mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι : Type u_9
inst✝⁷ : NormedField 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : NormedField 𝕜₂
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
inst✝ : TopologicalSpace F
q : SeminormFamily 𝕜₂ F ι
hq : WithSeminorms q
f : E →ₛₗ[σ₁₂] F
this✝ : IsTopologicalAddGroup F
x✝ : TopologicalSpace E := TopologicalSpace.induced (⇑f) inferInstance
this : IsTopologicalAddGroup E
⊢ ⨅ i, comap (⇑f) (comap (⇑(q i)) (𝓝 0)) = ⨅ i, comap (⇑(q.comp f i)) (𝓝 0)

Tactic:
refine iInf_congr fun i => ?_

Goal state after:
𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι : Type u_9
inst✝⁷ : NormedField 𝕜
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : NormedField 𝕜₂
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
inst✝ : TopologicalSpace F
q : SeminormFamily 𝕜₂ F ι
hq : WithSeminorms q
f : E →ₛₗ[σ₁₂] F
this✝ : IsTopologicalAddGroup F
x✝ : TopologicalSpace E := TopologicalSpace.induced (⇑f) inferInstance
this : IsTopologicalAddGroup E
i : ι
⊢ comap (⇑f) (comap (⇑(q i)) (𝓝 0)) = comap (⇑(q.comp f i)) (𝓝 0)
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι : Type u_9
inst✝⁸ : AddCommGroup E
inst✝⁷ : NormedField 𝕜₂
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜₂ F
inst✝⁴ : TopologicalSpace F
𝕜 : Type u_11
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : Module 𝕜 E
inst✝¹ : TopologicalSpace E
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
p : Seminorm 𝕜 E
hp : Continuous[inst✝¹, _] ⇑p
q : SeminormFamily 𝕜₂ F ι
hq : WithSeminorms q
f : E →ₛₗ[σ₁₂] F
hf : IsInducing ⇑f
s : Finset ι
C : ℝ≥0
hC : C ≠ 0
hqC : p ≤ (C • s.sup q).comp f
⊢ ∃ s C, C ≠ 0 ∧ p ≤ (C • s.sup q).comp f

Tactic:
exact ⟨s, C, hC, hqC⟩

Goal state after:
Module: Mathlib.Order.Filter.Germ.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
l : Filter α
f✝¹ g✝¹ h : α → β
M : Type u_5
N : Type u_6
R : Type u_7
inst✝² : Monoid M
inst✝¹ : AddMonoid N
inst✝ : DistribMulAction M N
c : M
f✝ g✝ : l.Germ N
f g : α → N
⊢ c • (↑f + ↑g) = c • ↑f + c • ↑g

Tactic:
by
      norm_cast
      simp [smul_add]

Goal state after:
Module: Mathlib.Order.Filter.Germ.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
l : Filter α
f✝¹ g✝¹ h : α → β
M : Type u_5
N : Type u_6
R : Type u_7
inst✝² : Monoid M
inst✝¹ : AddMonoid N
inst✝ : DistribMulAction M N
c✝ : l.Germ M
f✝ g✝ : l.Germ N
c : α → M
f g : α → N
⊢ ↑c • (↑f + ↑g) = ↑c • ↑f + ↑c • ↑g

Tactic:
by
      norm_cast
      simp [smul_add]

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: case mp
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
f : C(α, β)
⊢ (∀ (K : Set α),
      IsCompact K →
        ∀ (U : Set β), IsOpen[inst✝.toTopologicalSpace] U → MapsTo (⇑f) K U → ∀ᶠ (a : ι) in p, MapsTo (⇑(F a)) K U) →
    ∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a ↦ (F i) a) (⇑f) p K

Tactic:
intro h K hK
    -- Since `K` is compact, it suffices to prove locally uniform convergence
    rw [← tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hK]
    -- Now choose an entourage `U` in the codomain and a point `x ∈ K`.
    intro U hU x _
    -- Choose an open symmetric entourage `V` such that `V ○ V ⊆ U`.
    rcases comp_open_symm_mem_uniformity_sets hU with ⟨V, hV, hVo, hVsymm, hVU⟩
    -- Then choose a closed entourage `W ⊆ V`
    rcases mem_uniformity_isClosed hV with ⟨W, hW, hWc, hWU⟩
    -- Consider `s = {y ∈ K | (f x, f y) ∈ W}`
    set s := K ∩ f ⁻¹' ball (f x) W
    -- This is a neighbourhood of `x` within `K`, because `W` is an entourage.
    have hnhds : s ∈ 𝓝[K] x := inter_mem_nhdsWithin _ <| f.continuousAt _ (ball_mem_nhds _ hW)
    -- This set is compact because it is an intersection of `K`
    -- with a closed set `{y | (f x, f y) ∈ W} = f ⁻¹' UniformSpace.ball (f x) W`
    have hcomp : IsCompact s := hK.inter_right <| (isClosed_ball _ hWc).preimage f.continuous
    -- `f` maps `s` to the open set `ball (f x) V = {z | (f x, z) ∈ V}`
    have hmaps : MapsTo f s (ball (f x) V) := fun x hx ↦ hWU hx.2
    use s, hnhds
    -- Continuous maps `F i` in a neighbourhood of `f` map `s` to `ball (f x) V` as well.
    refine (h s hcomp _ (isOpen_ball _ hVo) hmaps).mono fun g hg y hy ↦ ?_
    -- Then for `y ∈ s` we have `(f y, f x) ∈ V` and `(f x, F i y) ∈ V`, thus `(f y, F i y) ∈ U`
    exact hVU ⟨f x, SetRel.symm V <| hmaps hy, hg hy⟩

Goal state after:
Module: Mathlib.MeasureTheory.Function.EssSup

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteLinearOrder β
f : α → β
h : ∀ ⦃x : α⦄ ⦃a : β⦄, a < f x → μ {y | a < f y} ≠ 0
i : α
b : β
hb : b ∈ {a | μ {x | a < f x} = 0}
⊢ f i ≤ b

Tactic:
exact not_lt.mp fun a ↦ h a hb

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
n : OuterMeasure ((i : ι) → α i)
⊢ n ≤ boundedBy (piPremeasure m) ↔ ∀ (s : (i : ι) → Set (α i)), (univ.pi s).Nonempty → n (univ.pi s) ≤ ∏ i, (m i) (s i)

Tactic:
le_boundedBy'

Goal state after:
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
n : OuterMeasure ((i : ι) → α i)
⊢ (∀ (s : Set ((i : ι) → α i)), s.Nonempty → n s ≤ piPremeasure m s) ↔
    ∀ (s : (i : ι) → Set (α i)), (univ.pi s).Nonempty → n (univ.pi s) ≤ ∏ i, (m i) (s i)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case mp
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
n : OuterMeasure ((i : ι) → α i)
⊢ (∀ (s : Set ((i : ι) → α i)), s.Nonempty → n s ≤ piPremeasure m s) →
    ∀ (s : (i : ι) → Set (α i)), (univ.pi s).Nonempty → n (univ.pi s) ≤ ∏ i, (m i) (s i)

Tactic:
intro h s hs

Goal state after:
case mp
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
n : OuterMeasure ((i : ι) → α i)
h : ∀ (s : Set ((i : ι) → α i)), s.Nonempty → n s ≤ piPremeasure m s
s : (i : ι) → Set (α i)
hs : (univ.pi s).Nonempty
⊢ n (univ.pi s) ≤ ∏ i, (m i) (s i)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case mpr
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
n : OuterMeasure ((i : ι) → α i)
⊢ (∀ (s : (i : ι) → Set (α i)), (univ.pi s).Nonempty → n (univ.pi s) ≤ ∏ i, (m i) (s i)) →
    ∀ (s : Set ((i : ι) → α i)), s.Nonempty → n s ≤ piPremeasure m s

Tactic:
intro h s hs

Goal state after:
case mpr
ι : Type u_1
α : ι → Type u_3
inst✝ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
n : OuterMeasure ((i : ι) → α i)
h : ∀ (s : (i : ι) → Set (α i)), (univ.pi s).Nonempty → n (univ.pi s) ≤ ∏ i, (m i) (s i)
s : Set ((i : ι) → α i)
hs : s.Nonempty
⊢ n s ≤ piPremeasure m s
Module: Mathlib.MeasureTheory.Function.SpecialFunctions.Basic

Goal state before:
pp: case refine_2
⊢ Measurable fun x ↦ if 0 < x then ∞ else if x = 0 then 1 else 0

Tactic:
refine Measurable.ite measurableSet_Ioi measurable_const ?_

Goal state after:
case refine_2
⊢ Measurable fun x ↦ if x = 0 then 1 else 0
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
⊢ eLpNorm 0 p μ < ∞

Tactic:
rw [eLpNorm_zero]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
⊢ 0 < ∞
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: case inr.inl.inr
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy✝ : x ≤ y
hxy : x < y
hxa : x < a
hya : a < y
⊢ (f x - f a) / (x - a) ≤ (f y - f a) / (y - a)

Tactic:
convert! hf.slope_mono_adjacent hx hy hxa hya using 1

Goal state after:
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy✝ : x ≤ y
hxy : x < y
hxa : x < a
hya : a < y
⊢ (f x - f a) / (x - a) = (f a - f x) / (a - x)
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_3
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
i : ι
hi : i ∈ s
⊢ z i ^ w i = rexp (log (z i) * w i)

Tactic:
rcases eq_or_lt_of_le (hz i hi) with hz | hz

Goal state after:
case e'_3.inl
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz✝ : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
i : ι
hi : i ∈ s
hz : 0 = z i
⊢ z i ^ w i = rexp (log (z i) * w i)

case e'_3.inr
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz✝ : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
i : ι
hi : i ∈ s
hz : 0 < z i
⊢ z i ^ w i = rexp (log (z i) * w i)
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_3
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
i : ι
hi : i ∈ s
⊢ z i ^ w i = rexp (log (z i) * w i)

pp: case e'_4
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
i : ι
hi : i ∈ s
⊢ w i * z i = w i * rexp (log (z i))

Tactic:
· rcases eq_or_lt_of_le (hz i hi) with hz | hz
      · simp [A i hi hz.symm]
      · exact rpow_def_of_pos hz _

Goal state after:
case e'_4
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
i : ι
hi : i ∈ s
⊢ w i * z i = w i * rexp (log (z i))
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_4
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
i : ι
hi : i ∈ s
⊢ w i * z i = w i * rexp (log (z i))

Tactic:
rcases eq_or_lt_of_le (hz i hi) with hz | hz

Goal state after:
case e'_4.inl
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz✝ : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
i : ι
hi : i ∈ s
hz : 0 = z i
⊢ w i * z i = w i * rexp (log (z i))

case e'_4.inr
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz✝ : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
i : ι
hi : i ∈ s
hz : 0 < z i
⊢ w i * z i = w i * rexp (log (z i))
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_4.inr
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz✝ : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
i : ι
hi : i ∈ s
hz : 0 < z i
⊢ w i * z i = w i * rexp (log (z i))

Tactic:
exp_log hz

Goal state after:
case e'_4.inr
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz✝ : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
i : ι
hi : i ∈ s
hz : 0 < z i
⊢ w i * z i = w i * z i
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_norm : ∫⁻ (a : α), f a ^ p ∂μ = 1
hg_norm : ∫⁻ (a : α), g a ^ q ∂μ = 1
⊢ (∫⁻ (a : α), f a ^ p ∂μ) * (ENNReal.ofReal p)⁻¹ + ∫⁻ (a : α), g a ^ q * (ENNReal.ofReal q)⁻¹ ∂μ = 1

Tactic:
lintegral_mul_const',

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_norm : ∫⁻ (a : α), f a ^ p ∂μ = 1
hg_norm : ∫⁻ (a : α), g a ^ q ∂μ = 1
⊢ (∫⁻ (a : α), f a ^ p ∂μ) * (ENNReal.ofReal p)⁻¹ + (∫⁻ (a : α), g a ^ q ∂μ) * (ENNReal.ofReal q)⁻¹ = 1

case hr
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_norm : ∫⁻ (a : α), f a ^ p ∂μ = 1
hg_norm : ∫⁻ (a : α), g a ^ q ∂μ = 1
⊢ (ENNReal.ofReal q)⁻¹ ≠ ∞
Module: Mathlib.Analysis.Convex.Mul

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁷ : CommRing 𝕜
inst✝¹⁶ : LinearOrder 𝕜
inst✝¹⁵ : IsStrictOrderedRing 𝕜
inst✝¹⁴ : CommRing E
inst✝¹³ : LinearOrder E
inst✝¹² : IsStrictOrderedRing E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : LinearOrder F
inst✝⁹ : IsOrderedAddMonoid F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜 G
inst✝⁶ : Module 𝕜 E
inst✝⁵ : Module 𝕜 F
inst✝⁴ : Module E F
inst✝³ : IsScalarTower 𝕜 E F
inst✝² : SMulCommClass 𝕜 E F
inst✝¹ : IsOrderedModule 𝕜 F
inst✝ : IsStrictOrderedModule E F
s : Set G
f : G → E
g : G → F
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
hf₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ f x
hg₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ g x
hfg : MonovaryOn f g s
x : G
hx : x ∈ s
y : G
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ f (a • x + b • y) • g (a • x + b • y) ≤ a • f x • g x + b • f y • g y

Tactic:
refine
    (smul_le_smul (hf.2 hx hy ha hb hab) (hg.2 hx hy ha hb hab) (hf₀ <| hf.1 hx hy ha hb hab) <|
      add_nonneg (smul_nonneg ha <| hg₀ hx) <| smul_nonneg hb <| hg₀ hy).trans ?_

Goal state after:
𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁷ : CommRing 𝕜
inst✝¹⁶ : LinearOrder 𝕜
inst✝¹⁵ : IsStrictOrderedRing 𝕜
inst✝¹⁴ : CommRing E
inst✝¹³ : LinearOrder E
inst✝¹² : IsStrictOrderedRing E
inst✝¹¹ : AddCommGroup F
inst✝¹⁰ : LinearOrder F
inst✝⁹ : IsOrderedAddMonoid F
inst✝⁸ : AddCommGroup G
inst✝⁷ : Module 𝕜 G
inst✝⁶ : Module 𝕜 E
inst✝⁵ : Module 𝕜 F
inst✝⁴ : Module E F
inst✝³ : IsScalarTower 𝕜 E F
inst✝² : SMulCommClass 𝕜 E F
inst✝¹ : IsOrderedModule 𝕜 F
inst✝ : IsStrictOrderedModule E F
s : Set G
f : G → E
g : G → F
hf : ConvexOn 𝕜 s f
hg : ConvexOn 𝕜 s g
hf₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ f x
hg₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ g x
hfg : MonovaryOn f g s
x : G
hx : x ∈ s
y : G
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ (a • f x + b • f y) • (a • g x + b • g y) ≤ a • f x • g x + b • f y • g y
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0_lt : 0 < p
f : α → ℝ≥0∞
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞
⊢ ∫⁻ (c : α), f c ^ p * (∫⁻ (c : α), f c ^ p ∂μ)⁻¹ ∂μ = 1

Tactic:
lintegral_mul_const',

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0_lt : 0 < p
f : α → ℝ≥0∞
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞
⊢ (∫⁻ (a : α), f a ^ p ∂μ) * (∫⁻ (c : α), f c ^ p ∂μ)⁻¹ = 1

case hr
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0_lt : 0 < p
f : α → ℝ≥0∞
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞
⊢ (∫⁻ (c : α), f c ^ p ∂μ)⁻¹ ≠ ∞
Module: Mathlib.Analysis.Convex.Mul

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
n : ℕ
⊢ ConvexOn 𝕜 (Ioi 0) fun x ↦ x ^ ↑n

Tactic:
simp_rw [zpow_natCast]

Goal state after:
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
n : ℕ
⊢ ConvexOn 𝕜 (Ioi 0) fun x ↦ x ^ n
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : ∑ i ∈ s, f i ^ p = 0
i : ι
his : i ∈ s
⊢ f i = 0

Tactic:
rw [sum_eq_zero_iff] at hf

Goal state after:
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : ∀ i ∈ s, f i ^ p = 0
i : ι
his : i ∈ s
⊢ f i = 0
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
f g : ι → ℝ≥0
p q r : ℝ
hpqr : p.HolderTriple q r
hf : Summable fun i ↦ f i ^ p
hg : Summable fun i ↦ g i ^ q
⊢ (Summable fun i ↦ (f i * g i) ^ r) ∧
    ∑' (i : ι), (f i * g i) ^ r ≤ (∑' (i : ι), f i ^ p) ^ (r / p) * (∑' (i : ι), g i ^ q) ^ (r / q)

Tactic:
by
  have H₁ : ∀ s : Finset ι,
      ∑ i ∈ s, (f i * g i) ^ r ≤ (∑' i, f i ^ p) ^ (r / p) * (∑' i, g i ^ q) ^ (r / q) := by
    intro s
    obtain ⟨hp, hq, hr⟩ := hpqr.all_pos
    refine le_trans (Lr_rpow_le_Lp_mul_Lq s f g hpqr) (mul_le_mul ?_ ?_ bot_le bot_le)
    · gcongr
      exact hf.sum_le_tsum _ (fun _ _ => zero_le)
    · gcongr
      exact hg.sum_le_tsum _ (fun _ _ => zero_le)
  have bdd : BddAbove (Set.range fun s => ∑ i ∈ s, (f i * g i) ^ r) := by
    refine ⟨(∑' i, f i ^ p) ^ (r / p) * (∑' i, g i ^ q) ^ (r / q), ?_⟩
    rintro a ⟨s, rfl⟩
    exact H₁ s
  have H₂ : Summable _ := (hasSum_of_isLUB _ (isLUB_ciSup bdd)).summable
  exact ⟨H₂, H₂.tsum_le_of_sum_le H₁⟩

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case inr.inr.inr.inr
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : p.HolderTriple q r
hpqr : 1 / r = 1 / p + 1 / q
hp : 0 < p.toReal
hq : 0 < q.toReal
hp₁ : 0 < p
hp₂ : p < ∞
hq₁ : 0 < q
hq₂ : q < ∞
hpqr' : 1 / r.toReal = 1 / p.toReal + 1 / q.toReal
⊢ eLpNorm (fun x ↦ b (f x) (g x)) r μ ≤ ↑c * eLpNorm f p μ * eLpNorm g q μ

Tactic:
have hr : 0 < r.toReal := one_div_pos.mp <| by rw [hpqr']; positivity

Goal state after:
case inr.inr.inr.inr
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : p.HolderTriple q r
hpqr : 1 / r = 1 / p + 1 / q
hp : 0 < p.toReal
hq : 0 < q.toReal
hp₁ : 0 < p
hp₂ : p < ∞
hq₁ : 0 < q
hq₂ : q < ∞
hpqr' : 1 / r.toReal = 1 / p.toReal + 1 / q.toReal
hr : 0 < r.toReal
⊢ eLpNorm (fun x ↦ b (f x) (g x)) r μ ≤ ↑c * eLpNorm f p μ * eLpNorm g q μ
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case h.inr
ι : Type u
s : Finset ι
f : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
A : p + q - q ≠ 0
B : ∀ (y : ℝ≥0), y * y ^ p / y = y ^ p
⊢ (∑ i ∈ s, f i ^ p) ^ (1 / q) ≠ 0

Tactic:
simpa [hpq.symm.ne_zero] using hf.ne'

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case inl
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
hp : 1 ≤ 1
⊢ (∑ i ∈ s, (f i + g i) ^ 1) ^ (1 / 1) ≤ (∑ i ∈ s, f i ^ 1) ^ (1 / 1) + (∑ i ∈ s, g i ^ 1) ^ (1 / 1)

Tactic:
simp [Finset.sum_add_distrib]

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

Goal state before:
pp: case inr.inl
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp1 : 1 ≤ p
hp_pos : 0 < p
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ = ∞
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

pp: case inr.inr
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp1 : 1 ≤ p
hp_pos : 0 < p
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)

Tactic:
· simp [hg_top, hp_pos]

Goal state after:
case inr.inr
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp1 : 1 ≤ p
hp_pos : 0 < p
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝² : SeminormedAddGroup E
ι : Type u_3
inst✝¹ : Nonempty ι
inst✝ : LinearOrder ι
f : ι → α → E
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
⊢ eLpNorm f_lim ∞ μ = essSup (fun x ↦ liminf (fun m ↦ ‖f m x‖ₑ) atTop) μ

Tactic:
rw [eLpNorm_exponent_top, eLpNormEssSup_eq_essSup_enorm]

Goal state after:
α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝² : SeminormedAddGroup E
ι : Type u_3
inst✝¹ : Nonempty ι
inst✝ : LinearOrder ι
f : ι → α → E
f_lim : α → E
h_lim : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (f_lim x))
⊢ essSup (fun x ↦ ‖f_lim x‖ₑ) μ = essSup (fun x ↦ liminf (fun m ↦ ‖f m x‖ₑ) atTop) μ
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
f : ↥(Lp E p μ)
C : ℝ
hC : 0 ≤ C
hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ C
⊢ ‖f‖ ≤ ↑(measureUnivNNReal μ) ^ p.toReal⁻¹ * C

Tactic:
lift C to ℝ≥0 using hC
  have := nnnorm_le_of_ae_bound hfC
  rwa [← NNReal.coe_le_coe, NNReal.coe_mul, NNReal.coe_rpow] at this

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → α
hf : MeasurePreserving f μ μ
n : ℕ
⊢ (⇑(compMeasurePreserving f hf))^[n] = ⇑(compMeasurePreserving f^[n] ⋯)

Tactic:
funext

Goal state after:
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
f : α → α
hf : MeasurePreserving f μ μ
n : ℕ
x✝ : ↥(Lp E p μ)
⊢ (⇑(compMeasurePreserving f hf))^[n] x✝ = (compMeasurePreserving f^[n] ⋯) x✝
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝² : PseudoEMetricSpace E
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
f : ι → α → E
g : α → E
hfg : TendstoInMeasure μ f u g
⊢ ∃ ns, Tendsto ns atTop u ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i ↦ f (ns i) x) atTop (𝓝 (g x))

Tactic:
obtain ⟨ms, hms1, hms2⟩ := hfg.exists_seq_tendstoInMeasure_atTop

Goal state after:
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝² : PseudoEMetricSpace E
u : Filter ι
inst✝¹ : u.NeBot
inst✝ : u.IsCountablyGenerated
f : ι → α → E
g : α → E
hfg : TendstoInMeasure μ f u g
ms : ℕ → ι
hms1 : Tendsto ms atTop u
hms2 : TendstoInMeasure μ (fun n ↦ f (ms n)) atTop g
⊢ ∃ ns, Tendsto ns atTop u ∧ ∀ᵐ (x : α) ∂μ, Tendsto (fun i ↦ f (ns i) x) atTop (𝓝 (g x))
Module: Mathlib.Analysis.Normed.Ring.Units

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
t : R
ht : ‖t‖ < 2⁻¹
⊢ ‖(1 - t)⁻¹ʳ‖ ≤ (‖1‖ + 1) * ‖1‖

Tactic:
have ht' : ‖t‖ < 1 := by linarith

Goal state after:
R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
t : R
ht : ‖t‖ < 2⁻¹
ht' : ‖t‖ < 1
⊢ ‖(1 - t)⁻¹ʳ‖ ≤ (‖1‖ + 1) * ‖1‖