case refine_2
ι : Type u
s : Finset ι
w₁ w₂ z₁ z₂ : ℝ≥0
hw' : w₁ + w₂ = 1
p : ℝ
hp : 1 ≤ p
h :
  (∑ i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons w₁ ![w₂] i * Matrix.vecCons z₁ ![z₂] i) ^ p ≤
    ∑ i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons w₁ ![w₂] i * Matrix.vecCons z₁ ![z₂] i ^ p
⊢ (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p

case refine_1
ι : Type u
s : Finset ι
w₁ w₂ z₁ z₂ : ℝ≥0
hw' : w₁ + w₂ = 1
p : ℝ
hp : 1 ≤ p
⊢ ∑ i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons w₁ ![w₂] i = 1
ι : Type u
s : Finset ι
w₁ w₂ z₁ z₂ : ℝ≥0
hw' : w₁ + w₂ = 1
p : ℝ
hp : 1 ≤ p
⊢ (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p
have h := rpow_arith_mean_le_arith_mean_rpow univ ![w₁, w₂] ![z₁, z₂] ?_ hp
no goals
case refine_2
ι : Type u
s : Finset ι
w₁ w₂ z₁ z₂ : ℝ≥0
hw' : w₁ + w₂ = 1
p : ℝ
hp : 1 ≤ p
h :
  (∑ i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons w₁ ![w₂] i * Matrix.vecCons z₁ ![z₂] i) ^ p ≤
    ∑ i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons w₁ ![w₂] i * Matrix.vecCons z₁ ![z₂] i ^ p
⊢ (w₁ * z₁ + w₂ * z₂) ^ p ≤ w₁ * z₁ ^ p + w₂ * z₂ ^ p
simpa [Fin.sum_univ_succ] using h
no goals
case refine_1
ι : Type u
s : Finset ι
w₁ w₂ z₁ z₂ : ℝ≥0
hw' : w₁ + w₂ = 1
p : ℝ
hp : 1 ≤ p
⊢ ∑ i : Fin (Nat.succ (Nat.succ 0)), Matrix.vecCons w₁ ![w₂] i = 1
simp [hw', Fin.sum_univ_succ]