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

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsTorsionFree R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : w i₁ + (w i₂ + w i₃) = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' :
  (Finset.affineCombination R Finset.univ t.points) w ∈
    line[R, (Finset.affineCombination R Finset.univ t.points) (Pi.single i₁ 1),
      (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)]
h₂' :
  (Finset.affineCombination R Finset.univ t.points) w ∈ line[R, t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂]
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hss : SignType.sign (w i₃) = 1
hs' : SignType.sign (∑ i, w i) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), 0 < w i
hle : ¬w i₁ < 1
⊢ False

Tactic:
not_lt

Goal state after:
R : Type u_1
V : Type u_2
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : LinearOrder R
inst✝⁴ : IsStrictOrderedRing R
inst✝³ : AddCommGroup V
inst✝² : Module R V
inst✝¹ : AddTorsor V P
inst✝ : IsTorsionFree R V
t : Affine.Triangle R P
i₁ i₂ i₃ : Fin 3
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
h3 : ∀ (i : Fin 3), i = i₁ ∨ i = i₂ ∨ i = i₃
hu : Finset.univ = {i₁, i₂, i₃}
r₁ : R
hr₁0 : 0 < r₁
hr₁1 : r₁ < 1
h₁ : Sbtw R (t.points i₂) ((lineMap (t.points i₂) (t.points i₃)) r₁) (t.points i₃)
r₂ : R
hr₂0 : 0 < r₂
hr₂1 : r₂ < 1
h₂ : Sbtw R (t.points i₁) ((lineMap (t.points i₁) (t.points i₃)) r₂) (t.points i₃)
w : Fin (2 + 1) → R
hw : w i₁ + (w i₂ + w i₃) = 1
hp : (Finset.affineCombination R Finset.univ t.points) w ∈ affineSpan R (Set.range t.points)
h₁' :
  (Finset.affineCombination R Finset.univ t.points) w ∈
    line[R, (Finset.affineCombination R Finset.univ t.points) (Pi.single i₁ 1),
      (Finset.affineCombination R Finset.univ t.points) (Finset.affineCombinationLineMapWeights i₂ i₃ r₁)]
h₂' :
  (Finset.affineCombination R Finset.univ t.points) w ∈ line[R, t.points i₂, (lineMap (t.points i₁) (t.points i₃)) r₂]
h₁s : SignType.sign (w i₂) = SignType.sign (w i₃)
h₂s : SignType.sign (w i₁) = SignType.sign (w i₃)
hss : SignType.sign (w i₃) = 1
hs' : SignType.sign (∑ i, w i) = SignType.sign (w i₃)
hs : ∀ (i : Fin 3), 0 < w i
hle : 1 ≤ w i₁
⊢ False
Module: Mathlib.Analysis.Convex.Cone.Dual

Goal state before:
pp: E : Type u_1
inst✝⁵ : TopologicalSpace E
inst✝⁴ : AddCommGroup E
inst✝³ : IsTopologicalAddGroup E
inst✝² : Module ℝ E
inst✝¹ : ContinuousSMul ℝ E
inst✝ : LocallyConvexSpace ℝ E
K : Set E
C : ProperCone ℝ E
hKconv : Convex ℝ K
hKcomp : IsCompact K
hKC : Disjoint K ↑C
⊢ ∃ f, (∀ x ∈ C, 0 ≤ f x) ∧ ∀ x ∈ K, f x < 0

Tactic:
obtain rfl | ⟨x₀, hx₀⟩ := K.eq_empty_or_nonempty
  · exact ⟨0, by simp⟩
  obtain ⟨f, u, v, hu, huv, hv⟩ :=
    geometric_hahn_banach_compact_closed hKconv hKcomp C.convex C.isClosed hKC
  have hv₀ : v < 0 := by simpa using hv 0 C.zero_mem
  refine ⟨f, fun x hx ↦ ?_, fun x hx ↦ (hu x hx).trans_le <| huv.le.trans hv₀.le⟩
  by_contra! hx₀
  simpa [hx₀.ne] using hv ((v * (f x)⁻¹) • x)
    (C.smul_mem hx <| le_of_lt <| mul_pos_of_neg_of_neg hv₀ <| inv_neg''.2 hx₀)

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

Goal state before:
pp: case inl
𝕜 : Type u_1
E : Type u_2
inst✝⁶ : TopologicalSpace 𝕜
inst✝⁵ : Ring 𝕜
inst✝⁴ : LinearOrder 𝕜
inst✝³ : IsStrictOrderedRing 𝕜
inst✝² : AddCommMonoid E
inst✝¹ : TopologicalSpace E
inst✝ : Module 𝕜 E
A : Set E
hA : Convex 𝕜 A
hAB : IsExposed 𝕜 A ∅
⊢ Convex 𝕜 ∅

Tactic:
exact convex_empty

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
hc : ConvexIndependent 𝕜 p
i j : ι
hij : p i = p j
⊢ p j ∈ (convexHull 𝕜) (p '' {j})

Tactic:
Set.image_singleton,

Goal state after:
𝕜 : Type u_1
E : Type u_2
ι : Type u_3
inst✝³ : Semiring 𝕜
inst✝² : PartialOrder 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : ι → E
hc : ConvexIndependent 𝕜 p
i j : ι
hij : p i = p j
⊢ p j ∈ (convexHull 𝕜) {p j}
Module: Mathlib.Analysis.Normed.Affine.Convex

Goal state before:
pp: case refine_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s t : Set E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht : t ∈ 𝓝ˢ s
U : Set E
hU₁ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hU₂ : s ⊆ U
hU₃ : U ⊆ t
V : Set E
hV₁ : V ∈ 𝓝 0
hV₂ : V + s ⊆ U
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
hb₁ : 0 ∈ interior ((convexHull ℝ) (range ⇑b))
hb₂ : (convexHull ℝ) (range ⇑b) ⊆ V
u : Finset E
hu₁ : ↑u ⊆ s
hu₃ : s ⊆ ⋃ i ∈ ↑u, interior ((convexHull ℝ) (range ⇑b)) + {i}
⊢ s ⊆ interior ((convexHull ℝ) (⇑b '' ↑Finset.univ + ↑u))

Tactic:
Finset.coe_univ,

Goal state after:
case refine_1
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s t : Set E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht : t ∈ 𝓝ˢ s
U : Set E
hU₁ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hU₂ : s ⊆ U
hU₃ : U ⊆ t
V : Set E
hV₁ : V ∈ 𝓝 0
hV₂ : V + s ⊆ U
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
hb₁ : 0 ∈ interior ((convexHull ℝ) (range ⇑b))
hb₂ : (convexHull ℝ) (range ⇑b) ⊆ V
u : Finset E
hu₁ : ↑u ⊆ s
hu₃ : s ⊆ ⋃ i ∈ ↑u, interior ((convexHull ℝ) (range ⇑b)) + {i}
⊢ s ⊆ interior ((convexHull ℝ) (⇑b '' univ + ↑u))
Module: Mathlib.Analysis.Normed.Affine.Convex

Goal state before:
pp: case refine_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s t : Set E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht : t ∈ 𝓝ˢ s
U : Set E
hU₁ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hU₂ : s ⊆ U
hU₃ : U ⊆ t
V : Set E
hV₁ : V ∈ 𝓝 0
hV₂ : V + s ⊆ U
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
hb₁ : 0 ∈ interior ((convexHull ℝ) (range ⇑b))
hb₂ : (convexHull ℝ) (range ⇑b) ⊆ V
u : Finset E
hu₁ : ↑u ⊆ s
hu₃ : s ⊆ ⋃ i ∈ ↑u, interior ((convexHull ℝ) (range ⇑b)) + {i}
⊢ (convexHull ℝ) (⇑b '' ↑Finset.univ + ↑u) ⊆ t

Tactic:
Finset.coe_univ,

Goal state after:
case refine_2
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : FiniteDimensional ℝ E
s t : Set E
hs₁ : Convex ℝ s
hs₂ : IsCompact s
ht : t ∈ 𝓝ˢ s
U : Set E
hU₁ : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] U
hU₂ : s ⊆ U
hU₃ : U ⊆ t
V : Set E
hV₁ : V ∈ 𝓝 0
hV₂ : V + s ⊆ U
b : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E
hb₁ : 0 ∈ interior ((convexHull ℝ) (range ⇑b))
hb₂ : (convexHull ℝ) (range ⇑b) ⊆ V
u : Finset E
hu₁ : ↑u ⊆ s
hu₃ : s ⊆ ⋃ i ∈ ↑u, interior ((convexHull ℝ) (range ⇑b)) + {i}
⊢ (convexHull ℝ) (⇑b '' univ + ↑u) ⊆ t
Module: Mathlib.Analysis.Convex.Measure

Goal state before:
pp: case inr
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s✝ : Set E
hs✝ : Convex ℝ s✝
x : E
hx✝ : x ∈ interior s✝
s : Set E
hs : Convex ℝ s
hx : x ∈ interior s
hb : μ (interior s) ≠ ∞
⊢ μ (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s) ≤ μ (interior s)

Tactic:
set d : ℕ := Module.finrank ℝ E

Goal state after:
case inr
E : Type u_1
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
μ : Measure E
inst✝ : μ.IsAddHaarMeasure
s✝ : Set E
hs✝ : Convex ℝ s✝
x : E
hx✝ : x ∈ interior s✝
s : Set E
hs : Convex ℝ s
hx : x ∈ interior s
hb : μ (interior s) ≠ ∞
d : ℕ := finrank ℝ E
⊢ μ (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s) ≤ μ (interior s)
Module: Mathlib.Geometry.Convex.ConvexSpace.AffineSpace

Goal state before:
pp: case neg.mpr
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
h : P → R → V := ⋯
h_rhs :
  ∑ a ∈ (∑ d ∈ f.weights.support, f.weights d • d.weights).support,
      h a ((∑ d ∈ f.weights.support, f.weights d • d.weights) a) =
    ∑ d ∈ f.weights.support, ∑ a ∈ (f.weights d • d.weights).support, h a ((f.weights d • d.weights) a)
d : StdSimplex R P
hd : ¬f.weights d = 0
p : P
⊢ ¬(f.weights d • d.weights) p = 0 → ¬d.weights p = 0

Tactic:
· 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.MetricSpace

Goal state before:
pp: X : Type u_2
inst✝³ : ConvexSpace ℝ X
inst✝² : MetricSpace X
inst✝¹ : IsConvexDist X
T : Type u_3
inst✝ : TopologicalSpace T
f : T → ℝ
hf : Continuous f
hf0 : ∀ (t : T), 0 ≤ f t
hf1 : ∀ (t : T), f t ≤ 1
x y : T → X
hx : ContinuousOn x (f ⁻¹' {0}ᶜ)
hy : ContinuousOn y (f ⁻¹' {1}ᶜ)
hx' : Bornology.IsBounded (Set.range x)
hy' : Bornology.IsBounded (Set.range y)
D : ℝ
hD' : 0 < D
hD : ∀ (t₁ t₂ : T), dist (x t₁) (y t₂) ≤ D
t : T
ht : f t ∈ Set.Ioo 0 1
⊢ ∀ (x : ↑(f ⁻¹' Set.Ioo 0 1)), ↑x ∈ f ⁻¹' {0}ᶜ

Tactic:
simp_all; grind

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

Goal state before:
pp: X : Type u_2
inst✝³ : ConvexSpace ℝ X
inst✝² : MetricSpace X
inst✝¹ : IsConvexDist X
T : Type u_3
inst✝ : TopologicalSpace T
f : T → ℝ
hf : Continuous f
hf0 : ∀ (t : T), 0 ≤ f t
hf1 : ∀ (t : T), f t ≤ 1
x y : T → X
hx : ContinuousOn x (f ⁻¹' {0}ᶜ)
hy : ContinuousOn y (f ⁻¹' {1}ᶜ)
hx' : Bornology.IsBounded (Set.range x)
hy' : Bornology.IsBounded (Set.range y)
D : ℝ
hD' : 0 < D
hD : ∀ (t₁ t₂ : T), dist (x t₁) (y t₂) ≤ D
t : T
ht : f t ∈ Set.Ioo 0 1
⊢ ∀ (x : ↑(f ⁻¹' Set.Ioo 0 1)), ↑x ∈ f ⁻¹' {1}ᶜ

Tactic:
simp_all; grind

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

Goal state before:
pp: case h
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 k : ℕ
h_card : finrank 𝕜 E + 1 ≤ k
hk :
  ∀ {ι : Type u_1} {F : ι → Set E} {s : Finset ι},
    (∀ i ∈ s, Convex 𝕜 (F i)) →
      (∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) → #s = k → (⋂ i ∈ s, F i).Nonempty
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 = k + 1
a : ↥s → E := fun i ↦ ⋯.some
h_ind : ¬AffineIndependent 𝕜 a
I : Set ↥s
p : E
hp_I : p ∈ (convexHull 𝕜) (a '' I)
hp_Ic : p ∈ (convexHull 𝕜) (a '' Iᶜ)
i : ↥s
J : Set ↥s
hi : i ∈ J
v : E
j : ↥s
hj : j ∈ Jᶜ
hj_v : a j = v
⊢ a j ∈ F ↑i

Tactic:
apply mem_of_subset_of_mem (s₁ := ⋂ k ∈ (s.erase j), F k)

Goal state after:
case h.h
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 k : ℕ
h_card : finrank 𝕜 E + 1 ≤ k
hk :
  ∀ {ι : Type u_1} {F : ι → Set E} {s : Finset ι},
    (∀ i ∈ s, Convex 𝕜 (F i)) →
      (∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) → #s = k → (⋂ i ∈ s, F i).Nonempty
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 = k + 1
a : ↥s → E := ⋯
h_ind : ¬AffineIndependent 𝕜 a
I : Set ↥s
p : E
hp_I : p ∈ (convexHull 𝕜) (a '' I)
hp_Ic : p ∈ (convexHull 𝕜) (a '' Iᶜ)
i : ↥s
J : Set ↥s
hi : i ∈ J
v : E
j : ↥s
hj : j ∈ Jᶜ
hj_v : a j = v
⊢ ⋂ k ∈ s.erase ↑j, F k ⊆ F ↑i

case h.a
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 k : ℕ
h_card : finrank 𝕜 E + 1 ≤ k
hk :
  ∀ {ι : Type u_1} {F : ι → Set E} {s : Finset ι},
    (∀ i ∈ s, Convex 𝕜 (F i)) →
      (∀ I ⊆ s, #I ≤ finrank 𝕜 E + 1 → (⋂ i ∈ I, F i).Nonempty) → #s = k → (⋂ i ∈ s, F i).Nonempty
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 = k + 1
a : ↥s → E := ⋯
h_ind : ¬AffineIndependent 𝕜 a
I : Set ↥s
p : E
hp_I : p ∈ (convexHull 𝕜) (a '' I)
hp_Ic : p ∈ (convexHull 𝕜) (a '' Iᶜ)
i : ↥s
J : Set ↥s
hi : i ∈ J
v : E
j : ↥s
hj : j ∈ Jᶜ
hj_v : a j = v
⊢ a j ∈ ⋂ k ∈ s.erase ↑j, F k
Module: Mathlib.Analysis.Convex.Radon

Goal state before:
pp: 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
F : Finset (Set E)
h_card : finrank 𝕜 E + 1 ≤ #F
h_convex : ∀ X ∈ F, Convex 𝕜 X
h_inter : ∀ G ⊆ F, #G = finrank 𝕜 E + 1 → (⋂₀ ↑G).Nonempty
I : Finset (Set E)
hI_ss : I ⊆ F
hI_card : #I ≤ finrank 𝕜 E + 1
J : Finset (Set E)
left✝ : I ⊆ J
hJ_ss : J ⊆ F
hJ_card : #J = finrank 𝕜 E + 1
⊢ ↑I ⊆ ↑J

Tactic:
simpa [hI_ss]

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

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : LinearOrder 𝕜
inst✝⁵ : IsStrictOrderedRing 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : FiniteDimensional 𝕜 E
inst✝¹ : TopologicalSpace E
inst✝ : T2Space E
F : Set (Set E)
h_card : ↑(finrank 𝕜 E) + 1 ≤ F.encard
h_convex : ∀ X ∈ F, Convex 𝕜 X
h_compact : ∀ X ∈ F, IsCompact X
h_inter : ∀ (G : Finset (Set E)), ↑G ⊆ F → #G = finrank 𝕜 E + 1 → (⋂₀ ↑G).Nonempty
I : Finset (Set E)
hI_ss : ↑I ⊆ F
hI_card : #I ≤ finrank 𝕜 E + 1
J : Set (Set E)
left✝ : ↑I ⊆ J
hJ_ss : J ⊆ F
hJ_card : J.encard = ↑(finrank 𝕜 E) + 1
⊢ ↑I ⊆ ?m.107

Tactic:
simpa [hI_ss]

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
s : Set V
⊢ G.edgeSet ⊆ s.sym2 ↔ G.support ⊆ s

Tactic:
by
  refine ⟨fun h u hu ↦ ?_, fun h e hadj ↦ ?_⟩
  · have ⟨v, huv⟩ := hu
    exact (Set.mk_mem_sym2_iff.mp <| h huv).left
  · cases e
    exact ⟨h hadj.mem_support_left, h hadj.mem_support_right⟩

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.SSameSide x y ↔ s.SSameSide y x

Tactic:
by
  rw [SSameSide, SSameSide, wSameSide_comm, and_comm (b := x ∉ s)]

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

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
inst✝⁹ : Field 𝕜
inst✝⁸ : LinearOrder 𝕜
inst✝⁷ : IsStrictOrderedRing 𝕜
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module 𝕜 V
s : Set V
x y : V
inst✝⁴ : TopologicalSpace 𝕜
inst✝³ : OrderTopology 𝕜
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul 𝕜 V
hsxy : IsVisible 𝕜 s x y
hy : y ∈ interior s
hxy : x ≠ y
⊢ s ∈ 𝓝 ((lineMap y x) 0)

Tactic:
by simpa using mem_interior_iff_mem_nhds.1 hy

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

Goal state before:
pp: V : Type u_2
inst✝⁴ : AddCommGroup V
inst✝³ : Module ℝ V
s : Set V
y : V
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddGroup V
inst✝ : ContinuousSMul ℝ V
hs : IsClosed[inst✝²] s
hy : y ∈ s
x : V
⊢ ∃ z ∈ s, Wbtw ℝ x z y ∧ IsVisible ℝ s x z

Tactic:
let t : Set ℝ := Ici 0 ∩ lineMap x y ⁻¹' s
  have ht₁ : 1 ∈ t := by simpa [t]
  have ht : BddBelow t := bddBelow_Ici.inter_of_left
  let δ : ℝ := sInf t
  have hδ₁ : δ ≤ 1 := csInf_le ht ht₁
  obtain ⟨hδ₀, hδ⟩ : 0 ≤ δ ∧ lineMap x y δ ∈ s :=
    (isClosed_Ici.inter <| hs.preimage lineMap_continuous).csInf_mem ⟨1, ht₁⟩ ht
  refine ⟨lineMap x y δ, hδ, wbtw_lineMap_iff.2 <| .inr ⟨hδ₀, hδ₁⟩, ?_⟩
  rintro _ hε ⟨⟨ε, ⟨hε₀, hε₁⟩, rfl⟩, -, h⟩
  replace hδ₀ : 0 < δ := hδ₀.lt_of_ne' <| by rintro hδ₀; simp [hδ₀] at h
  replace hε₁ : ε < 1 := hε₁.lt_of_ne <| by rintro rfl; simp at h
  rw [lineMap_lineMap_right] at hε
  exact (csInf_le ht ⟨mul_nonneg hε₀ hδ₀.le, hε⟩).not_gt <| mul_lt_of_lt_one_left hδ₀ hε₁

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

Goal state before:
pp: α : Type u_1
R : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : LinearOrder R
g : α → R
⊢ {t | μ {a | t ≤ g a} ≠ μ {a | t < g a}}.Countable

Tactic:
let F : R → ℝ≥0∞ := fun t ↦ μ {a : α | t ≤ g a}

Goal state after:
α : Type u_1
R : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : LinearOrder R
g : α → R
F : R → ℝ≥0∞ := fun t ↦ μ {a | t ≤ g a}
⊢ {t | μ {a | t ≤ g a} ≠ μ {a | t < g a}}.Countable
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case succ
R : Type u_3
E : Type u_5
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedRing R
inst✝ : NormedAlgebra ℝ R
f : E → R
hf : HasTemperateGrowth f
k : ℕ
IH : HasTemperateGrowth (f ^ k)
⊢ HasTemperateGrowth (f ^ (k + 1))

Tactic:
| succ k IH =>

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

Goal state before:
pp: case mp.refine_2
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
r : ℝ
hr : 0 < r
a : ℝ
x : E
hx : x ≠ 0
hxT : T.rayleighQuotient x = a
c : 𝕜 := ↑‖x‖⁻¹ * ↑r
this : c ≠ 0
⊢ T.rayleighQuotient (c • x) = a

Tactic:
· rw [T.rayleigh_smul x this]
      exact hxT

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

Goal state before:
pp: case h₂
H : Type u_8
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace ℝ H
r : ℝ
t : Set ℝ := ⋯
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
hx' : 1 / 2 < x
hx'' : 0 < x
⊢ |x| ^ (r - ↑n) ≤ (1 + |x|) ^ k

Tactic:
by_cases! h : 0 ≤ r - n

Goal state after:
case pos
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
hx' : 1 / 2 < x
hx'' : 0 < x
h : 0 ≤ r - ↑n
⊢ |x| ^ (r - ↑n) ≤ (1 + |x|) ^ k

case neg
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
hx' : 1 / 2 < x
hx'' : 0 < x
h : r - ↑n < 0
⊢ |x| ^ (r - ↑n) ≤ (1 + |x|) ^ k
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
hx' : 1 / 2 < x
hx'' : 0 < x
h✝ : r - ↑n < 0
h : 0 < ↑n - r
⊢ Real.exp (Real.log (3 / 2) * ↑k) ≤ (3 / 2) ^ k

Tactic:
rw [← Real.rpow_natCast, Real.rpow_def_of_pos]
      norm_num

Goal state after:
Module: Mathlib.LinearAlgebra.Matrix.Charpoly.Eigs

Goal state before:
pp: n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
R : Type u_2
inst✝¹ : CommRing R
B : Matrix n n R
inst✝ : IsDomain R
hAps : B.charpoly.Splits
⊢ 1 ^ B.charpoly.natDegree * B.charpoly.roots.prod = B.charpoly.roots.prod

Tactic:
one_pow,

Goal state after:
n : Type u_1
inst✝³ : Fintype n
inst✝² : DecidableEq n
R : Type u_2
inst✝¹ : CommRing R
B : Matrix n n R
inst✝ : IsDomain R
hAps : B.charpoly.Splits
⊢ 1 * B.charpoly.roots.prod = B.charpoly.roots.prod
Module: Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖
hK : ∀ K > 0, ∃ x, ‖(T - μ • 1) x‖ < K * ‖x‖
C : 𝕜
hC : 1 < ‖C‖
ε : ℝ
hε : ε > 0
x : X
hx : ‖(T - μ • 1) x‖ < ε * ‖x‖
this : x ≠ 0
η : 𝕜
hη : η ≠ 0
h₁ : ‖η • x‖ < 1
h₂ : 1 / ‖C‖ ≤ ‖η • x‖
h₃ : ‖η‖⁻¹ ≤ 1⁻¹ * ‖C‖ * ‖x‖
⊢ ‖(T - μ • 1) (η • x)‖ < ε

Tactic:
grw [map_smul, norm_smul, hx, mul_left_comm, ← norm_smul]

Goal state after:
𝕜 : Type u_1
X : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖
hK : ∀ K > 0, ∃ x, ‖(T - μ • 1) x‖ < K * ‖x‖
C : 𝕜
hC : 1 < ‖C‖
ε : ℝ
hε : ε > 0
x : X
hx : ‖(T - μ • 1) x‖ < ε * ‖x‖
this : x ≠ 0
η : 𝕜
hη : η ≠ 0
h₁ : ‖η • x‖ < 1
h₂ : 1 / ‖C‖ ≤ ‖η • x‖
h₃ : ‖η‖⁻¹ ≤ 1⁻¹ * ‖C‖ * ‖x‖
⊢ ε * ‖η • x‖ ≤ ε
Module: Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative

Goal state before:
pp: 𝕜 : Type u_1
X : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : NormedAddCommGroup X
inst✝¹ : NormedSpace 𝕜 X
T : X →L[𝕜] X
μ : 𝕜
inst✝ : CompleteSpace X
hT : IsCompactOperator ⇑T
hμ : μ ≠ 0
h₁ : ¬HasEigenvalue (↑T) μ
S : X →L[𝕜] X := T - μ • 1
K✝ : NNReal
hK✝ : AntilipschitzWith K✝ ⇑S
h₂ : ¬Function.Surjective ⇑S
c : 𝕜
hc : 1 < ‖c‖
f : ℕ → X
hf_norm_lower : ∀ (n : ℕ), 1 ≤ ‖f n‖
hf_norm_upper : ∀ (n : ℕ), ‖f n‖ ≤ ‖c‖ + 1
hf_far : ∀ (n : ℕ), ∀ y ∈ LinearMap.range (↑S ^ (n + 1)), 1 ≤ ‖f n - y‖
hf_mem : ∀ {n m : ℕ}, m ≤ n → f n ∈ (↑S ^ m).range
hf_mem' : ∀ {n m : ℕ}, m ≤ n → S (f n) ∈ (↑S ^ (m + 1)).range
hp : Pairwise fun x₁ x₂ ↦ ‖μ‖ ≤ ‖T (f x₁) - T (f x₂)‖
K : Set X
hK : IsCompact K
hK' : ⇑↑T '' Metric.closedBall 0 (‖c‖ + 1) ⊆ K
y : X
hyK : y ∈ K
ψ : ℕ → ℕ
hψ : StrictMono ψ
hψy : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (((fun n ↦ ↑T (f n)) ∘ ψ) n) (((fun n ↦ ↑T (f n)) ∘ ψ) N) < ε
N : ℕ
hN : ∀ n ≥ N, dist (((fun n ↦ ↑T (f n)) ∘ ψ) n) (((fun n ↦ ↑T (f n)) ∘ ψ) N) < ‖μ‖
⊢ ‖T (f (ψ (N + 1))) - T (f (ψ N))‖ < ‖μ‖

Tactic:
simpa [dist_eq_norm_sub] using hN (N + 1)

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
T : E →ₗ[𝕜] E
inst✝ : FiniteDimensional 𝕜 E
n : ℕ
hT : T.IsSymmetric
hn : finrank 𝕜 E = n
i : Fin n
v : E := (hT.unsortedEigenvectorBasis hn) i
μ : 𝕜 := ↑T (DirectSum.IsInternal.subordinateOrthonormalBasisIndex hn ⋯ i ⋯)
key : HasEigenvector T μ v
re_μ : ↑(RCLike.re μ) = μ
⊢ HasEigenvector T (↑(RCLike.re μ)) v

Tactic:
simpa [re_μ] using key

Goal state after: