module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Calculus.FDeriv.WithLp | {
"line": 61,
"column": 2
} | {
"line": 63,
"column": 5
} | {
"line": 65,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\nH : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝² : NormedSpace 𝕜 H\ninst✝¹ : Finite ι\np : ENNReal\ninst✝ : Fact (1 ≤ p)\nf : H →... | [] | have := Fintype.ofFinite ι
rw [← (PiLp.continuousLinearEquiv p 𝕜 E).comp_hasFDerivWithinAt_iff, hasFDerivWithinAt_pi']
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.WithLp | {
"line": 61,
"column": 2
} | {
"line": 63,
"column": 5
} | {
"line": 65,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\nH : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝³ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝² : NormedSpace 𝕜 H\ninst✝¹ : Finite ι\np : ENNReal\ninst✝ : Fact (1 ≤ p)\nf : H →... | [] | have := Fintype.ofFinite ι
rw [← (PiLp.continuousLinearEquiv p 𝕜 E).comp_hasFDerivWithinAt_iff, hasFDerivWithinAt_pi']
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 51
} | {
"line": 198,
"column": 2
} | [
{
"pp": "ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\nf' : ι → E → E →L[𝕜] G\nx : E\n... | [] | filter_upwards with n z _ using (by simp; abel) | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 274,
"column": 6
} | {
"line": 274,
"column": 41
} | {
"line": 274,
"column": 41
} | [
{
"pp": "ι : Type u_1\nl : Filter ι\nE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_6\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : TendstoUnifo... | [
"ι : Type u_1\nl : Filter ι\nE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_6\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : TendstoUniformlyOnFilter... | Metric.tendstoUniformlyOnFilter_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 277,
"column": 6
} | {
"line": 277,
"column": 41
} | {
"line": 277,
"column": 41
} | [
{
"pp": "ι : Type u_1\nl : Filter ι\nE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_6\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : TendstoUnifo... | [
"ι : Type u_1\nl : Filter ι\nE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_6\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : TendstoUniformlyOnFilter... | Metric.tendstoUniformlyOnFilter_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.SmoothTransition | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 32
} | {
"line": 230,
"column": 2
} | [
{
"pp": "case c0\nx y : ℝ\nhxy : x ≤ y\n⊢ 0 ≤ expNegInvGlue (1 - y)",
"ppTerm": "?c0",
"assigned": true,
"usedConstants": [
"Real",
"Real.instSub",
"HSub.hSub",
"expNegInvGlue.nonneg",
"Real.instOne",
"instHSub",
"One.toOfNat1",
"OfNat.ofNat"
],
... | [
"case b0\nx y : ℝ\nhxy : x ≤ y\n⊢ 0 ≤ expNegInvGlue y",
"case h₁\nx y : ℝ\nhxy : x ≤ y\n⊢ expNegInvGlue x ≤ expNegInvGlue y",
"case h₂\nx y : ℝ\nhxy : x ≤ y\n⊢ expNegInvGlue (1 - y) ≤ expNegInvGlue (1 - x)"
] | · exact expNegInvGlue.nonneg _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.SmoothTransition | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 32
} | {
"line": 231,
"column": 2
} | [
{
"pp": "case b0\nx y : ℝ\nhxy : x ≤ y\n⊢ 0 ≤ expNegInvGlue y",
"ppTerm": "?b0",
"assigned": true,
"usedConstants": [
"expNegInvGlue.nonneg"
],
"usedFVars": [
"y"
],
"usedGoals": []
},
{
"pp": "case h₁\nx y : ℝ\nhxy : x ≤ y\n⊢ expNegInvGlue x ≤ expNegInvGlue y",
... | [
"case h₁\nx y : ℝ\nhxy : x ≤ y\n⊢ expNegInvGlue x ≤ expNegInvGlue y",
"case h₂\nx y : ℝ\nhxy : x ≤ y\n⊢ expNegInvGlue (1 - y) ≤ expNegInvGlue (1 - x)"
] | · exact expNegInvGlue.nonneg _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.BumpFunction.InnerProduct | {
"line": 52,
"column": 4
} | {
"line": 54,
"column": 13
} | {
"line": 56,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nR : ℝ\nhR : 1 < R\n⊢ (Function.support fun x ↦ ((R - ‖x‖) / (R - 1)).smoothTransition) = Metric.ball 0 R",
"ppTerm": "?m.478",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRi... | [] | ext x
rw [mem_support, Ne, smoothTransition.zero_iff_nonpos, not_le, mem_ball_zero_iff]
simp [hR] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.BumpFunction.InnerProduct | {
"line": 52,
"column": 4
} | {
"line": 54,
"column": 13
} | {
"line": 56,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nR : ℝ\nhR : 1 < R\n⊢ (Function.support fun x ↦ ((R - ‖x‖) / (R - 1)).smoothTransition) = Metric.ball 0 R",
"ppTerm": "?m.478",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRi... | [] | ext x
rw [mem_support, Ne, smoothTransition.zero_iff_nonpos, not_le, mem_ball_zero_iff]
simp [hR] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 357,
"column": 8
} | {
"line": 357,
"column": 43
} | {
"line": 357,
"column": 43
} | [
{
"pp": "case refine_1\nι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : IsRCLikeNormedField 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ... | [
"case refine_1\nι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : IsRCLikeNormedField 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L... | Metric.tendstoUniformlyOnFilter_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 380,
"column": 10
} | {
"line": 380,
"column": 45
} | {
"line": 380,
"column": 45
} | [
{
"pp": "ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : IsRCLikeNormedField 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜... | [
"ι : Type u_1\nl : Filter ι\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\n𝕜 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : IsRCLikeNormedField 𝕜\ninst✝³ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E ... | Metric.tendstoUniformlyOnFilter_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ParametricIntegral | {
"line": 152,
"column": 12
} | {
"line": 152,
"column": 14
} | {
"line": 153,
"column": 4
} | [
{
"pp": "case pos.h_lim\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\nbou... | [
"case pos.h_lim\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 : α → ℝ\n... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 182,
"column": 8
} | {
"line": 182,
"column": 67
} | {
"line": 183,
"column": 4
} | [
{
"pp": "case refine_2\n𝕜 : 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... | [] | exact Subset.trans (ball_subset_ball (min_le_right _ _)) hδ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 182,
"column": 8
} | {
"line": 182,
"column": 67
} | {
"line": 183,
"column": 4
} | [
{
"pp": "case refine_2\n𝕜 : 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... | [] | exact Subset.trans (ball_subset_ball (min_le_right _ _)) hδ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 182,
"column": 8
} | {
"line": 182,
"column": 67
} | {
"line": 183,
"column": 4
} | [
{
"pp": "case refine_2\n𝕜 : 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... | [] | exact Subset.trans (ball_subset_ball (min_le_right _ _)) hδ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 520,
"column": 4
} | {
"line": 520,
"column": 46
} | {
"line": 521,
"column": 2
} | [
{
"pp": "ι : Type u_1\nl : Filter ι\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nG : Type u_3\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nf : ι → 𝕜 → G\ng : 𝕜 → G\nf' : ι → 𝕜 → G\ng' : 𝕜 → G\ninst✝¹ : IsRCLikeNormedField 𝕜\ninst✝ : l.NeBot\nhf' : TendstoUniformly f' g' l\nhf : ∀ᶠ (n ... | [] | filter_upwards [hf] with n h x _ using h x | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Convolution | {
"line": 738,
"column": 2
} | {
"line": 749,
"column": 9
} | {
"line": 750,
"column": 2
} | [
{
"pp": "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 : NontriviallyNormedField 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 : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace 𝕜 F\nL : E ... | have h2 : ∀ t, dist (L (f t) (g (x₀ - t))) (L (f t) z₀) ≤ ‖L (f t)‖ * ε := by
intro t; by_cases ht : t ∈ support f
· have h2t := hf ht
rw [mem_ball_zero_iff] at h2t
specialize hg (x₀ - t)
rw [sub_eq_add_neg, add_mem_ball_iff_norm, norm_neg, ← sub_eq_add_neg] at hg
refine ((L (f t)).dist_... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convolution | {
"line": 777,
"column": 2
} | {
"line": 778,
"column": 78
} | {
"line": 780,
"column": 0
} | [
{
"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... | [] | · simp_rw [Real.norm_of_nonneg (hnf _), hintf, mul_one]
exact (mul_le_mul_of_nonneg_right opNorm_lsmul_le hε).trans_eq (one_mul ε) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convolution | {
"line": 798,
"column": 2
} | {
"line": 798,
"column": 35
} | {
"line": 799,
"column": 2
} | [
{
"pp": "G : Type uG\nE' : Type uE'\ninst✝⁸ : NormedAddCommGroup 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'\ninst✝ : CompleteSpace E'\nι ... | [
"G : Type uG\nE' : Type uE'\ninst✝⁸ : NormedAddCommGroup 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'\ninst✝ : CompleteSpace E'\nι : Type u_1\n... | rw [Metric.tendsto_nhds] at hcg ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 60,
"column": 9
} | {
"line": 60,
"column": 21
} | {
"line": 60,
"column": 21
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nx : E\nn : ℕ∞\nhs : s ∈ 𝓝 x\nd : ℝ\nd_pos : 0 < d\nhd : Euclidean.closedBall x d ⊆ s\nc : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }\nf : E ... | [
"E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nx : E\nn : ℕ∞\nhs : s ∈ 𝓝 x\nd : ℝ\nd_pos : 0 < d\nhd : Euclidean.closedBall x d ⊆ s\nc : ContDiffBump (toEuclidean x) := { rIn := d / 2, rOut := d, rIn_pos := ⋯, rIn_lt_rOut := ⋯ }\nf : E → ℝ := ↑c ∘ ... | c.support_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 179,
"column": 4
} | {
"line": 179,
"column": 41
} | {
"line": 180,
"column": 4
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : MetricSpace α\nN : ℕ\nτ : ℝ\na : SatelliteConfig α N τ\ni : Fin N.succ\nH : last N ≤ i\n⊢ dist (a.c i) (a.c (last N)) ≤ a.r i + a.r (last N)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [
"case inr\nα : Type u_1\ninst✝ : MetricSpace α\nN : ℕ\nτ : ℝ\na : SatelliteConfig α N τ\ni : Fin N.succ\nH : last N ≤ i\nI : i = last N\n⊢ dist (a.c i) (a.c (last N)) ≤ a.r i + a.r (last N)"
] | have I : i = last N := top_le_iff.1 H | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 256,
"column": 2
} | {
"line": 258,
"column": 65
} | {
"line": 260,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\n⊢ Monotone p.iUnionUpTo",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Besicovitch.TauPackage.iUnionUpTo... | [] | intro i j hij
simp only [iUnionUpTo]
exact iUnion_mono' fun r => ⟨⟨r, r.2.trans_le hij⟩, Subset.rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 256,
"column": 2
} | {
"line": 258,
"column": 65
} | {
"line": 260,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\n⊢ Monotone p.iUnionUpTo",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Besicovitch.TauPackage.iUnionUpTo... | [] | intro i j hij
simp only [iUnionUpTo]
exact iUnion_mono' fun r => ⟨⟨r, r.2.trans_le hij⟩, Subset.rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 403,
"column": 4
} | {
"line": 403,
"column": 50
} | {
"line": 404,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\nN : ℕ\nhN : IsEmpty (SatelliteConfig α N p.τ)\ni : Ordinal.{u}\nIH : ∀ y < i, y < p.lastStep → p.color y < N\nhi : i < p.lastStep\nA : Set ℕ :=\n ⋃ j,\n ⋃ (_ :\n (closedBall (p.c (p.index ↑j)) (p.r (p.ind... | [
"α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\nN : ℕ\nhN : IsEmpty (SatelliteConfig α N p.τ)\ni : Ordinal.{u}\nIH : ∀ y < i, y < p.lastStep → p.color y < N\nhi : i < p.lastStep\nA : Set ℕ :=\n ⋃ j,\n ⋃ (_ :\n (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ cl... | have hb : (b : ℕ) ≤ N := Nat.lt_succ_iff.1 b.2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 435,
"column": 16
} | {
"line": 435,
"column": 18
} | {
"line": 436,
"column": 8
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\nN : ℕ\nhN : IsEmpty (SatelliteConfig α N p.τ)\ni : Ordinal.{u}\nIH : ∀ y < i, y < p.lastStep → p.color y < N\nhi : i < p.lastStep\nA : Set ℕ :=\n ⋃ j,\n ⋃ (_ :\n (closedBall (p.c (p.index ↑j)) (p.r (p.ind... | [
"α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\nN : ℕ\nhN : IsEmpty (SatelliteConfig α N p.τ)\ni : Ordinal.{u}\nIH : ∀ y < i, y < p.lastStep → p.color y < N\nhi : i < p.lastStep\nA : Set ℕ :=\n ⋃ j,\n ⋃ (_ :\n (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ cl... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 440,
"column": 16
} | {
"line": 440,
"column": 18
} | {
"line": 441,
"column": 8
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\nN : ℕ\nhN : IsEmpty (SatelliteConfig α N p.τ)\ni : Ordinal.{u}\nIH : ∀ y < i, y < p.lastStep → p.color y < N\nhi : i < p.lastStep\nA : Set ℕ :=\n ⋃ j,\n ⋃ (_ :\n (closedBall (p.c (p.index ↑j)) (p.r (p.ind... | [
"α : Type u_1\ninst✝¹ : MetricSpace α\nβ : Type u\ninst✝ : Nonempty β\np : TauPackage β α\nN : ℕ\nhN : IsEmpty (SatelliteConfig α N p.τ)\ni : Ordinal.{u}\nIH : ∀ y < i, y < p.lastStep → p.color y < N\nhi : i < p.lastStep\nA : Set ℕ :=\n ⋃ j,\n ⋃ (_ :\n (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ cl... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 963,
"column": 4
} | {
"line": 965,
"column": 63
} | {
"line": 966,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝¹² : TopologicalSpace G\ninst✝¹¹ : Group G\ninst✝¹⁰ : IsTopologicalGroup G\ninst✝⁹ : MeasurableSpace G\ninst✝⁸ : BorelSpace G\nH : Type u_2\ninst✝⁷ : Group H\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : IsTopologicalGroup H\ninst✝⁴ : CompactSpace H\ninst✝³ : MeasurableSpace H\ninst✝² : Bor... | [
"G : Type u_1\ninst✝¹² : TopologicalSpace G\ninst✝¹¹ : Group G\ninst✝¹⁰ : IsTopologicalGroup G\ninst✝⁹ : MeasurableSpace G\ninst✝⁸ : BorelSpace G\nH : Type u_2\ninst✝⁷ : Group H\ninst✝⁶ : TopologicalSpace H\ninst✝⁵ : IsTopologicalGroup H\ninst✝⁴ : CompactSpace H\ninst✝³ : MeasurableSpace H\ninst✝² : BorelSpace H\nμ... | have : C * ν univ = 1 * ν univ := by
rw [one_mul, ← smul_eq_mul, ← ENNReal.smul_def, ← smul_apply, ← hC,
map_apply hcont.measurable .univ, preimage_univ, huniv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 988,
"column": 2
} | {
"line": 990,
"column": 69
} | {
"line": 991,
"column": 2
} | [
{
"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.... | have μeq : μ = c ^ 2 • μ := by
rw [map_map continuous_inv.measurable continuous_inv.measurable] at this
simpa only [inv_involutive, Involutive.comp_self, Measure.map_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 1014,
"column": 2
} | {
"line": 1016,
"column": 69
} | {
"line": 1017,
"column": 2
} | [
{
"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... | have μeq : μ = c ^ 2 • μ := by
rw [map_map continuous_inv.measurable continuous_inv.measurable] at this
simpa only [inv_involutive, Involutive.comp_self, Measure.map_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 435,
"column": 12
} | {
"line": 437,
"column": 40
} | {
"line": 438,
"column": 6
} | [] | [] | a.r j - ‖a.c j - a.c i‖ ≤ τ * a.r i - a.r i := sub_le_sub H.2 H.1
_ = a.r i * (τ - 1) := by ring
_ ≤ s * (τ - 1) := by gcongr | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Convex.NNReal | {
"line": 42,
"column": 22
} | {
"line": 42,
"column": 24
} | {
"line": 42,
"column": 25
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : Module ℝ M\ns : Set M\nH : Convex ℝ≥0 s\nx✝ : M\nhx : x✝ ∈ s\ny✝ : M\nhy : y✝ ∈ s\na b : ℝ\n⊢ 0 ≤ a → 0 ≤ b → a + b = 1 → a • x✝ + b • y✝ ∈ s",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
... | [
"M : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : Module ℝ M\ns : Set M\nH : Convex ℝ≥0 s\nx✝ : M\nhx : x✝ ∈ s\ny✝ : M\nhy : y✝ ∈ s\na b : ℝ\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → a • x✝ + b • y✝ ∈ s"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 472,
"column": 23
} | {
"line": 472,
"column": 31
} | {
"line": 472,
"column": 32
} | [
{
"pp": "case inr.inl\nE : 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\nhδ2 : δ ≤ 1\nc' : Fin N.succ → E := fun i ↦ if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i... | [
"case inr.inl\nE : 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\nhδ2 : δ ≤ 1\nc' : Fin N.succ → E := fun i ↦ if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i\n... | if_true, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.MetricSpace.Holder | {
"line": 205,
"column": 2
} | {
"line": 205,
"column": 17
} | {
"line": 206,
"column": 2
} | [
{
"pp": "case inr\nX : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nr : ℝ≥0\nf : X → Y\nC D s : ℝ≥0\nA : Set X\nhA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D\nhf : HolderOnWith C r f A\nhsr : ↑s ≤ ↑r\nht : 0 < s\nhr : 0 < ↑r\nθ₁ : ℝ≥0 := NNReal.mk (↑s / ↑r) ⋯\nθ₂ : ℝ≥0 := NNReal.... | [
"case inr\nX : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nr : ℝ≥0\nf : X → Y\nC D s : ℝ≥0\nA : Set X\nhA : ∀ x ∈ A, ∀ y ∈ A, edist x y ≤ ↑D\nhf : HolderOnWith C r f A\nhsr : ↑s ≤ ↑r\nht : 0 < s\nhr : 0 < ↑r\nθ₁ : ℝ≥0 := NNReal.mk (↑s / ↑r) ⋯\nθ₂ : ℝ≥0 := NNReal.mk (1 - ↑s /... | rw [hθC, ← hθt] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.Darboux | {
"line": 82,
"column": 4
} | {
"line": 84,
"column": 19
} | {
"line": 85,
"column": 4
} | [
{
"pp": "case inl\nf f' : ℝ → ℝ\ns : Set ℝ\nhs : s.OrdConnected\nhf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : ℝ\nha : a ∈ s\nb : ℝ\nhb : b ∈ s\nm : ℝ\nhma : f' a < m\nhmb : m < f' b\nhab : a ≤ b\nthis : Icc a b ⊆ s\n⊢ m ∈ f' '' s",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Re... | [
"case inl\nf f' : ℝ → ℝ\ns : Set ℝ\nhs : s.OrdConnected\nhf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : ℝ\nha : a ∈ s\nb : ℝ\nhb : b ∈ s\nm : ℝ\nhma : f' a < m\nhmb : m < f' b\nhab : a ≤ b\nthis : Icc a b ⊆ s\nc : ℝ\ncmem : c ∈ Ioo a b\nhc : f' c = m\n⊢ m ∈ f' '' s"
] | rcases exists_hasDerivWithinAt_eq_of_gt_of_lt hab (fun x hx => (hf x <| this hx).mono this) hma
hmb with
⟨c, cmem, hc⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 188,
"column": 4
} | {
"line": 188,
"column": 14
} | {
"line": 189,
"column": 4
} | [
{
"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... | [
"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 : NormedAddCommGroup G\ninst✝ ... | rw [Bu_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 47
} | {
"line": 303,
"column": 4
} | [
{
"pp": "case insert\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ns : Set E\nι : Type u_2\nA' : Type u_4\ninst✝³ : NormedCommRing A'\ninst✝² : NormedAlgebra 𝕜 A'\ninst✝¹ : DecidableEq ι\ninst✝ : NormOneClass A'\nf : ι → E → A'\nN : ... | [
"case insert\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ns : Set E\nι : Type u_2\nA' : Type u_4\ninst✝³ : NormedCommRing A'\ninst✝² : NormedAlgebra 𝕜 A'\ninst✝¹ : DecidableEq ι\ninst✝ : NormOneClass A'\nf : ι → E → A'\nN : ℕ∞ω\nhs : Un... | rw [← Finset.sum_coe_sort (Finset.sym _ _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.ContDiff.Bounds | {
"line": 309,
"column": 4
} | {
"line": 309,
"column": 75
} | {
"line": 310,
"column": 4
} | [
{
"pp": "case insert\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ns : Set E\nι : Type u_2\nA' : Type u_4\ninst✝³ : NormedCommRing A'\ninst✝² : NormedAlgebra 𝕜 A'\ninst✝¹ : DecidableEq ι\ninst✝ : NormOneClass A'\nf : ι → E → A'\nN : ... | [
"case insert\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ns : Set E\nι : Type u_2\nA' : Type u_4\ninst✝³ : NormedCommRing A'\ninst✝² : NormedAlgebra 𝕜 A'\ninst✝¹ : DecidableEq ι\ninst✝ : NormOneClass A'\nf : ι → E → A'\nN : ℕ∞ω\nhs : Un... | simp +instances only [comp_apply, Finset.symInsertEquiv_symm_apply_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 32
} | {
"line": 71,
"column": 0
} | [
{
"pp": "case neg\nf : ℝ → ℝ\na b : ℝ\nh : ContinuousAt f a\nhd₀ : DifferentiableOn ℝ f (Ioo a b)\ng₀ : b ≤ a\n⊢ ContinuousOn f (Ico a b)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Real",
"congrArg",
"PseudoMetricSpace.toUniformSpace",
"continuousOn_empty._sim... | [] | · simp [Ico_eq_empty_of_le g₀] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 228,
"column": 2
} | {
"line": 230,
"column": 95
} | {
"line": 232,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhc : ContinuousAt f c\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 (Icc a c) b",
"ppTerm": "?m.72",
"assigned": t... | [] | refine isMinOn_Icc_of_anti_mono ?_ ?_
· apply antitoneOn_of_deriv_nonpos (convex_Icc a b) (continuousOn_Icc ha hb hd₀) <;> simp_all
· apply monotoneOn_of_deriv_nonneg (convex_Icc b c) (continuousOn_Icc hb hc hd₁) <;> simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 228,
"column": 2
} | {
"line": 230,
"column": 95
} | {
"line": 232,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\na b c : ℝ\nha : ContinuousAt f a\nhb : ContinuousAt f b\nhc : ContinuousAt f c\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 (Icc a c) b",
"ppTerm": "?m.72",
"assigned": t... | [] | refine isMinOn_Icc_of_anti_mono ?_ ?_
· apply antitoneOn_of_deriv_nonpos (convex_Icc a b) (continuousOn_Icc ha hb hd₀) <;> simp_all
· apply monotoneOn_of_deriv_nonneg (convex_Icc b c) (continuousOn_Icc hb hc hd₁) <;> simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.DerivativeTest | {
"line": 385,
"column": 23
} | {
"line": 391,
"column": 59
} | {
"line": 393,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\nx₀ : ℝ\nh : ContinuousAt f x₀\nhf : ∀ᶠ (x : ℝ) in 𝓝[≠] x₀, sign (deriv f x) = sign (x₀ - x)\n⊢ IsLocalMax f x₀",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instLE",
"Real",
"Set.Ioi",
"Semiring.toModule",
"Real.de... | [] | by
have hl := deriv_pos_left_of_sign_deriv hf
have hg := deriv_neg_right_of_sign_deriv hf
replace hf := (nhdsLT_sup_nhdsGT x₀) ▸
eventually_sup.mpr ⟨hl.mono fun x hx => hx.ne', hg.mono fun x hx => hx.ne⟩
exact isLocalMax_of_deriv h (hf.mono fun x hx ↦ differentiableAt_of_deriv_ne_zero hx)
(hl.mono fun _... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.VectorField | {
"line": 275,
"column": 2
} | {
"line": 276,
"column": 43
} | {
"line": 278,
"column": 0
} | [
{
"pp": "case hg\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nV W : E → E\ns : Set E\nx : E\nm n : ℕ∞ω\nhV : ContDiffWithinAt 𝕜 n V s x\nhW : ContDiffWithinAt 𝕜 n W s x\nhs : UniqueDiffOn 𝕜 s\nhmn : m + 1 ≤ n\nhx : x ∈ s\n⊢ ContDi... | [] | · exact ContDiffWithinAt.clm_apply (hV.fderivWithin_right hs hmn hx)
(hW.of_le (le_trans le_self_add hmn)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Gradient.Basic | {
"line": 161,
"column": 93
} | {
"line": 163,
"column": 30
} | {
"line": 165,
"column": 0
} | [
{
"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\nf : F → 𝕜\nf' x : F\n⊢ HasGradientWithinAt f f' univ x ↔ HasGradientAt f f' x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"LinearIso... | [] | by
rw [hasGradientWithinAt_iff_hasFDerivWithinAt, hasGradientAt_iff_hasFDerivAt]
exact hasFDerivWithinAt_univ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Complemented | {
"line": 85,
"column": 6
} | {
"line": 85,
"column": 57
} | {
"line": 85,
"column": 57
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace E\np q : Subspace 𝕜 E\nh : IsCompl p q\nhp : IsClosed ↑p\nhq : IsClosed ↑q\nthis✝ : CompleteSpace ↑↑p\nthis : CompleteSpace ↑↑q\n⊢ IsTopCompl p q",
"ppT... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : CompleteSpace E\np q : Subspace 𝕜 E\nh : IsCompl p q\nhp : IsClosed ↑p\nhq : IsClosed ↑q\nthis✝ : CompleteSpace ↑↑p\nthis : CompleteSpace ↑↑q\n⊢ Continuous ⇑(prodEquivOfIsCompl p q h... | isTopCompl_iff_continuous_symm_prodEquivOfIsCompl h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.IteratedDeriv.FaaDiBruno | {
"line": 141,
"column": 52
} | {
"line": 143,
"column": 72
} | {
"line": 145,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ng : 𝕜 → E\nf : 𝕜 → 𝕜\ns t : Set 𝕜\nx : 𝕜\nn : ℕ∞ω\ni : ℕ\nhg : ContDiffWithinAt 𝕜 n g t (f x)\nhf : ContDiffWithinAt 𝕜 n f s x\nht : UniqueDiffOn 𝕜 t\nhs : UniqueDiffOn 𝕜 ... | [] | by
rw [iteratedDerivWithin_vcomp_eq_sum_orderedFinpartition hg hf ht hs hx hst hi]
simp only [iteratedFDerivWithin_apply_eq_iteratedDerivWithin_mul_prod] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 35
} | {
"line": 361,
"column": 4
} | [
{
"pp": "case inr.hf\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⊢ Contin... | [
"case inr.hff'\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 ∈ Ioo b a, H... | · rwa [uIcc_of_ge hab.le] at hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 363,
"column": 4
} | {
"line": 363,
"column": 18
} | {
"line": 365,
"column": 0
} | [
{
"pp": "case inr.hab\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⊢ b ≤ a... | [] | · exact hab.le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 391,
"column": 4
} | {
"line": 391,
"column": 35
} | {
"line": 392,
"column": 4
} | [
{
"pp": "case inr.hf\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⊢ Contin... | [
"case inr.hff'\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 ∈ Ioo b a, H... | · rwa [uIcc_of_ge hab.le] at hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 33
} | {
"line": 293,
"column": 2
} | [
{
"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 : ENNReal.ofReal |A.det| < ↑m\n⊢ {x | (fun δ ↦ ∀ (s : Set E) (f : E → E), Appro... | [
"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 : ENNReal.ofReal |A.det| < ↑m\nd : ℝ≥0∞ := ENNReal.ofReal |A.det|\n⊢ {x | (fun δ ↦ ∀ (s : S... | let d := ENNReal.ofReal |A.det| | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 394,
"column": 4
} | {
"line": 394,
"column": 18
} | {
"line": 396,
"column": 0
} | [
{
"pp": "case inr.hab\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⊢ b ≤ a... | [] | · exact hab.le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 136,
"column": 58
} | {
"line": 136,
"column": 60
} | {
"line": 136,
"column": 61
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝¹ : PseudoMetricSpace α\ninst✝ : OpensMeasurableSpace α\nx : α\nr : ι → ℝ\nhr : Tendsto r l atTop\ny : α\na : ι\n⊢ a ∈ r ⁻¹' Ioi (dist x y) → y ∈ Metric.ball x (r a)",
"ppTerm": "?m.56",
"assigned": true,
... | [
"α : Type u_1\nι : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝¹ : PseudoMetricSpace α\ninst✝ : OpensMeasurableSpace α\nx : α\nr : ι → ℝ\nhr : Tendsto r l atTop\ny : α\na : ι\nha : a ∈ r ⁻¹' Ioi (dist x y)\n⊢ y ∈ Metric.ball x (r a)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 143,
"column": 58
} | {
"line": 143,
"column": 60
} | {
"line": 143,
"column": 61
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝¹ : PseudoMetricSpace α\ninst✝ : OpensMeasurableSpace α\nx : α\nr : ι → ℝ\nhr : Tendsto r l atTop\ny : α\na : ι\n⊢ a ∈ r ⁻¹' Ici (dist x y) → y ∈ Metric.closedBall x (r a)",
"ppTerm": "?m.55",
"assigned": ... | [
"α : Type u_1\nι : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\nl : Filter ι\ninst✝¹ : PseudoMetricSpace α\ninst✝ : OpensMeasurableSpace α\nx : α\nr : ι → ℝ\nhr : Tendsto r l atTop\ny : α\na : ι\nha : a ∈ r ⁻¹' Ici (dist x y)\n⊢ y ∈ Metric.closedBall x (r a)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.Jacobian | {
"line": 358,
"column": 48
} | {
"line": 358,
"column": 50
} | {
"line": 359,
"column": 4
} | [
{
"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 : ENNReal.ofReal |A.det| < ↑m\nd : ℝ≥0∞ := ENNReal.ofReal |A.det|\nε : ℝ\nhε : ... | [
"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 : ENNReal.ofReal |A.det| < ↑m\nd : ℝ≥0∞ := ENNReal.ofReal |A.det|\nε : ℝ\nhε : μ (closedBal... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 423,
"column": 4
} | {
"line": 423,
"column": 35
} | {
"line": 424,
"column": 4
} | [
{
"pp": "case inr.hf\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), f' x ≤ 0\nM : AntitoneOn f [[a, b]]\nhab : b < a\n⊢ Contin... | [
"case inr.hff'\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), f' x ≤ 0\nM : AntitoneOn f [[a, b]]\nhab : b < a\n⊢ ∀ x ∈ Ioo b a, H... | · rwa [uIcc_of_ge hab.le] at hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 426,
"column": 4
} | {
"line": 426,
"column": 18
} | {
"line": 428,
"column": 0
} | [
{
"pp": "case inr.hab\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), f' x ≤ 0\nM : AntitoneOn f [[a, b]]\nhab : b < a\n⊢ b ≤ a... | [] | · exact hab.le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 454,
"column": 4
} | {
"line": 454,
"column": 35
} | {
"line": 455,
"column": 4
} | [
{
"pp": "case inr.hf\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), f' x ≤ 0\nM : AntitoneOn f [[a, b]]\nhab : b < a\n⊢ Contin... | [
"case inr.hff'\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), f' x ≤ 0\nM : AntitoneOn f [[a, b]]\nhab : b < a\n⊢ ∀ x ∈ Ioo b a, H... | · rwa [uIcc_of_ge hab.le] at hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 457,
"column": 4
} | {
"line": 457,
"column": 18
} | {
"line": 459,
"column": 0
} | [
{
"pp": "case inr.hab\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), f' x ≤ 0\nM : AntitoneOn f [[a, b]]\nhab : b < a\n⊢ b ≤ a... | [] | · exact hab.le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 948,
"column": 2
} | {
"line": 948,
"column": 22
} | {
"line": 950,
"column": 0
} | [
{
"pp": "E : Type u_1\nf f' : ℝ → E\na : ℝ\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhderiv : ∀ x ∈ Iic a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Iic a) volume\ng : ℝ → E := f ∘ fun x ↦ -x\nhdg : ∀ x ∈ Ioi (-a), HasDerivAt g (-f' (-x)) x\nL : Tendsto g atTop (𝓝... | [] | simpa [g] using this | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.LocalExtr.Polynomial | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 73
} | {
"line": 45,
"column": 2
} | [
{
"pp": "case inr\np : ℝ[X]\nhp' : derivative p ≠ 0\nhp : p ≠ 0\n⊢ p.roots.toFinset.card ≤ ((derivative p).roots.toFinset \\ p.roots.toFinset).card + 1",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"Polynomial.derivative",
"Real",
"Polynomial.r... | [
"case inr\np : ℝ[X]\nhp' : derivative p ≠ 0\nhp : p ≠ 0\nx : ℝ\nhx : x ∈ p.roots.toFinset\ny : ℝ\nhy : y ∈ p.roots.toFinset\nhxy : x < y\nhxy' : ∀ z ∈ p.roots.toFinset, z ∉ Set.Ioo x y\n⊢ ∃ z ∈ (derivative p).roots.toFinset, x < z ∧ z < y"
] | refine Finset.card_le_sdiff_of_interleaved fun x hx y hy hxy hxy' => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.LocallyUniformLimit | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 14
} | {
"line": 57,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nz : ℂ\nM r : ℝ\nf : ℂ → E\nhr : 0 < r\nhf : ∀ w ∈ sphere z r, ‖f w‖ ≤ M\nhM : 0 ≤ M\n⊢ ∀ w ∈ sphere z r, ‖((w - z) ^ 2)⁻¹ • f w‖ ≤ M / r ^ 2",
"ppTerm": "?m.135",
"assigned": true,
"usedConstants": [
"NormedCommRing... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nz : ℂ\nM r : ℝ\nf : ℂ → E\nhr : 0 < r\nhf : ∀ w ∈ sphere z r, ‖f w‖ ≤ M\nhM : 0 ≤ M\nw : ℂ\nhw : w ∈ sphere z r\n⊢ ‖((w - z) ^ 2)⁻¹ • f w‖ ≤ M / r ^ 2"
] | intro w hw | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 111,
"column": 28
} | {
"line": 111,
"column": 51
} | {
"line": 113,
"column": 0
} | [
{
"pp": "H✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\nN O : StructureGroupoid H\nh : (fun s ↦ s.members) N = (fun s ↦ s.members) O\n⊢ N = O",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"PartialHomeomorph.toPartialEquiv",
"OpenPartial... | [] | cases N; cases O; congr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.StructureGroupoid | {
"line": 111,
"column": 28
} | {
"line": 111,
"column": 51
} | {
"line": 113,
"column": 0
} | [
{
"pp": "H✝ : Type u_1\ninst✝¹ : TopologicalSpace H✝\nH : Type u_2\ninst✝ : TopologicalSpace H\nN O : StructureGroupoid H\nh : (fun s ↦ s.members) N = (fun s ↦ s.members) O\n⊢ N = O",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"PartialHomeomorph.toPartialEquiv",
"OpenPartial... | [] | cases N; cases O; congr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 70
} | {
"line": 274,
"column": 2
} | [
{
"pp": "H : 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\n⊢ ∃ i, (i ∈ 𝓝 x ∧ IsPathConnected i) ∧ id i ⊆ s"... | [
"H : 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⊢ ∃ i, (i ∈ 𝓝 x ∧ IsPathConnected i) ∧ id i ⊆... | have ht : t ∈ 𝓝 x := Filter.inter_mem hs (chart_source_mem_nhds _ _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.ChartedSpace | {
"line": 313,
"column": 6
} | {
"line": 313,
"column": 75
} | {
"line": 314,
"column": 2
} | [
{
"pp": "case pos.refine_2\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⊢ ¬↑(chartAt H x) x = ↑(chartAt H x) y",
"ppTerm": "?pos.refine_2✝",
"assigned": true,
"u... | [] | exact (chartAt H x).injOn.ne (ChartedSpace.mem_chart_source x) hy hxy | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.HasGroupoid | {
"line": 242,
"column": 10
} | {
"line": 242,
"column": 56
} | {
"line": 243,
"column": 8
} | [
{
"pp": "H : Type u\ninst✝² : TopologicalSpace H\nα : Type u_5\ninst✝¹ : TopologicalSpace α\ne : OpenPartialHomeomorph α H\nh : e.source = univ\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\ne' e'' : OpenPartialHomeomorph α H\nhe' : e' ∈ atlas H α\nhe'' : e'' ∈ atlas H α\n⊢ e'.symm ≫ₕ e'' ∈ G",
... | [
"H : Type u\ninst✝² : TopologicalSpace H\nα : Type u_5\ninst✝¹ : TopologicalSpace α\ne : OpenPartialHomeomorph α H\nh : e.source = univ\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\ne' e'' : OpenPartialHomeomorph α H\nhe' : e' ∈ atlas H α\nhe'' : e'' ∈ atlas H α\n⊢ e.symm ≫ₕ e'' ∈ G"
] | e.singletonChartedSpace_mem_atlas_eq h e' he', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 314,
"column": 22
} | {
"line": 314,
"column": 24
} | {
"line": 314,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nh : IsRCLikeNormedField 𝕜\ns : Set E\nhs : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s\nthis✝ : RCLi... | [
"𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nh : IsRCLikeNormedField 𝕜\ns : Set E\nhs : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : ℝ⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ s\nthis✝ : RCLike 𝕜 := IsR... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 138,
"column": 6
} | {
"line": 138,
"column": 27
} | {
"line": 138,
"column": 28
} | [
{
"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\nx : M\nhx : x ∈ f.source\nh'x : ↑(... | [
"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\nx : M\nhx : x ∈ f.source\nh'x : ↑(f.extend I) ... | f.map_extend_nhds hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 143,
"column": 6
} | {
"line": 143,
"column": 27
} | {
"line": 143,
"column": 28
} | [
{
"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\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝ : I.Boundaryless\nx : M\nhx... | [
"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\nf : OpenPartialHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝ : I.Boundaryless\nx : M\nhx : x ∈ f.sou... | f.map_extend_nhds hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 171,
"column": 6
} | {
"line": 171,
"column": 22
} | {
"line": 171,
"column": 23
} | [
{
"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⊢ interior (f.extend I).target ⊆ i... | [
"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⊢ interior (↑I.symm ⁻¹' f.target ∩ range ↑I) ⊆... | f.extend_target, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 178,
"column": 6
} | {
"line": 178,
"column": 22
} | {
"line": 178,
"column": 23
} | [
{
"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\ny : H\nhy : y ∈ f.target\nhy' : ↑I... | [
"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\ny : H\nhy : y ∈ f.target\nhy' : ↑I y ∈ interio... | f.extend_target, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 16
} | {
"line": 197,
"column": 2
} | [
{
"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'\nP : (H → H') → Set H → H → Prop\nU : Opens M'\nf : M → ↥U\ns : Set M\nx ... | [
"case refine_1\nH : 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'\nP : (H → H') → Set H → H → Prop\nU : Opens M'\nf : M → ↥U\ns : Set M\... | congrm ?_ ∧ ?_ | Mathlib.Tactic._aux_Mathlib_Tactic_CongrM___elabRules_Mathlib_Tactic_congrM_1 | Mathlib.Tactic.congrM |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 341,
"column": 24
} | {
"line": 341,
"column": 26
} | {
"line": 341,
"column": 27
} | [
{
"pp": "case pos\n𝕜 : 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\nh : IsRCLikeNormedField 𝕜\nthis : RCLike 𝕜 := IsRCLikeNormedField.rcli... | [
"case pos\n𝕜 : 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\nh : IsRCLikeNormedField 𝕜\nthis : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜\nW :\n... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 365,
"column": 6
} | {
"line": 365,
"column": 31
} | {
"line": 365,
"column": 32
} | [
{
"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\nI : ModelWithCorners 𝕜 E H\ne e' : OpenPartialHomeomorph M H\n⊢ UniqueDiffOn 𝕜 (extendCoordC... | [
"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\nI : ModelWithCorners 𝕜 E H\ne e' : OpenPartialHomeomorph M H\n⊢ UniqueDiffOn 𝕜 (extendCoordChange e' e).... | ← extendCoordChange_symm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 532,
"column": 4
} | {
"line": 534,
"column": 52
} | {
"line": 535,
"column": 4
} | [
{
"pp": "case pos\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nι : Type v\ninst✝³ : Fintype ι\nE : ι → Type w\ninst✝² : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)\nH : ι → Type u'\ninst✝ : (i : ι) → TopologicalSpace (H i)\nI : (i : ι) → ModelWithCorners 𝕜 (E i) (H i)\nh :... | [
"case neg\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nι : Type v\ninst✝³ : Fintype ι\nE : ι → Type w\ninst✝² : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)\nH : ι → Type u'\ninst✝ : (i : ι) → TopologicalSpace (H i)\nI : (i : ι) → ModelWithCorners 𝕜 (E i) (H i)\nh : ¬IsRCLikeNo... | · let := h.rclike
let := fun i ↦ NormedSpace.restrictScalars ℝ 𝕜 (E i)
exact convex_pi fun i _hi ↦ (I i).convex_range | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 545,
"column": 2
} | {
"line": 545,
"column": 16
} | {
"line": 546,
"column": 2
} | [
{
"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... | [
"case refine_1\nH : 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') → Se... | congrm ?_ ∧ ?_ | Mathlib.Tactic._aux_Mathlib_Tactic_CongrM___elabRules_Mathlib_Tactic_congrM_1 | Mathlib.Tactic.congrM |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 16
} | {
"line": 555,
"column": 2
} | [
{
"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... | [
"case refine_1\nH : 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') → Se... | congrm ?_ ∧ ?_ | Mathlib.Tactic._aux_Mathlib_Tactic_CongrM___elabRules_Mathlib_Tactic_congrM_1 | Mathlib.Tactic.congrM |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 596,
"column": 6
} | {
"line": 608,
"column": 51
} | {
"line": 609,
"column": 4
} | [
{
"pp": "H : Type u_1\ninst✝¹ : TopologicalSpace H\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\n⊢ ∀ {s : Set H} {x : H} {u : Set H} {f : H → H},\n IsOpen[inst✝¹] u → x ∈ u → (G.IsLocalStructomorphWithinAt f s x ↔ G.IsLocalStructomorphWithinAt f (s ∩ u) x)",
"ppTerm": "?m.30",
"assigned... | [] | intro s x u f hu hux
constructor
· rintro h hx
rcases h hx.1 with ⟨e, heG, hef, hex⟩
have : s ∩ u ∩ e.source ⊆ s ∩ e.source := by mfld_set_tac
exact ⟨e, heG, hef.mono this, hex⟩
· rintro h hx
rcases h ⟨hx, hux⟩ with ⟨e, heG, hef, hex⟩
refine ⟨e.restr (interior u... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.LocalInvariantProperties | {
"line": 596,
"column": 6
} | {
"line": 608,
"column": 51
} | {
"line": 609,
"column": 4
} | [
{
"pp": "H : Type u_1\ninst✝¹ : TopologicalSpace H\nG : StructureGroupoid H\ninst✝ : ClosedUnderRestriction G\n⊢ ∀ {s : Set H} {x : H} {u : Set H} {f : H → H},\n IsOpen[inst✝¹] u → x ∈ u → (G.IsLocalStructomorphWithinAt f s x ↔ G.IsLocalStructomorphWithinAt f (s ∩ u) x)",
"ppTerm": "?m.30",
"assigned... | [] | intro s x u f hu hux
constructor
· rintro h hx
rcases h hx.1 with ⟨e, heG, hef, hex⟩
have : s ∩ u ∩ e.source ⊆ s ∩ e.source := by mfld_set_tac
exact ⟨e, heG, hef.mono this, hex⟩
· rintro h hx
rcases h ⟨hx, hux⟩ with ⟨e, heG, hef, hex⟩
refine ⟨e.restr (interior u... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 868,
"column": 78
} | {
"line": 870,
"column": 30
} | {
"line": 872,
"column": 0
} | [
{
"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✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\nI : ModelWithCorners 𝕜 E H\ninst✝⁵ : NormedAdd... | [] | by
simp only [mfld_simps]
rw [PartialEquiv.prod_trans] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 395,
"column": 2
} | {
"line": 395,
"column": 49
} | {
"line": 395,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_16\ninst✝¹ : TopologicalS... | [
"case h₁\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_16\ninst✝¹ : TopologicalSpac... | apply contDiffWithinAt_id.congr_of_eventuallyEq | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.ContMDiff.Constructions | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 49
} | {
"line": 410,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_16\ninst✝¹ : TopologicalS... | [
"case h₁\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁴ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nM' : Type u_16\ninst✝¹ : TopologicalSpac... | apply contDiffWithinAt_id.congr_of_eventuallyEq | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 297,
"column": 2
} | {
"line": 315,
"column": 10
} | {
"line": 317,
"column": 0
} | [
{
"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\nM : Type u_4\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nM' : Type u_7\ninst✝ : TopologicalSpa... | [] | refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· apply h.comp_of_eq
· exact (continuousAt_extChartAt_symm x).continuousWithinAt
· exact (mapsTo_preimage _ _).mono_left inter_subset_left
· exact extChartAt_to_inv x
· rw [← continuousWithinAt_inter (extChartAt_source_mem_nhds (I := I) x)]
have : ContinuousWithinAt (... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 297,
"column": 2
} | {
"line": 315,
"column": 10
} | {
"line": 317,
"column": 0
} | [
{
"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\nM : Type u_4\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nM' : Type u_7\ninst✝ : TopologicalSpa... | [] | refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· apply h.comp_of_eq
· exact (continuousAt_extChartAt_symm x).continuousWithinAt
· exact (mapsTo_preimage _ _).mono_left inter_subset_left
· exact extChartAt_to_inv x
· rw [← continuousWithinAt_inter (extChartAt_source_mem_nhds (I := I) x)]
have : ContinuousWithinAt (... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 53
} | {
"line": 134,
"column": 4
} | [
{
"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... | [
"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✝ : ChartedSpace HB B\nI... | have : q ∈ u p ×ˢ (univ : Set F) := ⟨hq, trivial⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 807,
"column": 2
} | {
"line": 808,
"column": 35
} | {
"line": 810,
"column": 0
} | [
{
"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... | [] | simp only [tangentMapWithin, mfld_simps]
rw [mfderivWithin_subset st hs h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 807,
"column": 2
} | {
"line": 808,
"column": 35
} | {
"line": 810,
"column": 0
} | [
{
"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... | [] | simp only [tangentMapWithin, mfld_simps]
rw [mfderivWithin_subset st hs h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 864,
"column": 55
} | {
"line": 870,
"column": 62
} | {
"line": 872,
"column": 0
} | [
{
"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
have : T1Space M := I.t1Space M
simp only [HasMFDerivWithinAt]
refine and_congr ?_ ?_
· exact continuousWithinAt_congr_set' _ h
· apply hasFDerivWithinAt_congr_set' (extChartAt I x x)
exact preimage_extChartAt_eventuallyEq_compl_singleton y h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.MFDeriv.Atlas | {
"line": 171,
"column": 26
} | {
"line": 177,
"column": 9
} | {
"line": 177,
"column": 10
} | [
{
"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... | [] | by
have :
(ContinuousLinearMap.id 𝕜 _ : TangentSpace I' (e x) →L[𝕜] TangentSpace I' (e x)) y = y :=
rfl
conv_rhs => rw [← this, ← he.comp_symm_deriv (e.map_source hx)]
rw [e.left_inv hx]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.PartitionOfUnity | {
"line": 199,
"column": 26
} | {
"line": 199,
"column": 90
} | {
"line": 199,
"column": 90
} | [
{
"pp": "ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nρ : PartitionOfUnity ι X s\nx₀ : X\nhx₀ : x₀ ∈ s\n⊢ ∑ i ∈ ρ.finsupport x₀, (ρ i) x₀ = ∑ᶠ (i : ι), (ρ i) x₀",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"ChainCompletePartialOrder.... | [
"ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nρ : PartitionOfUnity ι X s\nx₀ : X\nhx₀ : x₀ ∈ s\n⊢ ∑ i ∈ ρ.finsupport x₀, (ρ i) x₀ = ∑ i ∈ ρ.finsupport x₀, (ρ i) x₀"
] | finsum_eq_sum_of_support_subset _ (ρ.coe_finsupport x₀).superset | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.PartitionOfUnity | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 13
} | {
"line": 231,
"column": 2
} | [
{
"pp": "ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nρ : PartitionOfUnity ι X s\nx₀ : X\nt : Set X\nt_in : t ∈ 𝓝 x₀\nht : {i | ((fun i ↦ support ⇑(ρ i)) i ∩ t).Nonempty}.Finite\n⊢ {i | x₀ ∈ tsupport ⇑(ρ i)} ⊆ {i | ((fun i ↦ support ⇑(ρ i)) i ∩ t).Nonempty}",
"ppTerm": "?m.45",
"assig... | [
"ι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nρ : PartitionOfUnity ι X s\nx₀ : X\nt : Set X\nt_in : t ∈ 𝓝 x₀\nht : {i | ((fun i ↦ support ⇑(ρ i)) i ∩ t).Nonempty}.Finite\ni : ι\nhi : i ∈ {i | x₀ ∈ tsupport ⇑(ρ i)}\n⊢ i ∈ {i | ((fun i ↦ support ⇑(ρ i)) i ∩ t).Nonempty}"
] | rintro i hi | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 576,
"column": 2
} | {
"line": 576,
"column": 97
} | {
"line": 577,
"column": 2
} | [
{
"pp": "E : 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 : T2Space M\ninst✝ : SigmaCo... | [
"E : 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 : T2Space M\ninst✝ : SigmaCompactSpace M... | apply exists_isSubordinate _ isClosed_univ _ (fun i ↦ (chartAt H _).open_source) (fun x _ ↦ ?_) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 88,
"column": 16
} | {
"line": 88,
"column": 18
} | {
"line": 89,
"column": 4
} | [
{
"pp": "α : Type u_1\nE : Type u_2\n𝕜 : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : StrongDual 𝕜 E), (fun x ↦ c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : d.Countable\nhd... | [
"α : Type u_1\nE : Type u_2\n𝕜 : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nt : Set E\nf : α → E\nhf : ∀ (c : StrongDual 𝕜 E), (fun x ↦ c (f x)) =ᵐ[μ] 0\nh't : ∀ᵐ (x : α) ∂μ, f x ∈ t\nd : Set E\nd_count : d.Countable\nhd : t ⊆ closu... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 131,
"column": 85
} | {
"line": 149,
"column": 85
} | {
"line": 151,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable f μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → 0 ≤ ∫ (x : α) in s, f x ∂μ\n⊢ 0 ≤ᵐ[μ] f",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"NonUnitalNonAssocCommR... | [] | by
simp_rw [EventuallyLE, Pi.zero_apply]
rw [ae_const_le_iff_forall_lt_measure_zero]
intro b hb_neg
let s := {x | f x ≤ b}
have hs : NullMeasurableSet s μ := nullMeasurableSet_le hf.1.aemeasurable aemeasurable_const
have mus : μ s < ∞ := Integrable.measure_le_lt_top hf hb_neg
have h_int_gt : (∫ x in s, f ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 66
} | {
"line": 238,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nC : ℝ≥0\nf : E → ℝ\nμ : Measure E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nhf : LipschitzWith C f\nι : Type u_3\ns : Finset ι\na : ι → ℝ\nv : ι → E\ng : E → ℝ\ng... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nC : ℝ≥0\nf : E → ℝ\nμ : Measure E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nhf : LipschitzWith C f\nι : Type u_3\ns : Finset ι\na : ι → ℝ\nv : ι → E\ng : E → ℝ\ng_smooth : Co... | change Integrable (fun x ↦ a i * ((L ∘ (fderiv ℝ g)) x * f x)) μ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope | {
"line": 37,
"column": 41
} | {
"line": 37,
"column": 56
} | {
"line": 37,
"column": 56
} | [
{
"pp": "f : ℝ → ℝ\na b c : ℝ\nhf : IntervalIntegrable f volume a (b + c)\nhab : a ≤ b\nhc : 0 ≤ c\n⊢ uIcc a b ⊆ uIcc (a - c) (b + c - c)",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope.0.IntervalIntegrable.intervalInteg... | [] | by grind [uIcc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope | {
"line": 37,
"column": 79
} | {
"line": 37,
"column": 94
} | {
"line": 37,
"column": 94
} | [
{
"pp": "f : ℝ → ℝ\na b c : ℝ\nhf : IntervalIntegrable f volume a (b + c)\nhab : a ≤ b\nhc : 0 ≤ c\n⊢ uIcc a b ⊆ uIcc a (b + c)",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope.0.IntervalIntegrable.intervalIntegrable_slop... | [] | by grind [uIcc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope | {
"line": 60,
"column": 43
} | {
"line": 60,
"column": 58
} | {
"line": 60,
"column": 58
} | [
{
"pp": "f : ℝ → ℝ\na b c : ℝ\nhf : MonotoneOn f (uIcc a (b + c))\nhab : a ≤ b\nhc✝ : 0 ≤ c\nhc : 0 < c\nhf' : IntervalIntegrable f volume a (b + c)\n⊢ uIcc a b ⊆ uIcc (a - c) (b + c - c)",
"ppTerm": "?m.134",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Integral.Interv... | [] | by grind [uIcc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope | {
"line": 61,
"column": 23
} | {
"line": 61,
"column": 38
} | {
"line": 61,
"column": 38
} | [
{
"pp": "f : ℝ → ℝ\na b c : ℝ\nhf : MonotoneOn f (uIcc a (b + c))\nhab : a ≤ b\nhc✝ : 0 ≤ c\nhc : 0 < c\nhf' : IntervalIntegrable f volume a (b + c)\n⊢ uIcc a b ⊆ uIcc a (b + c)",
"ppTerm": "?m.145",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Integral.IntervalIntegral... | [] | by grind [uIcc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope | {
"line": 64,
"column": 23
} | {
"line": 64,
"column": 38
} | {
"line": 64,
"column": 38
} | [
{
"pp": "f : ℝ → ℝ\na b c : ℝ\nhf : MonotoneOn f (uIcc a (b + c))\nhab : a ≤ b\nhc✝ : 0 ≤ c\nhc : 0 < c\nhf' : IntervalIntegrable f volume a (b + c)\n⊢ uIcc (a + c) (b + c) ⊆ uIcc a (b + c)",
"ppTerm": "?m.177",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Integral.Inte... | [] | by grind [uIcc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Slope | {
"line": 65,
"column": 23
} | {
"line": 65,
"column": 38
} | {
"line": 65,
"column": 38
} | [
{
"pp": "f : ℝ → ℝ\na b c : ℝ\nhf : MonotoneOn f (uIcc a (b + c))\nhab : a ≤ b\nhc✝ : 0 ≤ c\nhc : 0 < c\nhf' : IntervalIntegrable f volume a (b + c)\n⊢ uIcc a b ⊆ uIcc a (b + c)",
"ppTerm": "?m.188",
"assigned": true,
"usedConstants": [
"_private.Mathlib.MeasureTheory.Integral.IntervalIntegral... | [] | by grind [uIcc] | [anonymous] | Lean.Parser.Term.byTactic |
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