l3lab/ntp-mathlib huggingface dataset:
theorem add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) : a ^ p + b ^ p ≤ (a + b) ^ p := by
  have hp_pos : 0 < p := by positivity
  by_cases h_zero : a + b = 0
  · simp [add_eq_zero_iff.mp h_zero, hp_pos.ne']
  have h_nonzero : ¬(a = 0 ∧ b = 0) := by rwa [add_eq_zero_iff] at h_zero
  have h_add : a / (a + b) + b / (a + b) = 1 := by rw [div_add_div_same, div_self h_zero]
  have h := add_rpow_le_one_of_add_le_one (a / (a + b)) (b / (a + b)) h_add.le hp1
  rw [div_rpow a (a + b), div_rpow b (a + b)] at h
  have hab_0 : (a + b) ^ p ≠ 0 := by simp [hp_pos, h_nonzero]
  have hab_0' : 0 < (a + b) ^ p := zero_lt_iff.mpr hab_0
  

Proof state:
case neg
ι : Type u
s : Finset ι
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
hab_0' : 0 < (a + b) ^ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ p

Next tactic:
have h_mul : (a + b) ^ p * (a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p) ≤ (a + b) ^ p := by
    nth_rw 4 [← mul_one ((a + b) ^ p)]
    exact (mul_le_mul_left hab_0').mpr h
theorem add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) : a ^ p + b ^ p ≤ (a + b) ^ p := by
  have hp_pos : 0 < p := by positivity
  by_cases h_zero : a + b = 0
  · simp [add_eq_zero_iff.mp h_zero, hp_pos.ne']
  have h_nonzero : ¬(a = 0 ∧ b = 0) := by rwa [add_eq_zero_iff] at h_zero
  have h_add : a / (a + b) + b / (a + b) = 1 := by rw [div_add_div_same, div_self h_zero]
  have h := add_rpow_le_one_of_add_le_one (a / (a + b)) (b / (a + b)) h_add.le hp1
  rw [div_rpow a (a + b), div_rpow b (a + b)] at h
  have hab_0 : (a + b) ^ p ≠ 0 := by simp [hp_pos, h_nonzero]
  have hab_0' : 0 < (a + b) ^ p := zero_lt_iff.mpr hab_0
  have h_mul : (a + b) ^ p * (a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p) ≤ (a + b) ^ p := by
    

Proof state:
ι : Type u
s : Finset ι
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
hab_0' : 0 < (a + b) ^ p
⊢ (a + b) ^ p * (a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p) ≤ (a + b) ^ p

Next tactic:
nth_rw 4 [← mul_one ((a + b) ^ p)]
theorem add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) : a ^ p + b ^ p ≤ (a + b) ^ p := by
  have hp_pos : 0 < p := by positivity
  by_cases h_zero : a + b = 0
  · simp [add_eq_zero_iff.mp h_zero, hp_pos.ne']
  have h_nonzero : ¬(a = 0 ∧ b = 0) := by rwa [add_eq_zero_iff] at h_zero
  have h_add : a / (a + b) + b / (a + b) = 1 := by rw [div_add_div_same, div_self h_zero]
  have h := add_rpow_le_one_of_add_le_one (a / (a + b)) (b / (a + b)) h_add.le hp1
  rw [div_rpow a (a + b), div_rpow b (a + b)] at h
  have hab_0 : (a + b) ^ p ≠ 0 := by simp [hp_pos, h_nonzero]
  have hab_0' : 0 < (a + b) ^ p := zero_lt_iff.mpr hab_0
  have h_mul : (a + b) ^ p * (a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p) ≤ (a + b) ^ p := by
    nth_rw 4 [← mul_one ((a + b) ^ p)]
    

Proof state:
ι : Type u
s : Finset ι
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
hab_0' : 0 < (a + b) ^ p
⊢ (a + b) ^ p * (a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p) ≤ (a + b) ^ p * 1

Next tactic:
exact (mul_le_mul_left hab_0').mpr h
theorem add_rpow_le_rpow_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) : a ^ p + b ^ p ≤ (a + b) ^ p := by
  have hp_pos : 0 < p := by positivity
  by_cases h_zero : a + b = 0
  · simp [add_eq_zero_iff.mp h_zero, hp_pos.ne']
  have h_nonzero : ¬(a = 0 ∧ b = 0) := by rwa [add_eq_zero_iff] at h_zero
  have h_add : a / (a + b) + b / (a + b) = 1 := by rw [div_add_div_same, div_self h_zero]
  have h := add_rpow_le_one_of_add_le_one (a / (a + b)) (b / (a + b)) h_add.le hp1
  rw [div_rpow a (a + b), div_rpow b (a + b)] at h
  have hab_0 : (a + b) ^ p ≠ 0 := by simp [hp_pos, h_nonzero]
  have hab_0' : 0 < (a + b) ^ p := zero_lt_iff.mpr hab_0
  have h_mul : (a + b) ^ p * (a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p) ≤ (a + b) ^ p := by
    nth_rw 4 [← mul_one ((a + b) ^ p)]
    exact (mul_le_mul_left hab_0').mpr h
  

Proof state:
case neg
ι : Type u
s : Finset ι
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
hab_0' : 0 < (a + b) ^ p
h_mul : (a + b) ^ p * (a ^ p / (a + b) ^ p + b ^ p / (a + b) ^ p) ≤ (a + b) ^ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ p

Next tactic:
rwa [div_eq_mul_inv, div_eq_mul_inv, mul_add, mul_comm (a ^ p), mul_comm (b ^ p), ← mul_assoc, ←
    mul_assoc, mul_inv_cancel hab_0, one_mul, one_mul] at h_mul
theorem rpow_add_rpow_le_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) :
    (a ^ p + b ^ p) ^ (1 / p) ≤ a + b := by
  

Proof state:
ι : Type u
s : Finset ι
p : ℝ
a b : ℝ≥0
hp1 : 1 ≤ p
⊢ (a ^ p + b ^ p) ^ (1 / p) ≤ a + b

Next tactic:
rw [← @NNReal.le_rpow_one_div_iff _ _ (1 / p) (by simp [lt_of_lt_of_le zero_lt_one hp1])]
theorem rpow_add_rpow_le_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) :
    (a ^ p + b ^ p) ^ (1 / p) ≤ a + b := by
  rw [← @NNReal.le_rpow_one_div_iff _ _ (1 / p) (by 

Proof state:
ι : Type u
s : Finset ι
p : ℝ
a b : ℝ≥0
hp1 : 1 ≤ p
⊢ 0 < 1 / p

Next tactic:
simp [lt_of_lt_of_le zero_lt_one hp1]
theorem rpow_add_rpow_le_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) :
    (a ^ p + b ^ p) ^ (1 / p) ≤ a + b := by
  rw [← @NNReal.le_rpow_one_div_iff _ _ (1 / p) (by simp [lt_of_lt_of_le zero_lt_one hp1])]
  

Proof state:
ι : Type u
s : Finset ι
p : ℝ
a b : ℝ≥0
hp1 : 1 ≤ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ (1 / (1 / p))

Next tactic:
rw [one_div_one_div]
theorem rpow_add_rpow_le_add {p : ℝ} (a b : ℝ≥0) (hp1 : 1 ≤ p) :
    (a ^ p + b ^ p) ^ (1 / p) ≤ a + b := by
  rw [← @NNReal.le_rpow_one_div_iff _ _ (1 / p) (by simp [lt_of_lt_of_le zero_lt_one hp1])]
  rw [one_div_one_div]
  

Proof state:
ι : Type u
s : Finset ι
p : ℝ
a b : ℝ≥0
hp1 : 1 ≤ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ p

Next tactic:
exact add_rpow_le_rpow_add _ _ hp1
theorem rpow_add_rpow_le {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) := by
  

Proof state:
ι : Type u
s : Finset ι
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)

Next 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, _root_.mul_inv_cancel hp_pos.ne.symm,
      one_mul]
theorem rpow_add_rpow_le {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) := by
  have h_rpow : ∀ a : ℝ≥0, a ^ q = (a ^ p) ^ (q / p) := fun a => by
    

Proof state:
ι : Type u
s : Finset ι
p q : ℝ
a✝ b : ℝ≥0
hp_pos : 0 < p
hpq : p ≤ q
a : ℝ≥0
⊢ a ^ q = (a ^ p) ^ (q / p)

Next tactic:
rw [← NNReal.rpow_mul, div_eq_inv_mul, ← mul_assoc, _root_.mul_inv_cancel hp_pos.ne.symm,
      one_mul]
theorem rpow_add_rpow_le {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) := by
  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, _root_.mul_inv_cancel hp_pos.ne.symm,
      one_mul]
  

Proof state:
ι : Type u
s : Finset ι
p q : ℝ
a b : ℝ≥0
hp_pos : 0 < p
hpq : p ≤ q
h_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)
⊢ (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)

Next tactic:
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]
theorem rpow_add_rpow_le {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) := by
  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, _root_.mul_inv_cancel 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
    

Proof state:
ι : Type u
s : Finset ι
p q : ℝ
a b : ℝ≥0
hp_pos : 0 < p
hpq : p ≤ q
h_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)
⊢ ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p

Next tactic:
refine' rpow_add_rpow_le_add (a ^ p) (b ^ p) _
theorem rpow_add_rpow_le {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) := by
  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, _root_.mul_inv_cancel 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) _
    

Proof state:
ι : Type u
s : Finset ι
p q : ℝ
a b : ℝ≥0
hp_pos : 0 < p
hpq : p ≤ q
h_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)
⊢ 1 ≤ q / p

Next tactic:
rwa [one_le_div hp_pos]
theorem rpow_add_rpow_le {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) := by
  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, _root_.mul_inv_cancel 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]
  

Proof state:
ι : Type u
s : Finset ι
p q : ℝ
a b : ℝ≥0
hp_pos : 0 < p
hpq : p ≤ q
h_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)
h_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p
⊢ (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)

Next tactic:
rw [h_rpow a, h_rpow b, NNReal.le_rpow_one_div_iff hp_pos, ← NNReal.rpow_mul, mul_comm,
    mul_one_div]
theorem rpow_add_rpow_le {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) := by
  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, _root_.mul_inv_cancel 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, NNReal.le_rpow_one_div_iff hp_pos, ← NNReal.rpow_mul, mul_comm,
    mul_one_div]
  

Proof state:
ι : Type u
s : Finset ι
p q : ℝ
a b : ℝ≥0
hp_pos : 0 < p
hpq : p ≤ q
h_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)
h_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p
⊢ ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (p / q) ≤ a ^ p + b ^ p

Next tactic:
rwa [one_div_div] at h_rpow_add_rpow_le_add
theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    (a + b) ^ p ≤ a ^ p + b ^ p := by
  

Proof state:
ι : Type u
s : Finset ι
p : ℝ
a b : ℝ≥0
hp : 0 ≤ p
hp1 : p ≤ 1
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Next tactic:
rcases hp.eq_or_lt with (rfl | hp_pos)
theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    (a + b) ^ p ≤ a ^ p + b ^ p := by
  rcases hp.eq_or_lt with (rfl | hp_pos)
  · 

Proof state:
case inl
ι : Type u
s : Finset ι
a b : ℝ≥0
hp : 0 ≤ 0
hp1 : 0 ≤ 1
⊢ (a + b) ^ 0 ≤ a ^ 0 + b ^ 0

Next tactic:
simp
theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    (a + b) ^ p ≤ a ^ p + b ^ p := by
  rcases hp.eq_or_lt with (rfl | hp_pos)
  · simp
  

Proof state:
case inr
ι : Type u
s : Finset ι
p : ℝ
a b : ℝ≥0
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Next tactic:
have h := rpow_add_rpow_le a b hp_pos hp1
theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    (a + b) ^ p ≤ a ^ p + b ^ p := by
  rcases hp.eq_or_lt with (rfl | hp_pos)
  · simp
  have h := rpow_add_rpow_le a b hp_pos hp1
  

Proof state:
case inr
ι : Type u
s : Finset ι
p : ℝ
a b : ℝ≥0
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : (a ^ 1 + b ^ 1) ^ (1 / 1) ≤ (a ^ p + b ^ p) ^ (1 / p)
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Next tactic:
rw [one_div_one] at h
theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    (a + b) ^ p ≤ a ^ p + b ^ p := by
  rcases hp.eq_or_lt with (rfl | hp_pos)
  · simp
  have h := rpow_add_rpow_le a b hp_pos hp1
  rw [one_div_one] at h
  

Proof state:
case inr
ι : Type u
s : Finset ι
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) ^ (1 / p)
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Next tactic:
repeat' rw [NNReal.rpow_one] at h
theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    (a + b) ^ p ≤ a ^ p + b ^ p := by
  rcases hp.eq_or_lt with (rfl | hp_pos)
  · simp
  have h := rpow_add_rpow_le a b hp_pos hp1
  rw [one_div_one] at h
  repeat' 

Proof state:
case inr
ι : Type u
s : Finset ι
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) ^ (1 / p)
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Next tactic:
rw [NNReal.rpow_one] at h
theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    (a + b) ^ p ≤ a ^ p + b ^ p := by
  rcases hp.eq_or_lt with (rfl | hp_pos)
  · simp
  have h := rpow_add_rpow_le a b hp_pos hp1
  rw [one_div_one] at h
  repeat' 

Proof state:
case inr
ι : Type u
s : Finset ι
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) ^ (1 / p)
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Next tactic:
rw [NNReal.rpow_one] at h
theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    (a + b) ^ p ≤ a ^ p + b ^ p := by
  rcases hp.eq_or_lt with (rfl | hp_pos)
  · simp
  have h := rpow_add_rpow_le a b hp_pos hp1
  rw [one_div_one] at h
  repeat' 

Proof state:
case inr
ι : Type u
s : Finset ι
p : ℝ
a b : ℝ≥0
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a + b ^ 1 ≤ (a ^ p + b ^ p) ^ (1 / p)
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Next tactic:
rw [NNReal.rpow_one] at h
theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    (a + b) ^ p ≤ a ^ p + b ^ p := by
  rcases hp.eq_or_lt with (rfl | hp_pos)
  · simp
  have h := rpow_add_rpow_le a b hp_pos hp1
  rw [one_div_one] at h
  repeat' 

Proof state:
case inr
ι : Type u
s : Finset ι
p : ℝ
a b : ℝ≥0
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a + b ≤ (a ^ p + b ^ p) ^ (1 / p)
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Next tactic:
rw [NNReal.rpow_one] at h
theorem rpow_add_le_add_rpow {p : ℝ} (a b : ℝ≥0) (hp : 0 ≤ p) (hp1 : p ≤ 1) :
    (a + b) ^ p ≤ a ^ p + b ^ p := by
  rcases hp.eq_or_lt with (rfl | hp_pos)
  · simp
  have h := rpow_add_rpow_le a b hp_pos hp1
  rw [one_div_one] at h
  repeat' rw [NNReal.rpow_one] at h
  

Proof state:
case inr
ι : Type u
s : Finset ι
p : ℝ
a b : ℝ≥0
hp : 0 ≤ p
hp1 : p ≤ 1
hp_pos : 0 < p
h : a + b ≤ (a ^ p + b ^ p) ^ (1 / p)
⊢ (a + b) ^ p ≤ a ^ p + b ^ p

Next tactic:
exact (NNReal.le_rpow_one_div_iff hp_pos).mp h
/-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0∞`-valued
functions and real exponents. -/
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i in s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p := by
  

Proof state:
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i in s, w i = 1
p : ℝ
hp : 1 ≤ p
⊢ (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p

Next tactic:
have hp_pos : 0 < p
/-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0∞`-valued
functions and real exponents. -/
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i in s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p := by
  have hp_pos : 0 < p
  

Proof state:
case hp_pos
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i in s, w i = 1
p : ℝ
hp : 1 ≤ p
⊢ 0 < p
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i in s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
⊢ (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p

Next tactic:
positivity
/-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0∞`-valued
functions and real exponents. -/
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i in s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p := by
  have hp_pos : 0 < p
  positivity
  

Proof state:
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i in s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
⊢ (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p

Next tactic:
have hp_nonneg : 0 ≤ p
/-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0∞`-valued
functions and real exponents. -/
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i in s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p := by
  have hp_pos : 0 < p
  positivity
  have hp_nonneg : 0 ≤ p
  

Proof state:
case hp_nonneg
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i in s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
⊢ 0 ≤ p
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i in s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
hp_nonneg : 0 ≤ p
⊢ (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p

Next tactic:
positivity
/-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0∞`-valued
functions and real exponents. -/
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i in s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p := by
  have hp_pos : 0 < p
  positivity
  have hp_nonneg : 0 ≤ p
  positivity
  

Proof state:
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i in s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
hp_nonneg : 0 ≤ p
⊢ (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p

Next tactic:
have hp_not_neg : ¬p < 0 := by simp [hp_nonneg]
/-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0∞`-valued
functions and real exponents. -/
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i in s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p := by
  have hp_pos : 0 < p
  positivity
  have hp_nonneg : 0 ≤ p
  positivity
  have hp_not_neg : ¬p < 0 := by 

Proof state:
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i in s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
hp_nonneg : 0 ≤ p
⊢ ¬p < 0

Next tactic:
simp [hp_nonneg]
/-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0∞`-valued
functions and real exponents. -/
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i in s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p := by
  have hp_pos : 0 < p
  positivity
  have hp_nonneg : 0 ≤ p
  positivity
  have hp_not_neg : ¬p < 0 := by simp [hp_nonneg]
  

Proof state:
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i in s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
hp_nonneg : 0 ≤ p
hp_not_neg : ¬p < 0
⊢ (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p

Next tactic:
have h_top_iff_rpow_top : ∀ (i : ι), i ∈ s → (w i * z i = ⊤ ↔ w i * z i ^ p = ⊤) := by
    simp [ENNReal.mul_eq_top, hp_pos, hp_nonneg, hp_not_neg]
/-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0∞`-valued
functions and real exponents. -/
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i in s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p := by
  have hp_pos : 0 < p
  positivity
  have hp_nonneg : 0 ≤ p
  positivity
  have hp_not_neg : ¬p < 0 := by simp [hp_nonneg]
  have h_top_iff_rpow_top : ∀ (i : ι), i ∈ s → (w i * z i = ⊤ ↔ w i * z i ^ p = ⊤) := by
    

Proof state:
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i in s, w i = 1
p : ℝ
hp : 1 ≤ p
hp_pos : 0 < p
hp_nonneg : 0 ≤ p
hp_not_neg : ¬p < 0
⊢ ∀ i ∈ s, w i * z i = ⊤ ↔ w i * z i ^ p = ⊤

Next tactic:
simp [ENNReal.mul_eq_top, hp_pos, hp_nonneg, hp_not_neg]
/-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0∞`-valued
functions and real exponents. -/
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i in s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p := by
  have hp_pos : 0 < p
  positivity
  have hp_nonneg : 0 ≤ p
  positivity
  have hp_not_neg : ¬p < 0 := by simp [hp_nonneg]
  have h_top_iff_rpow_top : ∀ (i : ι), i ∈ s → (w i * z i = ⊤ ↔ w i * z i ^ p = ⊤) := by
    simp [ENNReal.mul_eq_top, hp_pos, hp_nonneg, hp_not_neg]
  

Proof state:
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i in 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 = ⊤
⊢ (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p

Next tactic:
refine' le_of_top_imp_top_of_toNNReal_le _ _
/-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0∞`-valued
functions and real exponents. -/
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i in s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p := by
  have hp_pos : 0 < p
  positivity
  have hp_nonneg : 0 ≤ p
  positivity
  have hp_not_neg : ¬p < 0 := by simp [hp_nonneg]
  have h_top_iff_rpow_top : ∀ (i : ι), i ∈ s → (w i * z i = ⊤ ↔ w i * z i ^ p = ⊤) := by
    simp [ENNReal.mul_eq_top, hp_pos, hp_nonneg, hp_not_neg]
  refine' le_of_top_imp_top_of_toNNReal_le _ _
  · -- first, prove `(∑ i in s, w i * z i) ^ p = ⊤ → ∑ i in s, (w i * z i ^ p) = ⊤`
    

Proof state:
case refine'_1
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i in 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 = ⊤
⊢ (∑ i in s, w i * z i) ^ p = ⊤ → ∑ i in s, w i * z i ^ p = ⊤

Next tactic:
rw [rpow_eq_top_iff, sum_eq_top_iff, sum_eq_top_iff]
/-- Weighted generalized mean inequality, version for sums over finite sets, with `ℝ≥0∞`-valued
functions and real exponents. -/
theorem rpow_arith_mean_le_arith_mean_rpow (w z : ι → ℝ≥0∞) (hw' : ∑ i in s, w i = 1) {p : ℝ}
    (hp : 1 ≤ p) : (∑ i in s, w i * z i) ^ p ≤ ∑ i in s, w i * z i ^ p := by
  have hp_pos : 0 < p
  positivity
  have hp_nonneg : 0 ≤ p
  positivity
  have hp_not_neg : ¬p < 0 := by simp [hp_nonneg]
  have h_top_iff_rpow_top : ∀ (i : ι), i ∈ s → (w i * z i = ⊤ ↔ w i * z i ^ p = ⊤) := by
    simp [ENNReal.mul_eq_top, hp_pos, hp_nonneg, hp_not_neg]
  refine' le_of_top_imp_top_of_toNNReal_le _ _
  · -- first, prove `(∑ i in s, w i * z i) ^ p = ⊤ → ∑ i in s, (w i * z i ^ p) = ⊤`
    rw [rpow_eq_top_iff, sum_eq_top_iff, sum_eq_top_iff]
    

Proof state:
case refine'_1
ι : Type u
s : Finset ι
w z : ι → ℝ≥0∞
hw' : ∑ i in 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 = ⊤
⊢ ∑ i in s, w i * z i = 0 ∧ p < 0 ∨ (∃ a ∈ s, w a * z a = ⊤) ∧ 0 < p → ∃ a ∈ s, w a * z a ^ p = ⊤

Next tactic:
intro h