%
\begin{isabellebody}%
\def\isabellecontext{Pairs}%
%
\isamarkupsection{Pairs%
}
%
\begin{isamarkuptext}%
\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 \isa{fst} and
\isa{snd}. In any nontrivial application of pairs you will find that this
quickly leads to unreadable formulae involvings nests of projections. This
section is concerned with introducing some syntactic sugar to overcome this
problem: pattern matching with tuples.%
\end{isamarkuptext}%
%
\isamarkupsubsection{Notation%
}
%
\begin{isamarkuptext}%
It is possible to use (nested) tuples as patterns in $\lambda$-abstractions,
for example \isa{{\isasymlambda}{\isacharparenleft}x{\isacharcomma}y{\isacharcomma}z{\isacharparenright}{\isachardot}x{\isacharplus}y{\isacharplus}z} and \isa{{\isasymlambda}{\isacharparenleft}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isacharcomma}z{\isacharparenright}{\isachardot}x{\isacharplus}y{\isacharplus}z}. In fact,
tuple patterns can be used in most variable binding constructs. Here are
some typical examples:
\begin{quote}
\isa{let\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isacharequal}\ f\ z\ in\ {\isacharparenleft}y{\isacharcomma}\ x{\isacharparenright}}\\
\isa{case\ xs\ of\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymRightarrow}\ {\isadigit{0}}\ {\isacharbar}\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isacharhash}\ zs\ {\isasymRightarrow}\ x\ {\isacharplus}\ y}\\
\isa{{\isasymforall}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isasymin}A{\isachardot}\ x{\isacharequal}y}\\
\isa{{\isacharbraceleft}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}\ x{\isacharequal}y{\isacharbraceright}}\\
\isa{{\isasymUnion}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isasymin}A{\isachardot}\ {\isacharbraceleft}x\ {\isacharplus}\ y{\isacharbraceright}}
\end{quote}%
\end{isamarkuptext}%
%
\begin{isamarkuptext}%
The intuitive meaning of this notations should be pretty obvious.
Unfortunately, we need to know in more detail what the notation really stands
for once we have to reason about it. The fact of the matter is that 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 \isa{fst} and \isa{snd} and write \isa{{\isasymlambda}p{\isachardot}\ fst\ p\ {\isacharplus}\ snd\ p}
instead of \isa{{\isasymlambda}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}\ x{\isacharplus}y} (which denote the same function but are quite
different terms).
Internally, \isa{{\isasymlambda}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardot}\ t} becomes \isa{split\ {\isacharparenleft}{\isasymlambda}x\ y{\isachardot}\ t{\isacharparenright}}, where
\isa{split} is the uncurrying function of type \isa{{\isacharparenleft}{\isacharprime}a\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}c{\isacharparenright}\ {\isasymRightarrow}\ {\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}c} defined as
\begin{center}
\isa{split\ {\isasymequiv}\ {\isasymlambda}c\ p{\isachardot}\ c\ {\isacharparenleft}fst\ p{\isacharparenright}\ {\isacharparenleft}snd\ p{\isacharparenright}}
\hfill(\isa{split{\isacharunderscore}def})
\end{center}
Pattern matching in
other variable binding constructs is translated similarly. Thus we need to
understand how to reason about such constructs.%
\end{isamarkuptext}%
%
\isamarkupsubsection{Theorem proving%
}
%
\begin{isamarkuptext}%
The most obvious approach is the brute force expansion of \isa{split}:%
\end{isamarkuptext}%
\isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}{\isasymlambda}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}x{\isacharparenright}\ p\ {\isacharequal}\ fst\ p{\isachardoublequote}\isanewline
\isacommand{by}{\isacharparenleft}simp\ add{\isacharcolon}split{\isacharunderscore}def{\isacharparenright}%
\begin{isamarkuptext}%
This works well if rewriting with \isa{split{\isacharunderscore}def} finishes the
proof, as in the above lemma. But if it doesn't, you end up with exactly what
we are trying to avoid: nests of \isa{fst} and \isa{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 \isa{p} with some concrete pair \isa{{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}}, in which case both sides of the
equation would simplify to \isa{a} because of the simplification rules
\isa{split\ c\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isacharequal}\ c\ a\ b} and \isa{fst\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isacharequal}\ a}. 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 \isa{split\ f\ p} this is easy: the split
rule \isa{split{\isacharunderscore}split} replaces \isa{p} by a pair:%
\end{isamarkuptext}%
\isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}{\isasymlambda}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}y{\isacharparenright}\ p\ {\isacharequal}\ snd\ p{\isachardoublequote}\isanewline
\isacommand{apply}{\isacharparenleft}simp\ only{\isacharcolon}\ split{\isacharcolon}split{\isacharunderscore}split{\isacharparenright}%
\begin{isamarkuptxt}%
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ {\isasymforall}x\ y{\isachardot}\ p\ {\isacharequal}\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymlongrightarrow}\ y\ {\isacharequal}\ snd\ p%
\end{isabelle}
This subgoal is easily proved by simplification. The \isa{only{\isacharcolon}} above
merely serves to show the effect of splitting and to avoid solving the goal
outright.
Let us look at a second example:%
\end{isamarkuptxt}%
\isacommand{lemma}\ {\isachardoublequote}let\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isacharequal}\ p\ in\ fst\ p\ {\isacharequal}\ x{\isachardoublequote}\isanewline
\isacommand{apply}{\isacharparenleft}simp\ only{\isacharcolon}Let{\isacharunderscore}def{\isacharparenright}%
\begin{isamarkuptxt}%
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardot}\ fst\ p\ {\isacharequal}\ x{\isacharparenright}\ p%
\end{isabelle}
A paired \isa{let} reduces to a paired $\lambda$-abstraction, which
can be split as above. The same is true for paired set comprehension:%
\end{isamarkuptxt}%
\isacommand{lemma}\ {\isachardoublequote}p\ {\isasymin}\ {\isacharbraceleft}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}\ x{\isacharequal}y{\isacharbraceright}\ {\isasymlongrightarrow}\ fst\ p\ {\isacharequal}\ snd\ p{\isachardoublequote}\isanewline
\isacommand{apply}\ simp%
\begin{isamarkuptxt}%
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ split\ op\ {\isacharequal}\ p\ {\isasymlongrightarrow}\ fst\ p\ {\isacharequal}\ snd\ p%
\end{isabelle}
Again, simplification produces a term suitable for \isa{split{\isacharunderscore}split}
as above. If you are worried about the funny form of the premise:
\isa{split\ op\ {\isacharequal}} is the same as \isa{{\isasymlambda}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}\ x{\isacharequal}y}.
The same procedure works for%
\end{isamarkuptxt}%
\isacommand{lemma}\ {\isachardoublequote}p\ {\isasymin}\ {\isacharbraceleft}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}\ x{\isacharequal}y{\isacharbraceright}\ {\isasymLongrightarrow}\ fst\ p\ {\isacharequal}\ snd\ p{\isachardoublequote}%
\begin{isamarkuptxt}%
\noindent
except that we now have to use \isa{split{\isacharunderscore}split{\isacharunderscore}asm}, because
\isa{split} occurs in the assumptions.
However, splitting \isa{split} is not always a solution, as no \isa{split}
may be present in the goal. Consider the following function:%
\end{isamarkuptxt}%
\isacommand{consts}\ swap\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymtimes}\ {\isacharprime}a{\isachardoublequote}\isanewline
\isacommand{primrec}\isanewline
\ \ {\isachardoublequote}swap\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}y{\isacharcomma}x{\isacharparenright}{\isachardoublequote}%
\begin{isamarkuptext}%
\noindent
Note that the above \isacommand{primrec} definition is admissible
because \isa{{\isasymtimes}} is a datatype. When we now try to prove%
\end{isamarkuptext}%
\isacommand{lemma}\ {\isachardoublequote}swap{\isacharparenleft}swap\ p{\isacharparenright}\ {\isacharequal}\ p{\isachardoublequote}%
\begin{isamarkuptxt}%
\noindent
simplification will do nothing, because the defining equation for \isa{swap}
expects a pair. Again, we need to turn \isa{p} into a pair first, but this
time there is no \isa{split} in sight. In this case the only thing we can do
is to split the term by hand:%
\end{isamarkuptxt}%
\isacommand{apply}{\isacharparenleft}case{\isacharunderscore}tac\ p{\isacharparenright}%
\begin{isamarkuptxt}%
\noindent
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}a\ b{\isachardot}\ p\ {\isacharequal}\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymLongrightarrow}\ swap\ {\isacharparenleft}swap\ p{\isacharparenright}\ {\isacharequal}\ p%
\end{isabelle}
Again, \isa{case{\isacharunderscore}tac} is applicable because \isa{{\isasymtimes}} is a datatype.
The subgoal is easily proved by \isa{simp}.
In case the term to be split is a quantified variable, there are more options.
You can split \emph{all} \isa{{\isasymAnd}}-quantified variables in a goal
with the rewrite rule \isa{split{\isacharunderscore}paired{\isacharunderscore}all}:%
\end{isamarkuptxt}%
\isacommand{lemma}\ {\isachardoublequote}{\isasymAnd}p\ q{\isachardot}\ swap{\isacharparenleft}swap\ p{\isacharparenright}\ {\isacharequal}\ q\ {\isasymlongrightarrow}\ p\ {\isacharequal}\ q{\isachardoublequote}\isanewline
\isacommand{apply}{\isacharparenleft}simp\ only{\isacharcolon}split{\isacharunderscore}paired{\isacharunderscore}all{\isacharparenright}%
\begin{isamarkuptxt}%
\noindent
\begin{isabelle}%
\ {\isadigit{1}}{\isachardot}\ {\isasymAnd}a\ b\ aa\ ba{\isachardot}\ swap\ {\isacharparenleft}swap\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}aa{\isacharcomma}\ ba{\isacharparenright}\ {\isasymlongrightarrow}\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}aa{\isacharcomma}\ ba{\isacharparenright}%
\end{isabelle}%
\end{isamarkuptxt}%
\isacommand{apply}\ simp\isanewline
\isacommand{done}%
\begin{isamarkuptext}%
\noindent
Note that we have intentionally included only \isa{split{\isacharunderscore}paired{\isacharunderscore}all}
in the first simplification step. This time the reason was not merely
pedagogical:
\isa{split{\isacharunderscore}paired{\isacharunderscore}all} may interfere with certain congruence
rules of the simplifier, i.e.%
\end{isamarkuptext}%
\isacommand{apply}{\isacharparenleft}simp\ add{\isacharcolon}split{\isacharunderscore}paired{\isacharunderscore}all{\isacharparenright}%
\begin{isamarkuptext}%
\noindent
may fail (here it does not) where the above two stages succeed.
Finally, all \isa{{\isasymforall}} and \isa{{\isasymexists}}-quantified variables are split
automatically by the simplifier:%
\end{isamarkuptext}%
\isacommand{lemma}\ {\isachardoublequote}{\isasymforall}p{\isachardot}\ {\isasymexists}q{\isachardot}\ swap\ p\ {\isacharequal}\ swap\ q{\isachardoublequote}\isanewline
\isacommand{apply}\ simp\isanewline
\isacommand{done}%
\begin{isamarkuptext}%
\noindent
In case you would like to turn off this automatic splitting, just disable the
responsible simplification rules:
\begin{center}
\isa{{\isacharparenleft}{\isasymforall}x{\isachardot}\ {\isacharquery}P\ x{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isasymforall}a\ b{\isachardot}\ {\isacharquery}P\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}{\isacharparenright}}
\hfill
(\isa{split{\isacharunderscore}paired{\isacharunderscore}All})\\
\isa{{\isacharparenleft}{\isasymexists}x{\isachardot}\ {\isacharquery}P\ x{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isasymexists}a\ b{\isachardot}\ {\isacharquery}P\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}{\isacharparenright}}
\hfill
(\isa{split{\isacharunderscore}paired{\isacharunderscore}Ex})
\end{center}%
\end{isamarkuptext}%
\end{isabellebody}%
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