module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 580,
"column": 2
} | {
"line": 582,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nA : E →ₗ[𝕜] F\nB : F →ₗ[𝕜] E\n⊢ A = adjoint B... | refine ⟨fun h x y => by rw [h, adjoint_inner_left], fun h => ?_⟩
ext x
exact ext_inner_right 𝕜 fun y => by simp only [adjoint_inner_left, h x y] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 580,
"column": 2
} | {
"line": 582,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : FiniteDimensional 𝕜 E\ninst✝ : FiniteDimensional 𝕜 F\nA : E →ₗ[𝕜] F\nB : F →ₗ[𝕜] E\n⊢ A = adjoint B... | refine ⟨fun h x y => by rw [h, adjoint_inner_left], fun h => ?_⟩
ext x
exact ext_inner_right 𝕜 fun y => by simp only [adjoint_inner_left, h x y] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1094,
"column": 50
} | {
"line": 1094,
"column": 66
} | [
{
"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⊢ (‖↑(lp.single (↑p) i x) i‖ ^ (↑p).toReal) ^ (1 / (↑p).toReal) = ‖x‖",
"usedConstants": [
... | lp.coeFn_single, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 1098,
"column": 10
} | {
"line": 1098,
"column": 26
} | [
{
"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\nj : α\nhji : j ≠ i\n⊢ ‖↑(lp.single (↑p) i x) j‖ ^ (↑p).toReal = 0",
"usedConstants": [
"No... | lp.coeFn_single, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ParametricIntegral | {
"line": 133,
"column": 4
} | {
"line": 134,
"column": 44
} | [
{
"pp": "case h.hf\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : ... | exact ((hF_meas _ (hε x_in)).sub (hF_meas _ (hε x₀_in))).sub
(hF'_meas.apply_continuousLinearMap _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ParametricIntegral | {
"line": 149,
"column": 4
} | {
"line": 149,
"column": 32
} | [
{
"pp": "case pos.h_bound\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nb... | simp only [← div_eq_inv_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ParametricIntegral | {
"line": 179,
"column": 2
} | {
"line": 179,
"column": 78
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\nμ : Measure α\n𝕜 : Type u_2\ninst✝⁵ : RCLike 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedSpace 𝕜 E\nH : Type u_4\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nx₀ : H\nbound : α → ℝ\ns : ... | apply hasFDerivAt_integral_of_dominated_loc_of_lip' (ball_mem_nhds x₀ δ_pos) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.EverywherePos | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 88
} | [
{
"pp": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nx : α\nn : Set α\nns : n ∈ 𝓝[s] x\nhx : μ n = 0\nv : Set α\nvx : v ∈ 𝓝 x\nhv : v ∩ s ⊆ n\nw : Set α\nwv : w ⊆ v\nw_open : IsOpen[inst✝¹] w\nxw : x ∈ w\ny : α\nyw : y ∈ w\n⊢ y ∈ {x | ∃ n ∈ 𝓝[s] x, μ n = 0}... | refine ⟨s ∩ w, inter_mem_nhdsWithin _ (w_open.mem_nhds yw), measure_mono_null ?_ hx⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.EverywherePos | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 37
} | [
{
"pp": "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 ... | choose W W_open mem_W hW using this | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.Analysis.Convolution | {
"line": 226,
"column": 38
} | {
"line": 230,
"column": 20
} | [
{
"pp": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup E'\ninst✝⁸ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedSpace 𝕜 ... | by
refine (h.const_mul ‖L‖).mono'
(hmf.convolution_integrand_snd' L hmg) (Eventually.of_forall fun x => ?_)
rw [mul_apply', ← mul_assoc]
apply L.le_opNorm₂ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 149,
"column": 4
} | {
"line": 150,
"column": 68
} | [
{
"pp": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\nP : Type uP\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : No... | refine hg.continuousOn.comp_continuous (.prodMk_right _) fun x => ?_
simpa only [prodMk_mem_set_prod_eq, mem_univ, and_true] using hp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 149,
"column": 4
} | {
"line": 150,
"column": 68
} | [
{
"pp": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\nP : Type uP\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : No... | refine hg.continuousOn.comp_continuous (.prodMk_right _) fun x => ?_
simpa only [prodMk_mem_set_prod_eq, mem_univ, and_true] using hp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 252,
"column": 6
} | {
"line": 252,
"column": 42
} | [
{
"pp": "𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\nP : Type uP\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : No... | exact (le_max_right _ _).trans_lt hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convolution | {
"line": 776,
"column": 47
} | {
"line": 776,
"column": 53
} | [
{
"pp": "case h.e'_3.h.e'_4\nG : Type uG\nE' : Type uE'\ninst✝⁸ : NormedAddCommGroup E'\ng : G → E'\ninst✝⁷ : MeasurableSpace G\nμ : Measure G\ninst✝⁶ : SeminormedAddCommGroup G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : SecondCountableTopology G\ninst✝³ : μ.IsAddLeftInvariant\ninst✝² : SFinite μ\ninst✝¹ : NormedSpace ℝ ... | hintf, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convolution | {
"line": 777,
"column": 42
} | {
"line": 777,
"column": 48
} | [
{
"pp": "case convert_2\nG : Type uG\nE' : Type uE'\ninst✝⁸ : NormedAddCommGroup E'\ng : G → E'\ninst✝⁷ : MeasurableSpace G\nμ : Measure G\ninst✝⁶ : SeminormedAddCommGroup G\ninst✝⁵ : BorelSpace G\ninst✝⁴ : SecondCountableTopology G\ninst✝³ : μ.IsAddLeftInvariant\ninst✝² : SFinite μ\ninst✝¹ : NormedSpace ℝ E'\n... | hintf, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Calculus.BumpFunction.SmoothApprox | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 83
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf : E → F\nε : ℝ\nhf : UniformContinuous f\nhε : 0 < ε\n⊢ ∃ g, ContDiff ℝ ∞ g ∧ ∀ (a : E), dist (g a) (f... | rcases Metric.uniformContinuous_iff.mp hf (ε / 2) (half_pos hε) with ⟨δ, hδ, hfδ⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 439,
"column": 25
} | {
"line": 439,
"column": 64
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : a.c (last N) = 0\nlastr : a.r (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\ni j : Fin N.succ\ninej : i ≠ j\nah : Pairwise fun i j ↦ a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ... | by gcongr; linarith only [δnonneg, hδ1] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.RestrictScalars | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 100
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_4\ninst✝³ : NormedAdd... | rw [fderiv_comp_fderivWithin _ (by fun_prop) (h.restrictScalars 𝕜) hs, ContinuousLinearMap.fderiv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 965,
"column": 2
} | {
"line": 965,
"column": 42
} | [
{
"pp": "case inv_eq_self\nG : 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\n⊢ μ.inv = μ",
"usedConstants": [
"EN... | let c : ℝ≥0∞ := haarScalarFactor μ.inv μ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 991,
"column": 2
} | {
"line": 991,
"column": 42
} | [
{
"pp": "case inv_eq_self\nG : 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\n⊢ μ.inv = μ",
"usedConstants": [
... | let c : ℝ≥0∞ := haarScalarFactor μ.inv μ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Calculus.ContDiff.RestrictScalars | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 37
} | [
{
"pp": "case h\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\ninst✝⁸ : NormedAlgebra 𝕜 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜' E\ninst✝⁴ : IsScalarTower 𝕜 𝕜' E\nF : Type u_4\ninst✝³ : N... | rw [← Filter.EventuallyEq] at h₁a | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 160,
"column": 4
} | {
"line": 163,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁷ : NormedAddCommGroup D\ninst✝⁶ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type uF\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type uG\ninst✝¹ : NormedAddCommGro... | calc
‖B (isoE y) (isoF x)‖ ≤ ‖B (isoE y)‖ * ‖isoF x‖ := ContinuousLinearMap.le_opNorm _ _
_ ≤ ‖B‖ * ‖isoE y‖ * ‖isoF x‖ := by gcongr; apply ContinuousLinearMap.le_opNorm
_ = ‖B‖ * ‖y‖ * ‖x‖ := by simp only [LinearIsometryEquiv.norm_map] | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 902,
"column": 2
} | {
"line": 907,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : MetricSpace α\ninst✝⁵ : SecondCountableTopology α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : OpensMeasurableSpace α\ninst✝² : HasBesicovitchCovering α\nμ : Measure α\ninst✝¹ : SFinite μ\ninst✝ : μ.OuterRegular\nε : ℝ≥0∞\nhε : ε ≠ 0\nf : α → Set ℝ\ns : Set α\nhf : ∀ x ∈ s, ∀ δ > 0, (f ... | let q : BallPackage s' α :=
{ c := fun x => x
r := fun x => r1 x
rpos := fun x => (hr1 x.1 x.2).1.2.1
r_bound := 1
r_le := fun x => (hr1 x.1 x.2).1.2.2.le } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Calculus.Deriv.Star | {
"line": 89,
"column": 16
} | {
"line": 89,
"column": 46
} | [
{
"pp": "𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : StarRing 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : StarAddMonoid F\ninst✝² : StarModule 𝕜 F\ninst✝¹ : ContinuousStar F\ninst✝ : NormedStarGroup 𝕜\nf : 𝕜 → F\nx : 𝕜\nf' : F\nhf : HasDerivAt f (star f'... | convert! hf.star_conj <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Calculus.Deriv.Star | {
"line": 89,
"column": 16
} | {
"line": 89,
"column": 46
} | [
{
"pp": "𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : StarRing 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : StarAddMonoid F\ninst✝² : StarModule 𝕜 F\ninst✝¹ : ContinuousStar F\ninst✝ : NormedStarGroup 𝕜\nf : 𝕜 → F\nx : 𝕜\nf' : F\nhf : HasDerivAt f (star f'... | convert! hf.star_conj <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Star | {
"line": 89,
"column": 16
} | {
"line": 89,
"column": 46
} | [
{
"pp": "𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : StarRing 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : StarAddMonoid F\ninst✝² : StarModule 𝕜 F\ninst✝¹ : ContinuousStar F\ninst✝ : NormedStarGroup 𝕜\nf : 𝕜 → F\nx : 𝕜\nf' : F\nhf : HasDerivAt f (star f'... | convert! hf.star_conj <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 22
} | [
{
"pp": "f : ℝ → ℝ\na : ℝ\nh : ContinuousAt f a\nhd₀ : DifferentiableOn ℝ f (Ioi a)\n⊢ ContinuousOn f (Ici a)",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"Set.Ioi",
"Set.Ici",
"congrArg",
"PartialOrder.toPreorder",
"PseudoMetricSpace.toUniformS... | ← Ioi_union_left | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 354,
"column": 64
} | {
"line": 442,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nFu Gu : Type u\ninst✝³ : NormedAddCommGroup Fu\ninst✝² : NormedSpace 𝕜 Fu\ninst✝¹ : NormedAddCommGroup Gu\ninst✝ : NormedSpace 𝕜 Gu\ng : Fu → Gu\nf : E → Fu\nn : ℕ\ns : Set E\nt ... | by
/- We argue by induction on `n`, using that `D^(n+1) (g ∘ f) = D^n (g ' ∘ f ⬝ f')`. The successive
derivatives of `g' ∘ f` are controlled thanks to the inductive assumption, and those of `f'` are
controlled by assumption.
As composition of linear maps is a bilinear map, one may use
`ContinuousLinea... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 39
} | [
{
"pp": "f : ℝ → ℝ\na b c : ℝ\nh : ContinuousAt f b\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\nhd₁ : DifferentiableOn ℝ f (Ioo b c)\nh₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0\nh₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x\n⊢ IsMinOn f (Ioo a c) b",
"usedConstants": [
"Real",
"isMinOn_Ioo_of_anti_mono",
"Real.inst... | refine isMinOn_Ioo_of_anti_mono ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 456,
"column": 2
} | {
"line": 493,
"column": 74
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type uF\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type uG\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\... | let Fu : Type max uF uG := ULift.{uG, uF} F
let Gu : Type max uF uG := ULift.{uF, uG} G
have isoF : Fu ≃ₗᵢ[𝕜] F := LinearIsometryEquiv.ulift 𝕜 F
have isoG : Gu ≃ₗᵢ[𝕜] G := LinearIsometryEquiv.ulift 𝕜 G
-- lift `f` and `g` to versions `fu` and `gu` on the lifted spaces.
let fu : E → Fu := isoF.symm ∘ f
l... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 456,
"column": 2
} | {
"line": 493,
"column": 74
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type uF\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type uG\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ng : F → G\nf : E → F\nn : ℕ\ns : Set E\... | let Fu : Type max uF uG := ULift.{uG, uF} F
let Gu : Type max uF uG := ULift.{uF, uG} G
have isoF : Fu ≃ₗᵢ[𝕜] F := LinearIsometryEquiv.ulift 𝕜 F
have isoG : Gu ≃ₗᵢ[𝕜] G := LinearIsometryEquiv.ulift 𝕜 G
-- lift `f` and `g` to versions `fu` and `gu` on the lifted spaces.
let fu : E → Fu := isoF.symm ∘ f
l... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns t : Set E\nf : E → F\nx : E\nh : IsSymmSndFDerivWithinAt 𝕜 f t x\nhst : t ∈ 𝓝[s] x\nhf : ContDiffWithinA... | exact h v w | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns t : Set E\nf : E → F\nx : E\nh : IsSymmSndFDerivWithinAt 𝕜 f s x\nhst : s =ᶠ[𝓝 x] t\nv w : E\n⊢ ((fderiv... | exact h v w | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.FDeriv.ContinuousMultilinearMap | {
"line": 83,
"column": 68
} | {
"line": 86,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : Type u_3\nF : ι → Type u_4\nG : ι → Type u_5\nH : Type u_6\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : (i : ι) → NormedAddCommGroup (F i)\ninst✝⁶ : (i : ι) → NormedSpace 𝕜 (F i)\ninst✝⁵ : (i : ι) → NormedAdd... | by
convert!
hasStrictFDerivAt_compContinuousLinearMap (f x, (g · x)) |>.hasFDerivAt |>.comp x
(hf.prodMk (hasFDerivAt_pi.2 hg)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.VectorField | {
"line": 507,
"column": 51
} | {
"line": 508,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nM : E ≃L[𝕜] F\nx : E\nhf : ↑M = fderivWithin 𝕜 f s x\nV : F → F\n⊢ pullbackWithin 𝕜... | by
simp [pullbackWithin, ← hf] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.VectorField | {
"line": 580,
"column": 67
} | {
"line": 580,
"column": 68
} | [
{
"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 ... | I | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Gradient.Basic | {
"line": 172,
"column": 7
} | {
"line": 172,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\ng : 𝕜 → 𝕜\ng' u : 𝕜\nL' : Filter 𝕜\nh : HasDerivAtFilter g g' (L' ×ˢ pure u)\nthis : ContinuousLinearMap.smulRight 1 g' = (toDual 𝕜 𝕜) ((starRingEnd 𝕜) g')\n⊢ HasGradientAtFilter g ((starRingEnd 𝕜) g') u L'",
"usedConstants": [
"Pure.pure",
"Lin... | HasGradientAtFilter, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Gradient.Basic | {
"line": 337,
"column": 6
} | {
"line": 337,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace F\nx : F\nL : Filter F\nc : 𝕜\n⊢ HasGradientAtFilter (fun x ↦ c) 0 x L",
"usedConstants": [
"Pure.pure",
"LinearIsometryEquiv.instEquivLike",
"No... | HasGradientAtFilter, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Implicit | {
"line": 512,
"column": 2
} | {
"line": 513,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : CompleteSpace 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CompleteSpace E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : FiniteDimensional 𝕜 F\nf : E → F\nf' : E →L[�... | refine ((hf.implicitToOpenPartialHomeomorph f f' hf').tendsto_symm
(hf.mem_implicitToOpenPartialHomeomorph_source hf')).comp ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 161,
"column": 6
} | {
"line": 161,
"column": 56
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x\nr : (... | simpa only [sub_pos] using mem_ball_iff_norm.mp hz | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 175,
"column": 8
} | {
"line": 175,
"column": 69
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : SecondCountableTopology F\nf : E → F\ns : Set E\nf' : E → E →L[ℝ] F\nhf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x\nr : (... | refine add_le_add (hδ ?_) (ContinuousLinearMap.le_opNorm _ _) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 406,
"column": 2
} | {
"line": 407,
"column": 84
} | [
{
"pp": "case a.inr\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nA : E →L[ℝ] E\nm : ℝ≥0\nhm : ↑m < ENNReal.ofReal |A.det|\nmpos : 0 < m\n⊢ {x | (fun δ ↦ ∀ (s :... | have hA : A.det ≠ 0 := by
intro h; simp only [h, ENNReal.not_lt_zero, ENNReal.ofReal_zero, abs_zero] at hm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 1078,
"column": 2
} | {
"line": 1080,
"column": 38
} | [
{
"pp": "case pos\nE : Type u_1\nf f' : ℝ → E\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhderiv : ∀ (x : ℝ), HasDerivAt f (f' x) x\nhf' : Integrable f' volume\nhf : Integrable f volume\nhE : CompleteSpace E\nA : Tendsto f atBot (𝓝 0)\n⊢ ∫ (x : ℝ), f' x = 0",
"usedConstants": [
"Real",
... | have B : Tendsto f atTop (𝓝 0) :=
tendsto_zero_of_hasDerivAt_of_integrableOn_Ioi (a := 0) (fun x _hx ↦ hderiv x)
hf'.integrableOn hf.integrableOn | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.LineDeriv.IntegrationByParts | {
"line": 158,
"column": 4
} | {
"line": 159,
"column": 80
} | [
{
"pp": "case pos.inr.hf\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\n... | have h2x : L.symm x ∈ tsupport g :=
(Set.ext_iff.mp (tsupport_comp_eq_preimage g L.symm.toHomeomorph) x).mp hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 604,
"column": 50
} | {
"line": 604,
"column": 99
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\nf : E → E\nf' : E → E →L[ℝ] E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nR : ℝ\nε : ℝ≥0\nεpos : 0 < ε\nhf' : ∀ x ∈ ∅, HasFDerivWithinAt f ... | · simp only [measure_empty, zero_le, image_empty] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 263,
"column": 4
} | {
"line": 264,
"column": 79
} | [
{
"pp": "case refine_2\nH : Type u\nM : Type u_2\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyConnectedSpace H\ne : M → OpenPartialHomeomorph M H := chartAt H\n⊢ ∀ (x : M) (i : Set H),\n (fun x s ↦ (IsOpen[inst✝³] s ∧ ↑(e x) x ∈ s ∧ IsConnected s) ∧ s ⊆... | rintro x s ⟨⟨-, -, hsconn⟩, hssubset⟩
exact hsconn.isPreconnected.image _ ((e x).continuousOn_symm.mono hssubset) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 263,
"column": 4
} | {
"line": 264,
"column": 79
} | [
{
"pp": "case refine_2\nH : Type u\nM : Type u_2\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyConnectedSpace H\ne : M → OpenPartialHomeomorph M H := chartAt H\n⊢ ∀ (x : M) (i : Set H),\n (fun x s ↦ (IsOpen[inst✝³] s ∧ ↑(e x) x ∈ s ∧ IsConnected s) ∧ s ⊆... | rintro x s ⟨⟨-, -, hsconn⟩, hssubset⟩
exact hsconn.isPreconnected.image _ ((e x).continuousOn_symm.mono hssubset) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.HasGroupoid | {
"line": 117,
"column": 2
} | {
"line": 119,
"column": 40
} | [
{
"pp": "H : Type u\nM : Type u_2\ninst✝² : TopologicalSpace H\ninst✝¹ : TopologicalSpace M\ninst✝ : ChartedSpace H M\nG : StructureGroupoid H\ne e' : OpenPartialHomeomorph M H\nhe : e ∈ maximalAtlas M G\nhe' : e' ∈ maximalAtlas M G\nx : H\nhx : x ∈ (e.symm ≫ₕ e').source\nf : OpenPartialHomeomorph M H := chartA... | have xs : x ∈ s := by
simp only [s, f, mem_inter_iff, mem_preimage, mem_chart_source, and_true]
exact ((mem_inter_iff _ _ _).1 hx).1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 27
} | [
{
"pp": "case inr.h\nH✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\ne : OpenPartialHomeomorph H H\nhe : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}\nh : e.source.Nonempty\n⊢ e ∈ {OpenPartialHomeomorph.refl... | rcases h with ⟨x, hx⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 96
} | [
{
"pp": "case neg\nH : Type u\nM : Type u_2\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : T1Space H\nx y : M\nhxy : x ≠ y\nhy : y ∉ (chartAt H x).source\n⊢ ∃ U, IsOpen[inst✝²] U ∧ x ∈ U ∧ y ∉ U",
"usedConstants": [
"chartAt",
"OpenPartialHomeomorph... | · exact ⟨(chartAt H x).source, (chartAt H x).open_source, ChartedSpace.mem_chart_source x, hy⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 233,
"column": 55
} | {
"line": 233,
"column": 70
} | [
{
"pp": "H✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\ne : OpenPartialHomeomorph H H\nhe : ∀ x ∈ e.source, ∃ s, IsOpen[inst✝] s ∧ x ∈ s ∧ e.restr s ∈ {OpenPartialHomeomorph.refl H} ∪ {e | e.source = ∅}\nx : H\nhx : x ∈ e.source\ns : Set H\nopen_s : IsOpen[inst✝] s\nxs : ... | univ_subset_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 242,
"column": 8
} | {
"line": 243,
"column": 49
} | [
{
"pp": "H✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\ne e' : OpenPartialHomeomorph H H\nhe'e : e' ≈ e\nhe : e ∈ {OpenPartialHomeomorph.refl H}\n⊢ e = e'",
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialEquiv.target",
"Set.univ",
"Memb... | refine eq_of_eqOnSource_univ (Setoid.symm he'e) ?_ ?_ <;>
rw [Set.mem_singleton_iff.1 he] <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 293,
"column": 24
} | {
"line": 298,
"column": 85
} | [
{
"pp": "H : Type u_1\ninst✝ : TopologicalSpace H\nPG : Pregroupoid H\ne e' : OpenPartialHomeomorph H H\nhe : e ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}\nhe' : e' ∈ {e | PG.property (↑e) e.source ∧ PG.property (↑e.symm) e.target}\n⊢ e ≫ₕ e' ∈ {e | PG.property (↑e) e.source ∧ PG.propert... | by
constructor
· apply PG.comp he.1 he'.1 e.open_source e'.open_source
apply e.continuousOn_toFun.isOpen_inter_preimage e.open_source e'.open_source
· apply PG.comp he'.2 he.2 e'.open_target e.open_target
apply e'.continuousOn_invFun.isOpen_inter_preimage e'.open_target e.open_target | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 265,
"column": 7
} | {
"line": 265,
"column": 41
} | [
{
"pp": "H : Type u_1\nH' : Type u_3\nM' : Type u_4\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace H'\ninst✝¹ : TopologicalSpace M'\ninst✝ : ChartedSpace H' M'\nG : StructureGroupoid H\nG' : StructureGroupoid H'\nf f' : OpenPartialHomeomorph M' H'\nP : (H → H') → Set H → H → Prop\nhG : G.LocalInvariant... | by simp only [xf, xf', mfld_simps] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 94,
"column": 64
} | {
"line": 95,
"column": 89
} | [
{
"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\n⊢ (f.extend I).target = ↑I '' f.ta... | by
rw [extend, PartialEquiv.trans_target'', I.source_eq, univ_inter, I.toPartialEquiv_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE✝ : Type u_2\ninst✝⁵ : NormedAddCommGroup E✝\ninst✝⁴ : NormedSpace 𝕜 E✝\nH✝ : Type u_3\ninst✝³ : TopologicalSpace H✝\nI : ModelWithCorners 𝕜 E✝ H✝\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nH : Type u_5\ninst✝ : Topolog... | simp only [instIsRCLikeNormedField, ↓reduceDIte, this] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 87,
"column": 2
} | {
"line": 88,
"column": 77
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nE' : Type u_5\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nn : ℕ∞ω\nf : E → H'\ns : Set... | simp_rw [ContDiffWithinAtProp, modelWithCornersSelf_coe, range_id, inter_univ,
modelWithCornersSelf_coe_symm, CompTriple.comp_eq, preimage_id_eq, id_eq] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 87,
"column": 2
} | {
"line": 88,
"column": 77
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nE' : Type u_5\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nn : ℕ∞ω\nf : E → H'\ns : Set... | simp_rw [ContDiffWithinAtProp, modelWithCornersSelf_coe, range_id, inter_univ,
modelWithCornersSelf_coe_symm, CompTriple.comp_eq, preimage_id_eq, id_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 87,
"column": 2
} | {
"line": 88,
"column": 77
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nE' : Type u_5\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace 𝕜 E'\nH' : Type u_6\ninst✝ : TopologicalSpace H'\nI' : ModelWithCorners 𝕜 E' H'\nn : ℕ∞ω\nf : E → H'\ns : Set... | simp_rw [ContDiffWithinAtProp, modelWithCornersSelf_coe, range_id, inter_univ,
modelWithCornersSelf_coe_symm, CompTriple.comp_eq, preimage_id_eq, id_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 694,
"column": 82
} | {
"line": 706,
"column": 59
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\n⊢ contDiffGroupoid 0 I = continuousGroupoid H",
"usedConstants": [
"Eq.mpr",
"Continuous.com... | by
apply le_antisymm le_top
intro u _
-- we have to check that every open partial homeomorphism belongs to `contDiffGroupoid 0 I`,
-- by unfolding its definition
change u ∈ contDiffGroupoid 0 I
rw [contDiffGroupoid, mem_groupoid_of_pregroupoid, contDiffPregroupoid]
simp only [contDiffOn_zero]
constructo... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 59,
"column": 2
} | {
"line": 60,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\ninst✝⁹ : NormedA... | rw [contMDiffWithinAt_iff] at *
exact ⟨hf.1.prodMk hg.1, hf.2.prodMk hg.2⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 59,
"column": 2
} | {
"line": 60,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\ninst✝⁹ : NormedA... | rw [contMDiffWithinAt_iff] at *
exact ⟨hf.1.prodMk hg.1, hf.2.prodMk hg.2⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 65,
"column": 2
} | {
"line": 66,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\nE' : Type u_5\ninst✝³ : NormedAddComm... | rw [contMDiffWithinAt_iff] at *
exact ⟨hf.1.prodMk hg.1, hf.2.prodMk hg.2⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 65,
"column": 2
} | {
"line": 66,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\nE' : Type u_5\ninst✝³ : NormedAddComm... | rw [contMDiffWithinAt_iff] at *
exact ⟨hf.1.prodMk hg.1, hf.2.prodMk hg.2⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 392,
"column": 6
} | {
"line": 392,
"column": 61
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\nE' : Type u_5\ninst✝⁶ : NormedAddC... | contMDiffWithinAt_iff_of_mem_maximalAtlas he he' hx hy, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.Algebra.Monoid | {
"line": 359,
"column": 2
} | {
"line": 359,
"column": 33
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝¹² : NontriviallyNormedField 𝕜\nn : ℕ∞ω\nH : Type u_3\ninst✝¹¹ : TopologicalSpace H\nE : Type u_4\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nG : Type u_5\ninst✝⁸ : CommMonoid G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Charte... | simp only [← Finset.prod_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.BumpFunction | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 55
} | [
{
"pp": "E : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type uH\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nc : M\nf : SmoothBumpFunction I c\ninst✝ : FiniteDimensional ℝ E\nr : ℝ\nhr : r ∈ Ioo 0 f.rOut... | simp only [support_eq_inter_preimage, updateRIn_rOut] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.BumpFunction | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 55
} | [
{
"pp": "E : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type uH\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nc : M\nf : SmoothBumpFunction I c\ninst✝ : FiniteDimensional ℝ E\nr : ℝ\nhr : r ∈ Ioo 0 f.rOut... | simp only [support_eq_inter_preimage, updateRIn_rOut] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.BumpFunction | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 55
} | [
{
"pp": "E : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nH : Type uH\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nc : M\nf : SmoothBumpFunction I c\ninst✝ : FiniteDimensional ℝ E\nr : ℝ\nhr : r ∈ Ioo 0 f.rOut... | simp only [support_eq_inter_preimage, updateRIn_rOut] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.VectorBundle.Basic | {
"line": 107,
"column": 17
} | {
"line": 107,
"column": 50
} | [
{
"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\ne✝ : Pretrivialization F TotalSpace.proj\nx : TotalSpace F E\nb✝ : B\ny : E b✝\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\ninst✝² : (x : B) → AddCommMonoid (E x... | by simp_rw [e.apply_mk_symm hb v] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.VectorBundle.Basic | {
"line": 292,
"column": 60
} | {
"line": 292,
"column": 81
} | [
{
"pp": "case a\nR : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁹ : Semiring R\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : TopologicalSpace (TotalSpace F E)\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module R F\ninst✝³ : (x : B) → AddCommMonoid (E x)\ninst✝² : (x : B) → Module... | LinearEquiv.symm_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.VectorBundle.Basic | {
"line": 643,
"column": 29
} | {
"line": 643,
"column": 81
} | [
{
"pp": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace F E)\nin... | by simp only [map_smul, localTriv_apply, mfld_simps] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 241,
"column": 2
} | {
"line": 243,
"column": 73
} | [
{
"pp": "n : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NormedAddCommGroup F\ninst✝¹¹ : NormedSpace 𝕜 F\ninst✝¹⁰ : TopologicalSpace (TotalSpace F E)\ninst✝⁹ : (x : B) → TopologicalSpace (E x)\nEB : Type u_7\ninst✝⁸ : NormedAddCommGroup EB\n... | filter_upwards [(trivializationAt F E x).open_baseSet.mem_nhds
(mem_baseSet_trivializationAt F E x)] with y hy
using congr_arg Prod.snd <| (trivializationAt F E x).zeroSection 𝕜 hy | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Topology.Compactness.Paracompact | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 61
} | [
{
"pp": "ι : Type u\nX : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : ParacompactSpace X\ns : Set X\nhs : IsClosed[inst✝¹] s\nu : ι → Set X\nuo : ∀ (i : ι), IsOpen[inst✝¹] (u i)\nus : s ⊆ ⋃ i, u i\nuc : ⋃ i, Option.elim' sᶜ u i = univ\nv : Option ι → Set X\nvo : ∀ (a : Option ι), IsOpen[inst✝¹] (v a)\nvc : ⋃ i,... | simp only [iUnion_option, ← compl_subset_iff_union] at vc | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 397,
"column": 4
} | {
"line": 397,
"column": 57
} | [
{
"pp": "n : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nF : Type u_4\nE : B → Type u_6\ninst✝¹⁶ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝¹⁵ : NormedAddCommGroup EB\ninst✝¹⁴ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝¹³ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝¹² : TopologicalSpace B\ninst✝¹¹ : ... | simp_all only [Trivialization.mem_target, mfld_simps] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 800,
"column": 30
} | {
"line": 802,
"column": 36
} | [
{
"pp": "𝕜 : 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 : NormedAddCom... | by
rw [← mdifferentiableOn_univ, ← hst]
exact hf.union_of_isOpen hf' hs ht | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 437,
"column": 6
} | {
"line": 438,
"column": 63
} | [
{
"pp": "case refine_2\nn : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝²⁰ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝¹⁹ : NormedAddCommGroup EB\ninst✝¹⁸ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝¹⁷ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB ... | rintro ⟨b, v⟩ hb
exact (e.apply_symm_apply_eq_coordChangeL e' hb.1 v).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 437,
"column": 6
} | {
"line": 438,
"column": 63
} | [
{
"pp": "case refine_2\nn : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝²⁰ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝¹⁹ : NormedAddCommGroup EB\ninst✝¹⁸ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝¹⁷ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB ... | rintro ⟨b, v⟩ hb
exact (e.apply_symm_apply_eq_coordChangeL e' hb.1 v).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 930,
"column": 2
} | {
"line": 932,
"column": 62
} | [
{
"pp": "case pos\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 : No... | · simp only [mfderivWithin, hx, (mdifferentiableWithinAt_congr_set' y h).1 hx, ↓reduceIte]
apply fderivWithin_congr_set' (extChartAt I x x)
exact preimage_extChartAt_eventuallyEq_compl_singleton y h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.EMetricSpace.Paracompact | {
"line": 77,
"column": 4
} | {
"line": 77,
"column": 30
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\npow_pos : ∀ (k : ℕ), 0 < 2⁻¹ ^ k\nhpow_le : ∀ {m n : ℕ}, m ≤ n → 2⁻¹ ^ n ≤ 2⁻¹ ^ m\nh2pow : ∀ (n : ℕ), 2 * 2⁻¹ ^ (n + 1) = 2⁻¹ ^ n\nι : Type u_1\ns : ι → Set α\nho : ∀ (a : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (s a)\nhcov : ∀ (x : ... | rw [Nat.strongRecOn'_beta] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.PartitionOfUnity | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 10
} | [
{
"pp": "case h\nι : Type u_1\nX : Type u_2\ninst✝ : EMetricSpace X\nK U : ι → Set X\nhK : ∀ (i : ι), IsClosed[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (K i)\nhU : ∀ (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (U i)\nhKU : ∀ (i : ι), K i ⊆ U i\nhfin : LocallyFinite K\nx : ... | apply hR | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.ShrinkingLemma | {
"line": 290,
"column": 6
} | {
"line": 290,
"column": 22
} | [
{
"pp": "case h\nι : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace X\nu : ι → Set X\ns : Set X\ninst✝¹ : T2Space X\ninst✝ : LocallyCompactSpace X\nv : PartialRefinement u s fun w ↦ IsCompact (closure w)\nhs : IsCompact s\ni : ι\nhi : i ∉ v.carrier\nsi : Set X := s ∩ (⋃ j, ⋃ (_ : j ≠ i), v.toFun j)ᶜ\nhsi : s... | rw [ne_eq] at hj | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp | {
"line": 51,
"column": 2
} | {
"line": 53,
"column": 90
} | [
{
"pp": "case refine_1\nα : Type u_1\nG : Type u_2\np : ℝ≥0∞\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup G\nf : α → G\nhf : MemLp f p μ\nhf_meas : StronglyMeasurable f\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ∞\nthis✝¹ : MeasurableSpace G := borel G\nthis✝ : BorelSpace G\nthis : SeparableSpac... | · have h_fs_Lp : ∀ n, MemLp (fs n) p μ :=
SimpleFunc.memLp_approxOn_range hf_meas.measurable hf
exact fun n => (fs n).measure_support_lt_top_of_memLp (h_fs_Lp n) hp_ne_zero hp_ne_top | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Distribution.AEEqOfIntegralContDiff | {
"line": 86,
"column": 8
} | {
"line": 89,
"column": 22
} | [
{
"pp": "case h.refine_1\nE : Type u_1\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedSpace ℝ E\ninst✝¹¹ : FiniteDimensional ℝ E\nF : Type u_2\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace ℝ F\ninst✝⁸ : CompleteSpace F\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type u_... | have : ‖g n x‖ ≤ 1 := by
have := g_range n (mem_range_self (f := g n) x)
rw [Real.norm_of_nonneg this.1]
exact this.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 261,
"column": 38
} | {
"line": 261,
"column": 62
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\ninst✝ : SigmaFinite μ\nf : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s... | ← iUnion_spanningSets μ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 95
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\ninst✝ : SigmaFinite μ\nf : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhf_zero : ∀ (s : Set α), MeasurableSet s... | exact ae_eq_zero_restrict_of_forall_setIntegral_eq_zero hf_int_finite hf_zero h_meas_n hμn.ne | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Haar.Disintegration | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 76
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : CompleteSpace 𝕜\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : MeasurableSpace E\ninst✝⁸ : BorelSpace E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : MeasurableSpace F\ninst✝⁴ : BorelSpa... | obtain ⟨y, z, hyz⟩ : ∃ (y : S) (z : T), M.symm x = (y, z) := ⟨_, _, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 810,
"column": 6
} | {
"line": 811,
"column": 24
} | [
{
"pp": "case pos\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompactSpac... | have : 0 < f x := lt_of_le_of_ne (f_pos x) (Ne.symm xs)
linarith [g_pos x] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 810,
"column": 6
} | {
"line": 811,
"column": 24
} | [
{
"pp": "case pos\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompactSpac... | have : 0 < f x := lt_of_le_of_ne (f_pos x) (Ne.symm xs)
linarith [g_pos x] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SuccPred.IntervalSucc | {
"line": 86,
"column": 24
} | {
"line": 86,
"column": 33
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝³ : LinearOrder α\ninst✝² : SuccOrder α\ninst✝¹ : IsSuccArchimedean α\ninst✝ : LinearOrder β\nf : α → β\nhf : Monotone f\nm n : α\nhmn : m ≤ n\nk : α\nhmk : m ≤ k\nihk : ⋃ i ∈ Ico m k, Ioc (f i) (f (succ i)) = Ioc (f m) (f k)\nhk : IsMax k\n⊢ ⋃ i ∈ Ico m k, Io... | Ioc_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Taylor | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 61
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nx₀ : ℝ\nn : ℕ\ns : Set ℝ\nhs : Convex ℝ s\nhx₀s : x₀ ∈ s\nhf : ContDiffOn ℝ (↑n) f s\n⊢ Filter.Tendsto (fun x ↦ ((x - x₀) ^ n)⁻¹ • (f x - taylorWithinEval f n s x₀ x)) (𝓝[s] x₀) (𝓝 0)",
"usedConstants": [
"tayl... | have h_isLittleO := (taylor_isLittleO hs hx₀s hf).norm_norm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.StrictConvexSpace | {
"line": 87,
"column": 18
} | {
"line": 87,
"column": 81
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : LinearMap.CompatibleSMul E E 𝕜 ℝ\nh : StrictConvex 𝕜 (closedBall 0 1)\nr : ℝ\nhr : 0 < r\n⊢ StrictConvex 𝕜 (closedBall 0 r)",
... | simpa only [smul_unitClosedBall_of_nonneg hr.le] using h.smul r | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Convex.StrictConvexSpace | {
"line": 87,
"column": 18
} | {
"line": 87,
"column": 81
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : LinearMap.CompatibleSMul E E 𝕜 ℝ\nh : StrictConvex 𝕜 (closedBall 0 1)\nr : ℝ\nhr : 0 < r\n⊢ StrictConvex 𝕜 (closedBall 0 r)",
... | simpa only [smul_unitClosedBall_of_nonneg hr.le] using h.smul r | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.StrictConvexSpace | {
"line": 87,
"column": 18
} | {
"line": 87,
"column": 81
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : LinearMap.CompatibleSMul E E 𝕜 ℝ\nh : StrictConvex 𝕜 (closedBall 0 1)\nr : ℝ\nhr : 0 < r\n⊢ StrictConvex 𝕜 (closedBall 0 r)",
... | simpa only [smul_unitClosedBall_of_nonneg hr.le] using h.smul r | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.StrictConvexSpace | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 54
} | [
{
"pp": "E : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : StrictConvexSpace ℝ E\nx y : E\nh : ¬SameRay ℝ x y\n⊢ ‖y‖ - ‖x‖ < ‖y - x‖",
"usedConstants": [
"lt_norm_sub_of_not_sameRay",
"Real.partialOrder",
"Real",
"AddCommGroup.toAddCommMonoid",
"Nor... | exact lt_norm_sub_of_not_sameRay (mt SameRay.symm h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun | {
"line": 261,
"column": 2
} | {
"line": 266,
"column": 70
} | [
{
"pp": "f g : ℝ → ℝ\na b : ℝ\nhf : AbsolutelyContinuousOnInterval f a b\nhg : AbsolutelyContinuousOnInterval g a b\n⊢ ∫ (x : ℝ) in a..b, f x * deriv g x = f b * g b - f a * g a - ∫ (x : ℝ) in a..b, deriv f x * g x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProduct... | rw [← AbsolutelyContinuousOnInterval.integral_deriv_mul_eq_sub hf hg,
← intervalIntegral.integral_sub]
· simp_rw [add_sub_cancel_left]
· exact (hf.intervalIntegrable_deriv.mul_continuousOn hg.continuousOn).add
(hg.intervalIntegrable_deriv.continuousOn_mul hf.continuousOn)
· exact hf.intervalIntegrable... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.AbsolutelyContinuousFun | {
"line": 261,
"column": 2
} | {
"line": 266,
"column": 70
} | [
{
"pp": "f g : ℝ → ℝ\na b : ℝ\nhf : AbsolutelyContinuousOnInterval f a b\nhg : AbsolutelyContinuousOnInterval g a b\n⊢ ∫ (x : ℝ) in a..b, f x * deriv g x = f b * g b - f a * g a - ∫ (x : ℝ) in a..b, deriv f x * g x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProduct... | rw [← AbsolutelyContinuousOnInterval.integral_deriv_mul_eq_sub hf hg,
← intervalIntegral.integral_sub]
· simp_rw [add_sub_cancel_left]
· exact (hf.intervalIntegrable_deriv.mul_continuousOn hg.continuousOn).add
(hg.intervalIntegrable_deriv.continuousOn_mul hf.continuousOn)
· exact hf.intervalIntegrable... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 72,
"column": 27
} | {
"line": 72,
"column": 54
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : CommRing β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\nf : α → β\ninst✝ : DecidablePred fun x ↦ f x ≤ 0\ns : Finset α\nh0 : Even #({x ∈ s | f x ≤ 0})\n⊢ (-1) ^ #({x ∈ s | f x ≤ 0}) * ∏ x ∈ s, f x = ∏ x ∈ s, f x",
"usedConstants": [
"Eq.mpr",
... | neg_one_pow_eq_pow_mod_two, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.AbsMax | {
"line": 390,
"column": 44
} | {
"line": 390,
"column": 63
} | [
{
"pp": "case inr\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Bornology.IsBounded U\nhne : U.Nonempty\nhd : DiffContOnCl ℂ f U\nhc : IsC... | exact ⟨w, hwU, hle⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Complex.AbsMax | {
"line": 390,
"column": 44
} | {
"line": 390,
"column": 63
} | [
{
"pp": "case inr\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Bornology.IsBounded U\nhne : U.Nonempty\nhd : DiffContOnCl ℂ f U\nhc : IsC... | exact ⟨w, hwU, hle⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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