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.Topology.Bornology.BoundedOperation | {
"line": 128,
"column": 2
} | {
"line": 133,
"column": 92
} | {
"line": 135,
"column": 0
} | [
{
"pp": "R : Type u_1\nX : Type u_2\ninst✝² : PseudoMetricSpace R\ninst✝¹ : Mul R\ninst✝ : BoundedMul R\nf g : X → R\nf_bdd : ∃ C, ∀ (x y : X), dist (f x) (f y) ≤ C\ng_bdd : ∃ C, ∀ (x y : X), dist (g x) (g y) ≤ C\n⊢ ∃ C, ∀ (x y : X), dist ((f * g) x) ((f * g) y) ≤ C",
"ppTerm": "?m.35",
"assigned": true... | [] | obtain ⟨C, hC⟩ := Metric.isBounded_iff.mp <|
isBounded_mul (Metric.isBounded_range_iff.mpr f_bdd) (Metric.isBounded_range_iff.mpr g_bdd)
use C
intro x y
exact hC (Set.mul_mem_mul (Set.mem_range_self (f := f) x) (Set.mem_range_self (f := g) x))
(Set.mul_mem_mul (Set.mem_range_self (f := f) y) (Set.m... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Bornology.BoundedOperation | {
"line": 128,
"column": 2
} | {
"line": 133,
"column": 92
} | {
"line": 135,
"column": 0
} | [
{
"pp": "R : Type u_1\nX : Type u_2\ninst✝² : PseudoMetricSpace R\ninst✝¹ : Mul R\ninst✝ : BoundedMul R\nf g : X → R\nf_bdd : ∃ C, ∀ (x y : X), dist (f x) (f y) ≤ C\ng_bdd : ∃ C, ∀ (x y : X), dist (g x) (g y) ≤ C\n⊢ ∃ C, ∀ (x y : X), dist ((f * g) x) ((f * g) y) ≤ C",
"ppTerm": "?m.35",
"assigned": true... | [] | obtain ⟨C, hC⟩ := Metric.isBounded_iff.mp <|
isBounded_mul (Metric.isBounded_range_iff.mpr f_bdd) (Metric.isBounded_range_iff.mpr g_bdd)
use C
intro x y
exact hC (Set.mul_mem_mul (Set.mem_range_self (f := f) x) (Set.mem_range_self (f := g) x))
(Set.mul_mem_mul (Set.mem_range_self (f := f) y) (Set.m... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1166,
"column": 8
} | {
"line": 1167,
"column": 21
} | {
"line": 1168,
"column": 6
} | [
{
"pp": "case hp_ne_zero\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_6\ninst✝ : NormedAddCommGroup E\nf g : α → E\np q : ℝ\nhpq : p.HolderConjugate q\nhf : MemLp f (ENNReal.ofReal p) μ\nhg : MemLp g (ENNReal.ofReal q) μ\nh_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((... | [] | rw [Ne, ENNReal.ofReal_eq_zero, not_le]
exact hpq.pos | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.Basic | {
"line": 1166,
"column": 8
} | {
"line": 1167,
"column": 21
} | {
"line": 1168,
"column": 6
} | [
{
"pp": "case hp_ne_zero\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nE : Type u_6\ninst✝ : NormedAddCommGroup E\nf g : α → E\np q : ℝ\nhpq : p.HolderConjugate q\nhf : MemLp f (ENNReal.ofReal p) μ\nhg : MemLp g (ENNReal.ofReal q) μ\nh_left : ∫⁻ (a : α), ENNReal.ofReal (‖f a‖ * ‖g a‖) ∂μ = ∫⁻ (a : α), ((... | [] | rw [Ne, ENNReal.ofReal_eq_zero, not_le]
exact hpq.pos | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 143,
"column": 2
} | {
"line": 146,
"column": 28
} | {
"line": 148,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf g : α →ᵇ β\n⊢ ∃ C, 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"le_max_right",
"Set.mem_range_self",
"Real.instLE",
"Real",
... | [] | rcases isBounded_iff.1 (f.isBounded_range.union g.isBounded_range) with ⟨C, hC⟩
refine ⟨max 0 C, le_max_left _ _, fun x => (hC ?_ ?_).trans (le_max_right _ _)⟩
<;> [left; right]
<;> apply mem_range_self | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 143,
"column": 2
} | {
"line": 146,
"column": 28
} | {
"line": 148,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : PseudoMetricSpace β\nf g : α →ᵇ β\n⊢ ∃ C, 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"le_max_right",
"Set.mem_range_self",
"Real.instLE",
"Real",
... | [] | rcases isBounded_iff.1 (f.isBounded_range.union g.isBounded_range) with ⟨C, hC⟩
refine ⟨max 0 C, le_max_left _ _, fun x => (hC ?_ ?_).trans (le_max_right _ _)⟩
<;> [left; right]
<;> apply mem_range_self | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.Bounded.Basic | {
"line": 183,
"column": 6
} | {
"line": 183,
"column": 17
} | {
"line": 184,
"column": 6
} | [
{
"pp": "case neg\nα : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\nf g : α →ᵇ β\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nh : ¬Nonempty α\n⊢ dist f g < C",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero"... | [
"α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : PseudoMetricSpace β\nf g : α →ᵇ β\nC : ℝ\ninst✝ : CompactSpace α\nC0 : 0 < C\nh : ¬Nonempty α\n⊢ dist f g = 0"
] | convert! C0 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 138,
"column": 4
} | {
"line": 141,
"column": 28
} | {
"line": 142,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : PseudoEMetricSpace α\nδseq : ℕ → ℝ\nδseq_lim : Tendsto δseq atTop (𝓝 0)\nE : Set α\nx : α\nx_mem_closure : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E\n⊢ Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop\n (𝓝 ((closure[PseudoEMetricS... | [] | simp_rw [thickenedIndicatorAux_one_of_mem_closure _ E x_mem_closure]
rw [show (indicator (closure E) fun _ => (1 : ℝ≥0∞)) x = 1 by
simp only [x_mem_closure, indicator_of_mem]]
exact tendsto_const_nhds | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 138,
"column": 4
} | {
"line": 141,
"column": 28
} | {
"line": 142,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : PseudoEMetricSpace α\nδseq : ℕ → ℝ\nδseq_lim : Tendsto δseq atTop (𝓝 0)\nE : Set α\nx : α\nx_mem_closure : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E\n⊢ Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop\n (𝓝 ((closure[PseudoEMetricS... | [] | simp_rw [thickenedIndicatorAux_one_of_mem_closure _ E x_mem_closure]
rw [show (indicator (closure E) fun _ => (1 : ℝ≥0∞)) x = 1 by
simp only [x_mem_closure, indicator_of_mem]]
exact tendsto_const_nhds | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.ThickenedIndicator | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 66
} | {
"line": 224,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ₁ δ₂ : ℝ\nδ₁_pos : 0 < δ₁\nδ₂_pos : 0 < δ₂\nhle : δ₁ ≤ δ₂\nE : Set α\nx : α\n⊢ (thickenedIndicator δ₁_pos E) x ≤ (thickenedIndicator δ₂_pos E) x",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"thickenedIndicatorA... | [
"α : Type u_1\ninst✝ : PseudoEMetricSpace α\nδ₁ δ₂ : ℝ\nδ₁_pos : 0 < δ₁\nδ₂_pos : 0 < δ₂\nhle : δ₁ ≤ δ₂\nE : Set α\nx : α\n⊢ thickenedIndicatorAux δ₁ E x ≤ thickenedIndicatorAux δ₂ E x"
] | apply (toNNReal_le_toNNReal (by finiteness) (by finiteness)).mpr | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1299,
"column": 2
} | {
"line": 1302,
"column": 30
} | {
"line": 1303,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_7\nl : Filter ι\ninst✝ : l.IsCo... | [
"α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_7\nl : Filter ι\ninst✝ : l.IsCountablyGener... | have h :
{ x : ι | (fun n => AEStronglyMeasurable (fs n) μ) x } ∩
{ x : ι | (fun n => ∀ᵐ a ∂μ, ‖fs n a‖ ≤ bound a) x } ∈ l :=
inter_mem hfs_meas h_bound | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 56
} | {
"line": 158,
"column": 2
} | [
{
"pp": "f : StieltjesFunction ℝ\nx : ℝ\n⊢ ⨅ r, ↑f ↑↑r = ⨅ r, ↑f ↑r",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Real.iInf_Ioi_eq_iInf_rat_gt",
"Real",
"Set.Ioi",
"iInf",
"Real.instRatCast",
"Rat",
"PseudoMetricSpace.toUniformSpace",
"Rea... | [
"f : StieltjesFunction ℝ\nx : ℝ\n⊢ BddBelow (↑f '' Ioi x)"
] | refine (Real.iInf_Ioi_eq_iInf_rat_gt _ ?_ f.mono).symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 408,
"column": 4
} | {
"line": 408,
"column": 63
} | {
"line": 409,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : LinearOrder R\ninst✝³ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝² : OrderTopology R\ninst✝¹ : CompactIccSpace R\ninst✝ : DenselyOrdered R\na b : R\nhab : a < b\ns : ℕ → Set R\nhs : Ioc a b ⊆ ⋃ i, s i\nε : ℝ≥0\nεpos : 0 < ε\nh : ∑' (i : ℕ), f.length (s i) < ∞\nδ : ℝ≥0 :=... | [
"R : Type u_1\ninst✝⁴ : LinearOrder R\ninst✝³ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝² : OrderTopology R\ninst✝¹ : CompactIccSpace R\ninst✝ : DenselyOrdered R\na b : R\nhab : a < b\ns : ℕ → Set R\nhs : Ioc a b ⊆ ⋃ i, s i\nε : ℝ≥0\nεpos : 0 < ε\nh : ∑' (i : ℕ), f.length (s i) < ∞\nδ : ℝ≥0 := ε / 2\nδpos... | have : (𝓝[>] a).NeBot := nhdsGT_neBot_of_exists_gt ⟨b, hab⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 45
} | {
"line": 301,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\n⊢ ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
... | [
"𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\n⊢ ⟪x, x⟫ - ⟪y, x⟫ - (⟪x, y⟫ - ⟪y, y⟫) = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫"
] | simp only [inner_sub_left, inner_sub_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 698,
"column": 2
} | {
"line": 698,
"column": 61
} | {
"line": 699,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : LinearOrder R\ninst✝⁷ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁶ : OrderTopology R\ninst✝⁵ : CompactIccSpace R\ninst✝⁴ : MeasurableSpace R\ninst✝³ : BorelSpace R\ninst✝² : SecondCountableTopology R\ninst✝¹ : DenselyOrdered R\ninst✝ : Nonempty R\nl u : ℝ\nhfl : Tendsto ... | [
"R : Type u_1\ninst✝⁸ : LinearOrder R\ninst✝⁷ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁶ : OrderTopology R\ninst✝⁵ : CompactIccSpace R\ninst✝⁴ : MeasurableSpace R\ninst✝³ : BorelSpace R\ninst✝² : SecondCountableTopology R\ninst✝¹ : DenselyOrdered R\ninst✝ : Nonempty R\nl u : ℝ\nhfl : Tendsto (↑f) atBot (... | refine tendsto_nhds_unique (tendsto_measure_Iic_atTop _) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 100,
"column": 47
} | {
"line": 100,
"column": 62
} | {
"line": 100,
"column": 62
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : SeminormedAddCommGroup E\ninst✝⁶ : InnerProductSpace 𝕜 E\n𝕝 : Type u_4\ninst✝⁵ : CommSemiring 𝕝\ninst✝⁴ : StarRing 𝕝\ninst✝³ : Algebra 𝕝 𝕜\ninst✝² : Module 𝕝 E\ninst✝¹ : IsScalarTower 𝕝 𝕜 E\ninst✝ : StarModule 𝕝 𝕜\nx y : E\nr : 𝕝\n⊢ ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : SeminormedAddCommGroup E\ninst✝⁶ : InnerProductSpace 𝕜 E\n𝕝 : Type u_4\ninst✝⁵ : CommSemiring 𝕝\ninst✝⁴ : StarRing 𝕝\ninst✝³ : Algebra 𝕝 𝕜\ninst✝² : Module 𝕝 E\ninst✝¹ : IsScalarTower 𝕝 𝕜 E\ninst✝ : StarModule 𝕝 𝕜\nx y : E\nr : 𝕝\n⊢ r • ⟪x, y⟫ =... | inner_conj_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 362,
"column": 4
} | {
"line": 362,
"column": 79
} | {
"line": 363,
"column": 4
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\nt : ℝ\nhzero : ¬⟪x, y⟫ = 0\nhzero' : ‖⟪x, y⟫‖ ≠ 0\n⊢ 0 ≤ normSq x * (t * t) + 2 * ‖⟪x, y⟫‖ * t + normSq y",
"ppTerm": "?neg✝",
"assigned": true,
... | [
"case e'_4.e'_5\n𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\nt : ℝ\nhzero : ¬⟪x, y⟫ = 0\nhzero' : ‖⟪x, y⟫‖ ≠ 0\n⊢ normSq x * (t * t) = normSq (⟪x, y⟫ • x) * (t / ‖⟪x, y⟫‖) * (t / ‖⟪x, y⟫‖)",
"case e'_4.e'_6\n𝕜 : Type ... | convert! cauchy_schwarz_aux' (𝕜 := 𝕜) (⟪x, y⟫ • x) y (t / ‖⟪x, y⟫‖) using 3 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 266,
"column": 2
} | {
"line": 266,
"column": 45
} | {
"line": 266,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\n⊢ ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\n⊢ ⟪x, x⟫ - ⟪y, x⟫ - (⟪x, y⟫ - ⟪y, y⟫) = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫"
] | simp only [inner_sub_left, inner_sub_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 754,
"column": 4
} | {
"line": 754,
"column": 18
} | {
"line": 755,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\nl : ℝ\nhfg : f.meas... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 754,
"column": 4
} | {
"line": 754,
"column": 18
} | {
"line": 755,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\nl : ℝ\nhfg : f.meas... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 754,
"column": 4
} | {
"line": 754,
"column": 18
} | {
"line": 755,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\nl : ℝ\nhfg : f.meas... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 768,
"column": 6
} | {
"line": 768,
"column": 20
} | {
"line": 769,
"column": 4
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\nhf... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 768,
"column": 6
} | {
"line": 768,
"column": 20
} | {
"line": 769,
"column": 4
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\nhf... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 768,
"column": 6
} | {
"line": 768,
"column": 20
} | {
"line": 769,
"column": 4
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\nhf... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 776,
"column": 6
} | {
"line": 776,
"column": 20
} | {
"line": 777,
"column": 4
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\nhf... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 776,
"column": 6
} | {
"line": 776,
"column": 20
} | {
"line": 777,
"column": 4
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\nhf... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 776,
"column": 6
} | {
"line": 776,
"column": 20
} | {
"line": 777,
"column": 4
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝⁷ : LinearOrder R\ninst✝⁶ : TopologicalSpace R\nf : StieltjesFunction R\ninst✝⁵ : OrderTopology R\ninst✝⁴ : CompactIccSpace R\ninst✝³ : MeasurableSpace R\ninst✝² : BorelSpace R\ninst✝¹ : SecondCountableTopology R\ninst✝ : DenselyOrdered R\ng : StieltjesFunction R\ny : R\nhf... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 568,
"column": 49
} | {
"line": 571,
"column": 24
} | {
"line": 573,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nh : ⟪x, y⟫ = 0\n⊢ ‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm... | [] | by
rw [@norm_add_mul_self 𝕜, add_right_cancel_iff, add_eq_left, mul_eq_zero]
apply Or.inr
simp only [h, zero_re] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 639,
"column": 4
} | {
"line": 639,
"column": 13
} | {
"line": 639,
"column": 14
} | [
{
"pp": "F : Type u_3\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nι₁ : Type u_4\ns₁ : Finset ι₁\nw₁ : ι₁ → ℝ\nv₁ : ι₁ → F\nh₁ : ∑ i ∈ s₁, w₁ i = 0\nι₂ : Type u_5\ns₂ : Finset ι₂\nw₂ : ι₂ → ℝ\nv₂ : ι₂ → F\nh₂ : ∑ i ∈ s₂, w₂ i = 0\n⊢ ∑ x ∈ s₁, w₁ x * 0 + 0 - ∑ x ∈ s₁, w₁ x * ∑ i ∈ s₂, w₂ i ... | [
"F : Type u_3\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nι₁ : Type u_4\ns₁ : Finset ι₁\nw₁ : ι₁ → ℝ\nv₁ : ι₁ → F\nh₁ : ∑ i ∈ s₁, w₁ i = 0\nι₂ : Type u_5\ns₂ : Finset ι₂\nw₂ : ι₂ → ℝ\nv₂ : ι₂ → F\nh₂ : ∑ i ∈ s₂, w₂ i = 0\n⊢ ∑ x ∈ s₁, 0 + 0 - ∑ x ∈ s₁, w₁ x * ∑ i ∈ s₂, w₂ i * (‖v₁ x - v₂ i‖ * ... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 803,
"column": 4
} | {
"line": 803,
"column": 71
} | {
"line": 804,
"column": 2
} | [
{
"pp": "case mp\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : F\nh : ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = 1\nhx₀ : x ≠ 0\nhy₀ : y ≠ 0\n⊢ y = (‖y‖ / ‖x‖) • x",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Norm.norm",
"InnerProductSpace.toNormedSpace",
... | [] | exact ((inner_eq_norm_mul_iff_div hx₀).1 (eq_of_div_eq_one h)).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 864,
"column": 2
} | {
"line": 864,
"column": 45
} | {
"line": 865,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nhle : ‖x‖ ≤ ‖y‖\nh : re ⟪x, y⟫ = ‖y‖ ^ 2\nH₁ : ‖x‖ ^ 2 ≤ ‖y‖ ^ 2\nH₂ : re ⟪y, x⟫ = ‖y‖ ^ 2\n⊢ re ⟪x - y, x - y⟫ ≤ 0",
"ppTerm": "?m.131",
"assigned": true,
"usedConstants... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nhle : ‖x‖ ≤ ‖y‖\nh : re ⟪x, y⟫ = ‖y‖ ^ 2\nH₁ : ‖x‖ ^ 2 ≤ ‖y‖ ^ 2\nH₂ : re ⟪y, x⟫ = ‖y‖ ^ 2\n⊢ re (⟪x, x⟫ - ⟪y, x⟫ - (⟪x, y⟫ - ⟪y, y⟫)) ≤ 0"
] | simp only [inner_sub_left, inner_sub_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 908,
"column": 27
} | {
"line": 908,
"column": 78
} | {
"line": 909,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝ : RCLike 𝕜\nx : 𝕜\n⊢ ‖x‖ ^ 2 = re (x * star x)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"RCLike.star_def",
"Real",
"NonUnital... | [] | by rw [star_def, mul_conj, ← ofReal_pow, ofReal_re] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Subspace | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 48
} | {
"line": 115,
"column": 6
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nι : Type u_4\nG : ι → Type u_5\ninst✝³ : (i : ι) → NormedAddCommGroup (G i)\ninst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily �... | [
"case neg\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nι : Type u_4\nG : ι → Type u_5\ninst✝³ : (i : ι) → NormedAddCommGroup (G i)\ninst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily 𝕜 G V\ninst✝... | · simp only [LinearIsometry.inner_map_map] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 98,
"column": 21
} | {
"line": 98,
"column": 45
} | {
"line": 98,
"column": 45
} | [
{
"pp": "V : Type u_4\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℂ V\nS T : V →ₗ[ℂ] V\n⊢ (∀ (x : V), ⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ) ↔ S - T = 0",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Module.End.instRing",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
... | [
"V : Type u_4\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℂ V\nS T : V →ₗ[ℂ] V\n⊢ (∀ (x : V), ⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ) ↔ ∀ (x : V), ⟪(S - T) x, x⟫_ℂ = 0"
] | ← inner_map_self_eq_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 73,
"column": 47
} | {
"line": 73,
"column": 62
} | {
"line": 73,
"column": 62
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx y : E\n⊢ (starRingEnd 𝕜) ⟪x, T y⟫ = ⟪T y, x⟫",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProdu... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx y : E\n⊢ ⟪T y, x⟫ = ⟪T y, x⟫"
] | inner_conj_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 94,
"column": 80
} | {
"line": 95,
"column": 80
} | {
"line": 97,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_3\nT : ι → E →ₗ[𝕜] E\ns : Finset ι\nhT : ∀ i ∈ s, (T i).IsSymmetric\nx✝¹ x✝ : E\n⊢ ⟪(∑ i ∈ s, T i) x✝¹, x✝⟫ = ⟪x✝¹, (∑ i ∈ s, T i) x✝⟫",
"ppTerm": "?m.33",
"assigned":... | [] | by
simpa [sum_inner, inner_sum] using Finset.sum_congr rfl fun _ hi ↦ hT _ hi _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 212,
"column": 16
} | {
"line": 212,
"column": 25
} | {
"line": 212,
"column": 26
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx y : E\nh : I = 0\n⊢ ↑(re (↑(re ⟪T y, x⟫) + ↑(im ⟪T y, x⟫) * 0)) = ↑(re ⟪T y, x⟫) + ↑(im ⟪T y, x⟫) * 0",
"ppTerm": "?inl",
"assigned... | [
"case inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx y : E\nh : I = 0\n⊢ ↑(re (↑(re ⟪T y, x⟫) + 0)) = ↑(re ⟪T y, x⟫) + 0"
] | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 215,
"column": 66
} | {
"line": 215,
"column": 74
} | {
"line": 215,
"column": 75
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx y : E\nh : I * I = -1\n⊢ ⟪T x, y⟫ =\n (⟪T x, x⟫ + ⟪T x, y⟫ + (⟪T y, x⟫ + ⟪T y, y⟫) - (⟪T x, x⟫ - ⟪T x, y⟫ - (⟪T y, x⟫ - ⟪T y, y⟫)) -\n ... | [
"case inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx y : E\nh : I * I = -1\n⊢ ⟪T x, y⟫ =\n (⟪T x, x⟫ + ⟪T x, y⟫ + (⟪T y, x⟫ + ⟪T y, y⟫) - (⟪T x, x⟫ - ⟪T x, y⟫ - (⟪T y, x⟫ - ⟪T y, y⟫)) -\n (I *... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 365,
"column": 2
} | {
"line": 365,
"column": 91
} | {
"line": 367,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\np q : E →ₗ[𝕜] E\nhp : p.IsSymmetricProjection\nhq : q.IsSymmetricProjection\nhqp : q ∘ₗ p = p\nx y : E\n⊢ ⟪y, (p * q) x⟫ = ⟪y, p x⟫",
"ppTerm": "?m.130",
"assigned": true,
"usedC... | [] | simp_rw [Module.End.mul_apply, ← hp.isSymmetric _, ← hq.isSymmetric _, ← comp_apply, hqp] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.InnerProductSpace.Projection.Reflection | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 23
} | {
"line": 106,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nu v : E\n⊢ (𝕜 ∙ u).reflection v = 2 • (⟪u, v⟫ / ↑‖u‖ ^ 2) • u - v",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"LinearIsometryEquiv.instEquivLike",
"Norm... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nu v : E\n⊢ 2 • (𝕜 ∙ u).starProjection v - v = 2 • (⟪u, v⟫ / ↑‖u‖ ^ 2) • u - v"
] | reflection_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Projection.Reflection | {
"line": 110,
"column": 36
} | {
"line": 110,
"column": 53
} | {
"line": 110,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nx : E\n⊢ K.reflection x = x ↔ K.starProjection x = x",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Linea... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nx : E\n⊢ 2 • K.starProjection x - x = x ↔ K.starProjection x = x"
] | reflection_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 69
} | {
"line": 238,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+* 𝕜\nin... | [
"𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+* 𝕜\ninst✝⁴ : RingH... | have : f (x + w) = z := by rw [f.map_add, wim, fxy, add_sub_cancel] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 398,
"column": 4
} | {
"line": 399,
"column": 14
} | {
"line": 400,
"column": 2
} | [
{
"pp": "case hvm\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv w : E\n⊢ ⟪v, w⟫ • v ∈ 𝕜 ∙ v",
"ppTerm": "?hvm",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Submodule",
... | [] | rw [Submodule.mem_span_singleton]
use ⟪v, w⟫ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 398,
"column": 4
} | {
"line": 399,
"column": 14
} | {
"line": 400,
"column": 2
} | [
{
"pp": "case hvm\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv w : E\n⊢ ⟪v, w⟫ • v ∈ 𝕜 ∙ v",
"ppTerm": "?hvm",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Submodule",
... | [] | rw [Submodule.mem_span_singleton]
use ⟪v, w⟫ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 267,
"column": 4
} | {
"line": 269,
"column": 43
} | {
"line": 270,
"column": 4
} | [
{
"pp": "case mem\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝¹ : Fintype ι\nV : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace ↥(V i)\nhV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ\nx✝ : E\ni : ι\... | [
"case mem\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝¹ : Fintype ι\nV : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace ↥(V i)\nhV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ\nx✝¹ : E\ni : ι\nx : E\nhx ... | refine
(Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hij => ?_).trans
(starProjection_eq_self_iff.mpr hx) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 461,
"column": 2
} | {
"line": 461,
"column": 63
} | {
"line": 463,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nU V : Submodule 𝕜 E\ninst✝ : U.HasOrthogonalProjection\nh : U.orthogonalProjectionOnto ∘SL V.subtypeL = 0\nu : E\nhu : u ∈ U\nv : E\nhv : v ∈ V\nthis : U.orthogonalProjectionOnto v = 0\n⊢ v... | [] | rw [starProjection_apply, this, Submodule.coe_zero, sub_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 510,
"column": 67
} | {
"line": 513,
"column": 34
} | {
"line": 517,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\ninst✝⁴ : CompleteSpace E\nF : Type u_5\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : CompleteSpace F\nf : E →L[𝕜] F\nG : Submodule 𝕜 F\nh : IsCompl (↑f).ra... | [] | by
rw [coprodSubtypeLEquivOfIsCompl, ← ContinuousLinearEquiv.toLinearMap_toContinuousLinearMap,
ContinuousLinearEquiv.coe_ofBijective, coe_coprod, LinearMap.coprod_map_prod, Submodule.map_bot,
sup_bot_eq, Submodule.map_top] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 545,
"column": 2
} | {
"line": 545,
"column": 59
} | {
"line": 547,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CompleteSpace E\nF : Type u_5\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\ng : E →ₗ[𝕜] F\nhg : IsClosed[instTopologicalSpaceProd] ... | [] | exact (continuous_subtype_val.comp ψ.symm.continuous).snd | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 905,
"column": 86
} | {
"line": 905,
"column": 94
} | {
"line": 906,
"column": 8
} | [
{
"pp": "case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp✝ : Fact (1 ≤ p)\ninst✝⁶ : SeminormedAddCommGroup α\ninst✝⁵ : SeminormedAddCommGroup β\ninst✝⁴ : SeminormedRing 𝕜\ninst✝³ : Module 𝕜 α\ninst✝² : Module 𝕜 β\ninst✝¹ : IsBoundedSMul 𝕜 α\ninst✝ : IsBoundedSMul 𝕜 β\nc : 𝕜\nf : WithLp p ... | [
"case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp✝ : Fact (1 ≤ p)\ninst✝⁶ : SeminormedAddCommGroup α\ninst✝⁵ : SeminormedAddCommGroup β\ninst✝⁴ : SeminormedRing 𝕜\ninst✝³ : Module 𝕜 α\ninst✝² : Module 𝕜 β\ninst✝¹ : IsBoundedSMul 𝕜 α\ninst✝ : IsBoundedSMul 𝕜 β\nc : 𝕜\nf : WithLp p (α × β)\nhp ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 930,
"column": 86
} | {
"line": 930,
"column": 94
} | {
"line": 931,
"column": 8
} | [
{
"pp": "case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp✝ : Fact (1 ≤ p)\ninst✝⁶ : SeminormedAddCommGroup α\ninst✝⁵ : SeminormedAddCommGroup β\ninst✝⁴ : SeminormedRing 𝕜\ninst✝³ : Module 𝕜 α\ninst✝² : Module 𝕜 β\ninst✝¹ : NormSMulClass 𝕜 α\ninst✝ : NormSMulClass 𝕜 β\nc : 𝕜\nf : WithLp p ... | [
"case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp✝ : Fact (1 ≤ p)\ninst✝⁶ : SeminormedAddCommGroup α\ninst✝⁵ : SeminormedAddCommGroup β\ninst✝⁴ : SeminormedRing 𝕜\ninst✝³ : Module 𝕜 α\ninst✝² : Module 𝕜 β\ninst✝¹ : NormSMulClass 𝕜 α\ninst✝ : NormSMulClass 𝕜 β\nc : 𝕜\nf : WithLp p (α × β)\nhp ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 50
} | {
"line": 193,
"column": 0
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddCommGroup (β i)\np : ℝ≥0∞\ni : ι\na b : β i\n⊢ single p i (a - b) = single p i a - single p i b",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"WithLp",
... | [] | simp_rw [← toLp_single, Pi.single_sub, toLp_sub] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 50
} | {
"line": 193,
"column": 0
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddCommGroup (β i)\np : ℝ≥0∞\ni : ι\na b : β i\n⊢ single p i (a - b) = single p i a - single p i b",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"WithLp",
... | [] | simp_rw [← toLp_single, Pi.single_sub, toLp_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 50
} | {
"line": 193,
"column": 0
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddCommGroup (β i)\np : ℝ≥0∞\ni : ι\na b : β i\n⊢ single p i (a - b) = single p i a - single p i b",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"WithLp",
... | [] | simp_rw [← toLp_single, Pi.single_sub, toLp_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 672,
"column": 18
} | {
"line": 672,
"column": 88
} | {
"line": 672,
"column": 88
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : WithLp ∞ ((i : ι) → β i)\n⊢ edist x.ofLp y.ofLp ≤ edist x y",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"WithLp",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
... | [] | simpa only [ENNReal.coe_one, one_mul] using lipschitzWith_ofLp ∞ β x y | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 672,
"column": 18
} | {
"line": 672,
"column": 88
} | {
"line": 672,
"column": 88
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : WithLp ∞ ((i : ι) → β i)\n⊢ edist x.ofLp y.ofLp ≤ edist x y",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"WithLp",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
... | [] | simpa only [ENNReal.coe_one, one_mul] using lipschitzWith_ofLp ∞ β x y | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 672,
"column": 18
} | {
"line": 672,
"column": 88
} | {
"line": 672,
"column": 88
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : WithLp ∞ ((i : ι) → β i)\n⊢ edist x.ofLp y.ofLp ≤ edist x y",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"WithLp",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
... | [] | simpa only [ENNReal.coe_one, one_mul] using lipschitzWith_ofLp ∞ β x y | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 822,
"column": 34
} | {
"line": 822,
"column": 44
} | {
"line": 822,
"column": 44
} | [
{
"pp": "case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\nhp✝ : Fact (1 ≤ p)\ninst✝⁴ : Fintype ι\ninst✝³ : SeminormedRing 𝕜\ninst✝² : (i : ι) → SeminormedAddCommGroup (β i)\ninst✝¹ : (i : ι) → Module 𝕜 (β i)\ninst✝ : ∀ (i : ι), IsBoundedSMul 𝕜 (β i)\nc : 𝕜\nf : PiLp p β\n... | [
"case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\nhp✝ : Fact (1 ≤ p)\ninst✝⁴ : Fintype ι\ninst✝³ : SeminormedRing 𝕜\ninst✝² : (i : ι) → SeminormedAddCommGroup (β i)\ninst✝¹ : (i : ι) → Module 𝕜 (β i)\ninst✝ : ∀ (i : ι), IsBoundedSMul 𝕜 (β i)\nc : 𝕜\nf : PiLp p β\nhp : 1 ≤ p.t... | smul_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 166,
"column": 8
} | {
"line": 167,
"column": 70
} | {
"line": 168,
"column": 6
} | [
{
"pp": "case mpr.refine_1.inl\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\na x : ι → ℝ\ni : ι\nh : min 0 (a i) ≤ x i ∧ x i ≤ max 0 (a i)\nhai : a i ≤ 0\n⊢ 0 ≤ (fun i ↦ x i / a i) i ∧ (fun i ↦ x i / a i) i ≤ 1",
"ppTerm": "?mpr.refine_1.inl",
"assigned": true,
"usedConstants": [
"... | [] | rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai] at h
exact ⟨div_nonneg_of_nonpos h.2 hai, div_le_one_of_ge h.1 hai⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 166,
"column": 8
} | {
"line": 167,
"column": 70
} | {
"line": 168,
"column": 6
} | [
{
"pp": "case mpr.refine_1.inl\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\na x : ι → ℝ\ni : ι\nh : min 0 (a i) ≤ x i ∧ x i ≤ max 0 (a i)\nhai : a i ≤ 0\n⊢ 0 ≤ (fun i ↦ x i / a i) i ∧ (fun i ↦ x i / a i) i ≤ 1",
"ppTerm": "?mpr.refine_1.inl",
"assigned": true,
"usedConstants": [
"... | [] | rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai] at h
exact ⟨div_nonneg_of_nonpos h.2 hai, div_le_one_of_ge h.1 hai⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 77,
"column": 38
} | {
"line": 79,
"column": 34
} | {
"line": 81,
"column": 0
} | [
{
"pp": "E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁴ : NormedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\nF : Type u_4\ninst✝¹ : AddCommGroup F\ninst✝ : Module K F\nf : E ≃ₗ[K] F\n⊢ Submodule.map (↑(LinearEquiv.restrictScalars ℤ f)) (span ℤ (Set.range ⇑b)) = span ℤ (Se... | [] | by
simp_rw [Submodule.map_span, LinearEquiv.coe_coe, LinearEquiv.restrictScalars_apply,
Basis.coe_map, Set.range_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 28
} | {
"line": 101,
"column": 29
} | [
{
"pp": "E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁵ : NormedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : LinearOrder K\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace K F\nf : E ≃ₗ[K] F\nx : F\n⊢ x ∈ ⇑f '' fundamentalDomain b ↔ x ∈ fundament... | [
"E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁵ : NormedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : LinearOrder K\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace K F\nf : E ≃ₗ[K] F\nx : F\n⊢ x ∈ ⇑f '' fundamentalDomain b ↔ ∀ (i : ι), ((b.map f).rep... | mem_fundamentalDomain, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 344,
"column": 6
} | {
"line": 344,
"column": 62
} | {
"line": 344,
"column": 62
} | [
{
"pp": "case intro\nE : Type u_1\nι : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nb : Basis ι ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : Finite ι\nval✝ : Fintype ι\nthis : FiniteDimensional ℝ E\nD : Set (ι → ℝ) := Set.univ.pi fun x ↦ Set.Ico 0 1\n⊢ Measurab... | [
"case intro\nE : Type u_1\nι : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nb : Basis ι ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : Finite ι\nval✝ : Fintype ι\nthis : FiniteDimensional ℝ E\nD : Set (ι → ℝ) := Set.univ.pi fun x ↦ Set.Ico 0 1\n⊢ MeasurableSet (⇑↑b.e... | (_ : fundamentalDomain b = b.equivFun.toLinearMap ⁻¹' D) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 416,
"column": 4
} | {
"line": 416,
"column": 26
} | {
"line": 416,
"column": 27
} | [
{
"pp": "E : Type u_1\nι : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nb : Basis ι ℝ E\ninst✝³ : Fintype ι\ninst✝² : MeasurableSpace E\nμ : Measure E\ninst✝¹ : BorelSpace E\ninst✝ : μ.IsAddHaarMeasure\nthis : FiniteDimensional ℝ E\nx : E\nhx : (∀ (i : ι), 0 ≤ (b.repr x) i ∧ (b.repr x) i ≤... | [
"E : Type u_1\nι : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nb : Basis ι ℝ E\ninst✝³ : Fintype ι\ninst✝² : MeasurableSpace E\nμ : Measure E\ninst✝¹ : BorelSpace E\ninst✝ : μ.IsAddHaarMeasure\nthis : FiniteDimensional ℝ E\nx : E\nhx : (∀ (i : ι), 0 ≤ (b.repr x) i ∧ (b.repr x) i ≤ 1) ∧ ¬∀ (i ... | mem_fundamentalDomain, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 280,
"column": 67
} | {
"line": 280,
"column": 81
} | {
"line": 282,
"column": 0
} | [
{
"pp": "ι : Type u_1\nI J : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → Prepartition J\n⊢ J ∈ π.biUnion πi ↔ ∃ J' ∈ π, J ∈ πi J'",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"BoxIntegral.Prepartition.biUnion._proof_4",
"BoxIntegral.Prepartition",
"congrArg",
"... | [] | simp [biUnion] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 280,
"column": 67
} | {
"line": 280,
"column": 81
} | {
"line": 282,
"column": 0
} | [
{
"pp": "ι : Type u_1\nI J : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → Prepartition J\n⊢ J ∈ π.biUnion πi ↔ ∃ J' ∈ π, J ∈ πi J'",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"BoxIntegral.Prepartition.biUnion._proof_4",
"BoxIntegral.Prepartition",
"congrArg",
"... | [] | simp [biUnion] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 280,
"column": 67
} | {
"line": 280,
"column": 81
} | {
"line": 282,
"column": 0
} | [
{
"pp": "ι : Type u_1\nI J : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → Prepartition J\n⊢ J ∈ π.biUnion πi ↔ ∃ J' ∈ π, J ∈ πi J'",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"BoxIntegral.Prepartition.biUnion._proof_4",
"BoxIntegral.Prepartition",
"congrArg",
"... | [] | simp [biUnion] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 707,
"column": 4
} | {
"line": 707,
"column": 25
} | {
"line": 708,
"column": 4
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (�... | [
"case inl\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0)\nt : Se... | filter_upwards with r | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 767,
"column": 4
} | {
"line": 767,
"column": 28
} | {
"line": 768,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) ... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (�... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 185,
"column": 28
} | {
"line": 185,
"column": 76
} | {
"line": 185,
"column": 76
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\ninst✝ : AddCommMonoid M\nI : Box ι\ni : ι\nx : ℝ\nf : Box ι → M\n⊢ ∑ J ∈ {I.splitLower i x, I.splitUpper i x}, Option.elim' 0 f J =\n Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x)",
"ppTerm": "?m.39",
"assigned": true,
"usedConstant... | [
"ι : Type u_1\nM : Type u_3\ninst✝ : AddCommMonoid M\nI : Box ι\ni : ι\nx : ℝ\nf : Box ι → M\n⊢ Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x) =\n Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x)"
] | Finset.sum_pair (I.splitLower_ne_splitUpper i x) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Partition.Measure | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 38
} | {
"line": 88,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Finite ι\nI : Box ι\nπ : Prepartition I\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\n⊢ ∀ b ∈ π.boxes, MeasurableSet ↑b",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"BoxIntegral.Box.measurableSet_coe",
"Finset",
"Membership.mem... | [] | exact fun J _ => J.measurableSet_coe | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Oscillation | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 71
} | {
"line": 100,
"column": 0
} | [
{
"pp": "E : Type u\nF : Type v\ninst✝¹ : PseudoEMetricSpace F\ninst✝ : TopologicalSpace E\nf : E → F\nx : E\n⊢ oscillationWithin f univ x = 0 ↔ ContinuousWithinAt f univ x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.univ",
"Set.mem_univ",
"OscillationWithin.eq_z... | [] | exact OscillationWithin.eq_zero_iff_continuousWithinAt f (mem_univ x) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Oscillation | {
"line": 120,
"column": 8
} | {
"line": 120,
"column": 49
} | {
"line": 120,
"column": 50
} | [
{
"pp": "E : Type u\nF : Type v\ninst✝¹ : PseudoEMetricSpace F\ninst✝ : PseudoEMetricSpace E\nK : Set E\nf : E → F\nD : Set E\nε : ℝ≥0∞\ncomp : IsCompact K\nhK : ∀ x ∈ K, oscillationWithin f D x < ε\nS : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}\nr : ℝ\nx✝¹ : r > 0\nx... | [
"E : Type u\nF : Type v\ninst✝¹ : PseudoEMetricSpace F\ninst✝ : PseudoEMetricSpace E\nK : Set E\nf : E → F\nD : Set E\nε : ℝ≥0∞\ncomp : IsCompact K\nhK : ∀ x ∈ K, oscillationWithin f D x < ε\nS : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}\nr : ℝ\nx✝¹ : r > 0\nx : E\nx✝ : x... | ← ofReal_add (by linarith) (by linarith), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Partition.Additive | {
"line": 122,
"column": 2
} | {
"line": 137,
"column": 65
} | {
"line": 139,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nn : ℕ\nN : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\nI₀✝ : WithTop (Box ι)\nI : Box ι\ni : ι\ninst✝ : Finite ι\nf : Box ι → M\nI₀ : WithTop (Box ι)\nhf :\n ∀ (I : Box ι),\n ↑I ≤ I₀ →\n ∀ {i : ι} {x : ℝ},\n x ∈ Set.Ioo (I.lower i) (I.upper ... | [] | refine ⟨f, ?_⟩
replace hf (I : Box ι) (hI : ↑I ≤ I₀) (s) : ∑ J ∈ (splitMany I s).boxes, f J = f I := by
induction s using Finset.induction_on with
| empty => simp
| insert a s _ ihs =>
rw [splitMany_insert, inf_split, ← ihs, biUnion_boxes, sum_biUnion_boxes]
refine Finset.sum_congr rfl fun J' ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Partition.Additive | {
"line": 122,
"column": 2
} | {
"line": 137,
"column": 65
} | {
"line": 139,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nn : ℕ\nN : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\nI₀✝ : WithTop (Box ι)\nI : Box ι\ni : ι\ninst✝ : Finite ι\nf : Box ι → M\nI₀ : WithTop (Box ι)\nhf :\n ∀ (I : Box ι),\n ↑I ≤ I₀ →\n ∀ {i : ι} {x : ℝ},\n x ∈ Set.Ioo (I.lower i) (I.upper ... | [] | refine ⟨f, ?_⟩
replace hf (I : Box ι) (hI : ↑I ≤ I₀) (s) : ∑ J ∈ (splitMany I s).boxes, f J = f I := by
induction s using Finset.induction_on with
| empty => simp
| insert a s _ ihs =>
rw [splitMany_insert, inf_split, ← ihs, biUnion_boxes, sum_biUnion_boxes]
refine Finset.sum_congr rfl fun J' ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 395,
"column": 17
} | {
"line": 395,
"column": 26
} | {
"line": 395,
"column": 27
} | [
{
"pp": "ι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ns : Set (ι → ℝ)\nF : (ι → ℝ) → ℝ\ninst✝ : Fintype ι\nB : Box ι\nhB : hasIntegralVertices B\nhs₀ : s ≤ ↑B\nthis : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))\n⊢ ∑ i ∈ (s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))).toFinset, F i / ↑n ^... | [
"ι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ns : Set (ι → ℝ)\nF : (ι → ℝ) → ℝ\ninst✝ : Fintype ι\nB : Box ι\nhB : hasIntegralVertices B\nhs₀ : s ≤ ↑B\nthis : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))\n⊢ ∑ i ∈ (s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))).toFinset, F i / ↑n ^ card ι =\n ... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1149,
"column": 2
} | {
"line": 1150,
"column": 68
} | {
"line": 1152,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : RCLike 𝕜\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : Fintype ι\ninst✝¹ : FiniteDimensional 𝕜 E\nn : ℕ\nhn : finrank 𝕜 E = n\ninst✝ : DecidableEq ι\nV : ι → Submodule 𝕜 E\nhV : IsInternal V\nhV' : OrthogonalFamily 𝕜 (... | [] | apply Finset.card_eq_of_equiv_fin
simpa using hV.subordinateOrthonormalBasisIndexFiberEquiv hn hV' i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1149,
"column": 2
} | {
"line": 1150,
"column": 68
} | {
"line": 1152,
"column": 0
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : RCLike 𝕜\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : Fintype ι\ninst✝¹ : FiniteDimensional 𝕜 E\nn : ℕ\nhn : finrank 𝕜 E = n\ninst✝ : DecidableEq ι\nV : ι → Submodule 𝕜 E\nhV : IsInternal V\nhV' : OrthogonalFamily 𝕜 (... | [] | apply Finset.card_eq_of_equiv_fin
simpa using hV.subordinateOrthonormalBasisIndexFiberEquiv hn hV' i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Integrability | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 44
} | {
"line": 141,
"column": 44
} | [
{
"pp": "ι : Type u\nE : Type v\ninst✝³ : Fintype ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nl : IntegrationParams\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhl : l.bRiemann = false\nε : ℝ≥0\nε0 : 0 < ε\nδ : ℕ → ℝ≥0\nδ0 : ∀ (i : ℕ), 0 < δ i\nc✝ : ℝ≥0\nhδ... | [
"ι : Type u\nE : Type v\ninst✝³ : Fintype ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nl : IntegrationParams\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhl : l.bRiemann = false\nε : ℝ≥0\nε0 : 0 < ε\nδ : ℕ → ℝ≥0\nδ0 : ∀ (i : ℕ), 0 < δ i\nc✝ : ℝ≥0\nhδc : HasSum δ... | refine (norm_sum_le_of_le _ this).trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 95,
"column": 2
} | {
"line": 101,
"column": 25
} | {
"line": 103,
"column": 0
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI : Box ι\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ : TaggedPrepartition I\nπi : (J : Box ι) → Prepartition J\nhπi : ∀ J ∈ π, (πi J).IsPartition\nJ : ... | [] | calc
(∑ J' ∈ (πi J).boxes, vol J' (f (π.tag <| π.toPrepartition.biUnionIndex πi J'))) =
∑ J' ∈ (πi J).boxes, vol J' (f (π.tag J)) :=
sum_congr rfl fun J' hJ' => by rw [Prepartition.biUnionIndex_of_mem _ hJ hJ']
_ = vol J (f (π.tag J)) :=
(vol.map ⟨⟨fun g : E →L[ℝ] F => g (f (π.tag J)), rfl⟩,... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1217,
"column": 8
} | {
"line": 1217,
"column": 90
} | {
"line": 1217,
"column": 90
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝¹⁰ : RCLike 𝕜\nE✝ : Type u_4\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : InnerProductSpace 𝕜 E✝\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace ℝ F\nF' : Type u_6\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : InnerProductSpace ℝ F'\n... | [
"ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝¹⁰ : RCLike 𝕜\nE✝ : Type u_4\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : InnerProductSpace 𝕜 E✝\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace ℝ F\nF' : Type u_6\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : InnerProductSpace ℝ F'\ninst✝³ : Fin... | norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (L (p1 x)) (L3 (p2 x)) Mx_orth | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 331,
"column": 2
} | {
"line": 332,
"column": 64
} | {
"line": 334,
"column": 0
} | [
{
"pp": "case neg\nι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nc : ℝ\nhc : c ≠ 0\nhf : ¬Integrable I l f vol\n... | [] | · have : ¬Integrable I l (fun x => c • f x) vol := mt (fun h => h.of_smul hc) hf
rw [integral, integral, dif_neg hf, dif_neg this, smul_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 23
} | {
"line": 90,
"column": 2
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\na b : ι\nh₀ : a ≠ b\nthis : ∀ (a b : ι), a < b → ⟪gramSchmidt 𝕜 f a, gr... | [] | exact this _ _ hb | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 19
} | {
"line": 410,
"column": 19
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nh : Integrable I l f vol\nε : ℝ\nc : ℝ≥0\n⊢ l.RCond (h.co... | [
"ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nh : Integrable I l f vol\nε : ℝ\nc : ℝ≥0\n⊢ l.RCond ((if hε : 0 < ε t... | rw [convergenceR] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 305,
"column": 2
} | {
"line": 309,
"column": 82
} | {
"line": 311,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\nh₀ : LinearIndependent 𝕜 f\n⊢ LinearIndependent 𝕜 (gramSchmidtNormed 𝕜 f)",
... | [] | unfold gramSchmidtNormed
have (i : ι) : IsUnit (‖gramSchmidt 𝕜 f i‖⁻¹ : 𝕜) :=
isUnit_iff_ne_zero.mpr (by simp [gramSchmidt_ne_zero i h₀])
let w : ι → 𝕜ˣ := fun i ↦ (this i).unit
apply (gramSchmidt_linearIndependent h₀).units_smul (w := fun i ↦ (this i).unit) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 305,
"column": 2
} | {
"line": 309,
"column": 82
} | {
"line": 311,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\nh₀ : LinearIndependent 𝕜 f\n⊢ LinearIndependent 𝕜 (gramSchmidtNormed 𝕜 f)",
... | [] | unfold gramSchmidtNormed
have (i : ι) : IsUnit (‖gramSchmidt 𝕜 f i‖⁻¹ : 𝕜) :=
isUnit_iff_ne_zero.mpr (by simp [gramSchmidt_ne_zero i h₀])
let w : ι → 𝕜ˣ := fun i ↦ (this i).unit
apply (gramSchmidt_linearIndependent h₀).units_smul (w := fun i ↦ (this i).unit) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 238,
"column": 4
} | {
"line": 239,
"column": 36
} | {
"line": 241,
"column": 0
} | [
{
"pp": "case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn✝ : ℕ\n_i : Fact (finrank ℝ E = n✝ + 1)\no : Orientation ℝ E (Fin (n✝ + 1))\nb : OrthonormalBasis (Fin (n✝ + 1)) ℝ E\nv : Fin (n✝ + 1) → E\n⊢ |o.volumeForm v| = |b.toBasis.det v|",
"ppTerm": "?succ",
"assign... | [] | rw [o.volumeForm_robust (b.adjustToOrientation o) (b.orientation_adjustToOrientation o),
b.abs_det_adjustToOrientation] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 238,
"column": 4
} | {
"line": 239,
"column": 36
} | {
"line": 241,
"column": 0
} | [
{
"pp": "case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn✝ : ℕ\n_i : Fact (finrank ℝ E = n✝ + 1)\no : Orientation ℝ E (Fin (n✝ + 1))\nb : OrthonormalBasis (Fin (n✝ + 1)) ℝ E\nv : Fin (n✝ + 1) → E\n⊢ |o.volumeForm v| = |b.toBasis.det v|",
"ppTerm": "?succ",
"assign... | [] | rw [o.volumeForm_robust (b.adjustToOrientation o) (b.orientation_adjustToOrientation o),
b.abs_det_adjustToOrientation] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 238,
"column": 4
} | {
"line": 239,
"column": 36
} | {
"line": 241,
"column": 0
} | [
{
"pp": "case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn✝ : ℕ\n_i : Fact (finrank ℝ E = n✝ + 1)\no : Orientation ℝ E (Fin (n✝ + 1))\nb : OrthonormalBasis (Fin (n✝ + 1)) ℝ E\nv : Fin (n✝ + 1) → E\n⊢ |o.volumeForm v| = |b.toBasis.det v|",
"ppTerm": "?succ",
"assign... | [] | rw [o.volumeForm_robust (b.adjustToOrientation o) (b.orientation_adjustToOrientation o),
b.abs_det_adjustToOrientation] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Integrability | {
"line": 332,
"column": 4
} | {
"line": 332,
"column": 74
} | {
"line": 333,
"column": 4
} | [
{
"pp": "case left\nι : Type u\nE : Type v\ninst✝⁴ : Fintype ι\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nI : Box ι\nhb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nhc : ∀ᵐ (x : ι → ℝ) ∂μ, ContinuousAt f x\nl ... | [
"case left\nι : Type u\nE : Type v\ninst✝⁴ : Fintype ι\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nI : Box ι\nhb : ∃ C, ∀ x ∈ Box.Icc I, ‖f x‖ ≤ C\nhc : ∀ᵐ (x : ι → ℝ) ∂μ, ContinuousAt f x\nl : Integratio... | refine measure_eq_measure_of_null_sdiff s.inter_subset_left ?_ |>.symm | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 51
} | {
"line": 77,
"column": 51
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\np q : ℕ\nhpq : q < p\n⊢ Tendsto (fun x ↦ x ^ p / x ^ q) atTop atTop",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"congrArg",
"PartialOrder.to... | [
"𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\np q : ℕ\nhpq : q < p\n⊢ Tendsto (fun x ↦ x ^ (↑p - ↑q)) atTop atTop"
] | tendsto_congr' pow_div_pow_eventuallyEq_atTop | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 83,
"column": 6
} | {
"line": 83,
"column": 51
} | {
"line": 83,
"column": 51
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np q : ℕ\nhpq : p < q\n⊢ Tendsto (fun x ↦ x ^ p / x ^ q) atTop (𝓝 0)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\np q : ℕ\nhpq : p < q\n⊢ Tendsto (fun x ↦ x ^ (↑p - ↑q)) atTop (𝓝 0)"
] | tendsto_congr' pow_div_pow_eventuallyEq_atTop | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.PSeries | {
"line": 295,
"column": 6
} | {
"line": 296,
"column": 12
} | {
"line": 298,
"column": 2
} | [
{
"pp": "case inl.h_mono\np : ℝ\nhp : 0 ≤ p\n⊢ ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → (↑n ^ p)⁻¹ ≤ (↑m ^ p)⁻¹",
"ppTerm": "?inl.h_mono",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instPow",
"Real.partialOrder",
"Real.rpow_pos_of_pos",
"Real",
"Preorder.toLT",
... | [] | intro m n hm hmn
gcongr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.PSeries | {
"line": 295,
"column": 6
} | {
"line": 296,
"column": 12
} | {
"line": 298,
"column": 2
} | [
{
"pp": "case inl.h_mono\np : ℝ\nhp : 0 ≤ p\n⊢ ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → (↑n ^ p)⁻¹ ≤ (↑m ^ p)⁻¹",
"ppTerm": "?inl.h_mono",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instPow",
"Real.partialOrder",
"Real.rpow_pos_of_pos",
"Real",
"Preorder.toLT",
... | [] | intro m n hm hmn
gcongr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Cardinal | {
"line": 38,
"column": 49
} | {
"line": 38,
"column": 82
} | {
"line": 40,
"column": 0
} | [
{
"pp": "R : Type u\nM' : Type v\ninst✝² : Semiring R\ninst✝¹ : Infinite M'\ninst✝ : Nontrivial R\n⊢ #R[M'] = max (lift.{v, u} #R) (lift.{u, v} #M')",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Cardinal.mk_finsupp_lift_of_infinite",
"Cardina... | [] | by simp [MonoidAlgebra, max_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MonoidAlgebra.Cardinal | {
"line": 48,
"column": 49
} | {
"line": 48,
"column": 82
} | {
"line": 50,
"column": 0
} | [
{
"pp": "R : Type u\nM' : Type v\ninst✝² : Semiring R\ninst✝¹ : Nonempty M'\ninst✝ : Infinite R\n⊢ #R[M'] = max (lift.{v, u} #R) (lift.{u, v} #M')",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Cardinal",
"congrArg",
"Cardinal.lift",
... | [] | by simp [MonoidAlgebra, max_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MonoidAlgebra.Grading | {
"line": 155,
"column": 26
} | {
"line": 155,
"column": 30
} | {
"line": 156,
"column": 4
} | [
{
"pp": "case refine_2\nM : Type u_1\nι : Type u_2\nR : Type u_3\ninst✝³ : AddMonoid M\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : CommSemiring R\nf : M →+ ι\ni : ι\nx : R[M]\nm : M\nb : R\ny : M →₀ R\nhmy : m ∉ y.support\nhb : b ≠ 0\nih : ∀ (hx : y ∈ gradeBy R (⇑f) i), (decomposeAux f) ↑⟨y, hx⟩ = (D... | [
"case refine_2\nM : Type u_1\nι : Type u_2\nR : Type u_3\ninst✝³ : AddMonoid M\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : CommSemiring R\nf : M →+ ι\ni : ι\nx : R[M]\nm : M\nb : R\ny : M →₀ R\nhmy : m ∉ y.support\nhb : b ≠ 0\nih : ∀ (hx : y ∈ gradeBy R (⇑f) i), (decomposeAux f) ↑⟨y, hx⟩ = (DirectSum.of ... | hmby | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Data.Nat.Factorial.DoubleFactorial | {
"line": 64,
"column": 8
} | {
"line": 64,
"column": 16
} | {
"line": 64,
"column": 17
} | [
{
"pp": "n : ℕ\n⊢ (2 * (n + 1))‼ = 2 ^ (n + 1) * (n + 1)!",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"Nat.doubleFactorial",
"id",
"instMulNat",
"instO... | [
"n : ℕ\n⊢ (2 * n + 2 * 1)‼ = 2 ^ (n + 1) * (n + 1)!"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Comap | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 8
} | {
"line": 83,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_4\ninst✝ : CommSemiring R\nf : MvPolynomial σ R →ₐ[R] MvPolynomial σ R\nhf : ∀ (φ : MvPolynomial σ R), f φ = φ\nx : σ → R\n⊢ f = AlgHom.id R (MvPolynomial σ R)",
"ppTerm": "?m.130",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidA... | [
"σ : Type u_1\nR : Type u_4\ninst✝ : CommSemiring R\nf : MvPolynomial σ R →ₐ[R] MvPolynomial σ R\nhf : ∀ (φ : MvPolynomial σ R), f φ = φ\nx : σ → R\nφ : σ\n⊢ f (X φ) = (AlgHom.id R (MvPolynomial σ R)) (X φ)"
] | ext1 φ | Lean.Elab.Tactic.Ext._aux_Init_Ext___macroRules_Lean_Elab_Tactic_Ext_tacticExt1____1 | Lean.Elab.Tactic.Ext.tacticExt1___ |
Mathlib.Algebra.MvPolynomial.Polynomial | {
"line": 24,
"column": 2
} | {
"line": 29,
"column": 13
} | {
"line": 31,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nσ : Type u_3\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nx : S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np : MvPolynomial σ R\n⊢ Polynomial.eval x (eval₂ f g p) = eval₂ ((Polynomial.evalRingHom x).comp f) (fun s ↦ Polynomial.eval x (g s)) p",
"ppTerm": "?m.30"... | [] | apply induction_on p
· simp
· intro p q hp hq
simp [hp, hq]
· intro p n hp
simp [hp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Polynomial | {
"line": 24,
"column": 2
} | {
"line": 29,
"column": 13
} | {
"line": 31,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nσ : Type u_3\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nx : S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np : MvPolynomial σ R\n⊢ Polynomial.eval x (eval₂ f g p) = eval₂ ((Polynomial.evalRingHom x).comp f) (fun s ↦ Polynomial.eval x (g s)) p",
"ppTerm": "?m.30"... | [] | apply induction_on p
· simp
· intro p q hp hq
simp [hp, hq]
· intro p n hp
simp [hp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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