# `ShotUn.Fragment`
[🔗](https://github.com/jcschuster/ShotUn/blob/v0.2.1/lib/shot_un/fragment.ex#L1)

Predicates that classify a unification problem into one of the decidable
fragments handled by `ShotUn.Pattern` (Miller patterns) and
`ShotUn.Matching` (Huet second-order matching), plus a helper for
computing the order of a simple type.

# `bounded_order?`

```elixir
@spec bounded_order?(ShotDs.Data.Term.term_id(), non_neg_integer()) :: boolean()
```

Returns `true` when every free variable, constant and bound variable
appearing in the term has a type of order at most `max_order`.

# `ground?`

```elixir
@spec ground?(ShotDs.Data.Term.term_id()) :: boolean()
```

Returns `true` when the term has no free variables.

# `matching_problem?`

```elixir
@spec matching_problem?([term_pair()]) :: boolean()
```

Returns `true` when every right-hand side is ground and every type in
the problem has order at most 2 (Huet's second-order fragment).

# `outer_bvar_index`

```elixir
@spec outer_bvar_index(ShotDs.Data.Term.term_id()) :: pos_integer() | :not_bvar
```

If the term is the eta-long primitive of a single bound variable whose
binder lies in the surrounding context (i.e. above the term's own
abstractions), returns the index of that binder relative to the
surrounding context. Otherwise returns `:not_bvar`.

# `pattern?`

```elixir
@spec pattern?(ShotDs.Data.Term.term_id()) :: boolean()
```

Returns `true` when the term is a higher-order pattern: every
occurrence of a free variable is applied to a list of pairwise distinct
bound-variable arguments.

# `pattern_problem?`

```elixir
@spec pattern_problem?([term_pair()]) :: boolean()
```

Returns `true` when every side of every pair is a higher-order pattern.

# `type_order`

```elixir
@spec type_order(ShotDs.Data.Type.t()) :: non_neg_integer()
```

Returns the order of a simple type.

$$\text{order}(\beta) = 0 \quad\text{for base } \beta$$
$$\text{order}(\tau_1 \to \dots \to \tau_n \to \beta) =
  \max_{i}(\text{order}(\tau_i) + 1)$$

---

*Consult [api-reference.md](api-reference.md) for complete listing*