module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 770,
"column": 4
} | {
"line": 770,
"column": 48
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nhd : DiffContOnCl ℂ f {z | 0 < z.re}\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖\nC : ℝ\nhC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\ng ... | simp only [g, norm_smul, norm_pow, norm_exp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 770,
"column": 4
} | {
"line": 770,
"column": 48
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nhd : DiffContOnCl ℂ f {z | 0 < z.re}\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖\nC : ℝ\nhC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\ng ... | simp only [g, norm_smul, norm_pow, norm_exp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 770,
"column": 4
} | {
"line": 770,
"column": 48
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nhd : DiffContOnCl ℂ f {z | 0 < z.re}\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖\nC : ℝ\nhC : ∀ (x : ℝ), ‖f (↑x * I)‖ ≤ C\ng ... | simp only [g, norm_smul, norm_pow, norm_exp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.Basic | {
"line": 168,
"column": 34
} | {
"line": 168,
"column": 51
} | [
{
"pp": "z : ℍ\nx : ℝ\n⊢ (↑z).im ≠ (↑x).im",
"usedConstants": [
"False",
"Real",
"_private.Mathlib.Analysis.Complex.UpperHalfPlane.Basic.0.UpperHalfPlane.ne_ofReal._simp_1_1",
"Real.instZero",
"congrArg",
"UpperHalfPlane.im",
"True",
"of_eq_true",
"Zero.... | simp [im_ne_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.UpperHalfPlane.Basic | {
"line": 168,
"column": 34
} | {
"line": 168,
"column": 51
} | [
{
"pp": "z : ℍ\nx : ℝ\n⊢ (↑z).im ≠ (↑x).im",
"usedConstants": [
"False",
"Real",
"_private.Mathlib.Analysis.Complex.UpperHalfPlane.Basic.0.UpperHalfPlane.ne_ofReal._simp_1_1",
"Real.instZero",
"congrArg",
"UpperHalfPlane.im",
"True",
"of_eq_true",
"Zero.... | simp [im_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.UpperHalfPlane.Basic | {
"line": 168,
"column": 34
} | {
"line": 168,
"column": 51
} | [
{
"pp": "z : ℍ\nx : ℝ\n⊢ (↑z).im ≠ (↑x).im",
"usedConstants": [
"False",
"Real",
"_private.Mathlib.Analysis.Complex.UpperHalfPlane.Basic.0.UpperHalfPlane.ne_ofReal._simp_1_1",
"Real.instZero",
"congrArg",
"UpperHalfPlane.im",
"True",
"of_eq_true",
"Zero.... | simp [im_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 796,
"column": 27
} | {
"line": 796,
"column": 32
} | [
{
"pp": "case refine_3\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nhd : DiffContOnCl ℂ f {z | 0 < z.re}\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖\nC : ℝ\nhC : ∀ (x : ℝ), ‖f (↑... | I_re, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 796,
"column": 33
} | {
"line": 796,
"column": 42
} | [
{
"pp": "case refine_3\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nhd : DiffContOnCl ℂ f {z | 0 < z.re}\nhexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhre : SuperpolynomialDecay atTop expR fun x ↦ ‖f ↑x‖\nC : ℝ\nhC : ∀ (x : ℝ), ‖f (↑... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 44,
"column": 2
} | {
"line": 45,
"column": 8
} | [
{
"pp": "case inr.refine_2\nthis : Set.univ = Set.Iio 0 ∪ Set.Ioi 0 ∪ {0}\n⊢ Filter.Tendsto (fun x ↦ log x * x) (pure 0) (𝓝 0)",
"usedConstants": [
"Pure.pure",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAsso... | · convert tendsto_pure_nhds (fun x ↦ log x * x) 0
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.Poisson | {
"line": 123,
"column": 62
} | {
"line": 123,
"column": 72
} | [
{
"pp": "θ φ r R : ℝ\nh₁ : 0 < r\nh₂ : r < R\nkey : (-(↑R * cexp (↑θ * I) * (starRingEnd ℂ) (↑r * cexp (↑φ * I)))).re ≤ R * r\n⊢ R ^ 2 + r ^ 2 + -(2 * (↑R * cexp (↑θ * I) * (starRingEnd ℂ) (↑r * cexp (↑φ * I))).re) ≤ R ^ 2 + r ^ 2 + 2 * (R * r)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
... | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Poisson | {
"line": 203,
"column": 50
} | {
"line": 203,
"column": 78
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : ℂ → E\nR : ℝ\nw : ℂ\ninst✝ : CompleteSpace E\nhf : DiffContOnCl ℂ f (ball 0 R)\nhw : w ∈ ball 0 R\nhR : 0 < R\nh₁w : w ≠ 0\nW : ℂ := ↑R * cexp (↑w.arg * I)\nq : ℝ := ‖w‖ / R\nh₁q : 0 < q\nh₂q : q < 1\nη₀ : ∀ {x : ℂ}, ‖x‖ ≤ R → ↑... | aesop (add simp sub_eq_zero) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 250,
"column": 6
} | {
"line": 250,
"column": 94
} | [
{
"pp": "a b : ℝ\nc : ℂ\nhc : c ≠ 0\nD : ∀ (x : ℝ), HasDerivAt (fun y ↦ Complex.exp (c * ↑y) / c) (Complex.exp (c * ↑x)) x\n⊢ ∫ (x : ℝ) in a..b, Complex.exp (c * ↑x) = (Complex.exp (c * ↑b) - Complex.exp (c * ↑a)) / c",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"Inne... | integral_deriv_eq_sub' _ (funext fun x => (D x).deriv) fun x _ => (D x).differentiableAt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 344,
"column": 2
} | {
"line": 347,
"column": 13
} | [
{
"pp": "a b : ℝ\n⊢ ∫ (x : ℝ) in a..b, 1 / (1 + x ^ 2) = arctan b - arctan a",
"usedConstants": [
"Real.instIsOrderedRing",
"Not.intro",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.toNormedSpace",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | refine integral_deriv_eq_sub' _ Real.deriv_arctan (fun _ _ => differentiableAt_arctan _)
(continuous_const.div ?_ fun x => ?_).continuousOn
· fun_prop
· nlinarith | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 344,
"column": 2
} | {
"line": 347,
"column": 13
} | [
{
"pp": "a b : ℝ\n⊢ ∫ (x : ℝ) in a..b, 1 / (1 + x ^ 2) = arctan b - arctan a",
"usedConstants": [
"Real.instIsOrderedRing",
"Not.intro",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.toNormedSpace",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | refine integral_deriv_eq_sub' _ Real.deriv_arctan (fun _ _ => differentiableAt_arctan _)
(continuous_const.div ?_ fun x => ?_).continuousOn
· fun_prop
· nlinarith | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 214,
"column": 4
} | {
"line": 224,
"column": 73
} | [
{
"pp": "a c : ℂ\nR : ℝ\nhR : R ≠ 0\n⊢ (fun z ↦ log ‖↑R * (z + ↑R⁻¹ * (c - a))‖) =ᶠ[codiscreteWithin (sphere 0 |1|)] fun z ↦\n log ‖R‖ + log ‖z + ↑R⁻¹ * (c - a)‖",
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
... | have : {z | ‖z + ↑R⁻¹ * (c - a)‖ ≠ 0} ∈ codiscreteWithin (Metric.sphere (0 : ℂ) |1|) := by
apply codiscreteWithin_iff_locallyFiniteComplementWithin.2
intro z hz
use Set.univ
simp only [univ_mem, abs_one, Complex.ofReal_inv, ne_eq, norm_eq_zero, Set.univ_inter,
true_and]
apply Set.S... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 23
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : AddZeroClass β\ninst✝ : ContinuousAdd β\nl : Filter α\nf g : α → β\nhf : l.ZeroAtFilter f\nhg : l.ZeroAtFilter g\n⊢ l.ZeroAtFilter (f + g)",
"usedConstants": [
"congrArg",
"nhds",
"AddZeroClass.toAddZero",
"Eq... | simpa using hf.add hg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 23
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : AddZeroClass β\ninst✝ : ContinuousAdd β\nl : Filter α\nf g : α → β\nhf : l.ZeroAtFilter f\nhg : l.ZeroAtFilter g\n⊢ l.ZeroAtFilter (f + g)",
"usedConstants": [
"congrArg",
"nhds",
"AddZeroClass.toAddZero",
"Eq... | simpa using hf.add hg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 23
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : TopologicalSpace β\ninst✝¹ : AddZeroClass β\ninst✝ : ContinuousAdd β\nl : Filter α\nf g : α → β\nhf : l.ZeroAtFilter f\nhg : l.ZeroAtFilter g\n⊢ l.ZeroAtFilter (f + g)",
"usedConstants": [
"congrArg",
"nhds",
"AddZeroClass.toAddZero",
"Eq... | simpa using hf.add hg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 23
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝ : SeminormedAddCommGroup β\nl : Filter α\nf g : α → β\nhf : l.BoundedAtFilter f\nhg : l.BoundedAtFilter g\n⊢ l.BoundedAtFilter (f + g)",
"usedConstants": [
"Real",
"Asymptotics.IsBigO.add",
"Real.instOne",
"Pi.instOne",
"One.toOfNat1",... | simpa using hf.add hg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 23
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝ : SeminormedAddCommGroup β\nl : Filter α\nf g : α → β\nhf : l.BoundedAtFilter f\nhg : l.BoundedAtFilter g\n⊢ l.BoundedAtFilter (f + g)",
"usedConstants": [
"Real",
"Asymptotics.IsBigO.add",
"Real.instOne",
"Pi.instOne",
"One.toOfNat1",... | simpa using hf.add hg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.ZeroAndBoundedAtFilter | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 23
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝ : SeminormedAddCommGroup β\nl : Filter α\nf g : α → β\nhf : l.BoundedAtFilter f\nhg : l.BoundedAtFilter g\n⊢ l.BoundedAtFilter (f + g)",
"usedConstants": [
"Real",
"Asymptotics.IsBigO.add",
"Real.instOne",
"Pi.instOne",
"One.toOfNat1",... | simpa using hf.add hg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Periodic | {
"line": 70,
"column": 40
} | {
"line": 70,
"column": 48
} | [
{
"pp": "h : ℝ\nhh : h ≠ 0\nz : ℂ\nm : ℤ\nhm : log (cexp (2 * ↑π * I * z / ↑h)) = 2 * ↑π * I * z / ↑h + ↑m * (2 * ↑π * I)\n⊢ ↑h * (z / ↑h + ↑m) = z + ↑m * ↑h",
"usedConstants": [
"Distrib.leftDistribClass",
"Int.cast",
"Eq.mpr",
"Semigroup.toMul",
"instHDiv",
"NonUnitalCo... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 183,
"column": 6
} | {
"line": 183,
"column": 34
} | [
{
"pp": "h : ℝ\nf : ℂ → ℂ\nhh : 0 < h\nhf : Periodic f ↑h\nh_hol : ∀ᶠ (z : ℂ) in I∞, DifferentiableAt ℂ f z\nq : ℂ\nhq : q ∈ {0}ᶜ\nh_diff : DifferentiableAt ℂ f (invQParam h q)\n⊢ DifferentiableAt ℂ (cuspFunction h f) q",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | ← qParam_right_inv hh.ne' hq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Polynomial.GaussLucas | {
"line": 99,
"column": 59
} | {
"line": 106,
"column": 22
} | [
{
"pp": "P : ℂ[X]\nh₀ : 0 < P.degree\n⊢ (derivative P).rootSet ℂ ⊆ (convexHull ℝ) (P.rootSet ℂ)",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Multiset.toFinset",
"Polynomial.derivative",
"Eq.mpr",
"Polynomial.eval",
"InnerProductSpace.toNormedSpace",
"No... | by
intro z hz
rw [mem_rootSet, coe_aeval_eq_eval] at hz
rw [eq_centerMass_of_eval_derivative_eq_zero h₀ hz.2]
apply Finset.centerMass_mem_convexHull
· simp [derivRootWeight_nonneg]
· apply sum_derivRootWeight_pos h₀
· simp [mem_rootSet] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.RCLike.Sqrt | {
"line": 60,
"column": 93
} | {
"line": 65,
"column": 7
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\na : 𝕜\n⊢ sqrt a = ↑√((‖a‖ + re a) / 2) + (if 0 ≤ im a then 1 else -1) * ↑√((‖a‖ - re a) / 2) * I",
"usedConstants": [
"Complex.div_ofReal_re",
"NormedCommRing.toNormedRing",
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"Grou... | by
rw [sqrt, Complex.sqrt_eq_real_add_ite]
obtain (h | h) := I_eq_zero_or_im_I_eq_one (K := 𝕜)
· rw [← re_add_im a]
simp [h, im_eq_zero]
aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 123,
"column": 2
} | {
"line": 129,
"column": 30
} | [
{
"pp": "g g' : GL (Fin 2) ℝ\nz : ℂ\n⊢ (σ (g * g')) z = (σ g) ((σ g') z)",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Units.val",
"Eq.mpr",
"mul_neg_of_neg_of_pos",
"NormedCommRing.toSeminormedCommRing",
"MonoidHom.instMonoidHomClass",
... | simp only [σ, map_mul, Units.val_mul]
rcases g.det_ne_zero.lt_or_gt with (h | h) <;>
rcases g'.det_ne_zero.lt_or_gt with (h' | h')
· simp [mul_pos_of_neg_of_neg h h', h.not_gt, h'.not_gt]
· simp [(mul_neg_of_neg_of_pos h h').not_gt, h.not_gt, h']
· simp [(mul_neg_of_pos_of_neg h h').not_gt, h, h'.not_gt]
· ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 123,
"column": 2
} | {
"line": 129,
"column": 30
} | [
{
"pp": "g g' : GL (Fin 2) ℝ\nz : ℂ\n⊢ (σ (g * g')) z = (σ g) ((σ g') z)",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Units.val",
"Eq.mpr",
"mul_neg_of_neg_of_pos",
"NormedCommRing.toSeminormedCommRing",
"MonoidHom.instMonoidHomClass",
... | simp only [σ, map_mul, Units.val_mul]
rcases g.det_ne_zero.lt_or_gt with (h | h) <;>
rcases g'.det_ne_zero.lt_or_gt with (h' | h')
· simp [mul_pos_of_neg_of_neg h h', h.not_gt, h'.not_gt]
· simp [(mul_neg_of_neg_of_pos h h').not_gt, h.not_gt, h']
· simp [(mul_neg_of_pos_of_neg h h').not_gt, h, h'.not_gt]
· ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.Measure | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 59
} | [
{
"pp": "g : GL (Fin 2) ℝ\ns : Set ℍ\nhs : MeasurableSet s\nhinj : Set.InjOn (fun z ↦ ↑(g • ↑ofComplex z)) (UpperHalfPlane.coe '' s)\nmain :\n ∫⁻ (x : ℂ) in (fun z ↦ ↑(g • ↑ofComplex z)) '' (UpperHalfPlane.coe '' s), ↑((1 / ‖x.im‖₊) ^ 2) =\n ∫⁻ (x : ℂ) in UpperHalfPlane.coe '' s,\n ENNReal.ofReal |(smu... | rcases g.det_ne_zero.lt_or_gt with h | h <;> simp [h] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Geometry.Euclidean.Inversion.Basic | {
"line": 187,
"column": 19
} | {
"line": 187,
"column": 28
} | [
{
"pp": "case inr.inr.inl\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nb c d : P\nhb : b ≠ d\nhc : c ≠ d\n⊢ dist d c * dist b d ≤ dist d b * dist c d + dist b c * 0",
"usedConstants": [
"Eq.mpr",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp | {
"line": 420,
"column": 25
} | {
"line": 420,
"column": 63
} | [
{
"pp": "case inl\nx : ℝ\nh✝ : x ≤ 0\n⊢ |sinh x| = sinh |x|",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"Real.lattice",
"Real.instZero",
"abs",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.t... | simp [abs_of_nonneg, abs_of_nonpos, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Trigonometric.DerivHyp | {
"line": 420,
"column": 25
} | {
"line": 420,
"column": 63
} | [
{
"pp": "case inr\nx : ℝ\nh✝ : 0 ≤ x\n⊢ |sinh x| = sinh |x|",
"usedConstants": [
"Real.instLE",
"Real",
"Real.lattice",
"Real.instZero",
"abs",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Real.sinh",
"Real.instAddGroup",
"LE.le",... | simp [abs_of_nonneg, abs_of_nonpos, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 333,
"column": 8
} | {
"line": 333,
"column": 52
} | [
{
"pp": "z w : ℍ\nr : ℝ\ng : SL(2, ℝ)\ny₁ y₂ : ℍ\n⊢ dist (ModularGroup.S • y₁) (ModularGroup.S • y₂) = dist y₁ y₂",
"usedConstants": [
"Real.partialOrder",
"Real.instLE",
"Real",
"UpperHalfPlane.im_pos",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
... | have h₁ : 0 ≤ im y₁ * im y₂ := by positivity | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Asymptotic | {
"line": 136,
"column": 26
} | {
"line": 136,
"column": 73
} | [
{
"pp": "case mpr\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : ProperSpace 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nh : Meromorphic f\nh₁f : AnalyticOnNhd 𝕜 (toMeromorphicNFOn f univ) univ\n⊢ 0 ≤ MeromorphicOn.divisor f univ",
"usedConstants": ... | ← h.meromorphicOn.divisor_of_toMeromorphicNFOn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Triplewise | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 70
} | [
{
"pp": "case cons.refine_3\nα : Type u_1\nl : List α\np : α → α → α → Prop\nhead : α\ntail : List α\nih : Triplewise p tail ↔ ∀ (i j k : Nat) (hij : i < j) (hjk : j < k) (hk : k < tail.length), p tail[i] tail[j] tail[k]\nh :\n ∀ (i j k : Nat) (hij : i < j) (hjk : j < k) (hk : k < tail.length + 1),\n p (hea... | simpa using h (i + 1) (j + 1) (k + 1) (by lia) (by lia) (by lia) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.List.Triplewise | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 70
} | [
{
"pp": "case cons.refine_3\nα : Type u_1\nl : List α\np : α → α → α → Prop\nhead : α\ntail : List α\nih : Triplewise p tail ↔ ∀ (i j k : Nat) (hij : i < j) (hjk : j < k) (hk : k < tail.length), p tail[i] tail[j] tail[k]\nh :\n ∀ (i j k : Nat) (hij : i < j) (hjk : j < k) (hk : k < tail.length + 1),\n p (hea... | simpa using h (i + 1) (j + 1) (k + 1) (by lia) (by lia) (by lia) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Triplewise | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 70
} | [
{
"pp": "case cons.refine_3\nα : Type u_1\nl : List α\np : α → α → α → Prop\nhead : α\ntail : List α\nih : Triplewise p tail ↔ ∀ (i j k : Nat) (hij : i < j) (hjk : j < k) (hk : k < tail.length), p tail[i] tail[j] tail[k]\nh :\n ∀ (i j k : Nat) (hij : i < j) (hjk : j < k) (hk : k < tail.length + 1),\n p (hea... | simpa using h (i + 1) (j + 1) (k + 1) (by lia) (by lia) (by lia) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Hall.Finite | {
"line": 69,
"column": 6
} | {
"line": 69,
"column": 36
} | [
{
"pp": "case pos\nι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nx : ι\na : α\ns' : Finset ↑{x' | x' ≠ x}\nthis : DecidableEq ι\nha : s'.Nonempty → image (fun z ↦ ↑z) s' ≠ univ → #s' < #((image (fun z ↦ ↑z) s').biUnion t)\nhe : s'.Nonempty\nha' : #s' < #(s'.biUnion fun x_1... | exact Nat.le_sub_one_of_lt ha' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.CategoryTheory.CofilteredSystem | {
"line": 168,
"column": 4
} | {
"line": 170,
"column": 25
} | [
{
"pp": "J : Type u\ninst✝ : Category.{v_1, u} J\nF : J ⥤ Type v\ni j k : J\ns : Set (F.obj i)\nX✝ Y✝ : J\ng : X✝ ⟶ Y✝\nx : F.obj X✝\nh : x ∈ ⋂ f, F.map f ⁻¹' s\n⊢ F.map g x ∈ ⋂ f, F.map f ⁻¹' s",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"c... | rw [mem_iInter] at h ⊢
intro f
simpa using h (g ≫ f) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.CofilteredSystem | {
"line": 168,
"column": 4
} | {
"line": 170,
"column": 25
} | [
{
"pp": "J : Type u\ninst✝ : Category.{v_1, u} J\nF : J ⥤ Type v\ni j k : J\ns : Set (F.obj i)\nX✝ Y✝ : J\ng : X✝ ⟶ Y✝\nx : F.obj X✝\nh : x ∈ ⋂ f, F.map f ⁻¹' s\n⊢ F.map g x ∈ ⋂ f, F.map f ⁻¹' s",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"c... | rw [mem_iInter] at h ⊢
intro f
simpa using h (g ≫ f) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Between | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 44
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nx y z : V\n⊢ Wbtw R (-x) (-y) (-z) ↔ Wbtw R x y z",
"usedConstants": [
"NegZeroClass.toNeg",
"congrArg",
"AddMonoid.toAddZeroClass",
"Wbtw",
"HSub.hSub",
... | simp only [← zero_sub, wbtw_const_sub_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.Between | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 44
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nx y z : V\n⊢ Wbtw R (-x) (-y) (-z) ↔ Wbtw R x y z",
"usedConstants": [
"NegZeroClass.toNeg",
"congrArg",
"AddMonoid.toAddZeroClass",
"Wbtw",
"HSub.hSub",
... | simp only [← zero_sub, wbtw_const_sub_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Between | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 44
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : PartialOrder R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nx y z : V\n⊢ Wbtw R (-x) (-y) (-z) ↔ Wbtw R x y z",
"usedConstants": [
"NegZeroClass.toNeg",
"congrArg",
"AddMonoid.toAddZeroClass",
"Wbtw",
"HSub.hSub",
... | simp only [← zero_sub, wbtw_const_sub_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Between | {
"line": 363,
"column": 82
} | {
"line": 365,
"column": 41
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁴ : Ring R\ninst✝³ : PartialOrder R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : Wbtw R x y z\n⊢ y ∈ affineSpan R {x, z}",
"usedConstants": [
"Semiring.toModule",
"AffineMap.instFunLike",
"Ring.... | by
rcases h with ⟨r, ⟨-, rfl⟩⟩
exact lineMap_mem_affineSpan_pair _ _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Approximation | {
"line": 163,
"column": 20
} | {
"line": 163,
"column": 68
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : LocallyConvexSpace ℝ E\ninst✝ :... | by congr with i x; exact congrFun (hlc i).symm x | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Approximation | {
"line": 230,
"column": 20
} | {
"line": 230,
"column": 68
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nφ : E → ℝ\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : TopologicalSpace E\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module ℝ E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : LocallyConvexSpace ℝ E\ninst✝ : Hereditari... | by congr with i x; exact congrFun (hlc i).symm x | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 96,
"column": 57
} | {
"line": 102,
"column": 38
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nM : Matrix n n R\nx : n → R\nhM : M ∈ rowStochastic R n\nhx : ∀ (i : n), 0 ≤ x i\n⊢ ∀ (j : n), 0 ≤ (M *ᵥ x) j",
"usedConstants": [
"Eq.mpr",
"Is... | by
intro j
simp only [Matrix.mulVec, dotProduct]
apply Finset.sum_nonneg
intro k _
refine Left.mul_nonneg ?_ (hx k)
exact nonneg_of_mem_rowStochastic hM | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 204,
"column": 82
} | {
"line": 207,
"column": 78
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\n⊢ Convex R ↑(colStochastic R n)",
"usedConstants": [
"Matrix.colStochastic",
"Eq.mpr",
"Matrix.smul",
"NonAssocSemiring.toAddCommMon... | by
intro x hx y hy a b ha hb h
simp only [SetLike.mem_coe, mem_colStochastic_iff_sum] at hx hy ⊢
simp [add_nonneg, ha, hb, mul_nonneg, hx, hy, sum_add_distrib, ← mul_sum, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Caratheodory | {
"line": 79,
"column": 62
} | {
"line": 79,
"column": 71
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : DecidableEq E\nt : Finset E\nf : E → 𝕜\nfpos : ∀ y ∈ t, 0 ≤ f y\nfsum : ∑ y ∈ t, f y = 1\ng : E → 𝕜\ngcombo : ∑ e ∈ t, g e • e = 0\ngsum : ∑ e... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Between | {
"line": 869,
"column": 6
} | {
"line": 869,
"column": 87
} | [
{
"pp": "case refine_2.inr\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y : P\nr : R\nhr : 1 ≤ r\n⊢ Wbtw R x y ((lineMap x y) r)",
"usedConstants": [
"Iff.mpr"... | refine ⟨r⁻¹, ⟨inv_nonneg.2 (zero_le_one.trans hr), inv_le_one_of_one_le₀ hr⟩, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Convex.Cone.Dual | {
"line": 112,
"column": 62
} | {
"line": 112,
"column": 72
} | [
{
"pp": "case smul\nR : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\nM : Type u_2\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ns : Set M\nx : N\nhx : x ∈ dual p s\ny✝ : M\nt : R≥0\ny : ... | smul_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Independent | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nι : Type u_3\ninst✝⁴ : Semiring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : Subsingleton ι\np : ι → E\ns : Set ι\nx : ι\nhx : p x ∈ (convexHull 𝕜) (p '' s)\nthis : ((convexHull 𝕜) (p '' s)).Nonempty\n⊢ x ∈ s",
"usedConstants": ... | rw [convexHull_nonempty_iff, Set.image_nonempty] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Intrinsic | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝⁴ : Ring 𝕜\ninst✝³ : AddCommGroup V\ninst✝² : Module 𝕜 V\ninst✝¹ : TopologicalSpace P\ninst✝ : AddTorsor V P\ns : Set P\nh : IsInducing Subtype.val\n⊢ s ⊆ ↑(affineSpan 𝕜 s)",
"usedConstants": [
"subset_affineSpan"
]
}
] | apply subset_affineSpan | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.LinearAlgebra.ConvexSpace | {
"line": 94,
"column": 4
} | {
"line": 95,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝² : PartialOrder R\ninst✝¹ : Semiring R\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝ : IsStrictOrderedRing R\nx y : M\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\n⊢ ((Finsupp.single x s + Finsupp.single y t).sum fun x r ↦ r) = 1",
"usedConstants": [
"Finsupp.instFunLi... | classical
rw [Finsupp.sum_add_index] <;> simp [h] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.LinearAlgebra.ConvexSpace | {
"line": 94,
"column": 4
} | {
"line": 95,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝² : PartialOrder R\ninst✝¹ : Semiring R\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝ : IsStrictOrderedRing R\nx y : M\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\n⊢ ((Finsupp.single x s + Finsupp.single y t).sum fun x r ↦ r) = 1",
"usedConstants": [
"Finsupp.instFunLi... | classical
rw [Finsupp.sum_add_index] <;> simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.ConvexSpace | {
"line": 94,
"column": 4
} | {
"line": 95,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝² : PartialOrder R\ninst✝¹ : Semiring R\nM : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝ : IsStrictOrderedRing R\nx y : M\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\n⊢ ((Finsupp.single x s + Finsupp.single y t).sum fun x r ↦ r) = 1",
"usedConstants": [
"Finsupp.instFunLi... | classical
rw [Finsupp.sum_add_index] <;> simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Piecewise | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_3\ninst✝¹¹ : Semiring 𝕜\ninst✝¹⁰ : PartialOrder 𝕜\ninst✝⁹ : AddCommMonoid E\ninst✝⁸ : LinearOrder E\ninst✝⁷ : IsOrderedAddMonoid E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : PosSMulMono 𝕜 E\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\... | obtain hx | hx := le_or_gt x e | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Convex.SimplicialComplex.Basic | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 37
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns t : Finset E\nhs : s ∈ K.faces\nht : t ∈ K.faces\nh :\n ¬Disjoint ((convexHull 𝕜) ↑s) ((convexHull 𝕜) ↑t) ∧\n ∀ u ∈ K.faces, (convexHull 𝕜) ↑s ∩ (c... | not_disjoint_iff_nonempty_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.SimplicialComplex.Basic | {
"line": 184,
"column": 94
} | {
"line": 192,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns : Finset E\nhs : s ∈ K.faces\n⊢ s ∉ K.facets ↔ ∃ t ∈ K.faces, s ⊂ t",
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Mathlib.Tac... | by
refine ⟨fun hs' : ¬(_ ∧ _) => ?_, ?_⟩
· push Not at hs'
obtain ⟨t, ht⟩ := hs' hs
exact ⟨t, ht.1, ⟨ht.2.1, fun hts => ht.2.2 (Subset.antisymm ht.2.1 hts)⟩⟩
· rintro ⟨t, ht⟩ ⟨hs, hs'⟩
have := hs' ht.1 ht.2.1
rw [this] at ht
exact ht.2.2 (Subset.refl t) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.StrictConvexBetween | {
"line": 42,
"column": 4
} | {
"line": 46,
"column": 98
} | [
{
"pp": "case inr.inr\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : StrictConvexSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : p₂ -ᵥ p ∈ openSegment ℝ (p₁ -ᵥ p) (p₃ -ᵥ p)\nhp₂p₁ : p₂ ≠ p₁\nhp₂p... | rw [openSegment_eq_image, Set.mem_image] at h
rcases h with ⟨r, ⟨hr0, hr1⟩, hr⟩
simp_rw [@dist_eq_norm_vsub V, ← hr]
exact
norm_combo_lt_of_ne (le_max_left _ _) (le_max_right _ _) hp₁p₃ (sub_pos.2 hr1) hr0 (by abel) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.StrictConvexBetween | {
"line": 42,
"column": 4
} | {
"line": 46,
"column": 98
} | [
{
"pp": "case inr.inr\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : StrictConvexSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\np p₁ p₂ p₃ : P\nhp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p\nh : p₂ -ᵥ p ∈ openSegment ℝ (p₁ -ᵥ p) (p₃ -ᵥ p)\nhp₂p₁ : p₂ ≠ p₁\nhp₂p... | rw [openSegment_eq_image, Set.mem_image] at h
rcases h with ⟨r, ⟨hr0, hr1⟩, hr⟩
simp_rw [@dist_eq_norm_vsub V, ← hr]
exact
norm_combo_lt_of_ne (le_max_left _ _) (le_max_right _ _) hp₁p₃ (sub_pos.2 hr1) hr0 (by abel) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.StoneSeparation | {
"line": 44,
"column": 2
} | {
"line": 46,
"column": 72
} | [
{
"pp": "case inr.inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np q u x y : E\nhu : u ∈ segment 𝕜 x p\naz bz : 𝕜\nhaz : 0 ≤ az\nhbz : 0 ≤ bz\nhabz : az + bz = 1\nhaz' : 0 < az\nbv : 𝕜\nhbv : 0 ≤ bv\... | · rw [zero_add] at habv
rw [zero_smul, zero_add, habv, one_smul]
exact ⟨q, right_mem_segment _ _ _, subset_convexHull _ _ <| by simp⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.StrictCombination | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 95
} | [
{
"pp": "case pos\nR : Type u_1\nV : Type u_2\nι : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : TopologicalSpace V\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ns : Set V\nw : ι → R\nz : ι → V\nhs : StrictConvex R s\ni : ι\nt : Finset ι\nhi✝ : i ∉ t\nht :\n (∀ i ... | simp only [hzi, ← add_smul, ← add_div, ne_eq, hwi, not_false_eq_true, div_self, one_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.Side | {
"line": 716,
"column": 6
} | {
"line": 717,
"column": 37
} | [
{
"pp": "case h.mp.inr.inl\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx p : P\nhx : x ∉ s\nhp : p ∈ s\ny : P\nhy : y ∉ s\np₂ : P\nhp₂ : p₂ ∈ s\nh... | rw [vsub_eq_zero_iff_eq] at h
exact False.elim (hy (h ▸ hp₂)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Side | {
"line": 716,
"column": 6
} | {
"line": 717,
"column": 37
} | [
{
"pp": "case h.mp.inr.inl\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx p : P\nhx : x ∉ s\nhp : p ∈ s\ny : P\nhy : y ∉ s\np₂ : P\nhp₂ : p₂ ∈ s\nh... | rw [vsub_eq_zero_iff_eq] at h
exact False.elim (hy (h ▸ hp₂)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Distribution.TestFunction | {
"line": 469,
"column": 2
} | {
"line": 469,
"column": 19
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nΩ₁ Ω₂ : Opens E\nF : Type u_4\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedSpace 𝕜 F\nn₁ n₂ : ℕ∞\ninst✝¹ : Algebra ℝ 𝕜\ninst✝ : IsScalarTower ℝ 𝕜 F\n... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Distribution.TestFunction | {
"line": 505,
"column": 2
} | {
"line": 505,
"column": 19
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nΩ : Opens E\nF : Type u_4\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedSpace 𝕜 F\nn k : ℕ∞\ninst✝¹ : Algebra ℝ 𝕜\ninst✝ : IsScalarTower ℝ 𝕜 F\nf : 𝓓... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 426,
"column": 4
} | {
"line": 426,
"column": 40
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ᶠ[ae μ] f\nf_mble : AEMeasurable f μ\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t\nG : ℝ → ℝ\nG_mble : Measurable G\nG_nn : 0 ≤ G\ng_eq_G : g =ᶠ[ae (volum... | filter_upwards [f_eq_F] with ω fω_nn | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 193,
"column": 65
} | {
"line": 193,
"column": 80
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\n𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nx✝¹ x✝ : ↥(Lp E 2 μ)\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError ... | inner_conj_symm | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 81,
"column": 2
} | {
"line": 89,
"column": 48
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nhf_temperate : HasTemperateGrowth f\nn : ℕ\n⊢ ∃ k C, 0 ≤ C ∧ ∀ N ≤ n, ∀ (x : E), ‖iteratedFDeriv ℝ N f x‖ ≤ C * (1 + ‖x‖) ^ k",
"usedConstants": [
... | rcases hf_temperate.isBigO_uniform n with ⟨k, hk⟩
set F := fun x (N : Fin (n + 1)) ↦ iteratedFDeriv ℝ N f x
have : F =O[⊤] (fun x ↦ (1 + ‖x‖) ^ k) := by
simp_rw [F, isBigO_pi, Fin.forall_iff, Nat.lt_succ_iff]
exact hk
rcases this.exists_nonneg with ⟨C, C_nonneg, hC⟩
simp (discharger := positivity) only ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 81,
"column": 2
} | {
"line": 89,
"column": 48
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nhf_temperate : HasTemperateGrowth f\nn : ℕ\n⊢ ∃ k C, 0 ≤ C ∧ ∀ N ≤ n, ∀ (x : E), ‖iteratedFDeriv ℝ N f x‖ ≤ C * (1 + ‖x‖) ^ k",
"usedConstants": [
... | rcases hf_temperate.isBigO_uniform n with ⟨k, hk⟩
set F := fun x (N : Fin (n + 1)) ↦ iteratedFDeriv ℝ N f x
have : F =O[⊤] (fun x ↦ (1 + ‖x‖) ^ k) := by
simp_rw [F, isBigO_pi, Fin.forall_iff, Nat.lt_succ_iff]
exact hk
rcases this.exists_nonneg with ⟨C, C_nonneg, hC⟩
simp (discharger := positivity) only ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 70
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nh'f : HasTemperateGrowth (fderiv ℝ f)\nhf : Differentiable ℝ f\nk : ℕ\nC : ℝ\nh : ∀ (x : E), ‖f x‖ ≤ C * (1 + ‖x‖) ^ k\n⊢ HasTemperateGrowth f",
"u... | refine ⟨contDiff_succ_iff_fderiv.2 ⟨hf, by simp, h'f.1⟩, fun n ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 218,
"column": 25
} | {
"line": 218,
"column": 39
} | [
{
"pp": "ι : Type u_1\nE : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ι → E → F\na : ι\ns : Finset ι\nhas : a ∉ s\nih : (∀ i ∈ s, HasTemperateGrowth (f i)) → HasTemperateGrowth fun x ↦ ∑ i ∈ s, f i x\nhf : ∀ i ∈ in... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 218,
"column": 25
} | {
"line": 218,
"column": 39
} | [
{
"pp": "ι : Type u_1\nE : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ι → E → F\na : ι\ns : Finset ι\nhas : a ∉ s\nih : (∀ i ∈ s, HasTemperateGrowth (f i)) → HasTemperateGrowth fun x ↦ ∑ i ∈ s, f i x\nhf : ∀ i ∈ in... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 218,
"column": 25
} | {
"line": 218,
"column": 39
} | [
{
"pp": "ι : Type u_1\nE : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ι → E → F\na : ι\ns : Finset ι\nhas : a ∉ s\nih : (∀ i ∈ s, HasTemperateGrowth (f i)) → HasTemperateGrowth fun x ↦ ∑ i ∈ s, f i x\nhf : ∀ i ∈ in... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 154,
"column": 74
} | {
"line": 154,
"column": 83
} | [
{
"pp": "T : ℝ\nn : ℤ\n⊢ Complex.exp (2 * ↑π * Complex.I * ↑n * 0 / ↑T) = 1",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"Nat.instAtLeast... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 182,
"column": 6
} | {
"line": 182,
"column": 37
} | [
{
"pp": "T : ℝ\ninst✝ : Fact (0 < T)\nn : ℤ\n⊢ ‖fourier n‖ = 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"ContinuousMap.instNorm",
"congrArg",
"iSup",
"ContinuousMap",
"ContinuousMap.norm_eq_iSup_norm",
... | ContinuousMap.norm_eq_iSup_norm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 331,
"column": 30
} | {
"line": 331,
"column": 44
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nι : Type u_2\nf : ι → AddCircle T → E\na : ι\ns : Finset ι\nha : a ∉ s\niha : (∀ i ∈ s, Integrable (f i) haarAddCircle) → fourierCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, fourierCoeff (f i)\nhf : ∀ i ∈ insert a s, Int... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 331,
"column": 30
} | {
"line": 331,
"column": 44
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nι : Type u_2\nf : ι → AddCircle T → E\na : ι\ns : Finset ι\nha : a ∉ s\niha : (∀ i ∈ s, Integrable (f i) haarAddCircle) → fourierCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, fourierCoeff (f i)\nhf : ∀ i ∈ insert a s, Int... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 331,
"column": 30
} | {
"line": 331,
"column": 44
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nι : Type u_2\nf : ι → AddCircle T → E\na : ι\ns : Finset ι\nha : a ∉ s\niha : (∀ i ∈ s, Integrable (f i) haarAddCircle) → fourierCoeff (∑ i ∈ s, f i) = ∑ i ∈ s, fourierCoeff (f i)\nhf : ∀ i ∈ insert a s, Int... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 339,
"column": 4
} | {
"line": 339,
"column": 15
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : AddCircle T → E\nc : ℂ\nn : ℤ\n⊢ ∫ (t : AddCircle T), (c • (fourier (-n)) t) • f t ∂haarAddCircle =\n c • ∫ (t : AddCircle T), (fourier (-n)) t • f t ∂haarAddCircle",
"usedConstants": [
"Non... | smul_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 350,
"column": 2
} | {
"line": 350,
"column": 19
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn₁ n₂ : ℕ∞\nK₁ K₂ : Compacts E\nf : 𝓓^{n₁}_{K₁}(E, ... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 393,
"column": 2
} | {
"line": 393,
"column": 19
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn k : ℕ∞\nK : Compacts E\nf : 𝓓^{n}_{K}(E, F)\n⊢ ⇑(... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 449,
"column": 2
} | {
"line": 449,
"column": 19
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nn k : ℕ∞\nK : Compacts E\ni : ℕ\nf : 𝓓^{n}_{K}(E, F... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 50
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝³ : TopologicalSpace α\ninst✝² : BorelSpace α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns t : Set α\nφ : ι → α → ℝ\nhs : MeasurableSet s\nht : MeasurableSet t\nhts : t ⊆... | apply Tendsto.mono_left _ nhdsWithin_le_nhds | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 308,
"column": 12
} | {
"line": 308,
"column": 56
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : BorelSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝² : CompleteSpace E\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nhs : IsCo... | · exact (I n).mono inter_subset_right le_rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Lebesgue.Integral | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 53
} | [
{
"pp": "f : ℝ → ℝ\neq : ∫ (x : ℝ) in Ioi 0, f |x| = ∫ (x : ℝ) in Ioi 0, f x\n⊢ ∫ (x : ℝ), f |x| = 2 * ∫ (x : ℝ) in Ioi 0, f x",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Set.Ioi",
"HMul.hMul",
"Real.lattice",
... | by_cases hf : IntegrableOn (fun x => f |x|) (Ioi 0) | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Fourier.FourierTransform | {
"line": 468,
"column": 11
} | {
"line": 468,
"column": 25
} | [
{
"pp": "V : Type u_1\nE : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpace V\ninst✝ : FiniteDimensional ℝ V\nf : V → E\nw : V\n⊢ 𝓕⁻ f w = ∫ (v : V), Complex.exp (↑(2 * π * ⟪v, w⟫) *... | fourierInv_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 89
} | [
{
"pp": "a : ℂ\nha : a.re < -1\nc : ℝ\nhc : 0 < c\nthis : Tendsto (fun x ↦ (↑x ^ (a + 1) - ↑c ^ (a + 1)) / (a + 1)) atTop (𝓝 (-↑c ^ (a + 1) / (a + 1)))\n⊢ Tendsto (fun i ↦ ∫ (x : ℝ) in c..id i, ↑x ^ a) atTop (𝓝 (-↑c ^ (a + 1) / (a + 1)))",
"usedConstants": [
"instInnerProductSpaceRealComplex",
... | refine this.congr' ((eventually_gt_atTop 0).mp (Eventually.of_forall fun x hx => ?_)) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 388,
"column": 80
} | {
"line": 388,
"column": 89
} | [
{
"pp": "a : ℂ\nr : ℝ\nha : 0 < a.re\nhr : 0 < r\naux : (1 / ↑r) ^ a = 1 / ↑r * (1 / ↑r) ^ (a - 1)\n⊢ r⁻¹ • ∫ (x : ℝ) in Ioi (r * 0), (1 / ↑r) ^ (a - 1) * ↑x ^ (a - 1) * cexp (-↑x) =\n 1 / ↑r * ∫ (t : ℝ) in Ioi 0, (1 / ↑r) ^ (a - 1) * ↑t ^ (a - 1) * cexp (-↑t)",
"usedConstants": [
"instInnerProduct... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.PolarCoord | {
"line": 87,
"column": 4
} | {
"line": 87,
"column": 60
} | [
{
"pp": "case refine_1\nA : MapsTo (⇑Complex.equivRealProd.symm) ({q | 0 < q.1} ∪ {q | q.2 ≠ 0}) Complex.slitPlane\nz : ℂ\nhz : z ∈ Complex.slitPlane\n⊢ ContinuousWithinAt Complex.arg Complex.slitPlane z",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Complex.continuou... | · exact (Complex.continuousAt_arg hz).continuousWithinAt | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 41,
"column": 4
} | {
"line": 41,
"column": 28
} | [
{
"pp": "p b : ℝ\nhb : 0 < b\nhp : 1 < p\nthis : Tendsto (fun x ↦ x * (b * x ^ (p - 1) + -1)) atTop atTop\nx : ℝ\nhx : 0 < x\n⊢ x * (b * x ^ (p - 1) + -1) = -x - -b * x ^ p",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"HMul.hMul",
"Real.instZero",
... | rw [rpow_sub_one hx.ne'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 334,
"column": 4
} | {
"line": 334,
"column": 29
} | [
{
"pp": "case h.e'_2\n⊢ ∫ (x : ℝ) in Ioi 0, (2 * x ^ (2 - 1)) • (rexp (-x ^ 2) * (x ^ 2) ^ (1 / 2 - 1)) =\n 2 * ∫ (x : ℝ) in Ioi 0, rexp (-1 * x ^ 2)",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCom... | rw [← integral_const_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 225,
"column": 17
} | {
"line": 225,
"column": 43
} | [
{
"pp": "case h\nb : ℂ\nhb : 0 < b.re\nc : ℂ\nthis✝ : b ≠ 0\nh : (-↑π * b).re < 0\nt : ℝ\nthis :\n ∀ (x : ℝ),\n ↑(-2 * π * x * t) * I + -↑π * b * ↑x ^ 2 + 2 * ↑π * c * ↑x =\n -↑π * b * ↑x ^ 2 + (-2 * ↑π * I * ↑t + 2 * ↑π * c) * ↑x + 0\n⊢ ∫ (v : ℝ), cexp (-↑π * b * ↑v ^ 2 + (-2 * ↑π * I * ↑t + 2 * ↑π * ... | integral_cexp_quadratic h, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 235,
"column": 8
} | {
"line": 235,
"column": 35
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ :... | use f.seminormAux k n * ‖c‖ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 458,
"column": 2
} | {
"line": 458,
"column": 43
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nf : 𝓢(E, F)\nx₀ : E\n⊢ ‖f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 0 0) f... | have := norm_pow_mul_le_seminorm 𝕜 f 0 x₀ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 458,
"column": 2
} | {
"line": 459,
"column": 33
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nf : 𝓢(E, F)\nx₀ : E\n⊢ ‖f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 0 0) f... | have := norm_pow_mul_le_seminorm 𝕜 f 0 x₀
rwa [pow_zero, one_mul] at this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 458,
"column": 2
} | {
"line": 459,
"column": 33
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : SMulCommClass ℝ 𝕜 F\nf : 𝓢(E, F)\nx₀ : E\n⊢ ‖f x₀‖ ≤ (SchwartzMap.seminorm 𝕜 0 0) f... | have := norm_pow_mul_le_seminorm 𝕜 f 0 x₀
rwa [pow_zero, one_mul] at this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar | {
"line": 119,
"column": 2
} | {
"line": 120,
"column": 7
} | [
{
"pp": "V : Type u_1\nW : Type u_2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module ℝ V\ninst✝³ : TopologicalSpace V\ninst✝² : AddCommGroup W\ninst✝¹ : Module ℝ W\ninst✝ : TopologicalSpace W\ne : AddChar ℝ Circle\nL : V →ₗ[ℝ] W →ₗ[ℝ] ℝ\nhe : Continuous ⇑e\nhL : Continuous fun p ↦ (L p.1) p.2\nw : AddMonoidAlgebra ℂ W... | · simp only [coe_sum, coe_smul, charMonoidHom_apply, smul_eq_mul, Finset.sum_apply]
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.FiniteAbelian.Basic | {
"line": 100,
"column": 2
} | {
"line": 101,
"column": 63
} | [
{
"pp": "M : Type u\ninst✝² : AddCommGroup M\ninst✝¹ : Module ℤ M\ninst✝ : Module.Finite ℤ M\nhM : IsTorsion ℤ M\nι : Type\nw✝ : Fintype ι\np : ι → ℤ\nh : ∀ (i : ι), Irreducible (p i)\ne : ι → ℕ\nl : M ≃ₗ[ℤ] ⨁ (i : ι), ℤ ⧸ ℤ ∙ p i ^ e i\n⊢ Finite M",
"usedConstants": [
"Iff.mpr",
"Int.instIsStri... | haveI : ∀ i : ι, NeZero (p i ^ e i).natAbs := fun i =>
⟨Int.natAbs_ne_zero.mpr <| pow_ne_zero (e i) (h i).ne_zero⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.Analysis.Fourier.RiemannLebesgueLemma | {
"line": 74,
"column": 84
} | {
"line": 74,
"column": 92
} | [
{
"pp": "case h\nE : Type u_1\nV : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\nf : V → E\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : MeasurableSpace V\ninst✝² : BorelSpace V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\nw : V\nhw : w ≠ 0\nhiw : ⟪i w, w⟫ = 1 / 2\nv : V\n⊢ ce... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
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