mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: case pos
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
μ' μ ν : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : ν.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
hG : LocallyCompactSpace G
g : G → ℝ
g_cont : Continuous[inst✝⁸, _] g
g_comp : HasCompactSupport ⇑{ toFun := g, continuous_toFun := g_cont }
g_nonneg : 0 ≤ { toFun := g, continuous_toFun := g_cont }
g_one : { toFun := g, continuous_toFun := g_cont } 1 ≠ 0
⊢ μ'.haarScalarFactor ν = μ'.haarScalarFactor μ * μ.haarScalarFactor ν

Tactic:
have Z := integral_isMulLeftInvariant_eq_smul_of_hasCompactSupport μ' μ g_cont g_comp

Goal state after:
case pos
G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
μ' μ ν : Measure G
inst✝³ : μ.IsHaarMeasure
inst✝² : ν.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
hG : LocallyCompactSpace G
g : G → ℝ
g_cont : Continuous[inst✝⁸, _] g
g_comp : HasCompactSupport ⇑{ toFun := g, continuous_toFun := g_cont }
g_nonneg : 0 ≤ { toFun := g, continuous_toFun := g_cont }
g_one : { toFun := g, continuous_toFun := g_cont } 1 ≠ 0
Z : ∫ (x : G), g x ∂μ' = ∫ (x : G), g x ∂μ'.haarScalarFactor μ • μ
⊢ μ'.haarScalarFactor ν = μ'.haarScalarFactor μ * μ.haarScalarFactor ν
Module: Mathlib.Analysis.Calculus.ContDiff.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
μ : Measure G
f : G → E
n : ℕ∞
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ univ)
⊢ ContDiffOn 𝕜 (↑n) (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)

Tactic:
let eG : Type max uG uE' uF uP := ULift.{max uE' uF uP} G

Goal state after:
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup P
inst✝ : NormedSpace 𝕜 P
μ : Measure G
f : G → E
n : ℕ∞
L : E →L[𝕜] E' →L[𝕜] F
g : P → G → E'
s : Set P
k : Set G
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ univ)
eG : Type (max uG uE' uF uP) := ULift.{max uE' uF uP, uG} G
⊢ ContDiffOn 𝕜 (↑n) (fun q ↦ (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ)
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: case refine_2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
D_lt_one : D < 1
hx : ‖x‖ < 1 + D
⊢ 0 < μ (support fun i ↦ w D i * φ (x - i))

Tactic:
· set z := (D / (1 + D)) • x with hz
    have B : 0 < 1 + D := by linarith
    have C : ball z (D * (1 + D - ‖x‖) / (1 + D)) ⊆ support fun y : E => w D y * φ (x - y) := by
      intro y hy
      simp only [support_mul, w_support E Dpos]
      simp only [φ, mem_inter_iff, mem_support, Ne, indicator_apply_eq_zero,
        mem_closedBall_zero_iff, one_ne_zero, not_forall, not_false_iff, exists_prop, and_true]
      constructor
      · apply ball_subset_ball' _ hy
        simp only [hz, norm_smul, abs_of_nonneg Dpos.le, abs_of_nonneg B.le, dist_zero_right,
          Real.norm_eq_abs, abs_div]
        field_simp
        linarith only
      · have ID : ‖D / (1 + D) - 1‖ = 1 / (1 + D) := by
          rw [Real.norm_of_nonpos]
          · field
          · field_simp
            linarith only
        rw [← mem_closedBall_iff_norm']
        apply closedBall_subset_closedBall' _ (ball_subset_closedBall hy)
        rw [← one_smul ℝ x, dist_eq_norm, hz, ← sub_smul, one_smul, norm_smul, ID]
        field_simp
        nlinarith only [hx, D_lt_one]
    apply lt_of_lt_of_le _ (measure_mono C)
    apply measure_ball_pos
    exact div_pos (mul_pos Dpos (by linarith only [hx])) B

Goal state after:
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: case refine_1
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
hs : IsOpen (Ioo 0 1)
hk : IsCompact (closedBall 0 1)
p : ℝ
x : E
hp : p ∈ Ioo 0 1
hx : u (p⁻¹ • x) ≠ 0
this : ‖x‖ < p
⊢ x ∈ closedBall 0 1

Tactic:
mem_closedBall_zero_iff

Goal state after:
case refine_1
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
hs : IsOpen (Ioo 0 1)
hk : IsCompact (closedBall 0 1)
p : ℝ
x : E
hp : p ∈ Ioo 0 1
hx : u (p⁻¹ • x) ≠ 0
this : ‖x‖ < p
⊢ ‖x‖ ≤ 1
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: case pos
E✝ : Type u_1
inst✝⁵ : NormedAddCommGroup E✝
inst✝⁴ : NormedSpace ℝ E✝
inst✝³ : FiniteDimensional ℝ E✝
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
this✝¹ : MeasurableSpace E := borel E
this✝ : BorelSpace E
IR : ∀ (R : ℝ), 1 < R → 0 < (R - 1) / (R + 1)
R : ℝ
x : E
h : 1 < R
⊢ 0 ≤ y ((R - 1) / (R + 1)) (((R + 1) / 2)⁻¹ • x) ∧ y ((R - 1) / (R + 1)) (((R + 1) / 2)⁻¹ • x) ≤ 1

Tactic:
refine ⟨y_nonneg _ _, y_le_one _ (IR R h)⟩

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

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹⁰ : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
μ : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹ : μ.IsAddLeftInvariant
inst✝ : μ.IsNegInvariant
n : ℕ∞
hcf : HasCompactSupport f
hf : ContDiff 𝕜 (↑n) f
hg : LocallyIntegrable g μ
⊢ ContDiff 𝕜 (↑n) (g ⋆[L.flip, μ] f)

Tactic:
exact hcf.contDiff_convolution_right L.flip hg hf

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.ConformalLinearMap

Goal state before:
pp: case mpr
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ E
inst✝ : InnerProductSpace ℝ F
f : E →L[ℝ] F
c : ℝ
hc : 0 < c
hc₁ : 0 < c * c
huv : ∀ (u v : E), ⟪f u, f v⟫ = c * c * ⟪u, v⟫
⊢ IsConformalMap f

Tactic:
refine ⟨c, hc.ne', (c⁻¹ • f : E →ₗ[ℝ] F).isometryOfInner fun u v => ?_, ?_⟩

Goal state after:
case mpr.refine_1
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ E
inst✝ : InnerProductSpace ℝ F
f : E →L[ℝ] F
c : ℝ
hc : 0 < c
hc₁ : 0 < c * c
huv : ∀ (u v : E), ⟪f u, f v⟫ = c * c * ⟪u, v⟫
u v : E
⊢ ⟪(c⁻¹ • ↑f) u, (c⁻¹ • ↑f) v⟫ = ⟪u, v⟫

case mpr.refine_2
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedAddCommGroup F
inst✝¹ : InnerProductSpace ℝ E
inst✝ : InnerProductSpace ℝ F
f : E →L[ℝ] F
c : ℝ
hc : 0 < c
hc₁ : 0 < c * c
huv : ∀ (u v : E), ⟪f u, f v⟫ = c * c * ⟪u, v⟫
⊢ f = c • ((c⁻¹ • ↑f).isometryOfInner ⋯).toContinuousLinearMap
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure G := μ'.haarScalarFactor μ • μ
k : Set G
k_comp : IsCompact k
k_closed : IsClosed[inst✝⁷] k
k_mem : k ∈ 𝓝 1
one_k : 1 ∈ k
A : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x ↦ x • k}
c : Set (Set G)
⊢ c ⊆ A → IsChain (fun x1 x2 ↦ x1 ⊆ x2) c → ∃ ub ∈ A, ∀ s ∈ c, s ⊆ ub

Tactic:
cA

Goal state after:
G : Type u_1
inst✝⁷ : TopologicalSpace G
inst✝⁶ : Group G
inst✝⁵ : IsTopologicalGroup G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
inst✝² : LocallyCompactSpace G
μ' μ : Measure G
inst✝¹ : μ.IsHaarMeasure
inst✝ : μ'.IsHaarMeasure
s : Set G
hs : MeasurableSet s
h's : μ.IsEverywherePos s
ν : Measure G := μ'.haarScalarFactor μ • μ
k : Set G
k_comp : IsCompact k
k_closed : IsClosed[inst✝⁷] k
k_mem : k ∈ 𝓝 1
one_k : 1 ∈ k
A : Set (Set G) := {t | t ⊆ s ∧ t.PairwiseDisjoint fun x ↦ x • k}
c : Set (Set G)
cA : c ⊆ A
⊢ IsChain (fun x1 x2 ↦ x1 ⊆ x2) c → ∃ ub ∈ A, ∀ s ∈ c, s ⊆ ub
Module: Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
i j : Fin N.succ
inej : i ≠ j
hi : ‖a.c i‖ ≤ 2
hj : 2 < ‖a.c j‖
ah : Pairwise fun i j ↦ a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i ∨ a.r j ≤ ‖a.c j - a.c i‖ ∧ a.r i ≤ τ * a.r j
δnonneg : 0 ≤ δ
D : 0 ≤ 1 - δ / 4
hcrj : ‖a.c j‖ ≤ a.r j + 1
I : a.r i ≤ 2
J : (1 - δ / 4) * τ ≤ 1
H : a.r i ≤ ‖a.c i - a.c j‖ ∧ a.r j ≤ τ * a.r i
C : a.r j ≤ 4
⊢ a.r j - δ ≤ a.r j - a.r j / 4 * δ

Tactic:
gcongr _ - ?_
        exact mul_le_of_le_one_left δnonneg (by linarith only [C])

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

Goal state before:
pp: case hbc
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
I1 :
  ‖iteratedFDerivWithin 𝕜 n (fun y ↦ ((precompR Du B) (f y)) (fderivWithin 𝕜 g s y)) 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‖
⊢ ‖precompL Du B‖ ≤ ‖B‖

Tactic:
exact B.norm_precompL_le Du

Goal state after:
Module: Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace

Goal state before:
pp: case inr.inl
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
c' : Fin N.succ → E := fun i ↦ if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
norm_c'_le : ∀ (i : Fin N.succ), ‖c' i‖ ≤ 2
i j : Fin N.succ
inej : i ≠ j
hij : ‖a.c i‖ ≤ ‖a.c j‖
Hj : 2 < ‖a.c j‖
H'j : ‖a.c j‖ ≤ 2 ↔ False
Hi : ‖a.c i‖ ≤ 2
⊢ 1 - δ ≤ ‖c' i - c' j‖

pp: case inr.inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
c' : Fin N.succ → E := fun i ↦ if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
norm_c'_le : ∀ (i : Fin N.succ), ‖c' i‖ ≤ 2
i j : Fin N.succ
inej : i ≠ j
hij : ‖a.c i‖ ≤ ‖a.c j‖
Hj : 2 < ‖a.c j‖
H'j : ‖a.c j‖ ≤ 2 ↔ False
Hi : 2 < ‖a.c i‖
⊢ 1 - δ ≤ ‖c' i - c' j‖

Tactic:
· -- case `‖c i‖ ≤ 2`
      simp_rw [c', Hi, if_true, H'j, if_false]
      exact exists_normalized_aux2 a lastc lastr hτ δ hδ1 hδ2 i j inej Hi Hj

Goal state after:
case inr.inr
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
N : ℕ
τ : ℝ
a : SatelliteConfig E N τ
lastc : a.c (last N) = 0
lastr : a.r (last N) = 1
hτ : 1 ≤ τ
δ : ℝ
hδ1 : τ ≤ 1 + δ / 4
hδ2 : δ ≤ 1
c' : Fin N.succ → E := fun i ↦ if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i
norm_c'_le : ∀ (i : Fin N.succ), ‖c' i‖ ≤ 2
i j : Fin N.succ
inej : i ≠ j
hij : ‖a.c i‖ ≤ ‖a.c j‖
Hj : 2 < ‖a.c j‖
H'j : ‖a.c j‖ ≤ 2 ↔ False
Hi : 2 < ‖a.c i‖
⊢ 1 - δ ≤ ‖c' i - c' j‖
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:
HolderWith,

Goal state after:
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
⊢ (∀ (x y : X), edist ((a • f) x) ((a • f) y) ≤ ↑(C * ‖a‖₊) * edist x y ^ ↑r) ↔
    ∀ (x y : X), edist (f x) (f y) ≤ ↑C * edist x y ^ ↑r
Module: Mathlib.Analysis.Calculus.Darboux

Goal state before:
pp: f f' : ℝ → ℝ
s : Set ℝ
hs : Convex ℝ s
hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x
m b : ℝ
hb : b ∈ s
hmb : m ≤ f' b
a : ℝ
ha : a ∈ s
hma : f' a ≤ m
⊢ ∃ x ∈ s, f' x = m

Tactic:
exact (hs.ordConnected.image_hasDerivWithinAt hf).out (mem_image_of_mem f' ha)
    (mem_image_of_mem f' hb) ⟨hma, hmb⟩

Goal state after:
Module: Mathlib.Analysis.Calculus.Deriv.Star

Goal state before:
pp: case neg
𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : StarRing 𝕜
F : Type v
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 F
inst✝³ : StarAddMonoid F
inst✝² : StarModule 𝕜 F
inst✝¹ : ContinuousStar F
inst✝ : NormedStarGroup 𝕜
f : 𝕜 → F
z : 𝕜
hf : ¬DifferentiableAt 𝕜 f ((starRingEnd 𝕜) z)
⊢ deriv (star ∘ f ∘ ⇑(starRingEnd 𝕜)) z = (star ∘ deriv f ∘ ⇑(starRingEnd 𝕜)) z

Tactic:
· have := differentiableAt_star_conj_iff.not.2 hf
    simp_all [deriv_zero_of_not_differentiableAt]

Goal state after:
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: f : ℝ → ℝ
a b : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0
⊢ IsMaxOn f (Ioi a) b

Tactic:
refine isMaxOn_Ioi_of_mono_anti ?_ ?_

Goal state after:
case refine_1
f : ℝ → ℝ
a b : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0
⊢ MonotoneOn f (Ioc a b)

case refine_2
f : ℝ → ℝ
a b : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioi b, deriv f x ≤ 0
⊢ AntitoneOn f (Ici b)
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_1
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
⊢ AntitoneOn f (Ioc a b)

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 antitoneOn_of_deriv_nonpos (convex_Ioc a b) (continuousOn_Ioc hb hd₀) <;> simp_all

Goal state after:
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)
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_1
f : ℝ → ℝ
a b : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x
⊢ AntitoneOn f (Ioc a b)

pp: case refine_2
f : ℝ → ℝ
a b : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x
⊢ MonotoneOn f (Ici b)

Tactic:
· apply antitoneOn_of_deriv_nonpos (convex_Ioc a b) (continuousOn_Ioc hb hd₀) <;> simp_all

Goal state after:
case refine_2
f : ℝ → ℝ
a b : ℝ
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioi b)
h₀ : ∀ x ∈ Ioo a b, deriv f x ≤ 0
h₁ : ∀ x ∈ Ioi b, 0 ≤ deriv f x
⊢ MonotoneOn f (Ici b)
Module: Mathlib.Analysis.Calculus.ContDiff.Bounds

Goal state before:
pp: case h₂
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)
⊢ ‖iteratedFDerivWithin 𝕜 (n - i) (fderivWithin 𝕜 f s) s x‖ ≤ D ^ (n - i + 1)

Tactic:
exact J i

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 : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + (v + w) ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 ↦ ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
E1 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ
E2 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h < 1
h : ℝ
hδ : h * (‖v‖ + ‖w‖) < δ
h_lt_1 : h < 1
hpos : 0 < h
t : ℝ
ht : t ∈ Icc 0 1
this : x + h • v ∈ interior s
⊢ x + h • v + h • w = x + h • (v + w)

Tactic:
module

Goal state after:
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 : E → E
x : E
f : E → 𝕜
hf : DifferentiableAt 𝕜 f x
hV : DifferentiableAt 𝕜 V x
⊢ lieBracket 𝕜 (fun y ↦ f y • V y) W x = -(fderiv 𝕜 f x) (W x) • V x + f x • lieBracket 𝕜 V W x

Tactic:
rw [lieBracket_swap, lieBracket_smul_right hf hV, lieBracket_swap, add_comm]

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 : E → E
x : E
f : E → 𝕜
hf : DifferentiableAt 𝕜 f x
hV : DifferentiableAt 𝕜 V x
⊢ -(f x • -lieBracket 𝕜 V W x + (fderiv 𝕜 f x) (W x) • V x) = -(fderiv 𝕜 f x) (W x) • V x + f x • lieBracket 𝕜 V W x
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 : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 ↦ ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
E1 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ
E2 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h < 1
h : ℝ
hδ : h * (‖v‖ + ‖w‖) < δ
h_lt_1 : h < 1
hpos : 0 < h
xt_mem : ∀ t ∈ Icc 0 1, x + h • v + (t * h) • w ∈ interior s
g : ℝ → F :=
  fun t ↦ f (x + h • v + (t * h) • w) - (t * h) • (f' x) w - (t * h ^ 2) • (f'' v) w - ((t * h) ^ 2 / 2) • (f'' w) w
g' : ℝ → F :=
  fun t ↦ (f' (x + h • v + (t * h) • w)) (h • w) - h • (f' x) w - h ^ 2 • (f'' v) w - (t * h ^ 2) • (f'' w) w
hg' : g' = fun t ↦ (f' (x + h • v + (t * h) • w)) (h • w) - h • (f' x) w - h ^ 2 • (f'' v) w - (t * h ^ 2) • (f'' w) w
g_deriv : ∀ t ∈ Icc 0 1, HasDerivWithinAt g (g' t) (Icc 0 1) t
t : ℝ
ht : t ∈ Ico 0 1
I : ‖h • v + (t * h) • w‖ ≤ h * (‖v‖ + ‖w‖)
⊢ h • (f' (x + h • v + (t * h) • w)) w - h • (f' x) w - h ^ 2 • (f'' v) w - (t * h ^ 2) • (f'' w) w =
    h • ((f' (x + h • v + (t * h) • w)) w - (f' x) w - (h • (f'' v) w + (t * h) • (f'' w) w))

Tactic:
module

Goal state after:
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 : E → E
s : Set E
x : E
m n : ℕ∞ω
hV : ContDiffWithinAt 𝕜 n V s x
hW : ContDiffWithinAt 𝕜 n W s x
hs : UniqueDiffOn 𝕜 s
hmn : m + 1 ≤ n
hx : x ∈ s
⊢ ContDiffWithinAt 𝕜 m (lieBracketWithin 𝕜 V W s) s x

Tactic:
apply ContDiffWithinAt.sub
  · exact ContDiffWithinAt.clm_apply (hW.fderivWithin_right hs hmn hx)
      (hV.of_le (le_trans le_self_add hmn))
  · exact ContDiffWithinAt.clm_apply (hV.fderivWithin_right hs hmn hx)
      (hW.of_le (le_trans le_self_add hmn))

Goal state after: