mathlib-initiative/mathlib-tactics huggingface dataset:
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
f : α → E
inst✝ : IsFiniteMeasure μ
hs : StrictConvex ℝ s
hsc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hfs : ∀ᵐ (x : α) ∂μ, f x ∈ s
hfi : Integrable f μ
⊢ f =ᵐ[μ] const α (⨍ (x : α), f x ∂μ) ∨ ⨍ (x : α), f x ∂μ ∈ interior s

Tactic:
have : ∀ {t}, μ t ≠ 0 → (⨍ x in t, f x ∂μ) ∈ s := fun ht =>
    hs.convex.set_average_mem hsc ht (by finiteness) (ae_restrict_of_ae hfs) hfi.integrableOn
  refine (ae_eq_const_or_exists_average_ne_compl hfi).imp_right ?_
  rintro ⟨t, hm, h₀, h₀', hne⟩
  exact
    hs.openSegment_subset (this h₀) (this h₀') hne
      (average_mem_openSegment_compl_self hm.nullMeasurableSet h₀ h₀' hfi)

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

Goal state before:
pp: E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
C : Set E
f : E → ℝ
inst✝ : FiniteDimensional ℝ E
hC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C
hf : ConcaveOn ℝ C f
⊢ LocallyLipschitzOn C f

Tactic:
simpa using hf.neg.locallyLipschitzOn hC

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

Goal state before:
pp: case e_a
R : Type u_1
V : Type u_2
P : Type u_3
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
f : StdSimplex R (StdSimplex R P)
b : P
hL :
  (Finset.affineCombination R (StdSimplex.map convexCombination f).weights.support id)
      ⇑(StdSimplex.map convexCombination f).weights =
    ((StdSimplex.map convexCombination f).weights.support.weightedVSubOfPoint id b)
        ⇑(StdSimplex.map convexCombination f).weights +ᵥ
      b
hR :
  (Finset.affineCombination R f.join.weights.support id) ⇑f.join.weights =
    (f.join.weights.support.weightedVSubOfPoint id b) ⇑f.join.weights +ᵥ b
⊢ ((Finsupp.mapDomain convexCombination f.weights).sum fun x w ↦ w • (x -ᵥ b)) =
    ∑ x ∈ (f.weights.sum fun d r ↦ r • d.weights).support, (f.weights.sum fun d r ↦ r • d.weights) x • (x -ᵥ b)

Tactic:
Finsupp.sum_mapDomain_index (fun _ => by simp) (fun _ _ _ => by simp [add_smul])

Goal state after:
case e_a
R : Type u_1
V : Type u_2
P : Type u_3
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
f : StdSimplex R (StdSimplex R P)
b : P
hL :
  (Finset.affineCombination R (StdSimplex.map convexCombination f).weights.support id)
      ⇑(StdSimplex.map convexCombination f).weights =
    ((StdSimplex.map convexCombination f).weights.support.weightedVSubOfPoint id b)
        ⇑(StdSimplex.map convexCombination f).weights +ᵥ
      b
hR :
  (Finset.affineCombination R f.join.weights.support id) ⇑f.join.weights =
    (f.join.weights.support.weightedVSubOfPoint id b) ⇑f.join.weights +ᵥ b
⊢ (f.weights.sum fun a m ↦ m • (convexCombination a -ᵥ b)) =
    ∑ x ∈ (f.weights.sum fun d r ↦ r • d.weights).support, (f.weights.sum fun d r ↦ r • d.weights) x • (x -ᵥ b)
Module: Mathlib.Geometry.Convex.ConvexSpace.AffineSpace

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
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
f : StdSimplex R (StdSimplex R P)
⊢ convexCombination (StdSimplex.map convexCombination f) = convexCombination f.join

Tactic:
obtain ⟨b⟩ : Nonempty P := inferInstance
  -- Express both sides using weightedVSubOfPoint with base point b
  have hL := Finset.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one
    (f.map convexCombination).weights.support (f.map convexCombination).weights id
    (f.map convexCombination).total b
  have hR := Finset.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one
    f.join.weights.support f.join.weights id f.join.total b
  simp only [convexCombination, hL, hR]
  congr 1
  -- Now show the weighted vector sums are equal
  simp only [Finset.weightedVSubOfPoint_apply, StdSimplex.map, StdSimplex.join, id]
  -- Rewrite LHS using sum_mapDomain_index
  change (Finsupp.mapDomain convexCombination f.weights).sum (fun x w => w • (x -ᵥ b)) = _
  rw [Finsupp.sum_mapDomain_index (fun _ => by simp) (fun _ _ _ => by simp [add_smul])]
  simp only [Finsupp.sum, convexCombination]
  -- Expand convexCombination d using base point b
  conv_lhs =>
    congr; · skip
    ext d
    rw [d.weights.support.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one
        _ _ d.total b, vadd_vsub, Finset.weightedVSubOfPoint_apply]
    simp only [id]
  simp_rw [Finset.smul_sum, smul_smul]
  -- Expand RHS using sum_finsetSum_index
  let h : P → R → V := fun x w => w • (x -ᵥ b)
  have h_rhs : (∑ d ∈ f.weights.support, f.weights d • d.weights).sum h
      = ∑ d ∈ f.weights.support, (f.weights d • d.weights).sum h :=
    (Finsupp.sum_finsetSum_index (h := h) (fun _ => zero_smul _ _)
      (fun _ _ _ => add_smul _ _ _)).symm
  simp only [Finsupp.sum] at h_rhs ⊢
  rw [h_rhs]
  -- Both sides are now double sums; show the inner sums match
  congr 1
  ext d
  simp only [Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul]
  -- Show that d.support = (f.weights d • d.weights).support
  by_cases hd : f.weights d = 0
  · simp [hd]
  · refine Finset.sum_congr ?_ (fun _ _ => rfl)
    ext p
    simp only [Finsupp.mem_support_iff, ne_eq]
    constructor
    · intro hp
      exact (mul_pos ((f.nonneg d).lt_of_ne' hd) ((d.nonneg p).lt_of_ne' hp)).ne'
    · intro hp hp'
      simp only [Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul, hp', mul_zero,
        not_true_eq_false] at hp

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
b : β
⊢ Convex 𝕜 {x | x ∈ t ∧ f x ≤ b}

Tactic:
Set.sep_eq_inter_sep hst

Goal state after:
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
b : β
⊢ Convex 𝕜 (t ∩ {x | x ∈ s ∧ f x ≤ b})
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
b : β
⊢ Convex 𝕜 {x | x ∈ t ∧ b ≤ f x}

Tactic:
Set.sep_eq_inter_sep hst

Goal state after:
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
b : β
⊢ Convex 𝕜 (t ∩ {x | x ∈ s ∧ b ≤ f x})
Module: Mathlib.Analysis.Convex.Radon

Goal state before:
pp: case inr.base
𝕜 : Type u_2
E : Type u_3
inst✝⁵ : Field 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
ι : Type u_1
F : ι → Set E
s : Finset ι
h_convex : ∀ i ∈ s, Convex 𝕜 (F i)
h_inter : ∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty
hn : #s = finrank 𝕜 E + 1
⊢ (⋂ i ∈ s, F i).Nonempty

Tactic:
| base =>

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

Goal state before:
pp: ι : Sort u_1
V : Type u
G H : SimpleGraph V
a b c u v w : V
e : Sym2 V
G₁ G₂ : SimpleGraph V
inst✝² : DecidableEq V
inst✝¹ : Fintype ↑G₁.edgeSet
inst✝ : Fintype ↑G₂.edgeSet
⊢ Fintype ↑(G₁.edgeSet ∩ G₂.edgeSet)

Tactic:
exact Set.fintypeInter _ _

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
⊢ s.WSameSide x y ∧ x ∉ s ∧ y ∉ s ↔ s.WSameSide y x ∧ y ∉ s ∧ x ∉ s

Tactic:
wSameSide_comm,

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
⊢ s.WSameSide y x ∧ x ∉ s ∧ y ∉ s ↔ s.WSameSide y x ∧ y ∉ s ∧ x ∉ s
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
v : V
hv : v ∈ s.direction
⊢ s.WSameSide x (v +ᵥ y) ↔ s.WSameSide x y

Tactic:
wSameSide_comm,

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
v : V
hv : v ∈ s.direction
⊢ s.WSameSide (v +ᵥ y) x ↔ s.WSameSide x y
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
p₁ p₂ x : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
t : R
ht : t ≤ 0
⊢ SameRay R (- -t • (x -ᵥ p₁)) (p₁ -ᵥ x)

Tactic:
neg_smul,

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
p₁ p₂ x : P
hp₁ : p₁ ∈ s
hp₂ : p₂ ∈ s
t : R
ht : t ≤ 0
⊢ SameRay R (-(-t • (x -ᵥ p₁))) (p₁ -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 : Sbtw ℝ p₁ p₂ p₃
⊢ dist p₂ p < max (dist p₁ p) (dist p₃ p)

Tactic:
have hp₁p₃ : p₁ -ᵥ p ≠ p₃ -ᵥ p := by simpa using h.left_ne_right
  rw [Sbtw, ← wbtw_vsub_const_iff p, Wbtw, affineSegment_eq_segment, ← insert_endpoints_openSegment,
    Set.mem_insert_iff, Set.mem_insert_iff] at h
  rcases h with ⟨h | h | h, hp₂p₁, hp₂p₃⟩
  · rw [vsub_left_cancel_iff] at h
    exact False.elim (hp₂p₁ h)
  · rw [vsub_left_cancel_iff] at h
    exact False.elim (hp₂p₃ h)
  · rw [openSegment_eq_image, Set.mem_image] at h
    rcases h with ⟨r, ⟨hr0, hr1⟩, hr⟩
    simp_rw [@dist_eq_norm_vsub V, ← hr]
    exact
      norm_combo_lt_of_ne (le_max_left _ _) (le_max_right _ _) hp₁p₃ (sub_pos.2 hr1) hr0 (by abel)

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 : 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 p₂ : P
h : p₂ ∈ s
⊢ s.WSameSide x y ↔ y ∈ s ∨ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)

Tactic:
wSameSide_comm,

Goal state after:
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 p₂ : P
h : p₂ ∈ s
⊢ s.WSameSide y x ↔ y ∈ s ∨ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
Module: Mathlib.Analysis.Convex.StrictCombination

Goal state before:
pp: case pos
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
w : ι → R
z : ι → V
hs : StrictConvex R s
i : ι
t : Finset ι
hi✝ : i ∉ t
ht :
  (∀ i ∈ t, 0 ≤ w i) →
    ∀ (i j : ι), i ∈ t → j ∈ t → z i ≠ z j → w i ≠ 0 → w j ≠ 0 → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ interior s
h₀ : ∀ i_1 ∈ insert i t, 0 ≤ w i_1
i' j' : ι
hi' : i' ∈ insert i t
hj' : j' ∈ insert i t
hi'j' : z i' ≠ z j'
hi'0 : w i' ≠ 0
hj'0 : w j' ≠ 0
hmem : ∀ i_1 ∈ insert i t, z i_1 ∈ s
zi : z i ∈ s
hs₀ : ∀ j ∈ t, 0 ≤ w j
hsum_t : ¬∑ j ∈ t, w j = 0
hi : w i = 0
⊢ (w i / (w i + ∑ j ∈ t, w j)) • z i + ((∑ j ∈ t, w j) / (w i + ∑ j ∈ t, w j)) • t.centerMass w z ∈ interior s

Tactic:
simp only [hi, zero_add, zero_div, zero_smul, ne_eq, hsum_t, not_false_eq_true, div_self,
        one_smul]

Goal state after:
case pos
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
w : ι → R
z : ι → V
hs : StrictConvex R s
i : ι
t : Finset ι
hi✝ : i ∉ t
ht :
  (∀ i ∈ t, 0 ≤ w i) →
    ∀ (i j : ι), i ∈ t → j ∈ t → z i ≠ z j → w i ≠ 0 → w j ≠ 0 → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ interior s
h₀ : ∀ i_1 ∈ insert i t, 0 ≤ w i_1
i' j' : ι
hi' : i' ∈ insert i t
hj' : j' ∈ insert i t
hi'j' : z i' ≠ z j'
hi'0 : w i' ≠ 0
hj'0 : w j' ≠ 0
hmem : ∀ i_1 ∈ insert i t, z i_1 ∈ s
zi : z i ∈ s
hs₀ : ∀ j ∈ t, 0 ≤ w j
hsum_t : ¬∑ j ∈ t, w j = 0
hi : w i = 0
⊢ t.centerMass w z ∈ interior s
Module: Mathlib.Analysis.Convex.StrictCombination

Goal state before:
pp: case inr
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
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
hr : r ≠ 0
⊢ t.centerMass w z ∈ ball p r

Tactic:
rw [← interior_closedBall _ hr]

Goal state after:
case inr
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
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
hr : r ≠ 0
⊢ t.centerMass w z ∈ interior (closedBall p r)
Module: Mathlib.Analysis.Convex.Strong

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
φ : ℝ → ℝ
s : Set E
f : E → ℝ
hf : UniformConvexOn s φ f
hφ : ∀ (r : ℝ), r ≠ 0 → 0 < φ r
⊢ StrictConvexOn ℝ s f

Tactic:
by
  refine ⟨hf.1, fun x hx y hy hxy a b ha hb hab ↦ (hf.2 hx hy ha.le hb.le hab).trans_lt <|
    sub_lt_self _ ?_⟩
  rw [← sub_ne_zero, ← norm_pos_iff] at hxy
  positivity [hφ _ hxy.ne']

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

Goal state before:
pp: case refine_2
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 : P
h : s.SOppSide x y
p : P
hp : p ∈ s
hw : Wbtw R x p y
⊢ p ≠ y

Tactic:
· rintro rfl
    exact h.2.2 hp

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

Goal state before:
pp: case mp.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 p : P
hx : x ∉ s
hp : p ∈ s
y p₂ : P
hp₂ : p₂ ∈ s
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : 0 < r₂
h : r₁ • (x -ᵥ p) = r₂ • (p₂ -ᵥ y)
⊢ r₂⁻¹ • -r₁ • (x -ᵥ p) +ᵥ p₂ = y

Tactic:
neg_smul,

Goal state after:
case mp.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 p : P
hx : x ∉ s
hp : p ∈ s
y p₂ : P
hp₂ : p₂ ∈ s
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : 0 < r₂
h : r₁ • (x -ᵥ p) = r₂ • (p₂ -ᵥ y)
⊢ r₂⁻¹ • -(r₁ • (x -ᵥ p)) +ᵥ p₂ = y
Module: Mathlib.Analysis.Convex.Side

Goal state before:
pp: case mp.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 p : P
hx : x ∉ s
hp : p ∈ s
y : P
hy : y ∉ s
p₂ : P
hp₂ : p₂ ∈ s
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : 0 < r₂
h : r₁ • (x -ᵥ p) = r₂ • (p₂ -ᵥ y)
⊢ r₂⁻¹ • -r₁ • (x -ᵥ p) +ᵥ p₂ = y

Tactic:
neg_smul,

Goal state after:
case mp.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 p : P
hx : x ∉ s
hp : p ∈ s
y : P
hy : y ∉ s
p₂ : P
hp₂ : p₂ ∈ s
r₁ r₂ : R
hr₁ : 0 < r₁
hr₂ : 0 < r₂
h : r₁ • (x -ᵥ p) = r₂ • (p₂ -ᵥ y)
⊢ r₂⁻¹ • -(r₁ • (x -ᵥ p)) +ᵥ p₂ = y
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 ((Finset.affineCombination R Finset.univ s.points) w)
      (s.points i) ↔
    0 ≤ w i

Tactic:
wSameSide_comm,

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)
⊢ (affineSpan R (Set.range (s.faceOpposite i).points)).WSameSide (s.points i)
      ((Finset.affineCombination R Finset.univ s.points) w) ↔
    0 ≤ w i
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: case calc_1
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)
h_pos : ∀ (x : E), 0 ≤ (1 + ‖x‖) ^ (-r)
h_int : ∀ (t : ℝ), 0 < t → μ {a | t ≤ (1 + ‖a‖) ^ (-r)} = μ (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
f : ℝ → ℝ≥0∞ := fun t ↦ μ (Metric.closedBall 0 (t ^ (-r⁻¹) - 1))
mB : ℝ≥0∞ := μ (Metric.ball 0 1)
h_int' : ∀ t ∈ Ioc 0 1, f t = ENNReal.ofReal ((t ^ (-r⁻¹) - 1) ^ finrank ℝ E) * mB
⊢ (∫⁻ (a : ℝ) in Ioc 0 1, ENNReal.ofReal ((a ^ (-r⁻¹) - 1) ^ finrank ℝ E) ∂volume) * μ (Metric.ball 0 1) < ∞

Tactic:
exact ENNReal.mul_lt_top
      (finite_integral_rpow_sub_one_pow_aux (finrank ℝ E) hnr) measure_ball_lt_top

Goal state after:
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 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
t : ℝ
t_pos : t > 0
⊢ IntegrableOn G (Ioc t 0) volume

Tactic:
Ioc_eq_empty_of_le t_pos.lt.le

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
f_nn : 0 ≤ᵐ[μ] f
f_mble : AEMeasurable f μ
g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t
g_nn : ∀ᵐ (t : ℝ) ∂volume.restrict (Ioi 0), 0 ≤ g t
G : ℝ → ℝ
G_mble : Measurable G
G_nn : 0 ≤ G
g_eq_G : g =ᵐ[volume.restrict (Ioi 0)] G
g_eq_G_on : ∀ (t : ℝ), g =ᵐ[volume.restrict (Ioc 0 t)] G
t : ℝ
t_pos : t > 0
⊢ IntegrableOn G ∅ volume
Module: Mathlib.MeasureTheory.Integral.Layercake

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
M : ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᵐ[μ] f
f_bdd : f ≤ᵐ[μ] fun x ↦ M
t : ℝ
ht : t ∈ Ioi 0 \ Ioc 0 M
htM : M < t
⊢ μ {a | M < f a} = 0

Tactic:
rw [measure_eq_zero_iff_ae_notMem]

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f : α → ℝ
M : ℝ
f_intble : Integrable f μ
f_nn : 0 ≤ᵐ[μ] f
f_bdd : f ≤ᵐ[μ] fun x ↦ M
t : ℝ
ht : t ∈ Ioi 0 \ Ioc 0 M
htM : M < t
⊢ ∀ᵐ (a : α) ∂μ, a ∉ {a | M < f a}
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
⊢ HasTemperateGrowth fun x ↦ ‖x‖ ^ 2

Tactic:
apply _root_.Function.HasTemperateGrowth.of_fderiv (C := 1) (k := 2)
  · rw [fderiv_norm_sq]
    convert! (2 • innerSL ℝ).hasTemperateGrowth
  · exact .norm_sq ℝ differentiable_id
  · intro x
    rw [norm_pow, norm_norm, one_mul, add_pow_two]
    exact le_add_of_nonneg_left (by positivity)

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Compact.Basic

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜₁
inst✝⁷ : SeminormedRing 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁶ : TopologicalSpace M₁
inst✝⁵ : AddCommMonoid M₁
inst✝⁴ : TopologicalSpace M₂
inst✝³ : AddCommMonoid M₂
inst✝² : Module 𝕜₁ M₁
inst✝¹ : Module 𝕜₂ M₂
inst✝ : ContinuousConstSMul 𝕜₂ M₂
f : M₁ →ₛₗ[σ₁₂] M₂
hf : IsCompactOperator ⇑f
S : Set M₁
hS : IsVonNBounded 𝕜₁ S
K : Set M₂
hK : IsCompact K
hKf : ⇑f ⁻¹' K ∈ 𝓝 0
r : ℝ
hr : r > 0
hrS : ∀ (c : 𝕜₁), r ≤ ‖c‖ → S ⊆ c • ⇑f ⁻¹' K
c : 𝕜₁
hc : r < ‖c‖
this : c ≠ 0 := ne_zero_of_norm_ne_zero (Ne.symm (LT.lt.ne (LT.lt.trans hr hc)))
⊢ S ⊆ ⇑f ⁻¹' (σ₁₂ c • K)

Tactic:
this.isUnit.preimage_smul_setₛₗ σ₁₂

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜₁
inst✝⁷ : SeminormedRing 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁶ : TopologicalSpace M₁
inst✝⁵ : AddCommMonoid M₁
inst✝⁴ : TopologicalSpace M₂
inst✝³ : AddCommMonoid M₂
inst✝² : Module 𝕜₁ M₁
inst✝¹ : Module 𝕜₂ M₂
inst✝ : ContinuousConstSMul 𝕜₂ M₂
f : M₁ →ₛₗ[σ₁₂] M₂
hf : IsCompactOperator ⇑f
S : Set M₁
hS : IsVonNBounded 𝕜₁ S
K : Set M₂
hK : IsCompact K
hKf : ⇑f ⁻¹' K ∈ 𝓝 0
r : ℝ
hr : r > 0
hrS : ∀ (c : 𝕜₁), r ≤ ‖c‖ → S ⊆ c • ⇑f ⁻¹' K
c : 𝕜₁
hc : r < ‖c‖
this : c ≠ 0 := ne_zero_of_norm_ne_zero (Ne.symm (LT.lt.ne (LT.lt.trans hr hc)))
⊢ S ⊆ c • ⇑f ⁻¹' K
Module: Mathlib.Analysis.Normed.Operator.Compact.Basic

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜₁
inst✝⁷ : SeminormedRing 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁶ : TopologicalSpace M₁
inst✝⁵ : AddCommMonoid M₁
inst✝⁴ : TopologicalSpace M₂
inst✝³ : AddCommMonoid M₂
inst✝² : Module 𝕜₁ M₁
inst✝¹ : Module 𝕜₂ M₂
inst✝ : ContinuousConstSMul 𝕜₂ M₂
f : M₁ →ₛₗ[σ₁₂] M₂
hf : IsCompactOperator ⇑f
S : Set M₁
hS : IsVonNBounded 𝕜₁ S
K : Set M₂
hK : IsCompact K
hKf : ⇑f ⁻¹' K ∈ 𝓝 0
r : ℝ
hr : r > 0
hrS : ∀ (c : 𝕜₁), r ≤ ‖c‖ → S ⊆ c • ⇑f ⁻¹' K
c : 𝕜₁
hc : r < ‖c‖
this : c ≠ 0 := ne_zero_of_norm_ne_zero (Ne.symm (LT.lt.ne (LT.lt.trans hr hc)))
⊢ ⇑f '' S ⊆ σ₁₂ c • K

Tactic:
rw [image_subset_iff, this.isUnit.preimage_smul_setₛₗ σ₁₂]

Goal state after:
𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NontriviallyNormedField 𝕜₁
inst✝⁷ : SeminormedRing 𝕜₂
σ₁₂ : 𝕜₁ →+* 𝕜₂
M₁ : Type u_3
M₂ : Type u_4
inst✝⁶ : TopologicalSpace M₁
inst✝⁵ : AddCommMonoid M₁
inst✝⁴ : TopologicalSpace M₂
inst✝³ : AddCommMonoid M₂
inst✝² : Module 𝕜₁ M₁
inst✝¹ : Module 𝕜₂ M₂
inst✝ : ContinuousConstSMul 𝕜₂ M₂
f : M₁ →ₛₗ[σ₁₂] M₂
hf : IsCompactOperator ⇑f
S : Set M₁
hS : IsVonNBounded 𝕜₁ S
K : Set M₂
hK : IsCompact K
hKf : ⇑f ⁻¹' K ∈ 𝓝 0
r : ℝ
hr : r > 0
hrS : ∀ (c : 𝕜₁), r ≤ ‖c‖ → S ⊆ c • ⇑f ⁻¹' K
c : 𝕜₁
hc : r < ‖c‖
this : c ≠ 0 := ne_zero_of_norm_ne_zero (Ne.symm (LT.lt.ne (LT.lt.trans hr hc)))
⊢ S ⊆ c • ⇑f ⁻¹' K