mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
Du : Type u
inst✝⁷ : NormedAddCommGroup Du
inst✝⁶ : NormedSpace 𝕜 Du
s : Set Du
x : Du
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
IH :
  ∀ {Eu Fu Gu : Type u} [inst : NormedAddCommGroup Eu] [inst_1 : NormedSpace 𝕜 Eu] [inst_2 : NormedAddCommGroup Fu]
    [inst_3 : NormedSpace 𝕜 Fu] [inst_4 : NormedAddCommGroup Gu] [inst_5 : NormedSpace 𝕜 Gu] (B : Eu →L[𝕜] Fu →L[𝕜] Gu)
    {f : Du → Eu} {g : Du → Fu},
    ContDiffOn 𝕜 (↑n) f s →
      ContDiffOn 𝕜 (↑n) g s →
        ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (B (f y)) (g y)) s x‖ ≤
          ‖B‖ *
            ∑ i ∈ Finset.range (n + 1),
              ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
Eu Fu Gu : Type u
inst✝⁵ : NormedAddCommGroup Eu
inst✝⁴ : NormedSpace 𝕜 Eu
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
B : Eu →L[𝕜] Fu →L[𝕜] Gu
f : Du → Eu
g : Du → Fu
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hg : ContDiffOn 𝕜 (↑(n + 1)) g s
In : ↑n + 1 ≤ ↑n.succ
⊢ ‖B‖ *
      ∑ i ∈ Finset.range (n + 1),
        ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 g s) s x‖ =
    ‖B‖ *
      ∑ i ∈ Finset.range (n + 1),
        ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n + 1 - i) g s x‖

Tactic:
congr 1

Goal state after:
case e_a
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
Du : Type u
inst✝⁷ : NormedAddCommGroup Du
inst✝⁶ : NormedSpace 𝕜 Du
s : Set Du
x : Du
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
IH :
  ∀ {Eu Fu Gu : Type u} [inst : NormedAddCommGroup Eu] [inst_1 : NormedSpace 𝕜 Eu] [inst_2 : NormedAddCommGroup Fu]
    [inst_3 : NormedSpace 𝕜 Fu] [inst_4 : NormedAddCommGroup Gu] [inst_5 : NormedSpace 𝕜 Gu] (B : Eu →L[𝕜] Fu →L[𝕜] Gu)
    {f : Du → Eu} {g : Du → Fu},
    ContDiffOn 𝕜 (↑n) f s →
      ContDiffOn 𝕜 (↑n) g s →
        ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (B (f y)) (g y)) s x‖ ≤
          ‖B‖ *
            ∑ i ∈ Finset.range (n + 1),
              ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖
Eu Fu Gu : Type u
inst✝⁵ : NormedAddCommGroup Eu
inst✝⁴ : NormedSpace 𝕜 Eu
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
B : Eu →L[𝕜] Fu →L[𝕜] Gu
f : Du → Eu
g : Du → Fu
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hg : ContDiffOn 𝕜 (↑(n + 1)) g s
In : ↑n + 1 ≤ ↑n.succ
⊢ ∑ i ∈ Finset.range (n + 1),
      ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 g s) s x‖ =
    ∑ i ∈ Finset.range (n + 1),
      ↑(n.choose i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n + 1 - i) g s x‖
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
A : Type u_2
inst✝¹⁰ : Group G
inst✝⁹ : AddCommGroup A
inst✝⁸ : DistribMulAction G A
inst✝⁷ : MeasurableSpace A
inst✝⁶ : TopologicalSpace A
inst✝⁵ : BorelSpace A
inst✝⁴ : IsTopologicalAddGroup A
inst✝³ : LocallyCompactSpace A
inst✝² : ContinuousConstSMul G A
μ ν : Measure A
inst✝¹ : μ.IsAddHaarMeasure
inst✝ : ν.IsAddHaarMeasure
g : Gᵈᵐᵃ
f : A → ℝ
f_cont : Continuous[inst✝⁶, _] f
f_comp : HasCompactSupport ⇑{ toFun := f, continuous_toFun := f_cont }
f_nonneg : 0 ≤ { toFun := f, continuous_toFun := f_cont }
f_zero : { toFun := f, continuous_toFun := f_cont } 0 ≠ 0
int_f_ne_zero : ∫ (x : A), f x ∂g • ν ≠ 0
⊢ Continuous[inst✝⁶, _] fun x ↦ f ((DomMulAct.mk.symm g)⁻¹ • x)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.RestrictScalars

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NontriviallyNormedField 𝕜'
inst✝⁸ : NormedAlgebra 𝕜 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
inst✝⁵ : NormedSpace 𝕜' E
inst✝⁴ : IsScalarTower 𝕜 𝕜' E
F : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedSpace 𝕜' F
inst✝ : IsScalarTower 𝕜 𝕜' F
x : E
n : ℕ
s : Set E
φ : E → E [×n]→L[𝕜'] F
h : DifferentiableWithinAt 𝕜' φ s x
hs : UniqueDiffWithinAt 𝕜 s x
⊢ DifferentiableAt 𝕜 (⇑(restrictScalarsLinear 𝕜)) (φ x)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝⁴ : PseudoMetricSpace X
inst✝³ : SeminormedAddCommGroup Y
C r : ℝ≥0
f : X → Y
α : Type u_4
inst✝² : SeminormedRing α
inst✝¹ : Module α Y
inst✝ : NormSMulClass α Y
a : α
ha : ‖a‖₊ ≠ 0
⊢ HolderWith (C * ‖a‖₊) r (a • f) ↔ HolderWith C r f

Tactic:
simp_rw [HolderWith, ENNReal.coe_mul, Pi.smul_apply, edist_smul₀, ENNReal.smul_def, smul_eq_mul,
    mul_comm (C : ℝ≥0∞), mul_assoc,
    ENNReal.mul_le_mul_iff_right (ENNReal.coe_ne_zero.mpr ha) ENNReal.coe_ne_top, mul_comm]

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiffHolder.Pointwise

Goal state before:
pp: E : Type u_1
F : Type u_2
G : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
k : ℕ
α : ↑I
a : E
f : E → F →L[ℝ] G
g : E → F
hf : ContDiffPointwiseHolderAt k α f a
hg : ContDiffPointwiseHolderAt k α g a
⊢ DifferentiableAt ℝ (fun x ↦ x.1 x.2) (f a, g a)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞ω
hf : ContDiff 𝕜 N f
hg : ContDiff 𝕜 N g
x : D
n : ℕ
hn : ↑n ≤ N
⊢ ‖iteratedFDeriv 𝕜 n (fun y ↦ (B (f y)) (g y)) x‖ ≤
    ‖B‖ * ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖

Tactic:
← iteratedFDerivWithin_univ

Goal state after:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞ω
hf : ContDiff 𝕜 N f
hg : ContDiff 𝕜 N g
x : D
n : ℕ
hn : ↑n ≤ N
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (B (f y)) (g y)) univ x‖ ≤
    ‖B‖ *
      ∑ x_1 ∈ Finset.range (n + 1),
        ↑(n.choose x_1) * ‖iteratedFDerivWithin 𝕜 x_1 f univ x‖ * ‖iteratedFDerivWithin 𝕜 (n - x_1) g univ x‖
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞ω
hf : ContDiff 𝕜 N f
hg : ContDiff 𝕜 N g
x : D
n : ℕ
hn : ↑n ≤ N
hB : ‖B‖ ≤ 1
⊢ ‖iteratedFDeriv 𝕜 n (fun y ↦ (B (f y)) (g y)) x‖ ≤
    ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖

Tactic:
← iteratedFDerivWithin_univ

Goal state after:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞ω
hf : ContDiff 𝕜 N f
hg : ContDiff 𝕜 N g
x : D
n : ℕ
hn : ↑n ≤ N
hB : ‖B‖ ≤ 1
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y ↦ (B (f y)) (g y)) univ x‖ ≤
    ∑ x_1 ∈ Finset.range (n + 1),
      ↑(n.choose x_1) * ‖iteratedFDerivWithin 𝕜 x_1 f univ x‖ * ‖iteratedFDerivWithin 𝕜 (n - x_1) g univ x‖
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
A : Type u_3
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
f g : E → A
N : ℕ∞ω
hf : ContDiff 𝕜 N f
hg : ContDiff 𝕜 N g
x : E
n : ℕ
hn : ↑n ≤ N
⊢ ‖iteratedFDeriv 𝕜 n (fun y ↦ f y * g y) x‖ ≤
    ∑ i ∈ Finset.range (n + 1), ↑(n.choose i) * ‖iteratedFDeriv 𝕜 i f x‖ * ‖iteratedFDeriv 𝕜 (n - i) g x‖

Tactic:
← iteratedFDerivWithin_univ

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type uE
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
A : Type u_3
inst✝¹ : NormedRing A
inst✝ : NormedAlgebra 𝕜 A
f g : E → A
N : ℕ∞ω
hf : ContDiff 𝕜 N f
hg : ContDiff 𝕜 N g
x : E
n : ℕ
hn : ↑n ≤ N
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y ↦ f y * g y) univ x‖ ≤
    ∑ x_1 ∈ Finset.range (n + 1),
      ↑(n.choose x_1) * ‖iteratedFDerivWithin 𝕜 x_1 f univ x‖ * ‖iteratedFDerivWithin 𝕜 (n - x_1) g univ x‖
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_2
f : ℝ → ℝ
a b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ MonotoneOn f (Icc b c)

Tactic:
apply monotoneOn_of_deriv_nonneg (convex_Icc b c) (continuousOn_Icc hb hc hd₁)

Goal state after:
case refine_2.hf'
f : ℝ → ℝ
a b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ DifferentiableOn ℝ f (interior (Icc b c))

case refine_2.hf'_nonneg
f : ℝ → ℝ
a b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ ∀ x ∈ interior (Icc b c), 0 ≤ deriv f x
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_2
f : ℝ → ℝ
a b c : ℝ
ha : ContinuousAt f a
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ MonotoneOn f (Icc b c)

Tactic:
apply monotoneOn_of_deriv_nonneg (convex_Icc b c) (continuousOn_Icc hb hc hd₁)

Goal state after:
case refine_2.hf'
f : ℝ → ℝ
a b c : ℝ
ha : ContinuousAt f a
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ DifferentiableOn ℝ f (interior (Icc b c))

case refine_2.hf'_nonneg
f : ℝ → ℝ
a b c : ℝ
ha : ContinuousAt f a
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ ∀ x ∈ interior (Icc b c), 0 ≤ deriv f x
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n : ℕ
IH :
  ∀ m ≤ n,
    ∀ {Gu : Type u} [inst : NormedAddCommGroup Gu] [inst_1 : NormedSpace 𝕜 Gu] {g : Fu → Gu},
      ContDiffOn 𝕜 (↑m) g t →
        ContDiffOn 𝕜 (↑m) f s →
          (∀ i ≤ m, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) →
            (∀ (i : ℕ), 1 ≤ i → i ≤ m → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) →
              ‖iteratedFDerivWithin 𝕜 m (g ∘ f) s x‖ ≤ ↑m ! * C * D ^ m
Gu : Type u
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
g : Fu → Gu
hg : ContDiffOn 𝕜 (↑(n + 1)) g t
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hC : ∀ i ≤ n + 1, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n + 1 → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i
M : ↑n < ↑n.succ
Cnonneg : 0 ≤ C
Dnonneg : 0 ≤ D
I : ∀ i ∈ Finset.range (n + 1), ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 g t ∘ f) s x‖ ≤ ↑i ! * C * D ^ i
J : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 (g ∘ f) s y) s x‖ =
    ‖iteratedFDerivWithin 𝕜 n
        (fun y ↦ ((ContinuousLinearMap.compL 𝕜 E Fu Gu) (fderivWithin 𝕜 g t (f y))) (fderivWithin 𝕜 f s y)) s x‖

Tactic:
congr 1

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n : ℕ
IH :
  ∀ m ≤ n,
    ∀ {Gu : Type u} [inst : NormedAddCommGroup Gu] [inst_1 : NormedSpace 𝕜 Gu] {g : Fu → Gu},
      ContDiffOn 𝕜 (↑m) g t →
        ContDiffOn 𝕜 (↑m) f s →
          (∀ i ≤ m, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) →
            (∀ (i : ℕ), 1 ≤ i → i ≤ m → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) →
              ‖iteratedFDerivWithin 𝕜 m (g ∘ f) s x‖ ≤ ↑m ! * C * D ^ m
Gu : Type u
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
g : Fu → Gu
hg : ContDiffOn 𝕜 (↑(n + 1)) g t
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hC : ∀ i ≤ n + 1, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n + 1 → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i
M : ↑n < ↑n.succ
Cnonneg : 0 ≤ C
Dnonneg : 0 ≤ D
I : ∀ i ∈ Finset.range (n + 1), ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 g t ∘ f) s x‖ ≤ ↑i ! * C * D ^ i
J : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)
⊢ iteratedFDerivWithin 𝕜 n (fun y ↦ fderivWithin 𝕜 (g ∘ f) s y) s x =
    iteratedFDerivWithin 𝕜 n
      (fun y ↦ ((ContinuousLinearMap.compL 𝕜 E Fu Gu) (fderivWithin 𝕜 g t (f y))) (fderivWithin 𝕜 f s y)) s x
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_2
f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ MonotoneOn f (Icc b c)

Tactic:
apply monotoneOn_of_deriv_nonneg (convex_Icc b c) (continuousOn_Icc hb hc hd₁)

Goal state after:
case refine_2.hf'
f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ DifferentiableOn ℝ f (interior (Icc b c))

case refine_2.hf'_nonneg
f : ℝ → ℝ
b c : ℝ
hb : ContinuousAt f b
hc : ContinuousAt f c
hd₀ : DifferentiableOn ℝ f (Iio b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Iio b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioo b c, 0 ≤ deriv f x
⊢ ∀ x ∈ interior (Icc b c), 0 ≤ deriv f x
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: case a.e'_6
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n : ℕ
IH :
  ∀ m ≤ n,
    ∀ {Gu : Type u} [inst : NormedAddCommGroup Gu] [inst_1 : NormedSpace 𝕜 Gu] {g : Fu → Gu},
      ContDiffOn 𝕜 (↑m) g t →
        ContDiffOn 𝕜 (↑m) f s →
          (∀ i ≤ m, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) →
            (∀ (i : ℕ), 1 ≤ i → i ≤ m → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) →
              ‖iteratedFDerivWithin 𝕜 m (g ∘ f) s x‖ ≤ ↑m ! * C * D ^ m
Gu : Type u
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
g : Fu → Gu
hg : ContDiffOn 𝕜 (↑(n + 1)) g t
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hC : ∀ i ≤ n + 1, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n + 1 → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i
M : ↑n < ↑n.succ
Cnonneg : 0 ≤ C
Dnonneg : 0 ≤ D
I : ∀ i ∈ Finset.range (n + 1), ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 g t ∘ f) s x‖ ≤ ↑i ! * C * D ^ i
J : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ D ^ (i + (n - i + 1)) = D ^ (n + 1)

Tactic:
congr 1

Goal state after:
case a.e'_6.e_a
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
Fu : Type u
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
f : E → Fu
s : Set E
t : Set Fu
x : E
ht : UniqueDiffOn 𝕜 t
hs : UniqueDiffOn 𝕜 s
hst : MapsTo f s t
hx : x ∈ s
C D : ℝ
n : ℕ
IH :
  ∀ m ≤ n,
    ∀ {Gu : Type u} [inst : NormedAddCommGroup Gu] [inst_1 : NormedSpace 𝕜 Gu] {g : Fu → Gu},
      ContDiffOn 𝕜 (↑m) g t →
        ContDiffOn 𝕜 (↑m) f s →
          (∀ i ≤ m, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C) →
            (∀ (i : ℕ), 1 ≤ i → i ≤ m → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i) →
              ‖iteratedFDerivWithin 𝕜 m (g ∘ f) s x‖ ≤ ↑m ! * C * D ^ m
Gu : Type u
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
g : Fu → Gu
hg : ContDiffOn 𝕜 (↑(n + 1)) g t
hf : ContDiffOn 𝕜 (↑(n + 1)) f s
hC : ∀ i ≤ n + 1, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n + 1 → ‖iteratedFDerivWithin 𝕜 i f s x‖ ≤ D ^ i
M : ↑n < ↑n.succ
Cnonneg : 0 ≤ C
Dnonneg : 0 ≤ D
I : ∀ i ∈ Finset.range (n + 1), ‖iteratedFDerivWithin 𝕜 i (fderivWithin 𝕜 g t ∘ f) s x‖ ≤ ↑i ! * C * D ^ i
J : ∀ (i : ℕ), ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)
i : ℕ
hi : i ∈ Finset.range (n + 1)
⊢ i + (n - i + 1) = n + 1
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
x : E
s : Set E
h : s ∈ 𝓝 x
⊢ fderivWithin 𝕜 (fderivWithin 𝕜 f s) s x = fderiv 𝕜 (fderiv 𝕜 f) x

Tactic:
simp only [← fderivWithin_univ]
  apply fderivWithin_fderivWithin_eq_of_eventuallyEq
  simp [h]

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
⊢ (fun h ↦
      f (x + h • v + h • w) - f (x + h • v) - h • (f' x) w - h ^ 2 • (f'' v) w - (h ^ 2 / 2) • (f'' w) w) =o[𝓝[>] 0]
    fun h ↦ h ^ 2

Tactic:
refine IsLittleO.trans_isBigO
    (isLittleO_iff.2 fun ε εpos => ?_) (isBigO_const_mul_self ((‖v‖ + ‖w‖) * ‖w‖) _ _)

Goal state after:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : HasFDerivWithinAt f' f'' (interior s) x
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
ε : ℝ
εpos : 0 < ε
⊢ ∀ᶠ (x_1 : ℝ) in 𝓝[>] 0,
    ‖f (x + x_1 • v + x_1 • w) - f (x + x_1 • v) - x_1 • (f' x) w - x_1 ^ 2 • (f'' v) w - (x_1 ^ 2 / 2) • (f'' w) w‖ ≤
      ε * ‖(‖v‖ + ‖w‖) * ‖w‖ * x_1 ^ 2‖
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
n : ℕ
N : ℕ∞ω
t : Set F
ht : range f ⊆ t
ht' : UniqueDiffOn 𝕜 t
hg : ContDiffOn 𝕜 N g t
hf : ContDiff 𝕜 N f
hn : ↑n ≤ N
x : E
C D : ℝ
hC : ∀ i ≤ n, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDeriv 𝕜 i f x‖ ≤ D ^ i
⊢ ‖iteratedFDeriv 𝕜 n (g ∘ f) x‖ ≤ ↑n ! * C * D ^ n

Tactic:
← iteratedFDerivWithin_univ

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
n : ℕ
N : ℕ∞ω
t : Set F
ht : range f ⊆ t
ht' : UniqueDiffOn 𝕜 t
hg : ContDiffOn 𝕜 N g t
hf : ContDiff 𝕜 N f
hn : ↑n ≤ N
x : E
C D : ℝ
hC : ∀ i ≤ n, ‖iteratedFDerivWithin 𝕜 i g t (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDerivWithin 𝕜 i f univ x‖ ≤ D ^ i
⊢ ‖iteratedFDerivWithin 𝕜 n (g ∘ f) univ x‖ ≤ ↑n ! * C * D ^ n
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
n : ℕ
N : ℕ∞ω
hg : ContDiff 𝕜 N g
hf : ContDiff 𝕜 N f
hn : ↑n ≤ N
x : E
C D : ℝ
hC : ∀ i ≤ n, ‖iteratedFDeriv 𝕜 i g (f x)‖ ≤ C
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDeriv 𝕜 i f x‖ ≤ D ^ i
⊢ ‖iteratedFDeriv 𝕜 n (g ∘ f) x‖ ≤ ↑n ! * C * D ^ n

Tactic:
← iteratedFDerivWithin_univ

Goal state after:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
g : F → G
f : E → F
n : ℕ
N : ℕ∞ω
hg : ContDiff 𝕜 N g
hf : ContDiff 𝕜 N f
hn : ↑n ≤ N
x : E
C D : ℝ
hD : ∀ (i : ℕ), 1 ≤ i → i ≤ n → ‖iteratedFDeriv 𝕜 i f x‖ ≤ D ^ i
hC : ∀ i ≤ n, ‖iteratedFDerivWithin 𝕜 i g univ (f x)‖ ≤ C
⊢ ‖iteratedFDeriv 𝕜 n (g ∘ f) x‖ ≤ ↑n ! * C * D ^ n
Module: Mathlib.Analysis.Calculus.DifferentialForm.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
r : WithTop ℕ∞
ω : E → E [⋀^Fin n]→L[𝕜] F
s : Set E
x : E
hω : ContDiffWithinAt 𝕜 r ω s x
hr : minSmoothness 𝕜 2 ≤ r
hs : UniqueDiffOn 𝕜 s
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)
h'x : x ∈ s
⊢ alternatizeUncurryFin (fderivWithin 𝕜 (fun y ↦ alternatizeUncurryFin (fderivWithin 𝕜 ω s y)) s x) =
    alternatizeUncurryFin (alternatizeUncurryFinCLM 𝕜 E F ∘SL fderivWithin 𝕜 (fderivWithin 𝕜 ω s) s x)

Tactic:
congr 1

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
n : ℕ
r : WithTop ℕ∞
ω : E → E [⋀^Fin n]→L[𝕜] F
s : Set E
x : E
hω : ContDiffWithinAt 𝕜 r ω s x
hr : minSmoothness 𝕜 2 ≤ r
hs : UniqueDiffOn 𝕜 s
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (interior s)
h'x : x ∈ s
⊢ fderivWithin 𝕜 (fun y ↦ alternatizeUncurryFin (fderivWithin 𝕜 ω s y)) s x =
    alternatizeUncurryFinCLM 𝕜 E F ∘SL fderivWithin 𝕜 (fderivWithin 𝕜 ω s) s x
Module: Mathlib.Analysis.Calculus.VectorField

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
V W V₁ : E → E
x : E
hV : DifferentiableAt 𝕜 V x
hV₁ : DifferentiableAt 𝕜 V₁ x
⊢ (fderiv 𝕜 W x) (V x) + (fderiv 𝕜 W x) (V₁ x) - (fderiv 𝕜 (V + V₁) x) (W x) =
    (fderiv 𝕜 W x) (V x) - (fderiv 𝕜 V x) (W x) + ((fderiv 𝕜 W x) (V₁ x) - (fderiv 𝕜 V₁ x) (W x))

Tactic:
rw [fderiv_add hV hV₁, add_apply]

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
V W V₁ : E → E
x : E
hV : DifferentiableAt 𝕜 V x
hV₁ : DifferentiableAt 𝕜 V₁ x
⊢ (fderiv 𝕜 W x) (V x) + (fderiv 𝕜 W x) (V₁ x) - ((fderiv 𝕜 V x) (W x) + (fderiv 𝕜 V₁ x) (W x)) =
    (fderiv 𝕜 W x) (V x) - (fderiv 𝕜 V x) (W x) + ((fderiv 𝕜 W x) (V₁ x) - (fderiv 𝕜 V₁ x) (W x))