module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic | {
"line": 1139,
"column": 8
} | {
"line": 1139,
"column": 41
} | {
"line": 1139,
"column": 41
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), μ (support ⇑(fs n)) < ∞\nh_approx : ∀ (x : α), Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → β\ninst✝² : Zero β\ninst✝¹ : TopologicalSpace β\ninst✝ : T2Space β\nfs : ℕ → α →ₛ β\nhT_lt_top : ∀ (n : ℕ), μ (support ⇑(fs n)) < ∞\nh_approx : ∀ (x : α), Tendsto (fun n ↦ (fs n) x) atTop (𝓝 (f x))\nT : ℕ → Set α := fun n ↦ ... | funext fun n => h_fs_zero n x hxt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence | {
"line": 146,
"column": 37
} | {
"line": 146,
"column": 39
} | {
"line": 147,
"column": 4
} | [
{
"pp": "α : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ\nhF_meas : AEMeasurable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a\nh_bound : ∀ᵐ ... | [
"α : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ\nhF_meas : AEMeasurable F μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a\nh_bound : ∀ᵐ (a : α) ∂μ, ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Normed.Group.FunctionSeries | {
"line": 95,
"column": 34
} | {
"line": 95,
"column": 87
} | {
"line": 97,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\nhfu : ∀ (n : α) (x : β), ‖f n x‖ ≤ u n\n⊢ TendstoUniformlyOn (fun t x ↦ ∑ n ∈ t, f n x) (fun x ↦ ∑' (n : α), f n x) atTop univ",
"ppTerm": "?m.40",
"assig... | [] | exact tendstoUniformlyOn_tsum hu fun n x _ => hfu n x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence | {
"line": 213,
"column": 36
} | {
"line": 213,
"column": 38
} | {
"line": 213,
"column": 39
} | [
{
"pp": "α : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤... | [
"α : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a\nh0 :... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence | {
"line": 225,
"column": 37
} | {
"line": 225,
"column": 39
} | {
"line": 226,
"column": 4
} | [
{
"pp": "α : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤... | [
"α : Type u_2\nmα : MeasurableSpace α\nf : ℕ → α → ℝ≥0∞\nF : α → ℝ≥0∞\nμ : Measure α\nhf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ\nhf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))\nhf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a\nh_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a\nh0 :... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.MetricSpace.Polish | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 72
} | {
"line": 109,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝ : Countable ι\nt : ι → TopologicalSpace α\nht₀ : ∃ t₀, T2Space α ∧ ∀ (i : ι), t i ≤ t₀\nu : ι → UniformSpace α\nhcomp : ∀ (i : ι), CompleteSpace α\nhcount : ∀ (i : ι), (𝓤 α).IsCountablyGenerated\nhut : ∀ (i : ι), (u i).toTopologicalSpace = t i\n⊢ ∃ u, CompleteSpace α ... | [
"α : Type u_1\nι : Type u_3\ninst✝ : Countable ι\nu : ι → UniformSpace α\nhcomp : ∀ (i : ι), CompleteSpace α\nhcount : ∀ (i : ι), (𝓤 α).IsCountablyGenerated\nht₀ : ∃ t₀, T2Space α ∧ ∀ (i : ι), (fun i ↦ (u i).toTopologicalSpace) i ≤ t₀\nhut : ∀ (i : ι), (u i).toTopologicalSpace = (fun i ↦ (u i).toTopologicalSpace) ... | obtain rfl : t = fun i ↦ (u i).toTopologicalSpace := (funext hut).symm | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 311,
"column": 4
} | {
"line": 311,
"column": 50
} | {
"line": 313,
"column": 0
} | [
{
"pp": "case inr.mpr\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhne : y ≠ x\nh : n ≤ firstDiff y x\ni : ℕ\nhi : i < n\n⊢ y i = x i",
"ppTerm": "?inr.mpr",
"assigned": true,
"usedConstants": [
"PiNat.firstDiff",
"Nat.instPreorder",
"LT.lt.trans_le",
"PiNat.apply_eq_of_lt_... | [] | exact apply_eq_of_lt_firstDiff (hi.trans_le h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Dynamics.Ergodic.MeasurePreserving | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 18
} | {
"line": 65,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμa : Measure α\nμb : Measure β\nf f' : α → β\nhf : MeasurePreserving f μa μb\nhf' : Measurable f'\nh : f =ᵐ[μa] f'\n⊢ MeasurePreserving f' μa μb",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμa : Measure α\nμb : Measure β\nf f' : α → β\nhf : MeasurePreserving f μa μb\nhf' : Measurable f'\nh : f =ᵐ[μa] f'\n⊢ map f' μa = μb"
] | refine ⟨hf', ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 304,
"column": 4
} | {
"line": 305,
"column": 40
} | {
"line": 306,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nx y : α\nh : ∀ (A : Set α), MeasurableSet A → (x ∈ A ↔ y ∈ A)\nA : Set α\nA_mble : MeasurableSet A\nx_in_A : x ∈ A\n⊢ (dirac x) A = (dirac y) A",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"... | [
"case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nx y : α\nh : ∀ (A : Set α), MeasurableSet A → (x ∈ A ↔ y ∈ A)\nA : Set α\nA_mble : MeasurableSet A\nx_in_A : x ∉ A\n⊢ (dirac x) A = (dirac y) A"
] | · simp only [Measure.dirac_apply' _ A_mble, x_in_A, indicator_of_mem, Pi.one_apply,
(h A A_mble).mp x_in_A] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.GiryMonad | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 41
} | {
"line": 64,
"column": 2
} | [
{
"pp": "α✝ : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α✝\nmβ : MeasurableSpace β\nα : Type u_3\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\n⊢ Measurable fun b ↦ (b.1 + b.2) s",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"MeasureTheory.Meas... | [
"α✝ : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α✝\nmβ : MeasurableSpace β\nα : Type u_3\nm : MeasurableSpace α\ns : Set α\nhs : MeasurableSet s\n⊢ Measurable fun b ↦ b.1 s + b.2 s"
] | simp_rw [Measure.coe_add, Pi.add_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Measure.GiryMonad | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 31
} | {
"line": 318,
"column": 0
} | [
{
"pp": "case hg\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nm : Measure α\nf : α → β\nhf : Measurable f\n⊢ Measurable dirac",
"ppTerm": "?hg",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.measurable_dirac"
],
"usedFVars": [
"β",
... | [] | exacts [measurable_dirac, hf] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 881,
"column": 6
} | {
"line": 881,
"column": 87
} | {
"line": 882,
"column": 6
} | [
{
"pp": "case right\nE : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\ni : ι\nε : ℝ≥0∞\nhε₀ : 0 < ε\nthis : 0 < 2⁻¹ ^ encode i\n⊢ {a | edist (a.1 i) (a.2 i) < ε} ∈ 𝓟 {p | edist p.1 p.2 < min (2⁻¹ ^ encode i) ε}",
"ppTerm": "?right",
"ass... | [
"case right\nE : ℕ → Type u_1\nι : Type u_2\ninst✝¹ : Encodable ι\nF : ι → Type u_3\ninst✝ : (i : ι) → PseudoEMetricSpace (F i)\ni : ι\nε : ℝ≥0∞\nhε₀ : 0 < ε\nthis : 0 < 2⁻¹ ^ encode i\n⊢ ∀ (a b : (i : ι) → F i), edist a b < 2⁻¹ ^ encode i → edist a b < ε → edist (a i) (b i) < ε"
] | simp only [and_imp, Prod.forall, ofPred_subset_ofPred, lt_min_iff, mem_principal] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 516,
"column": 61
} | {
"line": 523,
"column": 74
} | {
"line": 525,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : MeasurableSpace γ\nμ μ' : Measure α\nν✝ ν' : Measure β\nτ : Measure γ\ninst✝ : SFinite ν✝\nν : Measure β\nC : Set (Set α)\nD : Set (Set β)\nhμ : μ.FiniteSpanningSetsIn C\nhν : ν.FiniteSpanningSets... | [] | by
haveI := hν.sigmaFinite
refine
⟨fun n => hμ.set n.unpair.1 ×ˢ hν.set n.unpair.2, fun n =>
mem_image2_of_mem (hμ.set_mem _) (hν.set_mem _), fun n => ?_, ?_⟩
· rw [prod_prod]
exact mul_lt_top (hμ.finite _) (hν.finite _)
· simp_rw [iUnion_unpair_prod, hμ.spanning, hν.spanning, univ_prod_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 580,
"column": 2
} | {
"line": 580,
"column": 30
} | {
"line": 589,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nν : Measure β\nC : Set (Set α)\nD : Set (Set β)\nhC : generateFrom C = inst✝¹\nhD : generateFrom D = inst✝\nh2C : IsPiSystem C\nh2D : IsPiSystem D\nh3C : μ.FiniteSpanningSetsIn C\nh3D : ν.FiniteSpanningSet... | [] | rw [h₁ s hs t ht, prod_prod] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Complement | {
"line": 259,
"column": 6
} | {
"line": 259,
"column": 47
} | {
"line": 259,
"column": 47
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nf : G ⧸ H → G\nhf : ∀ (q : G ⧸ H), ↑(f q) = q\n⊢ IsComplement (range f) ↑H",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"QuotientGroup.mk",
"Set.Elem",
"Set.domRestrict",
"i... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nf : G ⧸ H → G\nhf : ∀ (q : G ⧸ H), ↑(f q) = q\n⊢ Bijective ((range f).domRestrict QuotientGroup.mk)"
] | isComplement_subgroup_right_iff_bijective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 27
} | {
"line": 269,
"column": 2
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝ : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure β\npa qa : Set α → Prop\npb qb : Set β → Prop\nH : μ.InnerRegularWRT pb qb\nf : α → β\nhf : MeasurableEmbedding f\nhAB : ∀ (U : Set α), qa U → qb (f '' U)\nhAB' : ∀ K ⊆ range f, pb K → pa (f ⁻¹' K)\nU : Set α\nhU... | [
"α : Type u_2\nβ : Type u_3\ninst✝ : MeasurableSpace α\nmβ : MeasurableSpace β\nμ : Measure β\npa qa : Set α → Prop\npb qb : Set β → Prop\nH : μ.InnerRegularWRT pb qb\nf : α → β\nhf : MeasurableEmbedding f\nhAB : ∀ (U : Set α), qa U → qb (f '' U)\nhAB' : ∀ K ⊆ range f, pb K → pa (f ⁻¹' K)\nU : Set α\nhU : qa U\nr :... | rw [hf.comap_apply] at hr | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 381,
"column": 2
} | {
"line": 381,
"column": 86
} | {
"line": 382,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nA : Set α\nμ : Measure α\ninst✝ : μ.OuterRegular\nε : ℝ≥0∞\nhε : ε ≠ 0\nH : μ A = ∞\n⊢ ∃ U ⊇ A, IsOpen[inst✝¹] U ∧ μ U ≤ μ A + ε",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",... | [
"case inr\nα : Type u_1\ninst✝² : MeasurableSpace α\ninst✝¹ : TopologicalSpace α\nA : Set α\nμ : Measure α\ninst✝ : μ.OuterRegular\nε : ℝ≥0∞\nhε : ε ≠ 0\nH : μ A ≠ ∞\n⊢ ∃ U ⊇ A, IsOpen[inst✝¹] U ∧ μ U ≤ μ A + ε"
] | · exact ⟨univ, subset_univ _, isOpen_univ, by simp only [H, _root_.top_add, le_top]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.Complement | {
"line": 372,
"column": 4
} | {
"line": 374,
"column": 48
} | {
"line": 376,
"column": 0
} | [
{
"pp": "case mpr.py₂\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nT : Set G\nhHT : IsComplement (↑H) T\ng₁ g₂ : G\nh : g₂ * g₁⁻¹ ∈ H\n⊢ g₁ * (↑(hHT.equiv g₂).2)⁻¹ ∈ ↑H",
"ppTerm": "?mpr.py₂",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"DivInvMonoid.toInv"... | [] | · rw [SetLike.mem_coe, ← mul_mem_cancel_left h]
-- This used to be `simp [...]` before https://github.com/leanprover/lean4/pull/2644
rw [equiv_snd_eq_inv_mul, mul_assoc]; simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.Complement | {
"line": 582,
"column": 6
} | {
"line": 582,
"column": 63
} | {
"line": 583,
"column": 6
} | [
{
"pp": "G : Type u_1\ninst✝³ : Group G\nH K : Subgroup G\nS T✝ : Set G\nF : Type u_2\ninst✝² : Group F\ninst✝¹ : MulAction F G\ninst✝ : QuotientAction F H\nf : F\nT : H.LeftTransversal\ng : G\nt : ↑↑T\nht1 : (↑t)⁻¹ * f⁻¹ • g ∈ ↑H\nht2 : ∀ (y : ↑↑T), (fun s ↦ (↑s)⁻¹ * f⁻¹ • g ∈ ↑H) y → y = t\n⊢ ∃! s, (↑s)⁻¹ * g... | [
"case refine_1\nG : Type u_1\ninst✝³ : Group G\nH K : Subgroup G\nS T✝ : Set G\nF : Type u_2\ninst✝² : Group F\ninst✝¹ : MulAction F G\ninst✝ : QuotientAction F H\nf : F\nT : H.LeftTransversal\ng : G\nt : ↑↑T\nht1 : (↑t)⁻¹ * f⁻¹ • g ∈ ↑H\nht2 : ∀ (y : ↑↑T), (fun s ↦ (↑s)⁻¹ * f⁻¹ • g ∈ ↑H) y → y = t\n⊢ (fun s ↦ (↑s)... | refine ⟨⟨f • (t : G), Set.smul_mem_smul_set t.2⟩, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 793,
"column": 14
} | {
"line": 793,
"column": 16
} | {
"line": 794,
"column": 6
} | [
{
"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✝¹ : UpgradedIsCompletelyMetrizableSpa... | [
"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✝¹ : UpgradedIsCompletelyMetrizableSpace γ := upgr... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 888,
"column": 29
} | {
"line": 888,
"column": 31
} | {
"line": 889,
"column": 2
} | [
{
"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\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 1011,
"column": 80
} | {
"line": 1011,
"column": 82
} | {
"line": 1012,
"column": 2
} | [
{
"pp": "α : 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)\na : α\n⊢ AEMeasurable f (map (Prod.mk a) ν) → ∫⁻ (x : α × β), f x ∂map (Prod.mk a) ν = ∫⁻ (y : β... | [
"α : 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)\na : α\nha : AEMeasurable f (map (Prod.mk a) ν)\n⊢ ∫⁻ (x : α × β), f x ∂map (Prod.mk a) ν = ∫⁻ (y : β), f (a,... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 654,
"column": 4
} | {
"line": 665,
"column": 45
} | {
"line": 666,
"column": 4
} | [
{
"pp": "case iUnion.refine_1\nα : Type u_1\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nH✝ : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]\nhfin : ∀ {s : Set α}, μ s ≠ ∞\ns : ℕ → Set α\nhsd : Pairwise (Function.onFun Disjoint s)\... | [
"case iUnion.refine_2\nα : Type u_1\ninst✝³ : MeasurableSpace α\ninst✝² : TopologicalSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nH✝ : μ.InnerRegularWRT IsClosed[inst✝²] IsOpen[inst✝²]\nhfin : ∀ {s : Set α}, μ s ≠ ∞\ns : ℕ → Set α\nhsd : Pairwise (Function.onFun Disjoint s)\nhsm : ∀ (i ... | · calc
(∑ k ∈ t, μ (s k)) + ε / 2 ≤ ((∑ k ∈ t, μ (F k)) + ∑ k ∈ t, δ k) + ε / 2 := by
rw [← sum_add_distrib]
gcongr
apply hF
_ ≤ (∑ k ∈ t, μ (F k)) + ε / 2 + ε / 2 := by
gcongr
exact (ENNReal.sum_le_tsum _).trans hδε.le
_ = μ (⋃ k ∈ t, F k) + ε :... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 1094,
"column": 60
} | {
"line": 1094,
"column": 73
} | {
"line": 1096,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\n⊢ fst 0 = 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"congrArg",
"MeasureTheory.Measure.instZero",
"Prod.fst",
"True",
"eq_self",
... | [] | by simp [fst] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 1159,
"column": 6
} | {
"line": 1159,
"column": 9
} | {
"line": 1159,
"column": 9
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝¹ : SFinite ν\nρ : Measure (α × β)\ninst✝ : SFinite ρ\n⊢ SFinite ρ.snd",
"ppTerm": "?m.17",
"assigned": true,
... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝¹ : SFinite ν\nρ : Measure (α × β)\ninst✝ : SFinite ρ\n⊢ SFinite (map Prod.snd ρ)"
] | snd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 1163,
"column": 6
} | {
"line": 1163,
"column": 9
} | {
"line": 1163,
"column": 9
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝¹ : SFinite ν\nρ : Measure (α × β)\ninst✝ : IsFiniteMeasure ρ\n⊢ IsFiniteMeasure ρ.snd",
"ppTerm": "?m.17",
"assi... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝¹ : SFinite ν\nρ : Measure (α × β)\ninst✝ : IsFiniteMeasure ρ\n⊢ IsFiniteMeasure (map Prod.snd ρ)"
] | snd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 1154,
"column": 2
} | {
"line": 1154,
"column": 28
} | {
"line": 1156,
"column": 0
} | [
{
"pp": "case neg\nα✝ : Type u_1\nι : Type u_2\nγ : Type u_3\nα : Type u_4\nβ✝ : Type u_5\ninst✝⁴ : MeasurableSpace β✝\nβ : Type u_6\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\ninst✝¹ : StandardBorelSpace α\ninst✝ : StandardBorelSpace β\ne : α ≃ β\nh : ¬Countable α\n⊢ ¬Countable β",
"ppTerm": "... | [] | rwa [e.countable_iff] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Group.Measure | {
"line": 826,
"column": 6
} | {
"line": 826,
"column": 99
} | {
"line": 827,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝¹⁰ : MeasurableSpace G\ninst✝⁹ : Group G\ninst✝⁸ : TopologicalSpace G\nμ : Measure G\ninst✝⁷ : μ.IsHaarMeasure\ninst✝⁶ : BorelSpace G\ninst✝⁵ : ContinuousMul G\nH : Type u_3\ninst✝⁴ : Group H\ninst✝³ : TopologicalSpace H\ninst✝² : MeasurableSpace H\ninst✝¹ : BorelSpace H\ninst✝ : IsT... | [] | exact IsCompact.measure_lt_top (g.isCompact_preimage_of_isClosed hK.closure isClosed_closure) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 277,
"column": 34
} | {
"line": 277,
"column": 36
} | {
"line": 278,
"column": 6
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : μ ({x | f x ≠ 0} ∩ s) = 0\nt : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)\nA : s ⊆ t ∪ {x | f x = 0}\ng : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᵐ[μ] g\na : α\n⊢ f a = g a → {x | f x = 0} a = {x | g x = 0} a",
"ppTer... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhs : μ ({x | f x ≠ 0} ∩ s) = 0\nt : Set α := toMeasurable μ ({x | f x ≠ 0} ∩ s)\nA : s ⊆ t ∪ {x | f x = 0}\ng : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᵐ[μ] g\na : α\nha : f a = g a\n⊢ {x | f x = 0} a = {x | g x = 0} a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Group.Prod | {
"line": 276,
"column": 49
} | {
"line": 281,
"column": 42
} | {
"line": 283,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\ns : Set G\ninst✝⁴ : MeasurableInv G\nμ' ν' : Measure G\ninst✝³ : SigmaFinite μ'\ninst✝² : SigmaFinite ν'\ninst✝¹ : μ'.IsMulLeftInvariant\ninst✝ : ν'.IsMulLeftInvariant\nh2s : ν' s ≠ 0\nh3s : ν' s ≠ ∞\n⊢ ∀ᵐ (x : G) ∂μ... | [] | by
refine (ae_measure_preimage_mul_right_lt_top ν' ν' h3s).filter_mono ?_
refine (absolutelyContinuous_of_isMulLeftInvariant μ' ν' ?_).ae_le
refine mt ?_ h2s
intro hν
rw [hν, Measure.coe_zero, Pi.zero_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 309,
"column": 34
} | {
"line": 309,
"column": 36
} | {
"line": 310,
"column": 6
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\np : α → Prop\nf g : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᵐ[μ] g\na : α\n⊢ f a = g a → {x | g x ≠ 0} a = {x | f x ≠ 0} a",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"ENNReal",
"Eq"
],
"usedFVars": [
... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\np : α → Prop\nf g : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᵐ[μ] g\na : α\nha : f a = g a\n⊢ {x | g x ≠ 0} a = {x | f x ≠ 0} a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.Regular | {
"line": 1147,
"column": 2
} | {
"line": 1147,
"column": 81
} | {
"line": 1148,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : BorelSpace α\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : BorelSpace β\nμ : Measure β\ninst✝ : μ.Regular\nf : α → β\nhf : IsOpenEmbedding f\nthis : (comap f μ).OuterRegular\n⊢ (comap f μ).Reg... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : BorelSpace α\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace β\ninst✝¹ : BorelSpace β\nμ : Measure β\ninst✝ : μ.Regular\nf : α → β\nhf : IsOpenEmbedding f\nthis✝ : (comap f μ).OuterRegular\nthis : IsFiniteMeasureOnComp... | have := IsFiniteMeasureOnCompacts.comap' μ hf.continuous hf.measurableEmbedding | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 312,
"column": 41
} | {
"line": 312,
"column": 43
} | {
"line": 312,
"column": 43
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\np : α → Prop\nf g : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᵐ[μ] g\na : α\nha : f a = g a\nh : f a ≠ 0\n⊢ {x | g x ≠ 0} a",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"congrArg",
"Eq.mp",
"Ne",
"ENNReal... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\np : α → Prop\nf g : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᵐ[μ] g\na : α\nha : f a = g a\nh : g a ≠ 0\n⊢ {x | g x ≠ 0} a"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 335,
"column": 42
} | {
"line": 335,
"column": 44
} | {
"line": 335,
"column": 45
} | [
{
"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\nhf'_m : Measurable f'\nhf'_ae : f =ᵐ[μ] f'\ng' : α → ℝ≥0∞\ng'meas : Measurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x\nA : MeasurableSet {x | f' x ≠ 0}\na : α\n⊢ (↑(f a) ≠ 0 → ... | [
"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\nhf'_m : Measurable f'\nhf'_ae : f =ᵐ[μ] f'\ng' : α → ℝ≥0∞\ng'meas : Measurable g'\nhg' : ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = g' x\nA : MeasurableSet {x | f' x ≠ 0}\na : α\nha : ↑(f a) ≠ 0 → g a = g' a... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 340,
"column": 37
} | {
"line": 340,
"column": 39
} | {
"line": 340,
"column": 40
} | [
{
"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\nhf'_m : Measurable f'\nhf'_ae : f =ᵐ[μ] f'\ng' : α → ℝ≥0∞\ng'meas : Measurable g'\nhg' : g =ᵐ[μ.withDensity fun x ↦ ↑(f x)] g'\nA : MeasurableSet {x | f' x ≠ 0}\na : α\n⊢ f a = f' a → a ... | [
"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\nhf'_m : Measurable f'\nhf'_ae : f =ᵐ[μ] f'\ng' : α → ℝ≥0∞\ng'meas : Measurable g'\nhg' : g =ᵐ[μ.withDensity fun x ↦ ↑(f x)] g'\nA : MeasurableSet {x | f' x ≠ 0}\na : α\nha : f a = f' a\n⊢ a ∈ {x | f... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 343,
"column": 10
} | {
"line": 343,
"column": 12
} | {
"line": 343,
"column": 12
} | [
{
"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\nhf'_m : Measurable f'\nhf'_ae : f =ᵐ[μ] f'\ng' : α → ℝ≥0∞\ng'meas : Measurable g'\nhg' : g =ᵐ[μ.withDensity fun x ↦ ↑(f x)] g'\nA : MeasurableSet {x | f' x ≠ 0}\na : α\nha : f a = 0\nha_... | [
"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\nhf'_m : Measurable f'\nhf'_ae : f =ᵐ[μ] f'\ng' : α → ℝ≥0∞\ng'meas : Measurable g'\nhg' : g =ᵐ[μ.withDensity fun x ↦ ↑(f x)] g'\nA : MeasurableSet {x | f' x ≠ 0}\na : α\nha : f a = 0\nha_null✝ : a ∈ ... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.QuadraticDiscriminant | {
"line": 93,
"column": 2
} | {
"line": 96,
"column": 6
} | {
"line": 98,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c : K\nha : a ≠ 0\nh : ∃ s, discrim a b c = s * s\n⊢ ∃ x, a * (x * x) + b * x + c = 0",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instHDiv",
"HMul.hMul",
"AddGr... | [] | rcases h with ⟨s, hs⟩
use (-b + s) / (2 * a)
rw [quadratic_eq_zero_iff ha hs]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.QuadraticDiscriminant | {
"line": 93,
"column": 2
} | {
"line": 96,
"column": 6
} | {
"line": 98,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NeZero 2\na b c : K\nha : a ≠ 0\nh : ∃ s, discrim a b c = s * s\n⊢ ∃ x, a * (x * x) + b * x + c = 0",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instHDiv",
"HMul.hMul",
"AddGr... | [] | rcases h with ⟨s, hs⟩
use (-b + s) / (2 * a)
rw [quadratic_eq_zero_iff ha hs]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.L1Space.HasFiniteIntegral | {
"line": 122,
"column": 40
} | {
"line": 122,
"column": 42
} | {
"line": 123,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup β\ninst✝² : Lattice β\ninst✝¹ : HasSolidNorm β\ninst✝ : AddLeftMono β\nf g : α → β\nhg : HasFiniteIntegral g μ\nhnonneg : ∀ᵐ (a : α) ∂μ, 0 ≤ f a\nh : ∀ᵐ (a : α) ∂μ, f a ≤ g a\na : α\nhn : 0 ≤ f a\n⊢ f a ≤ g a ... | [
"α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup β\ninst✝² : Lattice β\ninst✝¹ : HasSolidNorm β\ninst✝ : AddLeftMono β\nf g : α → β\nhg : HasFiniteIntegral g μ\nhnonneg : ∀ᵐ (a : α) ∂μ, 0 ≤ f a\nh : ∀ᵐ (a : α) ∂μ, f a ≤ g a\na : α\nhn : 0 ≤ f a\nha : f a ≤ g a\n⊢ ‖f a‖ ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Algebra.QuadraticDiscriminant | {
"line": 128,
"column": 2
} | {
"line": 133,
"column": 70
} | {
"line": 135,
"column": 2
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\na b c : K\nh : ∀ (x : K), 0 ≤ a * (x * x) + b * x + c\nha : a < 0\n⊢ b * b - 4 * a * c ≤ 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"add_mul",
"Eq.mpr",
"Semigr... | [
"case inr.inl\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nb c : K\nh : ∀ (x : K), 0 ≤ 0 * (x * x) + b * x + c\n⊢ b * b - 4 * 0 * c ≤ 0",
"case inr.inr\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\na b c : K\nh : ∀ (x : K), 0 ≤ a... | · have : Tendsto (fun x => (a * x + b) * x + c) atTop atBot :=
tendsto_atBot_add_const_right _ c <|
(tendsto_atBot_add_const_right _ b (tendsto_id.const_mul_atTop_of_neg ha)).atBot_mul_atTop₀
tendsto_id
rcases (this.eventually (eventually_lt_atBot 0)).exists with ⟨x, hx⟩
exact False.elim... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Real.Sqrt | {
"line": 178,
"column": 47
} | {
"line": 178,
"column": 71
} | {
"line": 180,
"column": 0
} | [
{
"pp": "x : ℝ\nh : 0 ≤ x\n⊢ √x ^ 2 = x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Real.mul_self_sqrt",
"id",
"MulOne.toMul",
"instOfNatNat",
"sq",
"NP... | [] | rw [sq, mul_self_sqrt h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Real.Sqrt | {
"line": 178,
"column": 47
} | {
"line": 178,
"column": 71
} | {
"line": 180,
"column": 0
} | [
{
"pp": "x : ℝ\nh : 0 ≤ x\n⊢ √x ^ 2 = x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Real.mul_self_sqrt",
"id",
"MulOne.toMul",
"instOfNatNat",
"sq",
"NP... | [] | rw [sq, mul_self_sqrt h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Real.Sqrt | {
"line": 178,
"column": 47
} | {
"line": 178,
"column": 71
} | {
"line": 180,
"column": 0
} | [
{
"pp": "x : ℝ\nh : 0 ≤ x\n⊢ √x ^ 2 = x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Real.mul_self_sqrt",
"id",
"MulOne.toMul",
"instOfNatNat",
"sq",
"NP... | [] | rw [sq, mul_self_sqrt h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Real.Sqrt | {
"line": 374,
"column": 44
} | {
"line": 375,
"column": 79
} | {
"line": 377,
"column": 0
} | [
{
"pp": "x : ℝ\n⊢ √x⁻¹ = (√x)⁻¹",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instInv",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal.instInv",
"Real.sqrt.eq_1",
"id",
"Real.toNNReal_inv"... | [] | by
rw [Real.sqrt, Real.toNNReal_inv, NNReal.sqrt_inv, NNReal.coe_inv, Real.sqrt] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Norm | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 30
} | {
"line": 169,
"column": 0
} | [
{
"pp": "x y : ℝ\n⊢ ‖↑x + ↑y * I‖ = √(x ^ 2 + y ^ 2)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"HMul.hMul",
"congrArg",
"MonoidWithZeroHom.funLike",
"Complex.instMul",
"Complex.instNorm",
"Real.semir... | [] | rw [← normSq_add_mul_I]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Norm | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 30
} | {
"line": 169,
"column": 0
} | [
{
"pp": "x y : ℝ\n⊢ ‖↑x + ↑y * I‖ = √(x ^ 2 + y ^ 2)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"HMul.hMul",
"congrArg",
"MonoidWithZeroHom.funLike",
"Complex.instMul",
"Complex.instNorm",
"Real.semir... | [] | rw [← normSq_add_mul_I]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Norm | {
"line": 225,
"column": 32
} | {
"line": 225,
"column": 66
} | {
"line": 225,
"column": 67
} | [
{
"pp": "z : ℂ\nhz : ¬z = 0\n⊢ |z.re| / ‖z‖ ≤ 1",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"div_le_iff₀",
"n... | [
"z : ℂ\nhz : ¬z = 0\n⊢ |z.re| ≤ 1 * ‖z‖"
] | div_le_iff₀ (norm_pos_iff.mpr hz), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Complex.Norm | {
"line": 230,
"column": 39
} | {
"line": 230,
"column": 73
} | {
"line": 230,
"column": 74
} | [
{
"pp": "z : ℂ\nhz : ¬z = 0\n⊢ |z.im| / ‖z‖ ≤ 1",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"div_le_iff₀",
"n... | [
"z : ℂ\nhz : ¬z = 0\n⊢ |z.im| ≤ 1 * ‖z‖"
] | div_le_iff₀ (norm_pos_iff.mpr hz), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 530,
"column": 31
} | {
"line": 530,
"column": 94
} | {
"line": 530,
"column": 94
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhf : AEMeasurable f (μ.restrict s)\ng : α → ℝ≥0∞\nhs : MeasurableSet s\nh'f : ∀ᵐ (x : α) ∂μ.restrict s, f x < ∞\n⊢ ∫⁻ (a : α), g a ∂(μ.restrict s).withDensity f = ∫⁻ (a : α) in s, (f * g) a ∂μ",
"ppTerm": "?m.46",
"as... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\ns : Set α\nhf : AEMeasurable f (μ.restrict s)\ng : α → ℝ≥0∞\nhs : MeasurableSet s\nh'f : ∀ᵐ (x : α) ∂μ.restrict s, f x < ∞\n⊢ ∫⁻ (a : α) in s, (f * g) a ∂μ = ∫⁻ (a : α) in s, (f * g) a ∂μ"
] | lintegral_withDensity_eq_lintegral_mul_non_measurable₀ _ hf h'f | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 536,
"column": 31
} | {
"line": 536,
"column": 94
} | {
"line": 536,
"column": 94
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\ns : Set α\nhf : AEMeasurable f (μ.restrict s)\ng : α → ℝ≥0∞\nh'f : ∀ᵐ (x : α) ∂μ.restrict s, f x < ∞\n⊢ ∫⁻ (a : α), g a ∂(μ.restrict s).withDensity f = ∫⁻ (a : α) in s, (f * g) a ∂μ",
"ppTerm": "?m.46",
"assig... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\ns : Set α\nhf : AEMeasurable f (μ.restrict s)\ng : α → ℝ≥0∞\nh'f : ∀ᵐ (x : α) ∂μ.restrict s, f x < ∞\n⊢ ∫⁻ (a : α) in s, (f * g) a ∂μ = ∫⁻ (a : α) in s, (f * g) a ∂μ"
] | lintegral_withDensity_eq_lintegral_mul_non_measurable₀ _ hf h'f | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 455,
"column": 93
} | {
"line": 457,
"column": 6
} | {
"line": 459,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz w : K\n⊢ normSq (z + w) = normSq z + normSq w + 2 * re (z * (starRingEnd K) w)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"RCLike.conj_re",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
... | [] | by
simp only [normSq_apply, map_add, rclike_simps]
ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.RCLike.Basic | {
"line": 516,
"column": 71
} | {
"line": 519,
"column": 53
} | {
"line": 521,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : K\n⊢ ∃ c, ‖c‖ = 1 ∧ ↑‖x‖ = c * x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing.toSeminormedCommRing",
... | [] | by
obtain rfl | hx := eq_or_ne x 0
· exact ⟨1, by simp⟩
· exact ⟨‖x‖ / x, by simp [norm_ne_zero_iff.2, hx]⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.PartialHomeomorph.Basic | {
"line": 197,
"column": 4
} | {
"line": 198,
"column": 61
} | {
"line": 199,
"column": 4
} | [
{
"pp": "X : Type u_1\nX' : Type u_2\nY : Type u_3\nY' : Type u_4\nZ : Type u_5\nZ' : Type u_6\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace X'\ninst✝³ : TopologicalSpace Y\ninst✝² : TopologicalSpace Y'\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\ne : PartialHomeomorph X Y\nf : PartialEq... | [
"X : Type u_1\nX' : Type u_2\nY : Type u_3\nY' : Type u_4\nZ : Type u_5\nZ' : Type u_6\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : TopologicalSpace X'\ninst✝³ : TopologicalSpace Y\ninst✝² : TopologicalSpace Y'\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace Z'\ne : PartialHomeomorph X Y\nf : PartialEquiv X Y\nh :... | rw [continuousOn_iff_continuous_domRestrict,
← continuous_codRestrict_iff (s := f.source) (by simp)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 36
} | {
"line": 128,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝⁴ : Norm E\ninst✝³ : Norm F\nf : α → E\ninst✝² : One F\ninst✝¹ : NormOneClass F\ninst✝ : TopologicalSpace α\na : α\nc : ℝ\nhc : ∀ᶠ (a : α) in 𝓝[≠] a, ‖f a‖ ≤ c\nb : α\nhb : b ∈ {a}ᶜ → ‖f b‖ ≤ c\n⊢ ‖f b‖ ≤ max c ‖f a‖",
"ppTerm": "?m.92",
"assigned... | [
"case inl\nα : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝⁴ : Norm E\ninst✝³ : Norm F\nf : α → E\ninst✝² : One F\ninst✝¹ : NormOneClass F\ninst✝ : TopologicalSpace α\nc : ℝ\nb : α\nhc : ∀ᶠ (a : α) in 𝓝[≠] b, ‖f a‖ ≤ c\nhb : b ∈ {b}ᶜ → ‖f b‖ ≤ c\n⊢ ‖f b‖ ≤ max c ‖f b‖",
"case inr\nα : Type u_1\nE : Type u_3\nF : ... | rcases eq_or_ne b a with rfl | hb' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1436,
"column": 4
} | {
"line": 1436,
"column": 26
} | {
"line": 1437,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nE' : Type u_6\ninst✝ : SeminormedAddCommGroup E'\nl : Filter α\nι : Type u_18\nA : ι → α → E'\nC : ι → ℝ\nB : ι → α → ℝ\nhAB : ∀ i ∈ ∅, IsBigOWith (C i) l (A i) (B i)\n⊢ IsBigOWith (sSup (C '' ↑∅)) l (fun H ↦ ∑ i ∈ ∅, A i H) fun H ↦ ∑ i ∈ ∅, ‖B i H‖",
"ppTerm": "?inl",
"... | [] | simp [isBigOWith_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1436,
"column": 4
} | {
"line": 1436,
"column": 26
} | {
"line": 1437,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nE' : Type u_6\ninst✝ : SeminormedAddCommGroup E'\nl : Filter α\nι : Type u_18\nA : ι → α → E'\nC : ι → ℝ\nB : ι → α → ℝ\nhAB : ∀ i ∈ ∅, IsBigOWith (C i) l (A i) (B i)\n⊢ IsBigOWith (sSup (C '' ↑∅)) l (fun H ↦ ∑ i ∈ ∅, A i H) fun H ↦ ∑ i ∈ ∅, ‖B i H‖",
"ppTerm": "?inl",
"... | [] | simp [isBigOWith_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1436,
"column": 4
} | {
"line": 1436,
"column": 26
} | {
"line": 1437,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nE' : Type u_6\ninst✝ : SeminormedAddCommGroup E'\nl : Filter α\nι : Type u_18\nA : ι → α → E'\nC : ι → ℝ\nB : ι → α → ℝ\nhAB : ∀ i ∈ ∅, IsBigOWith (C i) l (A i) (B i)\n⊢ IsBigOWith (sSup (C '' ↑∅)) l (fun H ↦ ∑ i ∈ ∅, A i H) fun H ↦ ∑ i ∈ ∅, ‖B i H‖",
"ppTerm": "?inl",
"... | [] | simp [isBigOWith_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 86,
"column": 30
} | {
"line": 86,
"column": 49
} | {
"line": 87,
"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 : ... | [] | by simp [h] at hQε0 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 179,
"column": 22
} | {
"line": 179,
"column": 54
} | {
"line": 181,
"column": 0
} | [
{
"pp": "x : ℝ\n⊢ (starRingEnd ℂ) (cosh ↑x) = cosh ↑x",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Complex.cosh_conj",
"congrArg",
"CommSemiring.toSemiring",
"RingHom",
"id",
"Complex.ofReal",
"RingHom.instFunLike",
"Comple... | [] | by rw [← cosh_conj, conj_ofReal] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 333,
"column": 84
} | {
"line": 334,
"column": 60
} | {
"line": 336,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ sin z = sin ↑z.re * cosh ↑z.im + cos ↑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",
"Complex.instMul",
... | [] | by
convert! sin_add_mul_I z.re z.im; exact (re_add_im z).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 336,
"column": 91
} | {
"line": 337,
"column": 47
} | {
"line": 339,
"column": 0
} | [
{
"pp": "x y : ℂ\n⊢ cos (x + y * I) = cos x * cosh y - sin x * sinh y * I",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Complex.cos_mul_I",
"Semigroup.toMul",
"Complex.sinh",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Complex.sin",... | [] | by
rw [cos_add, cos_mul_I, sin_mul_I, mul_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 657,
"column": 33
} | {
"line": 657,
"column": 60
} | {
"line": 659,
"column": 0
} | [
{
"pp": "E : Type u_3\nE'' : Type u_9\ninst✝¹ : Norm E\ninst✝ : NormedAddCommGroup E''\nf : ℕ → E\ng'' : ℕ → E''\nh : f =O[atTop] g''\n⊢ f =O[cofinite] g''",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Asymptotics.IsBigO",
"id",
"Filte... | [] | rwa [Nat.cofinite_eq_atTop] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 657,
"column": 33
} | {
"line": 657,
"column": 60
} | {
"line": 659,
"column": 0
} | [
{
"pp": "E : Type u_3\nE'' : Type u_9\ninst✝¹ : Norm E\ninst✝ : NormedAddCommGroup E''\nf : ℕ → E\ng'' : ℕ → E''\nh : f =O[atTop] g''\n⊢ f =O[cofinite] g''",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Asymptotics.IsBigO",
"id",
"Filte... | [] | rwa [Nat.cofinite_eq_atTop] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 657,
"column": 33
} | {
"line": 657,
"column": 60
} | {
"line": 659,
"column": 0
} | [
{
"pp": "E : Type u_3\nE'' : Type u_9\ninst✝¹ : Norm E\ninst✝ : NormedAddCommGroup E''\nf : ℕ → E\ng'' : ℕ → E''\nh : f =O[atTop] g''\n⊢ f =O[cofinite] g''",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Asymptotics.IsBigO",
"id",
"Filte... | [] | rwa [Nat.cofinite_eq_atTop] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Exponential | {
"line": 679,
"column": 33
} | {
"line": 682,
"column": 49
} | {
"line": 684,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : 0 ≤ x\nhx' : x < 2\n⊢ rexp x ≤ (2 + x) / (2 - x)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"LE.le.eq_or_lt",
"False",
"Real.partialOrder",
"Real.instLE",
"Real",
"Preorder.toLT",
"i... | [] | by
obtain rfl | hx₀ := hx.eq_or_lt
· simp
· exact (exp_lt_two_add_div_two_sub hx₀ hx').le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 191,
"column": 4
} | {
"line": 196,
"column": 44
} | {
"line": 197,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : Ring β\nabv : β → α\ninst✝² : IsAbsoluteValue abv\ninst✝¹ : Archimedean α\ninst✝ : Nontrivial β\nx : β\nhx1 : abv x < 1\nhx1' : abv x ≠ 1\nthis : 0 < 1 - abv x\n⊢ ∀ n ≥ 0, |(1 -... | [] | intro n _
rw [abs_of_nonneg]
· gcongr
exact sub_le_self _ (abv_pow abv x n ▸ abv_nonneg _ _)
refine div_nonneg (sub_nonneg.2 ?_) (sub_nonneg.2 <| le_of_lt hx1)
exact pow_le_one₀ (by positivity) hx1.le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 191,
"column": 4
} | {
"line": 196,
"column": 44
} | {
"line": 197,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : Ring β\nabv : β → α\ninst✝² : IsAbsoluteValue abv\ninst✝¹ : Archimedean α\ninst✝ : Nontrivial β\nx : β\nhx1 : abv x < 1\nhx1' : abv x ≠ 1\nthis : 0 < 1 - abv x\n⊢ ∀ n ≥ 0, |(1 -... | [] | intro n _
rw [abs_of_nonneg]
· gcongr
exact sub_le_self _ (abv_pow abv x n ▸ abv_nonneg _ _)
refine div_nonneg (sub_nonneg.2 ?_) (sub_nonneg.2 <| le_of_lt hx1)
exact pow_le_one₀ (by positivity) hx1.le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 307,
"column": 14
} | {
"line": 307,
"column": 31
} | {
"line": 307,
"column": 31
} | [
{
"pp": "α : Type u_1\nx y z : ℝ\nl : Filter α\n⊢ Tendsto (fun x ↦ ⟨rexp x, ⋯⟩) atTop atTop",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Set.Ioi",
"Preorder.toLT",
... | [
"α : Type u_1\nx y z : ℝ\nl : Filter α\n⊢ Tendsto (fun x ↦ ↑⟨rexp x, ⋯⟩) atTop atTop"
] | tendsto_Ioi_atTop | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 354,
"column": 42
} | {
"line": 354,
"column": 64
} | {
"line": 354,
"column": 65
} | [
{
"pp": "a : ℝ\n⊢ Subtype.val '' ⇑expOrderIso '' Set.Iio a = Set.Ioo 0 ((Subtype.val ∘ ⇑expOrderIso) a)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Set.Ioi",
"Real.instZero",
"congrArg",
"Preorder.toLE",
... | [
"a : ℝ\n⊢ Subtype.val '' Set.Iio (expOrderIso a) = Set.Ioo 0 ((Subtype.val ∘ ⇑expOrderIso) a)"
] | expOrderIso.image_Iio, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.RCLike.Real | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 34
} | {
"line": 146,
"column": 0
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\nhr : r ≠ 0\n⊢ interior (closedBall x r) = ball x r",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"interior_closedBall",
"NormedAddCommGroup.toSeminorme... | [] | exact interior_closedBall x hr | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Module.RCLike.Real | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 34
} | {
"line": 146,
"column": 0
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\nhr : r ≠ 0\n⊢ interior (closedBall x r) = ball x r",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"interior_closedBall",
"NormedAddCommGroup.toSeminorme... | [] | exact interior_closedBall x hr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.RCLike.Real | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 34
} | {
"line": 146,
"column": 0
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\nhr : r ≠ 0\n⊢ interior (closedBall x r) = ball x r",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"interior_closedBall",
"NormedAddCommGroup.toSeminorme... | [] | exact interior_closedBall x hr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 833,
"column": 43
} | {
"line": 833,
"column": 54
} | {
"line": 833,
"column": 54
} | [
{
"pp": "θ : ℝ := π / 5\nhθ : θ = π / 5\nc : ℝ := cos θ\ns : ℝ := sin θ\nhs : s ≠ 0\n⊢ 2 * s * c = sin (2 * θ)",
"ppTerm": "?m.244",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.cos",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"i... | [
"θ : ℝ := π / 5\nhθ : θ = π / 5\nc : ℝ := cos θ\ns : ℝ := sin θ\nhs : s ≠ 0\n⊢ 2 * s * c = 2 * sin θ * cos θ"
] | sin_two_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 837,
"column": 63
} | {
"line": 837,
"column": 74
} | {
"line": 837,
"column": 74
} | [
{
"pp": "θ : ℝ := π / 5\nhθ : θ = π / 5\nc : ℝ := cos θ\ns : ℝ := sin θ\nhs : s ≠ 0\n⊢ sin (2 * θ) * c + cos (2 * θ) * s = 2 * s * c * c + cos (2 * θ) * s",
"ppTerm": "?m.341",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.cos",
"congrArg",
... | [
"θ : ℝ := π / 5\nhθ : θ = π / 5\nc : ℝ := cos θ\ns : ℝ := sin θ\nhs : s ≠ 0\n⊢ 2 * sin θ * cos θ * c + cos (2 * θ) * s = 2 * s * c * c + cos (2 * θ) * s"
] | sin_two_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 890,
"column": 2
} | {
"line": 891,
"column": 6
} | {
"line": 893,
"column": 0
} | [
{
"pp": "⊢ tan (π / 3) = √3",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Common.div_congr",
"Real.partialOrder",
"Real",
"instHDi... | [] | rw [tan_eq_sin_div_cos, sin_pi_div_three, cos_pi_div_three]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 890,
"column": 2
} | {
"line": 891,
"column": 6
} | {
"line": 893,
"column": 0
} | [
{
"pp": "⊢ tan (π / 3) = √3",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Common.div_congr",
"Real.partialOrder",
"Real",
"instHDi... | [] | rw [tan_eq_sin_div_cos, sin_pi_div_three, cos_pi_div_three]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Ball.Pointwise | {
"line": 316,
"column": 54
} | {
"line": 317,
"column": 87
} | {
"line": 319,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nδ ε : ℝ\nhε : 0 < ε\nhδ : 0 ≤ δ\nx : E\n⊢ thickening ε (Metric.closedBall x δ) = Metric.ball x (ε + δ)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"S... | [] | by
rw [← cthickening_singleton _ hδ, thickening_cthickening hε hδ, thickening_singleton] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Ball.Pointwise | {
"line": 349,
"column": 55
} | {
"line": 349,
"column": 65
} | {
"line": 349,
"column": 65
} | [
{
"pp": "E : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nδ ε : ℝ\nhε : 0 ≤ ε\nhδ : 0 < δ\na b : E\n⊢ Metric.ball (a + b) (δ + ε) = Metric.ball (a + b) (ε + δ)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Ad... | [
"E : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nδ ε : ℝ\nhε : 0 ≤ ε\nhδ : 0 < δ\na b : E\n⊢ Metric.ball (a + b) (ε + δ) = Metric.ball (a + b) (ε + δ)"
] | add_comm δ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Ball.Pointwise | {
"line": 358,
"column": 51
} | {
"line": 358,
"column": 61
} | {
"line": 358,
"column": 61
} | [
{
"pp": "E : Type u_2\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nδ ε : ℝ\ninst✝ : ProperSpace E\nhε : 0 ≤ ε\nhδ : 0 ≤ δ\na b : E\n⊢ Metric.closedBall (a + b) (δ + ε) = Metric.closedBall (a + b) (ε + δ)",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"E : Type u_2\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nδ ε : ℝ\ninst✝ : ProperSpace E\nhε : 0 ≤ ε\nhδ : 0 ≤ δ\na b : E\n⊢ Metric.closedBall (a + b) (ε + δ) = Metric.closedBall (a + b) (ε + δ)"
] | add_comm δ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 520,
"column": 2
} | {
"line": 522,
"column": 8
} | {
"line": 523,
"column": 2
} | [
{
"pp": "R : Type u_4\ninst✝¹ : NormedRing R\ninst✝ : HasSummableGeomSeries R\nx : R\nh : ‖x‖ < 1\n⊢ HasSum (fun n ↦ ↑n * x ^ n) (x * (1 - x)⁻¹ʳ ^ 2)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.choose",
"NormedRing.toRing",
"HMul.hMul",
"Add... | [
"R : Type u_4\ninst✝¹ : NormedRing R\ninst✝ : HasSummableGeomSeries R\nx : R\nh : ‖x‖ < 1\nA : HasSum (fun n ↦ (↑n + 1) * x ^ n) ((1 - x)⁻¹ʳ ^ 2)\n⊢ HasSum (fun n ↦ ↑n * x ^ n) (x * (1 - x)⁻¹ʳ ^ 2)"
] | have A : HasSum (fun (n : ℕ) ↦ (n + 1) * x ^ n) ((1 - x)⁻¹ʳ ^ 2) := by
convert! hasSum_choose_mul_geometric_of_norm_lt_one' 1 h with n
simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 579,
"column": 2
} | {
"line": 579,
"column": 90
} | {
"line": 580,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : SeminormedAddCommGroup α\nr C : ℝ\nf : ℕ → α\nhr : r < 1\nhf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n\na : α\nha : HasSum f a\nn : ℕ\n⊢ dist (∑ x ∈ Finset.range n, f x) a ≤ C * r ^ n / (1 - r)",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"AddCommGroup.toAddCom... | [
"α : Type u_1\ninst✝ : SeminormedAddCommGroup α\nr C : ℝ\nf : ℕ → α\nhr : r < 1\nhf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n\na : α\nha : HasSum f a\nn : ℕ\n⊢ Tendsto (fun n ↦ ∑ i ∈ Finset.range n, f i) atTop (𝓝 a)"
] | apply dist_le_of_le_geometric_of_tendsto r C hr (dist_partial_sum_le_of_le_geometric hf) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Connected.PathConnected | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 26
} | {
"line": 226,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nx y : X\nF : Set X\nh : JoinedIn F x y\n⊢ JoinedIn F y x",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"JoinedIn",
"Membership.mem",
"JoinedIn.mem",
"And.casesOn",
"And",
"Set.instMembership",
"Se... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nx y : X\nF : Set X\nh : JoinedIn F x y\nhx : x ∈ F\nhy : y ∈ F\n⊢ JoinedIn F y x"
] | obtain ⟨hx, hy⟩ := h.mem | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 662,
"column": 6
} | {
"line": 662,
"column": 22
} | {
"line": 662,
"column": 22
} | [
{
"pp": "α : Type u_4\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nr : ℝ\nhr : 1 < r\nhf : ∃ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0\nN₀ : ℕ\nhN₀ : ∀ (b : ℕ), N₀ ≤ b → r * ‖f b‖ ≤ ‖f (b + 1)‖\n⊢ ¬Summable f",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.instLE",
... | [
"α : Type u_4\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nr : ℝ\nhr : 1 < r\nhf : ∀ (a : ℕ), ∃ b, a ≤ b ∧ ‖f b‖ ≠ 0\nN₀ : ℕ\nhN₀ : ∀ (b : ℕ), N₀ ≤ b → r * ‖f b‖ ≤ ‖f (b + 1)‖\n⊢ ¬Summable f"
] | frequently_atTop | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Path | {
"line": 328,
"column": 4
} | {
"line": 328,
"column": 72
} | {
"line": 329,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\n⊢ ⇑(γ₁.trans γ₂).extend '' Iic (1 / 2) ∪ ⇑(γ₁.trans γ₂).extend '' Ici (1 / 2) = range ⇑γ₁ ∪ range ⇑γ₂",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\n⊢ (fun t ↦ γ₁.extend (2 * t)) '' Iic (1 / 2) ∪ ⇑(γ₁.trans γ₂).extend '' Ici (1 / 2) = range ⇑γ₁ ∪ range ⇑γ₂"
] | EqOn.image_eq fun t ht ↦ extend_trans_of_le_half _ _ (mem_Iic.1 ht), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 150,
"column": 83
} | {
"line": 150,
"column": 96
} | {
"line": 151,
"column": 4
} | [
{
"pp": "case inr\np x ε : ℝ\nhp : p ≠ 0\ny : ℝ\nn : ℤ\nhn : |p⁻¹ * (y - x) - p⁻¹ * (↑n * p)| ≤ |p⁻¹| * ε\n⊢ |p⁻¹ * (y - x) - p⁻¹ * (↑(round (p⁻¹ * (y - x))) * p)| ≤ |p⁻¹| * ε",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
... | [
"case inr\np x ε : ℝ\nhp : p ≠ 0\ny : ℝ\nn : ℤ\nhn : |p⁻¹ * (y - x) - p⁻¹ * (p * ↑n)| ≤ |p⁻¹| * ε\n⊢ |p⁻¹ * (y - x) - p⁻¹ * (p * ↑(round (p⁻¹ * (y - x))))| ≤ |p⁻¹| * ε"
] | mul_comm _ p, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Path | {
"line": 596,
"column": 51
} | {
"line": 600,
"column": 83
} | {
"line": 602,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\na b : X\nγ : Path a b\n⊢ γ.truncate 0 1 = γ.cast ⋯ ⋯",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"ContinuousMap.mk",
"ContinuousMap",
"Path.trunca... | [] | by
ext x
rw [cast_coe]
have : ↑x ∈ (Icc 0 1 : Set ℝ) := x.2
rw [truncate, coe_mk_mk, max_eq_left this.1, min_eq_left this.2, extend_extends'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 176,
"column": 8
} | {
"line": 177,
"column": 40
} | {
"line": 178,
"column": 6
} | [
{
"pp": "case inr.inr.inl.inl\np x ε : ℝ\ns : Set ℝ\nhε : ε < |p| / 2\nz : ℤ\nhs : s ⊆ Icc (x - |p| / 2) (x + |p| / 2)\nhz : z ≠ 0\ny : ℝ\nhy₀ : y ∈ s\nhy₁ : x + ↑z * p - ε ≤ y\nhy₂ : y ≤ x + ↑z * p + ε\nhy₃ : x - |p| / 2 ≤ y\nhy₄ : y ≤ x + |p| / 2\nhp : 0 < p\nhz' : ↑z ≤ -1\n⊢ False",
"ppTerm": "?inr.inr.i... | [] | have : ↑z * p ≤ -p := by nlinarith
linarith [abs_eq_self.mpr hp.le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 176,
"column": 8
} | {
"line": 177,
"column": 40
} | {
"line": 178,
"column": 6
} | [
{
"pp": "case inr.inr.inl.inl\np x ε : ℝ\ns : Set ℝ\nhε : ε < |p| / 2\nz : ℤ\nhs : s ⊆ Icc (x - |p| / 2) (x + |p| / 2)\nhz : z ≠ 0\ny : ℝ\nhy₀ : y ∈ s\nhy₁ : x + ↑z * p - ε ≤ y\nhy₂ : y ≤ x + ↑z * p + ε\nhy₃ : x - |p| / 2 ≤ y\nhy₄ : y ≤ x + |p| / 2\nhp : 0 < p\nhz' : ↑z ≤ -1\n⊢ False",
"ppTerm": "?inr.inr.i... | [] | have : ↑z * p ≤ -p := by nlinarith
linarith [abs_eq_self.mpr hp.le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 185,
"column": 8
} | {
"line": 185,
"column": 41
} | {
"line": 186,
"column": 8
} | [
{
"pp": "case inr.inr.inr.inr.inr\np x ε : ℝ\ns : Set ℝ\nhε : ε < |p| / 2\nz : ℤ\nhs : s ⊆ Icc (x - |p| / 2) (x + |p| / 2)\nhz : z ≠ 0\ny : ℝ\nhy₀ : y ∈ s\nhy₁ : x + ↑z * p - ε ≤ y\nhy₂ : y ≤ x + ↑z * p + ε\nhy₃ : x - |p| / 2 ≤ y\nhy₄ : y ≤ x + |p| / 2\nhp : p < 0\nhz' : 1 ≤ ↑z\n⊢ False",
"ppTerm": "?inr.in... | [
"case inr.inr.inr.inr.inr\np x ε : ℝ\ns : Set ℝ\nhε : ε < |p| / 2\nz : ℤ\nhs : s ⊆ Icc (x - |p| / 2) (x + |p| / 2)\nhz : z ≠ 0\ny : ℝ\nhy₀ : y ∈ s\nhy₁ : x + ↑z * p - ε ≤ y\nhy₂ : y ≤ x + ↑z * p + ε\nhy₃ : x - |p| / 2 ≤ y\nhy₄ : y ≤ x + |p| / 2\nhp : p < 0\nhz' : 1 ≤ ↑z\nthis : ↑z * p ≤ p\n⊢ False"
] | have : ↑z * p ≤ p := by nlinarith | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 194,
"column": 49
} | {
"line": 194,
"column": 62
} | {
"line": 194,
"column": 63
} | [
{
"pp": "p : ℝ\nhp : Fact (0 < p)\nm n : ℕ\n⊢ p⁻¹ * (↑m / ↑n * p) = ↑m / ↑n",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRin... | [
"p : ℝ\nhp : Fact (0 < p)\nm n : ℕ\n⊢ p⁻¹ * (p * (↑m / ↑n)) = ↑m / ↑n"
] | mul_comm _ p, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 65
} | {
"line": 202,
"column": 2
} | [
{
"pp": "p : ℝ\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nn : ℕ := addOrderOf u\n⊢ ∃ k, ‖u‖ = p * (↑k / ↑n)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"AddCircle.exists_gcd_eq_one_of_isOfFinAddOrder",
"Norm.norm",
"NormedCommRing.to... | [
"p : ℝ\nhp : Fact (0 < p)\nu : AddCircle p\nhu : IsOfFinAddOrder u\nn : ℕ := addOrderOf u\nm : ℕ\nhm : ↑(↑m / ↑(addOrderOf u) * p) = u\n⊢ ∃ k, ‖u‖ = p * (↑k / ↑n)"
] | obtain ⟨m, -, -, hm⟩ := exists_gcd_eq_one_of_isOfFinAddOrder hu | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 212,
"column": 37
} | {
"line": 212,
"column": 50
} | {
"line": 212,
"column": 51
} | [
{
"pp": "p : ℝ\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u ≠ 0\nn : ℕ\nhn : ‖u‖ = p * (↑n / ↑(addOrderOf u))\nhu : ↑(addOrderOf u) ≠ 0\n⊢ p * 1 ≤ ↑(addOrderOf u) * p * (↑n / ↑(addOrderOf u))",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAss... | [
"p : ℝ\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u ≠ 0\nn : ℕ\nhn : ‖u‖ = p * (↑n / ↑(addOrderOf u))\nhu : ↑(addOrderOf u) ≠ 0\n⊢ p * 1 ≤ p * ↑(addOrderOf u) * (↑n / ↑(addOrderOf u))"
] | mul_comm _ p, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 43
} | {
"line": 245,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ z.arg = 0 ↔ 0 ≤ z",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"Complex.im",
"Iff.rfl",
"PartialOrder.toPreorder",
"Complex.instZero",
"Preorder.toL... | [] | rw [arg_eq_zero_iff, eq_comm, nonneg_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 43
} | {
"line": 245,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ z.arg = 0 ↔ 0 ≤ z",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"Complex.im",
"Iff.rfl",
"PartialOrder.toPreorder",
"Complex.instZero",
"Preorder.toL... | [] | rw [arg_eq_zero_iff, eq_comm, nonneg_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 43
} | {
"line": 245,
"column": 0
} | [
{
"pp": "z : ℂ\n⊢ z.arg = 0 ↔ 0 ≤ z",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"Complex.im",
"Iff.rfl",
"PartialOrder.toPreorder",
"Complex.instZero",
"Preorder.toL... | [] | rw [arg_eq_zero_iff, eq_comm, nonneg_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 974,
"column": 48
} | {
"line": 974,
"column": 85
} | {
"line": 974,
"column": 85
} | [
{
"pp": "R : Type u_4\nK : Type u_5\ninst✝⁹ : NormedRing K\ninst✝⁸ : NormedRing R\ninst✝⁷ : Module K R\ninst✝⁶ : IsTorsionFree K R\ninst✝⁵ : NormSMulClass K R\ninst✝⁴ : NormSMulClass ℤ K\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : FloorSemiring K\ninst✝ : HasSolidNorm K\ng : ℕ → R\nt : R\n... | [] | by simpa only [nsmul_eq_mul] using hg | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 73,
"column": 31
} | {
"line": 73,
"column": 54
} | {
"line": 73,
"column": 54
} | [
{
"pp": "x : ℝ\nhx : 0 ≤ x\n⊢ 0 = (↑x).arg",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Complex.arg_ofReal_of_nonneg",
"Complex.arg",
"id",
"Complex.ofReal",
"Zero.toOfNat0",
"OfN... | [
"x : ℝ\nhx : 0 ≤ x\n⊢ 0 = 0"
] | arg_ofReal_of_nonneg hx | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 118,
"column": 27
} | {
"line": 118,
"column": 36
} | {
"line": 118,
"column": 37
} | [
{
"pp": "x : ℂ\n⊢ ↑(Real.log ‖x‖) + ↑((starRingEnd ℂ) x).arg * I =\n if x.arg = π then ↑(Real.log ‖x‖) + ↑x.arg * I else (starRingEnd ℂ) (↑(Real.log ‖x‖) + ↑x.arg * I)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"Real.pi",
... | [
"x : ℂ\n⊢ ↑(Real.log ‖x‖) + ↑(if x.arg = π then π else -x.arg) * I =\n if x.arg = π then ↑(Real.log ‖x‖) + ↑x.arg * I else (starRingEnd ℂ) (↑(Real.log ‖x‖) + ↑x.arg * I)"
] | arg_conj, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 15
} | {
"line": 160,
"column": 0
} | [
{
"pp": "x : ℂ\nhx : 0 ≤ x.im\nx✝ : ∃ n, x = ↑n * (2 * ↑π * I)\nn : ℕ\nhn : x = ↑↑n * (2 * ↑π * I)\nthis : 0 ≤ ↑↑n * (2 * π)\n⊢ ∃ n, x = ↑n * (2 * ↑π * I)",
"ppTerm": "?m.121",
"assigned": true,
"usedConstants": [
"Real.pi",
"HMul.hMul",
"Nat.instAtLeastTwoHAddOfNat",
"Comple... | [] | exact ⟨n, hn⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 200,
"column": 6
} | {
"line": 200,
"column": 72
} | {
"line": 201,
"column": 6
} | [
{
"pp": "case mp.inl\nθ ψ : ℝ\nn : ℤ\nhn : ↑n * π = (θ + ψ) / 2\n⊢ ↑θ = -↑ψ",
"ppTerm": "?mp.inl",
"assigned": true,
"usedConstants": [
"Int.cast",
"GroupWithZero.toMonoidWithZero",
"Real.partialOrder",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Grou... | [
"case mp.inl\nθ ψ : ℝ\nn : ℤ\nhn : ↑n * π * 2 - ψ = θ\n⊢ ↑θ = -↑ψ"
] | rw [eq_div_iff_mul_eq (two_ne_zero' ℝ), ← sub_eq_iff_eq_add] at hn | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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