module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.Typeclasses.Probability | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 45
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ : Measure α\ns : Set α\ninst✝ : IsZeroOrProbabilityMeasure μ\np : α → Prop\nf✝ : β → α\nf : α → β\nhf : ¬AEMeasurable f μ\n⊢ IsZeroOrProbabilityMeasure (Measure.map f μ)",
"usedConstants": [
"False",
... | · simp [isZeroOrProbabilityMeasure_iff, hf] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Typeclasses.Probability | {
"line": 230,
"column": 2
} | {
"line": 233,
"column": 79
} | [
{
"pp": "case inr\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : IsZeroOrProbabilityMeasure μ\np : ℝ≥0∞\nhμs : p < μ s\ns_mble : MeasurableSet s\nh : IsProbabilityMeasure μ\n⊢ μ sᶜ < 1 - p",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"Preorder.toLT",
... | · rw [prob_compl_eq_one_sub s_mble]
apply ENNReal.sub_lt_of_sub_lt prob_le_one (Or.inl one_ne_top)
convert hμs
exact ENNReal.sub_sub_cancel one_ne_top (lt_of_lt_of_le hμs prob_le_one).le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 96,
"column": 45
} | {
"line": 96,
"column": 63
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nc : β\ns : Set β\nhs : MeasurableSet s\n⊢ μ ((fun x ↦ c) ⁻¹' s) = μ univ * s.indicator 1 c",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"congrArg... | Set.preimage_const | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Count | {
"line": 60,
"column": 30
} | {
"line": 60,
"column": 79
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\ns : Set α\nhs : s.Finite\n⊢ count s = ↑(#hs.toFinset)",
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"Finset",
"instAddCommMonoidWithOneENNReal",
"id",
"AddMo... | by rw [← count_apply_finset, Finite.coe_toFinset] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 357,
"column": 2
} | {
"line": 358,
"column": 22
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nf : β → α\ns : Finset α\nμ : Measure β\nhf : AEMeasurable f μ\n⊢ (∀ᵐ (b : β) ∂μ, f b ∈ s) ↔ map f μ = ∑ a ∈ s, μ (f ⁻¹' {a}) • dirac a",
"usedConstants": [
"MeasureTheory.ae",
... | rw [← ae_map_iff hf (by measurability), ae_mem_finset_iff]
simp [map_apply₀ hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 357,
"column": 2
} | {
"line": 358,
"column": 22
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nf : β → α\ns : Finset α\nμ : Measure β\nhf : AEMeasurable f μ\n⊢ (∀ᵐ (b : β) ∂μ, f b ∈ s) ↔ map f μ = ∑ a ∈ s, μ (f ⁻¹' {a}) • dirac a",
"usedConstants": [
"MeasureTheory.ae",
... | rw [← ae_map_iff hf (by measurability), ae_mem_finset_iff]
simp [map_apply₀ hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ ∫⁻ (x : α) in {a}, f x ∂μ = f a * μ {a}",
"usedConstants": [
"MeasureTheory.Measure.restrict_singleton",
"instHSMul",
"MeasureTheory.Measure",
"MeasureTheory.lintegral_smul_meas... | simp [lintegral_dirac' _ hf, mul_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ ∫⁻ (x : α) in {a}, f x ∂μ = f a * μ {a}",
"usedConstants": [
"MeasureTheory.Measure.restrict_singleton",
"instHSMul",
"MeasureTheory.Measure",
"MeasureTheory.lintegral_smul_meas... | simp [lintegral_dirac' _ hf, mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ ∫⁻ (x : α) in {a}, f x ∂μ = f a * μ {a}",
"usedConstants": [
"MeasureTheory.Measure.restrict_singleton",
"instHSMul",
"MeasureTheory.Measure",
"MeasureTheory.lintegral_smul_meas... | simp [lintegral_dirac' _ hf, mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UnitInterval | {
"line": 499,
"column": 2
} | {
"line": 499,
"column": 86
} | [
{
"pp": "ι : Sort u_1\nc : ι → Set (↑I × ↑I)\nhc₁ : ∀ (i : ι), IsOpen (c i)\nhc₂ : univ ⊆ ⋃ i, c i\nδ : ℝ\nδ_pos : δ > 0\nball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i\nhδ : 0 < δ / 2\nh : 0 ≤ 1\nn m : ℕ\n⊢ ∃ i,\n Icc (addNSMul h (δ / 2) n) (addNSMul h (δ / 2) (n + 1)) ×ˢ Icc (addNSMul h (δ / 2) m) (a... | obtain ⟨i, hsub⟩ := ball_subset (addNSMul h (δ / 2) n, addNSMul h (δ / 2) m) trivial | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.OpenPos | {
"line": 119,
"column": 2
} | {
"line": 124,
"column": 53
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\nm : MeasurableSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\nU : Set X\nf g : X → Y\nhU : IsOpen U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : μ {a | a ∈ U ∧ ¬f a = g a} = 0\n⊢ EqOn f g... | have : IsOpen (U ∩ { a | f a ≠ g a }) := by
refine isOpen_iff_mem_nhds.mpr fun a ha => inter_mem (hU.mem_nhds ha.1) ?_
rcases ha with ⟨ha : a ∈ U, ha' : (f a, g a) ∈ (diagonal Y)ᶜ⟩
exact
(hf.continuousAt (hU.mem_nhds ha)).prodMk_nhds (hg.continuousAt (hU.mem_nhds ha))
(isClosed_diagonal.isOpen... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 129,
"column": 4
} | {
"line": 132,
"column": 57
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : SFinite ν\nm :\n ∀ {α : Type ?u.1545.10} {β : Type ?u.1545.9} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\n⊢ ∀ (c : ℝ≥0∞) ⦃s : Set (α × β)... | intro c s hs
simp only [← indicator_comp_right]
suffices Measurable fun x => c * ν (Prod.mk x ⁻¹' s) by simpa [lintegral_indicator (m hs)]
exact (measurable_measure_prodMk_left hs).const_mul _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 129,
"column": 4
} | {
"line": 132,
"column": 57
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : SFinite ν\nm :\n ∀ {α : Type ?u.1545.10} {β : Type ?u.1545.9} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\n⊢ ∀ (c : ℝ≥0∞) ⦃s : Set (α × β)... | intro c s hs
simp only [← indicator_comp_right]
suffices Measurable fun x => c * ν (Prod.mk x ⁻¹' s) by simpa [lintegral_indicator (m hs)]
exact (measurable_measure_prodMk_left hs).const_mul _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 663,
"column": 72
} | {
"line": 663,
"column": 85
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nγ : Type u_6\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ ν : Measure ((α × β) × γ)\ninst✝ : IsFiniteMeasure μ\n⊢ map (⇑MeasurableEquiv.prodAssoc) μ = map (⇑MeasurableEquiv.prodAssoc) ν ↔\n ∀ {s : Set α} {t : Set β} {u : Set γ},\n Measur... | ext_prod₃_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Group.Prod | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 26
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\nμ : Measure G\ninst✝² : SFinite μ\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsMulLeftInvariant\ng : G\n⊢ μ ≪ map (fun h ↦ g / h) μ",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"MeasureTheory.... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 686,
"column": 46
} | {
"line": 686,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\nthis :\n (sum fun i ↦ map Prod.swap ((sfiniteSeq μ i.1).prod (sfiniteSeq ν i.2))) =\n sum fun i ↦ map Prod.swap ((sfiniteSeq μ i.2).prod (sfiniteS... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 686,
"column": 56
} | {
"line": 686,
"column": 65
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\nthis :\n (sum fun i ↦ map Prod.swap ((sfiniteSeq μ i.1).prod (sfiniteSeq ν i.2))) =\n sum fun i ↦ map Prod.swap ((sfiniteSeq μ i.2).prod (sfiniteS... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 754,
"column": 66
} | {
"line": 754,
"column": 75
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ninst✝ : SFinite τ\nthis :\n (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 754,
"column": 76
} | {
"line": 754,
"column": 85
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ninst✝ : SFinite τ\nthis :\n (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 755,
"column": 63
} | {
"line": 755,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ninst✝ : SFinite τ\nthis :\n (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 755,
"column": 73
} | {
"line": 755,
"column": 82
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ninst✝ : SFinite τ\nthis :\n (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 772,
"column": 4
} | {
"line": 772,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\n⊢ (sum fun i ↦ (sfiniteSeq μ i).restrict s).prod (sum fun i ↦ (sfiniteSeq ν i).restrict t) =\n ((sum (sfiniteSeq μ)).prod (su... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 772,
"column": 14
} | {
"line": 772,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\n⊢ (sum fun p ↦ ((sfiniteSeq μ p.1).restrict s).prod ((sfiniteSeq ν p.2).restrict t)) =\n ((sum (sfiniteSeq μ)).prod (sum (sfi... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 806,
"column": 85
} | {
"line": 806,
"column": 94
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\nν' : Measure β\ninst✝ : SFinite ν'\n⊢ (sum (sfiniteSeq μ)).prod (sum fun n ↦ sfiniteSeq ν n + sfiniteSeq ν' n) =\n (sum (sfiniteSeq μ)).prod (sum ... | prod_sum, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 814,
"column": 85
} | {
"line": 814,
"column": 94
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\nμ' : Measure α\ninst✝ : SFinite μ'\n⊢ (sum fun n ↦ sfiniteSeq μ n + sfiniteSeq μ' n).prod (sum (sfiniteSeq ν)) =\n (sum (sfiniteSeq μ)).prod (sum ... | prod_sum, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 833,
"column": 29
} | {
"line": 833,
"column": 38
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nδ : Type u_4\ninst✝² : MeasurableSpace δ\nf : α → β\ng : γ → δ\nμa : Measure α\nμc : Measure γ\ninst✝¹ : SFinite μa\ninst✝ : SFinite μc\nhf : Measurable f\nhg : Measurable g\n⊢ ... | prod_sum, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Group.Prod | {
"line": 439,
"column": 2
} | {
"line": 439,
"column": 26
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\nμ : Measure G\ninst✝² : SFinite μ\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsMulLeftInvariant\ng : G\n⊢ QuasiMeasurePreserving (fun h ↦ g / h) μ μ",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 1007,
"column": 2
} | {
"line": 1007,
"column": 82
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝ : SFinite ν\nf : α × β → ℝ≥0∞\nhf : AEMeasurable f (μ.bind fun x ↦ map (Prod.mk x) ν)\n⊢ (fun a ↦ ∫⁻ (x : α × β), f x ∂map (Prod.mk a) ν) =ᶠ[ae μ] fun a ↦ ∫⁻ (y : β), f (a, y)... | filter_upwards [Measurable.map_prodMk_left.aemeasurable.ae_of_bind hf] with a ha | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 80,
"column": 39
} | {
"line": 80,
"column": 79
} | [
{
"pp": "case e_μ\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\ns : Set α\nt : Set α := toMeasurable μ s\n⊢ μ.restrict t = μ.restrict s",
"usedConstants": [
"MeasureTheory.Measure.restrict_toMeasurable_of_sFinite"
]
}
] | exact restrict_toMeasurable_of_sFinite s | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 30
} | [
{
"pp": "case h\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\n⊢ ((r • μ).withDensity f) s = (r • μ.withDensity f) s",
"usedConstants": [
"MeasureTheory.Measure.restrict_smul",
"MeasureTheory.Measure.withDensity",
"instHSMul",... | simp [withDensity_apply, hs] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 1124,
"column": 4
} | {
"line": 1126,
"column": 47
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nX : α → β\nY : α → γ\nμ : Measure α\nhY : AEMeasurable Y μ\nhX : ¬AEMeasurable X μ\n⊢ (map (fun a ↦ (X a, Y a)) μ).fst = map X μ",
"usedConstants": [
"AEMeasu... | have : ¬AEMeasurable (fun x ↦ (X x, Y x)) μ := by
contrapose! hX
exact measurable_fst.comp_aemeasurable hX | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 307,
"column": 36
} | {
"line": 307,
"column": 51
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\np : α → Prop\nf g : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᶠ[ae μ] g\na : α\nha : f a = g a\nh : g a ≠ 0\n⊢ {x | f x ≠ 0} a",
"usedConstants": [
"congrArg",
"Eq.mp",
"Ne",
"instZeroENNReal",
"ENNReal",
"Zero.toO... | rwa [← ha] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 307,
"column": 36
} | {
"line": 307,
"column": 51
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\np : α → Prop\nf g : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᶠ[ae μ] g\na : α\nha : f a = g a\nh : g a ≠ 0\n⊢ {x | f x ≠ 0} a",
"usedConstants": [
"congrArg",
"Eq.mp",
"Ne",
"instZeroENNReal",
"ENNReal",
"Zero.toO... | rwa [← ha] at h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 307,
"column": 36
} | {
"line": 307,
"column": 51
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\np : α → Prop\nf g : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᶠ[ae μ] g\na : α\nha : f a = g a\nh : g a ≠ 0\n⊢ {x | f x ≠ 0} a",
"usedConstants": [
"congrArg",
"Eq.mp",
"Ne",
"instZeroENNReal",
"ENNReal",
"Zero.toO... | rwa [← ha] at h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Group.Measure | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 26
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : DivisionMonoid G\ninst✝³ : MeasurableMul G\ninst✝² : MeasurableInv G\nμ : Measure G\ninst✝¹ : μ.IsInvInvariant\ninst✝ : μ.IsMulLeftInvariant\ng : G\n⊢ MeasurePreserving (fun t ↦ g / t) μ μ",
"usedConstants": [
"MeasureTheory.MeasurePreserving... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 647,
"column": 2
} | {
"line": 649,
"column": 35
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ≥0∞\nhfm : Measurable f\nμ : Measure α\ninst✝ : SFinite μ\nhμ : IsFiniteMeasure μ\ns : Set α := {x | f x = ∞}\nhs : MeasurableSet s\nkey : μ.withDensity f = μ.withDensity (sᶜ.indicator f) + sum fun x ↦ μ.withDensity (s.indicator 1)\nthis : ... | have : SigmaFinite (μ.withDensity (s.indicator 1)) := by
rw [withDensity_indicator hs]
exact SigmaFinite.withDensity 1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Complex.Basic | {
"line": 321,
"column": 16
} | {
"line": 321,
"column": 58
} | [
{
"pp": "⊢ ∀ (x : ℂ), 0 • x = 0",
"usedConstants": [
"Real",
"instHSMul",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"Complex.im",
"AddMonoid.toAddZeroClass",
"AddMonoid.toNSMul",
"Complex.smul_re",
"Complex.instZero",
"AddZeroClass.t... | by intros; ext <;> simp [smul_re, smul_im] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Complex.Basic | {
"line": 320,
"column": 17
} | {
"line": 320,
"column": 59
} | [
{
"pp": "⊢ ∀ (a : ℂ), 0 • a = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"Real.instZero",
"congrArg",
"Complex.im",
"Complex.smul_re",
... | by intros; ext <;> simp [smul_re, smul_im] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Norm | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 49
} | [
{
"pp": "x y : ℝ\nm : ℝ := max |x| |y|\nhm₀ : 0 ≤ m\n⊢ √(m ^ 2 + m ^ 2) = √2 * m",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"HMul.hMul",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"two_mul",
"Real.semiring",
"id",
... | rw [← two_mul, Real.sqrt_mul, Real.sqrt_sq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 296,
"column": 19
} | {
"line": 296,
"column": 27
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\nI' : A\nhf : I' * I' = -1\nx✝¹ x✝ : ℂ\nx₁ y₁ x₂ y₂ : ℝ\n⊢ (algebraMap ℝ A) (x₁ * x₂ - y₁ * y₂) + (x₁ * y₂ + y₁ * x₂) • I' =\n (algebraMap ℝ A) x₁ * ((algebraMap ℝ A) x₂ + y₂ • I') + y₁ • I' * ((algebraMap ℝ A) x₂ + y₂ • I')",
"usedConstants": [... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 296,
"column": 28
} | {
"line": 296,
"column": 36
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\nI' : A\nhf : I' * I' = -1\nx✝¹ x✝ : ℂ\nx₁ y₁ x₂ y₂ : ℝ\n⊢ (algebraMap ℝ A) (x₁ * x₂ - y₁ * y₂) + (x₁ * y₂ + y₁ * x₂) • I' =\n (algebraMap ℝ A) x₁ * (algebraMap ℝ A) x₂ + (algebraMap ℝ A) x₁ * y₂ • I' +\n y₁ • I' * ((algebraMap ℝ A) x₂ + y₂ • I... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 425,
"column": 79
} | {
"line": 427,
"column": 59
} | [
{
"pp": "A : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\nz : ℂ\na : A\n⊢ ℜ (z • a) = z.re • ℜ a - z.im • ℑ a",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"instTrivialStarReal",
... | by
have := by congrm (ℜ ($((re_add_im z).symm) • a))
simpa [-re_add_im, add_smul, ← smul_smul, sub_eq_add_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.OpenPartialHomeomorph.Continuity | {
"line": 110,
"column": 2
} | {
"line": 111,
"column": 96
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\nx : X\np : Y → Prop\ns : Set X\nhx : x ∈ e.source\n⊢ (∀ᶠ (y : Y) in 𝓝[↑e.symm ⁻¹' s] ↑e x, p y) ↔ ∀ᶠ (x : X) in 𝓝[s] x, p (↑e x)",
"usedConstants": [
"Eq.mpr",
"OpenPart... | refine Iff.trans ?_ eventually_map
rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.nhdsWithin_target_inter (e.mapsTo hx)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.OpenPartialHomeomorph.Continuity | {
"line": 110,
"column": 2
} | {
"line": 111,
"column": 96
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\nx : X\np : Y → Prop\ns : Set X\nhx : x ∈ e.source\n⊢ (∀ᶠ (y : Y) in 𝓝[↑e.symm ⁻¹' s] ↑e x, p y) ↔ ∀ᶠ (x : X) in 𝓝[s] x, p (↑e x)",
"usedConstants": [
"Eq.mpr",
"OpenPart... | refine Iff.trans ?_ eventually_map
rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.nhdsWithin_target_inter (e.mapsTo hx)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 349,
"column": 87
} | {
"line": 349,
"column": 97
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ (I * z + -I * (starRingEnd K) z) / 2 = -(I * ((starRingEnd K) z - z)) / 2",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NegZeroClass.toNeg",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"No... | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 358,
"column": 54
} | {
"line": 358,
"column": 63
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\nx✝ : (starRingEnd K) z = z\nh : (starRingEnd K) z = z := x✝\n⊢ I * 0 / 2 = ↑0",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"NormedRing.toR... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 561,
"column": 49
} | {
"line": 561,
"column": 95
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ ‖(starRingEnd K) z‖ = ‖z‖",
"usedConstants": [
"Norm.norm",
"Real",
"congrArg",
"CommSemiring.toSemiring",
"MonoidWithZeroHom.funLike",
"RingHom",
"NormedField.toField",
"Real.semiring",
"NormedField.toNo... | simp only [← sqrt_normSq_eq_norm, normSq_conj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.RCLike.Basic | {
"line": 561,
"column": 49
} | {
"line": 561,
"column": 95
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ ‖(starRingEnd K) z‖ = ‖z‖",
"usedConstants": [
"Norm.norm",
"Real",
"congrArg",
"CommSemiring.toSemiring",
"MonoidWithZeroHom.funLike",
"RingHom",
"NormedField.toField",
"Real.semiring",
"NormedField.toNo... | simp only [← sqrt_normSq_eq_norm, normSq_conj] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 561,
"column": 49
} | {
"line": 561,
"column": 95
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ ‖(starRingEnd K) z‖ = ‖z‖",
"usedConstants": [
"Norm.norm",
"Real",
"congrArg",
"CommSemiring.toSemiring",
"MonoidWithZeroHom.funLike",
"RingHom",
"NormedField.toField",
"Real.semiring",
"NormedField.toNo... | simp only [← sqrt_normSq_eq_norm, normSq_conj] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 849,
"column": 50
} | {
"line": 850,
"column": 38
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : ℝ\n⊢ 0 < ↑x ↔ 0 < x",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.instZero",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"Real.instLT",
... | by
rw [← ofReal_zero, ofReal_lt_ofReal] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.RCLike.Basic | {
"line": 853,
"column": 54
} | {
"line": 854,
"column": 38
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : ℝ\n⊢ ↑x < 0 ↔ x < 0",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.instZero",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"Real.instLT",
... | by
rw [← ofReal_zero, ofReal_lt_ofReal] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.RCLike.Basic | {
"line": 1157,
"column": 26
} | {
"line": 1157,
"column": 45
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx y : K\n⊢ ‖re x - re y‖ₑ = ‖re (x - y)‖ₑ",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"map_sub",
"AddMonoid.toAddSemigroup",
"Real.instAddMonoid",
... | rw [map_sub re x y] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 1157,
"column": 26
} | {
"line": 1157,
"column": 45
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx y : K\n⊢ ‖re x - re y‖ₑ = ‖re (x - y)‖ₑ",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"map_sub",
"AddMonoid.toAddSemigroup",
"Real.instAddMonoid",
... | rw [map_sub re x y] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 1157,
"column": 26
} | {
"line": 1157,
"column": 45
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx y : K\n⊢ ‖re x - re y‖ₑ = ‖re (x - y)‖ₑ",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
"map_sub",
"AddMonoid.toAddSemigroup",
"Real.instAddMonoid",
... | rw [map_sub re x y] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Exponential | {
"line": 100,
"column": 4
} | {
"line": 105,
"column": 10
} | [
{
"pp": "case h.e'_3.succ\nε : ℝ\nε0 : ε > 0\nj : ℕ\nhj : j + 1 ≥ 1\n⊢ ‖∑ m ∈ range (j + 1), 0 ^ m / ↑m.factorial - 1‖ = 0",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Nat.instCanonicallyOrderedAdd",
"MulOne.toOne",
"False",
"Real",... | induction j with
| zero => simp
| succ j ih =>
rw [← ih (by simp)]
simp only [sum_range_succ, pow_succ]
simp | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Complex.Exponential | {
"line": 365,
"column": 82
} | {
"line": 373,
"column": 78
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nn j : ℕ\nhn : 0 < n\n⊢ (↑n.factorial)⁻¹ * ∑ m ∈ range (j - n), (↑n.succ)⁻¹ ^ m =\n (↑n.succ - ↑n.succ * (↑n.succ)⁻¹ ^ (j - n)) / (↑n.factorial * ↑n)",
"usedConstants": [
"Iff.mpr",
"NonUnitalNonAss... | by
have h₁ : (n.succ : α) ≠ 1 :=
@Nat.cast_one α _ ▸ mt Nat.cast_inj.1 (mt Nat.succ.inj (pos_iff_ne_zero.1 hn))
have h₂ : (n.succ : α) ≠ 0 := by positivity
have h₃ : (n.factorial * n : α) ≠ 0 := by positivity
have h₄ : (n.succ - 1 : α) = n := by simp
rw [geom_sum_inv h₁ h₂, eq_div_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Exponential | {
"line": 518,
"column": 14
} | {
"line": 518,
"column": 40
} | [
{
"pp": "case e_a.refine_3.refine_2\nx : ℂ\nn j : ℕ\nhj : j ≥ n\nb : ℕ\nhb : b < j ∧ n ≤ b\n⊢ b - n + n = b",
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"Nat.instIsOrderedAddMonoid",
"congrArg",
"HSub.hSub",
"tsub_add_cancel_of_le",
"id",
"Nat.instStar... | tsub_add_cancel_of_le hb.2 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Tactic.NormNum.NatFactorial | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 81
} | [
{
"pp": "case out\nn x l y z : ℕ\nh₁ : IsNat n x\nh₂ : IsNat l y\nh₃ : x = z + y\na : ℕ\np : (z + 1).ascFactorial y = a\n⊢ n.descFactorial l = ↑a",
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"congrArg",
"Nat.ascFactorial",
"id",
"AddMonoidWithOne.toNatCast",... | simpa [h₁.out, h₂.out, ← p, h₃] using Nat.add_descFactorial_eq_ascFactorial _ _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 132,
"column": 6
} | {
"line": 132,
"column": 40
} | [
{
"pp": "x y : ℂ\n⊢ sinh (x + y) = sinh x * cosh y + cosh x * sinh y",
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"Field.isDomain",
"CharZero.NeZero.two",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
"Co... | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 132,
"column": 87
} | {
"line": 132,
"column": 95
} | [
{
"pp": "x y : ℂ\n⊢ 2 * (sinh x * cosh y + cosh x * sinh y) = cexp x * cexp y - cexp (-x) * cexp (-y)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NegZeroClass.toNeg",
"Complex.sinh",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"Comm... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 133,
"column": 50
} | {
"line": 133,
"column": 84
} | [
{
"pp": "x y : ℂ\n⊢ (cexp x - cexp (-x)) * cosh y + cosh x * (cexp y - cexp (-y)) = cexp x * cexp y - cexp (-x) * cexp (-y)",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"HMul.hMul",
"Field.isDomain",
"CommRing.toNonUnitalCommRing",
"Comple... | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 133,
"column": 85
} | {
"line": 133,
"column": 93
} | [
{
"pp": "x y : ℂ\n⊢ 2 * ((cexp x - cexp (-x)) * cosh y + cosh x * (cexp y - cexp (-y))) = 2 * (cexp x * cexp y - cexp (-x) * cexp (-y))",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"NonUnitalCommRing.toNonUnitalNonAssocComm... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 147,
"column": 6
} | {
"line": 147,
"column": 40
} | [
{
"pp": "x y : ℂ\n⊢ cosh (x + y) = cosh x * cosh y + sinh x * sinh y",
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"Field.isDomain",
"CharZero.NeZero.two",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
"Co... | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 147,
"column": 87
} | {
"line": 147,
"column": 95
} | [
{
"pp": "x y : ℂ\n⊢ 2 * (cosh x * cosh y + sinh x * sinh y) = cexp x * cexp y + cexp (-x) * cexp (-y)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NegZeroClass.toNeg",
"Complex.sinh",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"Comm... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 148,
"column": 48
} | {
"line": 148,
"column": 82
} | [
{
"pp": "x y : ℂ\n⊢ (cexp x + cexp (-x)) * cosh y + (cexp x - cexp (-x)) * sinh y = cexp x * cexp y + cexp (-x) * cexp (-y)",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"Complex.sinh",
"HMul.hMul",
"Field.isDomain",
"CommRing.toNonUnitalCo... | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 148,
"column": 83
} | {
"line": 148,
"column": 91
} | [
{
"pp": "x y : ℂ\n⊢ 2 * ((cexp x + cexp (-x)) * cosh y + (cexp x - cexp (-x)) * sinh y) = 2 * (cexp x * cexp y + cexp (-x) * cexp (-y))",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"Complex.sinh",
"NonUnitalCommRing.t... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 220,
"column": 6
} | {
"line": 220,
"column": 40
} | [
{
"pp": "x : ℂ\n⊢ cosh x + sinh x = cexp x",
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"Field.isDomain",
"CharZero.NeZero.two",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
"Complex.instZero",
"Is... | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 220,
"column": 41
} | {
"line": 220,
"column": 49
} | [
{
"pp": "x : ℂ\n⊢ 2 * (cosh x + sinh x) = 2 * cexp x",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Complex.sinh",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Complex.commRing",
"congrArg",
... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 235,
"column": 6
} | {
"line": 235,
"column": 40
} | [
{
"pp": "x : ℂ\n⊢ cosh x - sinh x = cexp (-x)",
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"Field.isDomain",
"CharZero.NeZero.two",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
"HSub.hSub",
"Comple... | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 288,
"column": 6
} | {
"line": 288,
"column": 40
} | [
{
"pp": "x : ℂ\n⊢ sinh (x * I) = sin x * I",
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"Field.isDomain",
"CharZero.NeZero.two",
"congrArg",
"Complex.sin",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
"Complex... | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 292,
"column": 6
} | {
"line": 292,
"column": 40
} | [
{
"pp": "x : ℂ\n⊢ cosh (x * I) = cos x",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Field.isDomain",
"CharZero.NeZero.two",
"Complex.cos",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOne",
"Complex.instZero",
"IsCance... | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 297,
"column": 51
} | {
"line": 297,
"column": 63
} | [
{
"pp": "x : ℂ\n⊢ cos (x * I) = cosh x",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Complex.instMul",
"id",
"Complex",
"Eq.symm",
"Eq",
"Complex.cosh_mul_I",
"Complex.cosh",
"instHMul",
"Complex.I"
]... | ← cosh_mul_I | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 69
} | [
{
"pp": "α : Type u_1\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : Archimedean α\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ n ≥ m, |f n| ≤ a\nhnm : ∀ n ≥ m, f n.succ ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ n ≥ m, a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ n ≥ m... | rw [abs_of_nonpos (sub_nonpos.2 hfij), neg_sub, sub_lt_iff_lt_add'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Ring.InfiniteSum | {
"line": 37,
"column": 4
} | {
"line": 37,
"column": 69
} | [
{
"pp": "ι : Type u_2\nι' : Type u_3\nf : ι → ℝ\ng : ι' → ℝ\nhf : Summable f\nhg : Summable g\nhf' : 0 ≤ f\nhg' : 0 ≤ g\n⊢ Summable fun x ↦ ∑' (y : ι'), f (x, y).1 * g (x, y).2",
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
... | by simpa only [hg.tsum_mul_left _] using hf.mul_right (∑' x, g x) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 904,
"column": 2
} | {
"line": 904,
"column": 36
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nl : Filter ι\nε : ι → 𝕜\nf : ι → E\nhε : Tendsto ε l (𝓝 0)\nhf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l (norm ∘ f)\n⊢ Tendsto (ε • f) l (𝓝 0... | rw [← isLittleO_one_iff 𝕜] at hε ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 628,
"column": 25
} | {
"line": 628,
"column": 43
} | [
{
"pp": "x y : ℝ\n⊢ ↑(cos x + cos y) = ↑(2 * cos ((x + y) / 2) * cos ((x - y) / 2))",
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
"Complex.cos",
"Real.cos",
"congrArg",
"Real.instDivInvMonoid",
"Real.instSub",
"Nat.instAtLeastTwoHAddOfNat",
... | simp [cos_add_cos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 628,
"column": 25
} | {
"line": 628,
"column": 43
} | [
{
"pp": "x y : ℝ\n⊢ ↑(cos x + cos y) = ↑(2 * cos ((x + y) / 2) * cos ((x - y) / 2))",
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
"Complex.cos",
"Real.cos",
"congrArg",
"Real.instDivInvMonoid",
"Real.instSub",
"Nat.instAtLeastTwoHAddOfNat",
... | simp [cos_add_cos] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 628,
"column": 25
} | {
"line": 628,
"column": 43
} | [
{
"pp": "x y : ℝ\n⊢ ↑(cos x + cos y) = ↑(2 * cos ((x + y) / 2) * cos ((x - y) / 2))",
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
"Complex.cos",
"Real.cos",
"congrArg",
"Real.instDivInvMonoid",
"Real.instSub",
"Nat.instAtLeastTwoHAddOfNat",
... | simp [cos_add_cos] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 132,
"column": 32
} | {
"line": 132,
"column": 40
} | [
{
"pp": "S : Type u_1\ninst✝ : Semiring S\nn : ℕ\nk : S\n⊢ eval k (ascPochhammer S n * (X + ↑n)) = eval k (ascPochhammer S n) * (k + ↑n)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 200,
"column": 34
} | {
"line": 200,
"column": 42
} | [
{
"pp": "case succ\nS : Type u_1\ninst✝² : Semiring S\ninst✝¹ : PartialOrder S\ninst✝ : IsStrictOrderedRing S\ns : S\nh : 0 < s\nn : ℕ\nih : 0 < eval s (ascPochhammer S n)\n⊢ 0 < eval s (ascPochhammer S n * (X + ↑n))",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Polynomial.ev... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 368,
"column": 8
} | {
"line": 368,
"column": 22
} | [
{
"pp": "case succ\nn : ℕ\nih : descPochhammer ℤ n = (ascPochhammer ℤ n).comp (X - ↑n + 1)\n⊢ descPochhammer ℤ (n + 1) = (ascPochhammer ℤ (n + 1)).comp (X - ↑(n + 1) + 1)",
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Polynomial.instOne",
"AddMonoid.toAddSemigroup",
"congrArg... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 376,
"column": 8
} | {
"line": 376,
"column": 22
} | [
{
"pp": "case succ\nR : Type u\ninst✝ : Ring R\nr : R\nn : ℕ\nih : eval r (descPochhammer R n) = eval (r - ↑n + 1) (ascPochhammer R n)\n⊢ eval r (descPochhammer R (n + 1)) = eval (r - ↑(n + 1) + 1) (ascPochhammer R (n + 1))",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"Nat.cast_succ",... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 395,
"column": 34
} | {
"line": 395,
"column": 42
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nr : R\nk : ℕ\n⊢ eval (-r) (ascPochhammer R k * (X + ↑k)) = (-1) ^ (k + 1) * eval r (descPochhammer R (k + 1))",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Polynomial.eval",
"NegZeroClass.toNeg",
"HMul.hMul",
"Ring.toNonA... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 409,
"column": 65
} | {
"line": 409,
"column": 74
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Ring R\nn k : ℕ\nih : eval (↑n) (descPochhammer R k) = ↑(n.descFactorial k)\nh : n < k\n⊢ (↑n - ↑k) * 0 = ↑((n - k) * 0)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"Ring.toNonAssocRing",
"MulZero... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 409,
"column": 75
} | {
"line": 409,
"column": 84
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Ring R\nn k : ℕ\nih : eval (↑n) (descPochhammer R k) = ↑(n.descFactorial k)\nh : n < k\n⊢ 0 = ↑((n - k) * 0)",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"HMul.hMul",
"Ring.toNonAssocRing",
"MulZeroClass.toMul",
"congrArg",... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 523,
"column": 14
} | {
"line": 523,
"column": 54
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionSemiring K\ninst✝ : CharZero K\na b : ℕ\n⊢ ↑(b ! * a.choose b) = eval (↑(a - (b - 1))) (ascPochhammer K b)",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.choose",
"HMul.hMul",
"Comm... | ← descFactorial_eq_factorial_mul_choose, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Pochhammer | {
"line": 533,
"column": 14
} | {
"line": 533,
"column": 54
} | [
{
"pp": "K : Type u_1\ninst✝¹ : DivisionRing K\ninst✝ : CharZero K\na b : ℕ\n⊢ ↑(b ! * a.choose b) = eval (↑a) (descPochhammer K b)",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.choose",
"HMul.hMul",
"CommSemiring.toNonU... | ← descFactorial_eq_factorial_mul_choose, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 252,
"column": 2
} | {
"line": 255,
"column": 27
} | [
{
"pp": "n : ℕ\nC : ℝ\nhC₁ : 1 ≤ C\nhC₀ : 0 < C\nthis : 0 < (rexp 1 * C)⁻¹\n⊢ ∃ ia, True ∧ ∀ x ∈ Set.Ioi ia, rexp x / x ^ n ∈ Set.Ici C",
"usedConstants": [
"Iff.mpr",
"gt_mem_nhds",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"Real.... | obtain ⟨N, hN⟩ : ∃ N : ℕ, ∀ k ≥ N, (↑k : ℝ) ^ n / exp 1 ^ k < (exp 1 * C)⁻¹ :=
eventually_atTop.1
((tendsto_pow_const_div_const_pow_of_one_lt n (one_lt_exp_iff.2 zero_lt_one)).eventually
(gt_mem_nhds this)) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 216,
"column": 31
} | {
"line": 218,
"column": 10
} | [
{
"pp": "⊢ cos π = -1",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Mathlib.Meta.NormNum.isInt_eq_true",
"CharZero.NeZero.two",
"MulZeroClass.toMul",
"Real.instZero",
"Real.cos",
"Mono... | by
rw [← mul_div_cancel_left₀ π two_ne_zero, mul_div_assoc, cos_two_mul, cos_pi_div_two]
norm_num | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Ball.Pointwise | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 84
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : c ≠ 0\nx : E\nr : ℝ\ny : E\n⊢ c⁻¹ • y ∈ ball (c⁻¹ • c • x) r ↔ y ∈ ball (c • x) (‖c‖ * r)",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid... | simp [← div_eq_inv_mul, div_lt_iff₀ (norm_pos_iff.2 hc), mul_comm _ r, dist_smul₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Connected.PathConnected | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 29
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y : X\nF : Set X\nf : X → Y\nhf : IsInducing f\nhx : x ∈ F\nhy : y ∈ F\nγ : Path (f x) (f y)\nhγ : ∀ (t : ↑I), γ t ∈ f '' F\n⊢ JoinedIn F x y",
"usedConstants": [
"Real",
"Membership.mem",
"Set.... | choose γ' hγ'F hγ' using hγ | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 416,
"column": 16
} | {
"line": 416,
"column": 40
} | [
{
"pp": "case h₁\nR : Type u_4\ninst✝ : NormedRing R\nk : ℕ\nr : R\nhr : ‖r‖ < 1\nu : ℕ → ℕ\nhu : (fun n ↦ ↑(u n)) =O[atTop] fun n ↦ ↑(n ^ k)\nr' : ℝ\nhrr' : ‖‖r‖‖ < r'\nh : r' < 1\nn : ℕ\nhn : ‖r ^ n‖ ≤ ‖r‖ ^ n\n⊢ ‖↑(u n)‖ ≤ ↑(u n) * ‖1‖",
"usedConstants": [
"NormedRing.toSeminormedRing",
"Nat.... | exact norm_cast_le (u n) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Instances.Sign | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 40
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝³ : Zero α\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\nh : a ≠ 0\nh_neg : a < 0\n⊢ ContinuousAt (⇑SignType.sign) a",
"usedConstants": [
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattice",
"LinearOrder.... | · exact continuousAt_sign_of_neg h_neg | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 43,
"column": 40
} | {
"line": 43,
"column": 48
} | [
{
"pp": "x : ℂ\nhx : x ≠ 0\n⊢ ↑‖x‖ * (↑(x.re / ‖x‖) + ↑(x.im / ‖x‖) * I) = x",
"usedConstants": [
"Distrib.leftDistribClass",
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"Complex.instNormedAddCommGro... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 281,
"column": 55
} | {
"line": 281,
"column": 69
} | [
{
"pp": "case succ.inr\nx : ℝ\nn : ℕ\nih : log (x ^ n) = ↑n * log x\nhx : x ≠ 0\n⊢ ↑n * log x + log x = ↑(n + 1) * log x",
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne"... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 70
} | [
{
"pp": "case ofNat\nx : ℝ\na✝ : ℕ\n⊢ log (x ^ Int.ofNat a✝) = ↑(Int.ofNat a✝) * log x",
"usedConstants": [
"zpow_natCast",
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"Real",
"HMul.hMul",
"congrArg",
"Real.instDivInvMonoid",
"AddGroupWithOne.toAddMonoidWi... | · rw [Int.ofNat_eq_natCast, zpow_natCast, log_pow, Int.cast_natCast] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 267,
"column": 51
} | {
"line": 267,
"column": 56
} | [
{
"pp": "case neg.mpr\ny : ℝ\nh₀ : ¬{ re := 0, im := y } = 0\nhy : 0 < y\n⊢ { re := 0, im := y }.arg = { re := y * I.re, im := y * I.im }.arg",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"Complex.im",
"Complex.arg",
"id",
... | I_re, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 267,
"column": 63
} | {
"line": 267,
"column": 72
} | [
{
"pp": "case neg.mpr\ny : ℝ\nh₀ : ¬{ re := 0, im := y } = 0\nhy : 0 < y\n⊢ { re := 0, im := y }.arg = { re := y * 0, im := y * 1 }.arg",
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"MulZeroClass.toMul",
"Real.instZer... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 57,
"column": 62
} | {
"line": 57,
"column": 80
} | [
{
"pp": "x : ℂ\nh : x ≠ 0\n⊢ 0 ^ x = 0",
"usedConstants": [
"Complex.log",
"HMul.hMul",
"eq_false",
"congrArg",
"Complex.instZero",
"Complex.instPow",
"Complex.instMul",
"ite_cond_eq_true",
"HPow.hPow",
"True",
"eq_self",
"Complex.exp",... | simp [cpow_def, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 57,
"column": 62
} | {
"line": 57,
"column": 80
} | [
{
"pp": "x : ℂ\nh : x ≠ 0\n⊢ 0 ^ x = 0",
"usedConstants": [
"Complex.log",
"HMul.hMul",
"eq_false",
"congrArg",
"Complex.instZero",
"Complex.instPow",
"Complex.instMul",
"ite_cond_eq_true",
"HPow.hPow",
"True",
"eq_self",
"Complex.exp",... | simp [cpow_def, *] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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