mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Defs

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
q : ℝ
inst✝ : ENorm ε
μ : Measure α
f : α → ε
hq0_lt : 0 < q
⊢ ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ = (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ 1

pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
q : ℝ
inst✝ : ENorm ε
μ : Measure α
f : α → ε
hq0_lt : 0 < q
⊢ q ≠ 0

Tactic:
ENNReal.rpow_one

Goal state after:
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
q : ℝ
inst✝ : ENorm ε
μ : Measure α
f : α → ε
hq0_lt : 0 < q
⊢ ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ = ∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ

α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
q : ℝ
inst✝ : ENorm ε
μ : Measure α
f : α → ε
hq0_lt : 0 < q
⊢ q ≠ 0
Module: Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov

Goal state before:
pp: case neg
α : Type u_1
ε' : Type u_3
m0 : MeasurableSpace α
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
p : ℝ≥0∞
μ : Measure α
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε'
hf : AEStronglyMeasurable f μ
ε : ℝ≥0∞
hε : ε ≠ 0
hmeas_top : ε = ∞ → μ {x | ‖f x‖ₑ = ∞} = 0
h : ¬ε = ∞
hεpow : ε ^ p.toReal ≠ 0
⊢ μ {x | ε ≤ ‖f x‖ₑ} ≤ ε⁻¹ ^ p.toReal * eLpNorm f p μ ^ p.toReal

Tactic:
have hεpow' : ε ^ p.toReal ≠ ∞ := by finiteness

Goal state after:
case neg
α : Type u_1
ε' : Type u_3
m0 : MeasurableSpace α
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
p : ℝ≥0∞
μ : Measure α
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε'
hf : AEStronglyMeasurable f μ
ε : ℝ≥0∞
hε : ε ≠ 0
hmeas_top : ε = ∞ → μ {x | ‖f x‖ₑ = ∞} = 0
h : ¬ε = ∞
hεpow : ε ^ p.toReal ≠ 0
hεpow' : ε ^ p.toReal ≠ ∞
⊢ μ {x | ε ≤ ‖f x‖ₑ} ≤ ε⁻¹ ^ p.toReal * eLpNorm f p μ ^ p.toReal
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

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
c : ε
s : Set α
hs : NullMeasurableSet s μ
hp : p ≠ 0
hp_top : p ≠ ∞
hp_pos : 0 < p.toReal
⊢ ‖c‖ₑ ^ 1 * μ s ^ (1 / p.toReal) = ‖c‖ₑ * μ s ^ (1 / p.toReal)

pp: case hz
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
c : ε
s : Set α
hs : NullMeasurableSet s μ
hp : p ≠ 0
hp_top : p ≠ ∞
hp_pos : 0 < p.toReal
⊢ 0 ≤ 1 / p.toReal

Tactic:
ENNReal.rpow_one

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
c : ε
s : Set α
hs : NullMeasurableSet s μ
hp : p ≠ 0
hp_top : p ≠ ∞
hp_pos : 0 < p.toReal
⊢ ‖c‖ₑ * μ s ^ (1 / p.toReal) = ‖c‖ₑ * μ s ^ (1 / p.toReal)

case hz
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
c : ε
s : Set α
hs : NullMeasurableSet s μ
hp : p ≠ 0
hp_top : p ≠ ∞
hp_pos : 0 < p.toReal
⊢ 0 ≤ 1 / p.toReal
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: case neg.inr.left
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
s : Set α
f : α → ε
inst✝ : DecidablePred fun x ↦ x ∈ s
g : α → ε
hs : MeasurableSet s
hf : MemLp f p (μ.restrict s)
hg : MemLp g p (μ.restrict sᶜ)
hp_zero : ¬p = 0
hp_top : p ≠ ∞
⊢ ∫⁻ (x : α) in s, ‖s.piecewise f g x‖ₑ ^ p.toReal ∂μ < ∞

Tactic:
have h (x) (hx : x ∈ s) : ‖Set.piecewise s f g x‖ₑ ^ p.toReal = ‖f x‖ₑ ^ p.toReal := by
      simp [hx]
    rw [setLIntegral_congr_fun hs h]
    exact lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top hp_zero hp_top hf.2

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

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : ENorm ε
f : α → ε
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
⊢ ∫⁻ (a : α), ‖f a‖ₑ ^ p.toReal ∂μ < ∞ → eLpNorm f p μ < ∞

Tactic:
intro h

Goal state after:
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : ENorm ε
f : α → ε
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
h : ∫⁻ (a : α), ‖f a‖ₑ ^ p.toReal ∂μ < ∞
⊢ eLpNorm f p μ < ∞
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
q : ℝ
μ : Measure α
inst✝ : ENorm ε
c : ε
hq_pos : 0 < q
hq_cancel : q * (1 / q) = 1
⊢ ‖c‖ₑ ^ 1 = ‖c‖ₑ

Tactic:
ENNReal.rpow_one

Goal state after:
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
q : ℝ
μ : Measure α
inst✝ : ENorm ε
c : ε
hq_pos : 0 < q
hq_cancel : q * (1 / q) = 1
⊢ ‖c‖ₑ = ‖c‖ₑ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: α : Type u_1
F : Type u_5
m0 : MeasurableSpace α
q : ℝ
μ : Measure α
inst✝¹ : NormedAddCommGroup F
inst✝ : IsFiniteMeasure μ
c : F
hc_ne_zero : c ≠ 0
hq_ne_zero : q ≠ 0
hp_cancel : q * (1 / q) = 1
⊢ ‖c‖ₑ ^ 1 = ‖c‖ₑ

Tactic:
ENNReal.rpow_one

Goal state after:
α : Type u_1
F : Type u_5
m0 : MeasurableSpace α
q : ℝ
μ : Measure α
inst✝¹ : NormedAddCommGroup F
inst✝ : IsFiniteMeasure μ
c : F
hc_ne_zero : c ≠ 0
hq_ne_zero : q ≠ 0
hp_cancel : q * (1 / q) = 1
⊢ ‖c‖ₑ = ‖c‖ₑ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: case mp
α : Type u_7
inst✝² : Semiring α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a b c : α
ha : 0 ≤ a
hb : b < 0
hc : 0 ≤ c
⊢ a ≤ b * c → a = 0 ∧ c = 0

Tactic:
intro h

Goal state after:
case mp
α : Type u_7
inst✝² : Semiring α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
a b c : α
ha : 0 ≤ a
hb : b < 0
hc : 0 ≤ c
h : a ≤ b * c
⊢ a = 0 ∧ c = 0
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity

Goal state before:
pp: case h₁
α : Type u_1
F : Type u_3
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup F
hp : p ≠ 0
hp' : p ≠ ∞
f : α → F
C : ℝ≥0
s : Set α
hs : MeasurableSet s
hf : ∀ᵐ (x : α) ∂μ, x ∈ s → C ≤ ‖f x‖₊
x : α
hxs : x ∈ s
hx : C ≤ ‖f x‖₊
⊢ ↑C ≤ ‖f x‖ₑ

Tactic:
rwa [coe_le_enorm]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
⊢ (f y - f x) / (y - x) < (f z - f x) / (z - x)

Tactic:
have hxy' : 0 < y - x := by linarith
  have hxz' : 0 < z - x := by linarith
  field_simp
  linarith only [hf.secant_strict_mono_aux1 hx hz hxy hyz]

Goal state after:
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: p : ℝ
hp : 1 < p
x y z : ℝ
⊢ x ∈ Ici 0 → z ∈ Ici 0 → x < y → y < z → (y ^ p - x ^ p) / (y - x) < (z ^ p - y ^ p) / (z - y)

Tactic:
(hx : 0 ≤ x)

Goal state after:
p : ℝ
hp : 1 < p
x y z : ℝ
hx : 0 ≤ x
⊢ z ∈ Ici 0 → x < y → y < z → (y ^ p - x ^ p) / (y - x) < (z ^ p - y ^ p) / (z - y)
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case inr
p : ℝ
hp✝ : 1 ≤ p
hp : 1 < p
⊢ ConvexOn ℝ (Ici 0) fun x ↦ x ^ p

Tactic:
exact (strictConvexOn_rpow hp).convexOn

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case pos.mp
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
i : ι
his : i ∈ s
hzi : z i = 0
hwi : w i ≠ 0
⊢ 0 = ∑ i ∈ s, w i * z i → ∀ j ∈ s, z j = ∑ i ∈ s, w i * z i

Tactic:
intro h

Goal state after:
case pos.mp
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
i : ι
his : i ∈ s
hzi : z i = 0
hwi : w i ≠ 0
h : 0 = ∑ i ∈ s, w i * z i
⊢ ∀ j ∈ s, z j = ∑ i ∈ s, w i * z i
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case pos.mpr
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
i : ι
his : i ∈ s
hzi : z i = 0
hwi : w i ≠ 0
⊢ (∀ j ∈ s, z j = ∑ i ∈ s, w i * z i) → 0 = ∑ i ∈ s, w i * z i

Tactic:
intro h

Goal state after:
case pos.mpr
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
i : ι
his : i ∈ s
hzi : z i = 0
hwi : w i ≠ 0
h : ∀ j ∈ s, z j = ∑ i ∈ s, w i * z i
⊢ 0 = ∑ i ∈ s, w i * z i
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case neg
p : ℝ
a b : ℝ≥0
hp1 : 1 ≤ p
hp_pos : 0 < p
h_zero : ¬a + b = 0
h_nonzero : ¬(a = 0 ∧ b = 0)
h_add : a / (a + b) + b / (a + b) = 1
h : a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p ≤ 1
hab_0 : (a + b) ^ p ≠ 0
h_mul : (a + b) ^ p * ((a + b) ^ p)⁻¹ * a ^ p + (a + b) ^ p * ((a + b) ^ p)⁻¹ * b ^ p ≤ (a + b) ^ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ p

Tactic:
mul_inv_cancel₀ hab_0,

Goal state after:
case neg
p : ℝ
a b : ℝ≥0
hp1 : 1 ≤ p
hp_pos : 0 < p
h_zero : ¬a + b = 0
h_nonzero : ¬(a = 0 ∧ b = 0)
h_add : a / (a + b) + b / (a + b) = 1
h : a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p ≤ 1
hab_0 : (a + b) ^ p ≠ 0
h_mul : 1 * a ^ p + 1 * b ^ p ≤ (a + b) ^ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ p
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: p q : ℝ
a b : ℝ≥0
hp_pos : 0 < p
hpq : p ≤ q
⊢ (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)

Tactic:
have h_rpow : ∀ a : ℝ≥0, a ^ q = (a ^ p) ^ (q / p) := fun a => by
    rw [← NNReal.rpow_mul, div_eq_inv_mul, ← mul_assoc, mul_inv_cancel₀ hp_pos.ne.symm,
      one_mul]
  have h_rpow_add_rpow_le_add :
    ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p := by
    refine rpow_add_rpow_le_add (a ^ p) (b ^ p) ?_
    rwa [one_le_div hp_pos]
  rw [h_rpow a, h_rpow b, one_div p, NNReal.le_rpow_inv_iff hp_pos, ← NNReal.rpow_mul, mul_comm,
    mul_one_div]
  rwa [one_div_div] at h_rpow_add_rpow_le_add

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_2.mp
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
hz' : ∀ i ∈ s, 0 < z i
this : ∏ x ∈ s, rexp (log (z x) * w x) = ∑ x ∈ s, w x * rexp (log (z x)) ↔ ∀ j ∈ s, log (z j) = ∑ x ∈ s, log (z x) * w x
⊢ (∀ j ∈ s, z j = ∑ i ∈ s, w i * z i) → ∀ j ∈ s, log (z j) = ∑ x ∈ s, log (z x) * w x

Tactic:
intro h

Goal state after:
case e'_2.mp
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
hz' : ∀ i ∈ s, 0 < z i
this : ∏ x ∈ s, rexp (log (z x) * w x) = ∑ x ∈ s, w x * rexp (log (z x)) ↔ ∀ j ∈ s, log (z j) = ∑ x ∈ s, log (z x) * w x
h : ∀ j ∈ s, z j = ∑ i ∈ s, w i * z i
⊢ ∀ j ∈ s, log (z j) = ∑ x ∈ s, log (z x) * w x
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case e'_2.mpr
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
hz' : ∀ i ∈ s, 0 < z i
this : ∏ x ∈ s, rexp (log (z x) * w x) = ∑ x ∈ s, w x * rexp (log (z x)) ↔ ∀ j ∈ s, log (z j) = ∑ x ∈ s, log (z x) * w x
⊢ (∀ j ∈ s, log (z j) = ∑ x ∈ s, log (z x) * w x) → ∀ j ∈ s, z j = ∑ i ∈ s, w i * z i

Tactic:
intro h

Goal state after:
case e'_2.mpr
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
hz' : ∀ i ∈ s, 0 < z i
this : ∏ x ∈ s, rexp (log (z x) * w x) = ∑ x ∈ s, w x * rexp (log (z x)) ↔ ∀ j ∈ s, log (z j) = ∑ x ∈ s, log (z x) * w x
h : ∀ j ∈ s, log (z j) = ∑ x ∈ s, log (z x) * w x
⊢ ∀ j ∈ s, z j = ∑ i ∈ s, w i * z i
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i ∈ s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
hp_nonneg : 0 ≤ p
hp_not_neg : ¬p < 0
h_top_iff_rpow_top : ∀ i ∈ s, w i * z i = ∞ ↔ w i * z i ^ p = ∞
h_top_rpow_sum : (∑ i ∈ s, w i * z i) ^ p ≠ ∞
a✝ : ∑ i ∈ s, w i * z i ^ p ≠ ∞
⊢ ∑ i ∈ s, w i * z i ≠ ∞

Tactic:
intro h

Goal state after:
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i ∈ s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
hp_nonneg : 0 ≤ p
hp_not_neg : ¬p < 0
h_top_iff_rpow_top : ∀ i ∈ s, w i * z i = ∞ ↔ w i * z i ^ p = ∞
h_top_rpow_sum : (∑ i ∈ s, w i * z i) ^ p ≠ ∞
a✝ : ∑ i ∈ s, w i * z i ^ p ≠ ∞
h : ∑ i ∈ s, w i * z i = ∞
⊢ False
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case inr.inl
α : Type u_2
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
p : ℝ
hp✝ : 0 ≤ p
hp : 0 < p
hq : 0 ≤ 0
hpq : p + 0 = 1
⊢ ∫⁻ (a : α), f a ^ p * g a ^ 0 ∂μ ≤ (∫⁻ (a : α), f a ∂μ) ^ p * (∫⁻ (a : α), g a ∂μ) ^ 0

Tactic:
rw [add_zero] at hpq

Goal state after:
case inr.inl
α : Type u_2
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
p : ℝ
hp✝ : 0 ≤ p
hp : 0 < p
hq : 0 ≤ 0
hpq : p = 1
⊢ ∫⁻ (a : α), f a ^ p * g a ^ 0 ∂μ ≤ (∫⁻ (a : α), f a ∂μ) ^ p * (∫⁻ (a : α), g a ∂μ) ^ 0
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: 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
h : ∀ j ∈ s, ∀ k ∈ s, z j = z k
⊢ s.Nonempty

Tactic:
grind [s.eq_empty_or_nonempty]

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case inr
p : ℝ
a b : ℝ≥0∞
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : (a ^ 1 + b ^ 1) ^ 1 ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Tactic:
ENNReal.rpow_one

Goal state after:
case inr
p : ℝ
a b : ℝ≥0∞
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a ^ 1 + b ^ 1 ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case inr
p : ℝ
a b : ℝ≥0∞
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a ^ 1 + b ^ 1 ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Tactic:
ENNReal.rpow_one

Goal state after:
case inr
p : ℝ
a b : ℝ≥0∞
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a + b ^ 1 ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case inr
p : ℝ
a b : ℝ≥0∞
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a + b ^ 1 ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Tactic:
ENNReal.rpow_one

Goal state after:
case inr
p : ℝ
a b : ℝ≥0∞
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a + b ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case inr
p : ℝ
a b : ℝ≥0∞
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a + b ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Tactic:
ENNReal.rpow_one

Goal state after:
case inr
p : ℝ
a b : ℝ≥0∞
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a + b ≤ (a ^ p + b ^ p) ^ p⁻¹
⊢ (a + b) ^ p ≤ a ^ p + b ^ p
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := fun i ↦ p i / (1 - p i₀)
⊢ (∑ i ∈ s, p i) / (1 - p i₀) = 1

Tactic:
← sum_insert_sub hi₀,

Goal state after:
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := fun i ↦ p i / (1 - p i₀)
⊢ (∑ x ∈ insert i₀ s, p x - p i₀) / (1 - p i₀) = 1
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case hnc
p : ℝ≥0∞
hp : 1 ≤ p
⊢ p ∉ Set.Ioo 0 1

Tactic:
intro h

Goal state after:
case hnc
p : ℝ≥0∞
hp : 1 ≤ p
h : p ∈ Set.Ioo 0 1
⊢ False
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: case hnc
⊢ 0 ∉ Set.Ioo 0 1

Tactic:
intro h

Goal state after:
case hnc
h : 0 ∈ Set.Ioo 0 1
⊢ False
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case mp
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
⊢ ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i → ∃ j ∈ s, ∃ k ∈ s, z j ≠ z k

Tactic:
intro h

Goal state after:
case mp
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
h : ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i
⊢ ∃ j ∈ s, ∃ k ∈ s, z j ≠ z k
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case neg
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq : p ≤ q
hf : AEStronglyMeasurable f μ
hq0_lt : 0 < q
hpq_eq : ¬p = q
⊢ eLpNorm' f p μ ≤ eLpNorm' f q μ * μ Set.univ ^ (1 / p - 1 / q)

Tactic:
have hpq : p < q := lt_of_le_of_ne hpq hpq_eq

Goal state after:
case neg
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p q : ℝ
hp0_lt : 0 < p
hpq✝ : p ≤ q
hf : AEStronglyMeasurable f μ
hq0_lt : 0 < q
hpq_eq : ¬p = q
hpq : p < q
⊢ eLpNorm' f p μ ≤ eLpNorm' f q μ * μ Set.univ ^ (1 / p - 1 / q)
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case mp
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
⊢ ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i → ∃ j ∈ s, ∃ k ∈ s, z j ≠ z k

Tactic:
intro h
    by_contra! h_contra
    rw [(geom_mean_eq_arith_mean_weighted_iff_of_pos' s w z hw hw' hz).mpr ?_] at h
    · exact (lt_self_iff_false _).mp h
    · intro j hjs
      rw [← arith_mean_weighted_of_constant s w (fun _ => z j) (z j) hw' fun _ _ => congrFun rfl]
      apply sum_congr rfl (fun x a => congrArg (HMul.hMul (w x)) (h_contra j hjs x a))

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

Goal state before:
pp: α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
μ : Measure α
p δ : ℝ≥0∞
hδ : δ ≠ 0
this : Tendsto (fun η ↦ p.LpAddConst * (η + η)) (𝓝[>] 0) (𝓝 0)
η : ℝ≥0∞
hη : p.LpAddConst * (η + η) < δ
ηpos : 0 < η
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
Hf : eLpNorm f p μ ≤ η
Hg : eLpNorm g p μ ≤ η
⊢ eLpNorm (f + g) p μ < δ

Tactic:
calc
    eLpNorm (f + g) p μ ≤ LpAddConst p * (eLpNorm f p μ + eLpNorm g p μ) :=
      eLpNorm_add_le' hf hg p
    _ ≤ LpAddConst p * (η + η) := by gcongr
    _ < δ := hη

Goal state after:
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 ≤ 1
hg : ∑ i ∈ s, g i ^ q ≤ 1
hp : 0 < p.toNNReal
⊢ ∑ i ∈ s, f i * g i ≤ 1

Tactic:
have hq : 0 < q.toNNReal := zero_lt_one.trans hpq.toNNReal.symm.lt

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 ≤ 1
hg : ∑ i ∈ s, g i ^ q ≤ 1
hp : 0 < p.toNNReal
hq : 0 < q.toNNReal
⊢ ∑ i ∈ s, f i * g i ≤ 1
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case inr.inr.inr.inr.hp_ne_zero
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
hr₁ : 0 < r
hr₂ : r < ∞
hrp : r.toReal < p.toReal
⊢ q ≠ 0

Tactic:
first | positivity | finiteness

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

Goal state before:
pp: case inr.inr.inr.inr.hp_ne_top
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
hr₁ : 0 < r
hr₂ : r < ∞
hrp : r.toReal < p.toReal
⊢ q ≠ ∞

Tactic:
first | positivity | finiteness

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

Goal state before:
pp: case inr.inr.inr.inr.hp_ne_zero
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
hr₁ : 0 < r
hr₂ : r < ∞
hrp : r.toReal < p.toReal
⊢ p ≠ 0

Tactic:
first | positivity | finiteness

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

Goal state before:
pp: case inr.inr.inr.inr.hp_ne_top
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
hr₁ : 0 < r
hr₂ : r < ∞
hrp : r.toReal < p.toReal
⊢ p ≠ ∞

Tactic:
first | positivity | finiteness

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

Goal state before:
pp: case inr.inr.inr.inr.hp_ne_zero
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
hr₁ : 0 < r
hr₂ : r < ∞
hrp : r.toReal < p.toReal
⊢ r ≠ 0

Tactic:
first | positivity | finiteness

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

Goal state before:
pp: case inr.inr.inr.inr.hp_ne_top
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
hr₁ : 0 < r
hr₂ : r < ∞
hrp : r.toReal < p.toReal
⊢ r ≠ ∞

Tactic:
first | positivity | finiteness

Goal state after: