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added simpler example to Lambda and Properties

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wadler 2018-06-30 13:50:47 -03:00
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18
src/plta/Lambda.lagda
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@ -669,6 +669,24 @@ above are isomorphic.
## Examples
We start with a simple example. The Church numeral two applied to the
successor function and zero yields the natural number two.
\begin{code}
_ : twoᶜ · sucᶜ · `zero ↠ `suc `suc `zero
_ =
begin
twoᶜ · sucᶜ · `zero
↦⟨ ξ-·₁ (β-ƛ V-ƛ) ⟩
(ƛ "z" ⇒ sucᶜ · (sucᶜ · ` "z")) · `zero
↦⟨ β-ƛ V-zero ⟩
sucᶜ · (sucᶜ · `zero)
↦⟨ ξ-·₂ V-ƛ (β-ƛ V-zero) ⟩
sucᶜ · `suc `zero
↦⟨ β-ƛ (V-suc V-zero) ⟩
`suc (`suc `zero)
∎
\end{code}
Here is a sample reduction demonstrating that two plus two is four.
\begin{code}
_ : four ↠ `suc `suc `suc `suc `zero

37
src/plta/Properties.lagda
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@ -1058,8 +1058,32 @@ _ : eval (gas 3) ⊢sucμ ≡
_ = refl
\end{code}
Similarly, we can use Agda to compute the reduction sequence for two plus two
given in the preceding chapter. Supplying 100 steps of gas is more than enough.
Similarly, we can use Agda to compute the reductions sequences given
in the previous chapter. We start with the Church numeral two
applied to successor and zero. Supplying 100 steps of gas is more than enough.
\begin{code}
_ : eval (gas 100) (⊢twoᶜ · ⊢sucᶜ · ⊢zero) ≡
steps
((ƛ "s" ⇒ (ƛ "z" ⇒ ` "s" · (` "s" · ` "z"))) · (ƛ "n" ⇒ `suc ` "n")
· `zero
↦⟨ ξ-·₁ (β-ƛ V-ƛ) ⟩
(ƛ "z" ⇒ (ƛ "n" ⇒ `suc ` "n") · ((ƛ "n" ⇒ `suc ` "n") · ` "z")) ·
`zero
↦⟨ β-ƛ V-zero ⟩
(ƛ "n" ⇒ `suc ` "n") · ((ƛ "n" ⇒ `suc ` "n") · `zero) ↦⟨
ξ-·₂ V-ƛ (β-ƛ V-zero) ⟩
(ƛ "n" ⇒ `suc ` "n") · `suc `zero ↦⟨ β-ƛ (V-suc V-zero) ⟩
`suc (`suc `zero) ∎)
(done (V-suc (V-suc V-zero)))
_ = refl
\end{code}
The example above was generated by using `C `N to normalise the
left-hand side of the equation and pasting in the result as the
right-hand side of the equation. The example reduction of the
previous chapter was derived from this result, reformatting and
writing `twoᶜ` and `sucᶜ` in place of their expansions.
Next, we show two plus two is four.
\begin{code}
_ : eval (gas 100) ⊢four ≡
steps
@ -1219,11 +1243,8 @@ _ : eval (gas 100) ⊢four ≡
(done (V-suc (V-suc (V-suc (V-suc V-zero)))))
_ = refl
\end{code}
The example above was generated by using `C `N to normalise the
left-hand side of the equation and pasting in the result as the
right-hand side of the equation. The example reduction of the
previous chapter was derived from this result, by writing `plus` and
`two` in place of their corresponding expansions and reformatting.
Again, the derivation in the previous chapter was derived by
editing the above.
Similarly, can evaluate the corresponding term for Church numerals.
\begin{code}
@ -1290,7 +1311,7 @@ _ : eval (gas 100) ⊢fourᶜ ≡
(done (V-suc (V-suc (V-suc (V-suc V-zero)))))
_ = refl
\end{code}
Again, the example in the previous section was derived by editing the
And again, the example in the previous section was derived by editing the
above.
## Well-typed terms don't get stuck