doc-src/TutorialI/Types/Pairs.thy
 author nipkow Wed Dec 13 09:39:53 2000 +0100 (2000-12-13) changeset 10654 458068404143 parent 10608 620647438780 child 10824 4a212e635318 permissions -rw-r--r--
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     1 (*<*)theory Pairs = Main:(*>*)

     2

     3 section{*Pairs*}

     4

     5 text{*\label{sec:products}

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

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

     8 @{term snd}. In any nontrivial application of pairs you will find that this

     9 quickly leads to unreadable formulae involvings nests of projections. This

    10 section is concerned with introducing some syntactic sugar to overcome this

    11 problem: pattern matching with tuples.

    12 *}

    13

    14 subsection{*Pattern matching with tuples*}

    15

    16 text{*

    17 It is possible to use (nested) tuples as patterns in $\lambda$-abstractions,

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

    19 tuple patterns can be used in most variable binding constructs. Here are

    20 some typical examples:

    21 \begin{quote}

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

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

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

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

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

    27 \end{quote}

    28 *}

    29

    30 text{*

    31 The intuitive meaning of this notations should be pretty obvious.

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

    33 for once we have to reason about it. The fact of the matter is that abstraction

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

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

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

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

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

    39 different terms).

    40

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

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

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

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

    45 \begin{center}

    46 @{thm split_def}

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

    48 \end{center}

    49 Pattern matching in

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

    51 understand how to reason about such constructs.

    52 *}

    53

    54 subsection{*Theorem proving*}

    55

    56 text{*

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

    58 *}

    59

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

    61 by(simp add:split_def)

    62

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

    64 proof, as in the above lemma. But if it doesn't, you end up with exactly what

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

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

    67 can be effective in small ones.

    68

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

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

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

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

    73 @{thm Product_Type.split[no_vars]} and @{thm fst_conv[no_vars]}.  This is the

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

    75 convert some atomic term into a pair.

    76

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

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

    79 *}

    80

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

    82 apply(split split_split);

    83

    84 txt{*

    85 @{subgoals[display,indent=0]}

    86 This subgoal is easily proved by simplification. The @{text"only:"} above

    87 merely serves to show the effect of splitting and to avoid solving the goal

    88 outright.

    89

    90 Let us look at a second example:

    91 *}

    92

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

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

    95 apply(simp only:Let_def)

    96

    97 txt{*

    98 @{subgoals[display,indent=0]}

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

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

   101 *}

   102

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

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

   105 apply simp

   106

   107 txt{*

   108 @{subgoals[display,indent=0]}

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

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

   111 @{term"split (op =)"} is the same as @{text"\<lambda>(x,y). x=y"}.

   112 The same procedure works for

   113 *}

   114

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

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

   117

   118 txt{*\noindent

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

   120 @{term split} occurs in the assumptions.

   121

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

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

   124 *}

   125

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

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

   128 primrec

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

   130

   131 text{*\noindent

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

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

   134 *}

   135

   136 lemma "swap(swap p) = p"

   137

   138 txt{*\noindent

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

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

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

   142 is to split the term by hand:

   143 *}

   144 apply(case_tac p)

   145

   146 txt{*\noindent

   147 @{subgoals[display,indent=0]}

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

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

   150

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

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

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

   154 *}

   155

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

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

   158 apply(simp only:split_paired_all)

   159

   160 txt{*\noindent

   161 @{subgoals[display,indent=0]}

   162 *}

   163

   164 apply simp

   165 done

   166

   167 text{*\noindent

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

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

   170 pedagogical:

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

   172 rules of the simplifier, i.e.

   173 *}

   174

   175 (*<*)

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

   177 (*>*)

   178 apply(simp add:split_paired_all)

   179 (*<*)done(*>*)

   180 text{*\noindent

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

   182

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

   184 automatically by the simplifier:

   185 *}

   186

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

   188 apply simp;

   189 done

   190

   191 text{*\noindent

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

   193 responsible simplification rules:

   194 \begin{center}

   195 @{thm split_paired_All[no_vars]}

   196 \hfill

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

   198 @{thm split_paired_Ex[no_vars]}

   199 \hfill

   200 (@{thm[source]split_paired_Ex})

   201 \end{center}

   202 *}

   203 (*<*)

   204 end

   205 (*>*)