 11
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nr : R\nn : ℕ\n⊢ ∃ j, r • I ^ j ≤ I ^ n",
"usedConstants": [
"Submodule.pointwiseDistribMulAction",
"Submodule.instAddCommMonoidWithOne",
"Algebra.to_smulCommClass",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRi... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 11
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : Ideal R\nn : ℕ\n⊢ ∃ j, ↑(I ^ j) * ↑(I ^ j) ⊆ ↑(I ^ n)",
"usedConstants": [
"Semiring.toModule",
"HMul.hMul",
"IsScalarTower.right",
"CommSemiring.toSemiring",
"Algebra.id",
"HasSubset.Subset",
"Ideal",
"instDistri... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology | {
"line": 215,
"column": 2
} | {
"line": 220,
"column": 35
} | [
{
"pp": "case mpr\nA : Type u_2\ninst✝² : CommRing A\ninst✝¹ : TopologicalSpace A\ninst✝ : IsTopologicalRing A\n⊢ DiscreteTopology A → (∀ (n : ℕ), IsOpen[inst✝¹] ↑(⊥ ^ n)) ∧ ∀ s ∈ 𝓝 0, ∃ n, ↑(⊥ ^ n) ⊆ s",
"usedConstants": [
"Filter.instMembership",
"Semiring.toModule",
"IsScalarTower.righ... | · intros
constructor
· simp
· intro U U_nhds
use 1
simp [mem_of_mem_nhds U_nhds] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 103,
"column": 2
} | {
"line": 105,
"column": 42
} | [
{
"pp": "case h\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Finite σ\np : MvPolynomial σ R\nn : ℕ\n⊢ ↑((AdicCompletion.of (MvPolynomial.idealOfVars σ R) (MvPolynomial σ R)) p) n = ↑((toAdicCompletion σ R) ↑p) n",
"usedConstants": [
"MvPowerSeries.truncTotal",
"Eq.mpr",
"Submo... | suffices p - (truncTotal n) p ∈ MvPolynomial.idealOfVars σ R ^ n by
simpa [toAdicCompletion, AdicCompletion.liftAlgHom, AdicCompletion.liftRingHom,
Ideal.Quotient.mk_eq_mk_iff_sub_mem] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.RingTheory.AdicCompletion.Noetherian | {
"line": 46,
"column": 42
} | {
"line": 46,
"column": 52
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝⁷ : CommRing R\nI : Ideal R\nM : Type u_2\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : IsNoetherianRing R\ninst✝³ : Module.Finite R M\nA : Type u_3\ninst✝² : CommRing A\ninst✝¹ : IsArtinianRing A\ninst✝ : IsLocalRing A\nf : ℕ → A\nn : ℕ\nhn : maximalIdeal A ^ n =... | SModEq.bot | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 475,
"column": 2
} | {
"line": 475,
"column": 24
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\ng h : G\na : ↑A\n⊢ single (1, h) ((A.ρ g⁻¹) a) - single (g, 1) a ∈ boundaries₂ A",
"usedConstants": [
"groupHomology.d₃₂",
"Rep.V",
"Representation",
"MonoidHom.instFunLike",
"InvOneClass.toOne",
"F... | use single (g, 1, h) a | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RepresentationTheory.Homological.GroupHomology.LowDegree | {
"line": 1032,
"column": 6
} | {
"line": 1032,
"column": 34
} | [
{
"pp": "k G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\ninst✝ : A.IsTrivial\ng : G\na : ↑A\n⊢ (H1AddEquivOfIsTrivial A)\n ((ConcreteCategory.hom (H1π A)) ((ConcreteCategory.hom (cycles₁IsoOfIsTrivial A).inv) (single g a))) =\n Additive.ofMul (Abelianization.of g) ⊗ₜ[ℤ] a",
"usedCo... | H1AddEquivOfIsTrivial_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 221,
"column": 2
} | {
"line": 222,
"column": 49
} | [
{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\n⊢ (chainsMap f φ).f 3 ≫ (chainsIso₃ B).hom = (chainsIso₃ A).hom ≫ chainsMap₃ f φ",
"usedConstants": [
"Finsupp.instFunLike",
"CategoryTheory.Category.assoc",
... | ext
simp [chainsMap_f, chainsIso₃, ← Fin.comp_tail] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 221,
"column": 2
} | {
"line": 222,
"column": 49
} | [
{
"pp": "k G H : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\ninst✝ : Group H\nA : Rep k G\nB : Rep k H\nf : G →* H\nφ : A ⟶ res f B\n⊢ (chainsMap f φ).f 3 ≫ (chainsIso₃ B).hom = (chainsIso₃ A).hom ≫ chainsMap₃ f φ",
"usedConstants": [
"Finsupp.instFunLike",
"CategoryTheory.Category.assoc",
... | ext
simp [chainsMap_f, chainsIso₃, ← Fin.comp_tail] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.ContentIdeal | {
"line": 91,
"column": 70
} | {
"line": 92,
"column": 38
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Semiring S\np : R[X]\nf : R →+* S\nthis : span (↑(map f p).coeffs ∪ {0}) = span (⇑f '' ↑p.coeffs ∪ {0})\n⊢ (map f p).contentIdeal = Ideal.map f p.contentIdeal",
"usedConstants": [
"Polynomial.contentIdeal",
"Eq.mpr",
"RingHo... | by
simpa [contentIdeal_def, map_span] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Congruence.Hom | {
"line": 344,
"column": 2
} | {
"line": 344,
"column": 59
} | [
{
"pp": "case h.e'_2.h.e'_6\nM : Type u_1\nN : Type u_2\ninst✝¹ : NonAssocSemiring M\ninst✝ : NonAssocSemiring N\nc : RingCon M\nf : N ≃+* M\nd : RingCon N\nhcd : d = c.comap f\nx : N\n⊢ ⟦f x⟧ = (c.comapQuotientEquivOfSurj ↑f ⋯ hcd) ↑x",
"usedConstants": [
"Eq.mpr",
"RingCon.toCon",
"RingE... | rw [comapQuotientEquivOfSurj_mk, RingEquiv.coe_toRingHom] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 76,
"column": 2
} | {
"line": 81,
"column": 16
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers J\nf : A →+* B\n⊢ hI.IsDPMorphism hJ f ↔ map f I ≤ J ∧ ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)",
"usedConstants": [
"Eq.mpr",
... | rw [isDPMorphism_def, and_congr_right_iff]
refine fun hIJ ↦ ⟨fun H n _ ↦ H, fun H n ↦ ?_⟩
by_cases hn : n = 0
· intro _ ha
rw [hn, hI.dpow_zero ha, hJ.dpow_zero (hIJ (mem_map_of_mem f ha)), map_one]
· exact H n hn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 76,
"column": 2
} | {
"line": 81,
"column": 16
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers J\nf : A →+* B\n⊢ hI.IsDPMorphism hJ f ↔ map f I ≤ J ∧ ∀ (n : ℕ), n ≠ 0 → ∀ a ∈ I, hJ.dpow n (f a) = f (hI.dpow n a)",
"usedConstants": [
"Eq.mpr",
... | rw [isDPMorphism_def, and_congr_right_iff]
refine fun hIJ ↦ ⟨fun H n _ ↦ H, fun H n ↦ ?_⟩
by_cases hn : n = 0
· intro _ ha
rw [hn, hI.dpow_zero ha, hJ.dpow_zero (hIJ (mem_map_of_mem f ha)), map_one]
· exact H n hn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 150,
"column": 6
} | {
"line": 150,
"column": 37
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nm k : ℕ\nhk : k ≠ 0\nhkm : m * k < n\nx : A\nhx : x ∈ I\nhmn : m < n\n⊢ dpow I m (dpow I k x) = ↑(m.uniformBell k) * dpow I (m * k) x",
"usedConstants": [
"Eq.mpr",
... | dpow_eq_of_mem (m := m * k) hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
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