module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.MetricSpace.Perfect | {
"line": 117,
"column": 37
} | {
"line": 117,
"column": 54
} | {
"line": 118,
"column": 4
} | [
{
"pp": "case true.false\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nhnonempty : C.Nonempty\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (nhds 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }\nC0 C1 : {... | [
"case true.false\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nhnonempty : C.Nonempty\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (nhds 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }\nC0 C1 : {C : Set α} →... | try contradiction | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.Topology.MetricSpace.Perfect | {
"line": 117,
"column": 37
} | {
"line": 117,
"column": 54
} | {
"line": 118,
"column": 4
} | [
{
"pp": "case true.true\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nhnonempty : C.Nonempty\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (nhds 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }\nC0 C1 : {C... | [] | try contradiction | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.MeasureTheory.Integral.Lebesgue.Map | {
"line": 137,
"column": 54
} | {
"line": 138,
"column": 85
} | {
"line": 140,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ng : α → β\nhg : MeasurePreserving g μ ν\nhge : MeasurableEmbedding g\nf : β → ℝ≥0∞\ns : Set α\n⊢ ∫⁻ (a : α) in s, f (g a) ∂μ = ∫⁻ (b : β) in g '' s, f b ∂ν",
"ppTerm": "?m.24",
"assi... | [] | by
rw [← hg.setLIntegral_comp_preimage_emb hge, Set.preimage_image_eq _ hge.injective] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.MutuallySingular | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 48
} | {
"line": 236,
"column": 0
} | [
{
"pp": "case h\nα : Type u_1\nm0 : MeasurableSpace α\nμ ν : Measure α\nh : μ ⟂ₘ ν\nh_bot_iff : ∀ (ξ : Measure α), ξ ≤ ⊥ ↔ ξ = 0\nξ : Measure α\nhξμ : ξ ≤ μ\nhξν : ξ ≤ ν\ns : Set α\nhs : MeasurableSet s\n⊢ MeasurableSet (s ∩ h.nullSetᶜ)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"C... | [] | · exact hs.inter h.measurableSet_nullSet.compl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 195,
"column": 18
} | {
"line": 195,
"column": 39
} | {
"line": 197,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : Countable α\ninst✝ : MeasurableSingletonClass α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\n⊢ (sum fun a ↦ μ {a} • dirac a) s = μ s",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Mea... | [] | rw [μ.sum_smul_dirac] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 195,
"column": 18
} | {
"line": 195,
"column": 39
} | {
"line": 197,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : Countable α\ninst✝ : MeasurableSingletonClass α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\n⊢ (sum fun a ↦ μ {a} • dirac a) s = μ s",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Mea... | [] | rw [μ.sum_smul_dirac] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 195,
"column": 18
} | {
"line": 195,
"column": 39
} | {
"line": 197,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : Countable α\ninst✝ : MeasurableSingletonClass α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\n⊢ (sum fun a ↦ μ {a} • dirac a) s = μ s",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Mea... | [] | rw [μ.sum_smul_dirac] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 779,
"column": 4
} | {
"line": 779,
"column": 56
} | {
"line": 780,
"column": 2
} | [
{
"pp": "α : Type u_2\ninst✝³ : MetricSpace α\ninst✝² : CompleteSpace α\ninst✝¹ : SecondCountableTopology α\ninst✝ : Nonempty α\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nI0 : 0 < 1 / 2\nI1 : 1 / 2 < 1\nu : ℕ → α\nhu : DenseRange u\ns : Set (ℕ → ℕ) := ⋯\ng : ↑s → α := ⋯\nA : ∀ (x : ↑s) (n : ℕ), dist (g x... | [] | simpa only [nonempty_coe_sort] using g_surj.nonempty | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 841,
"column": 21
} | {
"line": 841,
"column": 80
} | {
"line": 842,
"column": 4
} | [
{
"pp": "case h\nE : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\nx y z : (i : ι) → F i\nn : ι\n⊢ min (2⁻¹ ^ encode n) (edist (x n) (z n)) ≤\n min (2⁻¹ ^ encode n) (edist (x n) (y n)) + min (2⁻¹ ^ encode n) (edist (y n) (z n))",
"ppTerm":... | [] | grw [edist_triangle _ (y n), min_add_distrib, min_le_right] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 318,
"column": 4
} | {
"line": 319,
"column": 65
} | {
"line": 320,
"column": 4
} | [
{
"pp": "case add\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)\nh₁ :\n ∀ {L : ℝ≥0∞},\n L < ∫⁻ (x : α), ↑(f₁ x) ∂μ → ∃ g, (∀ (x : α), g x ≤ f₁ x) ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ∞ ∧ L < ∫⁻ (x : α), ↑(g x) ∂... | [
"case add\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)\nh₁ :\n ∀ {L : ℝ≥0∞},\n L < ∫⁻ (x : α), ↑(f₁ x) ∂μ → ∃ g, (∀ (x : α), g x ≤ f₁ x) ∧ ∫⁻ (x : α), ↑(g x) ∂μ < ∞ ∧ L < ∫⁻ (x : α), ↑(g x) ∂μ\nh₂ :\n ∀... | replace hL : L < ∫⁻ x, f₁ x ∂μ + ∫⁻ x, f₂ x ∂μ := by
rwa [← lintegral_add_left f₁.measurable.coe_nnreal_ennreal] | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.GroupTheory.Complement | {
"line": 435,
"column": 41
} | {
"line": 435,
"column": 53
} | {
"line": 437,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS T : Set G\nhST : IsComplement S T\nhs1 : 1 ∈ S\nht1 : 1 ∈ T\n⊢ 1 = hST.equiv.symm (⟨1, hs1⟩, ⟨1, ht1⟩)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"Equiv.instEquivLike",
"HMul.hMul",
"DivInvOneMonoid... | [] | simp [equiv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 685,
"column": 2
} | {
"line": 685,
"column": 54
} | {
"line": 688,
"column": 2
} | [
{
"pp": "γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝ : UpgradedIsCompletelyMetrizableSpac... | [
"γ : Type u_3\nβ : Type u_4\ninst✝⁵ : TopologicalSpace γ\ninst✝⁴ : PolishSpace γ\ninst✝³ : TopologicalSpace β\ninst✝² : T2Space β\ninst✝¹ : MeasurableSpace β\ninst✝ : OpensMeasurableSpace β\nf : γ → β\nf_cont : Continuous[inst✝⁵, inst✝³] f\nf_inj : Injective f\nthis✝ : UpgradedIsCompletelyMetrizableSpace γ := upgra... | let A := { p : b × b // Disjoint (p.1 : Set γ) p.2 } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 836,
"column": 2
} | {
"line": 842,
"column": 37
} | {
"line": 846,
"column": 0
} | [
{
"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⊢ ... | [] | simp_rw [← sum_sfiniteSeq μa, ← sum_sfiniteSeq μc, map_sum hf.aemeasurable,
map_sum hg.aemeasurable, prod_sum, map_sum (hf.prodMap hg).aemeasurable]
congr
ext1 i
refine prod_eq fun s t hs ht => ?_
rw [map_apply (hf.prodMap hg) (hs.prod ht), map_apply hf hs, map_apply hg ht]
exact prod_prod (f ⁻¹' s) (g ⁻¹... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 836,
"column": 2
} | {
"line": 842,
"column": 37
} | {
"line": 846,
"column": 0
} | [
{
"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⊢ ... | [] | simp_rw [← sum_sfiniteSeq μa, ← sum_sfiniteSeq μc, map_sum hf.aemeasurable,
map_sum hg.aemeasurable, prod_sum, map_sum (hf.prodMap hg).aemeasurable]
congr
ext1 i
refine prod_eq fun s t hs ht => ?_
rw [map_apply (hf.prodMap hg) (hs.prod ht), map_apply hf hs, map_apply hg ht]
exact prod_prod (f ⁻¹' s) (g ⁻¹... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 889,
"column": 12
} | {
"line": 889,
"column": 70
} | {
"line": 889,
"column": 71
} | [
{
"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\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ninst✝¹ : SFinite μa\ninst✝ : SFinite μc\nf : α → β\nhf : MeasurePreserv... | [
"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\nμa : Measure α\nμb : Measure β\nμc : Measure γ\nμd : Measure δ\ninst✝¹ : SFinite μa\ninst✝ : SFinite μc\nf : α → β\nhf : MeasurePreserving f μa μb\... | map_apply hgm.of_uncurry_left (measurable_prodMk_left hs), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Group.Measure | {
"line": 858,
"column": 4
} | {
"line": 858,
"column": 22
} | {
"line": 859,
"column": 4
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\nμ : Measure G\ninst✝⁶ : μ.IsHaarMeasure\nα : Type u_3\ninst✝⁵ : BorelSpace G\ninst✝⁴ : IsTopologicalGroup G\ninst✝³ : Group α\ninst✝² : MulAction α G\ninst✝¹ : SMulCommCla... | [
"G : Type u_1\nH : Type u_2\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : MeasurableSpace H\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\nμ : Measure G\ninst✝⁶ : μ.IsHaarMeasure\nα : Type u_3\ninst✝⁵ : BorelSpace G\ninst✝⁴ : IsTopologicalGroup G\ninst✝³ : Group α\ninst✝² : MulAction α G\ninst✝¹ : SMulCommClass α G G\nin... | rw [F.map_apply K] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 132,
"column": 2
} | {
"line": 133,
"column": 47
} | {
"line": 134,
"column": 2
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhr : r ≠ ∞\ns : Set α\nhs : MeasurableSet s\n⊢ (μ.withDensity (r • f)) s = (r • μ.withDensity f) s",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure.withDensity"... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\nhr : r ≠ ∞\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (a : α) in s, (r • f) a ∂μ = ∫⁻ (a : α) in s, r * f a ∂μ"
] | rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
smul_eq_mul, ← lintegral_const_mul' r f hr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Group.Prod | {
"line": 480,
"column": 2
} | {
"line": 488,
"column": 12
} | {
"line": 490,
"column": 0
} | [
{
"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.IsMulRightInvariant\ng : G\n⊢ QuasiMeasurePreserving (fun h ↦ g * h) μ μ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [] | have :=
(quasiMeasurePreserving_mul_right μ.inv g⁻¹).mono (inv_absolutelyContinuous μ.inv)
(absolutelyContinuous_inv μ.inv)
rw [μ.inv_inv] at this
have :=
(quasiMeasurePreserving_inv_of_right_invariant μ).comp
(this.comp (quasiMeasurePreserving_inv_of_right_invariant μ))
simp_rw [Function.comp... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Group.Prod | {
"line": 480,
"column": 2
} | {
"line": 488,
"column": 12
} | {
"line": 490,
"column": 0
} | [
{
"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.IsMulRightInvariant\ng : G\n⊢ QuasiMeasurePreserving (fun h ↦ g * h) μ μ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [] | have :=
(quasiMeasurePreserving_mul_right μ.inv g⁻¹).mono (inv_absolutelyContinuous μ.inv)
(absolutelyContinuous_inv μ.inv)
rw [μ.inv_inv] at this
have :=
(quasiMeasurePreserving_inv_of_right_invariant μ).comp
(this.comp (quasiMeasurePreserving_inv_of_right_invariant μ))
simp_rw [Function.comp... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 406,
"column": 4
} | {
"line": 408,
"column": 72
} | {
"line": 410,
"column": 0
} | [
{
"pp": "case iSup\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nh_mf : Measurable f\n⊢ ∀ ⦃f_1 : ℕ → α → ℝ≥0∞⦄,\n (∀ (n : ℕ), Measurable (f_1 n)) →\n Monotone f_1 →\n (∀ (n : ℕ), ∫⁻ (a : α), f_1 n a ∂μ.withDensity f = ∫⁻ (a : α), (f * f_1 n) a ∂μ) →\n ∫⁻ (a : α), (f... | [] | intro g h_mea_g h_mono_g h_ind
have : Monotone fun n a => f a * g n a := fun m n hmn x => by dsimp; grw [h_mono_g hmn x]
simp [lintegral_iSup, ENNReal.mul_iSup, h_mf.fun_mul (h_mea_g _), *] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 406,
"column": 4
} | {
"line": 408,
"column": 72
} | {
"line": 410,
"column": 0
} | [
{
"pp": "case iSup\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nh_mf : Measurable f\n⊢ ∀ ⦃f_1 : ℕ → α → ℝ≥0∞⦄,\n (∀ (n : ℕ), Measurable (f_1 n)) →\n Monotone f_1 →\n (∀ (n : ℕ), ∫⁻ (a : α), f_1 n a ∂μ.withDensity f = ∫⁻ (a : α), (f * f_1 n) a ∂μ) →\n ∫⁻ (a : α), (f... | [] | intro g h_mea_g h_mono_g h_ind
have : Monotone fun n a => f a * g n a := fun m n hmn x => by dsimp; grw [h_mono_g hmn x]
simp [lintegral_iSup, ENNReal.mul_iSup, h_mf.fun_mul (h_mea_g _), *] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 448,
"column": 6
} | {
"line": 448,
"column": 38
} | {
"line": 450,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f μ\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nhg : AEMeasurable g (μ.withDensity f')\nthis : μ.withDensity f = μ.withDensity f'\ng' : α → ℝ≥0∞ := AEMeasurable.mk g hg\nx : α\nhx : f x = AEMeasurable.mk f hf... | [] | simp only [f', hx, Pi.mul_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 474,
"column": 2
} | {
"line": 474,
"column": 61
} | {
"line": 475,
"column": 2
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nf_meas : Measurable f\ng i : α → ℝ≥0∞\ni_meas : Measurable i\nhi : i ≤ g\n⊢ ∫⁻ (a : α), i a ∂μ.withDensity f ≤ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ",
"ppTerm": "?m.46",
"assigned": true,
"... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nf_meas : Measurable f\ng i : α → ℝ≥0∞\ni_meas : Measurable i\nhi : i ≤ g\n⊢ ∫⁻ (a : α), (f * i) a ∂μ ≤ ⨆ g_1, ⨆ (_ : Measurable g_1), ⨆ (_ : g_1 ≤ f * g), ∫⁻ (a : α), g_1 a ∂μ"
] | rw [lintegral_withDensity_eq_lintegral_mul _ f_meas i_meas] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 255,
"column": 32
} | {
"line": 255,
"column": 80
} | {
"line": 257,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhfi : HasFiniteIntegral f μ\n⊢ HasFiniteIntegral (-f) μ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroCl... | [] | by simpa [hasFiniteIntegral_iff_enorm] using hfi | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 263,
"column": 36
} | {
"line": 263,
"column": 84
} | {
"line": 265,
"column": 0
} | [
{
"pp": "α : Type u_1\nε : Type u_4\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : ENorm ε\nf : α → ε\nhfi : HasFiniteIntegral f μ\n⊢ HasFiniteIntegral (fun x ↦ ‖f x‖ₑ) μ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegra... | [] | by simpa [hasFiniteIntegral_iff_enorm] using hfi | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 267,
"column": 44
} | {
"line": 267,
"column": 92
} | {
"line": 269,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup β\nf : α → β\nhfi : HasFiniteIntegral f μ\n⊢ HasFiniteIntegral (fun a ↦ ‖f a‖) μ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.to... | [] | by simpa [hasFiniteIntegral_iff_enorm] using hfi | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 524,
"column": 6
} | {
"line": 524,
"column": 38
} | {
"line": 526,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : AEMeasurable f μ\nh'f : ∀ᵐ (x : α) ∂μ, f x < ∞\ng : α → ℝ≥0∞\nf' : α → ℝ≥0∞ := AEMeasurable.mk f hf\nx : α\nhx : f x = AEMeasurable.mk f hf x\n⊢ (f' * g) x = (f * g) x",
"ppTerm": "?m.157",
"assigned": true,
"usedConsta... | [] | simp only [f', hx, Pi.mul_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Real.Sqrt | {
"line": 306,
"column": 59
} | {
"line": 308,
"column": 24
} | {
"line": 310,
"column": 0
} | [
{
"pp": "x : ℝ\n⊢ √x ≤ x ↔ x = 0 ∨ 1 ≤ x",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"Real.instZero",
"Monoid.toMulOneClass",
"AddGroupWithOne.toAddGroup",
... | [] | by
rw [sqrt_le_iff, ← sub_nonneg (a := x ^ 2), sq, ← mul_sub_one]
grind [mul_nonneg_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Real.Sqrt | {
"line": 419,
"column": 6
} | {
"line": 419,
"column": 23
} | {
"line": 419,
"column": 23
} | [
{
"pp": "a : ℕ\n⊢ √↑a < ↑a.sqrt + 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"congrArg",
"PartialOrder.toPreorder",
"Real.instLT... | [
"a : ℕ\n⊢ ↑a < (↑a.sqrt + 1) ^ 2",
"a : ℕ\n⊢ 0 ≤ ↑a.sqrt + 1"
] | sqrt_lt (by simp) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Complex.Basic | {
"line": 462,
"column": 18
} | {
"line": 462,
"column": 44
} | {
"line": 464,
"column": 0
} | [
{
"pp": "a b : ℂ\n⊢ { re := (a + b).re, im := -(a + b).im } = { re := a.re, im := -a.im } + { re := b.re, im := -b.im }",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"neg_add_rev",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"Complex... | [] | by ext <;> simp [add_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Norm | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 69
} | {
"line": 162,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ ‖z‖ ^ 2 - z.re ^ 2 = z.im ^ 2",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"add_sub_cancel_left",
"Complex.im",
"Real.instSub",
"... | [] | rw [Complex.sq_norm, normSq_apply, ← sq, ← sq, add_sub_cancel_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Norm | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 69
} | {
"line": 162,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ ‖z‖ ^ 2 - z.re ^ 2 = z.im ^ 2",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"add_sub_cancel_left",
"Complex.im",
"Real.instSub",
"... | [] | rw [Complex.sq_norm, normSq_apply, ← sq, ← sq, add_sub_cancel_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Norm | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 69
} | {
"line": 162,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ ‖z‖ ^ 2 - z.re ^ 2 = z.im ^ 2",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"add_sub_cancel_left",
"Complex.im",
"Real.instSub",
"... | [] | rw [Complex.sq_norm, normSq_apply, ← sq, ← sq, add_sub_cancel_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Norm | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 42
} | {
"line": 166,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ ‖z‖ ^ 2 - z.im ^ 2 = z.re ^ 2",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"Complex.im",
"Real.instSub",
"Complex.sq_norm_sub_sq_re",
"HSub.hSub",
"AddCommGroup.toAddGroup"... | [] | rw [← sq_norm_sub_sq_re, sub_sub_cancel] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Norm | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 42
} | {
"line": 166,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ ‖z‖ ^ 2 - z.im ^ 2 = z.re ^ 2",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"Complex.im",
"Real.instSub",
"Complex.sq_norm_sub_sq_re",
"HSub.hSub",
"AddCommGroup.toAddGroup"... | [] | rw [← sq_norm_sub_sq_re, sub_sub_cancel] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Norm | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 42
} | {
"line": 166,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ ‖z‖ ^ 2 - z.im ^ 2 = z.re ^ 2",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"congrArg",
"Complex.im",
"Real.instSub",
"Complex.sq_norm_sub_sq_re",
"HSub.hSub",
"AddCommGroup.toAddGroup"... | [] | rw [← sq_norm_sub_sq_re, sub_sub_cancel] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Complex.Basic | {
"line": 802,
"column": 42
} | {
"line": 802,
"column": 78
} | {
"line": 802,
"column": 78
} | [
{
"pp": "s s₁ t t₁ : Set ℝ\n⊢ ⇑equivRealProd ⁻¹' s ×ˢ t ⊆ s₁ ×ℂ t₁ ↔ s ×ˢ t ⊆ s₁ ×ˢ t₁",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"Real",
"Equiv.instEquivLike",
"SProd.sprod",
"congrArg",
"Complex.preimage_equivRealP... | [
"s s₁ t t₁ : Set ℝ\n⊢ ⇑equivRealProd ⁻¹' s ×ˢ t ⊆ ⇑equivRealProd ⁻¹' s₁ ×ˢ t₁ ↔ s ×ˢ t ⊆ s₁ ×ˢ t₁"
] | ← @preimage_equivRealProd_prod s₁ t₁ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 766,
"column": 2
} | {
"line": 767,
"column": 48
} | {
"line": 769,
"column": 0
} | [
{
"pp": "M : Type u_2\ninst✝² : Monoid M\ninst✝¹ : MeasurableSpace M\nμ ν : Measure M\ninst✝ : SFinite ν\ns : ℝ≥0∞\n⊢ μ ∗ₘ (s • ν) = s • μ ∗ₘ ν",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul",
"HM... | [] | unfold mconv
rw [Measure.prod_smul_right, Measure.map_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 766,
"column": 2
} | {
"line": 767,
"column": 48
} | {
"line": 769,
"column": 0
} | [
{
"pp": "M : Type u_2\ninst✝² : Monoid M\ninst✝¹ : MeasurableSpace M\nμ ν : Measure M\ninst✝ : SFinite ν\ns : ℝ≥0∞\n⊢ μ ∗ₘ (s • ν) = s • μ ∗ₘ ν",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul",
"HM... | [] | unfold mconv
rw [Measure.prod_smul_right, Measure.map_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Hom | {
"line": 371,
"column": 9
} | {
"line": 371,
"column": 18
} | {
"line": 371,
"column": 19
} | [
{
"pp": "V : Type u_1\ninst✝¹ : SeminormedAddCommGroup V\ninst✝ : NontrivialTopology V\nx : V\nhx : ‖x‖ ≠ 0\nthis : ‖(id V) x‖ / ‖x‖ ≤ ‖id V‖\n⊢ 1 ≤ ‖id V‖",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"NormedAddGroupHom",
... | [
"V : Type u_1\ninst✝¹ : SeminormedAddCommGroup V\ninst✝ : NontrivialTopology V\nx : V\nhx : ‖x‖ ≠ 0\nthis : ‖x‖ / ‖x‖ ≤ ‖id V‖\n⊢ 1 ≤ ‖id V‖"
] | id_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 568,
"column": 6
} | {
"line": 570,
"column": 10
} | {
"line": 572,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : NonUnitalNonAssocRing A\ninst✝⁴ : StarRing A\ninst✝³ : Module ℂ A\ninst✝² : IsScalarTower ℂ A A\ninst✝¹ : SMulCommClass ℂ A A\ninst✝ : StarModule ℂ A\na : A\na_eq : a = ↑(ℜ a) + I • ↑(ℑ a)\n⊢ star (↑(ℜ a) + I • ↑(ℑ a)) * (↑(ℜ a) + I • ↑(ℑ a)) + (↑(ℜ a) + I • ↑(ℑ a)) * star (↑(ℜ a... | [] | simp [mul_add, add_mul, smul_smul, mul_smul_comm,
smul_mul_assoc]
abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 568,
"column": 6
} | {
"line": 570,
"column": 10
} | {
"line": 572,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝⁵ : NonUnitalNonAssocRing A\ninst✝⁴ : StarRing A\ninst✝³ : Module ℂ A\ninst✝² : IsScalarTower ℂ A A\ninst✝¹ : SMulCommClass ℂ A A\ninst✝ : StarModule ℂ A\na : A\na_eq : a = ↑(ℜ a) + I • ↑(ℑ a)\n⊢ star (↑(ℜ a) + I • ↑(ℑ a)) * (↑(ℜ a) + I • ↑(ℑ a)) + (↑(ℜ a) + I • ↑(ℑ a)) * star (↑(ℜ a... | [] | simp [mul_add, add_mul, smul_smul, mul_smul_comm,
smul_mul_assoc]
abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 333,
"column": 31
} | {
"line": 333,
"column": 50
} | {
"line": 333,
"column": 51
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ ↑(re z) + ↑(im z) * I - (↑(re z) - ↑(im z) * I) = 2 * ↑(im z) * I",
"ppTerm": "?m.89",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"Real.instAddMonoid",
"AddGroup... | [
"K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ ↑(im z) * I + ↑(im z) * I = 2 * ↑(im z) * I"
] | add_sub_sub_cancel, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 742,
"column": 72
} | {
"line": 744,
"column": 53
} | {
"line": 746,
"column": 0
} | [
{
"pp": "α : Type u_1\nF : Type u_4\nE' : Type u_6\ninst✝¹ : Norm F\ninst✝ : SeminormedAddCommGroup E'\ng : α → F\nf' : α → E'\nl : Filter α\n⊢ (fun x ↦ ‖f' x‖) =o[l] g ↔ f' =o[l] g",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"Asym... | [] | by
simp only [IsLittleO_def]
exact forall₂_congr fun _ _ => isBigOWith_norm_left | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 885,
"column": 65
} | {
"line": 888,
"column": 86
} | {
"line": 890,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝¹ : Norm E\ninst✝ : Norm F\nl : Filter α\nf : α × β → E\ng : α × β → F\nl' : Filter β\n⊢ f =O[l ×ˢ l'] g → ∀ᶠ (a : α) in l, (fun x ↦ f (a, x)) =O[l'] fun x ↦ g (a, x)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
... | [] | by
simp only [isBigO_iff, eventually_iff, mem_prod_iff]
rintro ⟨c, t₁, ht₁, t₂, ht₂, ht⟩
exact mem_of_superset ht₁ fun _ ha ↦ ⟨c, mem_of_superset ht₂ fun _ hb ↦ ht ⟨ha, hb⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 32,
"column": 2
} | {
"line": 32,
"column": 16
} | {
"line": 33,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\na : ℕ → α\nn : ℕ\nhm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m\nhg : IsCauSeq abs fun n ↦ ∑ i ∈ range n, a i\nε : α\nε0 : ε > 0\ni : ℕ\n... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : ℕ → β\na : ℕ → α\nn : ℕ\nhm : ∀ (m : ℕ), n ≤ m → abv (f m) ≤ a m\nhg : IsCauSeq abs fun n ↦ ∑ i ∈ range n, a i\nε : α\nε0 : ε > 0\ni : ℕ\nhi : ∀ j ≥ i... | exists max n i | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticExists_,,_1» | Lean.Parser.Tactic.«tacticExists_,,» |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1223,
"column": 44
} | {
"line": 1224,
"column": 67
} | {
"line": 1226,
"column": 0
} | [
{
"pp": "α : Type u_1\nS : Type u_17\ninst✝¹ : NormedRing S\ninst✝ : NormMulClass S\nc : S\nhc : c ≠ 0\nf : α → S\nl : Filter α\n⊢ IsBigOWith ‖c‖⁻¹ l f fun x ↦ c * f x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
... | [] | by
simp [IsBigOWith, inv_mul_cancel_left₀ (norm_ne_zero_iff.mpr hc)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 288,
"column": 73
} | {
"line": 293,
"column": 34
} | {
"line": 295,
"column": 0
} | [
{
"pp": "α : Type u_1\nE' : Type u_6\nF' : Type u_7\nR : Type u_13\n𝕜' : Type u_16\ninst✝⁷ : SeminormedAddCommGroup E'\ninst✝⁶ : SeminormedAddCommGroup F'\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : NormedDivisionRing 𝕜'\nc c' : ℝ\nf' : α → E'\ng' : α → F'\nl : Filter α\ninst✝³ : Module R E'\ninst✝² : IsBoundedSMul ... | [] | by
simp only [IsBigOWith_def] at *
filter_upwards [h₁, h₂] with _ hx₁ hx₂
apply le_trans (norm_smul_le _ _)
convert! mul_le_mul hx₁ hx₂ (norm_nonneg _) (le_trans (norm_nonneg _) hx₁) using 1
rw [norm_smul, mul_mul_mul_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 350,
"column": 4
} | {
"line": 354,
"column": 55
} | {
"line": 356,
"column": 0
} | [
{
"pp": "case cons\nα : Type u_1\nR : Type u_13\n𝕜 : Type u_15\ninst✝¹ : SeminormedRing R\ninst✝ : NormedDivisionRing 𝕜\nl : Filter α\nι : Type u_17\nf : ι → α → R\ng : ι → α → 𝕜\ni : ι\nL : List ι\nihL :\n (∀ i ∈ L, f i =O[l] g i) →\n (∃ i ∈ L, f i =o[l] g i) →\n (fun x ↦ (List.map (fun x_1 ↦ f x_1... | [] | simp only [List.map_cons, List.prod_cons, List.forall_mem_cons, List.exists_mem_cons_iff]
at h₁ h₂ ⊢
cases h₂ with
| inl hi => exact hi.mul_isBigO <| .listProd h₁.2
| inr hL => exact h₁.1.mul_isLittleO <| ihL h₁.2 hL | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 350,
"column": 4
} | {
"line": 354,
"column": 55
} | {
"line": 356,
"column": 0
} | [
{
"pp": "case cons\nα : Type u_1\nR : Type u_13\n𝕜 : Type u_15\ninst✝¹ : SeminormedRing R\ninst✝ : NormedDivisionRing 𝕜\nl : Filter α\nι : Type u_17\nf : ι → α → R\ng : ι → α → 𝕜\ni : ι\nL : List ι\nihL :\n (∀ i ∈ L, f i =O[l] g i) →\n (∃ i ∈ L, f i =o[l] g i) →\n (fun x ↦ (List.map (fun x_1 ↦ f x_1... | [] | simp only [List.map_cons, List.prod_cons, List.forall_mem_cons, List.exists_mem_cons_iff]
at h₁ h₂ ⊢
cases h₂ with
| inl hi => exact hi.mul_isBigO <| .listProd h₁.2
| inr hL => exact h₁.1.mul_isLittleO <| ihL h₁.2 hL | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 51
} | {
"line": 86,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : ℕ → β\nha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n)\nhb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n\nε : α\nε0 : 0 < ε\nP : α\nhP : ... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : ℕ → β\nha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n)\nhb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n\nε : α\nε0 : 0 < ε\nP : α\nhP : ∀ (i : ℕ), |... | have two_mul_two : (4 : α) = 2 * 2 := by norm_num | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.Exponential | {
"line": 43,
"column": 78
} | {
"line": 43,
"column": 91
} | {
"line": 44,
"column": 8
} | [
{
"pp": "z : ℂ\nn : ℕ\nhn : ‖z‖ < ↑n\nhn0 : 0 < ↑n\nm : ℕ\nhm : n ≤ m\n⊢ ‖z * z ^ m / ↑(m.factorial * m.succ)‖ ≤ ‖z‖ / ↑n * ‖z ^ m / ↑m.factorial‖",
"ppTerm": "?m.128",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NonAssocSemiring.... | [
"z : ℂ\nn : ℕ\nhn : ‖z‖ < ↑n\nhn0 : 0 < ↑n\nm : ℕ\nhm : n ≤ m\n⊢ ‖z * z ^ m / (↑m.factorial * ↑m.succ)‖ ≤ ‖z‖ / ↑n * ‖z ^ m / ↑m.factorial‖"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 44,
"column": 54
} | {
"line": 44,
"column": 71
} | {
"line": 44,
"column": 72
} | [
{
"pp": "z : ℂ\nn : ℕ\nhn : ‖z‖ < ↑n\nhn0 : 0 < ↑n\nm : ℕ\nhm : n ≤ m\n⊢ ‖z / ↑m.succ * (z ^ m / ↑m.factorial)‖ ≤ ‖z‖ / ↑n * ‖z ^ m / ↑m.factorial‖",
"ppTerm": "?m.151",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NonAssocSemiring... | [
"z : ℂ\nn : ℕ\nhn : ‖z‖ < ↑n\nhn0 : 0 < ↑n\nm : ℕ\nhm : n ≤ m\n⊢ ‖z / ↑m.succ‖ * ‖z ^ m / ↑m.factorial‖ ≤ ‖z‖ / ↑n * ‖z ^ m / ↑m.factorial‖"
] | Complex.norm_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 120,
"column": 14
} | {
"line": 120,
"column": 27
} | {
"line": 120,
"column": 28
} | [
{
"pp": "x y : ℂ\nj m : ℕ\nx✝ : m ∈ range j\nI : ℕ\nhi : I ∈ range (m + 1)\nh₁ : ↑(m.choose I) ≠ 0\nh₂ : m.choose I * I.factorial * (m - I).factorial = m.factorial\n⊢ x ^ I * y ^ (m - I) * ↑(m.choose I) * (↑(m.choose I * I.factorial * (m - I).factorial))⁻¹ =\n x ^ I / ↑I.factorial * (y ^ (m - I) / ↑(m - I).f... | [
"x y : ℂ\nj m : ℕ\nx✝ : m ∈ range j\nI : ℕ\nhi : I ∈ range (m + 1)\nh₁ : ↑(m.choose I) ≠ 0\nh₂ : m.choose I * I.factorial * (m - I).factorial = m.factorial\n⊢ x ^ I * y ^ (m - I) * ↑(m.choose I) * (↑(m.choose I * I.factorial) * ↑(m - I).factorial)⁻¹ =\n x ^ I / ↑I.factorial * (y ^ (m - I) / ↑(m - I).factorial)"
... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 120,
"column": 28
} | {
"line": 120,
"column": 41
} | {
"line": 120,
"column": 42
} | [
{
"pp": "x y : ℂ\nj m : ℕ\nx✝ : m ∈ range j\nI : ℕ\nhi : I ∈ range (m + 1)\nh₁ : ↑(m.choose I) ≠ 0\nh₂ : m.choose I * I.factorial * (m - I).factorial = m.factorial\n⊢ x ^ I * y ^ (m - I) * ↑(m.choose I) * (↑(m.choose I * I.factorial) * ↑(m - I).factorial)⁻¹ =\n x ^ I / ↑I.factorial * (y ^ (m - I) / ↑(m - I).... | [
"x y : ℂ\nj m : ℕ\nx✝ : m ∈ range j\nI : ℕ\nhi : I ∈ range (m + 1)\nh₁ : ↑(m.choose I) ≠ 0\nh₂ : m.choose I * I.factorial * (m - I).factorial = m.factorial\n⊢ x ^ I * y ^ (m - I) * ↑(m.choose I) * (↑(m.choose I) * ↑I.factorial * ↑(m - I).factorial)⁻¹ =\n x ^ I / ↑I.factorial * (y ^ (m - I) / ↑(m - I).factorial)"... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 235,
"column": 70
} | {
"line": 235,
"column": 89
} | {
"line": 235,
"column": 90
} | [
{
"pp": "x : ℂ\n⊢ cexp x + cexp (-x) - (cexp x - cexp (-x)) = 2 * cexp (-x)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"congrArg",
"AddMonoid.toAddZeroClass",
"NonUnitalNonAssoc... | [
"x : ℂ\n⊢ cexp (-x) + cexp (-x) = 2 * cexp (-x)"
] | add_sub_sub_cancel, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 156,
"column": 9
} | {
"line": 156,
"column": 60
} | {
"line": 157,
"column": 2
} | [
{
"pp": "x : ℂ\n⊢ cexp (↑0 * x) = cexp x ^ 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"MulZeroClass.toMul",
"Monoid.toMulOneClass",
"congrArg",
"Complex.exp_zero",
"MulZeroClass.zero_mul",
"... | [] | by rw [Nat.cast_zero, zero_mul, exp_zero, pow_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 339,
"column": 84
} | {
"line": 340,
"column": 60
} | {
"line": 342,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ cos z = cos ↑z.re * cosh ↑z.im - sin ↑z.re * sinh ↑z.im * I",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"Complex.cos",
"HEq.refl",
"Complex.im",
"Complex.sin",
"HSub.hSub",
"C... | [] | by
convert! cos_add_mul_I z.re z.im; exact (re_add_im z).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Exponential | {
"line": 395,
"column": 15
} | {
"line": 395,
"column": 32
} | {
"line": 395,
"column": 33
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\nn : ℕ\nhn : 0 < n\nj : ℕ\nhj : j ≥ n\n⊢ ∑ m ∈ range j with n ≤ m, ‖x ^ n * (x ^ (m - n) / ↑m.factorial)‖ ≤\n ∑ m ∈ range j with n ≤ m, ‖x‖ ^ n * (1 / ↑m.factorial)",
"ppTerm": "?m.318",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Re... | [
"x : ℂ\nhx : ‖x‖ ≤ 1\nn : ℕ\nhn : 0 < n\nj : ℕ\nhj : j ≥ n\n⊢ ∑ x_1 ∈ range j with n ≤ x_1, ‖x ^ n‖ * ‖x ^ (x_1 - n) / ↑x_1.factorial‖ ≤\n ∑ m ∈ range j with n ≤ m, ‖x‖ ^ n * (1 / ↑m.factorial)"
] | Complex.norm_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 914,
"column": 2
} | {
"line": 915,
"column": 85
} | {
"line": 916,
"column": 0
} | [
{
"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 ⊢
simpa using! IsLittleO.smul_isBigO hε (hf.isBigO_const (one_ne_zero : (1 : 𝕜) ≠ 0)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 914,
"column": 2
} | {
"line": 915,
"column": 85
} | {
"line": 916,
"column": 0
} | [
{
"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 ⊢
simpa using! IsLittleO.smul_isBigO hε (hf.isBigO_const (one_ne_zero : (1 : 𝕜) ≠ 0)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Exponential | {
"line": 498,
"column": 15
} | {
"line": 498,
"column": 32
} | {
"line": 498,
"column": 33
} | [
{
"pp": "x : ℂ\nn j : ℕ\nhj : j ≥ n\n⊢ ∑ m ∈ Ico n j, ‖x ^ n * (x ^ (m - n) / ↑m.factorial)‖ ≤ ∑ m ∈ Ico n j, ‖x‖ ^ n * (‖x‖ ^ (m - n) / ↑(m - n).factorial)",
"ppTerm": "?m.284",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDi... | [
"x : ℂ\nn j : ℕ\nhj : j ≥ n\n⊢ ∑ x_1 ∈ Ico n j, ‖x ^ n‖ * ‖x ^ (x_1 - n) / ↑x_1.factorial‖ ≤\n ∑ m ∈ Ico n j, ‖x‖ ^ n * (‖x‖ ^ (m - n) / ↑(m - n).factorial)"
] | Complex.norm_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Field.Power | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 71
} | {
"line": 51,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na : α\nha : a < 0\nk : ℤ\n⊢ a ^ (2 * k) * a < 0",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"GroupWithZero.toDivInvM... | [] | exact mul_neg_of_pos_of_neg (Even.zpow_pos (even_two_mul _) ha.ne) ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Field.Power | {
"line": 80,
"column": 4
} | {
"line": 81,
"column": 47
} | {
"line": 82,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b : α\nn : ℤ\nm : ℕ\nhn : -[m+1] ≠ 0\n⊢ a ^ -[m+1] = b ^ -[m+1] ↔ a = b ∨ a = -b ∧ Even -[m+1]",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"zpow_natCast... | [
"α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b : α\nn : ℤ\nm : ℕ\nhn : ¬m.succ = 0\n⊢ a ^ m.succ = b ^ m.succ ↔ a = b ∨ a = -b ∧ Even m.succ"
] | simp only [← neg_ofNat_succ, ne_eq, neg_eq_zero, Nat.cast_eq_zero, zpow_neg, zpow_natCast,
inv_inj, even_neg, Int.even_coe_nat] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 474,
"column": 2
} | {
"line": 475,
"column": 12
} | {
"line": 477,
"column": 0
} | [
{
"pp": "x : ℂ\n⊢ cos (2 * x) = 2 * cos x ^ 2 - 1",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"sub_add",
"Iff.mpr",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"Complex.cos",
"AddGroupWithOne.toAddGroup",
"congrArg",... | [] | rw [cos_two_mul', eq_sub_iff_add_eq.2 (sin_sq_add_cos_sq x), ← sub_add, sub_add_eq_add_sub,
two_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 474,
"column": 2
} | {
"line": 475,
"column": 12
} | {
"line": 477,
"column": 0
} | [
{
"pp": "x : ℂ\n⊢ cos (2 * x) = 2 * cos x ^ 2 - 1",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"sub_add",
"Iff.mpr",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"Complex.cos",
"AddGroupWithOne.toAddGroup",
"congrArg",... | [] | rw [cos_two_mul', eq_sub_iff_add_eq.2 (sin_sq_add_cos_sq x), ← sub_add, sub_add_eq_add_sub,
two_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 474,
"column": 2
} | {
"line": 475,
"column": 12
} | {
"line": 477,
"column": 0
} | [
{
"pp": "x : ℂ\n⊢ cos (2 * x) = 2 * cos x ^ 2 - 1",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"sub_add",
"Iff.mpr",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"Complex.cos",
"AddGroupWithOne.toAddGroup",
"congrArg",... | [] | rw [cos_two_mul', eq_sub_iff_add_eq.2 (sin_sq_add_cos_sq x), ← sub_add, sub_add_eq_add_sub,
two_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 978,
"column": 78
} | {
"line": 978,
"column": 95
} | {
"line": 979,
"column": 4
} | [
{
"pp": "x : ℝ\n⊢ ‖cexp (I * ↑x) - 1‖ = ‖cexp (-↑(x / 2) * I) * (1 - cexp (↑x * I)) * I‖",
"ppTerm": "?m.154",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Semigroup.toMul",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [
"x : ℝ\n⊢ ‖cexp (I * ↑x) - 1‖ = ‖cexp (-↑(x / 2) * I) * (1 - cexp (↑x * I))‖ * ‖I‖"
] | Complex.norm_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 979,
"column": 21
} | {
"line": 979,
"column": 38
} | {
"line": 980,
"column": 4
} | [
{
"pp": "x : ℝ\n⊢ ‖cexp (I * ↑x) - 1‖ = ‖cexp (-↑(x / 2) * I) * (1 - cexp (↑x * I))‖",
"ppTerm": "?m.162",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Semigroup.toMul",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HM... | [
"x : ℝ\n⊢ ‖cexp (I * ↑x) - 1‖ = ‖cexp (-↑(x / 2) * I)‖ * ‖1 - cexp (↑x * I)‖"
] | Complex.norm_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 984,
"column": 37
} | {
"line": 984,
"column": 54
} | {
"line": 984,
"column": 55
} | [
{
"pp": "z : ℂ\n⊢ ‖cexp ↑z.re * (cos ↑z.im + sin ↑z.im * I)‖ = Real.exp z.re",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Complex.im",
"Complex.sin",
"Complex.ins... | [
"z : ℂ\n⊢ ‖cexp ↑z.re‖ * ‖cos ↑z.im + sin ↑z.im * I‖ = Real.exp z.re"
] | Complex.norm_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 289,
"column": 2
} | {
"line": 300,
"column": 19
} | {
"line": 302,
"column": 0
} | [
{
"pp": "b c : ℝ\nn : ℕ\nhb : 0 ≠ b\n⊢ Tendsto (fun x ↦ x ^ n / (b * rexp x + c)) atTop (𝓝 0)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Filter.Tendsto.neg",
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Eq.mpr",
... | [] | have H : ∀ d e, 0 < d → Tendsto (fun x : ℝ => x ^ n / (d * exp x + e)) atTop (𝓝 0) := by
intro b' c' h
convert! (tendsto_mul_exp_add_div_pow_atTop b' c' n h).inv_tendsto_atTop using 1
ext x
simp
rcases lt_or_gt_of_ne hb with h | h
· exact H b c h
· convert! (H (-b) (-c) (neg_pos.mpr h)).neg using... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 289,
"column": 2
} | {
"line": 300,
"column": 19
} | {
"line": 302,
"column": 0
} | [
{
"pp": "b c : ℝ\nn : ℕ\nhb : 0 ≠ b\n⊢ Tendsto (fun x ↦ x ^ n / (b * rexp x + c)) atTop (𝓝 0)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Filter.Tendsto.neg",
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Eq.mpr",
... | [] | have H : ∀ d e, 0 < d → Tendsto (fun x : ℝ => x ^ n / (d * exp x + e)) atTop (𝓝 0) := by
intro b' c' h
convert! (tendsto_mul_exp_add_div_pow_atTop b' c' n h).inv_tendsto_atTop using 1
ext x
simp
rcases lt_or_gt_of_ne hb with h | h
· exact H b c h
· convert! (H (-b) (-c) (neg_pos.mpr h)).neg using... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 354,
"column": 65
} | {
"line": 354,
"column": 91
} | {
"line": 355,
"column": 4
} | [
{
"pp": "a : ℝ\n⊢ Subtype.val '' Set.Iio (expOrderIso a) = Set.Ioo 0 ((Subtype.val ∘ ⇑expOrderIso) a)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Set.Ioi",
"Real.instZero",
"congrArg",
"Preorder.toLE",
"F... | [
"a : ℝ\n⊢ Set.Ioo 0 ↑(expOrderIso a) = Set.Ioo 0 ((Subtype.val ∘ ⇑expOrderIso) a)"
] | image_subtype_val_Ioi_Iio, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 417,
"column": 2
} | {
"line": 419,
"column": 53
} | {
"line": 421,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : x ∈ closure (Ioo 0 π)\n⊢ 0 ≤ sin x",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Real",
"closure_lt_subset_le",
"Real.pi",
"Real.lattice",
"Real.instZero",
"instHasSolidNormReal",
"continuous_const",
"PseudoMetricSpace... | [] | exact
closure_lt_subset_le continuous_const continuous_sin
(closure_mono (fun y => sin_pos_of_mem_Ioo) hx) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.CompactOpen | {
"line": 97,
"column": 54
} | {
"line": 97,
"column": 65
} | {
"line": 97,
"column": 66
} | [
{
"pp": "X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : C(X, Y)\ns : Set C(X, Y)\n⊢ s ∈ ⨅ K, ⨅ j, ⨅ (_ : IsCompact K), ⨅ (_ : IsOpen[inst✝] j), ⨅ (_ : MapsTo (⇑f) K j), 𝓟 {g | MapsTo (⇑g) K j} ↔\n ∃ i,\n (i.Finite ∧ ∀ (K : Set X) (U : Set Y), (K, U) ∈ i → IsCom... | [
"X : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : C(X, Y)\ns : Set C(X, Y)\n⊢ s ∈ ⨅ x, ⨅ (_ : IsCompact x.1), ⨅ (_ : IsOpen[inst✝] x.2), ⨅ (_ : MapsTo (⇑f) x.1 x.2), 𝓟 {g | MapsTo (⇑g) x.1 x.2} ↔\n ∃ i,\n (i.Finite ∧ ∀ (K : Set X) (U : Set Y), (K, U) ∈ i → IsCompact ... | iInf_prod', | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Connected.PathConnected | {
"line": 256,
"column": 58
} | {
"line": 256,
"column": 66
} | {
"line": 256,
"column": 66
} | [
{
"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)\nγ' : ↑I → X\nhγ'F : ∀ (t : ↑I), γ' t ∈ F\nhγ' : ∀ (t : ↑I), f (γ' t) = γ t\nh₀ : x ⤳ γ' 0\n⊢ γ 1 ⤳ f y",
"ppTerm": "?m... | [
"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)\nγ' : ↑I → X\nhγ'F : ∀ (t : ↑I), γ' t ∈ F\nhγ' : ∀ (t : ↑I), f (γ' t) = γ t\nh₀ : x ⤳ γ' 0\n⊢ f y ⤳ f y"
] | γ.target | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.CompactOpen | {
"line": 554,
"column": 4
} | {
"line": 556,
"column": 7
} | {
"line": 557,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : Unique X\nf : C(X, Y)\n⊢ const X ((fun f ↦ f default) f) = f",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Inhabited.de... | [] | ext x
rw [Unique.eq_default x]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.CompactOpen | {
"line": 554,
"column": 4
} | {
"line": 556,
"column": 7
} | {
"line": 557,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : Unique X\nf : C(X, Y)\n⊢ const X ((fun f ↦ f default) f) = f",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Inhabited.de... | [] | ext x
rw [Unique.eq_default x]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Path | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 54
} | {
"line": 76,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nx y : X\n⊢ ∀ {γ₁ γ₂ : Path x y}, ⇑γ₁ = ⇑γ₂ → γ₁ = γ₂",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real",
"Continuous",
"Set.Icc.instZero",
"ContinuousMa... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nx✝ y : X\nx : ↑I → X\nh11 : Continuous[_, inst✝] x\nh12 : { toFun := x, continuous_toFun := h11 }.toFun 0 = x✝\nh13 : { toFun := x, continuous_toFun := h11 }.toFun 1 = y\nh21 : Continuous[_, inst✝] ⇑{ toFun := x, continuous_toFun := h11, source' := h12, target' := h13 }\nh... | rintro ⟨⟨x, h11⟩, h12, h13⟩ ⟨⟨x, h21⟩, h22, h23⟩ rfl | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 811,
"column": 4
} | {
"line": 811,
"column": 47
} | {
"line": 812,
"column": 4
} | [
{
"pp": "E : Type u_4\ninst✝⁴ : Ring E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedRing E\ninst✝¹ : TopologicalSpace E\ninst✝ : OrderClosedTopology E\nl : E\nf : ℕ → E\nhfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)\nhfm : Monotone f\nk : ℕ\n⊢ Monotone fun n ↦ ∑ i ∈ Finset.range (2 * ... | [
"E : Type u_4\ninst✝⁴ : Ring E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedRing E\ninst✝¹ : TopologicalSpace E\ninst✝ : OrderClosedTopology E\nl : E\nf : ℕ → E\nhfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)\nhfm : Monotone f\nk n : ℕ\n⊢ ∑ i ∈ Finset.range (2 * n + 1), (-1) ^ i * f i ≤ ∑ ... | refine monotone_nat_of_le_succ (fun n ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 826,
"column": 4
} | {
"line": 826,
"column": 47
} | {
"line": 827,
"column": 4
} | [
{
"pp": "E : Type u_4\ninst✝⁴ : Ring E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedRing E\ninst✝¹ : TopologicalSpace E\ninst✝ : OrderClosedTopology E\nl : E\nf : ℕ → E\nhfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)\nhfa : Antitone f\nk : ℕ\n⊢ Monotone fun n ↦ ∑ i ∈ Finset.range (2 * ... | [
"E : Type u_4\ninst✝⁴ : Ring E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedRing E\ninst✝¹ : TopologicalSpace E\ninst✝ : OrderClosedTopology E\nl : E\nf : ℕ → E\nhfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)\nhfa : Antitone f\nk n : ℕ\n⊢ ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i ≤ ∑ i ∈ ... | refine monotone_nat_of_le_succ (fun n ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 328,
"column": 47
} | {
"line": 328,
"column": 72
} | {
"line": 330,
"column": 0
} | [
{
"pp": "⊢ arccos (-1) = π",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"instHDiv",
"Real.pi",
"CharZero.NeZero.two",
"add_halves",
"Real.arcsin",
"FloorRing.toFloorSemiring",
"congrArg",
"Real.instD... | [] | simp [arccos, add_halves] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 328,
"column": 47
} | {
"line": 328,
"column": 72
} | {
"line": 330,
"column": 0
} | [
{
"pp": "⊢ arccos (-1) = π",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"instHDiv",
"Real.pi",
"CharZero.NeZero.two",
"add_halves",
"Real.arcsin",
"FloorRing.toFloorSemiring",
"congrArg",
"Real.instD... | [] | simp [arccos, add_halves] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 328,
"column": 47
} | {
"line": 328,
"column": 72
} | {
"line": 330,
"column": 0
} | [
{
"pp": "⊢ arccos (-1) = π",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"instHDiv",
"Real.pi",
"CharZero.NeZero.two",
"add_halves",
"Real.arcsin",
"FloorRing.toFloorSemiring",
"congrArg",
"Real.instD... | [] | simp [arccos, add_halves] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 907,
"column": 48
} | {
"line": 907,
"column": 61
} | {
"line": 907,
"column": 62
} | [
{
"pp": "x : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\nB : ‖x‖ / (↑⌊‖x‖⌋₊ + 1) < 1\nn : ℕ\nhn : n ≥ ⌊‖x‖⌋₊\n⊢ ‖x * x ^ n / ↑((n + 1) * n !)‖ = ‖x‖ / (↑n + 1) * ‖x ^ n / ↑n !‖",
"ppTerm": "?m.282",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"x : ℝ\nA : 0 < ↑⌊‖x‖⌋₊ + 1\nB : ‖x‖ / (↑⌊‖x‖⌋₊ + 1) < 1\nn : ℕ\nhn : n ≥ ⌊‖x‖⌋₊\n⊢ ‖x * x ^ n / (↑(n + 1) * ↑n !)‖ = ‖x‖ / (↑n + 1) * ‖x ^ n / ↑n !‖"
] | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 260,
"column": 2
} | {
"line": 263,
"column": 42
} | {
"line": 265,
"column": 0
} | [
{
"pp": "x : ℝ\nhx1 : 0 < x\nhx2 : x ≠ 1\n⊢ log x < x - 1",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne... | [] | have h : log x ≠ 0 := by
rwa [← log_one, log_injOn_pos.ne_iff hx1]
exact mem_Ioi.mpr zero_lt_one
linarith [add_one_lt_exp h, exp_log hx1] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 260,
"column": 2
} | {
"line": 263,
"column": 42
} | {
"line": 265,
"column": 0
} | [
{
"pp": "x : ℝ\nhx1 : 0 < x\nhx2 : x ≠ 1\n⊢ log x < x - 1",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne... | [] | have h : log x ≠ 0 := by
rwa [← log_one, log_injOn_pos.ne_iff hx1]
exact mem_Ioi.mpr zero_lt_one
linarith [add_one_lt_exp h, exp_log hx1] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 39
} | {
"line": 43,
"column": 40
} | [
{
"pp": "x : ℂ\nhx : x ≠ 0\n⊢ ↑(rexp (Real.log ‖x‖)) * (↑(x.re / ‖x‖) + ↑(x.im / ‖x‖) * I) = x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"norm_pos_iff",
"Real",
"instHDiv",
... | [
"x : ℂ\nhx : x ≠ 0\n⊢ ↑‖x‖ * (↑(x.re / ‖x‖) + ↑(x.im / ‖x‖) * I) = x"
] | Real.exp_log (norm_pos_iff.mpr hx), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 427,
"column": 2
} | {
"line": 427,
"column": 58
} | {
"line": 428,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nf : α → ℝ\nh : ∀ (a : α), 0 < f a\nH : (support fun i ↦ log (f i)) = mulSupport f\nH' : HasFiniteMulSupport f ↔ HasFiniteSupport fun a ↦ log (f a)\nh' : HasFiniteMulSupport f\n⊢ log (if h : HasFiniteMulSupport f then ∏ i ∈ Finite.toFinset h, f i else 1) =\n if h : HasFiniteSu... | [
"case neg\nα : Type u_1\nf : α → ℝ\nh : ∀ (a : α), 0 < f a\nH : (support fun i ↦ log (f i)) = mulSupport f\nH' : HasFiniteMulSupport f ↔ HasFiniteSupport fun a ↦ log (f a)\nh' : ¬HasFiniteMulSupport f\n⊢ log (if h : HasFiniteMulSupport f then ∏ i ∈ Finite.toFinset h, f i else 1) =\n if h : HasFiniteSupport fun i... | · simp [h', log_prod (fun a _ ↦ (h a).ne'), H'.mp h', H] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 452,
"column": 74
} | {
"line": 456,
"column": 34
} | {
"line": 458,
"column": 0
} | [
{
"pp": "c : ℝ\n⊢ (fun x ↦ c) =o[atTop] log",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Filter.Tendsto.div_atTop",
"Real.instIsOrderedRing",
"False",
"Real.partialOrder",
"Real",
"tendsto_const_nhds_iff._simp_1",
"Preorder.toLT",
"Real.in... | [] | by
refine Asymptotics.isLittleO_of_tendsto' ?_
<| Tendsto.div_atTop (a := c) (by simp) tendsto_log_atTop
filter_upwards [eventually_gt_atTop 1] with x hx
aesop (add safe forward log_pos) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 126,
"column": 17
} | {
"line": 126,
"column": 29
} | {
"line": 126,
"column": 30
} | [
{
"pp": "⊢ ↑(-π / 2) + ↑(-π / 2) = ↑π",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"instHDiv",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"Real.instDivInvMonoid",
"AddMonoid.toAddZeroC... | [
"⊢ 2 • ↑(-π / 2) = ↑π"
] | ← two_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 138,
"column": 63
} | {
"line": 138,
"column": 75
} | {
"line": 138,
"column": 76
} | [
{
"pp": "⊢ ↑π + ↑π = 0",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"AddMonoid.toAddZeroClass",
"AddMonoid.toNSMul",
"... | [
"⊢ 2 • ↑π = 0"
] | ← two_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 164,
"column": 22
} | {
"line": 164,
"column": 34
} | {
"line": 164,
"column": 35
} | [
{
"pp": "θ : Angle\n⊢ θ + θ = 0 ↔ θ = 0 ∨ θ = ↑π",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddMonoid.toNSMul",
"AddCommGrou... | [
"θ : Angle\n⊢ 2 • θ = 0 ↔ θ = 0 ∨ θ = ↑π"
] | ← two_nsmul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 170,
"column": 32
} | {
"line": 170,
"column": 44
} | {
"line": 170,
"column": 45
} | [
{
"pp": "θ : Angle\n⊢ θ + θ = 0 ↔ θ = 0 ∨ θ = ↑π",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"instHSMul",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"AddMonoid.toAddZeroClass",
"... | [
"θ : Angle\n⊢ 2 • θ = 0 ↔ θ = 0 ∨ θ = ↑π"
] | ← two_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 190,
"column": 17
} | {
"line": 190,
"column": 29
} | {
"line": 190,
"column": 30
} | [
{
"pp": "θ : Angle\n⊢ θ + θ = ↑π ↔ θ = ↑(π / 2) ∨ θ = ↑(-π / 2)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"instHDiv",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"Real.instDivInvMonoid",
... | [
"θ : Angle\n⊢ 2 • θ = ↑π ↔ θ = ↑(π / 2) ∨ θ = ↑(-π / 2)"
] | ← two_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 315,
"column": 2
} | {
"line": 315,
"column": 61
} | {
"line": 316,
"column": 2
} | [
{
"pp": "case inl.inl\nx : ℂ\nhr : x.re < 0\nhi : x.im < 0\n⊢ (if 0 ≤ x.re then -Real.arcsin (x.im / ‖x‖)\n else if 0 ≤ -x.im then Real.arcsin (x.im / ‖x‖) + π else Real.arcsin (x.im / ‖x‖) - π) =\n if x.re < 0 ∧ x.im = 0 then π\n else\n -if 0 ≤ x.re then Real.arcsin (x.im / ‖x‖)\n else if ... | [
"case inl.inr.inl\nx : ℂ\nhr : x.re < 0\nhi : x.im = 0\n⊢ (if 0 ≤ x.re then -Real.arcsin (x.im / ‖x‖)\n else if 0 ≤ -x.im then Real.arcsin (x.im / ‖x‖) + π else Real.arcsin (x.im / ‖x‖) - π) =\n if x.re < 0 ∧ x.im = 0 then π\n else\n -if 0 ≤ x.re then Real.arcsin (x.im / ‖x‖)\n else if 0 ≤ x.im... | · simp [hr, hr.not_ge, hi.le, hi.ne, not_le.2 hi, add_comm] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 245,
"column": 32
} | {
"line": 245,
"column": 41
} | {
"line": 245,
"column": 42
} | [
{
"pp": "case inr.inr\nθ ψ : ℝ\nHcos : cos θ = cos ψ\nHsin : sin θ = sin ψ\nhc : ↑π = 0\nhs : ↑θ + ↑ψ = ↑π\nn : ℤ\nhn : ↑n * 2 - -1 = 0 ∨ False\n⊢ ↑θ = ↑ψ",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Int.cast",
"MulOne.toOne",
"False",
"Semigroup.toMul",
... | [
"case inr.inr\nθ ψ : ℝ\nHcos : cos θ = cos ψ\nHsin : sin θ = sin ψ\nhc : ↑π = 0\nhs : ↑θ + ↑ψ = ↑π\nn : ℤ\nhn : ↑n * 2 - -1 = 0\n⊢ ↑θ = ↑ψ"
] | or_false, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 40
} | {
"line": 284,
"column": 0
} | [
{
"pp": "case h\nθ : Angle\nx✝ : ℝ\n⊢ θ.cos = (↑x✝).cos ↔ θ = ↑x✝ ∨ θ = -↑x✝",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Real.Angle.cos_eq_real_cos_iff_eq_or_eq_neg"
],
"usedFVars": [
"θ",
"x✝"
],
"usedGoals": []
}
] | [] | exact cos_eq_real_cos_iff_eq_or_eq_neg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 461,
"column": 2
} | {
"line": 462,
"column": 30
} | {
"line": 464,
"column": 0
} | [
{
"pp": "θ : Angle\n⊢ -π < θ.toReal",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Real",
"Real.instArchimedean",
"Real.pi",
"HMul.hMul",
"Real.Angle",
"left_lt_toIocMod",
"Real.instLT",
"Real.two_pi_pos",
"Real.instAddCommGroup",
... | [] | induction θ using Real.Angle.induction_on
exact left_lt_toIocMod _ _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 461,
"column": 2
} | {
"line": 462,
"column": 30
} | {
"line": 464,
"column": 0
} | [
{
"pp": "θ : Angle\n⊢ -π < θ.toReal",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Real",
"Real.instArchimedean",
"Real.pi",
"HMul.hMul",
"Real.Angle",
"left_lt_toIocMod",
"Real.instLT",
"Real.two_pi_pos",
"Real.instAddCommGroup",
... | [] | induction θ using Real.Angle.induction_on
exact left_lt_toIocMod _ _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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