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

Goal state before:
pp: case pos.e_f
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}
f : ℕ → G
m_max : Maximal (fun x ↦ x ∈ A) (range f)
hms : range f ⊆ s
hdj : (range f).PairwiseDisjoint fun x ↦ x • k
hm : (range f).Nonempty
h'm : (range f).Countable
M : ∀ (i : ℕ), MeasurableSet (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) i)
sm : s ⊆ ⋃ y, f y • (k * k⁻¹)
s_eq : s = ⋃ n, s ∩ f n • (k * k⁻¹)
I : μ' s = ∑' (n : ℕ), μ' (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) n)
J : ν s = ∑' (n : ℕ), ν (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) n)
n : ℕ
⊢ μ' (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) n) = ν (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) n)

Tactic:
apply measure_isMulInvariant_eq_smul_of_isCompact_closure

Goal state after:
case pos.e_f
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 := ⋯
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) := ⋯
f : ℕ → G
m_max : Maximal (fun x ↦ x ∈ A) (range f)
hms : range f ⊆ s
hdj : (range f).PairwiseDisjoint fun x ↦ x • k
hm : (range f).Nonempty
h'm : (range f).Countable
M : ∀ (i : ℕ), MeasurableSet (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) i)
sm : s ⊆ ⋃ y, f y • (k * k⁻¹)
s_eq : s = ⋃ n, s ∩ f n • (k * k⁻¹)
I : μ' s = ∑' (n : ℕ), μ' (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) n)
J : ν s = ∑' (n : ℕ), ν (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) n)
n : ℕ
⊢ IsCompact (closure[inst✝⁷] (disjointed (fun n ↦ s ∩ f n • (k * k⁻¹)) n))
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}
m : Set G
m_max : Maximal (fun x ↦ x ∈ A) m
hms : m ⊆ s
hdj : m.PairwiseDisjoint fun x ↦ x • k
sm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)
hm : m.Nonempty
ρ : Measure G
hρ : ρ.IsEverywherePos s
M : ∀ (i : ↑m), MeasurableSet (s ∩ ↑i • k)
h'm : ρ s ≠ ∞
this : ∑' (x : ↑m), ρ (s ∩ ↑x • k) < ∞
⊢ m.Countable

Tactic:
have C : Set.Countable (support fun (i : m) ↦ ρ (s ∩ (i : G) • k)) :=
        Summable.countable_support_ennreal this.ne

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}
m : Set G
m_max : Maximal (fun x ↦ x ∈ A) m
hms : m ⊆ s
hdj : m.PairwiseDisjoint fun x ↦ x • k
sm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)
hm : m.Nonempty
ρ : Measure G
hρ : ρ.IsEverywherePos s
M : ∀ (i : ↑m), MeasurableSet (s ∩ ↑i • k)
h'm : ρ s ≠ ∞
this : ∑' (x : ↑m), ρ (s ∩ ↑x • k) < ∞
C : (support fun i ↦ ρ (s ∩ ↑i • k)).Countable
⊢ m.Countable
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}
m : Set G
m_max : Maximal (fun x ↦ x ∈ A) m
hms : m ⊆ s
hdj : m.PairwiseDisjoint fun x ↦ x • k
sm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)
hm : m.Nonempty
h'm : ¬m.Countable
H : ∀ (ρ : Measure G), ρ.IsEverywherePos s → ρ s = ∞
Hν : ν.IsEverywherePos s
x : G
_hx : x ∈ s
t : Set G
t_comp : IsCompact t
t_mem : t ∈ 𝓝 x
u : Set G
hu : u ⊆ t
⊢ μ' u = ν u

Tactic:
apply measure_isMulInvariant_eq_smul_of_isCompact_closure

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 := ⋯
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) := ⋯
m : Set G
m_max : Maximal (fun x ↦ x ∈ A) m
hms : m ⊆ s
hdj : m.PairwiseDisjoint fun x ↦ x • k
sm : s ⊆ ⋃ x ∈ m, x • (k * k⁻¹)
hm : m.Nonempty
h'm : ¬m.Countable
H : ∀ (ρ : Measure G), ρ.IsEverywherePos s → ρ s = ∞
Hν : ν.IsEverywherePos s
x : G
_hx : x ∈ s
t : Set G
t_comp : IsCompact t
t_mem : t ∈ 𝓝 x
u : Set G
hu : u ⊆ t
⊢ IsCompact (closure[inst✝⁷] u)
Module: Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace

Goal state before:
pp: case pos
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
i : Fin N.succ
h : ¬‖a.c i‖ ≤ 2
hi : ‖a.c i‖ = 0
⊢ ‖(2 / ‖a.c i‖) • a.c i‖ ≤ 2

pp: case neg
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
i : Fin N.succ
h : ¬‖a.c i‖ ≤ 2
hi : ¬‖a.c i‖ = 0
⊢ ‖(2 / ‖a.c i‖) • a.c i‖ ≤ 2

Tactic:
<;>

Goal state after:
case pos
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
i : Fin N.succ
h : ¬‖a.c i‖ ≤ 2
hi : ‖a.c i‖ = 0
⊢ ‖(2 / ‖a.c i‖) • a.c i‖ ≤ 2

case neg
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
i : Fin N.succ
h : ¬‖a.c i‖ ≤ 2
hi : ¬‖a.c i‖ = 0
⊢ ‖(2 / ‖a.c i‖) • a.c i‖ ≤ 2
Module: Mathlib.Topology.MetricSpace.Holder

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : PseudoMetricSpace X
inst✝ : MetricSpace Y
r : ℝ≥0
f : X → Y
⊢ HolderWith 0 r f ↔ ∀ (x₁ x₂ : X), f x₁ = f x₂

Tactic:
by
  refine ⟨fun h x₁ x₂ => ?_, fun h x₁ x₂ => h x₁ x₂ ▸ ?_⟩
  · specialize h x₁ x₂
    simp [ENNReal.coe_zero, zero_mul, nonpos_iff_eq_zero, edist_eq_zero] at h
    assumption
  · simp only [edist_self, ENNReal.coe_zero, zero_mul, le_refl]

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

Goal state before:
pp: case inl
x : ℝ
hx✝ : x ≠ 0
hx : x < 0
⊢ HasStrictDerivAt (fun x ↦ |x|) (↑(SignType.sign x)) x

pp: case inr
x : ℝ
hx✝ : x ≠ 0
hx : 0 < x
⊢ HasStrictDerivAt (fun x ↦ |x|) (↑(SignType.sign x)) x

Tactic:
· simpa [hx] using hasStrictDerivAt_abs_neg hx

Goal state after:
case inr
x : ℝ
hx✝ : x ≠ 0
hx : 0 < x
⊢ HasStrictDerivAt (fun x ↦ |x|) (↑(SignType.sign x)) x
Module: Mathlib.Analysis.Calculus.Deriv.Star

Goal state before:
pp: case e'_9
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
x : 𝕜
f' : F
hf : HasDerivAt f (star f') ((starRingEnd 𝕜) x)
⊢ f' = star (star f')

pp: case e'_10
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
x : 𝕜
f' : F
hf : HasDerivAt f (star f') ((starRingEnd 𝕜) x)
⊢ x = (starRingEnd 𝕜) ((starRingEnd 𝕜) x)

Tactic:
<;>

Goal state after:
case e'_9
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
x : 𝕜
f' : F
hf : HasDerivAt f (star f') ((starRingEnd 𝕜) x)
⊢ f' = star (star f')

case e'_10
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
x : 𝕜
f' : F
hf : HasDerivAt f (star f') ((starRingEnd 𝕜) x)
⊢ x = (starRingEnd 𝕜) ((starRingEnd 𝕜) x)
Module: Mathlib.Analysis.Normed.Module.Alternating.Curry

Goal state before:
pp: case zero
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
g : F [⋀^Fin (n + 1)]→L[𝕜] G
f : E →L[𝕜] F
x : E
v : Fin n → E
⊢ ((fun i ↦ ↑((fun x ↦ f) i)) 0) (Fin.cons x v 0) = Fin.cons (f x) (fun i ↦ ((fun i ↦ ↑((fun x ↦ f) i)) i) (v i)) 0

pp: case succ
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
g : F [⋀^Fin (n + 1)]→L[𝕜] G
f : E →L[𝕜] F
x : E
v : Fin n → E
i✝ : Fin n
⊢ ((fun i ↦ ↑((fun x ↦ f) i)) i✝.succ) (Fin.cons x v i✝.succ) =
    Fin.cons (f x) (fun i ↦ ((fun i ↦ ↑((fun x ↦ f) i)) i) (v i)) i✝.succ

Tactic:
<;>

Goal state after:
case zero
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
g : F [⋀^Fin (n + 1)]→L[𝕜] G
f : E →L[𝕜] F
x : E
v : Fin n → E
⊢ ((fun i ↦ ↑((fun x ↦ f) i)) 0) (Fin.cons x v 0) = Fin.cons (f x) (fun i ↦ ((fun i ↦ ↑((fun x ↦ f) i)) i) (v i)) 0

case succ
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
n : ℕ
g : F [⋀^Fin (n + 1)]→L[𝕜] G
f : E →L[𝕜] F
x : E
v : Fin n → E
i✝ : Fin n
⊢ ((fun i ↦ ↑((fun x ↦ f) i)) i✝.succ) (Fin.cons x v i✝.succ) =
    Fin.cons (f x) (fun i ↦ ((fun i ↦ ↑((fun x ↦ f) i)) i) (v i)) i✝.succ
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_1.hf'
f : ℝ → ℝ
a b c : ℝ
h : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ DifferentiableOn ℝ f (interior (Ioc a b))

pp: case refine_1.hf'_nonneg
f : ℝ → ℝ
a b c : ℝ
h : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ ∀ x ∈ interior (Ioc a b), 0 ≤ deriv f x

Tactic:
<;>

Goal state after:
case refine_1.hf'
f : ℝ → ℝ
a b c : ℝ
h : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ DifferentiableOn ℝ f (interior (Ioc a b))

case refine_1.hf'_nonneg
f : ℝ → ℝ
a b c : ℝ
h : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ ∀ x ∈ interior (Ioc a b), 0 ≤ deriv f x
Module: Mathlib.Analysis.Calculus.DerivativeTest

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

pp: case refine_2.hf'_nonpos
f : ℝ → ℝ
a b c : ℝ
h : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ ∀ x ∈ interior (Ico b c), deriv f x ≤ 0

Tactic:
<;>

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

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

Goal state before:
pp: case refine_1.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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ DifferentiableOn ℝ f (interior (Ioc a b))

pp: case refine_1.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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ ∀ x ∈ interior (Ioc a b), 0 ≤ deriv f x

Tactic:
<;>

Goal state after:
case refine_1.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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ DifferentiableOn ℝ f (interior (Ioc a b))

case refine_1.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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ ∀ x ∈ interior (Ioc a b), 0 ≤ deriv f x
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: 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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ DifferentiableOn ℝ f (interior (Icc b c))

pp: case refine_2.hf'_nonpos
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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ ∀ x ∈ interior (Icc b c), deriv f x ≤ 0

Tactic:
<;>

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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ DifferentiableOn ℝ f (interior (Icc b c))

case refine_2.hf'_nonpos
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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ ∀ x ∈ interior (Icc b c), deriv f x ≤ 0
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_1.hf'
f : ℝ → ℝ
a b c : ℝ
ha : ContinuousAt f a
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ DifferentiableOn ℝ f (interior (Icc a b))

pp: case refine_1.hf'_nonneg
f : ℝ → ℝ
a b c : ℝ
ha : ContinuousAt f a
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ ∀ x ∈ interior (Icc a b), 0 ≤ deriv f x

Tactic:
<;>

Goal state after:
case refine_1.hf'
f : ℝ → ℝ
a b c : ℝ
ha : ContinuousAt f a
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ DifferentiableOn ℝ f (interior (Icc a b))

case refine_1.hf'_nonneg
f : ℝ → ℝ
a b c : ℝ
ha : ContinuousAt f a
hb : ContinuousAt f b
hd₀ : DifferentiableOn ℝ f (Ioo a b)
hd₁ : DifferentiableOn ℝ f (Ioo b c)
h₀ : ∀ x ∈ Ioo a b, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ ∀ x ∈ interior (Icc a b), 0 ≤ deriv f x
Module: Mathlib.Analysis.Calculus.DerivativeTest

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

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

Tactic:
<;>

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

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

Goal state before:
pp: case refine_1.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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ DifferentiableOn ℝ f (interior (Icc a b))

pp: case refine_1.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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ ∀ x ∈ interior (Icc a b), 0 ≤ deriv f x

Tactic:
<;>

Goal state after:
case refine_1.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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ DifferentiableOn ℝ f (interior (Icc a b))

case refine_1.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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ ∀ x ∈ interior (Icc a b), 0 ≤ deriv f x
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: 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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ DifferentiableOn ℝ f (interior (Icc b c))

pp: case refine_2.hf'_nonpos
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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ ∀ x ∈ interior (Icc b c), deriv f x ≤ 0

Tactic:
<;>

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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ DifferentiableOn ℝ f (interior (Icc b c))

case refine_2.hf'_nonpos
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, 0 ≤ deriv f x
h₁ : ∀ x ∈ Ioo b c, deriv f x ≤ 0
⊢ ∀ x ∈ interior (Icc b c), deriv f x ≤ 0
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_1.hf'
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
⊢ DifferentiableOn ℝ f (interior (Ioc a b))

pp: case refine_1.hf'_nonneg
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
⊢ ∀ x ∈ interior (Ioc a b), 0 ≤ deriv f x

Tactic:
<;>

Goal state after:
case refine_1.hf'
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
⊢ DifferentiableOn ℝ f (interior (Ioc a b))

case refine_1.hf'_nonneg
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
⊢ ∀ x ∈ interior (Ioc a b), 0 ≤ deriv f x
Module: Mathlib.Analysis.Calculus.DerivativeTest

Goal state before:
pp: case refine_2.hf'
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
⊢ DifferentiableOn ℝ f (interior (Ici b))

pp: case refine_2.hf'_nonpos
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
⊢ ∀ x ∈ interior (Ici b), deriv f x ≤ 0

Tactic:
<;>

Goal state after:
case refine_2.hf'
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
⊢ DifferentiableOn ℝ f (interior (Ici b))

case refine_2.hf'_nonpos
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
⊢ ∀ x ∈ interior (Ici b), deriv f x ≤ 0