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.Basic | {
"line": 279,
"column": 71
} | {
"line": 280,
"column": 83
} | {
"line": 282,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : TopologicalSpace E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜 F\ninst✝¹ : TopologicalSpace F\nf : E → F\nf' : E →L[𝕜] F\nx : E\ns : Set E\ninst✝ : T1Space E\ny : E\... | [] | by
rw [← hasFDerivWithinAt_insert, insert_sdiff_singleton, hasFDerivWithinAt_insert] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Basic | {
"line": 523,
"column": 43
} | {
"line": 523,
"column": 63
} | {
"line": 523,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\nF : Type u_3\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 F\ninst✝ : TopologicalSpace F\nf : E → F\nx : E\ns : Set E\nh : s ∈ 𝓝 x\n⊢ fderivWithin 𝕜 f (univ ∩ s... | [
"𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\nF : Type u_3\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 F\ninst✝ : TopologicalSpace F\nf : E → F\nx : E\ns : Set E\nh : s ∈ 𝓝 x\n⊢ fderivWithin 𝕜 f univ x = fderivWithin... | fderivWithin_inter h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 701,
"column": 71
} | {
"line": 701,
"column": 73
} | {
"line": 702,
"column": 4
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\nl : Filter α\nf : α → E\ninst✝ : ContinuousSMul 𝕜 E\nhf : Tendsto f l (𝓝 0)\nU : Set E\nhU : U ∈ 𝓝 0\nε : ℝ≥0\nhε : ε ≠ 0\nc : 𝕜\nhcε : ‖c‖ < ↑... | [
"α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : TopologicalSpace E\ninst✝¹ : Module 𝕜 E\nl : Filter α\nf : α → E\ninst✝ : ContinuousSMul 𝕜 E\nhf : Tendsto f l (𝓝 0)\nU : Set E\nhU : U ∈ 𝓝 0\nε : ℝ≥0\nhε : ε ≠ 0\nc : 𝕜\nhcε : ‖c‖ < ↑ε\nhc₀ : c ≠... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 739,
"column": 47
} | {
"line": 739,
"column": 49
} | {
"line": 740,
"column": 2
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalSpace E\ninst✝² : Module 𝕜 E\nl : Filter α\nf : α → E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul 𝕜 E\nx : E\nh : Tendsto (fun x_1 ↦ f x_1 - x) l (𝓝 0)\nU : Set E\nhU₀ :... | [
"α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalSpace E\ninst✝² : Module 𝕜 E\nl : Filter α\nf : α → E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul 𝕜 E\nx : E\nh : Tendsto (fun x_1 ↦ f x_1 - x) l (𝓝 0)\nU : Set E\nhU₀ : U ∈ 𝓝 0\nh... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 810,
"column": 39
} | {
"line": 811,
"column": 67
} | {
"line": 812,
"column": 2
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nf : α → E\ng : α → F\nl : Filter α\nc : 𝕜\nhc : 1 < ‖c‖₊\nh : ∀ (i : ℝ), 0 < i → ∃ j,... | [] | by
simp [hr₀.ne', ENNReal.mul_div_right_comm, enorm_eq_nnnorm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 793,
"column": 57
} | {
"line": 828,
"column": 39
} | {
"line": 830,
"column": 0
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nf : α → E\ng : α → F\nl : Filter α\n⊢ f =O[𝕜; l] g ↔ f =O[l] g",
"ppTerm": "?m.20... | [] | by
rcases NormedField.exists_one_lt_norm 𝕜 with ⟨c, hc : 1 < ‖c‖₊⟩
constructor
· rw [nhds_basis_ball.isBigOTVS_iff nhds_basis_ball, isBigO_iff]
intro h
rcases h 1 one_pos with ⟨r, hr₀, hr⟩
lift r to ℝ≥0 using hr₀.le
norm_cast at hr₀
refine ⟨(‖c‖₊ / r : ℝ≥0), hr.mono fun x hx ↦ ?_⟩
suffice... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.ChangeOrigin | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 51
} | {
"line": 244,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhp : p (n + 1) = 0\ns : { s // s.card = n }\nhs : s ∈ Finset.univ... | [
"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\np : FormalMultilinearSeries 𝕜 E F\nn : ℕ\nhp : p (n + 1) = 0\ns : { s // s.card = n }\nhs : s ∈ Finset.univ\nthis : p (... | have : p (1 + n) = 0 := p.congr_zero (by abel) hp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Analytic.ConvergenceRadius | {
"line": 342,
"column": 2
} | {
"line": 345,
"column": 39
} | {
"line": 347,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\nF : Type u_4\nG : Type u_5\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 : FormalMultilinearSeries 𝕜 F G\n... | [] | rw [← ofReal_norm, ← ofReal_norm, ← ofReal_norm,
← ENNReal.ofReal_pow (by simp), ← ENNReal.ofReal_mul (by simp)]
gcongr
apply norm_compContinuousLinearMap_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.ConvergenceRadius | {
"line": 342,
"column": 2
} | {
"line": 345,
"column": 39
} | {
"line": 347,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\nF : Type u_4\nG : Type u_5\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 : FormalMultilinearSeries 𝕜 F G\n... | [] | rw [← ofReal_norm, ← ofReal_norm, ← ofReal_norm,
← ENNReal.ofReal_pow (by simp), ← ENNReal.ofReal_mul (by simp)]
gcongr
apply norm_compContinuousLinearMap_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.ChangeOrigin | {
"line": 277,
"column": 6
} | {
"line": 288,
"column": 12
} | {
"line": 289,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nx_mem_b... | [] | dsimp +instances only [f]
refine ContinuousMultilinearMap.hasSum_eval ?_ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k h.pos).hasSum x_mem_ball
rw [zero_add] at this
refine HasSum.sigma_of_hasSum this (fun l => ?_) ?_
· simp only [changeOriginSeries, sum_apply]
apply hasSum_fin... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.ChangeOrigin | {
"line": 277,
"column": 6
} | {
"line": 288,
"column": 12
} | {
"line": 289,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nh : ↑‖x‖₊ + ↑‖y‖₊ < p.radius\nx_mem_b... | [] | dsimp +instances only [f]
refine ContinuousMultilinearMap.hasSum_eval ?_ _
have := (p.hasFPowerSeriesOnBall_changeOrigin k h.pos).hasSum x_mem_ball
rw [zero_add] at this
refine HasSum.sigma_of_hasSum this (fun l => ?_) ?_
· simp only [changeOriginSeries, sum_apply]
apply hasSum_fin... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Multilinear.Curry | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 54
} | {
"line": 457,
"column": 2
} | [
{
"pp": "𝕜 : Type u\nG : Type wG\nG' : Type wG'\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : G [×0]→L[𝕜] G'\n⊢ ‖f 0‖ = ‖f‖",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
... | [
"𝕜 : Type u\nG : Type wG\nG' : Type wG'\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf : G [×0]→L[𝕜] G'\n⊢ ‖f‖ ≤ ‖f 0‖"
] | refine le_antisymm (by simpa using f.le_opNorm 0) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Analytic.ChangeOrigin | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 29
} | {
"line": 353,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx y : E\nr : ℝ≥0∞\nhf : ... | [] | exact this.analyticWithinAt | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Module.Multilinear.Curry | {
"line": 706,
"column": 4
} | {
"line": 706,
"column": 31
} | {
"line": 707,
"column": 4
} | [
{
"pp": "case h₁\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nEi : Fin n.succ → Type wEi\nG✝ : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝... | [
"case h₁\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nEi : Fin n.succ → Type wEi\nG✝ : Type wG\nG' : Type wG'\ninst✝¹⁴ : Fintype ι\ninst✝¹³ : Fintype ι'\ninst✝¹² : NontriviallyNormedField 𝕜\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝⁹ : (i : Fin... | apply (B.le_opNorm _).trans | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Analytic.Basic | {
"line": 632,
"column": 33
} | {
"line": 632,
"column": 35
} | {
"line": 632,
"column": 36
} | [
{
"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... | [
"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 : HasFPowerSeriesWithinOnBall f p s... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Analytic.Composition | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 17
} | {
"line": 114,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁰ : CommRing 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module 𝕜 F\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : ContinuousConstSMul 𝕜 E\ninst✝¹ : IsTopol... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹⁰ : CommRing 𝕜\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : AddCommGroup F\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : Module 𝕜 F\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : ContinuousConstSMul 𝕜 E\ninst✝¹ : IsTopologicalAddGro... | intro j hjn hj1 | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Analytic.Basic | {
"line": 765,
"column": 2
} | {
"line": 766,
"column": 43
} | {
"line": 768,
"column": 0
} | [
{
"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\nn : ℕ\nr : ℝ≥0∞\nhf : HasFPowerSeriesWithin... | [] | simpa [mul_pow, mul_div_assoc, mul_assoc, div_mul_eq_mul_div, div_pow]
using hp y hy.2 n (by simpa using hy.1) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Analytic.CPolynomial | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 49
} | {
"line": 85,
"column": 0
} | [
{
"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 g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr : ℝ≥0∞\nn m : ℕ\nhf : HasFiniteFPowerSeriesOnB... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Analytic.CPolynomial | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 49
} | {
"line": 85,
"column": 0
} | [
{
"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 g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr : ℝ≥0∞\nn m : ℕ\nhf : HasFiniteFPowerSeriesOnB... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.CPolynomial | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 49
} | {
"line": 85,
"column": 0
} | [
{
"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 g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr : ℝ≥0∞\nn m : ℕ\nhf : HasFiniteFPowerSeriesOnB... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.CPolynomial | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 49
} | {
"line": 90,
"column": 0
} | [
{
"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 g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nn m : ℕ\nhf : HasFiniteFPowerSeriesAt f pf x n\n... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Analytic.CPolynomial | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 49
} | {
"line": 90,
"column": 0
} | [
{
"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 g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nn m : ℕ\nhf : HasFiniteFPowerSeriesAt f pf x n\n... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.CPolynomial | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 49
} | {
"line": 90,
"column": 0
} | [
{
"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 g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nn m : ℕ\nhf : HasFiniteFPowerSeriesAt f pf x n\n... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.CPolynomial | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 49
} | {
"line": 94,
"column": 0
} | [
{
"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 g : E → F\nx : E\nhf : CPolynomialAt 𝕜 f x\nhg : CPolynomialAt 𝕜 g x\n⊢ CPolynomialAt 𝕜 (f - g) x",
... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Analytic.CPolynomial | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 49
} | {
"line": 94,
"column": 0
} | [
{
"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 g : E → F\nx : E\nhf : CPolynomialAt 𝕜 f x\nhg : CPolynomialAt 𝕜 g x\n⊢ CPolynomialAt 𝕜 (f - g) x",
... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.CPolynomial | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 49
} | {
"line": 94,
"column": 0
} | [
{
"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 g : E → F\nx : E\nhf : CPolynomialAt 𝕜 f x\nhg : CPolynomialAt 𝕜 g x\n⊢ CPolynomialAt 𝕜 (f - g) x",
... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.CPolynomial | {
"line": 121,
"column": 51
} | {
"line": 121,
"column": 66
} | {
"line": 121,
"column": 66
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nι : Type u_5\nEm : ι → Type u_6\ninst✝² : (i : ι) → NormedAddCommGroup (Em i)\ninst✝¹ : (i : ι) → NormedSpace 𝕜 (Em i)\ninst✝ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 Em F\ny... | [
"𝕜 : Type u_1\nF : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nι : Type u_5\nEm : ι → Type u_6\ninst✝² : (i : ι) → NormedAddCommGroup (Em i)\ninst✝¹ : (i : ι) → NormedSpace 𝕜 (Em i)\ninst✝ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 Em F\ny : (i : ι) →... | dif_neg ne.symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.CPolynomial | {
"line": 170,
"column": 64
} | {
"line": 170,
"column": 79
} | {
"line": 170,
"column": 79
} | [
{
"pp": "𝕜 : 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) → NormedSpa... | [
"𝕜 : 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) → NormedSpace 𝕜 (Em i)... | dif_neg ne.symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Inverse | {
"line": 105,
"column": 2
} | {
"line": 151,
"column": 81
} | {
"line": 153,
"column": 0
} | [
{
"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)... | [] | match n with
| 0 =>
simp only [comp_coeff_zero', leftInv_coeff_zero, ContinuousMultilinearMap.uncurry0_apply,
id_apply_zero]
| 1 =>
simp only [leftInv_coeff_one, comp_coeff_one, h, id_apply_one, ContinuousLinearEquiv.coe_apply,
ContinuousLinearEquiv.symm_apply_apply, continuousMultilinearCurryFi... | Lean.Elab.Tactic.evalMatch | Lean.Parser.Tactic.match |
Mathlib.Analysis.Analytic.Basic | {
"line": 931,
"column": 2
} | {
"line": 931,
"column": 17
} | {
"line": 932,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nhf : HasFPowerSeriesWithinOnBall ... | [
"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∞\nhf : HasFPowerSeriesWithinOnBall f p s x r\nu... | intro u hu y hy | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Analytic.Composition | {
"line": 652,
"column": 20
} | {
"line": 652,
"column": 22
} | {
"line": 653,
"column": 4
} | [
{
"pp": "case h\nm n : ℕ × ℕ\nhmn : m ≤ n\na : (n : ℕ) × Composition n\n⊢ a ∈ (fun p ↦ compPartialSumTarget 0 p.1 p.2) m → a ∈ (fun p ↦ compPartialSumTarget 0 p.1 p.2) n",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"FormalMultilinearSeries.comp... | [
"case h\nm n : ℕ × ℕ\nhmn : m ≤ n\na : (n : ℕ) × Composition n\nha : a ∈ (fun p ↦ compPartialSumTarget 0 p.1 p.2) m\n⊢ a ∈ (fun p ↦ compPartialSumTarget 0 p.1 p.2) n"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Analytic.Basic | {
"line": 992,
"column": 2
} | {
"line": 992,
"column": 43
} | {
"line": 994,
"column": 0
} | [
{
"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\nr : ℝ≥0∞\nhf : HasFPowerSeriesWithinOnBall f p univ x ... | [] | simpa using hf.tendstoLocallyUniformlyOn' | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Analytic.Inverse | {
"line": 505,
"column": 50
} | {
"line": 505,
"column": 52
} | {
"line": 505,
"column": 53
} | [
{
"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\nhp : 0 < p.radius\nC r : ℝ\nCpos : 0 < C\nrpos : ... | [
"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\nhp : 0 < p.radius\nC r : ℝ\nCpos : 0 < C\nrpos : 0 < r\nple :... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Analytic.Inverse | {
"line": 544,
"column": 2
} | {
"line": 549,
"column": 45
} | {
"line": 551,
"column": 0
} | [
{
"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\nhp : 0 < p.radius\nC r : ℝ\nCpos : 0 < C\nrpos : ... | [] | calc
‖p.rightInv i x n‖ * (a' : ℝ) ^ n = a ^ n * ‖p.rightInv i x n‖ := mul_comm _ _
_ ≤ ∑ k ∈ Ico 1 (n + 1), a ^ k * ‖p.rightInv i x k‖ :=
(haveI : ∀ k ∈ Ico 1 (n + 1), 0 ≤ a ^ k * ‖p.rightInv i x k‖ := fun k _ => by positivity
single_le_sum this (by simp [hn]))
_ ≤ (I + 1) * a := IRec (n + 1) (... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Analytic.Within | {
"line": 121,
"column": 4
} | {
"line": 130,
"column": 38
} | {
"line": 131,
"column": 4
} | [
{
"pp": "case mp.refine_1\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\... | [
"case mp.refine_2\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nh... | · intro y ⟨ys,yb⟩
simp only [mem_eball, edist_eq_enorm_sub] at yb
have e0 := p.hasSum (x := y - x) ?_
· have e1 := (h.hasSum (y := y - x) ?_ ?_)
· simp only [add_sub_cancel] at e1
exact e1.unique e0
· simpa only [add_sub_cancel]
· simpa only [mem_eball, edist_zero_rig... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Analytic.Within | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 16
} | {
"line": 165,
"column": 4
} | [
{
"pp": "case mpr.refine_1\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E... | [] | exact xy.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Analytic.Composition | {
"line": 920,
"column": 6
} | {
"line": 923,
"column": 13
} | {
"line": 924,
"column": 6
} | [
{
"pp": "case 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 : E ... | [
"case 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 : E → F\nq : For... | have : ∑ j : Fin c.length, c.blocksFun j = 0 := by
apply Finset.sum_eq_zero (fun j hj ↦ ?_)
have := j.2
grind | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Analytic.Constructions | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 49
} | {
"line": 169,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nhf : HasFPowerSeriesWithinO... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Analytic.Constructions | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 49
} | {
"line": 169,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nhf : HasFPowerSeriesWithinO... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Constructions | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 49
} | {
"line": 169,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nhf : HasFPowerSeriesWithinO... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Constructions | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 49
} | {
"line": 173,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f pf x r\nh... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Analytic.Constructions | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 49
} | {
"line": 173,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f pf x r\nh... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Constructions | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 49
} | {
"line": 173,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f pf x r\nh... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Constructions | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 49
} | {
"line": 178,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nhf : HasFPowerSeriesWithinAt f pf s x... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Analytic.Constructions | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 49
} | {
"line": 178,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nhf : HasFPowerSeriesWithinAt f pf s x... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Constructions | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 49
} | {
"line": 178,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nhf : HasFPowerSeriesWithinAt f pf s x... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Constructions | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 49
} | {
"line": 182,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nhf : HasFPowerSeriesAt f pf x\nhg : HasFPowerSer... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Analytic.Constructions | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 49
} | {
"line": 182,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nhf : HasFPowerSeriesAt f pf x\nhg : HasFPowerSer... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Constructions | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 49
} | {
"line": 182,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\npf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nhf : HasFPowerSeriesAt f pf x\nhg : HasFPowerSer... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Constructions | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 49
} | {
"line": 186,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\ns : Set E\nx : E\nhf : AnalyticWithinAt 𝕜 f s x\nhg : AnalyticWithinAt 𝕜 g s x\n⊢ AnalyticWit... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Analytic.Constructions | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 49
} | {
"line": 186,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\ns : Set E\nx : E\nhf : AnalyticWithinAt 𝕜 f s x\nhg : AnalyticWithinAt 𝕜 g s x\n⊢ AnalyticWit... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Constructions | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 49
} | {
"line": 186,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\ns : Set E\nx : E\nhf : AnalyticWithinAt 𝕜 f s x\nhg : AnalyticWithinAt 𝕜 g s x\n⊢ AnalyticWit... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Constructions | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 49
} | {
"line": 191,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\nx : E\nhf : AnalyticAt 𝕜 f x\nhg : AnalyticAt 𝕜 g x\n⊢ AnalyticAt 𝕜 (f - g) x",
"ppTerm"... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Analytic.Constructions | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 49
} | {
"line": 191,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\nx : E\nhf : AnalyticAt 𝕜 f x\nhg : AnalyticAt 𝕜 g x\n⊢ AnalyticAt 𝕜 (f - g) x",
"ppTerm"... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Constructions | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 49
} | {
"line": 191,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : E → F\nx : E\nhf : AnalyticAt 𝕜 f x\nhg : AnalyticAt 𝕜 g x\n⊢ AnalyticAt 𝕜 (f - g) x",
"ppTerm"... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Linear | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 58
} | {
"line": 84,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module 𝕜 E\ninst✝⁸ : TopologicalSpace E\nF : Type u_3\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜 F\ninst✝⁵ : TopologicalSpace F\nf : E →L[𝕜] F\nx : E\ns : Set E\ninst✝⁴ : ContinuousAdd E\ninst✝³ : ... | [
"𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module 𝕜 E\ninst✝⁸ : TopologicalSpace E\nF : Type u_3\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module 𝕜 F\ninst✝⁵ : TopologicalSpace F\nf : E →L[𝕜] F\nx : E\ns : Set E\ninst✝⁴ : ContinuousAdd E\ninst✝³ : ContinuousSM... | DifferentiableAt.fderivWithin f.differentiableAt hxs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Composition | {
"line": 1149,
"column": 12
} | {
"line": 1149,
"column": 22
} | {
"line": 1149,
"column": 23
} | [
{
"pp": "n : ℕ\na : Composition n\nb : Composition a.length\ni : ℕ\nIH :\n ∀ (hi : i < b.length),\n 0 < b.blocksFun ⟨i, hi⟩ →\n (take (take i b.blocks).sum a.blocks).sum = (take i (List.map sum (a.blocks.splitWrtComposition b))).sum\nhi : i + 1 < b.length\nhj : 0 < b.blocksFun ⟨i + 1, hi⟩\nA : i < b.le... | [
"n : ℕ\na : Composition n\nb : Composition a.length\ni : ℕ\nIH :\n ∀ (hi : i < b.length),\n 0 < b.blocksFun ⟨i, hi⟩ →\n (take (take i b.blocks).sum a.blocks).sum = (take i (List.map sum (a.blocks.splitWrtComposition b))).sum\nhi : i + 1 < b.length\nhj : 0 < b.blocksFun ⟨i + 1, hi⟩\nA : i < b.length\nB : i ... | take_take, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Comp | {
"line": 237,
"column": 61
} | {
"line": 239,
"column": 74
} | {
"line": 241,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\nf : E → E\nf' : E →L[𝕜] E\nhf : HasFDerivAt f f' x\nhx : f x = x\nn : ℕ\n⊢ HasFDerivAt f^[n] (f' ^ n) x",
"ppTerm": "?m.68",
"assigned": true,
"usedConstants": ... | [] | by
refine HasFDerivAtFilter.iterate hf ?_ n
simpa [hx] using hf.continuousAt.tendsto.prodMap (tendsto_pure_pure f x) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Composition | {
"line": 1142,
"column": 6
} | {
"line": 1155,
"column": 27
} | {
"line": 1156,
"column": 2
} | [
{
"pp": "case zero.succ\nn : ℕ\na : Composition n\nb : Composition a.length\ni : ℕ\nIH :\n ∀ (hi : i < b.length),\n 0 < b.blocksFun ⟨i, hi⟩ →\n (take (take i b.blocks).sum a.blocks).sum = (take i (List.map sum (a.blocks.splitWrtComposition b))).sum\nhi : i + 1 < b.length\nhj : 0 < b.blocksFun ⟨i + 1, h... | [] | have A : i < b.length := Nat.lt_of_succ_lt hi
have B : i < List.length (map List.sum (splitWrtComposition a.blocks b)) := by simp [A]
have C : 0 < blocksFun b ⟨i, A⟩ := Composition.blocks_pos' _ _ _
rw [sum_take_succ _ _ B, ← IH A C]
have :
take (sum (take i b.blocks)) a.blocks =
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Composition | {
"line": 1142,
"column": 6
} | {
"line": 1155,
"column": 27
} | {
"line": 1156,
"column": 2
} | [
{
"pp": "case zero.succ\nn : ℕ\na : Composition n\nb : Composition a.length\ni : ℕ\nIH :\n ∀ (hi : i < b.length),\n 0 < b.blocksFun ⟨i, hi⟩ →\n (take (take i b.blocks).sum a.blocks).sum = (take i (List.map sum (a.blocks.splitWrtComposition b))).sum\nhi : i + 1 < b.length\nhj : 0 < b.blocksFun ⟨i + 1, h... | [] | have A : i < b.length := Nat.lt_of_succ_lt hi
have B : i < List.length (map List.sum (splitWrtComposition a.blocks b)) := by simp [A]
have C : 0 < blocksFun b ⟨i, A⟩ := Composition.blocks_pos' _ _ _
rw [sum_take_succ _ _ B, ← IH A C]
have :
take (sum (take i b.blocks)) a.blocks =
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 614,
"column": 2
} | {
"line": 614,
"column": 49
} | {
"line": 616,
"column": 0
} | [
{
"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 g : E → F\nf' g' : E →L[𝕜] F\nL : Filter (E × E)\nhf : HasFDerivAtFilter f f' L\nhg : HasFDerivAtFilter g... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 614,
"column": 2
} | {
"line": 614,
"column": 49
} | {
"line": 616,
"column": 0
} | [
{
"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 g : E → F\nf' g' : E →L[𝕜] F\nL : Filter (E × E)\nhf : HasFDerivAtFilter f f' L\nhg : HasFDerivAtFilter g... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 614,
"column": 2
} | {
"line": 614,
"column": 49
} | {
"line": 616,
"column": 0
} | [
{
"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 g : E → F\nf' g' : E →L[𝕜] F\nL : Filter (E × E)\nhf : HasFDerivAtFilter f f' L\nhg : HasFDerivAtFilter g... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 856,
"column": 94
} | {
"line": 860,
"column": 89
} | {
"line": 862,
"column": 0
} | [
{
"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\nf' : E →L[𝕜] F\nx : E\ns : Set E\na : E\n⊢ HasFDerivWithinAt (fun x ↦ f (a + x)) f' s x ↔ HasFDe... | [] | by
have : map (a + ·) (𝓝[s] x) = 𝓝[a +ᵥ s] (a + x) := by
simp only [nhdsWithin, Filter.map_inf (add_right_injective a)]
simp [← Set.image_vadd]
simp [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleOTVS, ← this, Function.comp_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.Alternating.Basic | {
"line": 356,
"column": 19
} | {
"line": 356,
"column": 38
} | {
"line": 357,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\nι : Type u_6\ninst✝¹² : Semiring R\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : Module R M\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : AddCommMonoid M'\ninst✝⁷ : Module R M'\ninst✝⁶ : TopologicalSpace M'\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Mod... | [] | ext; simp [(· ∘ ·)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Alternating.Basic | {
"line": 356,
"column": 19
} | {
"line": 356,
"column": 38
} | {
"line": 357,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\nι : Type u_6\ninst✝¹² : Semiring R\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : Module R M\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : AddCommMonoid M'\ninst✝⁷ : Module R M'\ninst✝⁶ : TopologicalSpace M'\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Mod... | [] | ext; simp [(· ∘ ·)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Alternating.Basic | {
"line": 357,
"column": 20
} | {
"line": 357,
"column": 39
} | {
"line": 359,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\nι : Type u_6\ninst✝¹² : Semiring R\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : Module R M\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : AddCommMonoid M'\ninst✝⁷ : Module R M'\ninst✝⁶ : TopologicalSpace M'\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Mod... | [] | ext; simp [(· ∘ ·)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Alternating.Basic | {
"line": 357,
"column": 20
} | {
"line": 357,
"column": 39
} | {
"line": 359,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nM' : Type u_3\nN : Type u_4\nN' : Type u_5\nι : Type u_6\ninst✝¹² : Semiring R\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : Module R M\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : AddCommMonoid M'\ninst✝⁷ : Module R M'\ninst✝⁶ : TopologicalSpace M'\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Mod... | [] | ext; simp [(· ∘ ·)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Alternating.Topology | {
"line": 322,
"column": 4
} | {
"line": 328,
"column": 61
} | {
"line": 330,
"column": 0
} | [
{
"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... | [] | rw [ContinuousLinearMap.isEmbedding_postcomp
(ContinuousAlternatingMap.toContinuousMultilinearMapCLM 𝕜)
ContinuousAlternatingMap.isEmbedding_toContinuousMultilinearMap |>.continuous_iff]
exact map_continuous <|
(precomp (ContinuousMultilinearMap 𝕜 (fun _ : ι ↦ E) G)
((ContinuousAlternati... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.Alternating.Topology | {
"line": 322,
"column": 4
} | {
"line": 328,
"column": 61
} | {
"line": 330,
"column": 0
} | [
{
"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... | [] | rw [ContinuousLinearMap.isEmbedding_postcomp
(ContinuousAlternatingMap.toContinuousMultilinearMapCLM 𝕜)
ContinuousAlternatingMap.isEmbedding_toContinuousMultilinearMap |>.continuous_iff]
exact map_continuous <|
(precomp (ContinuousMultilinearMap 𝕜 (fun _ : ι ↦ E) G)
((ContinuousAlternati... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Completion | {
"line": 66,
"column": 74
} | {
"line": 71,
"column": 58
} | {
"line": 73,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx y : Completion α\n⊢ dist x y = dist y x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"UniformSpace.Completion.coe'",
"Real",
"TopologicalSpace.PseudoMetrizableSpace.toMetrizableSpace",
"congrArg",... | [] | by
refine induction_on₂ x y ?_ ?_
· exact isClosed_eq (Completion.continuous_dist continuous_fst continuous_snd)
(Completion.continuous_dist continuous_snd continuous_fst)
· intro a b
rw [Completion.dist_eq, Completion.dist_eq, dist_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Prod | {
"line": 513,
"column": 2
} | {
"line": 513,
"column": 72
} | {
"line": 514,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nn : ℕ\nF' : Fin n.succ → Type u_6\ninst✝¹ : (i : Fin n.succ) → NormedAddCommGroup (F' i)\ninst✝ : (i : Fin n.succ) → NormedSpace 𝕜 (F' i)\nφ : E → F' 0\nφs : E → (i : Fin n) → F'... | [
"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nn : ℕ\nF' : Fin n.succ → Type u_6\ninst✝¹ : (i : Fin n.succ) → NormedAddCommGroup (F' i)\ninst✝ : (i : Fin n.succ) → NormedSpace 𝕜 (F' i)\nφ : E → F' 0\nφs : E → (i : Fin n) → F' i.succ\nφ' ... | rw [hasFDerivAtFilter_pi', Fin.forall_fin_succ, hasFDerivAtFilter_pi'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 37
} | {
"line": 237,
"column": 2
} | [
{
"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\nx : E\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpToOn n f p s... | [
"case e'_12\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 : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpToOn n f p s... | convert! h.fderivWithin _ this x hx | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 273,
"column": 2
} | {
"line": 274,
"column": 70
} | {
"line": 275,
"column": 2
} | [
{
"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\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\n⊢ HasFTaylorSeriesUpToOn (↑n + 1) ... | [
"case mpr\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\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\n⊢ HasFTaylorSeriesUpToOn (↑n) f p s ∧\n ... | · exact fun h ↦ ⟨h.of_le (mod_cast Nat.le_succ n),
h.fderivWithin _ (mod_cast lt_add_one n), h.cont (n + 1) le_rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 60,
"column": 70
} | {
"line": 73,
"column": 87
} | {
"line": 75,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nE : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nG : Type u_1\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nx : 𝕜\nB : E →L[𝕜] F →L[𝕜] G\nu : 𝕜 → ... | [] | by
by_cases hxu : x ∈ tsupport u
· by_cases hxv : x ∈ tsupport v
· simpa using (B.hasFDerivAt_of_bilinear (hu hxv).hasFDerivAt (hv hxu).hasFDerivAt).hasDerivAt
· have hx : x ∉ tsupport fun x ↦ B (u x) (v x) :=
mt (closure_mono (fun x ↦ mt fun h ↦ by simp [h]) ·) hxv
convert! HasDerivAt.of_notM... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 158,
"column": 2
} | {
"line": 160,
"column": 57
} | {
"line": 162,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nx : 𝕜\ns : Set 𝕜\n𝕜' : Type u_2\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : NormedAlgebra 𝕜 𝕜'\ninst✝² : Module 𝕜' F\ninst✝¹ : IsBoundedSMul 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nc : 𝕜... | [] | by_cases hsx : UniqueDiffWithinAt 𝕜 s x
· exact (hc.hasDerivWithinAt.smul_const f).derivWithin hsx
· simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 158,
"column": 2
} | {
"line": 160,
"column": 57
} | {
"line": 162,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nx : 𝕜\ns : Set 𝕜\n𝕜' : Type u_2\ninst✝⁴ : NormedRing 𝕜'\ninst✝³ : NormedAlgebra 𝕜 𝕜'\ninst✝² : Module 𝕜' F\ninst✝¹ : IsBoundedSMul 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\nc : 𝕜... | [] | by_cases hsx : UniqueDiffWithinAt 𝕜 s x
· exact (hc.hasDerivWithinAt.smul_const f).derivWithin hsx
· simp [derivWithin_zero_of_not_uniqueDiffWithinAt hsx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Pow | {
"line": 169,
"column": 10
} | {
"line": 169,
"column": 12
} | {
"line": 170,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\n𝔸 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedCommRing 𝔸\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedSpace 𝕜 E\nf : E → 𝔸\nf' : E →L[𝕜] 𝔸\nx : E\nn a : ℕ\n⊢ a ∈ Finset.range n → f x ^ (a + (n.pred - a)) = f x ^ (n - ... | [
"𝕜 : Type u_1\n𝔸 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedCommRing 𝔸\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedSpace 𝕜 E\nf : E → 𝔸\nf' : E →L[𝕜] 𝔸\nx : E\nn a : ℕ\nha : a ∈ Finset.range n\n⊢ f x ^ (a + (n.pred - a)) = f x ^ (n - 1)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 329,
"column": 73
} | {
"line": 333,
"column": 42
} | {
"line": 335,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\na : 𝕜\n⊢ DifferentiableAt 𝕜 (fun x ↦ f (-x)) a ↔ DifferentiableAt 𝕜 f (-a)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"N... | [] | by
refine ⟨fun H ↦ ?_, fun H ↦ H.comp a differentiable_neg.differentiableAt⟩
convert! ((neg_neg a).symm ▸ H).comp (-a) differentiable_neg.differentiableAt
ext
simp only [Function.comp_apply, neg_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 49
} | {
"line": 350,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : 𝕜 → F\nf' g' : F\nL : Filter (𝕜 × 𝕜)\nhf : HasDerivAtFilter f f' L\nhg : HasDerivAtFilter g g' L\n⊢ HasDerivAtFilter (f - g) (f' - g') L",
"ppTerm": "?m.32",
"assigned... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 49
} | {
"line": 350,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : 𝕜 → F\nf' g' : F\nL : Filter (𝕜 × 𝕜)\nhf : HasDerivAtFilter f f' L\nhg : HasDerivAtFilter g g' L\n⊢ HasDerivAtFilter (f - g) (f' - g') L",
"ppTerm": "?m.32",
"assigned... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 49
} | {
"line": 350,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf g : 𝕜 → F\nf' g' : F\nL : Filter (𝕜 × 𝕜)\nhf : HasDerivAtFilter f f' L\nhg : HasDerivAtFilter g g' L\n⊢ HasDerivAtFilter (f - g) (f' - g') L",
"ppTerm": "?m.32",
"assigned... | [] | simpa only [sub_eq_add_neg] using hf.add hg.neg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 624,
"column": 27
} | {
"line": 624,
"column": 54
} | {
"line": 624,
"column": 55
} | [
{
"pp": "case inr.refine_3\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : Fintype ι\nf : ContinuousMultilinea... | [
"case inr.refine_3\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\... | (Equiv.injective _).eq_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Analytic.Uniqueness | {
"line": 88,
"column": 8
} | {
"line": 88,
"column": 26
} | {
"line": 90,
"column": 0
} | [
{
"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\ny : E\nc : ℝ\nc_pos : c > 0\nt : Set E\nt_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace]... | [] | simpa using! h₃.le | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Topology.Algebra.Module.PerfectSpace | {
"line": 29,
"column": 2
} | {
"line": 31,
"column": 61
} | {
"line": 32,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Nontrivial E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul 𝕜 E\nx : E\nhx : x ∈ univ\nr : 𝕜\nhr₀ : 0 < ‖r‖\nhr : ‖r‖ < 1\nc : E\nhc : c ≠ 0\nA : T... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : Nontrivial E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul 𝕜 E\nx : E\nhx : x ∈ univ\nr : 𝕜\nhr₀ : 0 < ‖r‖\nhr : ‖r‖ < 1\nc : E\nhc : c ≠ 0\nA : Tendsto (fun ... | have B : Tendsto (fun (n : ℕ) ↦ x + r ^ n • c) atTop (𝓝[univ \ {x}] x) := by
simp only [zero_smul, add_zero] at A
simp [tendsto_nhdsWithin_iff, A, hc, norm_pos_iff.mp hr₀] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.DSlope | {
"line": 71,
"column": 49
} | {
"line": 72,
"column": 48
} | {
"line": 74,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\na b : 𝕜\nf : 𝕜 → E\nh : b ≠ a\n⊢ dslope (fun x ↦ (x - a) • f x) a b = f b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",... | [] | by
rw [dslope_of_ne _ h, slope_sub_smul _ h.symm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.Deriv.Inv | {
"line": 42,
"column": 2
} | {
"line": 53,
"column": 90
} | {
"line": 55,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\nhx : x ≠ 0\n⊢ HasStrictDerivAt Inv.inv (-(x ^ 2)⁻¹) x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"NormedCommRing.toNormedRing",
"Set.instSProd... | [] | suffices
(fun p : 𝕜 × 𝕜 => (p.1 - p.2) * ((x * x)⁻¹ - (p.1 * p.2)⁻¹)) =o[𝓝 (x, x)] fun p =>
(p.1 - p.2) * 1 by
refine .of_isLittleO <| this.congr' ?_ (Eventually.of_forall fun _ => mul_one _)
refine Eventually.mono ((isOpen_ne.prod isOpen_ne).mem_nhds ⟨hx, hx⟩) ?_
rintro ⟨y, z⟩ ⟨hy, hz⟩
sim... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Inv | {
"line": 42,
"column": 2
} | {
"line": 53,
"column": 90
} | {
"line": 55,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\nhx : x ≠ 0\n⊢ HasStrictDerivAt Inv.inv (-(x ^ 2)⁻¹) x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"NormedCommRing.toNormedRing",
"Set.instSProd... | [] | suffices
(fun p : 𝕜 × 𝕜 => (p.1 - p.2) * ((x * x)⁻¹ - (p.1 * p.2)⁻¹)) =o[𝓝 (x, x)] fun p =>
(p.1 - p.2) * 1 by
refine .of_isLittleO <| this.congr' ?_ (Eventually.of_forall fun _ => mul_one _)
refine Eventually.mono ((isOpen_ne.prod isOpen_ne).mem_nhds ⟨hx, hx⟩) ?_
rintro ⟨y, z⟩ ⟨hy, hz⟩
sim... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Comp | {
"line": 333,
"column": 88
} | {
"line": 335,
"column": 55
} | {
"line": 337,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nf' : 𝕜\nhf : HasDerivWithinAt f f' s x\nhx : f x = x\nhs : MapsTo f s s\nn : ℕ\n⊢ HasDerivWithinAt f^[n] (f' ^ n) s x",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminorm... | [] | by
have := HasFDerivWithinAt.iterate hf hx hs n
rwa [ContinuousLinearMap.toSpanSingleton_pow] at this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 225,
"column": 29
} | {
"line": 225,
"column": 31
} | {
"line": 226,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nhf : AnalyticOnNhd 𝕜 f U\nhU : IsPreconnected U\nx : 𝕜\nhx : x ∈ U\nhx2 : (U \\ {x | ¬f x = 0})ᶜ ∉ 𝓝[≠] x\nnh : ∀ᶠ (x : 𝕜) in 𝓝[≠] x, ¬(fun z ↦ f z = 0... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nhf : AnalyticOnNhd 𝕜 f U\nhU : IsPreconnected U\nx : 𝕜\nhx : x ∈ U\nhx2 : (U \\ {x | ¬f x = 0})ᶜ ∉ 𝓝[≠] x\nnh : ∀ᶠ (x : 𝕜) in 𝓝[≠] x, ¬(fun z ↦ f z = 0) x\na : 𝕜\... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Calculus.FDeriv.Affine | {
"line": 66,
"column": 6
} | {
"line": 66,
"column": 58
} | {
"line": 66,
"column": 58
} | [
{
"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 →ᴬ[𝕜] F\nx : E\ns : Set E\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderivWithin 𝕜 (⇑f) s x = f.contLinear... | [
"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 →ᴬ[𝕜] F\nx : E\ns : Set E\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ fderiv 𝕜 (⇑f) x = f.contLinear"
] | DifferentiableAt.fderivWithin f.differentiableAt hxs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 214,
"column": 4
} | {
"line": 220,
"column": 47
} | {
"line": 221,
"column": 2
} | [
{
"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\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω... | [] | obtain ⟨u, hu, p, hp, h'p⟩ := hf
refine ⟨u, hu, _, hp.continuousLinearMap_comp g, fun i ↦ ?_⟩
change AnalyticOn 𝕜
(fun x ↦ (ContinuousLinearMap.compContinuousMultilinearMapL 𝕜
(fun _ : Fin i ↦ E) F G g) (p x i)) u
apply AnalyticOnNhd.comp_analyticOn _ (h'p i) (Set.mapsTo_univ _ _)
exact Co... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 214,
"column": 4
} | {
"line": 220,
"column": 47
} | {
"line": 221,
"column": 2
} | [
{
"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\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω... | [] | obtain ⟨u, hu, p, hp, h'p⟩ := hf
refine ⟨u, hu, _, hp.continuousLinearMap_comp g, fun i ↦ ?_⟩
change AnalyticOn 𝕜
(fun x ↦ (ContinuousLinearMap.compContinuousMultilinearMapL 𝕜
(fun _ : Fin i ↦ E) F G g) (p x i)) u
apply AnalyticOnNhd.comp_analyticOn _ (h'p i) (Set.mapsTo_univ _ _)
exact Co... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 229,
"column": 4
} | {
"line": 230,
"column": 56
} | {
"line": 232,
"column": 0
} | [
{
"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 f₁ : E → F\nx : E\nn✝ : ℕ∞ω\nh₁ : f₁ =ᶠ[𝓝[s] x] f\nhx : f₁ x = f x\nn : ℕ∞\nh : ContDiffWithinAt �... | [] | exact ⟨{ x ∈ u | f₁ x = f x }, Filter.inter_mem hu (mem_nhdsWithin_insert.2 ⟨hx, h₁⟩), p,
(H.mono (sep_subset _ _)).congr fun _ ↦ And.right⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 220,
"column": 2
} | {
"line": 230,
"column": 56
} | {
"line": 232,
"column": 0
} | [
{
"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 f₁ : E → F\nx : E\nn : ℕ∞ω\nh : ContDiffWithinAt 𝕜 n f s x\nh₁ : f₁ =ᶠ[𝓝[s] x] f\nhx : f₁ x = f x... | [] | match n with
| ω =>
obtain ⟨u, hu, p, H, H'⟩ := h
exact ⟨{x ∈ u | f₁ x = f x}, Filter.inter_mem hu (mem_nhdsWithin_insert.2 ⟨hx, h₁⟩), p,
(H.mono (sep_subset _ _)).congr fun _ ↦ And.right,
fun i ↦ (H' i).mono (sep_subset _ _)⟩
| (n : ℕ∞) =>
intro m hm
let ⟨u, hu, p, H⟩ := h m hm
exac... | Lean.Elab.Tactic.evalMatch | Lean.Parser.Tactic.match |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 220,
"column": 2
} | {
"line": 230,
"column": 56
} | {
"line": 232,
"column": 0
} | [
{
"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 f₁ : E → F\nx : E\nn : ℕ∞ω\nh : ContDiffWithinAt 𝕜 n f s x\nh₁ : f₁ =ᶠ[𝓝[s] x] f\nhx : f₁ x = f x... | [] | match n with
| ω =>
obtain ⟨u, hu, p, H, H'⟩ := h
exact ⟨{x ∈ u | f₁ x = f x}, Filter.inter_mem hu (mem_nhdsWithin_insert.2 ⟨hx, h₁⟩), p,
(H.mono (sep_subset _ _)).congr fun _ ↦ And.right,
fun i ↦ (H' i).mono (sep_subset _ _)⟩
| (n : ℕ∞) =>
intro m hm
let ⟨u, hu, p, H⟩ := h m hm
exac... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 220,
"column": 2
} | {
"line": 230,
"column": 56
} | {
"line": 232,
"column": 0
} | [
{
"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 f₁ : E → F\nx : E\nn : ℕ∞ω\nh : ContDiffWithinAt 𝕜 n f s x\nh₁ : f₁ =ᶠ[𝓝[s] x] f\nhx : f₁ x = f x... | [] | match n with
| ω =>
obtain ⟨u, hu, p, H, H'⟩ := h
exact ⟨{x ∈ u | f₁ x = f x}, Filter.inter_mem hu (mem_nhdsWithin_insert.2 ⟨hx, h₁⟩), p,
(H.mono (sep_subset _ _)).congr fun _ ↦ And.right,
fun i ↦ (H' i).mono (sep_subset _ _)⟩
| (n : ℕ∞) =>
intro m hm
let ⟨u, hu, p, H⟩ := h m hm
exac... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.OfCompLeft | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 53
} | {
"line": 92,
"column": 0
} | [
{
"pp": "case refine_2\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\ng : E → F\nf : F → G\... | [] | · exact hcomp.prodMap (hcomp.self_of_nhdsWithin ha) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.FDeriv.OfCompLeft | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 38
} | {
"line": 142,
"column": 2
} | [
{
"pp": "case refine_1\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\ng : E → F\nf : F → G\... | [
"case refine_2\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\ng : E → F\nf : F → G\nh : E → G\n... | · exact hg.tendsto.prodMap (by simp) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 476,
"column": 67
} | {
"line": 480,
"column": 36
} | {
"line": 482,
"column": 0
} | [
{
"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 : ℕ∞\nf' : E → FormalMultilinearSeries 𝕜 E F\nhf : HasFTaylorSeriesUpToOn (↑n) f f' s\n... | [] | by
intro x hx m hm
use s
simp only [Set.insert_eq_of_mem hx, self_mem_nhdsWithin, true_and]
exact ⟨f', hf.of_le (mod_cast hm)⟩ | [anonymous] | Lean.Parser.Term.byTactic |
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