mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
G₁ G₂ : SimpleGraph V
⊢ (G₁ ⊓ G₂).edgeSet = G₁.edgeSet ∩ G₂.edgeSet

Tactic:
ext ⟨x, y⟩

Goal state after:
case mk
V : Type u
G₁ G₂ : SimpleGraph V
x✝ : Sym2 V
x y : V
⊢ Quot.mk (Sym2.Rel V) (x, y) ∈ (G₁ ⊓ G₂).edgeSet ↔ Quot.mk (Sym2.Rel V) (x, y) ∈ G₁.edgeSet ∩ G₂.edgeSet
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
s : Set (SimpleGraph V)
⊢ (sSup s).edgeSet = ⋃₀ (edgeSet '' s)

Tactic:
ext ⟨x, y⟩

Goal state after:
case mk
V : Type u
s : Set (SimpleGraph V)
x✝ : Sym2 V
x y : V
⊢ Quot.mk (Sym2.Rel V) (x, y) ∈ (sSup s).edgeSet ↔ Quot.mk (Sym2.Rel V) (x, y) ∈ ⋃₀ (edgeSet '' s)
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
s : Set (SimpleGraph V)
h : s.Nonempty
⊢ (sInf s).edgeSet = ⋂₀ (edgeSet '' s)

Tactic:
ext ⟨x, y⟩

Goal state after:
case mk
V : Type u
s : Set (SimpleGraph V)
h : s.Nonempty
x✝ : Sym2 V
x y : V
⊢ Quot.mk (Sym2.Rel V) (x, y) ∈ (sInf s).edgeSet ↔ Quot.mk (Sym2.Rel V) (x, y) ∈ ⋂₀ (edgeSet '' s)
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
ι : Sort u_2
f : ι → SimpleGraph V
⊢ (⨆ i, f i).edgeSet = ⋃ i, (f i).edgeSet

Tactic:
ext ⟨x, y⟩

Goal state after:
case mk
V : Type u
ι : Sort u_2
f : ι → SimpleGraph V
x✝ : Sym2 V
x y : V
⊢ Quot.mk (Sym2.Rel V) (x, y) ∈ (⨆ i, f i).edgeSet ↔ Quot.mk (Sym2.Rel V) (x, y) ∈ ⋃ i, (f i).edgeSet
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
ι : Sort u_2
inst✝ : Nonempty ι
f : ι → SimpleGraph V
⊢ (⨅ i, f i).edgeSet = ⋂ i, (f i).edgeSet

Tactic:
ext ⟨x, y⟩

Goal state after:
case mk
V : Type u
ι : Sort u_2
inst✝ : Nonempty ι
f : ι → SimpleGraph V
x✝ : Sym2 V
x y : V
⊢ Quot.mk (Sym2.Rel V) (x, y) ∈ (⨅ i, f i).edgeSet ↔ Quot.mk (Sym2.Rel V) (x, y) ∈ ⋂ i, (f i).edgeSet
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
G₁ G₂ : SimpleGraph V
⊢ (G₁ \ G₂).edgeSet = G₁.edgeSet \ G₂.edgeSet

Tactic:
ext ⟨x, y⟩

Goal state after:
case mk
V : Type u
G₁ G₂ : SimpleGraph V
x✝ : Sym2 V
x y : V
⊢ Quot.mk (Sym2.Rel V) (x, y) ∈ (G₁ \ G₂).edgeSet ↔ Quot.mk (Sym2.Rel V) (x, y) ∈ G₁.edgeSet \ G₂.edgeSet
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
hJ_fin : Fintype ↑J
⊢ (⋂₀ J).Nonempty

Tactic:
let J' := J.toFinset

Goal state after:
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
hJ_fin : Fintype ↑J
J' : Finset (Set E) := J.toFinset
⊢ (⋂₀ J).Nonempty
Module: Mathlib.Combinatorics.SimpleGraph.Basic

Goal state before:
pp: V : Type u
s t : Set (Sym2 V)
⊢ fromEdgeSet (s ∪ t) = fromEdgeSet s ⊔ fromEdgeSet t

Tactic:
by
  ext v w
  simp [Set.mem_union, or_and_right]

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
v w : V
h : ∀ {a b : V}, G.Adj a b → a ≠ b
⊢ G.Adj v w ∨ v ≠ w ∧ ¬G.Adj v w ↔ w ∉ {v}

Tactic:
Set.mem_singleton_iff

Goal state after:
V : Type u
G : SimpleGraph V
v w : V
h : ∀ {a b : V}, G.Adj a b → a ≠ b
⊢ G.Adj v w ∨ v ≠ w ∧ ¬G.Adj v w ↔ ¬w = v
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.SOppSide x y ↔ x ∉ s ∧ y ∉ s ∧ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)

Tactic:
by
  rw [SOppSide, and_comm, wOppSide_iff_exists_right h, and_assoc, and_congr_right_iff,
    and_congr_right_iff]
  rintro _ hy
  rw [or_iff_right hy]

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

Goal state before:
pp: case neg
V : Type u_2
P : Type u_4
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace ℝ V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
s : AffineSubspace ℝ P
x p : P
hp : p ∈ ↑s
this : Nonempty ↥s
hx : x ∉ s
⊢ IsConnected {y | s.WOppSide x y}

Tactic:
· rw [setOfPred_wOppSide_eq_image2 hx hp, ← Set.image_prod]
    refine (isConnected_Iic.prod (isConnected_iff_connectedSpace.2 ?_)).image _
      ((continuous_fst.smul continuous_const).vadd continuous_snd).continuousOn
    convert! AddTorsor.connectedSpace s.direction s

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

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
Ω : Opens E
F : Type u_2
inst✝⁴ : AddCommGroup F
inst✝³ : Module ℝ F
inst✝² : TopologicalSpace F
n k : ℕ∞
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousSMul ℝ F
v₁ v₂ : E
⊢ lineDerivCLM (v₁ + v₂) = lineDerivCLM v₁ + lineDerivCLM v₂

Tactic:
by
  ext T f
  simp [lineDerivCLM_apply, TestFunction.lineDerivCLM_add, neg_add, -neg_add_rev]

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

Goal state before:
pp: case inr
α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
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
t : ℝ
ht : 0 ≤ t
h : 0 < t
⊢ IntervalIntegrable g volume 0 t

Tactic:
· exact g_intble t h

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

Goal state before:
pp: α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
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
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
⊢ ∫⁻ (y : ℝ), ∫⁻ (x : α), (Ioc 0 (f x)).indicator (fun t ↦ ENNReal.ofReal (g t)) y ∂μ =
    ∫⁻ (a : ℝ), (Ioi 0).indicator (fun t ↦ μ {a | t ≤ f a} * ENNReal.ofReal (g t)) a

Tactic:
apply congr_arg

Goal state after:
α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
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
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
⊢ (fun y ↦ ∫⁻ (x : α), (Ioc 0 (f x)).indicator (fun t ↦ ENNReal.ofReal (g t)) y ∂μ) = fun a ↦
    (Ioi 0).indicator (fun t ↦ μ {a | t ≤ f a} * ENNReal.ofReal (g t)) a
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: r : ℝ
n : ℕ
hnr : ↑n < r
hr : 0 < r
x : ℝ
hx : x ∈ Ioc 0 1
⊢ ENNReal.ofReal ((x ^ (-r⁻¹) - 1) ^ n) ≤ ENNReal.ofReal ((x ^ (-r⁻¹) - 0) ^ n)

Tactic:
by
        gcongr
        · rw [sub_nonneg]
          exact Real.one_le_rpow_of_pos_of_le_one_of_nonpos hx.1 hx.2 (by simpa using hr.le)
        · simp

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

Goal state before:
pp: case mk
α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
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
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
p_fst : α
p_snd : ℝ
⊢ Function.uncurry (fun x y ↦ (Ioc 0 (f x)).indicator (fun t ↦ ENNReal.ofReal (g t)) y) (p_fst, p_snd) =
    {p | p.2 ∈ Ioc 0 (f p.1)}.indicator (fun p ↦ ENNReal.ofReal (g p.2)) (p_fst, p_snd)

Tactic:
rw [Function.uncurry_apply_pair]

Goal state after:
case mk
α : Type u_1
inst✝¹ : MeasurableSpace α
f : α → ℝ
g : ℝ → ℝ
μ : Measure α
inst✝ : SFinite μ
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
g_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t
integrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)
p_fst : α
p_snd : ℝ
⊢ (Ioc 0 (f p_fst)).indicator (fun t ↦ ENNReal.ofReal (g t)) p_snd =
    {p | p.2 ∈ Ioc 0 (f p.1)}.indicator (fun p ↦ ENNReal.ofReal (g p.2)) (p_fst, p_snd)
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: case calc_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
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 ∈ Ioi 1, f t = 0
⊢ 0 * (volume.restrict (Ioi 1)) univ < ∞

Tactic:
zero_mul

Goal state after:
case calc_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
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 ∈ Ioi 1, f t = 0
⊢ 0 < ∞
Module: Mathlib.Analysis.SpecialFunctions.JapaneseBracket

Goal state before:
pp: case calc_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
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 ∈ Ioi 1, f t = 0
⊢ 0 < ∞

Tactic:
exact WithTop.top_pos

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
⊢ n ∈ Finset.range (N + 1)

Tactic:
simpa using hn

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 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:
norm_sub_sq (𝕜 := 𝕜),

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‖ ^ 2 - 2 * re ⟪(algebraMap ℝ 𝕜) t • x, T x⟫_𝕜 + ‖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.Distribution.TemperateGrowth

Goal state before:
pp: case h₁
C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
h'x : x ≤ 1
⊢ x ^ k ≤ ((1 + x) / 2) ^ (-↑l)

Tactic:
apply (pow_le_one₀ hx h'x).trans

Goal state after:
case h₁
C₁ C₂ : ℝ
k l : ℕ
x f : ℝ
hx : 0 ≤ x
hf : 0 ≤ f
h₁ : f ≤ C₁
h₂ : x ^ (k + l) * f ≤ C₂
this✝ : 0 ≤ C₂
this : 2 ^ l * (C₁ + C₂) * (1 + x) ^ (-↑l) = ((1 + x) / 2) ^ (-↑l) * (C₁ + C₂)
h'x : x ≤ 1
⊢ 1 ≤ ((1 + x) / 2) ^ (-↑l)
Module: Mathlib.Analysis.Distribution.TemperateGrowth

Goal state before:
pp: case h
E : Type u_5
F : Type u_6
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : NormedAddCommGroup F
μ : Measure E
inst✝ : μ.HasTemperateGrowth
f : E → F
C₁ C₂ : ℝ
k : ℕ
hf : ∀ (x : E), ‖f x‖ ≤ C₁
h'f : ∀ (x : E), ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂
v : E
⊢ ‖v‖ ^ k * ‖f v‖ ≤ 2 ^ μ.integrablePower * (1 + ‖v‖) ^ (-↑μ.integrablePower) * (C₁ + C₂)

Tactic:
exact (pow_mul_le_of_le_of_pow_mul_le (norm_nonneg _) (norm_nonneg _) (hf v) (h'f v)).trans
    (le_of_eq (by ring))

Goal state after:
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
n m : ℕ
h : m ≤ n
⊢ S (f n) ∈ Submodule.map (↑S) (LinearMap.range (↑S ^ m))

Tactic:
exact ⟨f n, hf_mem h, rfl⟩

Goal state after:
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
this : Std.Symm fun x₁ x₂ ↦ ‖μ‖ ≤ ‖T (f x₁) - T (f x₂)‖
m n : ℕ
hmn : m < n
⊢ ‖μ‖ ≤ ‖T (f m) - T (f n)‖

Tactic:
let u : X := μ⁻¹ • (S (f n) - S (f m) + μ • f n)

Goal state after:
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
this : Std.Symm fun x₁ x₂ ↦ ‖μ‖ ≤ ‖T (f x₁) - T (f x₂)‖
m n : ℕ
hmn : m < n
u : X := μ⁻¹ • (S (f n) - S (f m) + μ • f n)
⊢ ‖μ‖ ≤ ‖T (f m) - T (f n)‖
Module: Mathlib.Analysis.Matrix.Spectrum

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : RCLike 𝕜
n : Type u_2
inst✝¹ : Fintype n
A : Matrix n n 𝕜
inst✝ : DecidableEq n
hA : A.IsHermitian
j : n
⊢ ↑hA.eigenvectorUnitary *ᵥ Pi.single j 1 = (hA.eigenvectorBasis j).ofLp

Tactic:
simp_rw [mulVec_single_one, eigenvectorUnitary_col_eq]

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
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsCompactOperator ⇑T
hT' : (↑T).IsSymmetric
⊢ (∀ (μ : 𝕜), HasEigenvalue (↑T) μ → μ = 0) ↔ ‖T‖₊ = 0

Tactic:
← ENNReal.coe_eq_zero,

Goal state after:
𝕜 : Type u_1
inst✝³ : RCLike 𝕜
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
inst✝ : CompleteSpace E
T : E →L[𝕜] E
hT : IsCompactOperator ⇑T
hT' : (↑T).IsSymmetric
⊢ (∀ (μ : 𝕜), HasEigenvalue (↑T) μ → μ = 0) ↔ ↑‖T‖₊ = 0
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
f : α → E
hf : Integrable f μ
hf_int : ∀ (c : E), ∫ (x : α), ⟪c, f x⟫ ∂μ = 0
⊢ ∫ (x : α), f x ∂μ = 0

Tactic:
specialize hf_int (∫ x, f x ∂μ)

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_2
𝕜 : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
inst✝¹ : CompleteSpace E
inst✝ : NormedSpace ℝ E
f : α → E
hf : Integrable f μ
hf_int : ∫ (x : α), ⟪∫ (x : α), f x ∂μ, f x⟫ ∂μ = 0
⊢ ∫ (x : α), f x ∂μ = 0
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
inst✝¹ : RCLike 𝕜
μ : Measure α
inst✝ : IsFiniteMeasure μ
f g : α →ᵇ 𝕜
hf_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ⇑f
hg_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) =ᵐ[μ] ⇑g
⊢ (fun a ↦ ⟪↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) a, ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) a⟫) =ᵐ[μ]
    fun a ↦ g a * (starRingEnd 𝕜) (f a)

Tactic:
filter_upwards [hf_ae, hg_ae] with _ hf hg

Goal state after:
α : Type u_1
𝕜 : Type u_2
inst✝⁴ : TopologicalSpace α
inst✝³ : MeasurableSpace α
inst✝² : BorelSpace α
inst✝¹ : RCLike 𝕜
μ : Measure α
inst✝ : IsFiniteMeasure μ
f g : α →ᵇ 𝕜
hf_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) =ᵐ[μ] ⇑f
hg_ae : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) =ᵐ[μ] ⇑g
a✝ : α
hf : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝ = f a✝
hg : ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝ = g a✝
⊢ ⟪↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) f) a✝, ↑↑((BoundedContinuousFunction.toLp 2 μ 𝕜) g) a✝⟫ =
    g a✝ * (starRingEnd 𝕜) (f a✝)
Module: Mathlib.MeasureTheory.Function.L2Space

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : RCLike 𝕜
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : CompactSpace α
f g : C(α, 𝕜)
hf_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ⇑f
hg_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) =ᵐ[μ] ⇑g
⊢ (fun a ↦ ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a⟫) =ᵐ[μ] fun a ↦
    g a * (starRingEnd 𝕜) (f a)

Tactic:
filter_upwards [hf_ae, hg_ae] with _ hf hg

Goal state after:
α : Type u_1
𝕜 : Type u_2
inst✝⁵ : TopologicalSpace α
inst✝⁴ : MeasurableSpace α
inst✝³ : BorelSpace α
inst✝² : RCLike 𝕜
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : CompactSpace α
f g : C(α, 𝕜)
hf_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) =ᵐ[μ] ⇑f
hg_ae : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) =ᵐ[μ] ⇑g
a✝ : α
hf : ↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝ = f a✝
hg : ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝ = g a✝
⊢ ⟪↑↑((ContinuousMap.toLp 2 μ 𝕜) f) a✝, ↑↑((ContinuousMap.toLp 2 μ 𝕜) g) a✝⟫ = g a✝ * (starRingEnd 𝕜) (f a✝)
Module: Mathlib.Analysis.Normed.Lp.SmoothApprox

Goal state before:
pp: case inl
E : Type u_3
F : Type u_4
inst✝⁷ : MeasurableSpace E
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : BorelSpace E
inst✝¹ : NormedSpace ℝ F
μ : Measure E
inst✝ : IsFiniteMeasureOnCompacts μ
ε : ℝ
hε : 0 < ε
f : E → F
h₁ : HasCompactSupport f
h₂ : Continuous f
⊢ ∃ g, HasCompactSupport g ∧ ContDiff ℝ ∞ g ∧ eLpNorm (f - g) ∞ μ ≤ ENNReal.ofReal ε

Tactic:
obtain ⟨g, hg₁, hg₂, hg₃⟩ := h₂.exists_contDiff_approx ⊤ (ε := fun _ ↦ ε) (by fun_prop)
      (by intro; positivity)
    refine ⟨g, h₁.mono hg₃, hg₁, eLpNormEssSup_le_of_ae_bound (.of_forall fun x ↦ ?_)⟩
    simpa [← dist_eq_norm_sub'] using (hg₂ x).le

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Bases

Goal state before:
pp: case h
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
β : Type u_3
L : SummationFilter β
b : GeneralSchauderBasis β 𝕜 X L
A : Finset β
i : β
hi : i ∈ ↑A
⊢ (b.proj A) (↑b i) = ↑b i

Tactic:
proj_apply_basis_mem,

Goal state after:
case h
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
β : Type u_3
L : SummationFilter β
b : GeneralSchauderBasis β 𝕜 X L
A : Finset β
i : β
hi : i ∈ ↑A
⊢ (if i ∈ A then ↑b i else 0) = ↑b i
Module: Mathlib.Analysis.Normed.Module.Bases

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
X : Type u_2
inst✝¹ : NormedAddCommGroup X
inst✝ : NormedSpace 𝕜 X
P : ℕ → X →L[𝕜] X
hcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x
i j : ℕ
x : X
h : ¬i = j
⊢ (P (min i j + 1)) x - (P (min i (j + 1))) x - ((P (min (i + 1) j)) x - (P (min i j)) x) = 0

Tactic:
rcases Nat.lt_or_gt_of_ne h with h' | h'
    · rw [min_eq_left_of_lt h', min_eq_left (Nat.succ_le_of_lt h'),
        min_eq_left_of_lt (Nat.lt_succ_of_lt h')]
      abel
    · rw [min_eq_right_of_lt h', min_eq_right (Nat.succ_le_of_lt h'),
        min_eq_right_of_lt (Nat.lt_succ_of_lt h')]
      abel

Goal state after:
Module: Mathlib.Analysis.Fourier.AddCircle

Goal state before:
pp: T : ℝ
hT : Fact (0 < T)
E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : AddCircle T → E
⊢ ∫ (t : AddCircle T), f t ∂haarAddCircle = T⁻¹ • ∫ (t : AddCircle T), f t

Tactic:
rw [volume_eq_smul_haarAddCircle, integral_smul_measure, ENNReal.toReal_ofReal hT.out.le,
    inv_smul_smul₀ hT.out.ne']

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.l2Space

Goal state before:
pp: case e_f
ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : DecidableEq ι
i : ι
x : G i
j : ι
⊢ (V j) (↑(lp.single 2 i x) j) = if j = i then (V i) x else 0

Tactic:
rw [lp.single_apply]

Goal state after:
case e_f
ι : Type u_1
𝕜 : Type u_2
inst✝⁶ : RCLike 𝕜
E : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : InnerProductSpace 𝕜 E
G : ι → Type u_4
inst✝³ : (i : ι) → NormedAddCommGroup (G i)
inst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)
inst✝¹ : CompleteSpace E
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : DecidableEq ι
i : ι
x : G i
j : ι
⊢ (V j) (Pi.single i x j) = if j = i then (V i) x else 0