def X3DH_Alice.{u_1, u_2} {FType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. }
  [FieldField.{u} (K : Type u) : Type uA `Field` is a `CommRing` with multiplicative inverses for nonzero elements.

An instance of `Field K` includes maps `ratCast : ℚ → K` and `qsmul : ℚ → K → K`.
Those two fields are needed to implement the `DivisionRing K → Algebra ℚ K` instance since we need
to control the specific definitions for some special cases of `K` (in particular `K = ℚ` itself).
See also note [forgetful inheritance].

If the field has positive characteristic `p`, our division by zero convention forces
`ratCast (1 / p) = 1 / 0 = 0`.  FType u_1] {GType u_2 : Type u_2A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. }
  [AddCommGroupAddCommGroup.{u} (G : Type u) : Type uAn additive commutative group is an additive group with commutative `(+)`.  GType u_2] [ModuleModule.{u, v} (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] : Type (max u v)A module is a generalization of vector spaces to a scalar semiring.
It consists of a scalar semiring `R` and an additive monoid of "vectors" `M`,
connected by a "scalar multiplication" operation `r • x : M`
(where `r : R` and `x : M`) with some natural associativity and
distributivity axioms similar to those on a ring.  FType u_1 GType u_2]
  (ikₐF ekₐF : FType u_1) (IKᵦG SPKᵦG : GType u_2)
  (OPKᵦOption G : OptionOption.{u} (α : Type u) : Type uOptional values, which are either `some` around a value from the underlying type or `none`.

`Option` can represent nullable types or computations that might fail. In the codomain of a function
type, it can also represent partiality.
 GType u_2) : GType u_2 ×Prod.{u, v} (α : Type u) (β : Type v) : Type (max u v)The product type, usually written `α × β`. Product types are also called pair or tuple types.
Elements of this type are pairs in which the first element is an `α` and the second element is a
`β`.

Products nest to the right, so `(x, y, z) : α × β × γ` is equivalent to `(x, (y, z)) : α × (β × γ)`.


Conventions for notations in identifiers:

 * The recommended spelling of `×` in identifiers is `Prod`. GType u_2 ×Prod.{u, v} (α : Type u) (β : Type v) : Type (max u v)The product type, usually written `α × β`. Product types are also called pair or tuple types.
Elements of this type are pairs in which the first element is an `α` and the second element is a
`β`.

Products nest to the right, so `(x, y, z) : α × β × γ` is equivalent to `(x, (y, z)) : α × (β × γ)`.


Conventions for notations in identifiers:

 * The recommended spelling of `×` in identifiers is `Prod`. GType u_2 ×Prod.{u, v} (α : Type u) (β : Type v) : Type (max u v)The product type, usually written `α × β`. Product types are also called pair or tuple types.
Elements of this type are pairs in which the first element is an `α` and the second element is a
`β`.

Products nest to the right, so `(x, y, z) : α × β × γ` is equivalent to `(x, (y, z)) : α × (β × γ)`.


Conventions for notations in identifiers:

 * The recommended spelling of `×` in identifiers is `Prod`. GType u_2

def X3DH_Bob.{u_1, u_2} {FType u_1 : Type u_1A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. }
  [FieldField.{u} (K : Type u) : Type uA `Field` is a `CommRing` with multiplicative inverses for nonzero elements.

An instance of `Field K` includes maps `ratCast : ℚ → K` and `qsmul : ℚ → K → K`.
Those two fields are needed to implement the `DivisionRing K → Algebra ℚ K` instance since we need
to control the specific definitions for some special cases of `K` (in particular `K = ℚ` itself).
See also note [forgetful inheritance].

If the field has positive characteristic `p`, our division by zero convention forces
`ratCast (1 / p) = 1 / 0 = 0`.  FType u_1] {GType u_2 : Type u_2A type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. }
  [AddCommGroupAddCommGroup.{u} (G : Type u) : Type uAn additive commutative group is an additive group with commutative `(+)`.  GType u_2] [ModuleModule.{u, v} (R : Type u) (M : Type v) [Semiring R] [AddCommMonoid M] : Type (max u v)A module is a generalization of vector spaces to a scalar semiring.
It consists of a scalar semiring `R` and an additive monoid of "vectors" `M`,
connected by a "scalar multiplication" operation `r • x : M`
(where `r : R` and `x : M`) with some natural associativity and
distributivity axioms similar to those on a ring.  FType u_1 GType u_2]
  (ikᵦF spkᵦF : FType u_1) (opkᵦOption F : OptionOption.{u} (α : Type u) : Type uOptional values, which are either `some` around a value from the underlying type or `none`.

`Option` can represent nullable types or computations that might fail. In the codomain of a function
type, it can also represent partiality.
 FType u_1)
  (IKₐG EKₐG : GType u_2) : GType u_2 ×Prod.{u, v} (α : Type u) (β : Type v) : Type (max u v)The product type, usually written `α × β`. Product types are also called pair or tuple types.
Elements of this type are pairs in which the first element is an `α` and the second element is a
`β`.

Products nest to the right, so `(x, y, z) : α × β × γ` is equivalent to `(x, (y, z)) : α × (β × γ)`.


Conventions for notations in identifiers:

 * The recommended spelling of `×` in identifiers is `Prod`. GType u_2 ×Prod.{u, v} (α : Type u) (β : Type v) : Type (max u v)The product type, usually written `α × β`. Product types are also called pair or tuple types.
Elements of this type are pairs in which the first element is an `α` and the second element is a
`β`.

Products nest to the right, so `(x, y, z) : α × β × γ` is equivalent to `(x, (y, z)) : α × (β × γ)`.


Conventions for notations in identifiers:

 * The recommended spelling of `×` in identifiers is `Prod`. GType u_2 ×Prod.{u, v} (α : Type u) (β : Type v) : Type (max u v)The product type, usually written `α × β`. Product types are also called pair or tuple types.
Elements of this type are pairs in which the first element is an `α` and the second element is a
`β`.

Products nest to the right, so `(x, y, z) : α × β × γ` is equivalent to `(x, (y, z)) : α × (β × γ)`.


Conventions for notations in identifiers:

 * The recommended spelling of `×` in identifiers is `Prod`. GType u_2