module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 574,
"column": 4
} | {
"line": 575,
"column": 27
} | {
"line": 576,
"column": 2
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\nhmem1 : s.medial.points 0 ∈ affineSpan k (Set.range s.medial.points)\nhmem2 : s.medial.points 0 ∈ a... | [] | rw [this, Submodule.span_smul_eq_of_isUnit]
simpa using NeZero.ne n | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 870,
"column": 31
} | {
"line": 870,
"column": 55
} | {
"line": 870,
"column": 55
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nV : Type u_3\nW : Type u_4\nP : Type u_5\ninst✝¹⁴ : Ring R\ninst✝¹³ : Ring S\ninst✝¹² : AddCommGroup V\ninst✝¹¹ : Module R V\ninst✝¹⁰ : Module.Finite R V\ninst✝⁹ : Module.Free R V\ninst✝⁸ : AffineSpace V P\ninst✝⁷ : AddCommGroup W\ninst✝⁶ : Module R W\ninst✝⁵ : Module S W\ni... | [
"R : Type u_1\nS : Type u_2\nV : Type u_3\nW : Type u_4\nP : Type u_5\ninst✝¹⁴ : Ring R\ninst✝¹³ : Ring S\ninst✝¹² : AddCommGroup V\ninst✝¹¹ : Module R V\ninst✝¹⁰ : Module.Finite R V\ninst✝⁹ : Module.Free R V\ninst✝⁸ : AffineSpace V P\ninst✝⁷ : AddCommGroup W\ninst✝⁶ : Module R W\ninst✝⁵ : Module S W\ninst✝⁴ : Modu... | Module.finrank_linearMap | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.SmoothSeries | {
"line": 256,
"column": 41
} | {
"line": 287,
"column": 62
} | {
"line": 288,
"column": 0
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : IsRCLikeNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nv : ℕ → α → ℝ\nN : ℕ... | [] | by
classical
refine contDiff_iff_forall_nat_le.2 fun m hm => ?_
let t : Set α :=
{ i : α | ¬∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i }
have ht : Set.Finite t :=
haveI A :
∀ᶠ i in (Filter.cofinite : Filter α),
∀ k : ℕ, k ∈ Finset.range (m ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 560,
"column": 10
} | {
"line": 560,
"column": 61
} | {
"line": 561,
"column": 10
} | [
{
"pp": "case inr\n𝕜 : Type u_1\n𝕜' : Type u_2\nα : Type u_3\nE : α → Type u_4\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : Fact (1 ≤ p)\nf g : ↥(lp E p)\nhp' : 1 ≤ p.toReal\nhp'' : 0 < p.toReal\n⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
... | [
"case inr\n𝕜 : Type u_1\n𝕜' : Type u_2\nα : Type u_3\nE : α → Type u_4\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : Fact (1 ≤ p)\nf g : ↥(lp E p)\nhp' : 1 ≤ p.toReal\nhp'' : 0 < p.toReal\nhf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖\n⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖"
] | have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 559,
"column": 10
} | {
"line": 572,
"column": 27
} | {
"line": 573,
"column": 6
} | [
{
"pp": "case inr\n𝕜 : Type u_1\n𝕜' : Type u_2\nα : Type u_3\nE : α → Type u_4\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : Fact (1 ≤ p)\nf g : ↥(lp E p)\nhp' : 1 ≤ p.toReal\n⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRi... | [] | have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
have hf₂ := lp.hasSum_norm hp'' f
have hg₂ := lp.hasSum_norm hp'' g
-- apply Minkowski's inequality
obt... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 559,
"column": 10
} | {
"line": 572,
"column": 27
} | {
"line": 573,
"column": 6
} | [
{
"pp": "case inr\n𝕜 : Type u_1\n𝕜' : Type u_2\nα : Type u_3\nE : α → Type u_4\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : Fact (1 ≤ p)\nf g : ↥(lp E p)\nhp' : 1 ≤ p.toReal\n⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRi... | [] | have hp'' : 0 < p.toReal := zero_lt_one.trans_le hp'
have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _
have hg₁ : ∀ i, 0 ≤ ‖g i‖ := fun i => norm_nonneg _
have hf₂ := lp.hasSum_norm hp'' f
have hg₂ := lp.hasSum_norm hp'' g
-- apply Minkowski's inequality
obt... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 581,
"column": 2
} | {
"line": 581,
"column": 53
} | {
"line": 582,
"column": 2
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nhpq : p.toReal.HolderConjugate q.toReal\nf : ↥(lp E p)\ng : ↥(lp E q)\n⊢ (Summable fun i ↦ ‖↑f i‖ * ‖↑g i‖) ∧ ∑' (i : α), ‖↑f i‖ * ‖↑g i‖ ≤ ‖f‖ * ‖g‖",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants... | [
"α : Type u_3\nE : α → Type u_4\ninst✝ : (i : α) → NormedAddCommGroup (E i)\np q : ℝ≥0∞\nhpq : p.toReal.HolderConjugate q.toReal\nf : ↥(lp E p)\ng : ↥(lp E q)\nhf₁ : ∀ (i : α), 0 ≤ ‖↑f i‖\n⊢ (Summable fun i ↦ ‖↑f i‖ * ‖↑g i‖) ∧ ∑' (i : α), ‖↑f i‖ * ‖↑g i‖ ≤ ‖f‖ * ‖g‖"
] | have hf₁ : ∀ i, 0 ≤ ‖f i‖ := fun i => norm_nonneg _ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1114,
"column": 4
} | {
"line": 1115,
"column": 93
} | {
"line": 1117,
"column": 0
} | [
{
"pp": "case coe\nα : Type u_3\nE : α → Type u_4\np✝ : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : DecidableEq α\ni : α\nx : E i\nthis✝ : Nonempty α\np : ℝ≥0\nhp : 0 < ↑p\nthis : 0 < (↑p).toReal\n⊢ ∀ (b' : α), b' ≠ i → ‖↑(lp.single (↑p) i x) b'‖ ^ (↑p).toReal = 0",
"ppTerm": "?coe✝",
"as... | [] | · intro j hji
rw [lp.coeFn_single, Pi.single_eq_of_ne hji, _root_.norm_zero, Real.zero_rpow this.ne'] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1099,
"column": 2
} | {
"line": 1115,
"column": 93
} | {
"line": 1117,
"column": 0
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : DecidableEq α\nhp : 0 < p\ni : α\nx : E i\nthis : Nonempty α\n⊢ ‖lp.single p i x‖ = ‖x‖",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.... | [] | induction p with
| top =>
simp only [norm_eq_ciSup, lp.coeFn_single]
refine
ciSup_eq_of_forall_le_of_forall_lt_exists_gt (fun j => ?_) fun n hn => ⟨i, hn.trans_eq ?_⟩
· obtain rfl | hij := Decidable.eq_or_ne i j
· rw [Pi.single_eq_same]
· rw [Pi.single_eq_of_ne' hij, _root_.norm_zero]
... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.InnerProductSpace.Calculus | {
"line": 390,
"column": 2
} | {
"line": 392,
"column": 43
} | {
"line": 393,
"column": 2
} | [
{
"pp": "case pos\nn : ℕ∞\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nc : E\nr : ℝ\nh : 0 < r\n⊢ ContDiffOn ℝ (↑n) (↑(univUnitBall.trans' (unitBallBall c r h) ⋯).symm) (ball c r)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InnerPr... | [
"case neg\nn : ℕ∞\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nc : E\nr : ℝ\nh : ¬0 < r\n⊢ ContDiffOn ℝ (↑n) (↑(IsometryEquiv.vaddConst c).toHomeomorph.toOpenPartialHomeomorph.symm) (ball c r)"
] | · refine contDiffOn_univUnitBall_symm.comp (contDiff_unitBallBall_symm h).contDiffOn ?_
rw [← unitBallBall_source c r h, ← unitBallBall_target c r h]
apply OpenPartialHomeomorph.mapsTo_symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.ParametricIntegral | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 37
} | {
"line": 206,
"column": 0
} | [
{
"pp": "E : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\nx₀ : H\ns : Set H\ninst✝ : NormedSpace ℝ H\nμ : Measure ℝ\nF : H → ℝ → E\nF' : ℝ → H →L[ℝ] E\na b : ℝ\nbound : ℝ → ℝ\nhs : s ∈ 𝓝 x₀\nhF_int : IntervalIntegrable (F x₀) μ a b\nh_lip :\n ... | [] | exact ⟨⟨H₁.1, H₂.1⟩, H₁.2.sub H₂.2⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ParametricIntegral | {
"line": 296,
"column": 2
} | {
"line": 297,
"column": 43
} | {
"line": 298,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 E\nbound : α → ℝ\nF : 𝕜 → α → E\nx₀ : 𝕜\ns : Set 𝕜\nhs : s ∈ 𝓝 x₀\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeas... | [
"α : Type u_1\ninst✝⁴ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\nE : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace 𝕜 E\nbound : α → ℝ\nF : 𝕜 → α → E\nx₀ : 𝕜\ns : Set 𝕜\nhs : s ∈ 𝓝 x₀\nhF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x)... | have diff_x₀ : ∀ᵐ a ∂μ, HasDerivAt (F · a) (F' x₀ a) x₀ :=
h_diff.mono fun a ha ↦ ha x₀ (hε x₀_in) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.EverywherePos | {
"line": 67,
"column": 53
} | {
"line": 67,
"column": 96
} | {
"line": 67,
"column": 96
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nx : α\n⊢ x ∈ μ.everywherePosSubset s ↔ x ∈ s \\ {x | ∃ n ∈ 𝓝[s] x, μ n = 0}",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Filter.ins... | [] | simp [everywherePosSubset, pos_iff_ne_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 77
} | {
"line": 91,
"column": 2
} | [
{
"pp": "case inr.h_diff\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ... | [
"𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NormedSpace 𝕜 ... | · exact Eventually.of_forall fun t x _ => (L _).hasFDerivAt.comp x (h3 x t) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.EverywherePos | {
"line": 308,
"column": 4
} | {
"line": 308,
"column": 23
} | {
"line": 309,
"column": 4
} | [
{
"pp": "case refine_2\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompact... | [
"G : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompactLTTop\nK : Set G\nK_comp : ... | convert! hr using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 316,
"column": 4
} | {
"line": 316,
"column": 47
} | {
"line": 317,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\nx : β\nA : ∀ (z : β), p.c z ∈ p.iUnionUpTo p.lastStep ∨ p.τ * p.r z < p.R p.lastStep\nh : ¬p.c x ∈ p.iUnionUpTo p.lastStep\nH : p.τ * p.r x < p.R p.lastStep\nRpos : 0 < p.R p.lastStep\n⊢ p.τ⁻¹ * p.R p.lastStep < p... | [
"α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\nx : β\nA : ∀ (z : β), p.c z ∈ p.iUnionUpTo p.lastStep ∨ p.τ * p.r z < p.R p.lastStep\nh : ¬p.c x ∈ p.iUnionUpTo p.lastStep\nH : p.τ * p.r x < p.R p.lastStep\nRpos : 0 < p.R p.lastStep\n⊢ p.τ⁻¹ * p.R p.lastStep < 1 * p.R p.las... | conv_rhs => rw [← one_mul (p.R p.lastStep)] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 343,
"column": 2
} | {
"line": 354,
"column": 77
} | {
"line": 356,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : Group G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : IsFiniteMeasureOnCompacts μ'\ninst✝ : μ'.IsMulLeftInvariant\nc : ℝ≥0\nhc : ... | [] | have : IsHaarMeasure (c • μ) := IsHaarMeasure.nnreal_smul _ hc
obtain ⟨g, hg⟩ := exists_continuous_nonneg_pos (1 : G)
apply NNReal.coe_injective
calc
c * haarScalarFactor μ' (c • μ) = c * ((∫ x, g x ∂μ') / ∫ x, g x ∂(c • μ)) := by
rw [haarScalarFactor_eq_integral_div_of_continuous_nonneg_pos _ _ hg]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 343,
"column": 2
} | {
"line": 354,
"column": 77
} | {
"line": 356,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : Group G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : BorelSpace G\ninst✝³ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : IsFiniteMeasureOnCompacts μ'\ninst✝ : μ'.IsMulLeftInvariant\nc : ℝ≥0\nhc : ... | [] | have : IsHaarMeasure (c • μ) := IsHaarMeasure.nnreal_smul _ hc
obtain ⟨g, hg⟩ := exists_continuous_nonneg_pos (1 : G)
apply NNReal.coe_injective
calc
c * haarScalarFactor μ' (c • μ) = c * ((∫ x, g x ∂μ') / ∫ x, g x ∂(c • μ)) := by
rw [haarScalarFactor_eq_integral_div_of_continuous_nonneg_pos _ _ hg]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 414,
"column": 2
} | {
"line": 414,
"column": 45
} | {
"line": 415,
"column": 2
} | [
{
"pp": "case pos.h'f\nG : Type u_1\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : Group G\ninst✝⁴ : IsTopologicalGroup G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nφ : G ≃ₜ* G\nhG : LocallyCompactSpace G\nf : G → ℝ\nf_cont : Continuous[inst... | [
"case pos.int_nonzero\nG : Type u_1\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : Group G\ninst✝⁴ : IsTopologicalGroup G\ninst✝³ : MeasurableSpace G\ninst✝² : BorelSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\nφ : G ≃ₜ* G\nhG : LocallyCompactSpace G\nf : G → ℝ\nf_cont : Continuous[inst✝⁶, ... | · exact hf.1.comp_homeomorph φ.toHomeomorph | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 521,
"column": 4
} | {
"line": 521,
"column": 39
} | {
"line": 522,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ' μ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : IsFiniteMeasureOnCompacts μ'\ninst✝ : μ'.IsMulLeftInvariant\nf : G → ℝ\nhf : Continuous[inst✝⁷, _] f\nh'f :... | [
"case x_out\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ' μ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : IsFiniteMeasureOnCompacts μ'\ninst✝ : μ'.IsMulLeftInvariant\nf : G → ℝ\nhf : Continuous[inst✝⁷, _] f\nh'f :... | · simp only [ENNReal.toNNReal_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 187,
"column": 81
} | {
"line": 187,
"column": 93
} | {
"line": 188,
"column": 8
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nh :\n ∀ (δ : ℝ),\n 0 < δ → δ < 1 → ∃ s, (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card\nN : ℕ := multiplicity E + 1\nhN : N = multiplicity E + 1\nδ : ℝ\n... | [] | exact s_card | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 213,
"column": 4
} | {
"line": 214,
"column": 18
} | {
"line": 215,
"column": 4
} | [
{
"pp": "case refine_1\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nh :\n ∀ (δ : ℝ),\n 0 < δ → δ < 1 → ∃ s, (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card\nN : ℕ := multiplicity E + 1\nhN : N = multiplicity... | [
"case refine_2\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nh :\n ∀ (δ : ℝ),\n 0 < δ → δ < 1 → ∃ s, (∀ c ∈ s, ‖c‖ ≤ 2) ∧ (∀ c ∈ s, ∀ d ∈ s, c ≠ d → 1 - δ ≤ ‖c - d‖) ∧ multiplicity E < s.card\nN : ℕ := multiplicity E + 1\nhN : N = multiplicity E + 1\nF : ... | · simp only [pi_norm_le_iff_of_nonneg zero_le_two, mem_closedBall, dist_zero_right] at fmem
exact fmem i | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 759,
"column": 6
} | {
"line": 759,
"column": 70
} | {
"line": 760,
"column": 6
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEvery... | [
"case pos\nG : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s\n... | have ym : y ∈ m := m_max.mem_of_prop_insert (by simpa using h'y) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 994,
"column": 4
} | {
"line": 994,
"column": 24
} | {
"line": 995,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : CommGroup G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : LocallyCompactSpace G\ninst✝ : μ.Regular\nc : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)\nhc : μ.inv = c • μ\nthi... | [
"G : Type u_1\ninst✝⁷ : CommGroup G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : LocallyCompactSpace G\ninst✝ : μ.Regular\nc : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)\nhc : μ.inv = c • μ\nthis : map Inv.... | conv_rhs => rw [μeq] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 1020,
"column": 4
} | {
"line": 1020,
"column": 24
} | {
"line": 1021,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : CommGroup G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : LocallyCompactSpace G\ninst✝ : μ.InnerRegular\nc : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)\nhc : μ.inv = c • μ... | [
"G : Type u_1\ninst✝⁷ : CommGroup G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : LocallyCompactSpace G\ninst✝ : μ.InnerRegular\nc : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)\nhc : μ.inv = c • μ\nthis : map... | conv_rhs => rw [μeq] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.Analysis.Calculus.Darboux | {
"line": 55,
"column": 59
} | {
"line": 55,
"column": 72
} | {
"line": 55,
"column": 73
} | [
{
"pp": "a b : ℝ\nf f' : ℝ → ℝ\nhab : a ≤ b\nhf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : ℝ\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : ℝ → ℝ := fun x ↦ f x - m * x\nhg : ∀ x ∈ Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : b ∈ Icc a b\nhc : IsMinOn g (Icc a b) b\nhac : a < ... | [
"a b : ℝ\nf f' : ℝ → ℝ\nhab : a ≤ b\nhf : ∀ x ∈ Icc a b, HasDerivWithinAt f (f' x) (Icc a b) x\nm : ℝ\nhma : f' a < m\nhmb : m < f' b\nhab' : a < b\ng : ℝ → ℝ := fun x ↦ f x - m * x\nhg : ∀ x ∈ Icc a b, HasDerivWithinAt g (f' x - m) (Icc a b) x\ncmem : b ∈ Icc a b\nhc : IsMinOn g (Icc a b) b\nhac : a < b\n⊢ segment... | segment_symm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.ContinuousAlternatingMap | {
"line": 134,
"column": 2
} | {
"line": 135,
"column": 29
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case intro\n𝕜 : Type u_1\nι : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace �... | [] | exact hf.hasFDerivWithinAt.continuousAlternatingMapCompContinuousLinearMap hg.hasFDerivWithinAt
|>.differentiableWithinAt | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.FDeriv.ContinuousAlternatingMap | {
"line": 134,
"column": 2
} | {
"line": 135,
"column": 29
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case intro\n𝕜 : Type u_1\nι : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace �... | [] | exact hf.hasFDerivWithinAt.continuousAlternatingMapCompContinuousLinearMap hg.hasFDerivWithinAt
|>.differentiableWithinAt | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.ContinuousAlternatingMap | {
"line": 134,
"column": 2
} | {
"line": 135,
"column": 29
} | {
"line": 137,
"column": 0
} | [
{
"pp": "case intro\n𝕜 : Type u_1\nι : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\nH : Type u_6\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace �... | [] | exact hf.hasFDerivWithinAt.continuousAlternatingMapCompContinuousLinearMap hg.hasFDerivWithinAt
|>.differentiableWithinAt | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.DifferentialForm.VectorField | {
"line": 151,
"column": 2
} | {
"line": 152,
"column": 33
} | {
"line": 153,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nx : E\nω : E → E [⋀^Fin (n + 1)]→L[𝕜] F\nV : Fin (n + 2) → E → E\nhω : DifferentiableAt 𝕜 ω x\nhV :... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nx : E\nω : E → E [⋀^Fin (n + 1)]→L[𝕜] F\nV : Fin (n + 2) → E → E\nhω : DifferentiableWithinAt 𝕜 ω Set.univ x\nh... | simp only [← differentiableWithinAt_univ, ← extDerivWithin_univ, ← fderivWithin_univ,
← lieBracketWithin_univ] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.VectorField | {
"line": 307,
"column": 69
} | {
"line": 308,
"column": 49
} | {
"line": 310,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nV W : E → E\ns t : Set E\nx : E\nht : t ∈ 𝓝 x\n⊢ lieBracketWithin 𝕜 V W (s ∩ t) x = lieBracketWithin 𝕜 V W s x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants"... | [] | by
simp [lieBracketWithin, fderivWithin_inter, ht] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.LineDeriv.Basic | {
"line": 444,
"column": 2
} | {
"line": 445,
"column": 78
} | {
"line": 447,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nv : E\nf : E → F\nx₀ : E\ns : Set E\nhs : s ∈ 𝓝 x₀\nC : ℝ≥0\nhlip : LipschitzOnWith C f s\n⊢ ‖lineDeriv 𝕜 ... | [] | refine norm_lineDeriv_le_of_lip' 𝕜 C.coe_nonneg ?_
filter_upwards [hs] with x hx using hlip.norm_sub_le hx (mem_of_mem_nhds hs) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.LineDeriv.Basic | {
"line": 444,
"column": 2
} | {
"line": 445,
"column": 78
} | {
"line": 447,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nv : E\nf : E → F\nx₀ : E\ns : Set E\nhs : s ∈ 𝓝 x₀\nC : ℝ≥0\nhlip : LipschitzOnWith C f s\n⊢ ‖lineDeriv 𝕜 ... | [] | refine norm_lineDeriv_le_of_lip' 𝕜 C.coe_nonneg ?_
filter_upwards [hs] with x hx using hlip.norm_sub_le hx (mem_of_mem_nhds hs) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Norm | {
"line": 97,
"column": 16
} | {
"line": 97,
"column": 78
} | {
"line": 98,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : t ≠ 0\nh : HasStrictFDerivAt (fun x ↦ ‖x‖) f x\nh1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)\nh2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) x\n| Has... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : t ≠ 0\nh : HasStrictFDerivAt (fun x ↦ ‖x‖) f x\nh1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)\nh2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) x\n| t⁻¹ • t • x"
] | enter [3]; rw [← one_smul ℝ x, ← inv_mul_cancel₀ ht, mul_smul] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Norm | {
"line": 97,
"column": 16
} | {
"line": 97,
"column": 78
} | {
"line": 98,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : t ≠ 0\nh : HasStrictFDerivAt (fun x ↦ ‖x‖) f x\nh1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)\nh2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) x\n| Has... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : t ≠ 0\nh : HasStrictFDerivAt (fun x ↦ ‖x‖) f x\nh1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)\nh2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) x\n| t⁻¹ • t • x"
] | enter [3]; rw [← one_smul ℝ x, ← inv_mul_cancel₀ ht, mul_smul] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Analysis.Calculus.FDeriv.Norm | {
"line": 120,
"column": 16
} | {
"line": 120,
"column": 78
} | {
"line": 121,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : t ≠ 0\nh : HasFDerivAt (fun x ↦ ‖x‖) f x\nh1 : HasFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)\nh2 : HasFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) x\n| HasFDerivAt (fun y ↦ ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : t ≠ 0\nh : HasFDerivAt (fun x ↦ ‖x‖) f x\nh1 : HasFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)\nh2 : HasFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) x\n| t⁻¹ • t • x"
] | enter [3]; rw [← one_smul ℝ x, ← inv_mul_cancel₀ ht, mul_smul] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Norm | {
"line": 120,
"column": 16
} | {
"line": 120,
"column": 78
} | {
"line": 121,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : t ≠ 0\nh : HasFDerivAt (fun x ↦ ‖x‖) f x\nh1 : HasFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)\nh2 : HasFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) x\n| HasFDerivAt (fun y ↦ ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : t ≠ 0\nh : HasFDerivAt (fun x ↦ ‖x‖) f x\nh1 : HasFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)\nh2 : HasFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) x\n| t⁻¹ • t • x"
] | enter [3]; rw [← one_smul ℝ x, ← inv_mul_cancel₀ ht, mul_smul] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Analysis.Calculus.VectorField | {
"line": 564,
"column": 4
} | {
"line": 564,
"column": 24
} | {
"line": 565,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nx : E\nh'f : ContDiffWithinAt 𝕜 2 f s x\nhs : UniqueDiffOn ... | [
"𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace E\nf : E → F\ns : Set E\nx : E\nh'f : ContDiffWithinAt 𝕜 2 f s x\nhs : UniqueDiffOn 𝕜 s\nhx : x... | convert! this with y | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Calculus.FDeriv.Partial | {
"line": 81,
"column": 12
} | {
"line": 88,
"column": 37
} | {
"line": 89,
"column": 10
} | [
{
"pp": "𝕜 : Type u_1\nE₁ : Type u_2\nE₂ : Type u_3\nF : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E₁\ninst✝⁵ : NormedSpace 𝕜 E₁\ninst✝⁴ : NormedAddCommGroup E₂\ninst✝³ : NormedSpace 𝕜 E₂\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : IsRCLikeNormedField �... | [] | have h := tendsto_snd.prodMk <| tendsto_snd.comp <| tendsto_snd.comp <|
tendsto_fst (f := (𝓝 u.1 ×ˢ 𝓝 u.2) ×ˢ (𝓝 u.1 ×ˢ 𝓝 u.2)) (g := 𝓝 u.1)
let : NormedSpace ℝ E₁ := RestrictScalars.normedSpace ℝ 𝕜 E₁
apply isLittleO_sub_sub_fderiv (α := (E₁ × E₂) × (E₁ × E₂))
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Partial | {
"line": 81,
"column": 12
} | {
"line": 88,
"column": 37
} | {
"line": 89,
"column": 10
} | [
{
"pp": "𝕜 : Type u_1\nE₁ : Type u_2\nE₂ : Type u_3\nF : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E₁\ninst✝⁵ : NormedSpace 𝕜 E₁\ninst✝⁴ : NormedAddCommGroup E₂\ninst✝³ : NormedSpace 𝕜 E₂\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : IsRCLikeNormedField �... | [] | have h := tendsto_snd.prodMk <| tendsto_snd.comp <| tendsto_snd.comp <|
tendsto_fst (f := (𝓝 u.1 ×ˢ 𝓝 u.2) ×ˢ (𝓝 u.1 ×ˢ 𝓝 u.2)) (g := 𝓝 u.1)
let : NormedSpace ℝ E₁ := RestrictScalars.normedSpace ℝ 𝕜 E₁
apply isLittleO_sub_sub_fderiv (α := (E₁ × E₂) × (E₁ × E₂))
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Gradient.Basic | {
"line": 194,
"column": 37
} | {
"line": 194,
"column": 63
} | {
"line": 194,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\ng : 𝕜 → 𝕜\ng' u : 𝕜\nh : HasFDerivAt g ((toDual 𝕜 𝕜) g') u\n⊢ HasDerivAt g ((starRingEnd 𝕜) g') u",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"hasFDerivAt_iff_hasDerivAt",
"LinearIsometryEquiv.instEquivLike",
"IsModuleT... | [
"𝕜 : Type u_1\ninst✝ : RCLike 𝕜\ng : 𝕜 → 𝕜\ng' u : 𝕜\nh : HasDerivAt g (((toDual 𝕜 𝕜) g') 1) u\n⊢ HasDerivAt g ((starRingEnd 𝕜) g') u"
] | hasFDerivAt_iff_hasDerivAt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Gradient.Basic | {
"line": 198,
"column": 37
} | {
"line": 198,
"column": 63
} | {
"line": 198,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\ng : 𝕜 → 𝕜\ng' u : 𝕜\nh : HasDerivAt g g' u\n⊢ HasFDerivAt g ((toDual 𝕜 𝕜) ((starRingEnd 𝕜) g')) u",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"hasFDerivAt_iff_hasDerivAt",
"LinearIsometryEquiv.instEquivLike",
"IsModuleT... | [
"𝕜 : Type u_1\ninst✝ : RCLike 𝕜\ng : 𝕜 → 𝕜\ng' u : 𝕜\nh : HasDerivAt g g' u\n⊢ HasDerivAt g (((toDual 𝕜 𝕜) ((starRingEnd 𝕜) g')) 1) u"
] | hasFDerivAt_iff_hasDerivAt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 1028,
"column": 14
} | {
"line": 1028,
"column": 21
} | {
"line": 1028,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : MetricSpace α\nβ : Type u\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : HasBesicovitchCovering α\nμ : Measure α\ninst✝ : SFinite μ\ns : Set α\nf : α → Set (Set α)\n⊢ (∀ x ∈ s, f x ⊆ (fun r ↦ closedBall x r) '' Ioi 0) →\... | [
"α : Type u_1\ninst✝⁵ : MetricSpace α\nβ : Type u\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : HasBesicovitchCovering α\nμ : Measure α\ninst✝ : SFinite μ\ns : Set α\nf : α → Set (Set α)\nfsubset : ∀ x ∈ s, f x ⊆ (fun r ↦ closedBall x r) '' Ioi 0\n⊢ (∀ x ... | fsubset | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Calculus.ImplicitFunction.ProdDomain | {
"line": 59,
"column": 53
} | {
"line": 59,
"column": 63
} | {
"line": 59,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\nin... | [
"𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE₁ : Type u_2\ninst✝⁸ : NormedAddCommGroup E₁\ninst✝⁷ : NormedSpace 𝕜 E₁\ninst✝⁶ : CompleteSpace E₁\nE₂ : Type u_3\ninst✝⁵ : NormedAddCommGroup E₂\ninst✝⁴ : NormedSpace 𝕜 E₂\ninst✝³ : CompleteSpace E₂\nF : Type u_4\ninst✝² : NormedAddCommGroup F\ninst✝¹ : Norme... | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.LHopital | {
"line": 174,
"column": 5
} | {
"line": 174,
"column": 54
} | {
"line": 174,
"column": 54
} | [
{
"pp": "a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ x ∈ Iio a, HasDerivAt f (f' x) x\nhgg' : ∀ x ∈ Iio a, HasDerivAt g (g' x) x\nhg' : ∀ x ∈ Iio a, g' x ≠ 0\nhfbot : Tendsto f atBot (𝓝 0)\nhgbot : Tendsto g atBot (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) atBot l\nhdnf : ∀ x ∈ Ioi (-a), HasDerivAt (f ∘... | [] | by simpa using! hdiv.comp tendsto_neg_atTop_atBot | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 97,
"column": 4
} | {
"line": 98,
"column": 71
} | {
"line": 99,
"column": 4
} | [
{
"pp": "s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ... | [
"s : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x ∈ s₁ ∧ y ∈ s₁ ∧ x < y... | obtain ⟨v, hv, tv⟩ : ∃ v, OrdConnected v ∧ (s \ a) ∩ f ⁻¹' {z} = (s \ a) ∩ v :=
ordConnected_singleton.preimage_monotoneOn (hf.mono sdiff_subset) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Calculus.LHopital | {
"line": 280,
"column": 6
} | {
"line": 280,
"column": 38
} | {
"line": 280,
"column": 38
} | [
{
"pp": "a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[<] a) (𝓝 0)\nhga : Tendsto g (𝓝[<] a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[<] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[<] a\nhff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[<] a\nhgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y\ns... | [
"a : ℝ\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhfa : Tendsto f (𝓝[<] a) (𝓝 0)\nhga : Tendsto g (𝓝[<] a) (𝓝 0)\nhdiv : Tendsto (fun x ↦ f' x / g' x) (𝓝[<] a) l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ 𝓝[<] a\nhff' : ∀ y ∈ s₁, HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ 𝓝[<] a\nhgg' : ∀ y ∈ s₂, HasDerivAt g (g' y) y\ns₃ : Set ℝ\nh... | mem_nhdsLT_iff_exists_Ioo_subset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 357,
"column": 2
} | {
"line": 363,
"column": 18
} | {
"line": 365,
"column": 0
} | [
{
"pp": "case inr\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x\nM : MonotoneOn f [[a, b]]\nhab : b < a\n⊢ ∫ (x : ℝ)... | [] | · rw [integral_of_ge hab.le, ← integral_Icc_eq_integral_Ioc,
integral_Icc_deriv_smul_of_deriv_nonneg, integral_of_ge, ← integral_Icc_eq_integral_Ioc]
· apply M right_mem_uIcc left_mem_uIcc hab.le
· rwa [uIcc_of_ge hab.le] at hf
· grind
· grind
· exact hab.le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 485,
"column": 4
} | {
"line": 485,
"column": 68
} | {
"line": 486,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : l.IsCountablyGenerated\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nhfi : Integrable f μ\nh : Tendsto (fun i ↦ ∫ (x : α), (φ i).indicator... | [] | convert! h using 2; rw [integral_indicator (hφ.measurableSet _)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 485,
"column": 4
} | {
"line": 485,
"column": 68
} | {
"line": 486,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nE : Type u_3\ninst✝³ : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : l.IsCountablyGenerated\nφ : ι → Set α\nhφ : AECover μ l φ\nf : α → E\nhfi : Integrable f μ\nh : Tendsto (fun i ↦ ∫ (x : α), (φ i).indicator... | [] | convert! h using 2; rw [integral_indicator (hφ.measurableSet _)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 323,
"column": 8
} | {
"line": 323,
"column": 42
} | {
"line": 323,
"column": 42
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, 0 ≤ f' x\nhab : a ≤ b\nM : MonotoneOn f (Icc a b)\n⊢ Icc (f a) (f b) =ᵐ[volume] f '' Ioo a b",
"pp... | [
"F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, 0 ≤ f' x\nhab : a ≤ b\nM : MonotoneOn f (Icc a b)\n⊢ f '' Icc a b =ᵐ[volume] f '' Ioo a b"
] | ← hf.image_Icc_of_monotoneOn hab M | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 347,
"column": 8
} | {
"line": 347,
"column": 42
} | {
"line": 347,
"column": 42
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, 0 ≤ f' x\nhab : a ≤ b\nM : MonotoneOn f (Icc a b)\n⊢ Icc (f a) (f b) =ᵐ[volume] f '' Ioo a b",
"pp... | [
"F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf f' : ℝ → ℝ\na b : ℝ\ng : ℝ → F\nhf : ContinuousOn f (Icc a b)\nhff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo a b, 0 ≤ f' x\nhab : a ≤ b\nM : MonotoneOn f (Icc a b)\n⊢ f '' Icc a b =ᵐ[volume] f '' Ioo a b"
] | ← hf.image_Icc_of_monotoneOn hab M | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 347,
"column": 6
} | {
"line": 354,
"column": 83
} | {
"line": 357,
"column": 2
} | [] | [] | μ (f '' (s ∩ closedBall x r)) ≤ μ ({f x} + r • (A '' closedBall 0 1 + closedBall 0 ε)) :=
measure_mono K
_ = ENNReal.ofReal (r ^ finrank ℝ E) * μ (A '' closedBall 0 1 + closedBall 0 ε) := by
simp only [abs_of_nonneg r0, addHaar_smul, image_add_left, abs_pow, singleton_add,
measure_preima... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Calculus.LocalExtr.LineDeriv | {
"line": 33,
"column": 2
} | {
"line": 37,
"column": 49
} | {
"line": 39,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nf : E → ℝ\na b : E\nl : Filter E\nh : IsExtrFilter f l a\nh' : Tendsto (fun t ↦ a + t • b) (𝓝 0) l\n⊢ lineDeriv ℝ f a b = 0",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Real",
"LineDifferentiableAt",
... | [] | classical
exact if hd : LineDifferentiableAt ℝ f a b then
h.hasLineDerivAt_eq_zero hd.hasLineDerivAt h'
else
lineDeriv_zero_of_not_lineDifferentiableAt hd | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Analysis.Calculus.LocalExtr.LineDeriv | {
"line": 33,
"column": 2
} | {
"line": 37,
"column": 49
} | {
"line": 39,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nf : E → ℝ\na b : E\nl : Filter E\nh : IsExtrFilter f l a\nh' : Tendsto (fun t ↦ a + t • b) (𝓝 0) l\n⊢ lineDeriv ℝ f a b = 0",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Real",
"LineDifferentiableAt",
... | [] | classical
exact if hd : LineDifferentiableAt ℝ f a b then
h.hasLineDerivAt_eq_zero hd.hasLineDerivAt h'
else
lineDeriv_zero_of_not_lineDifferentiableAt hd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.LocalExtr.LineDeriv | {
"line": 33,
"column": 2
} | {
"line": 37,
"column": 49
} | {
"line": 39,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nf : E → ℝ\na b : E\nl : Filter E\nh : IsExtrFilter f l a\nh' : Tendsto (fun t ↦ a + t • b) (𝓝 0) l\n⊢ lineDeriv ℝ f a b = 0",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Real",
"LineDifferentiableAt",
... | [] | classical
exact if hd : LineDifferentiableAt ℝ f a b then
h.hasLineDerivAt_eq_zero hd.hasLineDerivAt h'
else
lineDeriv_zero_of_not_lineDifferentiableAt hd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1037,
"column": 2
} | {
"line": 1037,
"column": 22
} | {
"line": 1038,
"column": 2
} | [
{
"pp": "F : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nhf : ContDiff ℝ 1 f\nh'f : HasCompactSupport f\nx : ℝ\nI : F →L[ℝ] Completion F := Completion.toComplL\nf' : ℝ → Completion F := ⇑I ∘ f\nhf' : ContDiff ℝ 1 f'\nh'f' : HasCompactSupport f'\nthis : ‖f' x‖ₑ ≤ ∫⁻ (y : ℝ) in Ii... | [
"case e'_3\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nhf : ContDiff ℝ 1 f\nh'f : HasCompactSupport f\nx : ℝ\nI : F →L[ℝ] Completion F := Completion.toComplL\nf' : ℝ → Completion F := ⇑I ∘ f\nhf' : ContDiff ℝ 1 f'\nh'f' : HasCompactSupport f'\nthis : ‖f' x‖ₑ ≤ ∫⁻ (y : ℝ) in Iic... | convert! this with y | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 452,
"column": 16
} | {
"line": 452,
"column": 42
} | {
"line": 452,
"column": 42
} | [
{
"pp": "E : 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\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\nB : E ≃L[ℝ] E := A... | [
"E : 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\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\nhA : A.det ≠ 0\nB : E ≃L[ℝ] E := A.toContinuou... | ← ENNReal.coe_inv mpos.ne' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1134,
"column": 2
} | {
"line": 1135,
"column": 85
} | {
"line": 1136,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : ℝ → E\np : ℝ\nhp : p ≠ 0\nS : Set ℝ := Ioi 0\n⊢ ∫ (x : ℝ) in Ioi 0, (|p| * x ^ (p - 1)) • g (x ^ p) = ∫ (y : ℝ) in Ioi 0, g y",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinu... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : ℝ → E\np : ℝ\nhp : p ≠ 0\nS : Set ℝ := Ioi 0\na1 : ∀ x ∈ S, HasDerivWithinAt (fun t ↦ t ^ p) (p * x ^ (p - 1)) S x\n⊢ ∫ (x : ℝ) in Ioi 0, (|p| * x ^ (p - 1)) • g (x ^ p) = ∫ (y : ℝ) in Ioi 0, g y"
] | have a1 : ∀ x : ℝ, x ∈ S → HasDerivWithinAt (fun t : ℝ => t ^ p) (p * x ^ (p - 1)) S x :=
fun x hx => (hasDerivAt_rpow_const (Or.inl (mem_Ioi.mp hx).ne')).hasDerivWithinAt | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1140,
"column": 6
} | {
"line": 1141,
"column": 33
} | {
"line": 1142,
"column": 6
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : ℝ → E\np : ℝ\nhp : p ≠ 0\nS : Set ℝ := Ioi 0\na1 : ∀ x ∈ S, HasDerivWithinAt (fun t ↦ t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\nx : ℝ\nhx : x ∈ S\ny : ℝ\nhy : y ∈ S\nhxy : x < y\n⊢ (fun x ↦ x ^ p) y < (fun x ↦ x ^ p) x",... | [
"case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ng : ℝ → E\np : ℝ\nhp : p ≠ 0\nS : Set ℝ := Ioi 0\na1 : ∀ x ∈ S, HasDerivWithinAt (fun t ↦ t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\nx : ℝ\nhx : x ∈ S\ny : ℝ\nhy : y ∈ S\nhxy : x < y\n⊢ x ^ (-p) < y ^ (-p)"
] | rw [← inv_lt_inv₀ (rpow_pos_of_pos hx p) (rpow_pos_of_pos hy p), ← rpow_neg (le_of_lt hx),
← rpow_neg (le_of_lt hy)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 19
} | {
"line": 133,
"column": 2
} | [
{
"pp": "case pos.inl\nE : Type u_1\nF : Type u_2\nG : Type u_3\nW : Type u_4\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace ℝ E\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace ℝ F\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace ℝ G\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : NormedSpace ℝ W\nins... | [] | simp [Hf', Hg'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.RemovableSingularity | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 40
} | {
"line": 43,
"column": 4
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nc : ℂ\nhd : ∀ᶠ (z : ℂ) in 𝓝[≠] c, DifferentiableAt ℂ f z\nhc : ContinuousAt f c\nR : ℝ≥0\nhR0 : 0 < ↑R\nhRs : closedBall c ↑R ∩ {c}ᶜ ⊆ {x | (fun z ↦ DifferentiableAt ℂ f z) x}\nz : ℂ\nhz : z ∈ clos... | [
"case inl\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nf : ℂ → E\nR : ℝ≥0\nhR0 : 0 < ↑R\nz : ℂ\nhd : ∀ᶠ (z : ℂ) in 𝓝[≠] z, DifferentiableAt ℂ f z\nhc : ContinuousAt f z\nhRs : closedBall z ↑R ∩ {z}ᶜ ⊆ {x | (fun z ↦ DifferentiableAt ℂ f z) x}\nhz : z ∈ closedBall z ... | rcases eq_or_ne z c with (rfl | hne) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1188,
"column": 2
} | {
"line": 1189,
"column": 85
} | {
"line": 1190,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\np : ℝ\nhp : p ≠ 0\nS : Set ℝ := Ioi 0\n⊢ IntegrableOn (fun x ↦ (|p| * x ^ (p - 1)) • f (x ^ p)) (Ioi 0) volume ↔ IntegrableOn f (Ioi 0) volume",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Is... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\np : ℝ\nhp : p ≠ 0\nS : Set ℝ := Ioi 0\na1 : ∀ x ∈ S, HasDerivWithinAt (fun t ↦ t ^ p) (p * x ^ (p - 1)) S x\n⊢ IntegrableOn (fun x ↦ (|p| * x ^ (p - 1)) • f (x ^ p)) (Ioi 0) volume ↔ IntegrableOn f (Ioi 0) volume"
] | have a1 : ∀ x : ℝ, x ∈ S → HasDerivWithinAt (fun t : ℝ => t ^ p) (p * x ^ (p - 1)) S x :=
fun x hx => (hasDerivAt_rpow_const (Or.inl (mem_Ioi.mp hx).ne')).hasDerivWithinAt | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1192,
"column": 6
} | {
"line": 1196,
"column": 58
} | {
"line": 1197,
"column": 4
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\np : ℝ\nhp : p ≠ 0\nS : Set ℝ := Ioi 0\na1 : ∀ x ∈ S, HasDerivWithinAt (fun t ↦ t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\n⊢ InjOn (fun x ↦ x ^ p) S",
"ppTerm": "?inl",
"assigned": true,
"usedConstants":... | [] | apply StrictAntiOn.injOn
intro x hx y hy hxy
rw [← inv_lt_inv₀ (rpow_pos_of_pos hx p) (rpow_pos_of_pos hy p), ← rpow_neg (le_of_lt hx), ←
rpow_neg (le_of_lt hy)]
exact rpow_lt_rpow (le_of_lt hx) hxy (neg_pos.mpr h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1192,
"column": 6
} | {
"line": 1196,
"column": 58
} | {
"line": 1197,
"column": 4
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\np : ℝ\nhp : p ≠ 0\nS : Set ℝ := Ioi 0\na1 : ∀ x ∈ S, HasDerivWithinAt (fun t ↦ t ^ p) (p * x ^ (p - 1)) S x\nh : p < 0\n⊢ InjOn (fun x ↦ x ^ p) S",
"ppTerm": "?inl",
"assigned": true,
"usedConstants":... | [] | apply StrictAntiOn.injOn
intro x hx y hy hxy
rw [← inv_lt_inv₀ (rpow_pos_of_pos hx p) (rpow_pos_of_pos hy p), ← rpow_neg (le_of_lt hx), ←
rpow_neg (le_of_lt hy)]
exact rpow_lt_rpow (le_of_lt hx) hxy (neg_pos.mpr h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.LocallyUniformLimit | {
"line": 174,
"column": 59
} | {
"line": 178,
"column": 48
} | {
"line": 180,
"column": 0
} | [
{
"pp": "E : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nU : Set ℂ\nF : ι → ℂ → E\ninst✝ : CompleteSpace E\nu : ι → ℝ\nhu : Summable u\nhf : ∀ (i : ι), DifferentiableOn ℂ (F i) U\nhU : IsOpen U\nhF_le : ∀ (i : ι), ∀ w ∈ U, ‖F i w‖ ≤ u i\n⊢ DifferentiableOn ℂ (fun w ↦ ∑' (i :... | [] | by
classical
have hc := (tendstoUniformlyOn_tsum hu hF_le).tendstoLocallyUniformlyOn
refine hc.differentiableOn (Eventually.of_forall fun s => ?_) hU
exact DifferentiableOn.fun_sum fun i _ => hf i | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 29
} | {
"line": 250,
"column": 2
} | [
{
"pp": "H : Type u\nM : Type u_2\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyCompactSpace H\nthis :\n ∀ (x : M),\n (𝓝 x).HasBasis (fun s ↦ s ∈ 𝓝 (↑(chartAt H x) x) ∧ IsCompact s ∧ s ⊆ (chartAt H x).target) fun s ↦\n ↑(chartAt H x).symm '' s\n⊢... | [
"H : Type u\nM : Type u_2\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyCompactSpace H\nthis :\n ∀ (x : M),\n (𝓝 x).HasBasis (fun s ↦ s ∈ 𝓝 (↑(chartAt H x) x) ∧ IsCompact s ∧ s ⊆ (chartAt H x).target) fun s ↦\n ↑(chartAt H x).symm '' s\n⊢ ∀ (x : M) (... | refine .of_hasBasis this ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 276,
"column": 4
} | {
"line": 276,
"column": 45
} | {
"line": 277,
"column": 4
} | [
{
"pp": "case refine_1\nH : Type u\nM : Type u_2\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyPathConnectedSpace H\nx : M\ns : Set M\nhs : s ∈ 𝓝 x\ne : OpenPartialHomeomorph M H := chartAt H x\nt : Set M := s ∩ e.source\nht : t ∈ 𝓝 x\n⊢ ↑e.symm '' pathCo... | [
"case refine_1\nH : Type u\nM : Type u_2\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyPathConnectedSpace H\nx : M\ns : Set M\nhs : s ∈ 𝓝 x\ne : OpenPartialHomeomorph M H := ⋯\nt : Set M := ⋯\nht : t ∈ 𝓝 x\n⊢ pathComponentIn (↑e '' t) (↑e x) ∈ 𝓝 (↑e x)"
] | apply e.symm.image_mem_nhds (by simp [e]) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 518,
"column": 6
} | {
"line": 520,
"column": 54
} | {
"line": 521,
"column": 4
} | [
{
"pp": "case inl\nH : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : TopologicalSpace M'\ncm : ChartedSpace H M\ncm' : ChartedSpace H M'\ninst✝ : Nonempty H\nx : M\n⊢ Sum.elim (fun x ↦ (ChartedSpace.chartAt x).lift_openEmbe... | [] | rw [Sum.elim_inl]
left
use ChartedSpace.chartAt x, cm.chart_mem_atlas x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 518,
"column": 6
} | {
"line": 520,
"column": 54
} | {
"line": 521,
"column": 4
} | [
{
"pp": "case inl\nH : Type u\nH' : Type u_1\nM : Type u_2\nM' : Type u_3\nM'' : Type u_4\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : TopologicalSpace M'\ncm : ChartedSpace H M\ncm' : ChartedSpace H M'\ninst✝ : Nonempty H\nx : M\n⊢ Sum.elim (fun x ↦ (ChartedSpace.chartAt x).lift_openEmbe... | [] | rw [Sum.elim_inl]
left
use ChartedSpace.chartAt x, cm.chart_mem_atlas x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 396,
"column": 11
} | {
"line": 400,
"column": 52
} | {
"line": 401,
"column": 2
} | [
{
"pp": "H : Type u_1\ninst✝ : TopologicalSpace H\n⊢ ∀ e ∈ {e | ∃ s, ∃ (h : IsOpen[inst✝] s), e ≈ ofSet s h}, e.symm ∈ {e | ∃ s, ∃ (h : IsOpen[inst✝] s), e ≈ ofSet s h}",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"OpenPartialHomeomorph.ofSet",
"congrArg",
... | [] | by
rintro e ⟨s, hs, hse⟩
refine ⟨s, hs, ?_⟩
rw [← ofSet_symm]
exact OpenPartialHomeomorph.EqOnSource.symm' hse | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.HasGroupoid | {
"line": 454,
"column": 2
} | {
"line": 461,
"column": 69
} | {
"line": 462,
"column": 2
} | [
{
"pp": "H : Type u\nM : Type u_2\ninst✝⁴ : TopologicalSpace H\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nG : StructureGroupoid H\ne : OpenPartialHomeomorph M H\nhe : e ∈ atlas H M\nhs : Nonempty ↑e.source\ninst✝¹ : HasGroupoid M G\ninst✝ : ClosedUnderRestriction G\ns : Opens M := { carrier := e.s... | [
"H : Type u\nM : Type u_2\ninst✝⁴ : TopologicalSpace H\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nG : StructureGroupoid H\ne : OpenPartialHomeomorph M H\nhe : e ∈ atlas H M\nhs : Nonempty ↑e.source\ninst✝¹ : HasGroupoid M G\ninst✝ : ClosedUnderRestriction G\ns : Opens M := { carrier := e.source, is_op... | have : goal ≈ e.subtypeRestr (s := s) hs :=
(goal.eqOnSource_iff (e.subtypeRestr (s := s) hs)).mpr
⟨by
simp only [trans_toPartialEquiv, PartialEquiv.trans_source,
Homeomorph.toOpenPartialHomeomorph_source, toFun_eq_coe,
Homeomorph.toOpenPartialHomeomorph_apply, Opens.openPartialHom... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 244,
"column": 2
} | {
"line": 245,
"column": 59
} | {
"line": 247,
"column": 0
} | [
{
"pp": "H : Type u_1\nM : Type u_2\nH' : Type u_3\nM' : Type u_4\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\ninst✝ : ChartedSpace H' M'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nP : (H → H') → Set H... | [] | exact OpenPartialHomeomorph.preimage_eventuallyEq_target_inter_preimage_inter hf
(mem_chart_source H x) (chart_source_mem_nhds H' (f x)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 21
} | {
"line": 219,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : TopologicalSpace H\ninst✝ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ns : Set M\ny : M\nhy : y ∈ f.sourc... | [
"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : TopologicalSpace H\ninst✝ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ns : Set M\ny : M\nhy : y ∈ f.source\ne : Parti... | set e := f.extend I | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 24
} | {
"line": 263,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorn... | [
"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\nE' : Type u_5\nM' : Type u_6\nH' : Type u_7\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\n... | simp only [comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 342,
"column": 2
} | {
"line": 353,
"column": 68
} | {
"line": 355,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\ninst✝ : NormedSpace ℝ E\n⊢ Convex ℝ (range ↑I)",
"ppTerm": "?m.17",
"assigned": true,
"usedCons... | [] | by_cases h : IsRCLikeNormedField 𝕜
· letI : RCLike 𝕜 := h.rclike
have W := I.convex_range'
simp only [h, ↓reduceDIte, toPartialEquiv_coe] at W
simp only [Convex, StarConvex] at W ⊢
intro u hu v hv a b ha hb hab
convert! W hu hv ha hb hab using 2
· rw [← @algebraMap_smul (R := ℝ) (A := 𝕜)]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 342,
"column": 2
} | {
"line": 353,
"column": 68
} | {
"line": 355,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\ninst✝ : NormedSpace ℝ E\n⊢ Convex ℝ (range ↑I)",
"ppTerm": "?m.17",
"assigned": true,
"usedCons... | [] | by_cases h : IsRCLikeNormedField 𝕜
· letI : RCLike 𝕜 := h.rclike
have W := I.convex_range'
simp only [h, ↓reduceDIte, toPartialEquiv_coe] at W
simp only [Convex, StarConvex] at W ⊢
intro u hu v hv a b ha hb hab
convert! W hu hv ha hb hab using 2
· rw [← @algebraMap_smul (R := ℝ) (A := 𝕜)]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 454,
"column": 2
} | {
"line": 454,
"column": 29
} | {
"line": 455,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\ninst✝ : LocallyCompactSpace E\nI : ModelWithCorners 𝕜 E H\nthis : ∀ (x : H), (𝓝 x).HasBasis (fun s ↦ s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun ... | [
"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹ : TopologicalSpace H\ninst✝ : LocallyCompactSpace E\nI : ModelWithCorners 𝕜 E H\nthis : ∀ (x : H), (𝓝 x).HasBasis (fun s ↦ s ∈ 𝓝 (↑I x) ∧ IsCompact s) fun s ↦ ↑I.symm ... | refine .of_hasBasis this ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 542,
"column": 2
} | {
"line": 543,
"column": 78
} | {
"line": 545,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\nI : ModelWithCorners 𝕜 E H\ns : Set M\ninst✝¹ : ChartedSpace H M\ninst✝ : I.Boundaryless\nx ... | [] | rw [extChartAt]
exact extend_image_nhds_mem_nhds_of_boundaryless _ (mem_chart_source H x) hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 542,
"column": 2
} | {
"line": 543,
"column": 78
} | {
"line": 545,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\nI : ModelWithCorners 𝕜 E H\ns : Set M\ninst✝¹ : ChartedSpace H M\ninst✝ : I.Boundaryless\nx ... | [] | rw [extChartAt]
exact extend_image_nhds_mem_nhds_of_boundaryless _ (mem_chart_source H x) hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 662,
"column": 17
} | {
"line": 662,
"column": 19
} | {
"line": 663,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : TopologicalSpace H\ninst✝¹ : TopologicalSpace M\nI : ModelWithCorners 𝕜 E H\ns : Set M\ninst✝ : ChartedSpace H M\nx₀ x : M\nhx : x ∈ closure[... | [
"𝕜 : Type u_1\nE : Type u_2\nM : Type u_3\nH : Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : TopologicalSpace H\ninst✝¹ : TopologicalSpace M\nI : ModelWithCorners 𝕜 E H\ns : Set M\ninst✝ : ChartedSpace H M\nx₀ x : M\nhx : x ∈ closure[inst✝¹] (int... | ho | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 1011,
"column": 6
} | {
"line": 1012,
"column": 37
} | {
"line": 1013,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) ... | [] | gcongr
exact (hδ (A _)).2.2 _ _ (ht _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 1011,
"column": 6
} | {
"line": 1012,
"column": 37
} | {
"line": 1013,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ns : Set E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) ... | [] | gcongr
exact (hδ (A _)).2.2 _ _ (ht _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 359,
"column": 98
} | {
"line": 360,
"column": 98
} | {
"line": 362,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁸ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : ChartedSpace H M\nE' : Type u_5\ninst✝⁵ : NormedAddCo... | [] | by
simp_rw [ContMDiffAt, contMDiffWithinAt_iff_source_of_mem_source hx', preimage_univ, univ_inter] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 656,
"column": 4
} | {
"line": 656,
"column": 20
} | {
"line": 658,
"column": 0
} | [
{
"pp": "case some\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : N... | [] | exact h n le_rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 656,
"column": 4
} | {
"line": 656,
"column": 20
} | {
"line": 658,
"column": 0
} | [
{
"pp": "case some\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : N... | [] | exact h n le_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 656,
"column": 4
} | {
"line": 656,
"column": 20
} | {
"line": 658,
"column": 0
} | [
{
"pp": "case some\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : N... | [] | exact h n le_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.Algebra.Monoid | {
"line": 494,
"column": 6
} | {
"line": 494,
"column": 34
} | {
"line": 494,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ∞ω\n⊢ ContMDiff (𝓘(𝕜, E).prod 𝓘(𝕜, E)) 𝓘(𝕜, E) n fun p ↦ p.1 + p.2",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Prod.norm... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ∞ω\n⊢ ContMDiff 𝓘(𝕜, E × E) 𝓘(𝕜, E) n fun p ↦ p.1 + p.2"
] | ← modelWithCornersSelf_prod, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.VectorBundle.Basic | {
"line": 138,
"column": 71
} | {
"line": 139,
"column": 22
} | {
"line": 141,
"column": 0
} | [
{
"pp": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁷ : Semiring R\ninst✝⁶ : TopologicalSpace F\ninst✝⁵ : TopologicalSpace B\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\ninst✝² : (x : B) → AddCommMonoid (E x)\ninst✝¹ : (x : B) → Module R (E x)\ne : Pretrivialization F TotalSpace.proj\ni... | [] | by
rw [coe_linearMapAt] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.VectorBundle.Basic | {
"line": 238,
"column": 71
} | {
"line": 239,
"column": 22
} | {
"line": 241,
"column": 0
} | [
{
"pp": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁸ : Semiring R\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace (TotalSpace F E)\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\ninst✝² : (x : B) → AddCommMonoid (E x)\ninst✝¹ : (x : B) → Module R (E x)... | [] | by
rw [coe_linearMapAt] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear | {
"line": 209,
"column": 4
} | {
"line": 209,
"column": 25
} | {
"line": 210,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nEB : Type u_4\ninst✝³ : NormedAddCommGroup EB\ninst✝² : NormedSpace 𝕜 EB\nHB : Type u_5\ninst✝¹ : TopologicalSpace HB\ninst✝ : ChartedS... | [] | rw [hΦφ ⟨x, hx⟩ y hy] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.VectorBundle.Basic | {
"line": 980,
"column": 2
} | {
"line": 982,
"column": 98
} | {
"line": 984,
"column": 0
} | [
{
"pp": "B : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝¹⁹ : (x : B) → AddCommMonoid (E x)\ninst✝¹⁸ : NormedAddCommGroup F\ninst✝¹⁷ : TopologicalSpace B\ninst✝¹⁶ : (x : B) → TopologicalSpace (E x)\n𝕜₁ : Type u_5\n𝕜₂ : Type u_6\ninst✝¹⁵ : NontriviallyNormedField 𝕜₁\ninst✝¹⁴ : NontriviallyNormedField 𝕜₂\n... | [] | ext
simp_rw [inCoordinates, ContinuousLinearMap.coe_comp, ContinuousLinearEquiv.coe_coe,
Trivialization.coe_continuousLinearEquivAt_eq, Trivialization.symm_continuousLinearEquivAt_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.VectorBundle.Basic | {
"line": 980,
"column": 2
} | {
"line": 982,
"column": 98
} | {
"line": 984,
"column": 0
} | [
{
"pp": "B : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝¹⁹ : (x : B) → AddCommMonoid (E x)\ninst✝¹⁸ : NormedAddCommGroup F\ninst✝¹⁷ : TopologicalSpace B\ninst✝¹⁶ : (x : B) → TopologicalSpace (E x)\n𝕜₁ : Type u_5\n𝕜₂ : Type u_6\ninst✝¹⁵ : NontriviallyNormedField 𝕜₁\ninst✝¹⁴ : NontriviallyNormedField 𝕜₂\n... | [] | ext
simp_rw [inCoordinates, ContinuousLinearMap.coe_comp, ContinuousLinearEquiv.coe_coe,
Trivialization.coe_continuousLinearEquivAt_eq, Trivialization.symm_continuousLinearEquivAt_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 25
} | {
"line": 217,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nEB : Type u_4\ninst✝³ : NormedAddCommGroup EB\ninst✝² : NormedSpace 𝕜 EB\nHB : Type u_5\ninst✝¹ : TopologicalSpace HB\ninst✝ : ChartedS... | [] | rw [hΦφ ⟨x, hx⟩ y hy] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 200,
"column": 92
} | {
"line": 201,
"column": 72
} | {
"line": 203,
"column": 0
} | [
{
"pp": "n : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝¹⁵ : NontriviallyNormedField 𝕜\ninst✝¹⁴ : NormedAddCommGroup F\ninst✝¹³ : NormedSpace 𝕜 F\ninst✝¹² : TopologicalSpace (TotalSpace F E)\ninst✝¹¹ : (x : B) → TopologicalSpace (E x)\nEB : Type u_7\ninst✝¹⁰ : NormedA... | [] | by
simp_rw [← contMDiffWithinAt_univ]; exact contMDiffWithinAt_totalSpace | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection | {
"line": 160,
"column": 2
} | {
"line": 167,
"column": 50
} | {
"line": 169,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ : NormedAddC... | [] | classical
induction s using Finset.induction_on with
| empty =>
simpa only [Finset.sum_empty] using! contMDiffWithinAt_zeroSection ..
| insert i s hi h =>
simp only [Finset.sum_insert hi]
apply (hs _ (s.mem_insert_self i)).add_section
exact h fun i a ↦ hs _ (s.mem_insert_of_mem a) | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection | {
"line": 160,
"column": 2
} | {
"line": 167,
"column": 50
} | {
"line": 169,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ : NormedAddC... | [] | classical
induction s using Finset.induction_on with
| empty =>
simpa only [Finset.sum_empty] using! contMDiffWithinAt_zeroSection ..
| insert i s hi h =>
simp only [Finset.sum_insert hi]
apply (hs _ (s.mem_insert_self i)).add_section
exact h fun i a ↦ hs _ (s.mem_insert_of_mem a) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection | {
"line": 160,
"column": 2
} | {
"line": 167,
"column": 50
} | {
"line": 169,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ : NormedAddC... | [] | classical
induction s using Finset.induction_on with
| empty =>
simpa only [Finset.sum_empty] using! contMDiffWithinAt_zeroSection ..
| insert i s hi h =>
simp only [Finset.sum_insert hi]
apply (hs _ (s.mem_insert_self i)).add_section
exact h fun i a ↦ hs _ (s.mem_insert_of_mem a) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Paracompact | {
"line": 87,
"column": 4
} | {
"line": 88,
"column": 51
} | {
"line": 89,
"column": 2
} | [
{
"pp": "case refine_2\nι : Type u\nX : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : ParacompactSpace X\nu : ι → Set X\nuo : ∀ (a : ι), IsOpen[inst✝¹] (u a)\nuc : ⋃ i, u i = univ\nα : Type v\nt : α → Set X\nhto : ∀ (b : α), IsOpen[inst✝¹] (t b)\nind : α → ι\nhind : ∀ (b : α), t b ⊆ u (ind b)\nt_inv : X → α\nht_... | [] | simp only [eq_univ_iff_forall, mem_iUnion]
exact fun x ↦ ⟨ind (t_inv x), _, rfl, ht_inv _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Paracompact | {
"line": 87,
"column": 4
} | {
"line": 88,
"column": 51
} | {
"line": 89,
"column": 2
} | [
{
"pp": "case refine_2\nι : Type u\nX : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : ParacompactSpace X\nu : ι → Set X\nuo : ∀ (a : ι), IsOpen[inst✝¹] (u a)\nuc : ⋃ i, u i = univ\nα : Type v\nt : α → Set X\nhto : ∀ (b : α), IsOpen[inst✝¹] (t b)\nind : α → ι\nhind : ∀ (b : α), t b ⊆ u (ind b)\nt_inv : X → α\nht_... | [] | simp only [eq_univ_iff_forall, mem_iUnion]
exact fun x ↦ ⟨ind (t_inv x), _, rfl, ht_inv _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.