Trait ParametricMonad 

pub trait ParametricMonad<M: HKT3Unbound> {
    // Required methods
    fn pure<S, A>(value: A) -> M::Type<S, S, A>;
    fn ibind<S1, S2, S3, A, B, F>(
        m: M::Type<S1, S2, A>,
        f: F,
    ) -> M::Type<S1, S3, B>
       where F: FnMut(A) -> M::Type<S2, S3, B>;
}

The ParametricMonad (or Indexed Monad) trait allows for monadic computations where the type of the underlying state can change at each step.

Category Theory

An Indexed Monad is a generalization of a Monad. Instead of a single endofunctor $M: \mathcal{C} \to \mathcal{C}$, we have a family of functors $M_{ij}: \mathcal{C} \to \mathcal{C}$ indexed by states $i, j$.

• Bind: $M_{ij}(A) \to (A \to M_{jk}(B)) \to M_{ik}(B)$

Mathematical Definition

It models a category where objects are states and morphisms are state transitions carrying a value. The ibind operation composes these transitions: $(i \to j) \circ (j \to k) = (i \to k)$.

Use Cases

• Phase Transitions: Simulating a system evolving from Fluid -> Gas -> Plasma.
• Protocol State Machines: Enforcing correct ordering of operations (e.g., Unauthenticated -> Authenticated).
• Topology Rewrites: Changing the mesh type from Triangular to Hexagonal during a simulation step.

Required Methods

fn pure<S, A>(value: A) -> M::Type<S, S, A>

Injects a value into a computation that doesn’t change the state type ($S \to S$).

fn ibind<S1, S2, S3, A, B, F>(m: M::Type<S1, S2, A>, f: F) -> M::Type<S1, S3, B>

where F: FnMut(A) -> M::Type<S2, S3, B>,

Indexed Bind: Chains computations where the state type evolves.

Arguments

• m: The initial computation transitioning $S1 \to S2$.
• f: A function taking the result $A$ and producing a new computation $S2 \to S3$.

Returns

A new computation transitioning $S1 \to S3$.

Dyn Compatibility

This trait is not dyn compatible.

In older versions of Rust, dyn compatibility was called "object safety", so this trait is not object safe.

Implementors