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

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
x₁ : E
hx₁ : x₁ ∈ s
y₁ : E
hy₁ : y₁ ∈ t
a₁ b₁ : 𝕜
ha₁ : 0 ≤ a₁
hb₁ : 0 ≤ b₁
hab₁ : a₁ + b₁ = 1
x₂ : E
hx₂ : x₂ ∈ s
y₂ : E
hy₂ : y₂ ∈ t
a₂ b₂ : 𝕜
ha₂ : 0 ≤ a₂
hb₂ : 0 ≤ b₂
hab₂ : a₂ + b₂ = 1
p q : 𝕜
hp : 0 ≤ p
hq : 0 ≤ q
hpq : p + q = 1
x : E
hxs : x ∈ s
hx : (p * a₁ + q * a₂) • x = (p * a₁) • x₁ + (q * a₂) • x₂
⊢ ∃ i, ∃ (_ : i ∈ s), ∃ i_1, ∃ (_ : i_1 ∈ t), p • (a₁ • x₁ + b₁ • y₁) + q • (a₂ • x₂ + b₂ • y₂) ∈ [i -[𝕜] i_1]

Tactic:
rcases ht.exists_mem_add_smul_eq hy₁ hy₂ (mul_nonneg hp hb₁) (mul_nonneg hq hb₂) with ⟨y, hyt, hy⟩

Goal state after:
𝕜 : Type u_2
E : Type u_3
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
s t : Set E
hs : Convex 𝕜 s
ht : Convex 𝕜 t
x₁ : E
hx₁ : x₁ ∈ s
y₁ : E
hy₁ : y₁ ∈ t
a₁ b₁ : 𝕜
ha₁ : 0 ≤ a₁
hb₁ : 0 ≤ b₁
hab₁ : a₁ + b₁ = 1
x₂ : E
hx₂ : x₂ ∈ s
y₂ : E
hy₂ : y₂ ∈ t
a₂ b₂ : 𝕜
ha₂ : 0 ≤ a₂
hb₂ : 0 ≤ b₂
hab₂ : a₂ + b₂ = 1
p q : 𝕜
hp : 0 ≤ p
hq : 0 ≤ q
hpq : p + q = 1
x : E
hxs : x ∈ s
hx : (p * a₁ + q * a₂) • x = (p * a₁) • x₁ + (q * a₂) • x₂
y : E
hyt : y ∈ t
hy : (p * b₁ + q * b₂) • y = (p * b₁) • y₁ + (q * b₂) • y₂
⊢ ∃ i, ∃ (_ : i ∈ s), ∃ i_1, ∃ (_ : i_1 ∈ t), p • (a₁ • x₁ + b₁ • y₁) + q • (a₂ • x₂ + b₂ • y₂) ∈ [i -[𝕜] i_1]
Module: Mathlib.Analysis.Convex.Intrinsic

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_5
inst✝⁴ : Ring 𝕜
inst✝³ : AddCommGroup V
inst✝² : Module 𝕜 V
inst✝¹ : TopologicalSpace P
inst✝ : AddTorsor V P
⊢ intrinsicFrontier 𝕜 ∅ = ∅

Tactic:
simp [intrinsicFrontier]

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

Goal state before:
pp: case pos
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : μ t = 0
⊢ 0 = 0 • ⨍ (x : α), f x ∂μ

Tactic:
zero_smul

Goal state after:
case pos
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
inst✝ : IsFiniteMeasure μ
hfi : Integrable f μ
H : ∀ (t : Set α), MeasurableSet t → μ t ≠ 0 → μ tᶜ ≠ 0 → ⨍ (x : α) in t, f x ∂μ = ⨍ (x : α) in tᶜ, f x ∂μ
t : Set α
ht : MeasurableSet t
h₀ : μ t = 0
⊢ 0 = 0
Module: Mathlib.Analysis.Convex.Integral

Goal state before:
pp: α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
s : Set E
t : Set α
f : α → E
inst✝ : IsFiniteMeasure μ
hs : Convex ℝ s
h0 : μ t ≠ 0
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
ht : ⨍ (x : α) in t, f x ∂μ ∈ interior s
⊢ ⨍ (x : α), f x ∂μ ∈ interior s

Tactic:
by
  rw [← measure_toMeasurable] at h0; rw [← restrict_toMeasurable (by finiteness)] at ht
  by_cases h0' : μ (toMeasurable μ t)ᶜ = 0
  · rw [← ae_eq_univ] at h0'
    rwa [restrict_congr_set h0', restrict_univ] at ht
  exact hs.openSegment_interior_closure_subset_interior ht
      (hs.set_average_mem_closure h0' (by finiteness) (ae_restrict_of_ae hfs) hfi.integrableOn)
      (average_mem_openSegment_compl_self (measurableSet_toMeasurable μ t).nullMeasurableSet h0
        h0' hfi)

Goal state after:
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: R : Type u
inst✝² : PartialOrder R
inst✝¹ : Semiring R
M : Type u_9
N : Type u_10
P : Type u_11
inst✝ : IsStrictOrderedRing R
f : StdSimplex R M
g₁ : M → N
g₂ : N → P
a✝ : P
⊢ (map (g₂ ∘ g₁) f).weights a✝ = (map g₂ (map g₁ f)).weights a✝

Tactic:
simp [mapDomain_comp]

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

Goal state before:
pp: case inl
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
C : ℝ
inst✝ : StrictConvexSpace ℝ E
h_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
hC0 : C ≤ 0
⊢ f =ᵐ[μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C

pp: case inr
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
C : ℝ
inst✝ : StrictConvexSpace ℝ E
h_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
hC0 : 0 < C
⊢ f =ᵐ[μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C

Tactic:
· have : f =ᵐ[μ] 0 := h_le.mono fun x hx => norm_le_zero_iff.1 (hx.trans hC0)
    simp only [average_congr this, Pi.zero_apply, average_zero]
    exact Or.inl this

Goal state after:
case inr
α : Type u_1
E : Type u_2
m0 : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
μ : Measure α
f : α → E
C : ℝ
inst✝ : StrictConvexSpace ℝ E
h_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C
hC0 : 0 < C
⊢ f =ᵐ[μ] const α (⨍ (x : α), f x ∂μ) ∨ ‖⨍ (x : α), f x ∂μ‖ < C
Module: Mathlib.Geometry.Convex.ConvexSpace.Defs

Goal state before:
pp: R : Type u_1
M : Type u_3
I : Type u_6
inst✝³ : PartialOrder R
inst✝² : Semiring R
inst✝¹ : IsStrictOrderedRing R
inst✝ : ConvexSpace R M
J : I → Type u_9
s : StdSimplex R I
f : (i : I) → StdSimplex R (J i)
g : (i : I) → J i → M
⊢ sConvexComb (map (fun i ↦ map (g i) (f i)) s) =
    map (Sigma.uncurry g) (sConvexComb (map (fun i ↦ map (fun x ↦ ⟨i, x⟩) (f i)) s))

Tactic:
simp [map_sConvexComb, map_map, Sigma.uncurry]

Goal state after:
Module: Mathlib.Geometry.Convex.Set

Goal state before:
pp: R : Type u_3
X : Type u_5
Y : Type u_6
inst✝⁴ : Semiring R
inst✝³ : PartialOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : ConvexSpace R X
inst✝ : ConvexSpace R Y
f : X → Y
s : Set X
hf : IsAffineMap R f
hs : IsConvexSet R s
w : StdSimplex R Y
hw : ↑w.weights.support ⊆ f '' s
u : Finset X
hus : ↑u ⊆ s
hfu : InjOn f ↑u
huw : Finset.image f u = w.weights.support
⊢ ∑ x ∈ w.weights.support, w.weights x = 1

Tactic:
exact w.total

Goal state after:
Module: Mathlib.Geometry.Convex.ConvexSpace.AffineSpace

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
I : Type u_4
inst✝⁵ : Ring R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AffineSpace V P
s : StdSimplex R I
f : I → P
p : P := ⋯.some
⊢ ((Finsupp.mapDomain f s.weights).sum fun x r ↦ r • (x -ᵥ p)) = s.weights.sum fun x r ↦ r • (f x -ᵥ p)

Tactic:
simp [Finsupp.sum_mapDomain_index, add_smul]

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

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : SMul 𝕜 E
β : Type u_6
inst✝ : Preorder β
s : Set E
f : E → β
t : Set E
hf : QuasiconvexOn 𝕜 s f
hst : t ⊆ s
ht : Convex 𝕜 t
⊢ QuasiconvexOn 𝕜 t f

Tactic:
intro b
  rw [Set.sep_eq_inter_sep hst]
  exact Convex.inter ht (hf b)

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

Goal state before:
pp: 𝕜 : Type u_4
E : Type u_5
inst✝⁴ : Semiring 𝕜
inst✝³ : PartialOrder 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : SMul 𝕜 E
β : Type u_6
inst✝ : Preorder β
s : Set E
f : E → β
t : Set E
hf : QuasiconcaveOn 𝕜 s f
hst : t ⊆ s
ht : Convex 𝕜 t
⊢ QuasiconcaveOn 𝕜 t f

Tactic:
intro b
  rw [Set.sep_eq_inter_sep hst]
  exact Convex.inter ht (hf b)

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

Goal state before:
pp: case e_a
X : Type u_2
inst✝² : ConvexSpace ℝ X
inst✝¹ : MetricSpace X
inst✝ : IsConvexDist X
x y : X
s t s' t' : ℝ
hs : 0 ≤ s
ht : 0 ≤ t
h : s + t = 1
hs' : 0 ≤ s'
ht' : 0 ≤ t'
h' : s' + t' = 1
hss' : s' ≤ s
f : StdSimplex ℝ (Fin 3) := { weights := Finsupp.equivFunOnFinite.symm ![s', s - s', t], nonneg := ⋯, total := ⋯ }
⊢ convexCombPair s' t' hs' ht' h' x y = iConvexComb f ![x, y, y]

Tactic:
· delta convexCombPair
    congr 1
    ext a
    simp [StdSimplex.duple, StdSimplex.map, Finsupp.mapDomain,
      Finsupp.sum_fintype, Fin.sum_univ_succ, f, show t' = s - s' + t by grind]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v w x : V
h : u = w ∧ v = x ∨ u = x ∧ v = w
⊢ G.Adj u v ↔ G.Adj w x

Tactic:
rcases h with hl | hr

Goal state after:
case inl
V : Type u
G : SimpleGraph V
u v w x : V
hl : u = w ∧ v = x
⊢ G.Adj u v ↔ G.Adj w x

case inr
V : Type u
G : SimpleGraph V
u v w x : V
hr : u = x ∧ v = w
⊢ G.Adj u v ↔ G.Adj w x
Module: Mathlib.Analysis.Convex.SpecificFunctions.Pow

Goal state before:
pp: p : ℝ
hp₀ : 0 < p
hp₁ : p < 1
⊢ StrictConcaveOn ℝ≥0 univ fun x ↦ x ^ p

Tactic:
by
  have hp₀' : 0 < 1 / p := div_pos zero_lt_one hp₀
  have hp₁' : 1 < 1 / p := by rw [one_lt_div hp₀]; exact hp₁
  let f := NNReal.orderIsoRpow (1 / p) hp₀'
  have h₁ : StrictConvexOn ℝ≥0 univ f := by
    refine ⟨convex_univ, fun x _ y _ hxy a b ha hb hab => ?_⟩
    exact (strictConvexOn_rpow hp₁').2 x.2 y.2 (by simp [hxy]) ha hb (by simp; norm_cast)
  have h₂ : ∀ x, f.symm x = x ^ p := by simp [f, NNReal.orderIsoRpow_symm_eq]
  refine ⟨convex_univ, fun x mx y my hxy a b ha hb hab => ?_⟩
  simp only [← h₂]
  exact (f.strictConcaveOn_symm h₁).2 mx my hxy ha hb hab

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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y : P
hx : x ∈ s
⊢ SameRay R (x -ᵥ x) (y -ᵥ x)

Tactic:
vsub_self

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y : P
hx : x ∈ s
⊢ SameRay R 0 (y -ᵥ x)
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y : P
hx : x ∈ s
⊢ SameRay R (x -ᵥ x) (x -ᵥ y)

Tactic:
vsub_self

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x y : P
hx : x ∈ s
⊢ SameRay R 0 (x -ᵥ y)
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case inl.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x z : P
t : R
ht0 : 0 ≤ t
ht1 : t ≤ 1
hy : (lineMap x z) t ∈ s
ht1' : t < 1
ht0' : 0 < t
⊢ (1 - t) • (x -ᵥ x - t • (z -ᵥ x)) = t • (t • (z -ᵥ x) + -(z -ᵥ x))

Tactic:
vsub_self

Goal state after:
case inl.inl
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : PartialOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : AddCommGroup V
inst✝¹ : Module R V
inst✝ : AddTorsor V P
s : AffineSubspace R P
x z : P
t : R
ht0 : 0 ≤ t
ht1 : t ≤ 1
hy : (lineMap x z) t ∈ s
ht1' : t < 1
ht0' : 0 < t
⊢ (1 - t) • (0 - t • (z -ᵥ x)) = t • (t • (z -ᵥ x) + -(z -ᵥ x))
Module: Mathlib.Analysis.Convex.StrictConvexBetween

Goal state before:
pp: V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : StrictConvexSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p p₁ p₂ p₃ : P
h : Wbtw ℝ (p₁ -ᵥ p) (p₂ -ᵥ p) (p₃ -ᵥ p) ∧ p₂ ≠ p₁ ∧ p₂ ≠ p₃
hp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p
⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)

Tactic:
Wbtw,

Goal state after:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : NormedSpace ℝ V
inst✝² : StrictConvexSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
p p₁ p₂ p₃ : P
h : p₂ -ᵥ p ∈ affineSegment ℝ (p₁ -ᵥ p) (p₃ -ᵥ p) ∧ p₂ ≠ p₁ ∧ p₂ ≠ p₃
hp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p
⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)
Module: Mathlib.Analysis.Convex.StrictConvexBetween

Goal state before:
pp: case inr
E : Type u_3
PE : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : StrictConvexSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
r : ℝ
x y z : PE
hxy : dist x y = r * dist x z
hyz : dist y z = (1 - r) * dist x z
this : y -ᵥ x ∈ [0 -[ℝ] z -ᵥ x]
hne : x ≠ z
⊢ y = (AffineMap.lineMap x z) r

Tactic:
rw [← dist_ne_zero] at hne

Goal state after:
case inr
E : Type u_3
PE : Type u_5
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : StrictConvexSpace ℝ E
inst✝¹ : MetricSpace PE
inst✝ : NormedAddTorsor E PE
r : ℝ
x y z : PE
hxy : dist x y = r * dist x z
hyz : dist y z = (1 - r) * dist x z
this : y -ᵥ x ∈ [0 -[ℝ] z -ᵥ x]
hne : dist x z ≠ 0
⊢ y = (AffineMap.lineMap x z) r
Module: Mathlib.Analysis.Convex.StrictCombination

Goal state before:
pp: R : Type u_1
V : Type u_2
ι : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module R V
s : Set V
t : Finset ι
w : ι → R
z : ι → V
hs : StrictConvex R s
h0 : ∀ i ∈ t, 0 ≤ w i
h1 : ∑ i ∈ t, w i = 1
i j : ι
hi : i ∈ t
hj : j ∈ t
hij : z i ≠ z j
hi0 : w i ≠ 0
hj0 : w j ≠ 0
hz : ∀ i ∈ t, z i ∈ s
⊢ ∑ k ∈ t, w k • z k ∈ interior s

Tactic:
← t.centerMass_eq_of_sum_1 _ h1

Goal state after:
R : Type u_1
V : Type u_2
ι : Type u_4
inst✝⁵ : Field R
inst✝⁴ : LinearOrder R
inst✝³ : IsStrictOrderedRing R
inst✝² : TopologicalSpace V
inst✝¹ : AddCommGroup V
inst✝ : Module R V
s : Set V
t : Finset ι
w : ι → R
z : ι → V
hs : StrictConvex R s
h0 : ∀ i ∈ t, 0 ≤ w i
h1 : ∑ i ∈ t, w i = 1
i j : ι
hi : i ∈ t
hj : j ∈ t
hij : z i ≠ z j
hi0 : w i ≠ 0
hj0 : w j ≠ 0
hz : ∀ i ∈ t, z i ∈ s
⊢ t.centerMass w z ∈ interior s
Module: Mathlib.Analysis.Convex.StrictCombination

Goal state before:
pp: V : Type u_2
ι : Type u_4
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : StrictConvexSpace ℝ V
t : Finset ι
w : ι → ℝ
p : V
r : ℝ
z : ι → V
h0 : ∀ i ∈ t, 0 ≤ w i
h1 : ∑ i ∈ t, w i = 1
i j : ι
hi : i ∈ t
hj : j ∈ t
hij : z i ≠ z j
hi0 : w i ≠ 0
hj0 : w j ≠ 0
hz : ∀ i ∈ t, z i ∈ closedBall p r
⊢ ∑ k ∈ t, w k • z k ∈ ball p r

Tactic:
← t.centerMass_eq_of_sum_1 _ h1

Goal state after:
V : Type u_2
ι : Type u_4
inst✝² : NormedAddCommGroup V
inst✝¹ : NormedSpace ℝ V
inst✝ : StrictConvexSpace ℝ V
t : Finset ι
w : ι → ℝ
p : V
r : ℝ
z : ι → V
h0 : ∀ i ∈ t, 0 ≤ w i
h1 : ∑ i ∈ t, w i = 1
i j : ι
hi : i ∈ t
hj : j ∈ t
hij : z i ≠ z j
hi0 : w i ≠ 0
hj0 : w j ≠ 0
hz : ∀ i ∈ t, z i ∈ closedBall p r
⊢ t.centerMass w z ∈ ball p r
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
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
hw₁ : ∀ i_1 ∈ t.erase i, w i_1 = 0
hi : i ∈ t
hwi✝ : 0 < w i
hwi : w i = 1
j : ι
hj : j ∈ t
hji : j ≠ i
⊢ 0 • a j = 0

Tactic:
zero_smul

Goal state after:
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
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
hw₁ : ∀ i_1 ∈ t.erase i, w i_1 = 0
hi : i ∈ t
hwi✝ : 0 < w i
hwi : w i = 1
j : ι
hj : j ∈ t
hji : j ≠ i
⊢ 0 = 0
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case mp
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 p : P
hx : x ∉ s
hp : p ∈ s
y : P
⊢ s.SSameSide x y → ∃ a, 0 < a ∧ ∃ b ∈ ↑s, a • (x -ᵥ p) +ᵥ b = y

pp: case mpr
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 p : P
hx : x ∉ s
hp : p ∈ s
y : P
⊢ (∃ a, 0 < a ∧ ∃ b ∈ ↑s, a • (x -ᵥ p) +ᵥ b = y) → s.SSameSide x y

Tactic:
· rw [sSameSide_iff_exists_left hp]
    rintro ⟨-, hy, p₂, hp₂, h | h | ⟨r₁, r₂, hr₁, hr₂, h⟩⟩
    · rw [vsub_eq_zero_iff_eq] at h
      exact False.elim (hx (h.symm ▸ hp))
    · rw [vsub_eq_zero_iff_eq] at h
      exact False.elim (hy (h.symm ▸ hp₂))
    · refine ⟨r₁ / r₂, div_pos hr₁ hr₂, p₂, hp₂, ?_⟩
      rw [div_eq_inv_mul, ← smul_smul, h, smul_smul, inv_mul_cancel₀ hr₂.ne.symm, one_smul,
        vsub_vadd]

Goal state after:
case mpr
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 p : P
hx : x ∉ s
hp : p ∈ s
y : P
⊢ (∃ a, 0 < a ∧ ∃ b ∈ ↑s, a • (x -ᵥ p) +ᵥ b = y) → s.SSameSide x y
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case mpr
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 p : P
hx : x ∉ s
hp : p ∈ s
y : P
⊢ (∃ a, 0 < a ∧ ∃ b ∈ ↑s, a • (x -ᵥ p) +ᵥ b = y) → s.SSameSide x y

Tactic:
rintro ⟨t, ht, p', hp', rfl⟩
    exact sSameSide_smul_vsub_vadd_right hx hp hp' ht

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

Goal state before:
pp: 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
w : Fin (n + 1) → R
hw : ∑ j, w j = 1
i : Fin (n + 1)
⊢ (affineSpan R (Set.range (s.faceOpposite i).points)).WSameSide (s.points i)
      ((Finset.affineCombination R Finset.univ s.points) w) ↔
    0 ≤ w i

Tactic:
s.wSameSide_affineSpan_faceOpposite_point_left_iff hw

Goal state after:
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
w : Fin (n + 1) → R
hw : ∑ j, w j = 1
i : Fin (n + 1)
⊢ 0 ≤ w i ↔ 0 ≤ w i
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in Ioc 0 s, g x ∂volume
h's : μ {a | s < f a} = ∞
⊢ ∞ = ∫⁻ (t : ℝ) in Ioc 0 s, ∞ * ENNReal.ofReal (g t)

Tactic:
have I_pos : ∫⁻ (a : ℝ) in Ioc 0 s, ENNReal.ofReal (g a) ≠ 0 := by
            rw [← ofReal_integral_eq_lintegral_ofReal (g_intble s s_pos).1]
            · simpa only [not_lt, ne_eq, ENNReal.ofReal_eq_zero, not_le] using hs
            · filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht using g_nn _ ht.1

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ f
f_mble : Measurable f
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_mble : Measurable g
g_nn : ∀ t > 0, 0 ≤ g t
f_nonneg : ∀ (ω : α), 0 ≤ f ω
H1 : ¬g =ᵐ[volume.restrict (Ioi 0)] 0
s : ℝ
s_pos : s > 0
hs : 0 < ∫ (x : ℝ) in Ioc 0 s, g x ∂volume
h's : μ {a | s < f a} = ∞
I_pos : ∫⁻ (a : ℝ) in Ioc 0 s, ENNReal.ofReal (g a) ≠ 0
⊢ ∞ = ∫⁻ (t : ℝ) in Ioc 0 s, ∞ * ENNReal.ofReal (g t)
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
⊢ HasTemperateGrowth f ↔ ContDiff ℝ ∞ f ∧ ∀ (n : ℕ), ∃ k, iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k

Tactic:
by
  simp_rw [Asymptotics.isBigO_top]
  congrm ContDiff ℝ ∞ f ∧ (∀ n, ∃ k C, ∀ x, _ ≤ C * ?_)
  rw [norm_pow, Real.norm_of_nonneg (by positivity)]

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

Goal state before:
pp: case neg
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
t : ℝ
ht : 0 < t
hT' : (algebraMap ℝ 𝕜) t ∈ resolventSet 𝕜 T
hT0 : ¬T = 0
c : ℝ
hc0 : c > 0
x : E
hc : (c * ‖x‖) ^ 2 ≤ ‖((algebraMap 𝕜 (E →L[𝕜] E)) ((algebraMap ℝ 𝕜) t)) x - T x‖ ^ 2
hx : ¬x = 0
⊢ re ⟪T x, x⟫_𝕜 / ‖x‖ ^ 2 ≤ (t ^ 2 + ‖T‖ ^ 2) / (2 * t) - c ^ 2 / (2 * t)

Tactic:
algebraMap_apply,

Goal state after:
case neg
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
t : ℝ
ht : 0 < t
hT' : (algebraMap ℝ 𝕜) t ∈ resolventSet 𝕜 T
hT0 : ¬T = 0
c : ℝ
hc0 : c > 0
x : E
hc : (c * ‖x‖) ^ 2 ≤ ‖(algebraMap ℝ 𝕜) t • x - T x‖ ^ 2
hx : ¬x = 0
⊢ re ⟪T x, x⟫_𝕜 / ‖x‖ ^ 2 ≤ (t ^ 2 + ‖T‖ ^ 2) / (2 * t) - c ^ 2 / (2 * t)
Module: Mathlib.Analysis.InnerProductSpace.Rayleigh

Goal state before:
pp: case e'_12
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
T : F →L[ℝ] F
hT : (↑T).IsSymmetric
x₀ : F
e_8✝ : Real.instAddCommGroup = Real.instRCLike.toDivisionRing.toAddCommGroup
e_9✝ : Semiring.toModule ≍ toInnerProductSpaceReal.toModule
y : F
⊢ 2 • ⟪T y, x₀⟫_ℝ = ⟪T y, x₀⟫_ℝ + ⟪(T y, y).1, (T x₀, x₀).2⟫_ℝ

Tactic:
two_smul

Goal state after:
case e'_12
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
T : F →L[ℝ] F
hT : (↑T).IsSymmetric
x₀ : F
e_8✝ : Real.instAddCommGroup = Real.instRCLike.toDivisionRing.toAddCommGroup
e_9✝ : Semiring.toModule ≍ toInnerProductSpaceReal.toModule
y : F
⊢ ⟪T y, x₀⟫_ℝ + ⟪T y, x₀⟫_ℝ = ⟪T y, x₀⟫_ℝ + ⟪(T y, y).1, (T x₀, x₀).2⟫_ℝ
Module: Mathlib.Analysis.InnerProductSpace.Rayleigh

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : Nontrivial E
hT : T.IsSymmetric
this✝ : ProperSpace E
T' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint
x : E
hx : x ≠ 0
H₁ : IsCompact (sphere 0 ‖x‖)
H₂ : (sphere 0 ‖x‖).Nonempty
x₀ : E
hx₀' : x₀ ∈ sphere 0 ‖x‖
hTx₀ : IsMaxOn (↑T').reApplyInnerSelf (sphere 0 ‖x‖) x₀
hx₀ : ‖x₀‖ = ‖x‖
this : IsMaxOn (↑T').reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
⊢ HasEigenvalue T ↑(⨆ x, re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)

Tactic:
have hx₀_ne : x₀ ≠ 0 := by
    have : ‖x₀‖ ≠ 0 := by simp only [hx₀, norm_eq_zero, hx, Ne, not_false_iff]
    simpa [← norm_eq_zero, Ne]

Goal state after:
𝕜 : Type u_1
inst✝⁴ : RCLike 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : FiniteDimensional 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : Nontrivial E
hT : T.IsSymmetric
this✝ : ProperSpace E
T' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint
x : E
hx : x ≠ 0
H₁ : IsCompact (sphere 0 ‖x‖)
H₂ : (sphere 0 ‖x‖).Nonempty
x₀ : E
hx₀' : x₀ ∈ sphere 0 ‖x‖
hTx₀ : IsMaxOn (↑T').reApplyInnerSelf (sphere 0 ‖x‖) x₀
hx₀ : ‖x₀‖ = ‖x‖
this : IsMaxOn (↑T').reApplyInnerSelf (sphere 0 ‖x₀‖) x₀
hx₀_ne : x₀ ≠ 0
⊢ HasEigenvalue T ↑(⨆ x, re ⟪T ↑x, ↑x⟫_𝕜 / ‖↑x‖ ^ 2)