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 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Complex.PhragmenLindelof | {
"line": 824,
"column": 4
} | {
"line": 824,
"column": 41
} | {
"line": 825,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : ℂ → E\nhfd : DiffContOnCl ℂ f {z | 0 < z.re}\nhgd : DiffContOnCl ℂ g {z | 0 < z.re}\nhfexp : ∃ c < 2, ∃ B, f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z ↦ expR (B * ‖z‖ ^ c)\nhgexp : ∃ c < 2, ∃ B, g =O[cobounded ℂ ... | [] | exact isBigO_sub_exp_rpow hfexp hgexp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Polynomial.UnitTrinomial | {
"line": 34,
"column": 4
} | {
"line": 34,
"column": 36
} | {
"line": 35,
"column": 4
} | [
{
"pp": "q : ℤ[X]\nthis : ¬0 < q.natDegree\np : ℤ[X]\nh : ∀ (z : ℂ), ¬((aeval z) (q * p) = 0 ∧ (aeval z) (q * p).mirror = 0)\nhq' : q ∣ (q * p).mirror\nhp : IsUnit q.leadingCoeff\n⊢ IsUnit q",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Preorder.toLT",
... | [
"q : ℤ[X]\nthis : q.natDegree = 0\np : ℤ[X]\nh : ∀ (z : ℂ), ¬((aeval z) (q * p) = 0 ∧ (aeval z) (q * p).mirror = 0)\nhq' : q ∣ (q * p).mirror\nhp : IsUnit q.leadingCoeff\n⊢ IsUnit q"
] | rw [not_lt, Nat.le_zero] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Monotone.Union | {
"line": 37,
"column": 44
} | {
"line": 37,
"column": 57
} | {
"line": 37,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\nf : α → β\ns t : Set α\nc : α\nh₁ : StrictMonoOn f s\nh₂ : StrictMonoOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : α\nhxc : x ≤ c\nh✝ : x ∈ t\nh'x : x < c\n⊢ x ∈ t",
"ppTerm": "?m.110",
"assigned": true,
"usedConst... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Monotone.Union | {
"line": 85,
"column": 44
} | {
"line": 85,
"column": 57
} | {
"line": 85,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\nf : α → β\ns t : Set α\nc : α\nh₁ : MonotoneOn f s\nh₂ : MonotoneOn f t\nhs : IsGreatest s c\nht : IsLeast t c\nx : α\nhxc : x ≤ c\nh✝ : x ∈ t\nh'x : x < c\n⊢ x ∈ t",
"ppTerm": "?m.110",
"assigned": true,
"usedConstants... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 345,
"column": 6
} | {
"line": 345,
"column": 31
} | {
"line": 346,
"column": 6
} | [
{
"pp": "c : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := closedBall c |R|\nh₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₃f : (divisor f CB).support.Finite\ng : ℂ → ℂ\nh₁g : AnalyticOnNhd ℂ g (closedBall c |R|)\nh₂g : ∀ (u : ↑(closedBall c |R|)), g ↑u ≠ 0\nh₃g : f =ᶠ[... | [
"c : ℂ\nR : ℝ\nf : ℂ → ℂ\nhR : R ≠ 0\nh₁f : MeromorphicOn f (closedBall c |R|)\nCB : Set ℂ := closedBall c |R|\nh₂f : ∀ u ∈ CB, meromorphicOrderAt f u ≠ ⊤\nh₃f : (divisor f CB).support.Finite\ng : ℂ → ℂ\nh₁g : AnalyticOnNhd ℂ g (closedBall c |R|)\nh₂g : ∀ (u : ↑(closedBall c |R|)), g ↑u ≠ 0\nh₃g : f =ᶠ[codiscreteWi... | rw [← finsum_sub_distrib] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.Exp | {
"line": 44,
"column": 51
} | {
"line": 45,
"column": 86
} | {
"line": 46,
"column": 0
} | [
{
"pp": "τ : ℍ\n⊢ ‖cexp (2 * ↑π * Complex.I * ↑τ)‖ < 1",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Function.Periodic.qParam",
"instHDiv... | [] | by
simpa [Function.Periodic.norm_qParam, Complex.norm_exp] using τ.norm_qParam_lt_one 1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.TietzeExtension | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 39
} | {
"line": 177,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\ne : C(X, Y)\nhe : IsClosedEmbedding ⇑e\n⊢ ∃ g, ‖g‖ ≤ ‖f‖ / 3 ∧ dist (g.compContinuous e) f ≤ 2 / 3 * ‖f‖",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"... | [
"X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\ne : C(X, Y)\nhe : IsClosedEmbedding ⇑e\nh3 : 0 < 3\n⊢ ∃ g, ‖g‖ ≤ ‖f‖ / 3 ∧ dist (g.compContinuous e) f ≤ 2 / 3 * ‖f‖"
] | have h3 : (0 : ℝ) < 3 := by norm_num1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 158,
"column": 4
} | {
"line": 158,
"column": 74
} | {
"line": 159,
"column": 2
} | [
{
"pp": "case inr\ng : GL (Fin 2) ℝ\nz : ℍ\nhpos : 0 < (↑g).det\nhell : g.IsElliptic\nthis : ∀ {g : GL (Fin 2) ℝ}, 0 < (↑g).det → ∀ (hell : g.IsElliptic), 0 < ↑g 1 0 → (g • z = z ↔ z = fixedPt g hell)\nhc : ↑g 1 0 < 0\n⊢ g • z = z ↔ z = fixedPt g hell",
"ppTerm": "?inr",
"assigned": true,
"usedConst... | [] | simpa using @this (-g) (by simpa [Matrix.det_neg]) hell.neg (by simpa) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.TietzeExtension | {
"line": 304,
"column": 4
} | {
"line": 304,
"column": 51
} | {
"line": 305,
"column": 2
} | [
{
"pp": "case refine_1\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : X →ᵇ ℝ\na b : ℝ\ne : X → Y\nhf : ∀ (x : X), f x ∈ Icc a b\nhle : a ≤ b\nhe : IsClosedEmbedding e\ng : Y →ᵇ ℝ\nhgf : ‖g‖ = ‖f - const X ((a + b) / 2)‖\nhge : ⇑g ∘ e = ⇑(f - con... | [] | simpa only [Real.Icc_eq_closedBall] using! hf x | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.Complex | {
"line": 22,
"column": 6
} | {
"line": 22,
"column": 40
} | {
"line": 22,
"column": 41
} | [
{
"pp": "z : ℂ\ni j : Fin 2\n⊢ (leftMulMatrix basisOneI) z i j = !![z.re, -z.im; z.im, z.re] i j",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Complex.instAlgebraOfReal",
"Real",
"Semiring.toModule",
"Equiv.instEquivLi... | [
"z : ℂ\ni j : Fin 2\n⊢ (basisOneI.repr (z * basisOneI j)) i = !![z.re, -z.im; z.im, z.re] i j"
] | Algebra.leftMulMatrix_eq_repr_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 121,
"column": 2
} | {
"line": 126,
"column": 40
} | {
"line": 128,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nn : Type u_4\nm : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Matrix m m S\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\nk : m\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nf : S →+* Matrix n n R\n⊢ eval 0 ((Matrix.comp m m n n R[X]) ((cornerAddX M k).map ⇑f.polyToM... | [] | simp_rw [← coe_evalRingHom, RingHom.map_det, ← compRingEquiv_apply, ← RingEquiv.coe_toRingHom,
← RingHom.mapMatrix_apply, ← RingHom.comp_apply, ← RingHom.comp_assoc,
evalRingHom_mapMatrix_comp_compRingEquiv, RingHom.comp_assoc, RingHom.mapMatrix_comp,
evalRingHom_mapMatrix_comp_polyToMatrix, ← RingHom.mapMa... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 121,
"column": 2
} | {
"line": 126,
"column": 40
} | {
"line": 128,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nn : Type u_4\nm : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Matrix m m S\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\nk : m\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nf : S →+* Matrix n n R\n⊢ eval 0 ((Matrix.comp m m n n R[X]) ((cornerAddX M k).map ⇑f.polyToM... | [] | simp_rw [← coe_evalRingHom, RingHom.map_det, ← compRingEquiv_apply, ← RingEquiv.coe_toRingHom,
← RingHom.mapMatrix_apply, ← RingHom.comp_apply, ← RingHom.comp_assoc,
evalRingHom_mapMatrix_comp_compRingEquiv, RingHom.comp_assoc, RingHom.mapMatrix_comp,
evalRingHom_mapMatrix_comp_polyToMatrix, ← RingHom.mapMa... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 22
} | {
"line": 58,
"column": 0
} | [
{
"pp": "z w : ℍ\n⊢ 0 ≤ z.im * w.im",
"ppTerm": "?m.71",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"UpperHalfPlane.im_pos",
"HMul.hMul",
"Real.instZero",
"PartialOrder.toPreorder",
"le_of_lt",
"Real.semiring",
"mul_pos",
... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 78,
"column": 2
} | {
"line": 79,
"column": 15
} | {
"line": 79,
"column": 16
} | [
{
"pp": "case e_a.e_a\na b c : ℍ\n⊢ √(a.im * b.im) * √(b.im * c.im) = √(a.im * c.im) * b.im",
"ppTerm": "?e_a.e_a✝",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommR... | [
"case e_a.e_a\na b c : ℍ\n⊢ 0 ≤ b.im",
"case e_a.e_a.hx\na b c : ℍ\n⊢ 0 ≤ a.im",
"case e_a.e_a.hx\na b c : ℍ\n⊢ 0 ≤ b.im",
"case e_a.e_a.hx\na b c : ℍ\n⊢ 0 ≤ a.im"
] | rw [sqrt_mul, sqrt_mul, sqrt_mul, mul_comm (√a.im), mul_mul_mul_comm, mul_self_sqrt,
mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 22
} | {
"line": 98,
"column": 0
} | [
{
"pp": "z w : ℍ\nr : ℝ\nhr : 0 ≤ r\n⊢ 0 ≤ z.im * w.im",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"UpperHalfPlane.im_pos",
"HMul.hMul",
"Real.instZero",
"PartialOrder.toPreorder",
"le_of_lt",
"Real.semiring",... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 149,
"column": 2
} | {
"line": 150,
"column": 55
} | {
"line": 152,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nn : Type u_4\nm : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Matrix m m S\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\nk : m\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nf : S →+* Matrix n n R\nih :\n ∀ (M : Matrix { a // (a = k) = False } { a // (a = k) = False... | [] | rw [sub_mul, comp_det_mul_pow, ← det_pow, ← map_pow, ← det_mul, ← map_mul,
det_mul_corner_pow, map_mul, det_mul, ih, sub_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 149,
"column": 2
} | {
"line": 150,
"column": 55
} | {
"line": 152,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nn : Type u_4\nm : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Matrix m m S\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\nk : m\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nf : S →+* Matrix n n R\nih :\n ∀ (M : Matrix { a // (a = k) = False } { a // (a = k) = False... | [] | rw [sub_mul, comp_det_mul_pow, ← det_pow, ← map_pow, ← det_mul, ← map_mul,
det_mul_corner_pow, map_mul, det_mul, ih, sub_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 149,
"column": 2
} | {
"line": 150,
"column": 55
} | {
"line": 152,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nn : Type u_4\nm : Type u_5\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\nM : Matrix m m S\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\nk : m\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nf : S →+* Matrix n n R\nih :\n ∀ (M : Matrix { a // (a = k) = False } { a // (a = k) = False... | [] | rw [sub_mul, comp_det_mul_pow, ← det_pow, ← map_pow, ← det_mul, ← map_mul,
det_mul_corner_pow, map_mul, det_mul, ih, sub_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.UpperHalfPlane.Metric | {
"line": 333,
"column": 44
} | {
"line": 333,
"column": 73
} | {
"line": 334,
"column": 8
} | [
{
"pp": "z w : ℍ\nr : ℝ\ng : SL(2, ℝ)\ny₁ y₂ : ℍ\nh₁ : 0 ≤ y₁.im * y₂.im\n⊢ ‖↑y₁ * ↑y₂‖ ≠ 0",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"False",
"Real",
"HMul.hMul",
"C... | [] | simp [y₁.ne_zero, y₂.ne_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Category.CompHaus.Basic | {
"line": 99,
"column": 6
} | {
"line": 101,
"column": 36
} | {
"line": 102,
"column": 2
} | [
{
"pp": "X : TopCat\nY : CompHaus\nf : ↑(stoneCechObj X).toTop → ↑Y.toTop\nhf : Continuous f\nx : ↑(of (fun x ↦ True) (StoneCech ↑X)).toTop\n⊢ Set.EqOn (stoneCechExtend ⋯) f (Set.range stoneCechUnit)",
"ppTerm": "?m.162",
"assigned": true,
"usedConstants": [
"continuous_stoneCechUnit",
"... | [] | rintro _ ⟨y, rfl⟩
apply congr_fun (stoneCechExtend_extends (hf.comp _)) y
apply continuous_stoneCechUnit | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.CompHaus.Basic | {
"line": 99,
"column": 6
} | {
"line": 101,
"column": 36
} | {
"line": 102,
"column": 2
} | [
{
"pp": "X : TopCat\nY : CompHaus\nf : ↑(stoneCechObj X).toTop → ↑Y.toTop\nhf : Continuous f\nx : ↑(of (fun x ↦ True) (StoneCech ↑X)).toTop\n⊢ Set.EqOn (stoneCechExtend ⋯) f (Set.range stoneCechUnit)",
"ppTerm": "?m.162",
"assigned": true,
"usedConstants": [
"continuous_stoneCechUnit",
"... | [] | rintro _ ⟨y, rfl⟩
apply congr_fun (stoneCechExtend_extends (hf.comp _)) y
apply continuous_stoneCechUnit | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Category.CompHaus.Basic | {
"line": 142,
"column": 8
} | {
"line": 142,
"column": 32
} | {
"line": 143,
"column": 8
} | [
{
"pp": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ CompHaus\nFF : J ⥤ TopCat := F ⋙ compHausToTop\n⊢ IsCompact {u | ∀ {i j : J} (f : i ⟶ j), (ConcreteCategory.hom (F.map f)) (u i) = u j}",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"Pi.topologicalSpace",
"CategoryTheory.... | [
"J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ CompHaus\nFF : J ⥤ TopCat := ⋯\n⊢ IsClosed {u | ∀ {i j : J} (f : i ⟶ j), (ConcreteCategory.hom (F.map f)) (u i) = u j}"
] | apply IsClosed.isCompact | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Category.CompHaus.Basic | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 85
} | {
"line": 197,
"column": 4
} | [
{
"pp": "case mp\nX Y : CompHaus\nf : X ⟶ Y\ny : ↑Y.toTop\nhy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y\nhf : Epi f\nC : Set ((fun X ↦ ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)\nhC : IsClosed C\nD : Set ↑Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nφ : C((fun X ↦ ↑X.toTop) Y, ℝ)\nhφ0 ... | [
"case mp\nX Y : CompHaus\nf : X ⟶ Y\ny : ↑Y.toTop\nhy : ∀ (a : ↑X.toTop), (ConcreteCategory.hom f) a ≠ y\nhf : Epi f\nC : Set ((fun X ↦ ↑X.toTop) Y) := Set.range ⇑(ConcreteCategory.hom f)\nhC : IsClosed C\nD : Set ↑Y.toTop := {y}\nhD : IsClosed D\nhCD : Disjoint C D\nφ : C((fun X ↦ ↑X.toTop) Y, ℝ)\nhφ0 : Set.EqOn (... | haveI : T2Space (ULift.{u} <| Set.Icc (0 : ℝ) 1) := Homeomorph.ulift.symm.t2Space | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 48
} | {
"line": 218,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\nS : Set (Set (OnePoint X))\nho : ∀ t ∈ S, (∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' t)\n⊢ IsOpen[inst✝] (⋃ t ∈ S, some ⁻¹' t)",
"ppTerm": "?m.135",
"assigned": true,
"usedConstants": [
"OnePoint.infty",
"C... | [] | exact isOpen_biUnion fun s hs => (ho s hs).2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 57
} | {
"line": 325,
"column": 0
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\n⊢ Tendsto some (coclosedCompact X) (map some (coclosedCompact X) ⊔ pure ∞)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Lattice.toSemilatticeSup",
"OnePoint.infty",
"le_sup_left",
"Filter.map"... | [] | exact Filter.Tendsto.mono_right tendsto_map le_sup_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Compactness.CompactlyGeneratedSpace | {
"line": 369,
"column": 2
} | {
"line": 374,
"column": 30
} | {
"line": 376,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : CompactlyCoherentSpace X\n⊢ CompactlyGeneratedSpace X",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CompactlyCoherentSpace.isClos... | [] | apply compactlyGeneratedSpace_of_isClosed_of_t2
intro s hs
rw [CompactlyCoherentSpace.isClosed_iff]
intro K hK
rw [← Subtype.preimage_coe_inter_self]
exact (hs K hK).preimage_val | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.CompactlyGeneratedSpace | {
"line": 369,
"column": 2
} | {
"line": 374,
"column": 30
} | {
"line": 376,
"column": 0
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space X\ninst✝ : CompactlyCoherentSpace X\n⊢ CompactlyGeneratedSpace X",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CompactlyCoherentSpace.isClos... | [] | apply compactlyGeneratedSpace_of_isClosed_of_t2
intro s hs
rw [CompactlyCoherentSpace.isClosed_iff]
intro K hK
rw [← Subtype.preimage_coe_inter_self]
exact (hs K hK).preimage_val | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.AmpleSet | {
"line": 72,
"column": 2
} | {
"line": 79,
"column": 53
} | {
"line": 81,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝² : AddCommGroup F\ninst✝¹ : Module ℝ F\ninst✝ : TopologicalSpace F\ns t : Set F\nhs : AmpleSet s\nht : AmpleSet t\nx : F\nhx : x ∈ s ∪ t\n⊢ (convexHull ℝ) (connectedComponentIn (s ∪ t) x) = univ",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.univ_su... | [] | rcases hx with (h | h) <;>
-- The connected component of `x ∈ s` in `s ∪ t` contains the connected component of `x` in `s`,
-- hence is also full; similarly for `t`.
[have hx := hs x h; have hx := ht x h] <;>
rw [← Set.univ_subset_iff, ← hx] <;>
apply convexHull_mono <;>
apply connectedComponentIn_mono <;>
... | Batteries.Tactic._aux_Batteries_Tactic_SeqFocus___macroRules_Batteries_Tactic_seq_focus_1 | Batteries.Tactic.seq_focus |
Mathlib.Topology.Semicontinuity.Lindelof | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 60
} | {
"line": 64,
"column": 2
} | [
{
"pp": "X : Type u_1\nE : Type u_2\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : HereditarilyLindelofSpace X\ninst✝⁴ : LinearOrder E\ninst✝³ : TopologicalSpace E\ninst✝² : OrderClosedTopology E\ninst✝¹ : DenselyOrdered E\ninst✝ : SeparableSpace E\ns : X → E\n𝓕 : Set (X → E)\nh𝓕_cont : ∀ f ∈ 𝓕, UpperSemicontinuous ... | [
"X : Type u_1\nE : Type u_2\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : HereditarilyLindelofSpace X\ninst✝⁴ : LinearOrder E\ninst✝³ : TopologicalSpace E\ninst✝² : OrderClosedTopology E\ninst✝¹ : DenselyOrdered E\ninst✝ : SeparableSpace E\ns : X → E\n𝓕 : Set (X → E)\nh𝓕_cont : ∀ f ∈ 𝓕, UpperSemicontinuous f\nh𝓕 : ∀ (... | rcases exists_countable_dense E with ⟨D, D_count, D_dense⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.ConstantSpeed | {
"line": 148,
"column": 4
} | {
"line": 150,
"column": 25
} | {
"line": 152,
"column": 0
} | [
{
"pp": "case inr\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\nl : ℝ≥0\nx y z : ℝ\nhfs : HasConstantSpeedOnWith f (Icc x y) l\nhft : HasConstantSpeedOnWith f (Icc y z) l\nyx : y ≤ x\nu : ℝ\nxu : x ≤ u\nuz : u ≤ z\nv : ℝ\nxv : x ≤ v\nvz : v ≤ z\n⊢ eVariationOn f (Icc x z ∩ Icc u v) = ENNReal.ofReal (↑... | [] | rw [Icc_inter_Icc, sup_of_le_right xu, inf_of_le_right vz, ←
hft ⟨yx.trans xu, uz⟩ ⟨yx.trans xv, vz⟩, Icc_inter_Icc, sup_of_le_right (yx.trans xu),
inf_of_le_right vz] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 37
} | {
"line": 241,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf₁ f₂ : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f₂ - f₁\nhr : 1 ≤ r\n⊢ 0 ≤ logCounting f₂ r - logCounting f₁ r",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
... | [] | simpa using logCounting_nonneg h hr | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Convex.BetweenList | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 46
} | {
"line": 90,
"column": 0
} | [
{
"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\np₁ p₂ p₃ p₄ : P\n⊢ List.Wbtw R [p₁, p₂, p₃, p₄] ↔ Wbtw R p₁ p₂ p₃ ∧ Wbtw R p₁ p₂ p₄ ∧ Wbtw R p₁ p₃ p₄ ∧ Wbtw R p₂ p₃ p₄",
"ppTerm": "?m.47",
"... | [] | simp [List.Wbtw, triplewise_cons, and_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.BetweenList | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 46
} | {
"line": 90,
"column": 0
} | [
{
"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\np₁ p₂ p₃ p₄ : P\n⊢ List.Wbtw R [p₁, p₂, p₃, p₄] ↔ Wbtw R p₁ p₂ p₃ ∧ Wbtw R p₁ p₂ p₄ ∧ Wbtw R p₁ p₃ p₄ ∧ Wbtw R p₂ p₃ p₄",
"ppTerm": "?m.47",
"... | [] | simp [List.Wbtw, triplewise_cons, and_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.BetweenList | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 46
} | {
"line": 90,
"column": 0
} | [
{
"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\np₁ p₂ p₃ p₄ : P\n⊢ List.Wbtw R [p₁, p₂, p₃, p₄] ↔ Wbtw R p₁ p₂ p₃ ∧ Wbtw R p₁ p₂ p₄ ∧ Wbtw R p₁ p₃ p₄ ∧ Wbtw R p₂ p₃ p₄",
"ppTerm": "?m.47",
"... | [] | simp [List.Wbtw, triplewise_cons, and_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Hall.Finite | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 74
} | {
"line": 90,
"column": 2
} | [
{
"pp": "ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = n + 1\nht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), #s' ≤ #(s'.biUnio... | [
"ι : Type u\nα : Type v\ninst✝¹ : DecidableEq α\nt : ι → Finset α\ninst✝ : Fintype ι\nn : ℕ\nhn : Fintype.card ι = n + 1\nht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)\nih :\n ∀ {ι' : Type u} [inst : Fintype ι'] (t' : ι' → Finset α),\n Fintype.card ι' ≤ n →\n (∀ (s' : Finset ι'), #s' ≤ #(s'.biUnion t')) → ∃ f... | haveI : Nonempty ι := Fintype.card_pos_iff.mp (hn.symm ▸ Nat.succ_pos _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.Analysis.Convex.Between | {
"line": 95,
"column": 40
} | {
"line": 95,
"column": 78
} | {
"line": 95,
"column": 78
} | [
{
"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 : P\n⊢ p -ᵥ z ∈ (fun x ↦ p -ᵥ x) '' affineSegment R x y ↔ z ∈ affineSegment R x y",
"ppTerm": "?m.48",
"assigned": true,
"usedCons... | [
"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 : P\n⊢ z ∈ affineSegment R x y ↔ z ∈ affineSegment R x y"
] | (vsub_right_injective p).mem_set_image | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.BetweenList | {
"line": 297,
"column": 4
} | {
"line": 298,
"column": 64
} | {
"line": 299,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : Nontrivial P\nl : List P\nh : List.Sbtw R l\nhl : l = []\n⊢ ∃ p₁ p₂, p₁ ≠ p₂ ∧ ∃ l', l'.SortedLT ∧ ... | [
"case neg\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : Nontrivial P\nl : List P\nh : List.Sbtw R l\nhl : ¬l = []\n⊢ ∃ p₁ p₂, p₁ ≠ p₂ ∧ ∃ l', l'.SortedLT ∧ map (⇑(line... | · rcases exists_pair_ne P with ⟨p₁, p₂, hp₁p₂⟩
exact ⟨p₁, p₂, hp₁p₂, by simp [hl, sortedLT_iff_pairwise]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.CofilteredSystem | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 86
} | {
"line": 74,
"column": 2
} | [
{
"pp": "J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofilteredOrEmpty J\nF : J ⥤ Type u\nhf : ∀ (j : J), Finite (F.obj j)\nhne : ∀ (j : J), Nonempty (F.obj j)\nF' : J ⥤ TopCat := F ⋙ TopCat.discrete\nthis✝¹ : ∀ (j : J), DiscreteTopology ↑(F'.obj j)\nthis✝ : ∀ (j : J), Finite ↑(F'.obj j)\nthis : ∀ (j : J), ... | [
"J : Type u\ninst✝¹ : SmallCategory J\ninst✝ : IsCofilteredOrEmpty J\nF : J ⥤ Type u\nhf : ∀ (j : J), Finite (F.obj j)\nhne : ∀ (j : J), Nonempty (F.obj j)\nF' : J ⥤ TopCat := F ⋙ TopCat.discrete\nthis✝¹ : ∀ (j : J), DiscreteTopology ↑(F'.obj j)\nthis✝ : ∀ (j : J), Finite ↑(F'.obj j)\nthis : ∀ (j : J), Nonempty ↑(F... | obtain ⟨⟨u, hu⟩⟩ := TopCat.nonempty_limitCone_of_compact_t2_cofiltered_system.{u} F' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.CofilteredSystem | {
"line": 157,
"column": 23
} | {
"line": 157,
"column": 33
} | {
"line": 157,
"column": 33
} | [
{
"pp": "case a\nJ : Type u\ninst✝ : Category.{v_1, u} J\nF : J ⥤ Type v\ni j k : J\nf : i ⟶ j\ng : j ⟶ k\nh : F.eventualRange k = range ⇑(ConcreteCategory.hom (F.map g))\n⊢ range ⇑(ConcreteCategory.hom (F.map f ≫ F.map g)) ⊆ range ⇑(ConcreteCategory.hom (F.map g))",
"ppTerm": "?a✝",
"assigned": true,
... | [
"case a\nJ : Type u\ninst✝ : Category.{v_1, u} J\nF : J ⥤ Type v\ni j k : J\nf : i ⟶ j\ng : j ⟶ k\nh : F.eventualRange k = range ⇑(ConcreteCategory.hom (F.map g))\n⊢ range (⇑(ConcreteCategory.hom (F.map g)) ∘ ⇑(ConcreteCategory.hom (F.map f))) ⊆\n range ⇑(ConcreteCategory.hom (F.map g))"
] | types_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.CofilteredSystem | {
"line": 200,
"column": 22
} | {
"line": 200,
"column": 32
} | {
"line": 200,
"column": 32
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v_1, u} J\nF : J ⥤ Type v\ni j : J\ninst✝ : IsCofilteredOrEmpty J\nf : i ⟶ j\nh : ∀ ⦃k : J⦄ (g : k ⟶ i), range ⇑(ConcreteCategory.hom (F.map f)) ⊆ range ⇑(ConcreteCategory.hom (F.map (g ≫ f)))\nj' : J\nf' : j' ⟶ j\nk : J\ng : k ⟶ i\ng' : k ⟶ j'\nhe : g ≫ f = g' ≫ f'\n⊢ ra... | [
"J : Type u\ninst✝¹ : Category.{v_1, u} J\nF : J ⥤ Type v\ni j : J\ninst✝ : IsCofilteredOrEmpty J\nf : i ⟶ j\nh : ∀ ⦃k : J⦄ (g : k ⟶ i), range ⇑(ConcreteCategory.hom (F.map f)) ⊆ range ⇑(ConcreteCategory.hom (F.map (g ≫ f)))\nj' : J\nf' : j' ⟶ j\nk : J\ng : k ⟶ i\ng' : k ⟶ j'\nhe : g ≫ f = g' ≫ f'\n⊢ range (⇑(Concr... | types_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Hall.Basic | {
"line": 137,
"column": 6
} | {
"line": 145,
"column": 62
} | {
"line": 146,
"column": 6
} | [
{
"pp": "case mp.refine_2\nι : Type u\nα : Type v\ninst✝ : DecidableEq α\nt : ι → Finset α\nh : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)\nthis✝ : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')\nthis : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')\nu : (j : (Finset ι)ᵒᵖ) → (hall... | [
"case mp.refine_3\nι : Type u\nα : Type v\ninst✝ : DecidableEq α\nt : ι → Finset α\nh : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)\nthis✝ : ∀ (ι' : (Finset ι)ᵒᵖ), Nonempty ((hallMatchingsFunctor t).obj ι')\nthis : ∀ (ι' : (Finset ι)ᵒᵖ), Finite ((hallMatchingsFunctor t).obj ι')\nu : (j : (Finset ι)ᵒᵖ) → (hallMatchingsFun... | · -- Show that it is injective
intro i i'
have subi : ({i} : Finset ι) ⊆ {i, i'} := by simp
have subi' : ({i'} : Finset ι) ⊆ {i, i'} := by simp
rw [← Finset.le_iff_subset] at subi subi'
simp only
rw [← hu (CategoryTheory.homOfLE subi).op, ← hu (CategoryTheory.homOfLE subi... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Hall.Basic | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 69
} | {
"line": 207,
"column": 2
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : DecidableRel r\nthis : DecidableEq β\nr' : α → Finset β := fun a ↦ {b | r a b}\nh : ∀ (A : Finset α), {b | ∃ a ∈ A, r a b} = A.biUnion r'\n⊢ (∀ (A : Finset α), #A ≤ #{b | ∃ a ∈ A, r a b}) ↔ ∃ f, Injective f ∧ ∀ (x : α), r x (f x)",
... | [
"α : Type u\nβ : Type v\ninst✝¹ : Fintype β\nr : α → β → Prop\ninst✝ : DecidableRel r\nthis : DecidableEq β\nr' : α → Finset β := fun a ↦ {b | r a b}\nh : ∀ (A : Finset α), {b | ∃ a ∈ A, r a b} = A.biUnion r'\nh' : ∀ (f : α → β) (x : α), r x (f x) ↔ f x ∈ r' x\n⊢ (∀ (A : Finset α), #A ≤ #{b | ∃ a ∈ A, r a b}) ↔ ∃ f... | have h' : ∀ (f : α → β) (x), r x (f x) ↔ f x ∈ r' x := by simp [r'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Caratheodory | {
"line": 91,
"column": 10
} | {
"line": 91,
"column": 46
} | {
"line": 92,
"column": 10
} | [
{
"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... | [
"case ha\n𝕜 : 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 ∈ ... | apply mul_nonpos_of_nonneg_of_nonpos | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 129,
"column": 6
} | {
"line": 129,
"column": 31
} | {
"line": 130,
"column": 6
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nh✝ : Nonempty n\nd : ℕ\nih :\n ∀ m < d,\n ∀ (M : Matrix n n R) (s : R),\n 0 ≤ s →\n (∃ M' ∈ doublyStochastic R n, M = s • M') →\n #{i... | [
"R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nh✝ : Nonempty n\nd : ℕ\nih :\n ∀ m < d,\n ∀ (M : Matrix n n R) (s : R),\n 0 ≤ s →\n (∃ M' ∈ doublyStochastic R n, M = s • M') →\n #{i | M i.1 i.2... | rintro ⟨i', j'⟩ _ hN' hM' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Normed.Affine.Convex | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 36
} | {
"line": 79,
"column": 2
} | [
{
"pp": "E✝ : Type u_1\ninst✝⁴ : NormedAddCommGroup E✝\ninst✝³ : NormedSpace ℝ E✝\ns✝ : Set E✝\nx : E✝\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhs : s ∈ 𝓝 0\nb : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E\nc : AffineBasis (Fin (finrank ℝ E +... | [
"E✝ : Type u_1\ninst✝⁴ : NormedAddCommGroup E✝\ninst✝³ : NormedSpace ℝ E✝\ns✝ : Set E✝\nx : E✝\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhs : s ∈ 𝓝 0\nb : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E\nc : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := ... | have hε' : 0 < ε' := by positivity | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Continuous | {
"line": 48,
"column": 31
} | {
"line": 48,
"column": 49
} | {
"line": 49,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nε r M : ℝ\nhf : ConvexOn ℝ (ball x₀ r) f\nhε : 0 < ε\nhM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M\nK : ℝ := 2 * M / ε\nhK : K = 2 * M / ε\nx y : E\nhx : x ∈ ball x₀ (r - ε)\nhy : y ∈ ball x₀ (r - ε)\nhx₀r : ball x₀ (... | [] | simp [field, a, b] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.Extrema | {
"line": 60,
"column": 2
} | {
"line": 62,
"column": 49
} | {
"line": 63,
"column": 2
} | [
{
"pp": "E : Type u_1\nβ : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : Module ℝ E\ninst✝⁷ : IsTopologicalAddGroup E\ninst✝⁶ : ContinuousSMul ℝ E\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module ℝ β\ninst✝¹ : IsOrderedModule ℝ β\ninst... | [
"E : Type u_1\nβ : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : TopologicalSpace E\ninst✝⁸ : Module ℝ E\ninst✝⁷ : IsTopologicalAddGroup E\ninst✝⁶ : ContinuousSMul ℝ E\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module ℝ β\ninst✝¹ : IsOrderedModule ℝ β\ninst✝ : PosSMulR... | have h_maps : MapsTo g (Icc 0 1) s := by
simpa only [g, mapsTo_iff_image_subset, ← segment_eq_image_lineMap]
using h_conv.1.segment_subset a_in_s x_in_s | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.GaugeRescale | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 26
} | {
"line": 170,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℝ E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : T1Space E\ns t : Set E\nhsc : Convex ℝ s\nhsb : IsVonNBounded ℝ s\nhst : Convex ℝ t\nhtne : (interior t).Nonempty\nhtb : IsVonNBounded ℝ t\nx :... | [
"E : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℝ E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : T1Space E\ns t : Set E\nhsc : Convex ℝ s\nhsb : IsVonNBounded ℝ s\nhst : Convex ℝ t\nhtb : IsVonNBounded ℝ t\nx : E\nhx : x ∈ interior s\ny : E\nhy : y ∈ i... | rcases htne with ⟨y, hy⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Convex.Integral | {
"line": 76,
"column": 4
} | {
"line": 79,
"column": 14
} | {
"line": 80,
"column": 2
} | [
{
"pp": "case refine_2\nα : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ninst✝ : IsProbabilityMeasure μ\nhs : Convex ℝ s\nhsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace]... | [] | simp_rw [measureReal_def]
rw [← ENNReal.toReal_sum, (G n).sum_range_measure_preimage_singleton, measure_univ,
ENNReal.toReal_one]
finiteness | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Integral | {
"line": 76,
"column": 4
} | {
"line": 79,
"column": 14
} | {
"line": 80,
"column": 2
} | [
{
"pp": "case refine_2\nα : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\ns : Set E\nf : α → E\ninst✝ : IsProbabilityMeasure μ\nhs : Convex ℝ s\nhsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace]... | [] | simp_rw [measureReal_def]
rw [← ENNReal.toReal_sum, (G n).sum_range_measure_preimage_singleton, measure_univ,
ENNReal.toReal_one]
finiteness | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Integral | {
"line": 169,
"column": 2
} | {
"line": 170,
"column": 69
} | {
"line": 172,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nhg : ConcaveOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpa... | [] | simpa only [mem_setOf_eq, Pi.neg_apply, average_neg, neg_le_neg_iff] using
hg.neg.set_average_mem_epigraph hgc.neg hsc h0 ht hfs hfi hgi.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Convex.Integral | {
"line": 169,
"column": 2
} | {
"line": 170,
"column": 69
} | {
"line": 172,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nhg : ConcaveOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpa... | [] | simpa only [mem_setOf_eq, Pi.neg_apply, average_neg, neg_le_neg_iff] using
hg.neg.set_average_mem_epigraph hgc.neg hsc h0 ht hfs hfi hgi.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Integral | {
"line": 169,
"column": 2
} | {
"line": 170,
"column": 69
} | {
"line": 172,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nμ : Measure α\ns : Set E\nt : Set α\nf : α → E\ng : E → ℝ\nhg : ConcaveOn ℝ s g\nhgc : ContinuousOn g s\nhsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpa... | [] | simpa only [mem_setOf_eq, Pi.neg_apply, average_neg, neg_le_neg_iff] using
hg.neg.set_average_mem_epigraph hgc.neg hsc h0 ht hfs hfi hgi.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Integral | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 87
} | {
"line": 225,
"column": 87
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nf : α → E\ninst✝ : IsFiniteMeasure μ\nhfi : Integrable f μ\nH : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) ... | [
"α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nμ : Measure α\nf : α → E\ninst✝ : IsFiniteMeasure μ\nhfi : Integrable f μ\nH : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂... | refine hfi.ae_eq_of_forall_setIntegral_eq _ _ (integrable_const _) fun t ht ht' => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Convex.ConvexSpace.AffineSpace | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 59
} | {
"line": 60,
"column": 2
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AffineSpace V P\nf : StdSimplex R (StdSimplex R P)\nb : P\nhL :\n (Finset.affineCombination R (StdSimplex.map convexCombination f).w... | [
"R : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AffineSpace V P\nf : StdSimplex R (StdSimplex R P)\nb : P\nhL :\n (Finset.affineCombination R (StdSimplex.map convexCombination f).weights.suppo... | have hR := Finset.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one
f.join.weights.support f.join.weights id f.join.total b | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Convex.Set | {
"line": 174,
"column": 89
} | {
"line": 177,
"column": 96
} | {
"line": 179,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsStrictOrderedRing R\nX : ι → Type u_7\ninst✝ : (i : ι) → ConvexSpace R (X i)\ns : Set ι\nt : (i : ι) → Set (X i)\nht : ∀ i ∈ s, IsConvexSet R (t i)\n⊢ IsConvexSet R (s.pi t)",
"ppTerm": "?m.33",
"assigned": tru... | [] | by
classical
refine fun w hw i hi ↦ ht i hi ?_
grw [StdSimplex.weights_map, mapDomain_support, Finset.coe_image, hw, eval_image_pi_subset hi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Piecewise | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 38
} | {
"line": 104,
"column": 38
} | [
{
"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\... | [
"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\ninst✝¹ : Mo... | h_piecewise_Ici_eq_piecewise_Iic | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.SpecificFunctions.Pow | {
"line": 91,
"column": 6
} | {
"line": 91,
"column": 30
} | {
"line": 91,
"column": 30
} | [
{
"pp": "⊢ StrictConcaveOn ℝ (Ici 0) fun x ↦ √x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"StrictConcaveOn",
"Real.partialOrder",
"Real",
"instHDiv",
"instSMulOfMul",
"Set.Ici",
"Real.instZero",
"co... | [
"⊢ StrictConcaveOn ℝ (Ici 0) fun x ↦ (fun x ↦ x ^ (1 / 2)) x"
] | funext Real.sqrt_eq_rpow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Strict.Extreme | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 96
} | {
"line": 46,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℝ E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : Nontrivial E\nS : Set E\nx : E\nx✝ : x ∈ interior S ∩ extremePoints ℝ S\nx_int : S ∈ 𝓝 x\nx_ext : x ∈ extremePoints ℝ S\nh₁ : ∀ᶠ (v : E) in 𝓝... | [
"E : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module ℝ E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : Nontrivial E\nS : Set E\nx : E\nx✝ : x ∈ interior S ∩ extremePoints ℝ S\nx_int : S ∈ 𝓝 x\nx_ext : x ∈ extremePoints ℝ S\nh₁ : ∀ᶠ (v : E) in 𝓝[≠] 0, x - v... | obtain ⟨v, ⟨hv₁, hv₂⟩, (v_ne : v ≠ 0)⟩ := h₁.and h₂ |>.and eventually_mem_nhdsWithin |>.exists | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Convex.StoneSeparation | {
"line": 104,
"column": 8
} | {
"line": 105,
"column": 54
} | {
"line": 105,
"column": 54
} | [
{
"pp": "𝕜 : 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\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhsC : s ⊆ C\nhmax ... | [
"𝕜 : 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\ns t : Set E\nhs : Convex 𝕜 s\nht : Convex 𝕜 t\nhst : Disjoint s t\nS : Set (Set E) := {C | Convex 𝕜 C ∧ Disjoint C t}\nC : Set E\nhsC : s ⊆ C\nhmax : Maximal (f... | hmax.eq_of_subset ⟨convex_convexHull _ _, h⟩ <|
(subset_insert ..).trans <| subset_convexHull .. | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Side | {
"line": 702,
"column": 6
} | {
"line": 703,
"column": 79
} | {
"line": 704,
"column": 2
} | [
{
"pp": "case mp.inr.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\ns : AffineSubspace R P\nx p : P\nhx : x ∉ s\nhp : p ∈ s\ny p₂ : P\nhp₂ : p₂ ∈ s\nr₁ r₂ : R\nhr₁ : 0 <... | [] | rw [div_eq_inv_mul, ← smul_smul, neg_smul, h, smul_neg, smul_smul,
inv_mul_cancel₀ hr₂.ne.symm, one_smul, neg_vsub_eq_vsub_rev, vsub_vadd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Side | {
"line": 719,
"column": 6
} | {
"line": 720,
"column": 79
} | {
"line": 721,
"column": 2
} | [
{
"pp": "case mp.inr.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\ns : AffineSubspace R P\nx p : P\nhx : x ∉ s\nhp : p ∈ s\ny : P\nhy : y ∉ s\np₂ : P\nhp₂ : p₂ ∈ s\nr₁ ... | [] | rw [div_eq_inv_mul, ← smul_smul, neg_smul, h, smul_neg, smul_smul,
inv_mul_cancel₀ hr₂.ne.symm, one_smul, neg_vsub_eq_vsub_rev, vsub_vadd] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Side | {
"line": 954,
"column": 6
} | {
"line": 954,
"column": 83
} | {
"line": 956,
"column": 0
} | [
{
"pp": "case neg\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw₁ w₂ : Fin (n + 1) → R\nhw₁ : ∑ j, w₁ j = 1\nhw₂ : ∑ j, w₂ j... | [] | exact (s.sSameSide_affineSpan_faceOpposite_of_sign_eq hw₁ hw₂ h h0).wSameSide | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.JapaneseBracket | {
"line": 120,
"column": 4
} | {
"line": 128,
"column": 82
} | {
"line": 129,
"column": 2
} | [
{
"pp": "case calc_1\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nr : ℝ\nhnr : ↑(finrank ℝ E) < r\nhr : 0 < r\nh_meas : Measurable fun ω ↦ (1 + ‖ω‖) ^ (-r)\nh_... | [] | have h_int' : ∀ t ∈ Ioc (0 : ℝ) 1,
f t = ENNReal.ofReal ((t ^ (-r⁻¹) - 1) ^ finrank ℝ E) * mB := fun t ht ↦ by
refine μ.addHaar_closedBall (0 : E) ?_
rw [sub_nonneg]
exact Real.one_le_rpow_of_pos_of_le_one_of_nonpos ht.1 ht.2 (by simp [hr.le])
rw [setLIntegral_congr_fun measurableSet_Ioc h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.JapaneseBracket | {
"line": 120,
"column": 4
} | {
"line": 128,
"column": 82
} | {
"line": 129,
"column": 2
} | [
{
"pp": "case calc_1\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nr : ℝ\nhnr : ↑(finrank ℝ E) < r\nhr : 0 < r\nh_meas : Measurable fun ω ↦ (1 + ‖ω‖) ^ (-r)\nh_... | [] | have h_int' : ∀ t ∈ Ioc (0 : ℝ) 1,
f t = ENNReal.ofReal ((t ^ (-r⁻¹) - 1) ^ finrank ℝ E) * mB := fun t ht ↦ by
refine μ.addHaar_closedBall (0 : E) ?_
rw [sub_nonneg]
exact Real.one_le_rpow_of_pos_of_le_one_of_nonpos ht.1 ht.2 (by simp [hr.le])
rw [setLIntegral_congr_fun measurableSet_Ioc h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 228,
"column": 6
} | {
"line": 242,
"column": 48
} | {
"line": 246,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\nf_nonneg : ∀ (ω : α), 0 ≤ f ω\nH1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0\ns : ℝ\ns_pos : s ... | [] | calc
∞ = ∫⁻ t in Ioc 0 s, ∞ * ENNReal.ofReal (g t) := by
have I_pos : ∫⁻ (a : ℝ) in Ioc 0 s, ENNReal.ofReal (g a) ≠ 0 := by
rw [← ofReal_integral_eq_lintegral_ofReal (g_intble s s_pos).1]
· simpa only [not_lt, ne_eq, ENNReal.ofReal_eq_zero, not_le] using hs
· filter_u... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Distribution.TestFunction | {
"line": 570,
"column": 4
} | {
"line": 571,
"column": 81
} | {
"line": 572,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : 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... | [] | have hk' : 0 < (n : ℕ∞ω) := mod_cast (add_pos_of_right zero_lt_one k).trans_le hk
rw [(f.contDiff.differentiable hk'.ne').differentiableAt.lineDeriv_eq_fderiv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.TestFunction | {
"line": 570,
"column": 4
} | {
"line": 571,
"column": 81
} | {
"line": 572,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : 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... | [] | have hk' : 0 < (n : ℕ∞ω) := mod_cast (add_pos_of_right zero_lt_one k).trans_le hk
rw [(f.contDiff.differentiable hk'.ne').differentiableAt.lineDeriv_eq_fderiv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 17
} | {
"line": 185,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →L[𝕜] E\ninst✝ : CompleteSpace E\nhT : IsSelfAdjoint T\n⊢ spectralRadius 𝕜 T = ↑‖T‖₊",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Nontrivial",
"... | [
"𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →L[𝕜] E\ninst✝ : CompleteSpace E\nhT : IsSelfAdjoint T\na✝ : Nontrivial E\n⊢ spectralRadius 𝕜 T = ↑‖T‖₊"
] | nontriviality E | Mathlib.Tactic.Nontriviality.elabNontriviality | Mathlib.Tactic.Nontriviality.nontriviality |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 533,
"column": 10
} | {
"line": 533,
"column": 67
} | {
"line": 533,
"column": 67
} | [
{
"pp": "case h\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0\nhp : ↑p ≠ 0\nh_one_add : ∀ (x : E), 0 < 1 + ‖x‖\nhp_pos : 0 < ↑p\nl : ℕ\nhl : Integrable (fun x ↦ (1 + ‖x‖) ^ (-↑l)) μ\nk : ℕ := ⌈↑l / ↑p⌉₊\nhlk : ↑l ≤ ↑k * ↑p\nx : E\n⊢ ‖(... | [
"case h\nE : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : MeasurableSpace E\nμ : Measure E\nhμ : μ.HasTemperateGrowth\np : ℝ≥0\nhp : ↑p ≠ 0\nh_one_add : ∀ (x : E), 0 < 1 + ‖x‖\nhp_pos : 0 < ↑p\nl : ℕ\nhl : Integrable (fun x ↦ (1 + ‖x‖) ^ (-↑l)) μ\nk : ℕ := ⌈↑l / ↑p⌉₊\nhlk : ↑l ≤ ↑k * ↑p\nx : E\n⊢ (1 + ‖x‖) ^ (-... | Real.norm_of_nonneg (Real.rpow_nonneg (h_one_add x).le _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Fin.Tuple.Sort | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 30
} | {
"line": 174,
"column": 0
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\n⊢ Monotone f → ∀ (i j : Fin n), i < j → f i = f j → i < j",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Monotone",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLa... | [] | exact fun _ _ _ hij _ => hij | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Fin.Tuple.Sort | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 42
} | {
"line": 204,
"column": 0
} | [
{
"pp": "n : ℕ\nσ : Equiv.Perm (Fin n)\ni✝ j✝ : Fin n\nhij : i✝ < j✝\nh : σ (σ⁻¹ i✝) = σ (σ⁻¹ j✝)\n⊢ σ⁻¹ i✝ < σ⁻¹ j✝",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Equiv.apply_symm_apply",
"Equiv.instEquivLike",
"Equiv.Perm.instInv",
"congrArg",
"False.elim"... | [] | exact (hij.ne (by simpa using h)).elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 81,
"column": 32
} | {
"line": 81,
"column": 79
} | {
"line": 83,
"column": 0
} | [
{
"pp": "m : Type um\nn : Type un\nR : Type uR\ninst✝² : Semiring R\ninst✝¹ : StrongRankCondition R\ninst✝ : Fintype n\nA : Matrix m n R\n⊢ #↑(range Aᵀ) ≤ ↑(Fintype.card n)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"PartialOrd... | [] | simpa using Cardinal.mk_range_le_lift (f := Aᵀ) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 81,
"column": 32
} | {
"line": 81,
"column": 79
} | {
"line": 83,
"column": 0
} | [
{
"pp": "m : Type um\nn : Type un\nR : Type uR\ninst✝² : Semiring R\ninst✝¹ : StrongRankCondition R\ninst✝ : Fintype n\nA : Matrix m n R\n⊢ #↑(range Aᵀ) ≤ ↑(Fintype.card n)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"PartialOrd... | [] | simpa using Cardinal.mk_range_le_lift (f := Aᵀ) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 81,
"column": 32
} | {
"line": 81,
"column": 79
} | {
"line": 83,
"column": 0
} | [
{
"pp": "m : Type um\nn : Type un\nR : Type uR\ninst✝² : Semiring R\ninst✝¹ : StrongRankCondition R\ninst✝ : Fintype n\nA : Matrix m n R\n⊢ #↑(range Aᵀ) ≤ ↑(Fintype.card n)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"PartialOrd... | [] | simpa using Cardinal.mk_range_le_lift (f := Aᵀ) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 333,
"column": 2
} | {
"line": 333,
"column": 57
} | {
"line": 334,
"column": 2
} | [
{
"pp": "m : Type um\nR : Type uR\ninst✝¹ : Field R\ninst✝ : DecidableEq m\nw : m → R\n⊢ (diagonal w).cRank = lift.{uR, um} #{ i // w i ≠ 0 }",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Subtype",
"Ne",
"Field.toSemifield",
"Semifield.toDivisionSemiring",
... | [
"m : Type um\nR : Type uR\ninst✝¹ : Field R\ninst✝ : DecidableEq m\nw : m → R\nw' : { i // w i ≠ 0 } → m → R := fun i ↦ diagonal w ↑i\n⊢ (diagonal w).cRank = lift.{uR, um} #{ i // w i ≠ 0 }"
] | set w' : {i // (w i) ≠ 0} → _ := fun i ↦ (diagonal w) i | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 52
} | {
"line": 208,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : SMulCommClass ℝ 𝕜 F\ninst✝³ : NormedAddCommGroup F'\nins... | [] | exact f.zero_on_compl.comp_left₂ g.zero_on_compl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 52
} | {
"line": 220,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : SMulCommClass ℝ 𝕜 F\ninst✝³ : NormedAddCommGroup F'\nins... | [] | exact f.zero_on_compl.comp_left₂ g.zero_on_compl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 356,
"column": 2
} | {
"line": 357,
"column": 19
} | {
"line": 359,
"column": 0
} | [
{
"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, ... | [] | rw [monoLM]
split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 356,
"column": 2
} | {
"line": 357,
"column": 19
} | {
"line": 359,
"column": 0
} | [
{
"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, ... | [] | rw [monoLM]
split_ifs <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 389,
"column": 19
} | {
"line": 394,
"column": 10
} | {
"line": 396,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type u_4\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace ℝ E\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace ℝ F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : SMulCommClass ℝ 𝕜 F\ninst✝³ : NormedAddCommGroup F'\nins... | [] | by
split_ifs with hk
· have hk' : 0 < (n : ℕ∞ω) := mod_cast (add_pos_of_right zero_lt_one k).trans_le hk
ext
simp [fderiv_const_smul (f.contDiff.differentiable hk'.ne').differentiableAt]
· simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 675,
"column": 4
} | {
"line": 675,
"column": 54
} | {
"line": 676,
"column": 2
} | [
{
"pp": "case neg\n𝕜 : 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 : ℕ∞\nK : Compacts E\nC : ℝ\nhC : 0 ≤ C\... | [] | · simp [hx, f.iteratedFDeriv_zero_on_compl hx, hC] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Lp.SmoothApprox | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 36
} | {
"line": 60,
"column": 2
} | [
{
"pp": "case neg\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nε : ℝ\... | [
"case neg\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nε : ℝ\nhε : 0 < ε\... | have hε' : 0 < ε' := by positivity | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Lp.SmoothApprox | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 35
} | {
"line": 106,
"column": 2
} | [
{
"pp": "case h\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠... | [
"case right\nE : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠ ∞\nhp₂ ... | use ⟨g, hg₄.coeFn_toLp, hg₁, hg₂⟩ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 332,
"column": 6
} | {
"line": 334,
"column": 68
} | {
"line": 337,
"column": 0
} | [
{
"pp": "case insert\nT : 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 ∈ in... | [] | obtain ⟨hf₁, hf₂⟩ := by simpa using hf
rw [s.sum_insert ha, s.sum_insert ha,
fourierCoeff.add hf₁ (integrable_finsetSum' s hf₂), iha hf₂] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 332,
"column": 6
} | {
"line": 334,
"column": 68
} | {
"line": 337,
"column": 0
} | [
{
"pp": "case insert\nT : 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 ∈ in... | [] | obtain ⟨hf₁, hf₂⟩ := by simpa using hf
rw [s.sum_insert ha, s.sum_insert ha,
fourierCoeff.add hf₁ (integrable_finsetSum' s hf₂), iha hf₂] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 566,
"column": 52
} | {
"line": 566,
"column": 62
} | {
"line": 566,
"column": 63
} | [
{
"pp": "a b : ℝ\nhab : a < b\nf f' : ℝ → ℂ\nn : ℤ\nhn : n ≠ 0\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : IntervalIntegrable f' volume a b\nhT : Fact (0 < b - a)\n⊢ (1 / (b - a)) • ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x * f x =\n 1 / (-2 * ↑π... | [
"a b : ℝ\nhab : a < b\nf f' : ℝ → ℂ\nn : ℤ\nhn : n ≠ 0\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : IntervalIntegrable f' volume a b\nhT : Fact (0 < b - a)\n⊢ ↑(1 / (b - a)) * ∫ (x : ℝ) in a..b, (fourier (-n)) ↑x * f x =\n 1 / (-2 * ↑π * I * ↑n) ... | real_smul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 179,
"column": 18
} | {
"line": 179,
"column": 28
} | {
"line": 179,
"column": 29
} | [
{
"pp": "case e'_12\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\nv : V\nw : W\nha : HasFDerivAt (fun w' ↦ (L v... | [
"case e'_12\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\nv : V\nw : W\nha : HasFDerivAt (fun w' ↦ (L v) w') (L v) ... | real_smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 179,
"column": 29
} | {
"line": 179,
"column": 39
} | {
"line": 179,
"column": 40
} | [
{
"pp": "case e'_12\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\nv : V\nw : W\nha : HasFDerivAt (fun w' ↦ (L v... | [
"case e'_12\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\nv : V\nw : W\nha : HasFDerivAt (fun w' ↦ (L v) w') (L v) ... | real_smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 212,
"column": 2
} | {
"line": 213,
"column": 40
} | {
"line": 214,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : BorelSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns : Set α\nφ : ι → α → ℝ\na : E\ninst✝ : CompleteSpace E\nhs : MeasurableSet s\n... | [
"α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : BorelSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\ns : Set α\nφ : ι → α → ℝ\na : E\ninst✝ : CompleteSpace E\nhs : MeasurableSet s\nt : Set α\nh... | rw [integral_sub hi, setIntegral_indicator ht, inter_eq_right.mpr hts,
integral_smul_const, sub_add_cancel] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 232,
"column": 21
} | {
"line": 232,
"column": 47
} | {
"line": 232,
"column": 47
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : BorelSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\nφ : ι → α → ℝ\na : E\ninst✝ : CompleteSpace E\nt : Set α\nht : MeasurableSet t\n... | [] | by simpa [nhdsWithin_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 234,
"column": 27
} | {
"line": 234,
"column": 53
} | {
"line": 234,
"column": 53
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nι : Type u_3\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : BorelSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ng : α → E\nl : Filter ι\nx₀ : α\nφ : ι → α → ℝ\na : E\ninst✝ : CompleteSpace E\nt : Set α\nht : MeasurableSet t\n... | [] | by simpa [nhdsWithin_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 322,
"column": 34
} | {
"line": 322,
"column": 56
} | {
"line": 322,
"column": 56
} | [
{
"pp": "case h\nι : 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✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : 𝓢(E, F)\nk n : ℕ\nx : E\n⊢ ‖x‖ ^ k * ‖iteratedFD... | [
"case h\nι : 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✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : 𝓢(E, F)\nk n : ℕ\nx : E\n⊢ @SchwartzMap.seminormAux E F inst... | f.le_seminormAux k n x | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 610,
"column": 10
} | {
"line": 617,
"column": 97
} | {
"line": 618,
"column": 2
} | [
{
"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✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace ℝ E\ninst✝¹² : NormedAddCommGroup F\ninst✝¹¹ : NormedSpace ℝ F\ninst✝¹⁰ : NormedField 𝕜\ninst✝⁹ : NormedField 𝕜'\ninst... | [] | by
change Continuous (mkLM A hadd hsmul hsmooth hbound : 𝓢(D, E) →ₛₗ[σ] 𝓢(F, G))
refine
WithSeminorms.continuous_of_isBounded (schwartz_withSeminorms 𝕜 D E)
(schwartz_withSeminorms 𝕜' F G) _ fun n => ?_
rcases hbound n with ⟨s, C, hC, h⟩
refine ⟨s, ⟨C, hC⟩, fun f => ?_⟩
exact (mkLM... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 259,
"column": 24
} | {
"line": 259,
"column": 38
} | {
"line": 259,
"column": 38
} | [
{
"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)))\nx : ℝ\nhx : 0 < x\n⊢ a.re ≠ -Complex.re 1",
"ppTerm": "?m.188",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg... | [
"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)))\nx : ℝ\nhx : 0 < x\n⊢ a.re ≠ -1"
] | Complex.one_re | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 267,
"column": 76
} | {
"line": 267,
"column": 90
} | {
"line": 267,
"column": 90
} | [
{
"pp": "a : ℂ\nha : a.re < -1\nc : ℝ\nhc : 0 < c\nx : ℝ\nhx : 0 < x\n⊢ x ^ (a.re + 1) = x ^ (a.re + Complex.re 1)",
"ppTerm": "?m.268",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real",
"Real.instAdd",
"Complex.re",
"Real.instOne",
"instHAdd",
"H... | [] | Complex.one_re | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 769,
"column": 2
} | {
"line": 770,
"column": 47
} | {
"line": 772,
"column": 0
} | [
{
"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 : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAlgebra ℝ 𝕜\ninst✝ : NormedSpace 𝕜 F\ng₁ g₂ : E → 𝕜\nhg₁ : Function.HasTemperateGrowth g₁\n... | [] | ext1 f
exact smulLeftCLM_smulLeftCLM_apply hg₁ hg₂ f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 769,
"column": 2
} | {
"line": 770,
"column": 47
} | {
"line": 772,
"column": 0
} | [
{
"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 : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAlgebra ℝ 𝕜\ninst✝ : NormedSpace 𝕜 F\ng₁ g₂ : E → 𝕜\nhg₁ : Function.HasTemperateGrowth g₁\n... | [] | ext1 f
exact smulLeftCLM_smulLeftCLM_apply hg₁ hg₂ f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 867,
"column": 6
} | {
"line": 901,
"column": 14
} | {
"line": 903,
"column": 0
} | [
{
"pp": "case right\nι : 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✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : N... | [] | calc
_ ≤ ‖x‖ ^ k * ∑ i ∈ Finset.range (n + 1), (n.choose i) *
‖iteratedFDeriv ℝ i L x‖ * ‖iteratedFDeriv ℝ (n - i) f x‖ := by
gcongr 1
exact norm_iteratedFDeriv_le_of_bilinear_of_le_one (smulRightL ℝ G F)
(by fun_prop) (f.smooth ⊤) x (mod_cast le_top) norm_smulRightL_... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 386,
"column": 8
} | {
"line": 386,
"column": 18
} | {
"line": 386,
"column": 19
} | [
{
"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 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)",
"ppTerm": "?m.458",
"assigned": true,
"u... | [
"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 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)"
] | real_smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
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