theorem aokiRolewicz_pow_equiv_self.{u_1}aokiRolewicz_pow_equiv_self.{u_1} {α : Type u_1} [AddCommMonoid α] {A : EQuasinorm α} {p : ℝ} (hp : (2 * A.C) ^ p = 2) :
  (A.aokiRolewicz p hp).pow p ≈ A
  {αType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. } [AddCommMonoidAddCommMonoid.{u} (M : Type u) : Type uAn additive commutative monoid is an additive monoid with commutative `(+)`.  αType u_1]
  {AEQuasinorm α : EQuasinormEQuasinorm.{u_1} (α : Type u_1) [AddMonoid α] : Type u_1A quasinorm on a monoid $α$ is a function $α → [0,∞]$ and a finite constant
$C$ that sends $0 : α$ to zero and is $C$-subadditive.  αType u_1} {pℝ : ℝReal : TypeThe type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational
numbers. }
  (hp(2 * A.C) ^ p = 2 : (HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`.2 *HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`. AEQuasinorm α.CEQuasinorm.C.{u_1} {α : Type u_1} [AddMonoid α] (self : EQuasinorm α) : ℝ≥0∞The subadditivity constant. )HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`. ^HPow.hPow.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HPow α β γ] : α → β → γ`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`. pℝ =Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)

example : a = d :=
  Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
        (h1 : a = b) (h2 : p a) : p b :=
  Eq.subst h1 h2

example (α : Type) (a b : α) (p : α → Prop)
    (h1 : a = b) (h2 : p a) : p b :=
  h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)


Conventions for notations in identifiers:

 * The recommended spelling of `=` in identifiers is `eq`. 2) :
  (AEQuasinorm α.aokiRolewiczEQuasinorm.aokiRolewicz.{u_1} {α : Type u_1} [AddCommMonoid α] (A : EQuasinorm α) (p : ℝ) (hp : (2 * A.C) ^ p = 2) :
  ESeminorm α pℝ hp(2 * A.C) ^ p = 2).powEQuasinorm.pow.{u_1} {α : Type u_1} [AddMonoid α] (A : EQuasinorm α) (p : ℝ) : EQuasinorm αThe quasinorm raised to a power $p$, as a quasinorm.  pℝ ≈HasEquiv.Equiv.{u, v} {α : Sort u} [self : HasEquiv α] : α → α → Sort v`x ≈ y` says that `x` and `y` are equivalent. Because this is a typeclass,
the notion of equivalence is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `≈` in identifiers is `equiv`. AEQuasinorm α