mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.Deriv.MeanValue

Goal state before:
pp: f f' : ℝ → ℝ
hf : ∀ (x : ℝ), HasDerivAt f (f' x) x
hf' : f' ≤ 0
x : ℝ
⊢ deriv f x ≤ 0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
w : α → ℝ≥0
wpos : ∀ (x : α), 0 < w x
wmeas : Measurable w
wint : ∫⁻ (x : α), ↑(w x) ∂μ < ε / 2
f' : α → ℝ≥0∞ := fun x ↦ ↑(f x + w x)
g : α → ℝ≥0∞
le_g : ∀ (x : α), f' x ≤ g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), f' x ∂μ + ε / 2
⊢ ∫⁻ (x : α), ↑(f x) ∂μ + ε / 2 + ε / 2 = ∫⁻ (x : α), ↑(f x) ∂μ + ε

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
x y : E
f' : E → StrongDual ℝ E
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t ↦ (AffineMap.lineMap x y) t
I : Set ℝ := Icc 0 1
hsub : Ioo 0 1 ⊆ I
hmaps : MapsTo g I s
hfg : ∀ t ∈ I, HasDerivWithinAt (f ∘ g) ((f' (g t)) (y - x)) I t
⊢ 0 < 1

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : Measurable f
ε : ℝ≥0∞
ε0 : ε ≠ 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
L : E →L[𝕜] F
r ε : ℝ
x : E
r' : ℝ
r'_mem : r' ∈ Ioc (r / 2) r
hr' : ∀ y ∈ ball x r', ∀ z ∈ ball x r', ‖f z - f y - L (z - y)‖ < ε * r
s : ℝ
s_gt : r / 2 < s
s_lt : s < r'
this : s ∈ Ioc (r / 2) r
⊢ r' - s > 0

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
x y : E
f' : E → StrongDual ℝ E
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
g : ℝ → E := fun t ↦ (AffineMap.lineMap x y) t
I : Set ℝ := Icc 0 1
hsub : Ioo 0 1 ⊆ I
hmaps : MapsTo g I s
hfg : ∀ t ∈ I, HasDerivWithinAt (f ∘ g) ((f' (g t)) (y - x)) I t
⊢ ∃ t ∈ Ioo 0 1, (f' (g t)) (y - x) = (f (g 1) - f (g 0)) / (1 - 0)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
L : E →L[𝕜] F
r ε : ℝ
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (A f L r ε)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
K : Set (E →L[𝕜] F)
r s ε : ℝ
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B f K r s ε)

Tactic:
by

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

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : E → ℝ
s : Set E
x y : E
f' : E → StrongDual ℝ E
hf : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x
hs : Convex ℝ s
xs : x ∈ s
ys : y ∈ s
⊢ ∃ z ∈ segment ℝ x y, f y - f x = (f' z) (y - x)

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
L : E →L[𝕜] F
r ε δ : ℝ
h : ε ≤ δ
⊢ A f L r ε ⊆ A f L r δ

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
r ε : ℝ
L : E →L[𝕜] F
x : E
hx : x ∈ A f L r ε
y z : E
hy : y ∈ closedBall x (r / 2)
hz : z ∈ closedBall x (r / 2)
⊢ ‖f z - f y - L (z - y)‖ ≤ ε * r

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : SMul R M
C C₁✝ C₂✝ : ConvexCone R M
s : Set M
c : R
x : M
C₁ C₂ : ConvexCone R M
h : C₁.carrier = C₂.carrier
⊢ C₁ = C₂

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : SMul R M
C✝ C₁ C₂ : ConvexCone R M
s✝ : Set M
c : R
x : M
C : ConvexCone R M
s : Set M
hs : s = ↑C
⊢ ∀ ⦃c : R⦄, 0 < c → ∀ ⦃x : M⦄, x ∈ s → c • x ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
M : Type u_4
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : SMul R M
ι : Sort u_7
f : ι → ConvexCone R M
⊢ ↑(iInf f) = ⋂ i, ↑(f i)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
M : Type u_4
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : SMul R M
x : M
ι : Sort u_7
f : ι → ConvexCone R M
⊢ (∀ i ∈ Set.range f, x ∈ ↑i) ↔ ∀ (i : ι), x ∈ f i

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
M : Type u_4
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : SMul R M
s : Set M
⊢ s ⊆ ↑(hull R s)

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
M : Type u_4
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : SMul R M
C : ConvexCone R M
⊢ ↑C = ∅ ↔ C = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
M : Type u_4
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : SMul R M
C₁ C₂ : ConvexCone R M
⊢ Disjoint ↑C₁ ↑C₂ ↔ Disjoint C₁ C₂

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝⁷ : Semiring R
inst✝⁶ : PartialOrder R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : AddCommMonoid O
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R O
f : M →ₗ[R] N
C : ConvexCone R N
c : R
hc : 0 < c
x : M
hx : x ∈ ⇑f ⁻¹' ↑C
⊢ c • x ∈ ⇑f ⁻¹' ↑C

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
ε : ℝ
hε : 0 < ε
x : E
hx : DifferentiableAt 𝕜 f x
δ : ℝ := ε / 2 / 2
⊢ 0 < δ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x ↦ f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x ↦ ∞) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x ↦ ∞) x ∂μ + ε / 2
x : α
h : x ∉ s
⊢ f x = AEMeasurable.mk f fmeas x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝⁷ : Semiring R
inst✝⁶ : PartialOrder R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : AddCommMonoid N
inst✝³ : AddCommMonoid O
inst✝² : Module R M
inst✝¹ : Module R N
inst✝ : Module R O
f : M →ₗ[R] N
C : ConvexCone R N
x : M
hx : x ∈ ⇑f ⁻¹' ↑C
y : M
hy : y ∈ ⇑f ⁻¹' ↑C
⊢ x + y ∈ ⇑f ⁻¹' ↑C

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
ε : ℝ
hε : 0 < ε
x : E
hx : DifferentiableAt 𝕜 f x
δ : ℝ := ε / 2 / 2
R : ℝ
R_pos : R > 0
hR : ∀ y ∈ ball x R, ‖f y - f x - (fderiv 𝕜 f x) (y - x)‖ ≤ δ * ‖y - x‖
r : ℝ
hr : r ∈ Ioo 0 R
this : r ∈ Ioc (r / 2) r
y : E
hy : y ∈ ball x r
z : E
hz : z ∈ ball x r
⊢ ‖f z - f y - (fderiv 𝕜 f x) (z - y)‖ = ‖f z - f x - (fderiv 𝕜 f x) (z - x) - (f y - f x - (fderiv 𝕜 f x) (y - x))‖

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
M : Type u_4
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : SMul R M
C : ConvexCone R M
⊢ C.Pointed ↔ ¬C.Blunt

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
G : Type u_3
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommGroup G
inst✝ : SMul R G
C : ConvexCone R G
⊢ C.Salient ↔ ¬C.Flat

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
ε : ℝ
hε : 0 < ε
x : E
hx : DifferentiableAt 𝕜 f x
δ : ℝ := ε / 2 / 2
R : ℝ
R_pos : R > 0
hR : ∀ y ∈ ball x R, ‖f y - f x - (fderiv 𝕜 f x) (y - x)‖ ≤ δ * ‖y - x‖
r : ℝ
hr : r ∈ Ioo 0 R
this : r ∈ Ioc (r / 2) r
y : E
hy : y ∈ ball x r
z : E
hz : z ∈ ball x r
⊢ δ * ‖z - x‖ + δ * ‖y - x‖ ≤ δ * r + δ * r

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
G : Type u_3
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommGroup G
inst✝ : SMul R G
C : ConvexCone R G
hC : C.Flat
⊢ C.Pointed

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
ε : ℝ
hε : 0 < ε
x : E
hx : DifferentiableAt 𝕜 f x
δ : ℝ := ε / 2 / 2
R : ℝ
R_pos : R > 0
hR : ∀ y ∈ ball x R, ‖f y - f x - (fderiv 𝕜 f x) (y - x)‖ ≤ δ * ‖y - x‖
r : ℝ
hr : r ∈ Ioo 0 R
this : r ∈ Ioc (r / 2) r
y : E
hy : y ∈ ball x r
z : E
hz : z ∈ ball x r
⊢ δ * r + δ * r = ε / 2 * r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
hx : 0 < x
⊢ HasStrictDerivAt log x⁻¹ x

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
G : Type u_3
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommGroup G
inst✝ : SMul R G
C : ConvexCone R G
⊢ C.Blunt → C.Salient

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommGroup G
inst✝ : SMul R G
C✝ C₁ C₂ C : ConvexCone R G
h₁ : C.Pointed
x : G
⊢ x - x ∈ C

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommGroup G
inst✝ : SMul R G
C✝ C₁ C₂ C : ConvexCone R G
h₁ : C.Pointed
x y z : G
xy : y - x ∈ C
zy : z - y ∈ C
⊢ z - x ∈ C

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
ε : ℝ
hε : 0 < ε
x : E
hx : DifferentiableAt 𝕜 f x
δ : ℝ := ε / 2 / 2
R : ℝ
R_pos : R > 0
hR : ∀ y ∈ ball x R, ‖f y - f x - (fderiv 𝕜 f x) (y - x)‖ ≤ δ * ‖y - x‖
r : ℝ
hr : r ∈ Ioo 0 R
this : r ∈ Ioc (r / 2) r
y : E
hy : y ∈ ball x r
z : E
hz : z ∈ ball x r
⊢ ε / 2 * r < ε * r

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
ε : ℝ
hε : 0 < ε
x : E
hx : DifferentiableAt 𝕜 f x
⊢ ∃ R > 0, ∀ r ∈ Ioo 0 R, x ∈ A f (fderiv 𝕜 f x) r ε

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
c : 𝕜
hc : 1 < ‖c‖
r ε : ℝ
hε : 0 < ε
hr : 0 < r
x : E
L₁ L₂ : E →L[𝕜] F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
⊢ 0 ≤ 4 * ‖c‖ * ε

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommGroup G
inst✝ : SMul R G
C✝ C₁ C₂ C : ConvexCone R G
h₁ : C.Pointed
h₂ : C.Salient
⊢ ∀ (a b : G), a ≤ b → b ≤ a → a = b

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
C₁ C₂ : ConvexCone R M
x : M
x✝¹ : R
x✝ : 0 < x✝¹
⊢ ∀ ⦃x : M⦄, x ∈ 0 → x✝¹ • x ∈ 0

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
C₁ C₂ : ConvexCone R M
x x✝ : M
⊢ x✝ ∈ 0 → ∀ ⦃y : M⦄, y ∈ 0 → x✝ + y ∈ 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
c : 𝕜
hc : 1 < ‖c‖
r ε : ℝ
hε : 0 < ε
hr : 0 < r
x : E
L₁ L₂ : E →L[𝕜] F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
y : E
ley : r / (2 * ‖c‖) ≤ ‖y‖
ylt : ‖y‖ < r / 2
⊢ 0 < 2 * ‖c‖

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
M : Type u_4
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ Pointed 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
hx : x ≠ 0
⊢ HasStrictDerivAt log x⁻¹ x

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x ↦ f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x ↦ ∞) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x ↦ ∞) x ∂μ + ε / 2
⊢ ∫⁻ (x : α), g0 x ∂μ + ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε / 2 + (0 + ε / 2)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: x : ℝ
hx : x = 0
⊢ deriv log x = x⁻¹

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
c : 𝕜
hc : 1 < ‖c‖
r ε : ℝ
hε : 0 < ε
hr : 0 < r
x : E
L₁ L₂ : E →L[𝕜] F
h₁ : x ∈ A f L₁ r ε
h₂ : x ∈ A f L₂ r ε
y : E
ley : r ≤ 2 * ‖c‖ * ‖y‖
ylt : ‖y‖ < r / 2
⊢ ‖(L₁ - L₂) y‖ = ‖f (x + y) - f x - L₂ (x + y - x) - (f (x + y) - f x - L₁ (x + y - x))‖

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
C₁✝ C₂✝ : ConvexCone R M
x : M
C₁ C₂ : ConvexCone R M
⊢ ∀ ⦃c : R⦄, 0 < c → ∀ ⦃x : M⦄, x ∈ ↑C₁ + ↑C₂ → c • x ∈ ↑C₁ + ↑C₂

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
this : ε / 2 ≠ 0
g0 : α → ℝ≥0∞
f_lt_g0 : ∀ (x : α), ↑(AEMeasurable.mk f fmeas x) < g0 x
g0_cont : LowerSemicontinuous g0
g0_int : ∫⁻ (x : α), g0 x ∂μ ≤ ∫⁻ (x : α), ↑(AEMeasurable.mk f fmeas x) ∂μ + ε / 2
s : Set α
hs : {x | (fun x ↦ f x = AEMeasurable.mk f fmeas x) x}ᶜ ⊆ s
smeas : MeasurableSet s
μs : μ s = 0
g1 : α → ℝ≥0∞
le_g1 : ∀ (x : α), s.indicator (fun _x ↦ ∞) x ≤ g1 x
g1_cont : LowerSemicontinuous g1
g1_int : ∫⁻ (x : α), g1 x ∂μ ≤ ∫⁻ (x : α), s.indicator (fun _x ↦ ∞) x ∂μ + ε / 2
⊢ ∫⁻ (x : α), ↑(f x) ∂μ + ε / 2 + (0 + ε / 2) = ∫⁻ (x : α), ↑(f x) ∂μ + ε

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: n : ℕ∞ω
x : ℝ
hx : x ≠ 0
y : ℝ
hy : 0 < y
⊢ y ∈ expPartialHomeomorph.target

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
C₁✝ C₂✝ : ConvexCone R M
x : M
C₁ C₂ : ConvexCone R M
⊢ ∀ ⦃x : M⦄, x ∈ ↑C₁ + ↑C₂ → ∀ ⦃y : M⦄, y ∈ ↑C₁ + ↑C₂ → x + y ∈ ↑C₁ + ↑C₂

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
C₁ C₂ : ConvexCone R M
x : M
x✝ : ConvexCone R M
⊢ 0 + x✝ = x✝

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: 𝕜 : Type u_1
R : Type u_2
G : Type u_3
M : Type u_4
N : Type u_5
O : Type u_6
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
C₁ C₂ : ConvexCone R M
x : M
x✝ : ConvexCone R M
⊢ x✝ + 0 = x✝

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
M : Type u_4
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ ⊤.IsGenerating

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
M : Type u_4
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
⊢ ⊥.IsGenerating ↔ Subsingleton M

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fmeas : AEMeasurable f μ
ε : ℝ≥0∞
ε0 : ε ≠ 0
⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ε

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable (fun x ↦ ↑(f x)) μ
ε : ℝ
εpos : 0 < ε
⊢ AEMeasurable f μ

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_2
M : Type u_4
inst✝³ : Semiring R
inst✝² : PartialOrder R
inst✝¹ : AddCommMonoid M
inst✝ : Module R M
C₁ C₂ : ConvexCone R M
h : C₁ ≤ C₂
hgen : C₁.IsGenerating
⊢ C₂.IsGenerating

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_7
M : Type u_8
inst✝³ : Ring R
inst✝² : PartialOrder R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
C : ConvexCone R M
h : C.IsReproducing
⊢ Submodule.span R ↑C = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: n : ℕ∞ω
x : ℝ
hx : x ≠ 0
y : ℝ
hy : 0 < y
⊢ ContDiffAt ℝ n log y

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_7
M : Type u_8
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
inst✝³ : AddLeftStrictMono R
inst✝² : AddCommGroup M
inst✝¹ : Nontrivial M
inst✝ : Module R M
C : ConvexCone R M
h : Submodule.span R ↑C = ⊤
x : M
h' : ↑C = ∅
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_7
M : Type u_8
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
inst✝³ : AddLeftStrictMono R
inst✝² : AddCommGroup M
inst✝¹ : Nontrivial M
inst✝ : Module R M
C : ConvexCone R M
h : Submodule.span R ↑C = ⊤
x : M
hne : (↑C).Nonempty
⊢ ∀ {a b : M}, a ∈ ↑C - ↑C → b ∈ ↑C - ↑C → a + b ∈ ↑C - ↑C

Tactic:
by

Goal state after:
Module: Mathlib.Geometry.Convex.Cone.Basic

Goal state before:
pp: R : Type u_7
M : Type u_8
inst✝⁵ : Ring R
inst✝⁴ : LinearOrder R
inst✝³ : AddLeftStrictMono R
inst✝² : AddCommGroup M
inst✝¹ : Nontrivial M
inst✝ : Module R M
C : ConvexCone R M
h : Submodule.span R ↑C = ⊤
x : M
hne : (↑C).Nonempty
⊢ 0 ∈ ↑C - ↑C

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory

Goal state before:
pp: α : Type u_1
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
μ : Measure α
inst✝¹ : μ.WeaklyRegular
inst✝ : SigmaFinite μ
f : α → ℝ≥0
fint : Integrable (fun x ↦ ↑(f x)) μ
fmeas : AEMeasurable f μ
ε : ℝ≥0
εpos : 0 < ↑ε
δ : ℝ≥0
δpos : 0 < δ
hδε : δ < ε
int_f_ne_top : ∫⁻ (a : α), ↑(f a) ∂μ ≠ ∞
g : α → ℝ≥0∞
f_lt_g : ∀ (x : α), ↑(f x) < g x
gcont : LowerSemicontinuous g
gint : ∫⁻ (x : α), g x ∂μ ≤ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ
⊢ ∫⁻ (x : α), ↑(f x) ∂μ + ↑δ ≠ ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.Log.Deriv

Goal state before:
pp: n : ℕ∞ω
x : ℝ
hx✝ : x ≠ 0
A : ∀ (y : ℝ), 0 < y → ContDiffAt ℝ n log y
hx : x < 0
⊢ ContDiffAt ℝ n (log ∘ fun y ↦ -y) x

Tactic:
by

Goal state after: