(*<*)theory Pairs = Main:(*>*)

section{*Pairs*}

text{*\label{sec:products}

Pairs were already introduced in \S\ref{sec:pairs}, but only with a minimal

repertoire of operations: pairing and the two projections @{term fst} and

@{term snd}. In any non-trivial application of pairs you will find that this

quickly leads to unreadable formulae involving nests of projections. This

section is concerned with introducing some syntactic sugar to overcome this

problem: pattern matching with tuples.

*}

subsection{*Pattern Matching with Tuples*}

text{*

Tuples may be used as patterns in $\lambda$-abstractions,

for example @{text"\<lambda>(x,y,z).x+y+z"} and @{text"\<lambda>((x,y),z).x+y+z"}. In fact,

tuple patterns can be used in most variable binding constructs,

and they can be nested. Here are

some typical examples:

\begin{quote}

@{term"let (x,y) = f z in (y,x)"}\\

@{term"case xs of [] => 0 | (x,y)#zs => x+y"}\\

@{text"\<forall>(x,y)\<in>A. x=y"}\\

@{text"{(x,y,z). x=z}"}\\

@{term"\<Union>(x,y)\<in>A. {x+y}"}

\end{quote}

The intuitive meanings of these expressions should be obvious.

Unfortunately, we need to know in more detail what the notation really stands

for once we have to reason about it.  Abstraction

over pairs and tuples is merely a convenient shorthand for a more complex

internal representation.  Thus the internal and external form of a term may

differ, which can affect proofs. If you want to avoid this complication,

stick to @{term fst} and @{term snd} and write @{term"%p. fst p + snd p"}

instead of @{text"\<lambda>(x,y). x+y"} (which denote the same function but are quite

different terms).

Internally, @{term"%(x,y). t"} becomes @{text"split (\<lambda>x y. t)"}, where

@{term split}\indexbold{*split (constant)}

is the uncurrying function of type @{text"('a \<Rightarrow> 'b

\<Rightarrow> 'c) \<Rightarrow> 'a \<times> 'b \<Rightarrow> 'c"} defined as

\begin{center}

@{thm split_def}

\hfill(@{thm[source]split_def})

\end{center}

Pattern matching in

other variable binding constructs is translated similarly. Thus we need to

understand how to reason about such constructs.

*}

subsection{*Theorem Proving*}

text{*

The most obvious approach is the brute force expansion of @{term split}:

*}

lemma "(\<lambda>(x,y).x) p = fst p"

by(simp add:split_def)

text{* This works well if rewriting with @{thm[source]split_def} finishes the

proof, as it does above.  But if it does not, you end up with exactly what

we are trying to avoid: nests of @{term fst} and @{term snd}. Thus this

approach is neither elegant nor very practical in large examples, although it

can be effective in small ones.

If we step back and ponder why the above lemma presented a problem in the

first place, we quickly realize that what we would like is to replace @{term

p} with some concrete pair @{term"(a,b)"}, in which case both sides of the

equation would simplify to @{term a} because of the simplification rules

@{thm split_conv[no_vars]} and @{thm fst_conv[no_vars]}.  This is the

key problem one faces when reasoning about pattern matching with pairs: how to

convert some atomic term into a pair.

In case of a subterm of the form @{term"split f p"} this is easy: the split

rule @{thm[source]split_split} replaces @{term p} by a pair:

*}

lemma "(\<lambda>(x,y).y) p = snd p"

apply(split split_split);

txt{*

@{subgoals[display,indent=0]}

This subgoal is easily proved by simplification. Thus we could have combined

simplification and splitting in one command that proves the goal outright:

*}

(*<*)

by simp

lemma "(\<lambda>(x,y).y) p = snd p"(*>*)

by(simp split: split_split)

text{*

Let us look at a second example:

*}

lemma "let (x,y) = p in fst p = x";

apply(simp only: Let_def)

txt{*

@{subgoals[display,indent=0]}

A paired @{text let} reduces to a paired $\lambda$-abstraction, which

can be split as above. The same is true for paired set comprehension:

*}

(*<*)by(simp split:split_split)(*>*)

lemma "p \<in> {(x,y). x=y} \<longrightarrow> fst p = snd p"

apply simp

txt{*

@{subgoals[display,indent=0]}

Again, simplification produces a term suitable for @{thm[source]split_split}

as above. If you are worried about the strange form of the premise:

@{term"split (op =)"} is short for @{text"\<lambda>(x,y). x=y"}.

The same proof procedure works for

*}

(*<*)by(simp split:split_split)(*>*)

lemma "p \<in> {(x,y). x=y} \<Longrightarrow> fst p = snd p"

txt{*\noindent

except that we now have to use @{thm[source]split_split_asm}, because

@{term split} occurs in the assumptions.

However, splitting @{term split} is not always a solution, as no @{term split}

may be present in the goal. Consider the following function:

*}

(*<*)by(simp split:split_split_asm)(*>*)

consts swap :: "'a \<times> 'b \<Rightarrow> 'b \<times> 'a"

primrec

  "swap (x,y) = (y,x)"

text{*\noindent

Note that the above \isacommand{primrec} definition is admissible

because @{text"\<times>"} is a datatype. When we now try to prove

*}

lemma "swap(swap p) = p"

txt{*\noindent

simplification will do nothing, because the defining equation for @{term swap}

expects a pair. Again, we need to turn @{term p} into a pair first, but this

time there is no @{term split} in sight. In this case the only thing we can do

is to split the term by hand:

*}

apply(case_tac p)

txt{*\noindent

@{subgoals[display,indent=0]}

Again, @{text case_tac} is applicable because @{text"\<times>"} is a datatype.

The subgoal is easily proved by @{text simp}.

Splitting by @{text case_tac} also solves the previous examples and may thus

appear preferable to the more arcane methods introduced first. However, see

the warning about @{text case_tac} in \S\ref{sec:struct-ind-case}.

In case the term to be split is a quantified variable, there are more options.

You can split \emph{all} @{text"\<And>"}-quantified variables in a goal

with the rewrite rule @{thm[source]split_paired_all}:

*}

(*<*)by simp(*>*)

lemma "\<And>p q. swap(swap p) = q \<longrightarrow> p = q"

apply(simp only:split_paired_all)

txt{*\noindent

@{subgoals[display,indent=0]}

*}

apply simp

done

text{*\noindent

Note that we have intentionally included only @{thm[source]split_paired_all}

in the first simplification step. This time the reason was not merely

pedagogical:

@{thm[source]split_paired_all} may interfere with certain congruence

rules of the simplifier, i.e.

*}

(*<*)

lemma "\<And>p q. swap(swap p) = q \<longrightarrow> p = q"

(*>*)

apply(simp add:split_paired_all)

(*<*)done(*>*)

text{*\noindent

may fail (here it does not) where the above two stages succeed.

Finally, all @{text"\<forall>"} and @{text"\<exists>"}-quantified variables are split

automatically by the simplifier:

*}

lemma "\<forall>p. \<exists>q. swap p = swap q"

apply simp;

done

text{*\noindent

In case you would like to turn off this automatic splitting, just disable the

responsible simplification rules:

\begin{center}

@{thm split_paired_All[no_vars]}

\hfill

(@{thm[source]split_paired_All})\\

@{thm split_paired_Ex[no_vars]}

\hfill

(@{thm[source]split_paired_Ex})

\end{center}

*}

(*<*)

end

(*>*)