author | nipkow |

Mon Aug 28 10:16:58 2000 +0200 (2000-08-28) | |

changeset 9690 | 50f22b1b136a |

parent 9689 | 751fde5307e4 |

child 9691 | 88d8d45a4cc4 |

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doc-src/TutorialI/Recdef/Nested2.thy | file | annotate | diff | revisions | |

doc-src/TutorialI/Recdef/document/Nested2.tex | file | annotate | diff | revisions |

1.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000 1.2 +++ b/doc-src/TutorialI/Recdef/Nested2.thy Mon Aug 28 10:16:58 2000 +0200 1.3 @@ -0,0 +1,78 @@ 1.4 +(*<*) 1.5 +theory Nested2 = Nested0:; 1.6 +consts trev :: "('a,'b)term => ('a,'b)term"; 1.7 +(*>*) 1.8 + 1.9 +text{*\noindent 1.10 +The termintion condition is easily proved by induction: 1.11 +*}; 1.12 + 1.13 +lemma [simp]: "t \\<in> set ts \\<longrightarrow> size t < Suc(term_size ts)"; 1.14 +by(induct_tac ts, auto); 1.15 +(*<*) 1.16 +recdef trev "measure size" 1.17 + "trev (Var x) = Var x" 1.18 + "trev (App f ts) = App f (rev(map trev ts))"; 1.19 +(*>*) 1.20 +text{*\noindent 1.21 +By making this theorem a simplification rule, \isacommand{recdef} 1.22 +applies it automatically and the above definition of @{term"trev"} 1.23 +succeeds now. As a reward for our effort, we can now prove the desired 1.24 +lemma directly. The key is the fact that we no longer need the verbose 1.25 +induction schema for type \isa{term} but the simpler one arising from 1.26 +@{term"trev"}: 1.27 +*}; 1.28 + 1.29 +lemmas [cong] = map_cong; 1.30 +lemma "trev(trev t) = t"; 1.31 +apply(induct_tac t rule:trev.induct); 1.32 +txt{*\noindent 1.33 +This leaves us with a trivial base case @{term"trev (trev (Var x)) = Var x"} and the step case 1.34 +\begin{quote} 1.35 +@{term[display,margin=60]"ALL t. t : set ts --> trev (trev t) = t ==> trev (trev (App f ts)) = App f ts"} 1.36 +\end{quote} 1.37 +both of which are solved by simplification: 1.38 +*}; 1.39 + 1.40 +by(simp_all del:map_compose add:sym[OF map_compose] rev_map); 1.41 + 1.42 +text{*\noindent 1.43 +If this surprises you, see Datatype/Nested2...... 1.44 + 1.45 +The above definition of @{term"trev"} is superior to the one in \S\ref{sec:nested-datatype} 1.46 +because it brings @{term"rev"} into play, about which already know a lot, in particular 1.47 +@{prop"rev(rev xs) = xs"}. 1.48 +Thus this proof is a good example of an important principle: 1.49 +\begin{quote} 1.50 +\emph{Chose your definitions carefully\\ 1.51 +because they determine the complexity of your proofs.} 1.52 +\end{quote} 1.53 + 1.54 +Let us now return to the question of how \isacommand{recdef} can come up with sensible termination 1.55 +conditions in the presence of higher-order functions like @{term"map"}. For a start, if nothing 1.56 +were known about @{term"map"}, @{term"map trev ts"} might apply @{term"trev"} to arbitrary terms, 1.57 +and thus \isacommand{recdef} would try to prove the unprovable 1.58 +@{term"size t < Suc (term_size ts)"}, without any assumption about \isa{t}. 1.59 +Therefore \isacommand{recdef} has been supplied with the congruence theorem \isa{map\_cong}: 1.60 +\begin{quote} 1.61 +@{thm[display,margin=50]"map_cong"[no_vars]} 1.62 +\end{quote} 1.63 +Its second premise expresses (indirectly) that the second argument of @{term"map"} is only applied 1.64 +to elements of its third argument. Congruence rules for other higher-order functions on lists would 1.65 +look very similar but have not been proved yet because they were never needed. 1.66 +If you get into a situation where you need to supply \isacommand{recdef} with new congruence 1.67 +rules, you can either append the line 1.68 +\begin{ttbox} 1.69 +congs <congruence rules> 1.70 +\end{ttbox} 1.71 +to the specific occurrence of \isacommand{recdef} or declare them globally: 1.72 +\begin{ttbox} 1.73 +lemmas [????????] = <congruence rules> 1.74 +\end{ttbox} 1.75 + 1.76 +Note that \isacommand{recdef} feeds on exactly the same \emph{kind} of 1.77 +congruence rules as the simplifier (\S\ref{sec:simp-cong}) but that 1.78 +declaring a congruence rule for the simplifier does not make it 1.79 +available to \isacommand{recdef}, and vice versa. This is intentional. 1.80 +*}; 1.81 +(*<*)end;(*>*)

2.1 --- /dev/null Thu Jan 01 00:00:00 1970 +0000 2.2 +++ b/doc-src/TutorialI/Recdef/document/Nested2.tex Mon Aug 28 10:16:58 2000 +0200 2.3 @@ -0,0 +1,84 @@ 2.4 +\begin{isabelle}% 2.5 +% 2.6 +\begin{isamarkuptext}% 2.7 +\noindent 2.8 +The termintion condition is easily proved by induction:% 2.9 +\end{isamarkuptext}% 2.10 +\isacommand{lemma}\ [simp]:\ {"}t\ {\isasymin}\ set\ ts\ {\isasymlongrightarrow}\ size\ t\ <\ Suc(term\_size\ ts){"}\isanewline 2.11 +\isacommand{by}(induct\_tac\ ts,\ auto)% 2.12 +\begin{isamarkuptext}% 2.13 +\noindent 2.14 +By making this theorem a simplification rule, \isacommand{recdef} 2.15 +applies it automatically and the above definition of \isa{trev} 2.16 +succeeds now. As a reward for our effort, we can now prove the desired 2.17 +lemma directly. The key is the fact that we no longer need the verbose 2.18 +induction schema for type \isa{term} but the simpler one arising from 2.19 +\isa{trev}:% 2.20 +\end{isamarkuptext}% 2.21 +\isacommand{lemmas}\ [cong]\ =\ map\_cong\isanewline 2.22 +\isacommand{lemma}\ {"}trev(trev\ t)\ =\ t{"}\isanewline 2.23 +\isacommand{apply}(induct\_tac\ t\ rule:trev.induct)% 2.24 +\begin{isamarkuptxt}% 2.25 +\noindent 2.26 +This leaves us with a trivial base case \isa{trev\ (trev\ (Var\ \mbox{x}))\ =\ Var\ \mbox{x}} and the step case 2.27 +\begin{quote} 2.28 + 2.29 +\begin{isabelle}% 2.30 +{\isasymforall}\mbox{t}.\ \mbox{t}\ {\isasymin}\ set\ \mbox{ts}\ {\isasymlongrightarrow}\ trev\ (trev\ \mbox{t})\ =\ \mbox{t}\ {\isasymLongrightarrow}\isanewline 2.31 +trev\ (trev\ (App\ \mbox{f}\ \mbox{ts}))\ =\ App\ \mbox{f}\ \mbox{ts} 2.32 +\end{isabelle}% 2.33 + 2.34 +\end{quote} 2.35 +both of which are solved by simplification:% 2.36 +\end{isamarkuptxt}% 2.37 +\isacommand{by}(simp\_all\ del:map\_compose\ add:sym[OF\ map\_compose]\ rev\_map)% 2.38 +\begin{isamarkuptext}% 2.39 +\noindent 2.40 +If this surprises you, see Datatype/Nested2...... 2.41 + 2.42 +The above definition of \isa{trev} is superior to the one in \S\ref{sec:nested-datatype} 2.43 +because it brings \isa{rev} into play, about which already know a lot, in particular 2.44 +\isa{rev\ (rev\ \mbox{xs})\ =\ \mbox{xs}}. 2.45 +Thus this proof is a good example of an important principle: 2.46 +\begin{quote} 2.47 +\emph{Chose your definitions carefully\\ 2.48 +because they determine the complexity of your proofs.} 2.49 +\end{quote} 2.50 + 2.51 +Let us now return to the question of how \isacommand{recdef} can come up with sensible termination 2.52 +conditions in the presence of higher-order functions like \isa{map}. For a start, if nothing 2.53 +were known about \isa{map}, \isa{map\ trev\ \mbox{ts}} might apply \isa{trev} to arbitrary terms, 2.54 +and thus \isacommand{recdef} would try to prove the unprovable 2.55 +\isa{size\ \mbox{t}\ <\ Suc\ (term\_size\ \mbox{ts})}, without any assumption about \isa{t}. 2.56 +Therefore \isacommand{recdef} has been supplied with the congruence theorem \isa{map\_cong}: 2.57 +\begin{quote} 2.58 + 2.59 +\begin{isabelle}% 2.60 +{\isasymlbrakk}\mbox{xs}\ =\ \mbox{ys};\ {\isasymAnd}\mbox{x}.\ \mbox{x}\ {\isasymin}\ set\ \mbox{ys}\ {\isasymLongrightarrow}\ \mbox{f}\ \mbox{x}\ =\ \mbox{g}\ \mbox{x}{\isasymrbrakk}\isanewline 2.61 +{\isasymLongrightarrow}\ map\ \mbox{f}\ \mbox{xs}\ =\ map\ \mbox{g}\ \mbox{ys} 2.62 +\end{isabelle}% 2.63 + 2.64 +\end{quote} 2.65 +Its second premise expresses (indirectly) that the second argument of \isa{map} is only applied 2.66 +to elements of its third argument. Congruence rules for other higher-order functions on lists would 2.67 +look very similar but have not been proved yet because they were never needed. 2.68 +If you get into a situation where you need to supply \isacommand{recdef} with new congruence 2.69 +rules, you can either append the line 2.70 +\begin{ttbox} 2.71 +congs <congruence rules> 2.72 +\end{ttbox} 2.73 +to the specific occurrence of \isacommand{recdef} or declare them globally: 2.74 +\begin{ttbox} 2.75 +lemmas [????????] = <congruence rules> 2.76 +\end{ttbox} 2.77 + 2.78 +Note that \isacommand{recdef} feeds on exactly the same \emph{kind} of 2.79 +congruence rules as the simplifier (\S\ref{sec:simp-cong}) but that 2.80 +declaring a congruence rule for the simplifier does not make it 2.81 +available to \isacommand{recdef}, and vice versa. This is intentional.% 2.82 +\end{isamarkuptext}% 2.83 +\end{isabelle}% 2.84 +%%% Local Variables: 2.85 +%%% mode: latex 2.86 +%%% TeX-master: "root" 2.87 +%%% End: