# `ShotDs.Stt.Booleans`
[🔗](https://github.com/jcschuster/ShotDs/blob/v1.0.0/lib/shot_ds/stt/booleans.ex#L1)

Church's encoding of Booleans in simple type theory (STT).

Church defines Booleans as lambda-abstractions which take a value for truth
and a value for falsity. Conversely, the term denoting truth (`tt`) projects
the first value, the term for falsity (`ff`) projects the second.

> #### Note {: .info}
>
> Some standard encodings like _if-then-else_ are only definable as
> parameterized family of terms in STT, not as polymorphic statements.

# `b_type`

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

Returns the simple type of an encoded Boolean, i.e. $\iota\to\iota\to\iota$.

# `b_var`

```elixir
@spec b_var(ShotDs.Data.Declaration.var_name_t()) :: ShotDs.Data.Term.term_id()
```

Generates a variable term with the given name corresponding to a Church
Boolean.

# `b_and`

```elixir
@spec b_and(ShotDs.Data.Term.term_id(), ShotDs.Data.Term.term_id()) ::
  ShotDs.Data.Term.term_id()
```

Generates the Church Boolean corresponding to the logical conjunction of the
Boolean terms associated with the given IDs.

$$\mathtt{b\_and} := \lambda PQAB.\text{ }P\text{ }(Q\text{ }A\text{ }B)\text{ }B$$

# `b_not`

```elixir
@spec b_not(ShotDs.Data.Term.term_id()) :: ShotDs.Data.Term.term_id()
```

Generates the Church Boolean corresponding to the negation of the Boolean term
associated with the given ID.

$$\mathtt{b\_not} := \lambda PAB.\text{ }P\text{ }B\text{ }A$$

# `b_or`

```elixir
@spec b_or(ShotDs.Data.Term.term_id(), ShotDs.Data.Term.term_id()) ::
  ShotDs.Data.Term.term_id()
```

Generates the Church boolean corresponding to the logical disjunction of the
Boolean terms associated with the given IDs.

$$\mathtt{b\_or} := \lambda PQAB.\text{ }P\text{ }A\text{ }(Q\text{ }A\text{ }B)$$

# `ff`

```elixir
@spec ff() :: ShotDs.Data.Term.term_id()
```

Returns the ID of the term corresponding to the Boolean value _false_.

$$\mathtt{ff} := \lambda AB.\text{ }B$$

# `tt`

```elixir
@spec tt() :: ShotDs.Data.Term.term_id()
```

Returns the ID of the term corresponding to the Boolean value _true_.

$$\mathtt{tt} := \lambda AB.\text{ }A$$

---

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