added explanation of tabular reasoning

This commit is contained in:
wadler 2018-01-30 17:58:57 -02:00
parent b0d7c5f49c
commit 185d8981f0
2 changed files with 134 additions and 11 deletions

View file

@ -109,9 +109,10 @@ Once more, a useful exercise is to carry out an interactive development.
## Tabular reasoning
A few declarations allow us to support the form of tabular reasoning that
we have used throughout the book. We package the declarations into a module,
to match the format used in Agda's standard library.
A few declarations allow us to support the form of tabular reasoning
that we have used throughout the book. We package the declarations
into a module, named `≡-Reasoning`, to match the format used in Agda's
standard library.
\begin{code}
module ≡-Reasoning {} {A : Set } where
@ -133,15 +134,59 @@ module ≡-Reasoning {} {A : Set } where
open ≡-Reasoning
\end{code}
Opening the module makes all of the definitions
available in the current environment.
For example, here is a repeat of the definitions of naturals and
of addition, and of the proof that `zero` is a right identity
of addition. (We need to repeat all the definitions, rather than
importing some of them, because all of the relevant modules import
the definition of equivalence from the Agda standard library, and
hence importing any of them would report an attempt to redefine
equivalence.)
As a simple example, let's look at the proof of transitivity
rewritten in tabular form.
\begin{code}
trans : ∀ {} {A : Set } {x y z : A} → x ≡ y → y ≡ z → x ≡ z
trans {_} {_} {x} {y} {z} x≡y y≡z =
begin
x
≡⟨ x≡y ⟩
y
≡⟨ y≡z ⟩
z
\end{code}
Tabular proofs begin with `begin`, end with `∎`
(which is sometimes pronounced "qed" or "tombstone")
and consist of a string of equations. Due to the
fixity declarations, the body parses as follows.
begin (x ≡⟨ x≡y ⟩ (y ≡⟨ y≡z ⟩ (z ∎)))
The application of `begin` is purely cosmetic, as it simply returns
its argument. That argument consists of `_≡⟨_⟩_` applied to `x`,
`x≡y`, and `y ≡⟨ y≡z ⟩ (z ∎)`. The first argument is a term, `x`,
while the second and third arguments are both proofs of equations, in
particular proofs of `x ≡ y` and `y ≡ z` respectively, which are
combined by `trans` in the body of `_≡⟨_⟩_` to yield a proof of `x ≡
z`. The proof of `y ≡ z` consists of `_≡⟨_⟩_` applied to `y`, `y≡z`,
and `z ∎`. The first argument is a term, `y`, while the second and
third arguments are both proofs of equations, in particular proofs of
`y ≡ z` and `z ≡ z` respectively, which are combined by `trans` in the
body of `_≡⟨_⟩_` to yield a proof of `y ≡ z`. Finally, the proof of
`z ≡ z` consists of `_∎` applied to the term `z`, which yields `refl`.
After simplification, the body is equivalent to the following term.
trans x≡y (trans y≡z refl)
We could replace all uses of tabular reasoning by a chain of
applications of `trans`, but the result would be far less perspicuous.
Also note that the syntactic trick behind `∎` means that the chain
always ends in the form `trans e refl` even though `e` alone would do,
where `e` is a proof of an equivalence.
As an example, we consider the proof that zero is a right identity for
addition. We first repeat the definitions of naturals and addition.
We cannot import them, because the relevant modules import equivalence
from the Agda standard library, and hence importing would report an
attempt to redefine equivalence.
\begin{code}
data : Set where
zero :
@ -150,7 +195,10 @@ data : Set where
_+_ :
zero + n = n
(suc m) + n = suc (m + n)
\end{code}
We now repeat the proof that zero is a right identity.
\begin{code}
+-identity : ∀ (m : ) → m + zero ≡ m
+-identity zero =
begin
@ -167,6 +215,53 @@ zero + n = n
suc m
\end{code}
Tabular proofs begin with `begin`, end with `∎`
(which is sometimes pronounced "qed" or "tombstone")
and consist of a string of equations. Let's begin by
considering the second proof. Due to the relevant
infix declarations, it parses as follows.
begin (... ≡⟨⟩ (... ≡⟨ ... ⟩ (... ∎)))
The first argument to `_≡⟨⟩_` is a term (in this
case, `suc m + zero`, and the second is the proof of
an equation (in this case, of `suc (m + zero) ≡ suc m`).
As far as Agda is concerned, `suc m + zero` and
`suc (m + zero)` are identical, because the first
simplifies to the second, so no justification is required.
Similarly, the first argument to `_≡⟨_⟩_` is a term
(in this case, `suc (m + zero)`) and the third argument
is the proof of an equation (in this case, of `suc m ≡ suc m`),
and the justification is also a proof of an equation
(in this case,
The form
`_≡⟨⟩_` is used for an equation that requires no justification,
because both sides simplify to the same term.
The form `_≡⟨_⟩_` gives
Let's
Each case involves equational reasoning set out in
tabular form, beginning with `begin`, ending with `∎`
(sometimes pronounced "qed" or "tombstone"), and with
a sequence of equations and justifications. The form
`≡⟨⟩` is used for an equation that needs no justification,
and the
pWhen the argument is zero,
we must
show `zero + zero ≡ zero`, which follows by simple
computation
## Rewriting by an equation
@ -182,7 +277,7 @@ data even : → Set where
even-id : ∀ (m : ) → even (m + zero) → even m
even-id m ev with m + zero | +-identity m
... | .m | refl = ev
... | .m | refl = ev
\end{code}
Including the lines

View file

@ -4,6 +4,9 @@ open Eq.≡-Reasoning
open import Data.Nat using (; zero; suc; _+_; _*_)
open import Data.Product using (∃; _,_)
+-assoc : (m n p : ) m + (n + p) (m + n) + p
+-assoc m n p = {!!}
+-identity : (m : ) m + zero m
+-identity zero =
begin
@ -60,6 +63,31 @@ open import Data.Product using (∃; _,_)
suc n + m
*-distrib-+ : (m n p : ) (m + n) * p m * p + n * p
*-distrib-+ zero n p =
begin
(zero + n) * p
≡⟨⟩
n * p
≡⟨⟩
zero * p + n * p
*-distrib-+ (suc m) n p =
begin
(suc m + n) * p
≡⟨⟩
p + (m + n) * p
≡⟨ cong (_+_ p) (*-distrib-+ m n p)
p + (m * p + n * p)
≡⟨ +-assoc p (m * p) (n * p)
(p + m * p) + n * p
≡⟨⟩
suc m * p + n * p
data even : Set where
ev0 : even zero
ev+2 : {n : } even n even (suc (suc n))