Loogle!

Result

Found 1502 declarations whose name contains "fderiv". Of these, 15 have a name containing "fderiv" and "mono".

• HasFDerivWithinAt.mono 📋 Mathlib.Analysis.Calculus.FDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s t : Set E} (h : HasFDerivWithinAt f f' t x) (hst : s ⊆ t) : HasFDerivWithinAt f f' s x
• HasFDerivAtFilter.mono 📋 Mathlib.Analysis.Calculus.FDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {L₁ L₂ : Filter E} (h : HasFDerivAtFilter f f' x L₂) (hst : L₁ ≤ L₂) : HasFDerivAtFilter f f' x L₁
• HasFDerivWithinAt.congr_mono 📋 Mathlib.Analysis.Calculus.FDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : E → F} {f' : E →L[𝕜] F} {x : E} {s t : Set E} (h : HasFDerivWithinAt f f' s x) (ht : Set.EqOn f₁ f t) (hx : f₁ x = f x) (h₁ : t ⊆ s) : HasFDerivWithinAt f₁ f' t x
• HasFDerivWithinAt.mono_of_mem 📋 Mathlib.Analysis.Calculus.FDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s t : Set E} (h : HasFDerivWithinAt f f' t x) (hst : t ∈ nhdsWithin x s) : HasFDerivWithinAt f f' s x
• HasFDerivWithinAt.mono_of_mem_nhdsWithin 📋 Mathlib.Analysis.Calculus.FDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s t : Set E} (h : HasFDerivWithinAt f f' t x) (hst : t ∈ nhdsWithin x s) : HasFDerivWithinAt f f' s x
• DifferentiableWithinAt.fderivWithin_congr_mono 📋 Mathlib.Analysis.Calculus.FDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f f₁ : E → F} {x : E} {s t : Set E} (h : DifferentiableWithinAt 𝕜 f s x) (hs : Set.EqOn f₁ f t) (hx : f₁ x = f x) (hxt : UniqueDiffWithinAt 𝕜 t x) (h₁ : t ⊆ s) : fderivWithin 𝕜 f₁ t x = fderivWithin 𝕜 f s x
• FDerivMeasurableAux.A_mono 📋 Mathlib.Analysis.Calculus.FDeriv.Measurable
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {F : Type u_3} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} (L : E →L[𝕜] F) (r : ℝ) {ε δ : ℝ} (h : ε ≤ δ) : FDerivMeasurableAux.A f L r ε ⊆ FDerivMeasurableAux.A f L r δ
• IsSymmSndFDerivWithinAt.mono_of_mem_nhdsWithin 📋 Mathlib.Analysis.Calculus.FDeriv.Symmetric
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} {F : Type u_3} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {s t : Set E} {f : E → F} {x : E} (h : IsSymmSndFDerivWithinAt 𝕜 f t x) (hst : t ∈ nhdsWithin x s) (hf : ContDiffWithinAt 𝕜 2 f t x) (hs : UniqueDiffOn 𝕜 s) (ht : UniqueDiffOn 𝕜 t) (hx : x ∈ s) : IsSymmSndFDerivWithinAt 𝕜 f s x
• HasMFDerivWithinAt.mono 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s t : Set M} {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : HasMFDerivWithinAt I I' f t x f') (hst : s ⊆ t) : HasMFDerivWithinAt I I' f s x f'
• HasMFDerivWithinAt.mono_of_mem 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s t : Set M} {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : HasMFDerivWithinAt I I' f s x f') (ht : s ∈ nhdsWithin x t) : HasMFDerivWithinAt I I' f t x f'
• HasMFDerivWithinAt.mono_of_mem_nhdsWithin 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s t : Set M} {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : HasMFDerivWithinAt I I' f s x f') (ht : s ∈ nhdsWithin x t) : HasMFDerivWithinAt I I' f t x f'
• HasMFDerivWithinAt.congr_mono 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f f₁ : M → M'} {x : M} {s t : Set M} {f' : TangentSpace I x →L[𝕜] TangentSpace I' (f x)} (h : HasMFDerivWithinAt I I' f s x f') (ht : ∀ x ∈ t, f₁ x = f x) (hx : f₁ x = f x) (h₁ : t ⊆ s) : HasMFDerivWithinAt I I' f₁ t x f'
• MDifferentiableWithinAt.mfderivWithin_mono 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s t : Set M} (h : MDifferentiableWithinAt I I' f s x) (hxt : UniqueMDiffWithinAt I t x) (h₁ : t ⊆ s) : mfderivWithin I I' f t x = mfderivWithin I I' f s x
• MDifferentiableWithinAt.mfderivWithin_mono_of_mem_nhdsWithin 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f : M → M'} {x : M} {s t : Set M} (h : MDifferentiableWithinAt I I' f s x) (hxt : UniqueMDiffWithinAt I t x) (h₁ : s ∈ nhdsWithin x t) : mfderivWithin I I' f t x = mfderivWithin I I' f s x
• MDifferentiableWithinAt.mfderivWithin_congr_mono 📋 Mathlib.Geometry.Manifold.MFDeriv.Basic
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {E' : Type u_5} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_6} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [TopologicalSpace M'] [ChartedSpace H' M'] {f f₁ : M → M'} {x : M} {s t : Set M} (h : MDifferentiableWithinAt I I' f s x) (hs : ∀ x ∈ t, f₁ x = f x) (hx : f₁ x = f x) (hxt : UniqueMDiffWithinAt I t x) (h₁ : t ⊆ s) : mfderivWithin I I' f₁ t x = mfderivWithin I I' f s x

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using (by default) Ctrl-K Ctrl-S. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO.

This is Loogle revision b340a5b serving mathlib revision 846a90c