module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Analytic.ConvergenceRadius | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 42
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSerie... | refine le_radius_of_bound _ C fun n ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Analytic.Basic | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nhf : HasFPowerSeriesAt f p x\n⊢ HasFPowerSe... | rw [← hasFPowerSeriesWithinAt_univ] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Basic | {
"line": 627,
"column": 4
} | {
"line": 627,
"column": 96
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\ny : E\nhf : HasFPowerSeriesWithin... | have : ContinuousAt (fun z ↦ p.partialSum k z) y := (p.partialSum_continuous k).continuousAt | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Analytic.Uniqueness | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 52
} | [
{
"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\nf : E → F\nU : Set E\nhf : AnalyticOnNhd 𝕜 f U\nhU : IsPreconnected U\nz₀ : E\nh₀ : z₀ ∈ U\nhfz₀ : f =ᶠ[𝓝 ... | exact UniformSpace.Completion.coe_injective F this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Analytic.Composition | {
"line": 485,
"column": 2
} | {
"line": 485,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nq : FormalMultilinearSeries 𝕜 F G\n... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Analytic.Composition | {
"line": 577,
"column": 2
} | {
"line": 577,
"column": 22
} | [
{
"pp": "m M N : ℕ\ni : (n : ℕ) × Composition n\nhi : i ∈ compPartialSumTargetSet m M N\n⊢ ∃ j, ∃ (hj : j ∈ compPartialSumSource m M N), compChangeOfVariables m M N j hj = i",
"usedConstants": []
}
] | rcases i with ⟨n, c⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Analytic.Basic | {
"line": 772,
"column": 2
} | {
"line": 772,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\nx : E\nhf : HasFPowerSeriesAt f p x\nn : ℕ\n⊢ (fun y ↦ f (x +... | rw [← hasFPowerSeriesWithinAt_univ] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Basic | {
"line": 893,
"column": 2
} | {
"line": 893,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\nx : E\nhf : HasFPowerSeriesAt f p x\n⊢ (fun y ↦ f y.1 - f y.2... | rw [← hasFPowerSeriesWithinAt_univ] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Composition | {
"line": 914,
"column": 4
} | {
"line": 914,
"column": 25
} | [
{
"pp": "case h.H.inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nm n : ℕ\ng : F → G\nf ... | simp [hg.finite _ hc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Analytic.Composition | {
"line": 914,
"column": 4
} | {
"line": 914,
"column": 25
} | [
{
"pp": "case h.H.inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nm n : ℕ\ng : F → G\nf ... | simp [hg.finite _ hc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Composition | {
"line": 914,
"column": 4
} | {
"line": 914,
"column": 25
} | [
{
"pp": "case h.H.inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nm n : ℕ\ng : F → G\nf ... | simp [hg.finite _ hc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Composition | {
"line": 1176,
"column": 4
} | {
"line": 1176,
"column": 88
} | [
{
"pp": "case succ\nn : ℕ\na : Composition n\nb : Composition a.length\ni : ℕ\nhi : i < b.length\nj : ℕ\nIHj :\n j < b.blocksFun ⟨i, hi⟩ →\n a.sizeUpTo (b.sizeUpTo i + j) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j\nhj : j + 1 < b.blocksFun ⟨i, hi⟩\nA : j < b.blocksFun ⟨i, hi⟩\nB... | rw [getElem_of_eq (getElem_splitWrtComposition _ _ _ _), getElem_drop, getElem_take] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.TangentCone.Basic | {
"line": 64,
"column": 2
} | {
"line": 71,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : SMul 𝕜 E\ninst✝¹ : TopologicalSpace E\ns t : Set E\nx : E\ninst✝ : ContinuousAdd E\nh : 𝓝[s] x ≤ 𝓝[t] x\n⊢ tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 t x",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Set.Map... | simp only [tangentConeAt_def, setOf_subset_setOf]
refine fun y hy ↦ hy.mono ?_
gcongr _ • ?_
rw [nhdsWithin_le_iff]
suffices Tendsto (x + ·) (𝓝[(x + ·) ⁻¹' s] 0) (𝓝[s] x) from
this.mono_right h |> tendsto_nhdsWithin_iff.mp |>.2
refine .inf ?_ (mapsTo_preimage _ _).tendsto
exact (continuous_const_add x... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.TangentCone.Basic | {
"line": 64,
"column": 2
} | {
"line": 71,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : SMul 𝕜 E\ninst✝¹ : TopologicalSpace E\ns t : Set E\nx : E\ninst✝ : ContinuousAdd E\nh : 𝓝[s] x ≤ 𝓝[t] x\n⊢ tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 t x",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Set.Map... | simp only [tangentConeAt_def, setOf_subset_setOf]
refine fun y hy ↦ hy.mono ?_
gcongr _ • ?_
rw [nhdsWithin_le_iff]
suffices Tendsto (x + ·) (𝓝[(x + ·) ⁻¹' s] 0) (𝓝[s] x) from
this.mono_right h |> tendsto_nhdsWithin_iff.mp |>.2
refine .inf ?_ (mapsTo_preimage _ _).tendsto
exact (continuous_const_add x... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Constructions | {
"line": 487,
"column": 25
} | {
"line": 487,
"column": 44
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_9\ninst✝² : Fintype ι\nFm : ι → Type u_10\ninst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)\np : (i : ι) → FormalMultilinearSeries ... | simp [radius_pi_le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Analytic.Constructions | {
"line": 487,
"column": 25
} | {
"line": 487,
"column": 44
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_9\ninst✝² : Fintype ι\nFm : ι → Type u_10\ninst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)\np : (i : ι) → FormalMultilinearSeries ... | simp [radius_pi_le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Constructions | {
"line": 487,
"column": 25
} | {
"line": 487,
"column": 44
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_9\ninst✝² : Fintype ι\nFm : ι → Type u_10\ninst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)\np : (i : ι) → FormalMultilinearSeries ... | simp [radius_pi_le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Constructions | {
"line": 957,
"column": 2
} | {
"line": 965,
"column": 15
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\n𝕝 : Type u_8\ninst✝⁴ : NormedDivisionRing 𝕝\ninst✝³ : NormedAlgebra 𝕜 𝕝\ninst✝² : Module 𝕝 F\ninst✝¹ :... | constructor
· exact fun a ↦ h₁f.smul a
· intro hprod
rw [analyticAt_congr (g := (f⁻¹ • f) • g), smul_assoc]
· exact (h₁f.inv h₂f).fun_smul hprod
· filter_upwards [h₁f.continuousAt.preimage_mem_nhds (compl_singleton_mem_nhds_iff.2 h₂f)]
intro y hy
rw [Set.preimage_compl, Set.mem_compl_iff, Se... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Constructions | {
"line": 957,
"column": 2
} | {
"line": 965,
"column": 15
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\n𝕝 : Type u_8\ninst✝⁴ : NormedDivisionRing 𝕝\ninst✝³ : NormedAlgebra 𝕜 𝕝\ninst✝² : Module 𝕝 F\ninst✝¹ :... | constructor
· exact fun a ↦ h₁f.smul a
· intro hprod
rw [analyticAt_congr (g := (f⁻¹ • f) • g), smul_assoc]
· exact (h₁f.inv h₂f).fun_smul hprod
· filter_upwards [h₁f.continuousAt.preimage_mem_nhds (compl_singleton_mem_nhds_iff.2 h₂f)]
intro y hy
rw [Set.preimage_compl, Set.mem_compl_iff, Se... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Constructions | {
"line": 993,
"column": 2
} | {
"line": 993,
"column": 26
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\n𝕝 : Type u_8\ninst✝¹ : NormedDivisionRing 𝕝\ninst✝ : NormedAlgebra 𝕜 𝕝\nf g : E → 𝕝\ns : Set E\nx : E\nfa : AnalyticWithinAt 𝕜 f s x\nga : AnalyticWithinAt 𝕜 g s x\ng0 : g ... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Analytic.Constructions | {
"line": 1000,
"column": 2
} | {
"line": 1000,
"column": 26
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\n𝕝 : Type u_8\ninst✝¹ : NormedDivisionRing 𝕝\ninst✝ : NormedAlgebra 𝕜 𝕝\nf g : E → 𝕝\nx : E\nfa : AnalyticAt 𝕜 f x\nga : AnalyticAt 𝕜 g x\ng0 : g x ≠ 0\n⊢ AnalyticAt 𝕜 (f /... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Calculus.FDeriv.Basic | {
"line": 947,
"column": 70
} | {
"line": 949,
"column": 78
} | [
{
"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\nf : E → F\nx₀ : E\ns : Set E\nhs : s ∈ 𝓝 x₀\nC : ℝ≥0\nhlip : LipschitzOnWith C f s\n⊢ ‖fderiv 𝕜 f x₀‖ ≤ ↑C... | by
refine norm_fderiv_le_of_lip' 𝕜 C.coe_nonneg ?_
filter_upwards [hs] with x hx using hlip.norm_sub_le hx (mem_of_mem_nhds hs) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 806,
"column": 6
} | {
"line": 811,
"column": 67
} | [] | ‖f x‖ₑ ≤ egauge 𝕜 (ball 0 1) (f x) := le_egauge_ball_one ..
_ ≤ egauge 𝕜 (ball 0 r) (g x) := hx
_ ≤ ‖c‖ₑ * ‖g x‖ₑ / ↑r :=
egauge_ball_le_of_one_lt_norm hc <| .inl hr₀.ne'
_ = (‖c‖₊ / r : ℝ≥0) * ‖g x‖ₑ := by
simp [hr₀.ne', ENNReal.mul_div_right_comm, enorm_eq_nnnorm] | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Analytic.CPolynomial | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 25
} | [
{
"pp": "case hnc\n𝕜 : Type u_1\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\nEm : ι → Type u_6\ninst✝² : (i : ι) → NormedAddCommGroup (Em i)\ninst✝¹ : (i : ι) →... | exact Nat.ne_of_lt hm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Analytic.Inverse | {
"line": 249,
"column": 48
} | {
"line": 262,
"column": 85
} | [
{
"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\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nx : E\nh : p 1 = (continuousMultilinearCurryFin1 𝕜 E F)... | by
ext (n v)
match n with
| 0 =>
simp only [comp_coeff_zero', Matrix.zero_empty, id_apply_zero]
congr
ext i
exact i.elim0
| 1 =>
simp only [comp_coeff_one, h, rightInv_coeff_one, ContinuousLinearEquiv.apply_symm_apply,
id_apply_one, ContinuousLinearEquiv.coe_apply, continuousMultilinea... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Inverse | {
"line": 279,
"column": 4
} | {
"line": 279,
"column": 78
} | [
{
"pp": "case e_f.H\n𝕜 : 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\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nx : E\nn✝ n : ℕ\nhn : 2 ≤ n + 2\nv : Fin (n ... | simp [comp_rightInv_aux1 N, this, comp_rightInv_aux2, -Set.toFinset_setOf] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Analytic.Inverse | {
"line": 385,
"column": 8
} | {
"line": 385,
"column": 19
} | [
{
"pp": "case a.a\nn : ℕ\np : ℕ → ℝ\nhp : ∀ (k : ℕ), 0 ≤ p k\nr a : ℝ\nhr : 0 ≤ r\nha : 0 ≤ a\nk : ℕ\na✝¹ : k ∈ Ico 2 (n + 1)\nc : Composition k\na✝ : c ∈ {c | 1 < c.length}.toFinset\n⊢ a ^ k * (r ^ c.length * ∏ j, p (c.blocksFun j)) = (∏ x, r) * (a ^ k * ∏ x, p (c.blocksFun x))",
"usedConstants": [
"... | prod_const, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nf : E → F\nx : E\ns : Set E\nh : HasFPowerSeriesWithinAt f p s x\n⊢ Tendsto ... | apply Tendsto.mono_left _ nhdsWithin_le_nhds | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 315,
"column": 55
} | {
"line": 338,
"column": 5
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\nh : HasFPowerSeriesWithinOnBall f p s... | by
/- In the completion of the space, the derivative series is summable, and its sum is a derivative
of the function. Therefore, by uniqueness of derivatives, its sum is the image of `f'` under
the canonical embedding. As this is an embedding, it means that there was also convergence in
the original space, to `... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 406,
"column": 6
} | {
"line": 418,
"column": 51
} | [
{
"pp": "case mpr.refine_3\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : WithTop ℕ∞\nhn : n ≠ ∞\nh'n : n + 1 ≠ ∞\nu : Set E\nhu : u ∈... | intro h i
simp only [WithTop.add_eq_top, WithTop.one_ne_top, or_false] at h
match i with
| 0 =>
simp only [FormalMultilinearSeries.unshift]
apply AnalyticOnNhd.comp_analyticOn _ ((hf h).mono inter_subset_right)
(Set.mapsTo_univ _ _)
exact LinearIsometryEquiv.analyticO... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 406,
"column": 6
} | {
"line": 418,
"column": 51
} | [
{
"pp": "case mpr.refine_3\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : WithTop ℕ∞\nhn : n ≠ ∞\nh'n : n + 1 ≠ ∞\nu : Set E\nhu : u ∈... | intro h i
simp only [WithTop.add_eq_top, WithTop.one_ne_top, or_false] at h
match i with
| 0 =>
simp only [FormalMultilinearSeries.unshift]
apply AnalyticOnNhd.comp_analyticOn _ ((hf h).mono inter_subset_right)
(Set.mapsTo_univ _ _)
exact LinearIsometryEquiv.analyticO... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 746,
"column": 2
} | {
"line": 748,
"column": 22
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nn : WithTop ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpTo n f p\n⊢ Continuous ... | rw [← hasFTaylorSeriesUpToOn_univ_iff] at h
rw [← continuousOn_univ]
exact h.continuousOn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 746,
"column": 2
} | {
"line": 748,
"column": 22
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nn : WithTop ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpTo n f p\n⊢ Continuous ... | rw [← hasFTaylorSeriesUpToOn_univ_iff] at h
rw [← continuousOn_univ]
exact h.continuousOn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 613,
"column": 4
} | {
"line": 618,
"column": 40
} | [
{
"pp": "case mp\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nhx : x ∈ s\n⊢ ContDiffWithinAt 𝕜 0 f s x → ∃ u ∈ 𝓝[s] x, ContinuousOn f ... | intro h
obtain ⟨u, H, p, hp⟩ := h 0 le_rfl
refine ⟨u, ?_, ?_⟩
· simpa [hx] using H
· simp only [Nat.cast_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp
exact hp.1.mono inter_subset_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 613,
"column": 4
} | {
"line": 618,
"column": 40
} | [
{
"pp": "case mp\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nhx : x ∈ s\n⊢ ContDiffWithinAt 𝕜 0 f s x → ∃ u ∈ 𝓝[s] x, ContinuousOn f ... | intro h
obtain ⟨u, H, p, hp⟩ := h 0 le_rfl
refine ⟨u, ?_, ?_⟩
· simpa [hx] using H
· simp only [Nat.cast_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp
exact hp.1.mono inter_subset_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 502,
"column": 6
} | {
"line": 502,
"column": 37
} | [
{
"pp": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx y : (n : ℕ) → E n\nhx : x ∉ s\nhy : y ∈ s\nA : ∃ n, Disjoint s (cylinder x n)\nB : Nat.find A ≤ firstDiff x y\n⊢ ¬Disjoint s (cylinder x (Nat.find A))",
... | not_disjoint_iff_nonempty_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 628,
"column": 64
} | {
"line": 667,
"column": 70
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nn : WithTop ℕ∞\nh : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\n⊢ HasFTaylorSeriesUpToOn ... | by
constructor
· intro x _
simp only [ftaylorSeriesWithin, ContinuousMultilinearMap.curry0_apply,
iteratedFDerivWithin_zero_apply]
· intro m hm x hx
have : (m + 1 : ℕ) ≤ n := ENat.add_one_natCast_le_withTop_of_lt hm
rcases (h x hx).of_le this _ le_rfl with ⟨u, hu, p, Hp⟩
rw [insert_eq_of_mem... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.IteratedDeriv.ConvergenceOnBall | {
"line": 45,
"column": 2
} | {
"line": 45,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nf : 𝕜 → 𝕜\nx : 𝕜\nr : ENNReal\nhr_pos : 0 < r\nh : AnalyticOnNhd 𝕜 f (Metric.eball x r)\np : FormalMultilinearSeries 𝕜 𝕜 𝕜 := FormalMultilinearSeries.ofScalars 𝕜 fun n ↦ iteratedDeriv n f x / ↑n.factorial\nhr : r ≤ p.radius\ng : 𝕜 → 𝕜 := fun t ↦ p.sum (t - x)... | unfold Filter.EventuallyEq Filter.Eventually | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.Analysis.Calculus.FDeriv.Bilinear | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 62
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nb : E × F → G\nu : Set (E × F)\nh : ... | DifferentiableAt.fderivWithin (h.differentiableAt p) hxs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 76
} | [
{
"pp": "case inr\nk : Type u_1\nE : Type u_2\ninst✝⁷ : Field k\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module k E\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : PartialOrder E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : PosSMulMono k E\nf : k → E\nx y : k\ns : Set k\nhf : StrictMonoOn f s\nhx : x ∈ s... | exact slope_comm f x y ▸ (slope_pos_iff_of_le hxy.le).mpr (hf hy hx hxy) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 76
} | [
{
"pp": "case inr\nk : Type u_1\nE : Type u_2\ninst✝⁷ : Field k\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module k E\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : PartialOrder E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : PosSMulMono k E\nf : k → E\nx y : k\ns : Set k\nhf : StrictMonoOn f s\nhx : x ∈ s... | exact slope_comm f x y ▸ (slope_pos_iff_of_le hxy.le).mpr (hf hy hx hxy) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 76
} | [
{
"pp": "case inr\nk : Type u_1\nE : Type u_2\ninst✝⁷ : Field k\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module k E\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : PartialOrder E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : PosSMulMono k E\nf : k → E\nx y : k\ns : Set k\nhf : StrictMonoOn f s\nhx : x ∈ s... | exact slope_comm f x y ▸ (slope_pos_iff_of_le hxy.le).mpr (hf hy hx hxy) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Slope | {
"line": 167,
"column": 2
} | {
"line": 168,
"column": 70
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : OrderTopology 𝕜\ng : 𝕜 → 𝕜\ng' : 𝕜\nhx : AccPt x (𝓟 s)\nhd : HasDerivWithinAt g g' s x\nhg : MonotoneOn g s\nthis : (𝓝[s \\ {x}] x).NeBot\n⊢ 0 ≤ g'",
"usedCo... | have h'g : MonotoneOn g (insert x s) :=
hg.insert_of_continuousWithinAt hx.clusterPt hd.continuousWithinAt | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 178,
"column": 79
} | {
"line": 179,
"column": 40
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\na b : 𝕜\n⊢ DifferentiableAt 𝕜 f a ↔ DifferentiableAt 𝕜 (fun x ↦ f (x + b)) (a - b)",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"NonUnital... | by
simp [differentiableAt_comp_add_const] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.DSlope | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\na : 𝕜\ns : Set 𝕜\nh : s ∈ 𝓝 a\nhc : ContinuousOn f s\nhd : DifferentiableAt 𝕜 f a\nx : 𝕜\nhx : x ∈ s\n⊢ ContinuousWithinAt (dslope f a) s x",
"usedConstants": ... | rcases eq_or_ne x a with (rfl | hne) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 73
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\n⊢ (swap dslope z₀)^[p.order] f z₀ ≠ 0",
"usedConstants": [
"NormedComm... | ← coeff_zero (has_fpower_series_iterate_dslope_fslope p.order hp) 1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Module.Alternating.Topology | {
"line": 317,
"column": 14
} | {
"line": 320,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\ninst✝¹⁴ : NormedField 𝕜\ninst✝¹³ : AddCommGroup E\ninst✝¹² : Module 𝕜 E\ninst✝¹¹ : TopologicalSpace E\ninst✝¹⁰ : ContinuousSMul 𝕜 E\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : Module 𝕜 F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : IsTopologic... | by
rw [ContinuousAlternatingMap.isEmbedding_toContinuousMultilinearMap.continuous_iff]
exact (map_continuous <| compContinuousMultilinearMapL 𝕜 (fun _ : ι ↦ E) F G g).comp
ContinuousAlternatingMap.continuous_toContinuousMultilinearMap | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Alternating.Basic | {
"line": 586,
"column": 6
} | {
"line": 586,
"column": 47
} | [
{
"pp": "𝕜 : Type u\nn : ℕ\nE : Type wE\nF : Type wF\nG : Type wG\nι : Type v\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : SeminormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : SeminormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : ... | simp only [coe_mk, MultilinearMap.coe_mk] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 303,
"column": 6
} | {
"line": 303,
"column": 15
} | [
{
"pp": "case h.e'_9\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nc : 𝕜 → 𝔸\nc' : 𝔸\nhc : HasDerivWithinAt c c' s x\nd : 𝔸\n⊢ c' * d = c' * d + c x * 0",
"usedConstants": [
"Eq.mpr",
"NormedRing.toR... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 316,
"column": 6
} | {
"line": 316,
"column": 15
} | [
{
"pp": "case h.e'_9\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nc : 𝕜 → 𝔸\nc' : 𝔸\nhc : HasStrictDerivAt c c' x\nd : 𝔸\n⊢ c' * d = c' * d + c x * 0",
"usedConstants": [
"Eq.mpr",
"NormedRing.toRing",
"H... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Deriv.Inv | {
"line": 167,
"column": 75
} | {
"line": 169,
"column": 20
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝕜' : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nc d : 𝕜 → 𝕜'\nc' d' : 𝕜'\nhc : HasDerivAt c c' x\nhd : HasDerivAt d d' x\nhx : d x ≠ 0\n⊢ HasDerivAt (fun y ↦ c y / d y) ((c' * d x - c x * d') / d x ^ 2) x",... | by
rw [← hasDerivWithinAt_univ] at *
exact hc.div hd hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 600,
"column": 2
} | {
"line": 602,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_5\n𝔸' : Type u_7\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸'\nu : Finset ι\ng : ι → E → 𝔸'\ng' : ι → E →L[𝕜] 𝔸'\ninst✝ : DecidableEq ι\nx : E\nhg : ... | simpa [← Finset.prod_attach u] using .congr_fderiv
(hasStrictFDerivAt_finset_prod.comp x <| hasStrictFDerivAt_pi.mpr fun i ↦ hg i i.prop)
(by ext; simp [Finset.prod_erase_attach (g · x), ← u.sum_attach]) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 600,
"column": 2
} | {
"line": 602,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_5\n𝔸' : Type u_7\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸'\nu : Finset ι\ng : ι → E → 𝔸'\ng' : ι → E →L[𝕜] 𝔸'\ninst✝ : DecidableEq ι\nx : E\nhg : ... | simpa [← Finset.prod_attach u] using .congr_fderiv
(hasStrictFDerivAt_finset_prod.comp x <| hasStrictFDerivAt_pi.mpr fun i ↦ hg i i.prop)
(by ext; simp [Finset.prod_erase_attach (g · x), ← u.sum_attach]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 600,
"column": 2
} | {
"line": 602,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_5\n𝔸' : Type u_7\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸'\nu : Finset ι\ng : ι → E → 𝔸'\ng' : ι → E →L[𝕜] 𝔸'\ninst✝ : DecidableEq ι\nx : E\nhg : ... | simpa [← Finset.prod_attach u] using .congr_fderiv
(hasStrictFDerivAt_finset_prod.comp x <| hasStrictFDerivAt_pi.mpr fun i ↦ hg i i.prop)
(by ext; simp [Finset.prod_erase_attach (g · x), ← u.sum_attach]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.RCLike | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 58
} | [
{
"pp": "E : Type u_4\nF : Type u_5\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\ns : Set E\nx : E\nhf : ContDiffWithinAt ℝ 1 f s x\nhs : Convex ℝ s\nt : Set E\nhst : t ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries ℝ E F\nhp : ... | rcases Metric.mem_nhdsWithin_iff.mp hst with ⟨ε, ε0, hε⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 145,
"column": 21
} | {
"line": 145,
"column": 30
} | [
{
"pp": "case ha\nf : ℝ → ℝ\na b : ℝ\nhf : ContinuousOn f (Icc a b)\nB B' : ℝ → ℝ\nha : f a ≤ B a\nhB : ContinuousOn B (Icc a b)\nhB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x\nbound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r\nx : ℝ\nhx : x ∈ Icc a b\nr : ℝ\nhr : r ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 560,
"column": 2
} | {
"line": 561,
"column": 43
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\n𝕜 : Type u_3\nG : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E → G\ns : Set E\nx y : E\nhs : Convex ℝ s\nh... | have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by
simp only [hf' x hx, norm_zero, le_rfl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 816,
"column": 65
} | {
"line": 818,
"column": 28
} | [
{
"pp": "𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nH : Type u_5\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nf : G → H\nf' : G → G →L[𝕜] H\nx : G\nhder : ∀ᶠ (y : G) in 𝓝 x, HasFDerivAt f (f' y) y\nhcont : ContinuousAt f' x\nc : ℝ\nh... | by
rw [← dist_eq_norm]
exact le_of_lt (hε H').2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 703,
"column": 8
} | {
"line": 703,
"column": 22
} | [
{
"pp": "case succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx₀ : E\nn : WithTop ℕ∞\nhf : ContDiffWithinAt 𝕜 n f s x₀\nhs : UniqueDiffO... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 361,
"column": 4
} | {
"line": 365,
"column": 31
} | [
{
"pp": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕂\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedRing 𝔹\ninst✝² : NormedAlgebra 𝕂 𝔸\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx : 𝔸\nhx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius\n⊢ exp (-x) * exp x = 1",
"usedCon... | have hnx : -x ∈ Metric.eball (0 : 𝔸) (expSeries 𝕂 𝔸).radius := by
rw [Metric.mem_eball, ← neg_zero, edist_neg_neg]
exact hx
rw [← exp_add_of_commute_of_mem_ball (Commute.neg_left <| Commute.refl x) hnx hx,
neg_add_cancel, exp_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 361,
"column": 4
} | {
"line": 365,
"column": 31
} | [
{
"pp": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕂\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedRing 𝔹\ninst✝² : NormedAlgebra 𝕂 𝔸\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx : 𝔸\nhx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius\n⊢ exp (-x) * exp x = 1",
"usedCon... | have hnx : -x ∈ Metric.eball (0 : 𝔸) (expSeries 𝕂 𝔸).radius := by
rw [Metric.mem_eball, ← neg_zero, edist_neg_neg]
exact hx
rw [← exp_add_of_commute_of_mem_ball (Commute.neg_left <| Commute.refl x) hnx hx,
neg_add_cancel, exp_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.CauSeqFilter | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 27
} | [
{
"pp": "case right.a\nβ : Type v\ninst✝ : NormedField β\nf : CauSeq β norm\ns : Set (β × β)\nhs : s ∈ uniformity β\nε : ℝ\nhε : ε > 0\nhεs : ∀ ⦃a b : β⦄, dist a b < ε → (a, b) ∈ s\nN : ℕ\nhN : ∀ j ≥ N, ∀ k ≥ N, ‖↑f j - ↑f k‖ < ε\na b : β\na' : ℕ\nha'1 : a' ≥ N\nha'2 : ↑f a' = a\nb' : ℕ\nhb'1 : b' ≥ N\nhb'2 : ↑... | apply hN <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.ExtendFrom | {
"line": 86,
"column": 53
} | {
"line": 86,
"column": 67
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : RegularSpace Y\nf : X → Y\nA : Set X\nhA : Dense A\nhf : ∀ (x : X), ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)\n⊢ ∀ x ∈ univ, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)",
"usedConstants": [
"Eq.mpr",
"Set.mem_univ._... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.ExtendFrom | {
"line": 86,
"column": 53
} | {
"line": 86,
"column": 67
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : RegularSpace Y\nf : X → Y\nA : Set X\nhA : Dense A\nhf : ∀ (x : X), ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)\n⊢ ∀ x ∈ univ, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)",
"usedConstants": [
"Eq.mpr",
"Set.mem_univ._... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ExtendFrom | {
"line": 86,
"column": 53
} | {
"line": 86,
"column": 67
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : RegularSpace Y\nf : X → Y\nA : Set X\nhA : Dense A\nhf : ∀ (x : X), ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)\n⊢ ∀ x ∈ univ, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)",
"usedConstants": [
"Eq.mpr",
"Set.mem_univ._... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 41
} | [
{
"pp": "n : ℕ\nc : OrderedFinpartition n\ninst✝ : NeZero n\n⊢ c.emb (c.index 0) 0 ≤ 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"OrderedFinpartition.invEmbedding",
"PartialOrder.toPreorder",
"Preorder.toLE",
"OrderedFinpartition.emb",
"OrderedFinpartition.index",
... | conv_rhs => rw [← c.emb_invEmbedding 0] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 306,
"column": 14
} | {
"line": 306,
"column": 25
} | [
{
"pp": "case zero\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\nq :... | simp at hij | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 306,
"column": 14
} | {
"line": 306,
"column": 25
} | [
{
"pp": "case zero\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\nq :... | simp at hij | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 306,
"column": 14
} | {
"line": 306,
"column": 25
} | [
{
"pp": "case zero\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\nq :... | simp at hij | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.MeanValue | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 93
} | [
{
"pp": "f : ℝ → ℝ\na : ℝ\nhf : ∀ t ∈ atTop, ∃ i, a < i ∧ MapsTo (derivWithin f (Ioi a)) (Ioo a i) t\nhcont_at_a : ContinuousWithinAt f (Ici a) a\nhdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))\n⊢ ∃ i, a < i ∧ ∀ ⦃x : ℝ⦄, x ∈ Ioo a i → ∀ x_1 ∈ Ioc a x, max (derivWithin f (Ioi a) a + 1) 0 < ... | obtain ⟨b, hab, hb⟩ := hf (Ioi (max (derivWithin f (Ioi a) a + 1) 0)) (Ioi_mem_atTop _) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 541,
"column": 8
} | {
"line": 541,
"column": 47
} | [
{
"pp": "case neg.refine_2\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Se... | conv_rhs => rw [← c.emb_invEmbedding 0] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.Analysis.Calculus.Deriv.MeanValue | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 38
} | [
{
"pp": "D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : DifferentiableOn ℝ f (interior D)\nC : ℝ\nlt_hf' : ∀ x ∈ interior D, deriv f x < C\nx✝ : ℝ\nhx✝ : x✝ ∈ D\ny : ℝ\nhy : y ∈ D\nhxy : x✝ < y\nx : ℝ\nhx : x ∈ interior D\n⊢ -C < deriv (fun y ↦ -f y) x",
"usedConstants": [
"IsRigh... | rw [deriv.fun_neg, neg_lt_neg_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.Deriv.MeanValue | {
"line": 480,
"column": 92
} | {
"line": 482,
"column": 71
} | [
{
"pp": "D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : DifferentiableOn ℝ f (interior D)\nhf'_nonpos : ∀ x ∈ interior D, deriv f x ≤ 0\nx : ℝ\nhx : x ∈ D\ny : ℝ\nhy : y ∈ D\nhxy : x ≤ y\n⊢ f y ≤ f x",
"usedConstants": [
"Real.instLE",
"Real",
"NonUnitalCommRing.toNonU... | by
simpa only [zero_mul, sub_nonpos] using
hD.image_sub_le_mul_sub_of_deriv_le hf hf' hf'_nonpos x hx y hy hxy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 664,
"column": 10
} | {
"line": 664,
"column": 44
} | [
{
"pp": "case emb.inl\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\n... | refine (Fin.heq_fun_iff ?_).mpr ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 668,
"column": 10
} | {
"line": 668,
"column": 44
} | [
{
"pp": "case emb.inr\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\n... | refine (Fin.heq_fun_iff ?_).mpr ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 689,
"column": 8
} | {
"line": 689,
"column": 42
} | [
{
"pp": "case pos.emb\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\n... | refine (Fin.heq_fun_iff ?_).mpr ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 696,
"column": 12
} | {
"line": 696,
"column": 58
} | [
{
"pp": "case pos.emb.refine_2.zero\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set ... | simp only [cases_zero, cast_zero, val_eq_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 707,
"column": 8
} | {
"line": 707,
"column": 29
} | [
{
"pp": "case neg.emb\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\n... | refine hfunext rfl ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 773,
"column": 71
} | {
"line": 778,
"column": 38
} | [
{
"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\nn : ℕ\nc : OrderedFinpartition n\np : (i : Fin c.length) → ContinuousMultilinearMap 𝕜 (fun i ↦ E) F\nm : Fi... | by
ext d
by_cases h : d = m
· rw [h]
simp [applyOrderedFinpartition]
· simp [h, applyOrderedFinpartition] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 632,
"column": 37
} | {
"line": 632,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn : WithTop ℕ∞\nA : Type u_4\ninst✝⁴ : NormedRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : Module A F\ninst✝¹... | exact hf.smul_const v | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 947,
"column": 4
} | {
"line": 948,
"column": 49
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nα : Type u_5\nH : Ty... | have H₂ : ∀ i, (q₂ · (c.partSize i)) =O[l] (1 : α → ℝ) := fun i ↦
(hq₂_bdd _ <| c.partSize_le i).isBigO_one ℝ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 237,
"column": 12
} | {
"line": 237,
"column": 26
} | [
{
"pp": "case e_a.e_f.h\nx : ℝ\nh : |x| < 1\nn : ℕ\nF : ℝ → ℝ := fun x ↦ ∑ i ∈ Finset.range n, x ^ (i + 1) / (↑i + 1) + log (1 - x)\nF' : ℝ → ℝ := fun x ↦ -x ^ n / (1 - x)\ny : ℝ\nhy : y ∈ Set.Ioo (-1) 1\nthis : HasDerivAt F (∑ i ∈ Finset.range n, ↑(i + 1) * y ^ i / (↑i + 1) + -1 / (1 - y)) y\ni : ℕ\n⊢ y ^ i = ... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 18
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ns : Set E\nx : E\nhx : x ∈ s\ny : F\nhy : y ∈ B.polar s\n⊢ ‖(B.flip y) x‖ ≤ 1",
"usedConstants": [
"N... | B.flip_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 51
} | [
{
"pp": "case h.mp\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\nS : Type u_4\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nm : S\ny : F\nhy : y ∈ B.... | rw [← one_div, le_div_iff₀ (norm_pos_iff.2 hr)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 20
} | [
{
"pp": "case refine_2\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\nhdom : f.domain ≠ ⊤\ny : E\nhy : y ∉ f.domain\nc : ℝ\nle_c : ∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c\nc_le :... | simp only at hzs | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 16
} | [
{
"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\nf : E → F\nK : Set (E →L[𝕜] F)\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nc : 𝕜\nhc : 1 < ‖c‖\nx ... | intro e p hp | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 45
} | [
{
"pp": "x : ℝ\nh₀ : 0 ≤ x\nh : x < 1\nn : ℕ\nF : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)\nF' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)\nA : ∀ y ∈ Set.Icc 0 x, HasDerivAt F (F' y) y\ny : ℝ\nhy : y ∈ interior (Set.Icc 0 x)\n⊢ 0 ≤ F' y",
"used... | simp only [interior_Icc, Set.mem_Ioo] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 402,
"column": 2
} | {
"line": 416,
"column": 20
} | [
{
"pp": "a : ℝ\nh : 0 < a\n⊢ HasSum (fun k ↦ 2 * (1 / (2 * ↑k + 1)) * (1 / (2 * a + 1)) ^ (2 * k + 1)) (log (1 + a⁻¹))",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Real.instIsOrderedRing",
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Not.intro",
"Mathlib.Tactic.FieldSimp.NF.d... | have h₁ : |1 / (2 * a + 1)| < 1 := by
rw [abs_of_pos, div_lt_one]
· linarith
· linarith
· exact div_pos one_pos (by linarith)
convert hasSum_log_sub_log_of_abs_lt_one h₁ using 1
have h₂ : (2 : ℝ) * a + 1 ≠ 0 := by linarith
have h₃ := h.ne'
rw [← log_div]
· congr
simp [field]
ring
· f... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 402,
"column": 2
} | {
"line": 416,
"column": 20
} | [
{
"pp": "a : ℝ\nh : 0 < a\n⊢ HasSum (fun k ↦ 2 * (1 / (2 * ↑k + 1)) * (1 / (2 * a + 1)) ^ (2 * k + 1)) (log (1 + a⁻¹))",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Real.instIsOrderedRing",
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Not.intro",
"Mathlib.Tactic.FieldSimp.NF.d... | have h₁ : |1 / (2 * a + 1)| < 1 := by
rw [abs_of_pos, div_lt_one]
· linarith
· linarith
· exact div_pos one_pos (by linarith)
convert hasSum_log_sub_log_of_abs_lt_one h₁ using 1
have h₂ : (2 : ℝ) * a + 1 ≠ 0 := by linarith
have h₃ := h.ne'
rw [← log_div]
· congr
simp [field]
ring
· f... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Convex.Cone.Basic | {
"line": 533,
"column": 16
} | {
"line": 533,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\nM : Type u_4\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup M\ninst✝ : Module 𝕜 M\ns : Set M\nx : M\nhs : Convex 𝕜 s\nhx : x ∈ hull 𝕜 s\ny₁ : M\nr₁ : 𝕜\nhr₁ : 0 < r₁\nhy₁ : y₁ ∈ r₁ • s\ny₂ : M\nr₂ : 𝕜\nhr₂ : 0 < r₂\nhy₂ : y₂ ∈ r₂ ... | exact add_mem_add hy₁ hy₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 616,
"column": 4
} | {
"line": 616,
"column": 16
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nx : ℝ\nhx : x ∈ D f K\nn : ℕ → ℕ\nL : ℕ → ℕ → ℕ → F\nhn :\n ∀ (e p q : ℕ),\n n e ≤ p →\n n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / ... | intro e p hp | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Convex.Gauge | {
"line": 141,
"column": 4
} | {
"line": 148,
"column": 16
} | [
{
"pp": "case h.refine_1\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\na : ℝ\nhs₁ : Convex ℝ s\nhs₀ : 0 ∈ s\nhs₂ : Absorbent ℝ s\nha : 0 ≤ a\nx : E\nh : gauge s x ≤ a\nr : ℝ\nhr : a < r\n⊢ x ∈ r • s",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"... | have hr' := ha.trans_lt hr
rw [mem_smul_set_iff_inv_smul_mem₀ hr'.ne']
obtain ⟨δ, δ_pos, hδr, hδ⟩ := exists_lt_of_gauge_lt hs₂ (h.trans_lt hr)
suffices (r⁻¹ * δ) • δ⁻¹ • x ∈ s by rwa [smul_smul, mul_inv_cancel_right₀ δ_pos.ne'] at this
rw [mem_smul_set_iff_inv_smul_mem₀ δ_pos.ne'] at hδ
refine hs₁.s... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Gauge | {
"line": 141,
"column": 4
} | {
"line": 148,
"column": 16
} | [
{
"pp": "case h.refine_1\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\na : ℝ\nhs₁ : Convex ℝ s\nhs₀ : 0 ∈ s\nhs₂ : Absorbent ℝ s\nha : 0 ≤ a\nx : E\nh : gauge s x ≤ a\nr : ℝ\nhr : a < r\n⊢ x ∈ r • s",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"... | have hr' := ha.trans_lt hr
rw [mem_smul_set_iff_inv_smul_mem₀ hr'.ne']
obtain ⟨δ, δ_pos, hδr, hδ⟩ := exists_lt_of_gauge_lt hs₂ (h.trans_lt hr)
suffices (r⁻¹ * δ) • δ⁻¹ • x ∈ s by rwa [smul_smul, mul_inv_cancel_right₀ δ_pos.ne'] at this
rw [mem_smul_set_iff_inv_smul_mem₀ δ_pos.ne'] at hδ
refine hs₁.s... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Gauge | {
"line": 222,
"column": 2
} | {
"line": 223,
"column": 40
} | [
{
"pp": "case refine_1\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\na : ℝ\nhs₀ : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : ℝ⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s\nr : ℝ\nhr : r > 0\nb : ℝ\nhb : 0 < b\nx : E\nhx' : x ∈ s\nhs₂ : Absorbs ℝ s {(fun x ↦ b • x) x}\nhx : (fun x ↦ b • x) x ∉ a • s\nh : ∀ (c : ℝ), r ≤ ... | · rw [← div_eq_inv_mul]
exact div_le_one_of_le₀ hba.le ha.le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Gauge | {
"line": 225,
"column": 20
} | {
"line": 225,
"column": 47
} | [
{
"pp": "case refine_2\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\na : ℝ\nhs₀ : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : ℝ⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s\nr : ℝ\nhr : r > 0\nb : ℝ\nhb : 0 < b\nx : E\nhx' : x ∈ s\nhs₂ : Absorbs ℝ s {(fun x ↦ b • x) x}\nhx : (fun x ↦ b • x) x ∉ a • s\nh : ∀ (c : ℝ), r ≤ ... | mul_inv_cancel_left₀ ha.ne' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Gauge | {
"line": 250,
"column": 52
} | {
"line": 250,
"column": 63
} | [
{
"pp": "case inr.e_a.h.mp\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module ℝ E\nα : Type u_3\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : MulActionWithZero α ℝ\ninst✝² : IsStrictOrderedModule α ℝ\ninst✝¹ : MulActionWithZero α E\ninst✝ : IsScalarTower α ℝ (Set E)\ns... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Gauge | {
"line": 274,
"column": 20
} | {
"line": 274,
"column": 31
} | [
{
"pp": "case inr.h.e_a.h.mp\nE : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module ℝ E\nα : Type u_3\ninst✝⁸ : Field α\ninst✝⁷ : LinearOrder α\ninst✝⁶ : IsStrictOrderedRing α\ninst✝⁵ : MulActionWithZero α ℝ\ninst✝⁴ : IsStrictOrderedModule α ℝ\ninst✝³ : MulActionWithZero α E\ninst✝² : SMulCommClass α ℝ ℝ\nins... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Gauge | {
"line": 278,
"column": 19
} | {
"line": 278,
"column": 30
} | [
{
"pp": "case inr.h.e_a.h.mpr\nE : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module ℝ E\nα : Type u_3\ninst✝⁸ : Field α\ninst✝⁷ : LinearOrder α\ninst✝⁶ : IsStrictOrderedRing α\ninst✝⁵ : MulActionWithZero α ℝ\ninst✝⁴ : IsStrictOrderedModule α ℝ\ninst✝³ : MulActionWithZero α E\ninst✝² : SMulCommClass α ℝ ℝ\nin... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Gauge | {
"line": 546,
"column": 2
} | {
"line": 547,
"column": 48
} | [
{
"pp": "case inl\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nhx : ‖x‖ = 0\n⊢ sInf {r | r ∈ Ioi 0 ∧ (r = 0 ∨ ‖x‖ = 0)} = 0 x",
"usedConstants": [
"Set.ext",
"Real.instIsOrderedRing",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
... | · convert csInf_Ioi (a := (0 : ℝ))
exact Set.ext fun r ↦ and_iff_left (.inr hx) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Gauge | {
"line": 600,
"column": 2
} | {
"line": 601,
"column": 27
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nr : ℝ\nx : E\nhs : Absorbent ℝ s\nhr : 0 ≤ r\nhsr : s ⊆ closedBall 0 r\n⊢ ‖x‖ / r ≤ gauge s x",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"gauge",
... | rw [← gauge_closedBall hr]
exact gauge_mono hs hsr _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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