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.Distributions.Exponential | {
"line": 132,
"column": 4
} | {
"line": 133,
"column": 15
} | [
{
"pp": "r : ℝ\nhr : 0 < r\nx : ℝ\nh : 0 ≤ x\n⊢ ∫⁻ (y : ℝ) in Iic x, exponentialPDF r y = ENNReal.ofReal (1 - rexp (-(r * x)))",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"le_refl",
"Real",
"HMul.hMul",
"Real.instZero",
"ENNReal.ofReal",
"congrArg",
... | rw [lintegral_Iic_eq_lintegral_Iio_add_Icc _ h, lintegral_exponentialPDF_of_nonpos (le_refl 0),
zero_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 486,
"column": 2
} | {
"line": 486,
"column": 10
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\na : ℕ\nx₀ : (i : ↥(Iic a)) → X ↑i\nn : ℕ\n⊢ Measure.map (frestrictLe n) (trajFun κ a x₀) = inducedFamily (fun b ↦ (partialTraj κ a b) x₀) (Iic n... | ext s ms | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 532,
"column": 29
} | {
"line": 532,
"column": 41
} | [
{
"pp": "case h.h\nX : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\na b : ℕ\nx : (i : ↥(Iic a)) → X ↑i\ns✝ : Set ((i : ↥(Iic b)) → X ↑i)\na✝ : MeasurableSet s✝\n⊢ (Measure.map (frestrictLe b) (trajFun κ... | frestrictLe, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.ProductMeasure | {
"line": 592,
"column": 2
} | {
"line": 593,
"column": 27
} | [
{
"pp": "case hf\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)\ninst✝² : DecidableEq ι\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Finset ι\nf : ((i : ι) → X i) → E\nmf : StronglyMeasu... | exact mf.comp_measurable (measurable_updateFinset.mono le_rfl (piFinset.le s))
|>.aestronglyMeasurable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 657,
"column": 2
} | {
"line": 657,
"column": 10
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝ : ∀ (n : ℕ), IsMarkovKernel (κ n)\na b : ℕ\nhab : a ≤ b\nu : (i : ↥(Iic a)) → X ↑i\n⊢ (partialTraj κ a b) u ⊗ₘ traj κ b = Measure.map (fun x ↦ (frestrictLe b x, x)) ((traj κ a) u)... | ext s ms | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Probability.Distributions.Gaussian.Basic | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 52
} | [
{
"pp": "case h.e_z.e_a.e_a\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nh : ∀ (L : StrongDual ℝ E), charFunDual μ L = cexp ((∫ (x : E), ↑(L x) ∂μ) * I - ↑Var[⇑L; μ] / 2)\nL : StrongDual ℝ E\n... | rw [integral_const_mul, integral_complex_ofReal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Distributions.Gaussian.Basic | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 52
} | [
{
"pp": "case h.e_z.e_a.e_a\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nh : ∀ (L : StrongDual ℝ E), charFunDual μ L = cexp ((∫ (x : E), ↑(L x) ∂μ) * I - ↑Var[⇑L; μ] / 2)\nL : StrongDual ℝ E\n... | rw [integral_const_mul, integral_complex_ofReal] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Gaussian.Basic | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 52
} | [
{
"pp": "case h.e_z.e_a.e_a\nE : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : IsFiniteMeasure μ\nh : ∀ (L : StrongDual ℝ E), charFunDual μ L = cexp ((∫ (x : E), ↑(L x) ∂μ) * I - ↑Var[⇑L; μ] / 2)\nL : StrongDual ℝ E\n... | rw [integral_const_mul, integral_complex_ofReal] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.IonescuTulcea.Traj | {
"line": 725,
"column": 50
} | {
"line": 725,
"column": 74
} | [
{
"pp": "X : ℕ → Type u_1\ninst✝⁴ : (n : ℕ) → MeasurableSpace (X n)\nκ : (n : ℕ) → Kernel ((i : ↥(Iic n)) → X ↑i) (X (n + 1))\ninst✝³ : ∀ (n : ℕ), IsMarkovKernel (κ n)\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\na b : ℕ\nhab : a ≤ b\nx₀ : (i : ↥(Iic a)) → X ↑... | traj_map_frestrictLe _ _ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Fernique | {
"line": 280,
"column": 25
} | {
"line": 280,
"column": 64
} | [
{
"pp": "case refine_2\nc : ℝ≥0∞\nhc_gt : 1 / 2 < c\nhc_lt : c < 1\n⊢ 1 < 2 * c",
"usedConstants": [
"MulOne.toOne",
"False",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"NeZero.charZero_one",
... | ENNReal.div_lt_iff (by simp) (by simp), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Moments.CovarianceBilinDual | {
"line": 202,
"column": 4
} | {
"line": 215,
"column": 48
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : OpensMeasurableSpace E\nL₁ L₂ : StrongDual ℝ E\nh : MemLp id 2 μ\n⊢ ∫ (x : E), ‖L₁ x‖ * ‖L₂ x‖ ∂μ ≤ ∫ (x : E), ‖L₁‖ * ‖x‖ * ‖L₂‖ * ‖x‖ ∂μ",
"usedConstants": [
"NormedCommRing.... | refine integral_mono_ae ?_ ?_ (ae_of_all _ fun x ↦ ?_)
· simp_rw [← norm_mul]
exact (MemLp.integrable_mul (h.continuousLinearMap_comp L₁)
(h.continuousLinearMap_comp L₂)).norm
· simp_rw [mul_assoc]
refine Integrable.const_mul ?_ _
simp_rw [← mul_assoc, mul_comm _ (‖L₂‖), mul_assoc, ← p... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.CovarianceBilinDual | {
"line": 202,
"column": 4
} | {
"line": 215,
"column": 48
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : OpensMeasurableSpace E\nL₁ L₂ : StrongDual ℝ E\nh : MemLp id 2 μ\n⊢ ∫ (x : E), ‖L₁ x‖ * ‖L₂ x‖ ∂μ ≤ ∫ (x : E), ‖L₁‖ * ‖x‖ * ‖L₂‖ * ‖x‖ ∂μ",
"usedConstants": [
"NormedCommRing.... | refine integral_mono_ae ?_ ?_ (ae_of_all _ fun x ↦ ?_)
· simp_rw [← norm_mul]
exact (MemLp.integrable_mul (h.continuousLinearMap_comp L₁)
(h.continuousLinearMap_comp L₂)).norm
· simp_rw [mul_assoc]
refine Integrable.const_mul ?_ _
simp_rw [← mul_assoc, mul_comm _ (‖L₂‖), mul_assoc, ← p... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.CovarianceBilinDual | {
"line": 217,
"column": 4
} | {
"line": 219,
"column": 8
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : OpensMeasurableSpace E\nL₁ L₂ : StrongDual ℝ E\nh : MemLp id 2 μ\n⊢ ∫ (x : E), ‖L₁‖ * ‖x‖ * ‖L₂‖ * ‖x‖ ∂μ = ‖L₁‖ * ‖L₂‖ * ∫ (x : E), ‖x‖ ^ 2 ∂μ",
"usedConstants": [
"Mathlib.T... | rw [← integral_const_mul]
congr with x
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.CovarianceBilinDual | {
"line": 217,
"column": 4
} | {
"line": 219,
"column": 8
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : OpensMeasurableSpace E\nL₁ L₂ : StrongDual ℝ E\nh : MemLp id 2 μ\n⊢ ∫ (x : E), ‖L₁‖ * ‖x‖ * ‖L₂‖ * ‖x‖ ∂μ = ‖L₁‖ * ‖L₂‖ * ∫ (x : E), ‖x‖ ^ 2 ∂μ",
"usedConstants": [
"Mathlib.T... | rw [← integral_const_mul]
congr with x
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Moments.CovarianceBilinDual | {
"line": 195,
"column": 79
} | {
"line": 219,
"column": 8
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\nmE : MeasurableSpace E\nμ : Measure E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : OpensMeasurableSpace E\nL₁ L₂ : StrongDual ℝ E\n⊢ ‖((uncenteredCovarianceBilinDual μ) L₁) L₂‖ ≤ ‖L₁‖ * ‖L₂‖ * ∫ (x : E), ‖x‖ ^ 2 ∂μ",
"usedConstants": [
"Mathlib.Tactic.Ring.C... | by
by_cases h : MemLp id 2 μ
swap; · simp only [uncenteredCovarianceBilinDual_of_not_memLp h, norm_zero]; positivity
calc ‖uncenteredCovarianceBilinDual μ L₁ L₂‖
_ = ‖∫ x, L₁ x * L₂ x ∂μ‖ := by rw [uncenteredCovarianceBilinDual_apply h]
_ ≤ ∫ x, ‖L₁ x‖ * ‖L₂ x‖ ∂μ := (norm_integral_le_integral_norm _).trans (... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Fernique | {
"line": 315,
"column": 26
} | {
"line": 315,
"column": 65
} | [
{
"pp": "case pos\na : ℝ\nc : ℝ≥0∞\nhc_gt : 1 / 2 < c\nhc_lt : c < 1\nn : ℕ\nha : a = 0\n⊢ 1 ≤ 2 * c",
"usedConstants": [
"MulOne.toOne",
"False",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"N... | ENNReal.div_lt_iff (by simp) (by simp), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.CharFun | {
"line": 102,
"column": 8
} | {
"line": 102,
"column": 15
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : SecondCountableTopology E\ninst✝⁴ : CompleteSpace E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nμ : Measure E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : IsFiniteMeasure μ\nm : E\nf : StrongDual ℝ E →L[ℝ] StrongDual ℝ E →L[ℝ] ℝ\nhf : f.toBilinForm.Is... | this 0, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic | {
"line": 120,
"column": 4
} | {
"line": 122,
"column": 22
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nX : Ω → E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : IsFiniteMeasure P\nhX : AEMeasurable X P\nh :\n ∀ (L : StrongDual ℝ E), charFunDual (Measure.map X P) L = ce... | refine ⟨isGaussian_iff_charFunDual_eq.2 fun t ↦ ?_⟩
rw [h, integral_map, variance_map, integral_complex_ofReal, Function.comp_def]
all_goals fun_prop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic | {
"line": 120,
"column": 4
} | {
"line": 122,
"column": 22
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nX : Ω → E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : IsFiniteMeasure P\nhX : AEMeasurable X P\nh :\n ∀ (L : StrongDual ℝ E), charFunDual (Measure.map X P) L = ce... | refine ⟨isGaussian_iff_charFunDual_eq.2 fun t ↦ ?_⟩
rw [h, integral_map, variance_map, integral_complex_ofReal, Function.comp_def]
all_goals fun_prop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gaussian.Fernique | {
"line": 231,
"column": 12
} | {
"line": 231,
"column": 33
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\nμ : Measure E\ninst✝² : IsGaussian μ\ninst✝¹ : CompleteSpace E\ninst✝ : SecondCountableTopology E\nh : ∀ (x : E), μ ≠ Measure.dirac x\nx : E\nL : StrongDual ℝ E\nhL : Var[⇑L; μ] ≠ 0... | map_eq_gaussianReal L | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Fernique | {
"line": 527,
"column": 25
} | {
"line": 527,
"column": 64
} | [
{
"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\ninst✝ : IsProbabilityMeasure μ\nh_rot : Measure.map (⇑(ContinuousLinearMap.rotation (-(π / 4)))) (μ.prod μ) = μ.prod μ\na : ℝ\... | ENNReal.div_lt_iff (by simp) (by simp), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic | {
"line": 114,
"column": 26
} | {
"line": 114,
"column": 43
} | [
{
"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\nn : ℕ\nt : Fin (n + 1) → T\nm : ℝ\nx... | ext; simp; module | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Basic | {
"line": 114,
"column": 26
} | {
"line": 114,
"column": 43
} | [
{
"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\nn : ℕ\nt : Fin (n + 1) → T\nm : ℝ\nx... | ext; simp; module | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Gaussian.IsGaussianProcess.Independence | {
"line": 98,
"column": 70
} | {
"line": 99,
"column": 78
} | [
{
"pp": "T : Type u_1\nΩ : Type u_2\nE : Type u_3\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : SecondCountableTopology E\ninst✝¹ : CompleteSpace E\nS : T → Type u_4\nX : (t : T) → S t → Ω → E\ninst✝ : InnerProductSpace ℝ E\nhX... | by
simpa using h t₁ t₂ ht s₁ s₂ ((toDual ℝ E).symm L₁) ((toDual ℝ E).symm L₂) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 264,
"column": 2
} | {
"line": 267,
"column": 68
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\n⊢ Function.Injective toMeasure",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"PMF.toMeasure_apply_singleton",
"congrArg",
"PMF",
"MeasurableSingletonClass.measurableSet_single... | intro p q h
ext x
rw [← p.toMeasure_apply_singleton x <| measurableSet_singleton x,
← q.toMeasure_apply_singleton x <| measurableSet_singleton x, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProbabilityMassFunction.Basic | {
"line": 264,
"column": 2
} | {
"line": 267,
"column": 68
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\n⊢ Function.Injective toMeasure",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"PMF.toMeasure_apply_singleton",
"congrArg",
"PMF",
"MeasurableSingletonClass.measurableSet_single... | intro p q h
ext x
rw [← p.toMeasure_apply_singleton x <| measurableSet_singleton x,
← q.toMeasure_apply_singleton x <| measurableSet_singleton x, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Distributions.Poisson.Basic | {
"line": 154,
"column": 47
} | {
"line": 155,
"column": 90
} | [
{
"pp": "r : ℝ≥0\nt : ℝ\n⊢ ↑(rexp (-↑r)) * ∑' (a : ℕ), (↑↑r * cexp (↑t * I)) ^ a / ↑a ! = ↑(rexp (-↑r)) * cexp (↑↑r * cexp (↑t * I))",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | by
rw [(NormedSpace.expSeries_div_hasSum_exp (r * cexp (t * I))).tsum_eq, exp_eq_exp_ℂ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Poisson.Basic | {
"line": 156,
"column": 39
} | {
"line": 159,
"column": 13
} | [
{
"pp": "r : ℝ≥0\nt : ℝ\n⊢ ↑(rexp (-↑r)) * cexp (↑↑r * cexp (↑t * I)) = cexp (↑↑r * (cexp (↑t * I) - 1))",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOn... | by
rw [ofReal_exp, ← Complex.exp_add]
push_cast
ring_nf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Distributions.Poisson.Basic | {
"line": 234,
"column": 2
} | {
"line": 237,
"column": 70
} | [
{
"pp": "r : ℝ≥0\n⊢ PMF ℕ",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Iff.mpr",
"Eq.mpr",
"HasSum.toNNReal",
"Real",
"ENNReal.ofNNReal",
"ENNReal.instAddCommMonoid",
"ENNReal.ofReal",
"congrArg",
"id",
"NNReal",
"Real.toNNReal_... | refine ⟨fun n ↦ ENNReal.ofReal (poissonPMFReal r n), ?_⟩
apply ENNReal.hasSum_coe.mpr
rw [← toNNReal_one]
exact (poissonPMFRealSum r).toNNReal (fun n ↦ poissonPMFReal_nonneg) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Distributions.Poisson.Basic | {
"line": 234,
"column": 2
} | {
"line": 237,
"column": 70
} | [
{
"pp": "r : ℝ≥0\n⊢ PMF ℕ",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Iff.mpr",
"Eq.mpr",
"HasSum.toNNReal",
"Real",
"ENNReal.ofNNReal",
"ENNReal.instAddCommMonoid",
"ENNReal.ofReal",
"congrArg",
"id",
"NNReal",
"Real.toNNReal_... | refine ⟨fun n ↦ ENNReal.ofReal (poissonPMFReal r n), ?_⟩
apply ENNReal.hasSum_coe.mpr
rw [← toNNReal_one]
exact (poissonPMFRealSum r).toNNReal (fun n ↦ poissonPMFReal_nonneg) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 108,
"column": 37
} | {
"line": 108,
"column": 89
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nf : α → PMF β\n⊢ ∑' (b : α) (a : β), p b * (f b) a = 1",
"usedConstants": [
"PMF.tsum_coe",
"ENNReal.tsum_mul_left",
"HMul.hMul",
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"CommSemiring.toSemiring",... | simp only [ENNReal.tsum_mul_left, tsum_coe, mul_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 108,
"column": 37
} | {
"line": 108,
"column": 89
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nf : α → PMF β\n⊢ ∑' (b : α) (a : β), p b * (f b) a = 1",
"usedConstants": [
"PMF.tsum_coe",
"ENNReal.tsum_mul_left",
"HMul.hMul",
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"CommSemiring.toSemiring",... | simp only [ENNReal.tsum_mul_left, tsum_coe, mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.ProbabilityMassFunction.Monad | {
"line": 108,
"column": 37
} | {
"line": 108,
"column": 89
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\np : PMF α\nf : α → PMF β\n⊢ ∑' (b : α) (a : β), p b * (f b) a = 1",
"usedConstants": [
"PMF.tsum_coe",
"ENNReal.tsum_mul_left",
"HMul.hMul",
"ENNReal.instAddCommMonoid",
"congrArg",
"PMF",
"CommSemiring.toSemiring",... | simp only [ENNReal.tsum_mul_left, tsum_coe, mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ProbabilityMassFunction.Binomial | {
"line": 70,
"column": 26
} | {
"line": 70,
"column": 45
} | [
{
"pp": "k b : ℕ\nhb : k ≤ b\nx : ℝ≥0\nh : x ≤ 1\neq0 : k % (b + 1) = k\neq1 : 1 - ↑x = ENNReal.ofReal (1 - ↑x)\nthis : 1 - ↑x ≥ 0\n⊢ ENNReal.ofReal (↑(b.choose k) * ↑x ^ k * (1 - ↑x) ^ (b - k)) = (binomial x h b) ↑k",
"usedConstants": [
"Eq.mpr",
"PMF.binomial",
"instNeZeroNatHAdd_1",
... | PMF.binomial_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.HasLawExists | {
"line": 38,
"column": 2
} | {
"line": 44,
"column": 61
} | [
{
"pp": "ι : Type v\n𝓧 : ι → Type u\nm𝓧 : (i : ι) → MeasurableSpace (𝓧 i)\nμ : (i : ι) → Measure (𝓧 i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\n⊢ ∃ Ω x P X, (∀ (i : ι), Measurable (X i)) ∧ (∀ (i : ι), HasLaw (X i) (μ i) P) ∧ iIndepFun X P ∧ IsProbabilityMeasure P",
"usedConstants": [
"MeasureT... | use Π i, (𝓧 i), .pi, infinitePi μ, fun i ↦ Function.eval i
refine ⟨by fun_prop, fun i ↦ MeasurePreserving.hasLaw (measurePreserving_eval_infinitePi _ _),
?_, by infer_instance⟩
rw [iIndepFun_iff_map_fun_eq_infinitePi_map (by fun_prop), map_id']
congr
funext i
exact ((measurePreserving_eval_infinitePi μ i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.HasLawExists | {
"line": 38,
"column": 2
} | {
"line": 44,
"column": 61
} | [
{
"pp": "ι : Type v\n𝓧 : ι → Type u\nm𝓧 : (i : ι) → MeasurableSpace (𝓧 i)\nμ : (i : ι) → Measure (𝓧 i)\nhμ : ∀ (i : ι), IsProbabilityMeasure (μ i)\n⊢ ∃ Ω x P X, (∀ (i : ι), Measurable (X i)) ∧ (∀ (i : ι), HasLaw (X i) (μ i) P) ∧ iIndepFun X P ∧ IsProbabilityMeasure P",
"usedConstants": [
"MeasureT... | use Π i, (𝓧 i), .pi, infinitePi μ, fun i ↦ Function.eval i
refine ⟨by fun_prop, fun i ↦ MeasurePreserving.hasLaw (measurePreserving_eval_infinitePi _ _),
?_, by infer_instance⟩
rw [iIndepFun_iff_map_fun_eq_infinitePi_map (by fun_prop), map_id']
congr
funext i
exact ((measurePreserving_eval_infinitePi μ i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 61
} | [
{
"pp": "case neg\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nhκη : κ ≤ η\na : α\nx : γ\nhα : ¬Countable α\n⊢ 0 ≤ (κ.map fun a ↦ (a, ())).density η a x univ",
"usedConstants": [
"Or.resolve_left... | have := hαγ.countableOrCountablyGenerated.resolve_left hα | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 111,
"column": 4
} | {
"line": 111,
"column": 61
} | [
{
"pp": "case neg\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nκ η : Kernel α γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\ninst✝ : IsFiniteKernel η\nhκη : κ ≤ η\na : α\nx : γ\nhx_le_one : (∂κ a/∂η a) x ≤ 1 x\nhα : ¬Countable α\n⊢ (κ.map fun a ↦ (a, ())).density η a... | have := hαγ.countableOrCountablyGenerated.resolve_left hα | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 61
} | [
{
"pp": "case neg\nα : Type u_1\nγ : Type u_2\nmα : MeasurableSpace α\nmγ : MeasurableSpace γ\nhαγ : MeasurableSpace.CountableOrCountablyGenerated α γ\nκ η : Kernel α γ\nhα : ¬Countable α\n⊢ Measurable fun p ↦ (κ.map fun a ↦ (a, ())).density η p.1 p.2 univ",
"usedConstants": [
"Or.resolve_left",
... | have := hαγ.countableOrCountablyGenerated.resolve_left hα | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 61
} | [
{
"pp": "case neg\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\na : α\ns : Set γ\nhs : MeasurableSet s\nh_le : κ ≤ κ + η\nhα : ¬Countable α\n⊢ ∫⁻ (x : γ)... | have := hαγ.countableOrCountablyGenerated.resolve_left hα | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.RadonNikodym | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 49
} | [
{
"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\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\na : α\n⊢ Measurable fun b ↦\n ENNReal.ofReal (κ.rnDerivAux (κ + η) a b) - ENNReal.ofReal (1 - ... | · exact measurable_singularPart_fun_right κ η a | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.WithDensity | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 66
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nf : α → β → ℝ≥0∞\nκ : Kernel α β\ninst✝ : IsFiniteKernel κ\nhf_ne_top : ∀ (a : α) (b : β), f a b ≠ ∞\nhf : Measurable (Function.uncurry f)\nfs : ℕ → α → β → ℝ≥0∞ := fun n a b ↦ min (f a b) (↑n + 1) - min (f a b) ↑n\nh... | suffices IsFiniteKernel (withDensity κ (fs n)) by infer_instance | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Probability.Kernel.Condexp | {
"line": 237,
"column": 64
} | {
"line": 241,
"column": 49
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\ninst✝³ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝² : IsFiniteMeasure μ\nβ : Type u_3\nγ : Type u_4\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : StandardBorelSpace β\ninst✝ : Nonempty β\nX : Ω → β\nY : Ω → γ\nhX : Measurable X\nhY : Measurable Y\ns : ... | by
simp_rw [Kernel.map_apply' _ hX _ hs]
filter_upwards [condDistrib_ae_eq_condExp hY hX (μ := μ) hs,
condExpKernel_ae_eq_condExp hY.comap_le (μ := μ) (hX hs)] with a ha₁ ha₂
rw [← measureReal_eq_measureReal_iff, ha₁, ha₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Independence.Conditional | {
"line": 259,
"column": 4
} | {
"line": 259,
"column": 16
} | [
{
"pp": "Ω : Type u_1\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\ns t : Set Ω\nhs : MeasurableSet s\nht : MeasurableSet t\n⊢ ∀ s_1 ∈ {s}, MeasurableSet s_1",
"usedConstants": [
"Set"
]
}
] | intro s' hs' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Probability.Independence.Conditional | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 16
} | [
{
"pp": "Ω : Type u_1\nm' mΩ : MeasurableSpace Ω\ninst✝¹ : StandardBorelSpace Ω\nhm' : m' ≤ mΩ\nμ : Measure Ω\ninst✝ : IsFiniteMeasure μ\ns t : Set Ω\nhs : MeasurableSet s\nht : MeasurableSet t\n⊢ ∀ s ∈ {t}, MeasurableSet s",
"usedConstants": [
"Set"
]
}
] | intro s' hs' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Probability.Independence.ZeroOne | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 31
} | [
{
"pp": "α : 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 : SemilatticeInf ι\ninst✝¹ : NoMinOrder ι\ninst✝ : Nonempty ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s κ μα\n⊢ Indep (limsup s atBot) (lims... | let ns : ι → Set ι := Set.Ici | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Probability.Independence.Conditional | {
"line": 887,
"column": 6
} | {
"line": 887,
"column": 51
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | rw [compProd_map_condDistrib hg.aemeasurable] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Independence.Conditional | {
"line": 887,
"column": 6
} | {
"line": 887,
"column": 51
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | rw [compProd_map_condDistrib hg.aemeasurable] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Independence.Conditional | {
"line": 887,
"column": 6
} | {
"line": 887,
"column": 51
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | rw [compProd_map_condDistrib hg.aemeasurable] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Independence.Conditional | {
"line": 878,
"column": 4
} | {
"line": 887,
"column": 51
} | [
{
"pp": "Ω : Type u_1\nβ : Type u_3\nβ' : Type u_4\nmΩ : MeasurableSpace Ω\ninst✝⁵ : StandardBorelSpace Ω\nμ : Measure Ω\ninst✝⁴ : IsFiniteMeasure μ\nf : Ω → β\ng : Ω → β'\nγ : Type u_5\nmγ : MeasurableSpace γ\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\ninst✝³ : StandardBorelSpace β\ninst✝² : Nonempty β\... | calc ((Kernel.id ×ₖ condDistrib g k μ) ×ₖ condDistrib f k μ) ∘ₘ μ.map k
_ = (Kernel.id ×ₖ (condDistrib f k μ).prodMkRight _) ∘ₘ (μ.map k ⊗ₘ condDistrib g k μ) := by
rw [Measure.compProd_eq_comp_prod, Measure.comp_assoc]
congr 2
have h := Kernel.prod_prodMkRight_comp_deterministic_prod (condDistrib... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Probability.Kernel.Category.SFinKer | {
"line": 133,
"column": 30
} | {
"line": 143,
"column": 7
} | [
{
"pp": "X✝ Y✝ : SFinKer\nκ : X✝ ⟶ Y✝\n⊢ { hom := κ.hom ∥ₖ Kernel.id, property := ⋯ } ≫\n (let f₁ := fun x ↦ (x, PUnit.unit);\n have hf₁ := ⋯;\n have hf₂ := ⋯;\n { hom := { hom := Kernel.id.map Prod.fst, property := ⋯ }, inv := { hom := Kernel.id.map f₁, property := ⋯ },\n hom... | by
ext : 1; dsimp
rw [Kernel.id_map (by fun_prop), Kernel.id_map (by fun_prop)]
simp only [Kernel.deterministic_comp_eq_map, Kernel.comp_deterministic_eq_comap]
ext _ _ hs
have := κ.2
rw [Kernel.map_apply' _ (by fun_prop) _ hs, Kernel.comap_apply' _ (by fun_prop),
Kernel.parallelComp_apply... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 68
} | [
{
"pp": "Ω : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nι : Type u_3\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : FirstCountableTopology ι\nℱ : Filtration ι m\ninst✝¹... | rw [Set.inter_comm _ t, IsStoppingTime.measurableSet_inter_eq_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Martingale.OptionalSampling | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 70
} | [
{
"pp": "case pos\nΩ : Type u_1\nE : Type u_2\nm : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\nι : Type u_3\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\ninst✝² : FirstCountableTopology ι\nℱ : Filtration ι... | rw [Set.inter_comm _ t, IsStoppingTime.measurableSet_inter_eq_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Moments.Tilted | {
"line": 162,
"column": 10
} | {
"line": 162,
"column": 73
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nht : t ∈ interior (integrableExpSet X μ)\n⊢ AEMeasurable X (μ.tilted fun x ↦ t * X x)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"MeasureTheory.AEStronglyMeasurable.aemeasurable",
"Real",
... | exact (memLp_tilted_mul ht 1).aestronglyMeasurable.aemeasurable | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Moments.Tilted | {
"line": 162,
"column": 10
} | {
"line": 162,
"column": 73
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nht : t ∈ interior (integrableExpSet X μ)\n⊢ AEMeasurable X (μ.tilted fun x ↦ t * X x)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"MeasureTheory.AEStronglyMeasurable.aemeasurable",
"Real",
... | exact (memLp_tilted_mul ht 1).aestronglyMeasurable.aemeasurable | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Moments.Tilted | {
"line": 162,
"column": 10
} | {
"line": 162,
"column": 73
} | [
{
"pp": "Ω : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nX : Ω → ℝ\nt : ℝ\nht : t ∈ interior (integrableExpSet X μ)\n⊢ AEMeasurable X (μ.tilted fun x ↦ t * X x)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"MeasureTheory.AEStronglyMeasurable.aemeasurable",
"Real",
... | exact (memLp_tilted_mul ht 1).aestronglyMeasurable.aemeasurable | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Process.LocalProperty | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 83
} | [
{
"pp": "ι : Type u_1\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nP : Measure Ω\ninst✝⁵ : ConditionallyCompleteLinearOrderBot ι\ninst✝⁴ : TopologicalSpace ι\ninst✝³ : OrderTopology ι\n𝓕 : Filtration ι mΩ\ninst✝² : SecondCountableTopology ι\ninst✝¹ : IsFiniteMeasure P\ninst✝ : NoMaxOrder ι\nτ : ℕ → Ω → WithTop ι\nσ ... | refine ⟨nk, hnk, fun n ↦ (hτ.isStoppingTime n).min ((hσ _).isStoppingTime _), ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RepresentationTheory.Intertwining | {
"line": 486,
"column": 6
} | {
"line": 488,
"column": 40
} | [
{
"pp": "A : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nU : Type u_5\ninst✝⁷ : CommSemiring A\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommMonoid V\ninst✝⁴ : AddCommMonoid W\ninst✝³ : AddCommMonoid U\ninst✝² : Module A V\ninst✝¹ : Module A W\ninst✝ : Module A U\nρ : Representation A G V\nσ : Representation A G ... | induction n with
| zero => rfl
| succ n ih => simp [ih, pow_succ] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RepresentationTheory.Character | {
"line": 112,
"column": 48
} | {
"line": 113,
"column": 56
} | [
{
"pp": "k : Type u\ninst✝³ : Field k\nG : Type v\ninst✝² : Group G\ninst✝¹ : Fintype G\ninst✝ : Invertible ↑(Fintype.card G)\nV W : FDRep k G\n⊢ ↑(finrank k ↥(invariants (of (linHom V.ρ W.ρ)).ρ)) = ↑(finrank k (V ⟶ W))",
"usedConstants": [
"Eq.mpr",
"Submodule",
"CategoryTheory.CategorySt... | ← LinearEquiv.finrank_eq
(Representation.linHom.invariantsEquivFDRepHom V W), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.StrongLaw | {
"line": 730,
"column": 15
} | {
"line": 730,
"column": 16
} | [
{
"pp": "case hindep\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMe... | I | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.StrongLaw | {
"line": 734,
"column": 15
} | {
"line": 734,
"column": 16
} | [
{
"pp": "case hident\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMe... | I | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.StrongLaw | {
"line": 742,
"column": 2
} | {
"line": 742,
"column": 88
} | [
{
"pp": "case h\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\ninst✝⁵ : IsProbabilityMeasure μ\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nh' : StronglyMeasura... | obtain ⟨δ, δpos, hδ⟩ : ∃ δ, 0 < δ ∧ δ + δ + δ < ε := ⟨ε/4, by positivity, by linarith⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Probability.StrongLaw | {
"line": 819,
"column": 2
} | {
"line": 819,
"column": 19
} | [
{
"pp": "case e_a.e_f.h\nΩ : Type u_1\nmΩ : MeasurableSpace Ω\nμ : Measure Ω\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nX : ℕ → Ω → E\nhint : Integrable (X 0) μ\nhindep : Pairwise ((fun x1 x2 ↦ x1 ⟂ᵢ[μ] x2) ... | exact (h₁ i).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 129,
"column": 34
} | {
"line": 129,
"column": 84
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : ... | simpa using congr($((f.isIntertwining' g).symm) x) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 129,
"column": 34
} | {
"line": 129,
"column": 84
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : ... | simpa using congr($((f.isIntertwining' g).symm) x) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Coinvariants | {
"line": 129,
"column": 34
} | {
"line": 129,
"column": 84
} | [
{
"pp": "k : Type u_1\nG : Type u_2\nV : Type u_3\nW : Type u_4\nX : Type u_5\ninst✝⁷ : CommRing k\ninst✝⁶ : Monoid G\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module k V\ninst✝³ : AddCommGroup W\ninst✝² : Module k W\ninst✝¹ : AddCommGroup X\ninst✝ : Module k X\nρ : Representation k G V\nτ : Representation k G W\nυ : ... | simpa using congr($((f.isIntertwining' g).symm) x) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.FiniteCyclic | {
"line": 57,
"column": 2
} | {
"line": 61,
"column": 50
} | [
{
"pp": "case refine_1\nk : Type u_1\nG : Type u_2\ninst✝⁴ : CommRing k\ninst✝³ : Group G\nV : Type u_4\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nρ : Representation k G V\ng : G\ninst✝ : Finite G\nhg : ∀ (x : G), x ∈ Subgroup.zpowers g\n⊢ (Set.range fun gv ↦ (ρ gv.1) gv.2 - gv.2) ⊆ ↑(ρ g - LinearMap.id).ra... | · rintro a ⟨⟨γ, α⟩, rfl⟩
rcases mem_powers_iff_mem_zpowers.2 (hg γ) with ⟨i, rfl⟩
induction i with | zero => exact ⟨0, by simp⟩ | succ n _ =>
use (Fin.partialSum (fun (j : Fin (n + 1)) => ρ (g ^ (j : ℕ)) α) (Fin.last _))
simpa using ρ.apply_sub_id_partialSum_eq _ _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RepresentationTheory.Homological.GroupCohomology.LowDegree | {
"line": 885,
"column": 41
} | {
"line": 888,
"column": 17
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nA : Rep k G\nx : ↥(cocycles₂ A)\n⊢ (ConcreteCategory.hom (inhomogeneousCochains.d A 2)) ((ConcreteCategory.hom (cochainsIso₂ A).inv) ⇑x) = 0",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"inhomogeneousCochains.d",
"S... | by
rw [← LinearMap.comp_apply, ← ModuleCat.hom_comp, ← inhomogeneousCochains.d_def,
eq_d₂₃_comp_inv, ModuleCat.hom_comp, LinearMap.comp_apply, cocycles₂.d₂₃_apply x,
map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RepresentationTheory.Homological.GroupHomology.LongExactSequence | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 82
} | [
{
"pp": "k G : Type u\ninst✝¹ : CommRing k\ninst✝ : Group G\nX : ShortComplex (Rep k G)\nhX : X.ShortExact\nz : ↥(cycles₂ X.X₃)\ny : G × G →₀ ↑X.X₂\nhy : (mapRange.linearMap (Rep.Hom.hom X.g).toLinearMap) y = ↑z\nx : G →₀ ↑X.X₁\nhx : (mapRange.linearMap (Rep.Hom.hom X.f).toLinearMap) x = (ConcreteCategory.hom (... | conv_rhs => rw [← LinearMap.comp_apply, ← ModuleCat.hom_comp, eq_d₂₁_comp_inv] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.RepresentationTheory.Tannaka | {
"line": 116,
"column": 2
} | {
"line": 119,
"column": 25
} | [
{
"pp": "k G : Type u\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : Nontrivial k\n⊢ Function.Injective ⇑(equivHom k G)",
"usedConstants": [
"Pi.Function.module",
"CategoryTheory.Functor",
"False",
"MonoidHom.instFunLike",
"NonUnitalCommRing.toNonUnitalNonAs... | intro s t h
classical
apply_fun (fun x ↦ (x.hom.hom.app rightFDRep).hom (single t 1) 1) at h
simp_all [single_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RepresentationTheory.Tannaka | {
"line": 116,
"column": 2
} | {
"line": 119,
"column": 25
} | [
{
"pp": "k G : Type u\ninst✝³ : CommRing k\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : Nontrivial k\n⊢ Function.Injective ⇑(equivHom k G)",
"usedConstants": [
"Pi.Function.module",
"CategoryTheory.Functor",
"False",
"MonoidHom.instFunLike",
"NonUnitalCommRing.toNonUnitalNonAs... | intro s t h
classical
apply_fun (fun x ↦ (x.hom.hom.app rightFDRep).hom (single t 1) 1) at h
simp_all [single_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 59,
"column": 4
} | {
"line": 63,
"column": 34
} | [
{
"pp": "σ✝ : Type u_1\nR✝ : Type u_2\nn✝ : ℕ\ninst✝³ : CommRing R✝\ninst✝² : Finite σ✝\nσ : Type u_3\nR : Type u_4\ninst✝¹ : Finite σ\ninst✝ : CommRing R\nn : ℕ\n⊢ (Ideal.Quotient.mk (MvPolynomial.idealOfVars σ R ^ n)) ((truncTotal n) 1) = 1",
"usedConstants": [
"MvPowerSeries.truncTotal",
"Iff... | by_cases! h : n = 0
· have := Ideal.Quotient.subsingleton_iff.mpr
(show MvPolynomial.idealOfVars σ R ^ n = ⊤ by simp [h])
exact Subsingleton.allEq ..
rw [truncTotal_one h, map_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Equiv | {
"line": 59,
"column": 4
} | {
"line": 63,
"column": 34
} | [
{
"pp": "σ✝ : Type u_1\nR✝ : Type u_2\nn✝ : ℕ\ninst✝³ : CommRing R✝\ninst✝² : Finite σ✝\nσ : Type u_3\nR : Type u_4\ninst✝¹ : Finite σ\ninst✝ : CommRing R\nn : ℕ\n⊢ (Ideal.Quotient.mk (MvPolynomial.idealOfVars σ R ^ n)) ((truncTotal n) 1) = 1",
"usedConstants": [
"MvPowerSeries.truncTotal",
"Iff... | by_cases! h : n = 0
· have := Ideal.Quotient.subsingleton_iff.mpr
(show MvPolynomial.idealOfVars σ R ^ n = ⊤ by simp [h])
exact Subsingleton.allEq ..
rw [truncTotal_one h, map_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RepresentationTheory.Homological.GroupHomology.Functoriality | {
"line": 659,
"column": 4
} | {
"line": 659,
"column": 41
} | [
{
"pp": "case h\nk G : Type u\ninst✝² : CommRing k\ninst✝¹ : Group G\nA : Rep k G\nS : Subgroup G\ninst✝ : S.Normal\nx : ↥(cycles₁ A)\nhx :\n (ConcreteCategory.hom (H1π (A.quotientToCoinvariants S)))\n ((ConcreteCategory.hom (mapCycles₁ (QuotientGroup.mk' S) (A.toCoinvariantsMkQ S))) x) =\n 0\ny : ↑(Mo... | refine (H1π_eq_iff _ _).2 ⟨W + δ, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.DedekindDomain.SelmerGroup | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 45
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type v\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\nx✝¹ x✝ : Kˣ\n⊢ v.valuationOfNeZeroToFun (x✝¹ * x✝) = v.valuationOfNeZeroToFun x✝¹ * v.valuationOfNeZeroToFun x✝",
"usedConstants": [... | rw [← WithZero.coe_inj, WithZero.coe_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 23
} | [
{
"pp": "case succ\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\na : A\nhI : DividedPowers I\nha : a ∈ I\nn : ℕ\nih : ↑n ! * hI.dpow n a = a ^ n\n⊢ ↑n ! * (↑((n + 1).choose n) * hI.dpow (n + 1) a) = a ^ (n + 1)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | ← hI.mul_dpow ha, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.Basic | {
"line": 298,
"column": 6
} | {
"line": 298,
"column": 70
} | [
{
"pp": "case insert\nA : Type u_1\ninst✝² : CommSemiring A\nM : Type u_2\ninst✝¹ : AddCommMonoid M\nI : AddSubmonoid M\ndpow : ℕ → M → A\ndpow_zero : ∀ {x : M}, x ∈ I → dpow 0 x = 1\ndpow_eval_zero : ∀ {n : ℕ}, n ≠ 0 → dpow n 0 = 0\nι : Type u_3\ninst✝ : DecidableEq ι\nx : ι → M\ndpow_add : ∀ {n : ℕ} {x y : M}... | dpow_add (hx a (mem_insert_self a s)) (I.sum_mem fun i ↦ hx' i), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 59
} | [
{
"pp": "case neg.h.h\nA : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nn : ℕ\nhn_fac : IsUnit ↑(n - 1)!\nhnI : I ^ n = 0\nm : ℕ\nx : A\nhx : x ∈ I\ny : A\nhy : y ∈ I\nhmn : n ≤ m\nh_sub : I ^ m ≤ I ^ n\nhxy : (x + y) ^ m = 0\nk : ℕ × ℕ\nhk : k ∈ Finset.antidiagonal m\n⊢ ... | rw [← Finset.mem_antidiagonal.mp hk, add_comm, pow_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 117,
"column": 2
} | {
"line": 118,
"column": 56
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\na x : A\nhx : x ∈ I\n⊢ dpow I m (a * x) = a ^ m * dpow I m x",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semigroup.toMul",
"HMul.hMul",
"Monoi... | rw [dpow_eq_of_mem (Ideal.mul_mem_left I _ hx), dpow_eq_of_mem hx,
mul_pow, ← mul_assoc, mul_comm _ (a ^ m), mul_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 117,
"column": 2
} | {
"line": 118,
"column": 56
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\na x : A\nhx : x ∈ I\n⊢ dpow I m (a * x) = a ^ m * dpow I m x",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semigroup.toMul",
"HMul.hMul",
"Monoi... | rw [dpow_eq_of_mem (Ideal.mul_mem_left I _ hx), dpow_eq_of_mem hx,
mul_pow, ← mul_assoc, mul_comm _ (a ^ m), mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.RatAlgebra | {
"line": 117,
"column": 2
} | {
"line": 118,
"column": 56
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommSemiring A\nI : Ideal A\ninst✝ : DecidablePred fun x ↦ x ∈ I\nm : ℕ\na x : A\nhx : x ∈ I\n⊢ dpow I m (a * x) = a ^ m * dpow I m x",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Semigroup.toMul",
"HMul.hMul",
"Monoi... | rw [dpow_eq_of_mem (Ideal.mul_mem_left I _ hx), dpow_eq_of_mem hx,
mul_pow, ← mul_assoc, mul_comm _ (a ^ m), mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 142,
"column": 4
} | {
"line": 148,
"column": 28
} | [
{
"pp": "A✝ : Type u_1\nB✝ : Type u_2\ninst✝³ : CommSemiring A✝\ninst✝² : CommSemiring B✝\nI✝ : Ideal A✝\nJ✝ : Ideal B✝\nhI✝ : DividedPowers I✝\nhJ✝ : DividedPowers J✝\nA : Type u_3\nB : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers... | simp only [mem_setOf_eq, map_add] at hx hy ⊢
refine ⟨I.add_mem hx.1 hy.1, fun n ↦ ?_⟩
rw [hI.dpow_add hx.1 hy.1, map_sum,
hJ.dpow_add (hf (mem_map_of_mem f hx.1)) (hf (mem_map_of_mem f hy.1))]
apply congr_arg
ext k
rw [map_mul, hx.2, hy.2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.DPMorphism | {
"line": 142,
"column": 4
} | {
"line": 148,
"column": 28
} | [
{
"pp": "A✝ : Type u_1\nB✝ : Type u_2\ninst✝³ : CommSemiring A✝\ninst✝² : CommSemiring B✝\nI✝ : Ideal A✝\nJ✝ : Ideal B✝\nhI✝ : DividedPowers I✝\nhJ✝ : DividedPowers J✝\nA : Type u_3\nB : Type u_4\ninst✝¹ : CommSemiring A\ninst✝ : CommSemiring B\nI : Ideal A\nJ : Ideal B\nhI : DividedPowers I\nhJ : DividedPowers... | simp only [mem_setOf_eq, map_add] at hx hy ⊢
refine ⟨I.add_mem hx.1 hy.1, fun n ↦ ?_⟩
rw [hI.dpow_add hx.1 hy.1, map_sum,
hJ.dpow_add (hf (mem_map_of_mem f hx.1)) (hf (mem_map_of_mem f hy.1))]
apply congr_arg
ext k
rw [map_mul, hx.2, hy.2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 108,
"column": 6
} | {
"line": 113,
"column": 67
} | [
{
"pp": "case neg.hmn\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nhn : n ≠ 0\nx : ℤ_[p]\nhx : x ∈ Ideal.span {↑p}\nhx0 : ¬x = 0\nhlt : ↑(padicValNat p n !) < ↑n\nhnorm : 0 < ‖↑n !‖\n⊢ -↑x.valuation * ↑n < -(↑n !).valuation",
"usedConstants": [
"Int.instAddCommGroup",
"IsRightCancelAdd.addRightStrict... | simp only [neg_mul, Padic.valuation_natCast, neg_lt_neg_iff]
apply lt_of_lt_of_le hlt
conv_lhs => rw [← one_mul (n : ℤ)]
gcongr
norm_cast
rwa [← PadicInt.mem_span_pow_iff_le_valuation x hx0, pow_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.Padic | {
"line": 108,
"column": 6
} | {
"line": 113,
"column": 67
} | [
{
"pp": "case neg.hmn\np : ℕ\nhp : Fact (Nat.Prime p)\nn : ℕ\nhn : n ≠ 0\nx : ℤ_[p]\nhx : x ∈ Ideal.span {↑p}\nhx0 : ¬x = 0\nhlt : ↑(padicValNat p n !) < ↑n\nhnorm : 0 < ‖↑n !‖\n⊢ -↑x.valuation * ↑n < -(↑n !).valuation",
"usedConstants": [
"Int.instAddCommGroup",
"IsRightCancelAdd.addRightStrict... | simp only [neg_mul, Padic.valuation_natCast, neg_lt_neg_iff]
apply lt_of_lt_of_le hlt
conv_lhs => rw [← one_mul (n : ℤ)]
gcongr
norm_cast
rwa [← PadicInt.mem_span_pow_iff_le_valuation x hx0, pow_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 139,
"column": 8
} | {
"line": 139,
"column": 29
} | [
{
"pp": "case refine_1\nA : Type u_1\ninst✝ : CommRing A\nI : Ideal A\nhI : DividedPowers I\nJ : Ideal A\nhIJ : hI.IsSubDPIdeal (J ⊓ I)\nn : ℕ\na b : A\nha : a ∈ I\nhb : b ∈ I\nhab : a - b ∈ J\nhab' : a - b ∈ I\n⊢ hI.dpow n a - hI.dpow n b ∈ J",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
... | ← add_sub_cancel b a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 429,
"column": 4
} | {
"line": 437,
"column": 37
} | [
{
"pp": "case a\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\nJ : hI.SubDPIdeal := ⋯\n⊢ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier",
"usedConstants": [
"Ideal.span_le",
"E... | rw [le_iInf₂_iff]
intro K hK
have : S ≤ K := by
simp only [Set.mem_insert_iff, Set.mem_setOf_eq] at hK
rcases hK with rfl | hKS
exacts [hS, hKS]
rw [span_le]
rintro y ⟨n, hn, x, hx, rfl⟩
exact K.dpow_mem n hn x (this hx) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DividedPowers.SubDPIdeal | {
"line": 429,
"column": 4
} | {
"line": 437,
"column": 37
} | [
{
"pp": "case a\nA : Type u_1\ninst✝ : CommSemiring A\nI : Ideal A\nhI : DividedPowers I\nS : Set A\nhS : S ⊆ ↑I\nJ : hI.SubDPIdeal := ⋯\n⊢ span {y | ∃ n, ∃ (_ : n ≠ 0), ∃ x, ∃ (_ : x ∈ S), y = hI.dpow n x} ≤ ⨅ s ∈ insert ⊤ {J | S ⊆ ↑J.carrier}, s.carrier",
"usedConstants": [
"Ideal.span_le",
"E... | rw [le_iInf₂_iff]
intro K hK
have : S ≤ K := by
simp only [Set.mem_insert_iff, Set.mem_setOf_eq] at hK
rcases hK with rfl | hKS
exacts [hS, hKS]
rw [span_le]
rintro y ⟨n, hn, x, hx, rfl⟩
exact K.dpow_mem n hn x (this hx) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Frobenius | {
"line": 219,
"column": 91
} | {
"line": 237,
"column": 53
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nG : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : MulSemiringAction G S\ninst✝⁴ : SMulCommClass G R S\nQ : Ideal S\ninst✝³ : Finite G\ninst✝² : Algebra.IsInvariant R S G\ninst✝¹ : Q.IsPrime\ninst✝ : Finite (S ⧸ Q)\n⊢ ∃ σ... | by
let P := Q.under R
have := Algebra.IsInvariant.isIntegral R S G
have : Q.IsMaximal := Ideal.Quotient.maximal_of_isField _ (Finite.isField_of_domain (S ⧸ Q))
obtain ⟨p, hc⟩ := CharP.exists (R ⧸ P)
have : Finite (R ⧸ P) := .of_injective _ Ideal.algebraMap_quotient_injective
cases nonempty_fintype (R ⧸ P)
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.FormalGroup.Basic | {
"line": 103,
"column": 43
} | {
"line": 103,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ✝ : Type u_3\nτ : Type u_4\nσ : Type\nF : FormalGroup R\naux₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\naux₂ : failed to pretty print expression (use 'set_... | subst_add aux₂, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.FormalGroup.Basic | {
"line": 121,
"column": 59
} | {
"line": 121,
"column": 74
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Type u_2\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ✝ : Type u_3\nτ : Type u_4\nσ : Type\nF : FormalGroup R\naux₁ : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\naux₂ : failed to pretty print expression (use 'set_... | subst_add aux₂, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.HahnSeries.HEval | {
"line": 193,
"column": 2
} | {
"line": 194,
"column": 66
} | [
{
"pp": "case coeff.h\nΓ : Type u_1\nR : Type u_3\ninst✝³ : AddCommMonoid Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedCancelAddMonoid Γ\ninst✝ : CommRing R\nx : R⟦Γ⟧\nr : R\ng : Γ\n⊢ ((heval x) (C r)).coeff g = (r • 1).coeff g",
"usedConstants": [
"Eq.mpr",
"WithTop.decidableLT",
"HahnSer... | simp only [heval_apply, coeff_hsum, smulFamily_toFun, powers_toFun, HahnSeries.coeff_smul,
HahnSeries.coeff_one, smul_eq_mul, mul_ite, mul_one, mul_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Grassmannian | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 57
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nM : Type v\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nk : ℕ\nA : Type w\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nB : Type w\ninst✝³ : CommRing B\ninst✝² : Algebra R B\nf : A →ₐ[R] B\nC : Type w\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\ng : B →ₐ[R] C\nN : G(k, A ... | rw [map_toSubmodule g (map f N), map_toSubmodule f N] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 616,
"column": 4
} | {
"line": 616,
"column": 74
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝¹ : PartialOrder Γ\ninst✝ : AddCommMonoid R\ns : SummableFamily Γ R α\nf : α ↪ β\n⊢ (⋃ a, (if h : a ∈ Set.range ⇑f then s (Classical.choose h) else 0).support).IsPWO",
"usedConstants": [
"HahnSeries.supp... | refine s.isPWO_iUnion_support.mono (Set.iUnion_subset fun b g h => ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Henselian | {
"line": 148,
"column": 6
} | {
"line": 149,
"column": 30
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\ntfae_3_to_2 :\n (∀ {K : Type u} [inst : Field K] (φ : R →+* K),\n Surjective ⇑φ →\n ∀ (f : R[X]),\n f.Monic → ∀ (a₀ : K), eval₂ φ a₀ f = 0 → eval₂ φ a₀ (derivative f) ≠ 0 → ∃ a, f.IsRoot a ∧ φ a = a₀) →\n ∀ (f : R[X]),\n ... | rwa [← mem_nonunits_iff, ← mem_maximalIdeal, ← ker_eq_maximalIdeal φ hφ,
RingHom.mem_ker] at h₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.HahnSeries.Summable | {
"line": 890,
"column": 21
} | {
"line": 891,
"column": 73
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nV : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : AddCommGroup Γ\ninst✝² : LinearOrder Γ\ninst✝¹ : IsOrderedAddMonoid Γ\ninst✝ : Field R\nq : ℚ≥0\n⊢ ↑q = ↑q.num / ↑q.den",
"usedConstants": [
"Semiring.toNatCast",
"NonAssocSemiring.toAddCommMon... | by
simp [← single_zero_nnratCast, ← single_zero_natCast, NNRat.cast_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.IdealFilter.Topology | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 73
} | [
{
"pp": "A : Type u_1\ninst✝ : Ring A\nF : IdealFilter A\nI : Ideal A\nhI : I ∈ F\nJ : Ideal A\nhJ : J ∈ F\n⊢ ∃ z ∈ {x | ∃ I ∈ F, ↑I = x}, z ⊆ ↑I ∩ ↑J",
"usedConstants": [
"Semiring.toModule",
"Submodule.completeLattice",
"PartialOrder.toPreorder",
"setOf",
"Membership.mem",
... | exact ⟨I ⊓ J, ⟨I ⊓ J, Order.PFilter.inf_mem hI hJ, rfl⟩, fun _ h ↦ h⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 438,
"column": 6
} | {
"line": 438,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsNoetherianRing R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsNoetherianRing S\np : Ideal R\ninst✝² : p.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\ns : Finset R\nhp : p ∈ (span ↑s).minimalPrimes\nheq : ↑s.card = p... | use y, hx | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 438,
"column": 6
} | {
"line": 438,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsNoetherianRing R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsNoetherianRing S\np : Ideal R\ninst✝² : p.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\ns : Finset R\nhp : p ∈ (span ↑s).minimalPrimes\nheq : ↑s.card = p... | use y, hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
"line": 438,
"column": 6
} | {
"line": 438,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsNoetherianRing R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsNoetherianRing S\np : Ideal R\ninst✝² : p.IsPrime\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : P.LiesOver p\ns : Finset R\nhp : p ∈ (span ↑s).minimalPrimes\nheq : ↑s.card = p... | use y, hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.