module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 907,
"column": 2
} | {
"line": 907,
"column": 26
} | [
{
"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\nx a : E\n⊢ fderiv 𝕜 (fun x ↦ f (a + x)) x = fderiv 𝕜 f (a + x)",
"usedConstants": [
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 911,
"column": 2
} | {
"line": 911,
"column": 30
} | [
{
"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 (x - a)) f' s x ↔ HasFDe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 920,
"column": 2
} | {
"line": 920,
"column": 30
} | [
{
"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\nx : E\ns : Set E\na : E\n⊢ fderivWithin 𝕜 (fun x ↦ f (x - a)) s x = fderivWithin 𝕜 f (-a +ᵥ s) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Bilinear | {
"line": 64,
"column": 8
} | {
"line": 64,
"column": 56
} | [
{
"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\nh : IsBoundedBilinear... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.Alternating.Basic | {
"line": 641,
"column": 8
} | {
"line": 642,
"column": 15
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\nι : Type u_4\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : Module R M\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : TopologicalSpace N\ninst✝² : IsTopologicalAddGroup N\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 13
} | [
{
"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∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nhN : ∞ ≤ N\n⊢ HasFTaylorSeriesUpToOn N f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 674,
"column": 2
} | {
"line": 674,
"column": 26
} | [
{
"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\na : E\n⊢ iteratedFDerivWithin 𝕜 n (fun z ↦ f (z + a)) s = fun x ↦ iteratedFDerivWi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 686,
"column": 2
} | {
"line": 686,
"column": 30
} | [
{
"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\na : E\n⊢ iteratedFDerivWithin 𝕜 n (fun z ↦ f (z - a)) s = fun x ↦ iteratedFDerivWi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 719,
"column": 37
} | {
"line": 719,
"column": 48
} | [
{
"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 : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nH : HasFTaylorSeriesUpToOn n f p univ\n⊢ ∀ (x : E), ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 719,
"column": 67
} | {
"line": 719,
"column": 78
} | [
{
"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 : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nH : HasFTaylorSeriesUpToOn n f p univ\n⊢ ∀ (m : ℕ), ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 719,
"column": 37
} | {
"line": 719,
"column": 48
} | [
{
"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 : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nH : HasFTaylorSeriesUpTo n f p\n⊢ ∀ x ∈ univ, (p x 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 719,
"column": 67
} | {
"line": 719,
"column": 78
} | [
{
"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 : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nH : HasFTaylorSeriesUpTo n f p\n⊢ ∀ (m : ℕ), ↑m ≤ n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 720,
"column": 4
} | {
"line": 720,
"column": 15
} | [
{
"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\nf : E → F\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nH : HasFTaylorSeriesUpToOn n f p univ\n⊢ ∀ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 721,
"column": 4
} | {
"line": 721,
"column": 15
} | [
{
"pp": "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\nf : E → F\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nH : HasFTaylorSeriesUpTo n f p\n⊢ ∀ (m : ℕ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 96,
"column": 2
} | {
"line": 98,
"column": 9
} | [
{
"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\nh : HasFPowerSeriesAt f p x\n⊢ HasStrictFDerivAt f ((conti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 77
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\nhf : AnalyticAt 𝕜 f x\n⊢ HasStrictDerivAt f (deriv f x) x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 191,
"column": 6
} | {
"line": 191,
"column": 17
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 946,
"column": 2
} | {
"line": 946,
"column": 43
} | [
{
"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 : ℕ\na : E\n⊢ (iteratedFDeriv 𝕜 n fun z ↦ f (a + z)) = fun x ↦ iteratedFDeriv 𝕜 n f (a + x)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 956,
"column": 2
} | {
"line": 956,
"column": 26
} | [
{
"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 : ℕ\na : E\n⊢ (iteratedFDeriv 𝕜 n fun z ↦ f (z + a)) = fun x ↦ iteratedFDeriv 𝕜 n f (x + a)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 966,
"column": 2
} | {
"line": 966,
"column": 30
} | [
{
"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 : ℕ\na : E\n⊢ (iteratedFDeriv 𝕜 n fun z ↦ f (z - a)) = fun x ↦ iteratedFDeriv 𝕜 n f (x - a)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 15
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ninst✝ : CompleteSpace F\nh : Has... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 57
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ninst✝ : CompleteSpace F\nh : Has... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 233,
"column": 4
} | {
"line": 233,
"column": 15
} | [
{
"pp": "case refine_1.hf\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 237,
"column": 6
} | {
"line": 237,
"column": 61
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : HasFPow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 238,
"column": 6
} | {
"line": 238,
"column": 17
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nr : ℝ≥0∞\nf : E → F\nx : E\ns : Set E\ninst✝ : CompleteSpace F\nh : HasFPow... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Alternating.Basic | {
"line": 468,
"column": 88
} | {
"line": 482,
"column": 25
} | [
{
"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 : ... | by
intro dg v a b heq hne
trans ∑ i, f fun j ↦ Function.update (fun _ ↦ g) i dg j (v j)
· simp
· rw [← Finset.sum_add_sum_compl {a, b}, Finset.sum_pair hne, Finset.sum_eq_zero, add_zero]
· convert! f.map_add_swap _ hne with i
rcases eq_or_ne i a with rfl | hia
· simp [heq, hne, hne... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 326,
"column": 50
} | {
"line": 326,
"column": 80
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Alternating.Basic | {
"line": 527,
"column": 39
} | {
"line": 527,
"column": 50
} | [
{
"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 : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Alternating.Basic | {
"line": 527,
"column": 6
} | {
"line": 527,
"column": 61
} | [
{
"pp": "case h.hbc.hbc.hbc.h1\n𝕜 : 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 : Norm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Module.Alternating.Basic | {
"line": 583,
"column": 29
} | {
"line": 583,
"column": 40
} | [
{
"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 : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 348,
"column": 55
} | {
"line": 348,
"column": 85
} | [
{
"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\nhr : HasFPowerSeriesWithinOnBall f p ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 419,
"column": 2
} | {
"line": 419,
"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\nf : OpenPartialHomeomorph E F\na : F\ni : E ≃L[𝕜] F\nh0 : a ∈ f.target\nh : AnalyticAt 𝕜 (↑f) (↑f.symm a)\nh' ... | exact f.analyticAt_symm' (by simp [h0]) h h' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 455,
"column": 17
} | {
"line": 455,
"column": 79
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : 𝕜 → F\ns : Set 𝕜\ninst✝ : CompleteSpace F\nh : AnalyticOnNhd 𝕜 f s\nn : ℕ\nIH : AnalyticOnNhd 𝕜 (deriv^[n] f) s\n⊢ AnalyticOnNhd 𝕜 (deriv^[n + 1] f) s",
"use... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 466,
"column": 17
} | {
"line": 466,
"column": 79
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝³ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\ninst✝ : CompleteSpace F\nh : AnalyticAt 𝕜 f x\nn : ℕ\nIH : AnalyticAt 𝕜 (deriv^[n] f) x\n⊢ AnalyticAt 𝕜 (deriv^[n + 1] f) x",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 504,
"column": 4
} | {
"line": 504,
"column": 15
} | [
{
"pp": "case refine_1\n𝕜 : 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∞\nn : ℕ\nf : E → F\nx : E\nh : HasFiniteFPowerSeriesO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 57
} | [
{
"pp": "case refine_2\n𝕜 : 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∞\nn : ℕ\nf : E → F\nx : E\nh : HasFiniteFPowerSeriesO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 565,
"column": 17
} | {
"line": 565,
"column": 79
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\ns : Set 𝕜\nh : CPolynomialOn 𝕜 f s\nn : ℕ\nIH : CPolynomialOn 𝕜 (deriv^[n] f) s\n⊢ CPolynomialOn 𝕜 (deriv^[n + 1] f) s",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 696,
"column": 2
} | {
"line": 696,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\ninst✝²... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 706,
"column": 2
} | {
"line": 706,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝³ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\ninst✝²... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 741,
"column": 4
} | {
"line": 741,
"column": 11
} | [
{
"pp": "case fderiv\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝² : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝ : Fintype ι\nf : ContinuousMultilinearMap 𝕜... | ext v m | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 755,
"column": 10
} | {
"line": 755,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝² : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E F\nn : ℕ\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 822,
"column": 2
} | {
"line": 822,
"column": 13
} | [
{
"pp": "case H\n𝕜 : 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\nr : ℝ≥0∞\nh : HasFPowerSeriesOnBall f p x r\nx✝ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Pow | {
"line": 40,
"column": 85
} | {
"line": 46,
"column": 30
} | [
{
"pp": "𝕜 : Type u_1\n𝔸 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing 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 : ℕ\n⊢ f x •> ∑ i ∈ Finset.range (n + 1), f x ^ ((n + 1).pred - i) •> f'... | by
rw [Finset.sum_range_succ _ (n + 1), Finset.smul_sum]
simp only [Nat.pred_eq_sub_one, add_tsub_cancel_right, tsub_self, pow_zero, one_smul]
simp_rw [smul_comm (_ : 𝔸) (_ : 𝔸ᵐᵒᵖ), smul_smul, ← pow_succ']
congr! 5 with x hx
simp only [Finset.mem_range, Nat.lt_succ_iff] at hx
rw [tsub_add_eq_add_tsub hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Pow | {
"line": 53,
"column": 12
} | {
"line": 53,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\n𝔸 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedSpace 𝕜 E\nf : E → 𝔸\nf' : E →L[𝕜] 𝔸\nx : E\nh : HasStrictFDerivAt f f' x\nn : ℕ\n⊢ HasStrictFDerivAt (f ^ 0) (∑ i ∈ Fi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Pow | {
"line": 54,
"column": 12
} | {
"line": 54,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\n𝔸 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedSpace 𝕜 E\nf : E → 𝔸\nf' : E →L[𝕜] 𝔸\nx : E\nh : HasStrictFDerivAt f f' x\nn : ℕ\n⊢ HasStrictFDerivAt (f ^ 1) (∑ i ∈ Fi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Pow | {
"line": 70,
"column": 12
} | {
"line": 70,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\n𝔸 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedSpace 𝕜 E\nf : E → 𝔸\nf' : E →L[𝕜] 𝔸\nx : E\ns : Set E\nh : HasFDerivWithinAt f f' s x\nn : ℕ\n⊢ HasFDerivWithinAt (f ^... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Pow | {
"line": 71,
"column": 12
} | {
"line": 71,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\n𝔸 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedSpace 𝕜 E\nf : E → 𝔸\nf' : E →L[𝕜] 𝔸\nx : E\ns : Set E\nh : HasFDerivWithinAt f f' s x\nn : ℕ\n⊢ HasFDerivWithinAt (f ^... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Pow | {
"line": 86,
"column": 12
} | {
"line": 86,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\n𝔸 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedSpace 𝕜 E\nf : E → 𝔸\nf' : E →L[𝕜] 𝔸\nx : E\nh : HasFDerivAt f f' x\nn : ℕ\n⊢ HasFDerivAt (f ^ 0) (∑ i ∈ Finset.range 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.FDeriv.Pow | {
"line": 87,
"column": 12
} | {
"line": 87,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\n𝔸 : Type u_2\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedRing 𝔸\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAlgebra 𝕜 𝔸\ninst✝ : NormedSpace 𝕜 E\nf : E → 𝔸\nf' : E →L[𝕜] 𝔸\nx : E\nh : HasFDerivAt f f' x\nn : ℕ\n⊢ HasFDerivAt (f ^ 1) (∑ i ∈ Finset.range 1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Pow | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\n𝔸 : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nf : 𝕜 → 𝔸\nf' : 𝔸\nx : 𝕜\nh : HasStrictDerivAt f f' x\nn : ℕ\n⊢ HasStrictDerivAt (fun x ↦ f x ^ n) (∑ i ∈ Finset.range n, f x ^ (n.pred - i) * f' * f x ^ i) x",
"usedConstants... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Pow | {
"line": 47,
"column": 2
} | {
"line": 47,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\n𝔸 : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nf : 𝕜 → 𝔸\nf' : 𝔸\nx : 𝕜\ns : Set 𝕜\nh : HasDerivWithinAt f f' s x\nn : ℕ\n⊢ HasDerivWithinAt (fun x ↦ f x ^ n) (∑ i ∈ Finset.range n, f x ^ (n.pred - i) * f' * f x ^ i) s x",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Pow | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\n𝔸 : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nf : 𝕜 → 𝔸\nf' : 𝔸\nx : 𝕜\nh : HasDerivAt f f' x\nn : ℕ\n⊢ HasDerivAt (fun x ↦ f x ^ n) (∑ i ∈ Finset.range n, f x ^ (n.pred - i) * f' * f x ^ i) x",
"usedConstants": [
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Pow | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\n𝔸 : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedCommRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nf : 𝕜 → 𝔸\nf' : 𝔸\nx : 𝕜\nh : HasStrictDerivAt f f' x\nn : ℕ\n⊢ HasStrictDerivAt (fun x ↦ f x ^ n) (↑n * f x ^ (n - 1) * f') x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Pow | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\n𝔸 : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedCommRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nf : 𝕜 → 𝔸\nf' : 𝔸\nx : 𝕜\ns : Set 𝕜\nh : HasDerivWithinAt f f' s x\nn : ℕ\n⊢ HasDerivWithinAt (fun x ↦ f x ^ n) (↑n * f x ^ (n - 1) * f') s x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Pow | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\n𝔸 : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedCommRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nf : 𝕜 → 𝔸\nf' : 𝔸\nx : 𝕜\nh : HasDerivAt f f' x\nn : ℕ\n⊢ HasDerivAt (f ^ n) (↑n * f x ^ (n - 1) * f') x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Pow | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nn : ℕ\nx : 𝕜\n⊢ HasStrictDerivAt (fun x ↦ x ^ n) (↑n * x ^ (n - 1)) x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Pow | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\ns : Set 𝕜\nn : ℕ\nx : 𝕜\n⊢ HasDerivWithinAt (fun x ↦ x ^ n) (↑n * x ^ (n - 1)) s x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Pow | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nn : ℕ\nx : 𝕜\n⊢ HasDerivAt (fun x ↦ x ^ n) (↑n * x ^ (n - 1)) x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 13
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 195,
"column": 2
} | {
"line": 195,
"column": 13
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nL : Filter (𝕜 × 𝕜)\nι : Type u_1\nu : Finset ι\nA : ι → 𝕜 → F\nA' : ι → F\nh : ∀ i ∈ u, HasDerivAtFilter (A i) (A' i) L\n⊢ HasDerivAtFilter (fun y ↦ ∑ i ∈ u, A i y) (∑ i ∈ u, A' i) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 256,
"column": 40
} | {
"line": 256,
"column": 51
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\nL : Filter (𝕜 × 𝕜)\nh : HasDerivAtFilter f f' L\n⊢ HasDerivAtFilter (-f) (-f') L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 62
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\n⊢ deriv (-f) x = -deriv f x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
... | simp only [deriv, fderiv_neg, ContinuousLinearMap.neg_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 62
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\n⊢ deriv (-f) x = -deriv f x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
... | simp only [deriv, fderiv_neg, ContinuousLinearMap.neg_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 62
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nx : 𝕜\n⊢ deriv (-f) x = -deriv f x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
... | simp only [deriv, fderiv_neg, ContinuousLinearMap.neg_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 35
} | [
{
"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",
"usedConstants": [
"Eq.mpr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 424,
"column": 2
} | {
"line": 424,
"column": 35
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\nL : Filter (𝕜 × 𝕜)\nc : F\nhf : HasDerivAtFilter f f' L\n⊢ HasDerivAtFilter (fun x ↦ c - f x) (-f') L",
"usedConstants": [
"Eq.mpr",
"NegZeroClass... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 458,
"column": 2
} | {
"line": 458,
"column": 56
} | [
{
"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 (fun x ↦ f (x - b)) a ↔ DifferentiableAt 𝕜 f (a - b)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedCommRing.to... | simp [sub_eq_add_neg, differentiableAt_comp_add_const] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 458,
"column": 2
} | {
"line": 458,
"column": 56
} | [
{
"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 (fun x ↦ f (x - b)) a ↔ DifferentiableAt 𝕜 f (a - b)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedCommRing.to... | simp [sub_eq_add_neg, differentiableAt_comp_add_const] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 458,
"column": 2
} | {
"line": 458,
"column": 56
} | [
{
"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 (fun x ↦ f (x - b)) a ↔ DifferentiableAt 𝕜 f (a - b)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NormedCommRing.to... | simp [sub_eq_add_neg, differentiableAt_comp_add_const] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 470,
"column": 2
} | {
"line": 470,
"column": 56
} | [
{
"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.toNormedRing",
"NormedCommRing.to... | simp [sub_eq_add_neg, differentiableAt_comp_add_const] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 470,
"column": 2
} | {
"line": 470,
"column": 56
} | [
{
"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.toNormedRing",
"NormedCommRing.to... | simp [sub_eq_add_neg, differentiableAt_comp_add_const] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 470,
"column": 2
} | {
"line": 470,
"column": 56
} | [
{
"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.toNormedRing",
"NormedCommRing.to... | simp [sub_eq_add_neg, differentiableAt_comp_add_const] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Polynomial | {
"line": 56,
"column": 21
} | {
"line": 56,
"column": 32
} | [
{
"pp": "case add\n𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\np✝ : 𝕜[X]\nx : 𝕜\np q : 𝕜[X]\nhp : HasStrictDerivAt (fun x ↦ eval x p) (eval x (derivative p)) x\nhq : HasStrictDerivAt (fun x ↦ eval x q) (eval x (derivative q)) x\n⊢ HasStrictDerivAt (fun x ↦ eval x (p + q)) (eval x (derivative (p + q))) x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Polynomial | {
"line": 57,
"column": 20
} | {
"line": 58,
"column": 27
} | [
{
"pp": "case monomial\n𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\np : 𝕜[X]\nx : 𝕜\nn : ℕ\na : 𝕜\n⊢ HasStrictDerivAt (fun x ↦ eval x ((monomial n) a)) (eval x (derivative ((monomial n) a))) x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Polynomial.derivative",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Polynomial | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 65
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R 𝕜\nq : R[X]\nx : 𝕜\n⊢ HasStrictDerivAt (fun x ↦ (aeval x) q) ((aeval x) (derivative q)) x",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Polynomial.derivative",
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Polynomial | {
"line": 121,
"column": 2
} | {
"line": 121,
"column": 65
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\nR : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : Algebra R 𝕜\nq : R[X]\nhxs : UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin (fun x ↦ (aeval x) q) s x = (aeval x) (derivative q)",
"usedConstants": [
"NormedCommRing.toNormedRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Uniqueness | {
"line": 90,
"column": 57
} | {
"line": 90,
"column": 84
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np₁ p₂ : FormalMultilinearSeries 𝕜 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nh₁ : HasFPowerSeriesAt f p₁ x\nh₂ : HasFPowerSeriesAt f p₂ x\n⊢ HasFPowerSeriesAt 0 (p₁ - p₂) x",
"usedConstants": ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Uniqueness | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E → F\nU : Set E\nhf : AnalyticOnNhd 𝕜 f U\nhU : IsPreconnected U\nz₀ : E\nh₀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 13
} | [
{
"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\ns : Set 𝕜\nB : E →L[𝕜] F →L[𝕜] ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 65,
"column": 6
} | {
"line": 65,
"column": 17
} | [
{
"pp": "case pos\n𝕜 : 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\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Uniqueness | {
"line": 191,
"column": 29
} | {
"line": 191,
"column": 45
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Analytic.Uniqueness | {
"line": 202,
"column": 2
} | {
"line": 202,
"column": 27
} | [
{
"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\nU : Set E\nhf : AnalyticOnNhd 𝕜 f U\nhg : AnalyticOnNhd 𝕜 g U\nhU : IsPreconnected U\nz₀ : E\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 13
} | [
{
"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 → ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 13
} | [
{
"pp": "𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : 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✝ : IsScalarTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 13
} | [
{
"pp": "𝕜 : Type u\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\nx : 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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 13
} | [
{
"pp": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\nx : 𝕜\nR : Type u_2\ninst✝³ : Monoid R\ninst✝² : DistribMulAction R F\ninst✝¹ : SMulCommClass 𝕜 R F\ninst✝ : ContinuousConstSMul R F\nc : R\nhf : HasStrictDerivA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 13
} | [
{
"pp": "𝕜 : Type u\ninst✝⁶ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\nL : Filter (𝕜 × 𝕜)\nR : Type u_2\ninst✝³ : Monoid R\ninst✝² : DistribMulAction R F\ninst✝¹ : SMulCommClass 𝕜 R F\ninst✝ : ContinuousConstSMul R F\nc : R\nhf : H... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 13
} | [
{
"pp": "k : 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\nhxy : x ≤ y\n⊢ slope f x y ≤ 0 ↔ f y ≤ f x",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 13
} | [
{
"pp": "k : 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 : AntitoneOn f s\nhx : x ∈ s\nhy : y ∈ s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 13
} | [
{
"pp": "k : 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\nhxy : x ≤ y\n⊢ slope f x y < 0 ↔ f y < f x",
"used... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 13
} | [
{
"pp": "k : 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 : StrictAntiOn f s\nhx : x ∈ s\nhy : y ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Slope | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 55
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\nf' : F\nx : 𝕜\nL : Filter 𝕜\n⊢ Tendsto (fun y ↦ slope f x y - f') (L ⊓ 𝓟 {x}ᶜ) (𝓝 0) ↔ Tendsto (slope f x) (L ⊓ 𝓟 {x}ᶜ) (𝓝 f')",
"usedConstants": [
"AddGrou... | rw [← nhds_translation_sub f', tendsto_comap_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 24
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nc d : 𝕜 → 𝔸\nc' d' : 𝔸\nhc : HasDerivWithinAt c c' s x\nhd : HasDerivWithinAt d d' s x\n⊢ HasDerivWithinAt (c * d) (c' * d x + c x * d') s x",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 24
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nc d : 𝕜 → 𝔸\nc' d' : 𝔸\nhc : HasStrictDerivAt c c' x\nhd : HasStrictDerivAt d d' x\n⊢ HasStrictDerivAt (c * d) (c' * d x + c x * d') x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 311,
"column": 2
} | {
"line": 311,
"column": 28
} | [
{
"pp": "𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\nx c : 𝕜\n⊢ HasDerivAt (fun x ↦ x * c) c x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Slope | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 62
} | [
{
"pp": "case neg\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\ns t : Set 𝕜\nh : s ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (s ∩ t)\nx : 𝕜\nH : ¬UniqueDiffWithinAt 𝕜 s x\n⊢ derivWithin f s x ∈ closure[... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 346,
"column": 6
} | {
"line": 346,
"column": 49
} | [
{
"pp": "case neg.inr\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝕜' : Type u_2\ninst✝¹ : NormedDivisionRing 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nu : 𝕜 → 𝕜'\nv : 𝕜'\nhu : ¬DifferentiableAt 𝕜 u x\nhd : v ≠ 0\nH : DifferentiableAt 𝕜 (fun y ↦ u y * v) x\n⊢ DifferentiableAt 𝕜 u x",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 28
} | [
{
"pp": "𝕜 : Type u\ninst✝ : NontriviallyNormedField 𝕜\nx c : 𝕜\n⊢ HasDerivAt (fun y ↦ c * y) c x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 389,
"column": 2
} | {
"line": 389,
"column": 39
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝕜' : Type u_2\ninst✝¹ : NormedDivisionRing 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nv : 𝕜 → 𝕜'\nu : 𝕜'\n⊢ deriv (fun y ↦ u * v y) x = u * deriv v x",
"usedConstants": [
"Eq.mpr",
"NormedRing.toRing",
"HMul.hMul",
"congr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 413,
"column": 2
} | {
"line": 413,
"column": 77
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nx : 𝕜\nι : Type u_2\ninst✝² : DecidableEq ι\n𝔸' : Type u_3\ninst✝¹ : NormedCommRing 𝔸'\ninst✝ : NormedAlgebra 𝕜 𝔸'\nu : Finset ι\nf : ι → 𝕜 → 𝔸'\nf' : ι → 𝔸'\nhf : ∀ i ∈ u, HasDerivAt (f i) (f' i) x\n⊢ HasDerivAt (fun x ↦ ∏ i ∈ u, f i x) (∑ i ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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