mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Convex.StoneSeparation

Goal state before:
pp: case inr.inr.refine_2
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q x y : E
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤ bv
habv : av + bv = 1
hav' : 0 < av
au bu : 𝕜
hau : 0 ≤ au
hbu : 0 ≤ bu
habu : au + bu = 1
hab : 0 < az * av + bz * au
⊢ 0 ≤ bz * au / (az * av + bz * au)

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Convex.StoneSeparation

Goal state before:
pp: case inr.inr.refine_3
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p q x y : E
az bz : 𝕜
haz : 0 ≤ az
hbz : 0 ≤ bz
habz : az + bz = 1
haz' : 0 < az
av bv : 𝕜
hav : 0 ≤ av
hbv : 0 ≤ bv
habv : av + bv = 1
hav' : 0 < av
au bu : 𝕜
hau : 0 ≤ au
hbu : 0 ≤ bu
habu : au + bu = 1
hab : 0 < az * av + bz * au
⊢ az * av + bz * au ≠ 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Convex.StrictConvexBetween

Goal state before:
pp: case hx
V : Type u_1
P : Type u_2
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : NormedSpace ℝ V
inst✝⁹ : StrictConvexSpace ℝ V
E : Type u_3
F : Type u_4
PE : Type u_5
PF : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : StrictConvexSpace ℝ E
inst✝³ : MetricSpace PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor E PE
inst✝ : NormedAddTorsor F PF
r : ℝ
f : PF → PE
x✝ y✝ z : PE
hi : Isometry f
x y : PF
⊢ dist (f x) (f (midpoint ℝ x y)) = dist (f x) (f y) / 2

Tactic:
hi.dist_eq,

Goal state after:
case hx
V : Type u_1
P : Type u_2
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : NormedSpace ℝ V
inst✝⁹ : StrictConvexSpace ℝ V
E : Type u_3
F : Type u_4
PE : Type u_5
PF : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : StrictConvexSpace ℝ E
inst✝³ : MetricSpace PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor E PE
inst✝ : NormedAddTorsor F PF
r : ℝ
f : PF → PE
x✝ y✝ z : PE
hi : Isometry f
x y : PF
⊢ dist x (midpoint ℝ x y) = dist (f x) (f y) / 2
Module: Mathlib.Analysis.Convex.StrictConvexBetween

Goal state before:
pp: case hy
V : Type u_1
P : Type u_2
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : NormedSpace ℝ V
inst✝⁹ : StrictConvexSpace ℝ V
E : Type u_3
F : Type u_4
PE : Type u_5
PF : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : StrictConvexSpace ℝ E
inst✝³ : MetricSpace PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor E PE
inst✝ : NormedAddTorsor F PF
r : ℝ
f : PF → PE
x✝ y✝ z : PE
hi : Isometry f
x y : PF
⊢ dist (f (midpoint ℝ x y)) (f y) = dist (f x) (f y) / 2

Tactic:
hi.dist_eq,

Goal state after:
case hy
V : Type u_1
P : Type u_2
inst✝¹¹ : NormedAddCommGroup V
inst✝¹⁰ : NormedSpace ℝ V
inst✝⁹ : StrictConvexSpace ℝ V
E : Type u_3
F : Type u_4
PE : Type u_5
PF : Type u_6
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : StrictConvexSpace ℝ E
inst✝³ : MetricSpace PE
inst✝² : MetricSpace PF
inst✝¹ : NormedAddTorsor E PE
inst✝ : NormedAddTorsor F PF
r : ℝ
f : PF → PE
x✝ y✝ z : PE
hi : Isometry f
x y : PF
⊢ dist (midpoint ℝ x y) y = dist (f x) (f y) / 2
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : x -ᵥ p₁ = 0
⊢ ∃ p ∈ s, Wbtw R x p y

pp: case inr.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : p₂ -ᵥ y = 0
⊢ ∃ p ∈ s, Wbtw R x p y

pp: case inr.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : 0 < r₂
h : r₁ • (x -ᵥ p₁) = r₂ • (p₂ -ᵥ y)
⊢ ∃ p ∈ s, Wbtw R x p y

Tactic:
· rw [vsub_eq_zero_iff_eq] at h
    rw [h]
    exact ⟨p₁, hp₁, wbtw_self_left _ _ _⟩

Goal state after:
case inr.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : p₂ -ᵥ y = 0
⊢ ∃ p ∈ s, Wbtw R x p y

case inr.inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : 0 < r₂
h : r₁ • (x -ᵥ p₁) = r₂ • (p₂ -ᵥ y)
⊢ ∃ p ∈ s, Wbtw R x p y
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case inr.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : p₂ = y
⊢ ∃ p ∈ s, Wbtw R x p y

Tactic:
rw [← h]

Goal state after:
case inr.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
h : p₂ = y
⊢ ∃ p ∈ s, Wbtw R x p p₂
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y p₁ : P
hp₁ : p₁ ∈ s
p₂ : P
hp₂ : p₂ ∈ s
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : 0 < r₂
h : r₁ • (x -ᵥ p₁) = r₂ • (p₂ -ᵥ y)
⊢ 0 ≤ r₂ / (r₁ + r₂)

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Convex.Strong

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
f : E → ℝ
m : ℝ
hf : StrongConvexOn s m f
hm : 0 < m
r : ℝ
hr : r ≠ 0
⊢ 0 < m / 2 * r ^ 2

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Convex.Strong

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
s : Set E
f : E → ℝ
m : ℝ
hf : StrongConcaveOn s m f
hm : 0 < m
r : ℝ
hr : r ≠ 0
⊢ 0 < m / 2 * r ^ 2

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Convex.Visible

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this : 0 < 1 - ε
hwi : 0 < 1 - w i
⊢ 0 < (w i)⁻¹ / ((1 - ε) / ε + (w i)⁻¹)

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Convex.Visible

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this : 0 < 1 - ε
hwi : 0 < 1 - w i
⊢ 0 < (1 - ε) / ε + (w i)⁻¹

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Convex.Visible

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this : 0 < 1 - ε
hwi : 0 < 1 - w i
⊢ 0 ≤ (1 - w i) / w i / ((1 - ε) / ε + (1 - w i) / w i + 1)

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Convex.Visible

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this : 0 < 1 - ε
hwi : 0 < 1 - w i
⊢ 0 < (1 - ε) / ε + (1 - w i) / w i + 1

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Convex.Visible

Goal state before:
pp: case h₁
𝕜 : Type u_1
V : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this : 0 < 1 - ε
hwi : 0 < 1 - w i
⊢ 0 ≤ (1 - ε) / ε

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Convex.Visible

Goal state before:
pp: case h₂
𝕜 : Type u_1
V : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this : 0 < 1 - ε
hwi : 0 < 1 - w i
⊢ 0 ≤ 1

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Convex.Visible

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup V
inst✝ : Module 𝕜 V
s : Set V
x : V
ι : Type u_4
t : Finset ι
a : ι → V
w : ι → 𝕜
hw₀ : ∀ i ∈ t, 0 ≤ w i
hw₁ : ∑ i ∈ t, w i = 1
ha : ∀ i ∈ t, a i ∈ s
hx : x ∉ (convexHull 𝕜) s
hw : IsVisible 𝕜 ((convexHull 𝕜) s) x (∑ i ∈ t, w i • a i)
i : ι
hi : i ∈ t
hwi✝¹ : 0 < w i
hwi✝ : w i < 1
ε : 𝕜
hε : (lineMap x (a i)) ε ∈ (convexHull 𝕜) s
h : (lineMap x (a i)) ε ≠ x ∧ (lineMap x (a i)) ε ≠ a i
hε₀ : 0 < ε
hε₁ : ε < 1
this : 0 < 1 - ε
hwi : 0 < 1 - w i
⊢ 0 < (1 - ε) / ε

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case inr
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Field R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AffineSpace V P
n : ℕ
inst✝ : NeZero n
s : Simplex R P n
w₁ w₂ : Fin (n + 1) → R
hw₁ : ∑ j, w₁ j = 1
hw₂ : ∑ j, w₂ j = 1
i : Fin (n + 1)
hs : 0 < w₂ i
h0 : w₁ i ≠ 0
h0' : w₂ i ≠ 0
j : Fin (n + 1)
hj : j ≠ i
hj' : s.points j ∈ affineSpan R (Set.range (s.faceOpposite i).points)
w₃ : Fin (n + 1) → R := w₂ - w₂ i • (w₁ i)⁻¹ • (w₁ - Pi.single j 1)
hw₃1 : ∑ k, w₃ k = 1
hw₃i : w₃ i = 0
h : 0 < w₁ i
⊢ 0 ≤ w₂ i * (w₁ i)⁻¹

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.Distribution

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
Ω : Opens E
n : ℕ∞
x : E
hx : x ∉ Ω
⊢ delta x = 0

Tactic:
by
  ext f
  change f x = 0
  have hx_support : x ∉ tsupport f := by
    intro hx_mem
    exact hx (f.tsupport_subset hx_mem)
  exact image_eq_zero_of_notMem_tsupport hx_support

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
x : E
⊢ 0 ≤ 1 + ‖x‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: case h
E : Type u_1
inst✝ : NormedAddCommGroup E
r : ℝ
x : E
hr : 0 < r
⊢ √2 ^ r ≠ 0

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: case hx
E : Type u_1
inst✝ : NormedAddCommGroup E
r : ℝ
x : E
hr : 0 < r
⊢ 0 ≤ √(1 + ‖x‖ ^ 2)

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: case hx
E : Type u_1
inst✝ : NormedAddCommGroup E
r : ℝ
x : E
hr : 0 < r
⊢ 0 ≤ √2

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: case hy
E : Type u_1
inst✝ : NormedAddCommGroup E
r : ℝ
x : E
hr : 0 < r
⊢ 0 ≤ √(1 + ‖x‖ ^ 2)

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: case hx
E : Type u_1
inst✝ : NormedAddCommGroup E
r : ℝ
x : E
hr : 0 < r
⊢ 0 ≤ 2

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: case hx
E : Type u_1
inst✝ : NormedAddCommGroup E
r : ℝ
x : E
hr : 0 < r
⊢ 0 ≤ 1 + ‖x‖ ^ 2

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: case hx
E : Type u_1
inst✝ : NormedAddCommGroup E
r : ℝ
x : E
hr : 0 < r
⊢ 0 ≤ 1 + ‖x‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: E : Type u_1
inst✝ : NormedAddCommGroup E
r t : ℝ
hr : 0 < r
ht : 0 < t
x : E
⊢ 0 < 1 + ‖x‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
hr : 0 < r
h_meas : Measurable fun ω ↦ (1 + ‖ω‖) ^ (-r)
x : E
⊢ 0 ≤ (1 + ‖x‖) ^ (-r)

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
x✝ : E
⊢ 0 ≤ 1 + ‖x✝‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: case refine_2
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
r : ℝ
hnr : ↑(finrank ℝ E) < r
hr : 0 < r
x : E
⊢ 0 < (1 + ‖x‖ ^ 2) ^ (-r / 2)

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
n k : ℕ
C : ℝ
x : E
⊢ 0 ≤ 1 + ‖x‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
hf_temperate : HasTemperateGrowth f
N : ℕ
k : ℕ → ℕ
hk : ∀ (n : ℕ), iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k n
n : ℕ
hn : n ≤ N
x : E
⊢ 0 ≤ (1 + ‖x‖) ^ k n

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
hf_temperate : HasTemperateGrowth f
N : ℕ
k : ℕ → ℕ
hk : ∀ (n : ℕ), iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k n
n : ℕ
hn : n ≤ N
x : E
⊢ 0 ≤ (1 + ‖x‖) ^ (Finset.range (N + 1)).sup k

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
h'f : HasTemperateGrowth (fderiv ℝ f)
hf : Differentiable ℝ f
k : ℕ
C : ℝ
h : ∀ (x : E), ‖f x‖ ≤ C * (1 + ‖x‖) ^ k
⊢ HasTemperateGrowth f

Tactic:
refine ⟨contDiff_succ_iff_fderiv.2 ⟨hf, by simp, h'f.1⟩, fun n ↦ ?_⟩
  rcases n with rfl | m
  · exact ⟨k, C, fun x ↦ by simpa using h x⟩
  · rcases h'f.2 m with ⟨k', C', h'⟩
    refine ⟨k', C', ?_⟩
    simpa [iteratedFDeriv_succ_eq_comp_right] using h'

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f g : E → F
hf : ContDiff ℝ ∞ f ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
hg : ContDiff ℝ ∞ g ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n g =O[⊤] fun x ↦ (1 + ‖x‖) ^ k
n k₁ : ℕ
h₁ : iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k₁
k₂ : ℕ
h₂ : iteratedFDeriv ℝ n g =O[⊤] fun x ↦ (1 + ‖x‖) ^ k₂
a✝ : E
⊢ 0 ≤ ‖a✝‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: 𝕜 : Type u_1
𝕂 : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
Ω Ω₁ Ω₂ : Opens E
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
F' : Type u_5
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedSpace 𝕜 F'
n n₁ n₂ k : ℕ∞
inst✝¹ : Algebra ℝ 𝕜
inst✝ : IsScalarTower ℝ 𝕜 F
Φ : 𝓓^{n₁}(Ω₁, F) → 𝓓^{n₂}(Ω₂, F) :=
  fun f ↦
    if h : n₂ ≤ n₁ ∧ Ω₁ ≤ Ω₂ then { toFun := ⇑f, contDiff' := ⋯, hasCompactSupport' := ⋯, tsupport_subset' := ⋯ } else 0
x✝² : Compacts E
x✝¹ : ↑x✝² ⊆ ↑Ω₁
x✝ : 𝓓^{n₁}_{x✝²}(E, F)
⊢ Φ (ofSupportedIn x✝¹ x✝) =
    (if h : n₂ ≤ n₁ ∧ Ω₁ ≤ Ω₂ then ofSupportedInCLM 𝕜 ⋯ ∘SL ContDiffMapSupportedIn.monoCLM 𝕜 else 0) x✝

Tactic:
by ext; dsimp [Φ]; split_ifs with h <;> simp [h]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᵐ[μ] f
key : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ = ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a}
lhs_finite : ∫⁻ (ω : α), ENNReal.ofReal (f ω) ∂μ < ∞
rhs_finite : ∫⁻ (t : ℝ) in Ioi 0, μ {a | t < f a} < ∞
rhs_integrand_finite : ∀ t > 0, μ {a | t < f a} < ∞
x : ℝ
⊢ 0 x ≤ (fun t ↦ μ.real {a | t < f a}) x

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: 𝕜 : Type u_1
𝕂 : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
Ω Ω₁ Ω₂ : Opens E
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : NormedSpace 𝕜 F
F' : Type u_5
inst✝⁴ : NormedAddCommGroup F'
inst✝³ : NormedSpace ℝ F'
inst✝² : NormedSpace 𝕜 F'
n n₁ n₂ k : ℕ∞
inst✝¹ : Algebra ℝ 𝕜
inst✝ : IsScalarTower ℝ 𝕜 F
Φ : 𝓓^{n}(Ω, F) → 𝓓^{k}(Ω, E →L[ℝ] F) :=
  fun f ↦
    if hk : k + 1 ≤ n then { toFun := fderiv ℝ ⇑f, contDiff' := ⋯, hasCompactSupport' := ⋯, tsupport_subset' := ⋯ }
    else 0
x✝² : Compacts E
x✝¹ : ↑x✝² ⊆ ↑Ω
x✝ : 𝓓^{n}_{x✝²}(E, F)
⊢ Φ (ofSupportedIn x✝¹ x✝) = (ofSupportedInCLM 𝕜 x✝¹ ∘SL ContDiffMapSupportedIn.fderivCLM 𝕜 n k) x✝

Tactic:
by ext; dsimp [Φ]; split_ifs with h <;> simp [h]

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
x : H
⊢ 0 ≤ 1 ^ 2 + 2 * 1 * ‖x‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := {y | 1 / 2 < y}
y : H
⊢ 0 ≤ ‖y‖ ^ 2

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := {y | 1 / 2 < y}
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
⊢ 0 ≤ ∑ k ∈ Finset.range (N + 1), ‖Polynomial.eval r (descPochhammer ℝ k)‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := {y | 1 / 2 < y}
ht : (Set.range fun x ↦ 1 + ‖x‖ ^ 2) ⊆ t
hdiff : ContDiffOn ℝ ∞ (fun x ↦ x ^ r) t
hunique : UniqueDiffOn ℝ t
N k : ℕ
hk : max r ((↑N - r) * Real.log 2 / Real.log (3 / 2)) ≤ ↑k
hk₁ : r ≤ ↑k
hk₂ : Real.log 2 * (↑N - r) ≤ Real.log (3 / 2) * ↑k
n : ℕ
hn : n ≤ N
x : ℝ
hx : x ∈ t
this✝ : ContDiffAt ℝ (↑n) (fun x ↦ x ^ r) x
this : n ∈ Finset.range (N + 1)
x✝¹ : ℕ
x✝ : x✝¹ ∈ Finset.range (N + 1)
⊢ 0 ≤ ‖Polynomial.eval r (descPochhammer ℝ x✝¹)‖

Tactic:
positivity

Goal state after:
Module: Mathlib.Analysis.Distribution.TestFunction

Goal state before:
pp: E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
n : ℕ∞
m : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
μ : Measure E
H : LocallyIntegrableOn (fun x ↦ 1) (↑Ω) μ
f : 𝓓^{n}(Ω, F)
⊢ Integrable (⇑f) μ

Tactic:
rw [← integrableOn_iff_integrable_of_support_subset (subset_tsupport f)]

Goal state after:
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
Ω : Opens E
F : Type u_4
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
n : ℕ∞
m : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
μ : Measure E
H : LocallyIntegrableOn (fun x ↦ 1) (↑Ω) μ
f : 𝓓^{n}(Ω, F)
⊢ IntegrableOn (⇑f) (tsupport ⇑f) μ
Module: Mathlib.Analysis.InnerProductSpace.Rayleigh

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
T : E →L[𝕜] E
hT : (↑T).IsSymmetric
⊢ ‖T‖ = ⨆ x, |T.rayleighQuotient x|

Tactic:
by
  set M := ⨆ x, |T.rayleighQuotient x|
  have nonneg : 0 ≤ M := le_ciSup_of_le T.bddAbove_rayleighQuotient 0 (abs_nonneg _)
  have hM x : |re ⟪T x, x⟫| ≤ M * ‖x‖ ^ 2 := by
    have hM : |T.rayleighQuotient x| ≤ M := le_ciSup T.bddAbove_rayleighQuotient x
    by_cases hx : 0 < ‖x‖ ^ 2
    · rwa [rayleighQuotient, abs_div, abs_sq, reApplyInnerSelf, div_le_iff₀ hx] at hM
    · simp_all
  refine le_antisymm ?_ (ciSup_le T.rayleighQuotient_le_norm)
  refine opNorm_le_of_re_inner_le nonneg fun x y hx hy ↦ ?_
  transitivity M * (‖x + y‖ ^ 2 + ‖x - y‖ ^ 2) / 4
  · have key := congrArg re (add_conj ⟪T x, y⟫)
    rw [map_add, conj_inner_symm, ← coe_coe, ← hT, coe_coe, re_mul_ofReal, ofNat_re] at key
    grind [inner_add_left, inner_add_right, inner_sub_left, inner_sub_right]
  · rw [parallelogram_law_with_norm 𝕜 x y, hx, hy]
    grind

Goal state after: