3 \def\isabellecontext{Pairs}%
6 \isamarkupsection{Pairs and Tuples%
10 \begin{isamarkuptext}%
12 Ordered pairs were already introduced in \S\ref{sec:pairs}, but only with a minimal
13 repertoire of operations: pairing and the two projections \isa{fst} and
14 \isa{snd}. In any non-trivial application of pairs you will find that this
15 quickly leads to unreadable nests of projections. This
16 section introduces syntactic sugar to overcome this
17 problem: pattern matching with tuples.%
21 \isamarkupsubsection{Pattern Matching with Tuples%
25 \begin{isamarkuptext}%
26 Tuples may be used as patterns in $\lambda$-abstractions,
27 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,
28 tuple patterns can be used in most variable binding constructs,
29 and they can be nested. Here are
30 some typical examples:
32 \isa{let\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isacharequal}\ f\ z\ in\ {\isacharparenleft}y{\isacharcomma}\ x{\isacharparenright}}\\
33 \isa{case\ xs\ of\ {\isacharbrackleft}{\isacharbrackright}\ {\isasymRightarrow}\ {\isadigit{0}}\ {\isacharbar}\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isacharhash}\ zs\ {\isasymRightarrow}\ x\ {\isacharplus}\ y}\\
34 \isa{{\isasymforall}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isasymin}A{\isachardot}\ x{\isacharequal}y}\\
35 \isa{{\isacharbraceleft}{\isacharparenleft}x{\isacharcomma}y{\isacharcomma}z{\isacharparenright}{\isachardot}\ x{\isacharequal}z{\isacharbraceright}}\\
36 \isa{{\isasymUnion}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isasymin}A{\isachardot}\ {\isacharbraceleft}x\ {\isacharplus}\ y{\isacharbraceright}}
38 The intuitive meanings of these expressions should be obvious.
39 Unfortunately, we need to know in more detail what the notation really stands
40 for once we have to reason about it. Abstraction
41 over pairs and tuples is merely a convenient shorthand for a more complex
42 internal representation. Thus the internal and external form of a term may
43 differ, which can affect proofs. If you want to avoid this complication,
44 stick to \isa{fst} and \isa{snd} and write \isa{{\isasymlambda}p{\isachardot}\ fst\ p\ {\isacharplus}\ snd\ p}
45 instead of \isa{{\isasymlambda}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}\ x{\isacharplus}y}. These terms are distinct even though they
46 denote the same function.
48 Internally, \isa{{\isasymlambda}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardot}\ t} becomes \isa{split\ {\isacharparenleft}{\isasymlambda}x\ y{\isachardot}\ t{\isacharparenright}}, where
49 \cdx{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
51 \isa{split\ {\isasymequiv}\ {\isasymlambda}c\ p{\isachardot}\ c\ {\isacharparenleft}fst\ p{\isacharparenright}\ {\isacharparenleft}snd\ p{\isacharparenright}}
52 \hfill(\isa{split{\isacharunderscore}def})
55 other variable binding constructs is translated similarly. Thus we need to
56 understand how to reason about such constructs.%
60 \isamarkupsubsection{Theorem Proving%
64 \begin{isamarkuptext}%
65 The most obvious approach is the brute force expansion of \isa{split}:%
68 \isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}{\isasymlambda}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}x{\isacharparenright}\ p\ {\isacharequal}\ fst\ p{\isachardoublequote}\isanewline
70 \isacommand{by}{\isacharparenleft}simp\ add{\isacharcolon}\ split{\isacharunderscore}def{\isacharparenright}\isamarkupfalse%
72 \begin{isamarkuptext}%
73 This works well if rewriting with \isa{split{\isacharunderscore}def} finishes the
74 proof, as it does above. But if it does not, you end up with exactly what
75 we are trying to avoid: nests of \isa{fst} and \isa{snd}. Thus this
76 approach is neither elegant nor very practical in large examples, although it
77 can be effective in small ones.
79 If we consider why this lemma presents a problem,
80 we quickly realize that we need to replace the variable~\isa{p} by some pair \isa{{\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}}. Then both sides of the
81 equation would simplify to \isa{a} by the simplification rules
82 \isa{split\ c\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isacharequal}\ c\ a\ b} and \isa{fst\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isacharequal}\ a}.
83 To reason about tuple patterns requires some way of
84 converting a variable of product type into a pair.
86 In case of a subterm of the form \isa{split\ f\ p} this is easy: the split
87 rule \isa{split{\isacharunderscore}split} replaces \isa{p} by a pair:%
88 \index{*split (method)}%
91 \isacommand{lemma}\ {\isachardoublequote}{\isacharparenleft}{\isasymlambda}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}y{\isacharparenright}\ p\ {\isacharequal}\ snd\ p{\isachardoublequote}\isanewline
93 \isacommand{apply}{\isacharparenleft}split\ split{\isacharunderscore}split{\isacharparenright}\isamarkupfalse%
97 \ {\isadigit{1}}{\isachardot}\ {\isasymforall}x\ y{\isachardot}\ p\ {\isacharequal}\ {\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}\ {\isasymlongrightarrow}\ y\ {\isacharequal}\ snd\ p%
99 This subgoal is easily proved by simplification. Thus we could have combined
100 simplification and splitting in one command that proves the goal outright:%
105 \isacommand{by}{\isacharparenleft}simp\ split{\isacharcolon}\ split{\isacharunderscore}split{\isacharparenright}\isamarkupfalse%
107 \begin{isamarkuptext}%
108 Let us look at a second example:%
111 \isacommand{lemma}\ {\isachardoublequote}let\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isacharequal}\ p\ in\ fst\ p\ {\isacharequal}\ x{\isachardoublequote}\isanewline
113 \isacommand{apply}{\isacharparenleft}simp\ only{\isacharcolon}\ Let{\isacharunderscore}def{\isacharparenright}\isamarkupfalse%
115 \begin{isamarkuptxt}%
117 \ {\isadigit{1}}{\isachardot}\ {\isacharparenleft}{\isasymlambda}{\isacharparenleft}x{\isacharcomma}\ y{\isacharparenright}{\isachardot}\ fst\ p\ {\isacharequal}\ x{\isacharparenright}\ p%
119 A paired \isa{let} reduces to a paired $\lambda$-abstraction, which
120 can be split as above. The same is true for paired set comprehension:%
124 \isacommand{lemma}\ {\isachardoublequote}p\ {\isasymin}\ {\isacharbraceleft}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}\ x{\isacharequal}y{\isacharbraceright}\ {\isasymlongrightarrow}\ fst\ p\ {\isacharequal}\ snd\ p{\isachardoublequote}\isanewline
126 \isacommand{apply}\ simp\isamarkupfalse%
128 \begin{isamarkuptxt}%
130 \ {\isadigit{1}}{\isachardot}\ split\ op\ {\isacharequal}\ p\ {\isasymlongrightarrow}\ fst\ p\ {\isacharequal}\ snd\ p%
132 Again, simplification produces a term suitable for \isa{split{\isacharunderscore}split}
133 as above. If you are worried about the strange form of the premise:
134 \isa{split\ op\ {\isacharequal}} is short for \isa{{\isasymlambda}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}\ x{\isacharequal}y}.
135 The same proof procedure works for%
139 \isacommand{lemma}\ {\isachardoublequote}p\ {\isasymin}\ {\isacharbraceleft}{\isacharparenleft}x{\isacharcomma}y{\isacharparenright}{\isachardot}\ x{\isacharequal}y{\isacharbraceright}\ {\isasymLongrightarrow}\ fst\ p\ {\isacharequal}\ snd\ p{\isachardoublequote}\isamarkupfalse%
141 \begin{isamarkuptxt}%
143 except that we now have to use \isa{split{\isacharunderscore}split{\isacharunderscore}asm}, because
144 \isa{split} occurs in the assumptions.
146 However, splitting \isa{split} is not always a solution, as no \isa{split}
147 may be present in the goal. Consider the following function:%
151 \isacommand{consts}\ swap\ {\isacharcolon}{\isacharcolon}\ {\isachardoublequote}{\isacharprime}a\ {\isasymtimes}\ {\isacharprime}b\ {\isasymRightarrow}\ {\isacharprime}b\ {\isasymtimes}\ {\isacharprime}a{\isachardoublequote}\isanewline
153 \isacommand{primrec}\isanewline
154 \ \ {\isachardoublequote}swap\ {\isacharparenleft}x{\isacharcomma}y{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}y{\isacharcomma}x{\isacharparenright}{\isachardoublequote}\isamarkupfalse%
156 \begin{isamarkuptext}%
158 Note that the above \isacommand{primrec} definition is admissible
159 because \isa{{\isasymtimes}} is a datatype. When we now try to prove%
162 \isacommand{lemma}\ {\isachardoublequote}swap{\isacharparenleft}swap\ p{\isacharparenright}\ {\isacharequal}\ p{\isachardoublequote}\isamarkupfalse%
164 \begin{isamarkuptxt}%
166 simplification will do nothing, because the defining equation for \isa{swap}
167 expects a pair. Again, we need to turn \isa{p} into a pair first, but this
168 time there is no \isa{split} in sight. In this case the only thing we can do
169 is to split the term by hand:%
172 \isacommand{apply}{\isacharparenleft}case{\isacharunderscore}tac\ p{\isacharparenright}\isamarkupfalse%
174 \begin{isamarkuptxt}%
177 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}a\ b{\isachardot}\ p\ {\isacharequal}\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}\ {\isasymLongrightarrow}\ swap\ {\isacharparenleft}swap\ p{\isacharparenright}\ {\isacharequal}\ p%
179 Again, \methdx{case_tac} is applicable because \isa{{\isasymtimes}} is a datatype.
180 The subgoal is easily proved by \isa{simp}.
182 Splitting by \isa{case{\isacharunderscore}tac} also solves the previous examples and may thus
183 appear preferable to the more arcane methods introduced first. However, see
184 the warning about \isa{case{\isacharunderscore}tac} in \S\ref{sec:struct-ind-case}.
186 In case the term to be split is a quantified variable, there are more options.
187 You can split \emph{all} \isa{{\isasymAnd}}-quantified variables in a goal
188 with the rewrite rule \isa{split{\isacharunderscore}paired{\isacharunderscore}all}:%
192 \isacommand{lemma}\ {\isachardoublequote}{\isasymAnd}p\ q{\isachardot}\ swap{\isacharparenleft}swap\ p{\isacharparenright}\ {\isacharequal}\ q\ {\isasymlongrightarrow}\ p\ {\isacharequal}\ q{\isachardoublequote}\isanewline
194 \isacommand{apply}{\isacharparenleft}simp\ only{\isacharcolon}\ split{\isacharunderscore}paired{\isacharunderscore}all{\isacharparenright}\isamarkupfalse%
196 \begin{isamarkuptxt}%
199 \ {\isadigit{1}}{\isachardot}\ {\isasymAnd}a\ b\ aa\ ba{\isachardot}\isanewline
200 \isaindent{\ {\isadigit{1}}{\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}%
204 \isacommand{apply}\ simp\isanewline
206 \isacommand{done}\isamarkupfalse%
208 \begin{isamarkuptext}%
210 Note that we have intentionally included only \isa{split{\isacharunderscore}paired{\isacharunderscore}all}
211 in the first simplification step, and then we simplify again.
212 This time the reason was not merely
214 \isa{split{\isacharunderscore}paired{\isacharunderscore}all} may interfere with other functions
216 The following command could fail (here it does not)
217 where two separate \isa{simp} applications succeed.%
221 \isacommand{apply}{\isacharparenleft}simp\ add{\isacharcolon}\ split{\isacharunderscore}paired{\isacharunderscore}all{\isacharparenright}\isamarkupfalse%
224 \begin{isamarkuptext}%
226 Finally, the simplifier automatically splits all \isa{{\isasymforall}} and
227 \isa{{\isasymexists}}-quantified variables:%
230 \isacommand{lemma}\ {\isachardoublequote}{\isasymforall}p{\isachardot}\ {\isasymexists}q{\isachardot}\ swap\ p\ {\isacharequal}\ swap\ q{\isachardoublequote}\isanewline
232 \isacommand{by}\ simp\isamarkupfalse%
234 \begin{isamarkuptext}%
236 To turn off this automatic splitting, just disable the
237 responsible simplification rules:
239 \isa{{\isacharparenleft}{\isasymforall}x{\isachardot}\ P\ x{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isasymforall}a\ b{\isachardot}\ P\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}{\isacharparenright}}
241 (\isa{split{\isacharunderscore}paired{\isacharunderscore}All})\\
242 \isa{{\isacharparenleft}{\isasymexists}x{\isachardot}\ P\ x{\isacharparenright}\ {\isacharequal}\ {\isacharparenleft}{\isasymexists}a\ b{\isachardot}\ P\ {\isacharparenleft}a{\isacharcomma}\ b{\isacharparenright}{\isacharparenright}}
244 (\isa{split{\isacharunderscore}paired{\isacharunderscore}Ex})
252 %%% TeX-master: "root"