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doc-src/Ref/simplifier.tex

author | wenzelm |

Mon, 03 May 1999 19:03:35 +0200 | |

changeset 6569 | 66c941ea1f01 |

parent 5776 | 3bcc29d0c783 |

child 7620 | 8d721c3f4acb |

permissions | -rw-r--r-- |

tuned;

%% $Id$ \chapter{Simplification} \label{chap:simplification} \index{simplification|(} This chapter describes Isabelle's generic simplification package. It performs conditional and unconditional rewriting and uses contextual information (`local assumptions'). It provides several general hooks, which can provide automatic case splits during rewriting, for example. The simplifier is already set up for many of Isabelle's logics: \FOL, \ZF, \HOL, \HOLCF. The first section is a quick introduction to the simplifier that should be sufficient to get started. The later sections explain more advanced features. \section{Simplification for dummies} \label{sec:simp-for-dummies} Basic use of the simplifier is particularly easy because each theory is equipped with sensible default information controlling the rewrite process --- namely the implicit {\em current simpset}\index{simpset!current}. A suite of simple commands is provided that refer to the implicit simpset of the current theory context. \begin{warn} Make sure that you are working within the correct theory context. Executing proofs interactively, or loading them from ML files without associated theories may require setting the current theory manually via the \ttindex{context} command. \end{warn} \subsection{Simplification tactics} \label{sec:simp-for-dummies-tacs} \begin{ttbox} Simp_tac : int -> tactic Asm_simp_tac : int -> tactic Full_simp_tac : int -> tactic Asm_full_simp_tac : int -> tactic trace_simp : bool ref \hfill{\bf initially false} \end{ttbox} \begin{ttdescription} \item[\ttindexbold{Simp_tac} $i$] simplifies subgoal~$i$ using the current simpset. It may solve the subgoal completely if it has become trivial, using the simpset's solver tactic. \item[\ttindexbold{Asm_simp_tac}]\index{assumptions!in simplification} is like \verb$Simp_tac$, but extracts additional rewrite rules from the local assumptions. \item[\ttindexbold{Full_simp_tac}] is like \verb$Simp_tac$, but also simplifies the assumptions (without using the assumptions to simplify each other or the actual goal). \item[\ttindexbold{Asm_full_simp_tac}] is like \verb$Asm_simp_tac$, but also simplifies the assumptions. In particular, assumptions can simplify each other. \footnote{\texttt{Asm_full_simp_tac} used to process the assumptions from left to right. For backwards compatibilty reasons only there is now \texttt{Asm_lr_simp_tac} that behaves like the old \texttt{Asm_full_simp_tac}.} \item[set \ttindexbold{trace_simp};] makes the simplifier output internal operations. This includes rewrite steps, but also bookkeeping like modifications of the simpset. \end{ttdescription} \medskip As an example, consider the theory of arithmetic in \HOL. The (rather trivial) goal $0 + (x + 0) = x + 0 + 0$ can be solved by a single call of \texttt{Simp_tac} as follows: \begin{ttbox} context Arith.thy; Goal "0 + (x + 0) = x + 0 + 0"; {\out 1. 0 + (x + 0) = x + 0 + 0} by (Simp_tac 1); {\out Level 1} {\out 0 + (x + 0) = x + 0 + 0} {\out No subgoals!} \end{ttbox} The simplifier uses the current simpset of \texttt{Arith.thy}, which contains suitable theorems like $\Var{n}+0 = \Var{n}$ and $0+\Var{n} = \Var{n}$. \medskip In many cases, assumptions of a subgoal are also needed in the simplification process. For example, \texttt{x = 0 ==> x + x = 0} is solved by \texttt{Asm_simp_tac} as follows: \begin{ttbox} {\out 1. x = 0 ==> x + x = 0} by (Asm_simp_tac 1); \end{ttbox} \medskip \texttt{Asm_full_simp_tac} is the most powerful of this quartet of tactics but may also loop where some of the others terminate. For example, \begin{ttbox} {\out 1. ALL x. f x = g (f (g x)) ==> f 0 = f 0 + 0} \end{ttbox} is solved by \texttt{Simp_tac}, but \texttt{Asm_simp_tac} and {\tt Asm_simp_tac} loop because the rewrite rule $f\,\Var{x} = g\,(f\,(g\,\Var{x}))$ extracted from the assumption does not terminate. Isabelle notices certain simple forms of nontermination, but not this one. Because assumptions may simplify each other, there can be very subtle cases of nontermination. \begin{warn} \verb$Asm_full_simp_tac$ may miss opportunities to simplify an assumption $A@i$ using an assumption $A@j$ in case $A@j$ is to the right of $A@i$. For example, given the subgoal \begin{ttbox} {\out [| \dots f a \dots; P a; ALL x. P x --> f x = g x |] ==> \dots} \end{ttbox} \verb$Asm_full_simp_tac$ will not simplify the \texttt{f a} on the left. This problem can be overcome by reordering assumptions (see \S\ref{sec:reordering-asms}). Future versions of \verb$Asm_full_simp_tac$ will not suffer from this deficiency. \end{warn} \medskip Using the simplifier effectively may take a bit of experimentation. Set the \verb$trace_simp$\index{tracing!of simplification} flag to get a better idea of what is going on. The resulting output can be enormous, especially since invocations of the simplifier are often nested (e.g.\ when solving conditions of rewrite rules). \subsection{Modifying the current simpset} \begin{ttbox} Addsimps : thm list -> unit Delsimps : thm list -> unit Addsimprocs : simproc list -> unit Delsimprocs : simproc list -> unit Addcongs : thm list -> unit Delcongs : thm list -> unit Addsplits : thm list -> unit Delsplits : thm list -> unit \end{ttbox} Depending on the theory context, the \texttt{Add} and \texttt{Del} functions manipulate basic components of the associated current simpset. Internally, all rewrite rules have to be expressed as (conditional) meta-equalities. This form is derived automatically from object-level equations that are supplied by the user. Another source of rewrite rules are \emph{simplification procedures}, that is \ML\ functions that produce suitable theorems on demand, depending on the current redex. Congruences are a more advanced feature; see \S\ref{sec:simp-congs}. \begin{ttdescription} \item[\ttindexbold{Addsimps} $thms$;] adds rewrite rules derived from $thms$ to the current simpset. \item[\ttindexbold{Delsimps} $thms$;] deletes rewrite rules derived from $thms$ from the current simpset. \item[\ttindexbold{Addsimprocs} $procs$;] adds simplification procedures $procs$ to the current simpset. \item[\ttindexbold{Delsimprocs} $procs$;] deletes simplification procedures $procs$ from the current simpset. \item[\ttindexbold{Addcongs} $thms$;] adds congruence rules to the current simpset. \item[\ttindexbold{Delcongs} $thms$;] deletes congruence rules from the current simpset. \item[\ttindexbold{Addsplits} $thms$;] adds splitting rules to the current simpset. \item[\ttindexbold{Delsplits} $thms$;] deletes splitting rules from the current simpset. \end{ttdescription} When a new theory is built, its implicit simpset is initialized by the union of the respective simpsets of its parent theories. In addition, certain theory definition constructs (e.g.\ \ttindex{datatype} and \ttindex{primrec} in \HOL) implicitly augment the current simpset. Ordinary definitions are not added automatically! It is up the user to manipulate the current simpset further by explicitly adding or deleting theorems and simplification procedures. \medskip Good simpsets are hard to design. Rules that obviously simplify, like $\Var{n}+0 = \Var{n}$, should be added to the current simpset right after they have been proved. More specific ones (such as distributive laws, which duplicate subterms) should be added only for specific proofs and deleted afterwards. Conversely, sometimes a rule needs to be removed for a certain proof and restored afterwards. The need of frequent additions or deletions may indicate a badly designed simpset. \begin{warn} The union of the parent simpsets (as described above) is not always a good starting point for the new theory. If some ancestors have deleted simplification rules because they are no longer wanted, while others have left those rules in, then the union will contain the unwanted rules. After this union is formed, changes to a parent simpset have no effect on the child simpset. \end{warn} \section{Simplification sets}\index{simplification sets} The simplifier is controlled by information contained in {\bf simpsets}. These consist of several components, including rewrite rules, simplification procedures, congruence rules, and the subgoaler, solver and looper tactics. The simplifier should be set up with sensible defaults so that most simplifier calls specify only rewrite rules or simplification procedures. Experienced users can exploit the other components to streamline proofs in more sophisticated manners. \subsection{Inspecting simpsets} \begin{ttbox} print_ss : simpset -> unit rep_ss : simpset -> \{mss : meta_simpset, subgoal_tac: simpset -> int -> tactic, loop_tac : int -> tactic, finish_tac : thm list -> int -> tactic, unsafe_finish_tac : thm list -> int -> tactic\} \end{ttbox} \begin{ttdescription} \item[\ttindexbold{print_ss} $ss$;] displays the printable contents of simpset $ss$. This includes the rewrite rules and congruences in their internal form expressed as meta-equalities. The names of the simplification procedures and the patterns they are invoked on are also shown. The other parts, functions and tactics, are non-printable. \item[\ttindexbold{rep_ss} $ss$;] decomposes $ss$ as a record of its internal components, namely the meta_simpset, the subgoaler, the loop, and the safe and unsafe solvers. \end{ttdescription} \subsection{Building simpsets} \begin{ttbox} empty_ss : simpset merge_ss : simpset * simpset -> simpset \end{ttbox} \begin{ttdescription} \item[\ttindexbold{empty_ss}] is the empty simpset. This is not very useful under normal circumstances because it doesn't contain suitable tactics (subgoaler etc.). When setting up the simplifier for a particular object-logic, one will typically define a more appropriate ``almost empty'' simpset. For example, in \HOL\ this is called \ttindexbold{HOL_basic_ss}. \item[\ttindexbold{merge_ss} ($ss@1$, $ss@2$)] merges simpsets $ss@1$ and $ss@2$ by building the union of their respective rewrite rules, simplification procedures and congruences. The other components (tactics etc.) cannot be merged, though; they are taken from either simpset\footnote{Actually from $ss@1$, but it would unwise to count on that.}. \end{ttdescription} \subsection{Accessing the current simpset} \label{sec:access-current-simpset} \begin{ttbox} simpset : unit -> simpset simpset_ref : unit -> simpset ref simpset_of : theory -> simpset simpset_ref_of : theory -> simpset ref print_simpset : theory -> unit SIMPSET :(simpset -> tactic) -> tactic SIMPSET' :(simpset -> 'a -> tactic) -> 'a -> tactic \end{ttbox} Each theory contains a current simpset\index{simpset!current} stored within a private ML reference variable. This can be retrieved and modified as follows. \begin{ttdescription} \item[\ttindexbold{simpset}();] retrieves the simpset value from the current theory context. \item[\ttindexbold{simpset_ref}();] retrieves the simpset reference variable from the current theory context. This can be assigned to by using \texttt{:=} in ML. \item[\ttindexbold{simpset_of} $thy$;] retrieves the simpset value from theory $thy$. \item[\ttindexbold{simpset_ref_of} $thy$;] retrieves the simpset reference variable from theory $thy$. \item[\ttindexbold{print_simpset} $thy$;] prints the current simpset of theory $thy$ in the same way as \texttt{print_ss}. \item[\ttindexbold{SIMPSET} $tacf$, \ttindexbold{SIMPSET'} $tacf'$] are tacticals that make a tactic depend on the implicit current simpset of the theory associated with the proof state they are applied on. \end{ttdescription} \begin{warn} There is a subtle difference between \texttt{(SIMPSET'~$tacf$)} and \texttt{($tacf$~(simpset()))}. For example \texttt{(SIMPSET' simp_tac)} would depend on the theory of the proof state it is applied to, while \texttt{(simp_tac (simpset()))} implicitly refers to the current theory context. Both are usually the same in proof scripts, provided that goals are only stated within the current theory. Robust programs would not count on that, of course. \end{warn} \subsection{Rewrite rules} \begin{ttbox} addsimps : simpset * thm list -> simpset \hfill{\bf infix 4} delsimps : simpset * thm list -> simpset \hfill{\bf infix 4} \end{ttbox} \index{rewrite rules|(} Rewrite rules are theorems expressing some form of equality, for example: \begin{eqnarray*} Suc(\Var{m}) + \Var{n} &=& \Var{m} + Suc(\Var{n}) \\ \Var{P}\conj\Var{P} &\bimp& \Var{P} \\ \Var{A} \un \Var{B} &\equiv& \{x.x\in \Var{A} \disj x\in \Var{B}\} \end{eqnarray*} Conditional rewrites such as $\Var{m}<\Var{n} \Imp \Var{m}/\Var{n} = 0$ are also permitted; the conditions can be arbitrary formulas. Internally, all rewrite rules are translated into meta-equalities, theorems with conclusion $lhs \equiv rhs$. Each simpset contains a function for extracting equalities from arbitrary theorems. For example, $\neg(\Var{x}\in \{\})$ could be turned into $\Var{x}\in \{\} \equiv False$. This function can be installed using \ttindex{setmksimps} but only the definer of a logic should need to do this; see \S\ref{sec:setmksimps}. The function processes theorems added by \texttt{addsimps} as well as local assumptions. \begin{ttdescription} \item[$ss$ \ttindexbold{addsimps} $thms$] adds rewrite rules derived from $thms$ to the simpset $ss$. \item[$ss$ \ttindexbold{delsimps} $thms$] deletes rewrite rules derived from $thms$ from the simpset $ss$. \end{ttdescription} \begin{warn} The simplifier will accept all standard rewrite rules: those where all unknowns are of base type. Hence ${\Var{i}+(\Var{j}+\Var{k})} = {(\Var{i}+\Var{j})+\Var{k}}$ is OK. It will also deal gracefully with all rules whose left-hand sides are so-called {\em higher-order patterns}~\cite{nipkow-patterns}. \indexbold{higher-order pattern}\indexbold{pattern, higher-order} These are terms in $\beta$-normal form (this will always be the case unless you have done something strange) where each occurrence of an unknown is of the form $\Var{F}(x@1,\dots,x@n)$, where the $x@i$ are distinct bound variables. Hence $(\forall x.\Var{P}(x) \land \Var{Q}(x)) \bimp (\forall x.\Var{P}(x)) \land (\forall x.\Var{Q}(x))$ is also OK, in both directions. In some rare cases the rewriter will even deal with quite general rules: for example ${\Var{f}(\Var{x})\in range(\Var{f})} = True$ rewrites $g(a) \in range(g)$ to $True$, but will fail to match $g(h(b)) \in range(\lambda x.g(h(x)))$. However, you can replace the offending subterms (in our case $\Var{f}(\Var{x})$, which is not a pattern) by adding new variables and conditions: $\Var{y} = \Var{f}(\Var{x}) \Imp \Var{y}\in range(\Var{f}) = True$ is acceptable as a conditional rewrite rule since conditions can be arbitrary terms. There is basically no restriction on the form of the right-hand sides. They may not contain extraneous term or type variables, though. \end{warn} \index{rewrite rules|)} \subsection{*Simplification procedures} \begin{ttbox} addsimprocs : simpset * simproc list -> simpset delsimprocs : simpset * simproc list -> simpset \end{ttbox} Simplification procedures are {\ML} objects of abstract type \texttt{simproc}. Basically they are just functions that may produce \emph{proven} rewrite rules on demand. They are associated with certain patterns that conceptually represent left-hand sides of equations; these are shown by \texttt{print_ss}. During its operation, the simplifier may offer a simplification procedure the current redex and ask for a suitable rewrite rule. Thus rules may be specifically fashioned for particular situations, resulting in a more powerful mechanism than term rewriting by a fixed set of rules. \begin{ttdescription} \item[$ss$ \ttindexbold{addsimprocs} $procs$] adds the simplification procedures $procs$ to the current simpset. \item[$ss$ \ttindexbold{delsimprocs} $procs$] deletes the simplification procedures $procs$ from the current simpset. \end{ttdescription} For example, simplification procedures \ttindexbold{nat_cancel} of \texttt{HOL/Arith} cancel common summands and constant factors out of several relations of sums over natural numbers. Consider the following goal, which after cancelling $a$ on both sides contains a factor of $2$. Simplifying with the simpset of \texttt{Arith.thy} will do the cancellation automatically: \begin{ttbox} {\out 1. x + a + x < y + y + 2 + a + a + a + a + a} by (Simp_tac 1); {\out 1. x < Suc (a + (a + y))} \end{ttbox} \subsection{*Congruence rules}\index{congruence rules}\label{sec:simp-congs} \begin{ttbox} addcongs : simpset * thm list -> simpset \hfill{\bf infix 4} delcongs : simpset * thm list -> simpset \hfill{\bf infix 4} addeqcongs : simpset * thm list -> simpset \hfill{\bf infix 4} deleqcongs : simpset * thm list -> simpset \hfill{\bf infix 4} \end{ttbox} Congruence rules are meta-equalities of the form \[ \dots \Imp f(\Var{x@1},\ldots,\Var{x@n}) \equiv f(\Var{y@1},\ldots,\Var{y@n}). \] This governs the simplification of the arguments of~$f$. For example, some arguments can be simplified under additional assumptions: \[ \List{\Var{P@1} \bimp \Var{Q@1};\; \Var{Q@1} \Imp \Var{P@2} \bimp \Var{Q@2}} \Imp (\Var{P@1} \imp \Var{P@2}) \equiv (\Var{Q@1} \imp \Var{Q@2}) \] Given this rule, the simplifier assumes $Q@1$ and extracts rewrite rules from it when simplifying~$P@2$. Such local assumptions are effective for rewriting formulae such as $x=0\imp y+x=y$. The local assumptions are also provided as theorems to the solver; see \S~\ref{sec:simp-solver} below. \begin{ttdescription} \item[$ss$ \ttindexbold{addcongs} $thms$] adds congruence rules to the simpset $ss$. These are derived from $thms$ in an appropriate way, depending on the underlying object-logic. \item[$ss$ \ttindexbold{delcongs} $thms$] deletes congruence rules derived from $thms$. \item[$ss$ \ttindexbold{addeqcongs} $thms$] adds congruence rules in their internal form (conclusions using meta-equality) to simpset $ss$. This is the basic mechanism that \texttt{addcongs} is built on. It should be rarely used directly. \item[$ss$ \ttindexbold{deleqcongs} $thms$] deletes congruence rules in internal form from simpset $ss$. \end{ttdescription} \medskip Here are some more examples. The congruence rule for bounded quantifiers also supplies contextual information, this time about the bound variable: \begin{eqnarray*} &&\List{\Var{A}=\Var{B};\; \Forall x. x\in \Var{B} \Imp \Var{P}(x) = \Var{Q}(x)} \Imp{} \\ &&\qquad\qquad (\forall x\in \Var{A}.\Var{P}(x)) = (\forall x\in \Var{B}.\Var{Q}(x)) \end{eqnarray*} The congruence rule for conditional expressions can supply contextual information for simplifying the arms: \[ \List{\Var{p}=\Var{q};~ \Var{q} \Imp \Var{a}=\Var{c};~ \neg\Var{q} \Imp \Var{b}=\Var{d}} \Imp if(\Var{p},\Var{a},\Var{b}) \equiv if(\Var{q},\Var{c},\Var{d}) \] A congruence rule can also {\em prevent\/} simplification of some arguments. Here is an alternative congruence rule for conditional expressions: \[ \Var{p}=\Var{q} \Imp if(\Var{p},\Var{a},\Var{b}) \equiv if(\Var{q},\Var{a},\Var{b}) \] Only the first argument is simplified; the others remain unchanged. This can make simplification much faster, but may require an extra case split to prove the goal. \subsection{*The subgoaler}\label{sec:simp-subgoaler} \begin{ttbox} setsubgoaler : simpset * (simpset -> int -> tactic) -> simpset \hfill{\bf infix 4} prems_of_ss : simpset -> thm list \end{ttbox} The subgoaler is the tactic used to solve subgoals arising out of conditional rewrite rules or congruence rules. The default should be simplification itself. Occasionally this strategy needs to be changed. For example, if the premise of a conditional rule is an instance of its conclusion, as in $Suc(\Var{m}) < \Var{n} \Imp \Var{m} < \Var{n}$, the default strategy could loop. \begin{ttdescription} \item[$ss$ \ttindexbold{setsubgoaler} $tacf$] sets the subgoaler of $ss$ to $tacf$. The function $tacf$ will be applied to the current simplifier context expressed as a simpset. \item[\ttindexbold{prems_of_ss} $ss$] retrieves the current set of premises from simplifier context $ss$. This may be non-empty only if the simplifier has been told to utilize local assumptions in the first place, e.g.\ if invoked via \texttt{asm_simp_tac}. \end{ttdescription} As an example, consider the following subgoaler: \begin{ttbox} fun subgoaler ss = assume_tac ORELSE' resolve_tac (prems_of_ss ss) ORELSE' asm_simp_tac ss; \end{ttbox} This tactic first tries to solve the subgoal by assumption or by resolving with with one of the premises, calling simplification only if that fails. \subsection{*The solver}\label{sec:simp-solver} \begin{ttbox} setSolver : simpset * (thm list -> int -> tactic) -> simpset \hfill{\bf infix 4} addSolver : simpset * (thm list -> int -> tactic) -> simpset \hfill{\bf infix 4} setSSolver : simpset * (thm list -> int -> tactic) -> simpset \hfill{\bf infix 4} addSSolver : simpset * (thm list -> int -> tactic) -> simpset \hfill{\bf infix 4} \end{ttbox} The solver is a pair of tactics that attempt to solve a subgoal after simplification. Typically it just proves trivial subgoals such as \texttt{True} and $t=t$. It could use sophisticated means such as {\tt blast_tac}, though that could make simplification expensive. Rewriting does not instantiate unknowns. For example, rewriting cannot prove $a\in \Var{A}$ since this requires instantiating~$\Var{A}$. The solver, however, is an arbitrary tactic and may instantiate unknowns as it pleases. This is the only way the simplifier can handle a conditional rewrite rule whose condition contains extra variables. When a simplification tactic is to be combined with other provers, especially with the classical reasoner, it is important whether it can be considered safe or not. For this reason the solver is split into a safe and an unsafe part. The standard simplification strategy solely uses the unsafe solver, which is appropriate in most cases. For special applications where the simplification process is not allowed to instantiate unknowns within the goal, simplification starts with the safe solver, but may still apply the ordinary unsafe one in nested simplifications for conditional rules or congruences. \begin{ttdescription} \item[$ss$ \ttindexbold{setSSolver} $tacf$] installs $tacf$ as the \emph{safe} solver of $ss$. \item[$ss$ \ttindexbold{addSSolver} $tacf$] adds $tacf$ as an additional \emph{safe} solver; it will be tried after the solvers which had already been present in $ss$. \item[$ss$ \ttindexbold{setSolver} $tacf$] installs $tacf$ as the unsafe solver of $ss$. \item[$ss$ \ttindexbold{addSolver} $tacf$] adds $tacf$ as an additional unsafe solver; it will be tried after the solvers which had already been present in $ss$. \end{ttdescription} \medskip \index{assumptions!in simplification} The solver tactic is invoked with a list of theorems, namely assumptions that hold in the local context. This may be non-empty only if the simplifier has been told to utilize local assumptions in the first place, e.g.\ if invoked via \texttt{asm_simp_tac}. The solver is also presented the full goal including its assumptions in any case. Thus it can use these (e.g.\ by calling \texttt{assume_tac}), even if the list of premises is not passed. \medskip As explained in \S\ref{sec:simp-subgoaler}, the subgoaler is also used to solve the premises of congruence rules. These are usually of the form $s = \Var{x}$, where $s$ needs to be simplified and $\Var{x}$ needs to be instantiated with the result. Typically, the subgoaler will invoke the simplifier at some point, which will eventually call the solver. For this reason, solver tactics must be prepared to solve goals of the form $t = \Var{x}$, usually by reflexivity. In particular, reflexivity should be tried before any of the fancy tactics like \texttt{blast_tac}. It may even happen that due to simplification the subgoal is no longer an equality. For example $False \bimp \Var{Q}$ could be rewritten to $\neg\Var{Q}$. To cover this case, the solver could try resolving with the theorem $\neg False$. \medskip \begin{warn} If the simplifier aborts with the message \texttt{Failed congruence proof!}, then the subgoaler or solver has failed to prove a premise of a congruence rule. This should never occur under normal circumstances; it indicates that some congruence rule, or possibly the subgoaler or solver, is faulty. \end{warn} \subsection{*The looper}\label{sec:simp-looper} \begin{ttbox} setloop : simpset * (int -> tactic) -> simpset \hfill{\bf infix 4} addloop : simpset * (string * (int -> tactic)) -> simpset \hfill{\bf infix 4} delloop : simpset * string -> simpset \hfill{\bf infix 4} addsplits : simpset * thm list -> simpset \hfill{\bf infix 4} delsplits : simpset * thm list -> simpset \hfill{\bf infix 4} \end{ttbox} The looper is a list of tactics that are applied after simplification, in case the solver failed to solve the simplified goal. If the looper succeeds, the simplification process is started all over again. Each of the subgoals generated by the looper is attacked in turn, in reverse order. A typical looper is case splitting: the expansion of a conditional. Another possibility is to apply an elimination rule on the assumptions. More adventurous loopers could start an induction. \begin{ttdescription} \item[$ss$ \ttindexbold{setloop} $tacf$] installs $tacf$ as the only looper tactic of $ss$. \item[$ss$ \ttindexbold{addloop} $(name,tacf)$] adds $tacf$ as an additional looper tactic with name $name$; it will be tried after the looper tactics that had already been present in $ss$. \item[$ss$ \ttindexbold{delloop} $name$] deletes the looper tactic $name$ from $ss$. \item[$ss$ \ttindexbold{addsplits} $thms$] adds split tactics for $thms$ as additional looper tactics of $ss$. \item[$ss$ \ttindexbold{addsplits} $thms$] deletes the split tactics for $thms$ from the looper tactics of $ss$. \end{ttdescription} The splitter replaces applications of a given function; the right-hand side of the replacement can be anything. For example, here is a splitting rule for conditional expressions: \[ \Var{P}(if(\Var{Q},\Var{x},\Var{y})) \bimp (\Var{Q} \imp \Var{P}(\Var{x})) \conj (\lnot\Var{Q} \imp \Var{P}(\Var{y})) \] Another example is the elimination operator (which happens to be called~$split$) for Cartesian products: \[ \Var{P}(split(\Var{f},\Var{p})) \bimp (\forall a~b. \Var{p} = \langle a,b\rangle \imp \Var{P}(\Var{f}(a,b))) \] For technical reasons, there is a distinction between case splitting in the conclusion and in the premises of a subgoal. The former is done by \texttt{split_tac} with rules like \texttt{split_if}, which does not split the subgoal, while the latter is done by \texttt{split_asm_tac} with rules like \texttt{split_if_asm}, which splits the subgoal. The operator \texttt{addsplits} automatically takes care of which tactic to call, analyzing the form of the rules given as argument. \begin{warn} Due to \texttt{split_asm_tac}, the simplifier may split subgoals! \end{warn} Case splits should be allowed only when necessary; they are expensive and hard to control. Here is an example of use, where \texttt{split_if} is the first rule above: \begin{ttbox} by (simp_tac (simpset() addloop ("split if",split_tac [split_if])) 1); \end{ttbox} Users would usually prefer the following shortcut using \texttt{addsplits}: \begin{ttbox} by (simp_tac (simpset() addsplits [split_if]) 1); \end{ttbox} \section{The simplification tactics}\label{simp-tactics} \index{simplification!tactics}\index{tactics!simplification} \begin{ttbox} simp_tac : simpset -> int -> tactic asm_simp_tac : simpset -> int -> tactic full_simp_tac : simpset -> int -> tactic asm_full_simp_tac : simpset -> int -> tactic safe_asm_full_simp_tac : simpset -> int -> tactic \end{ttbox} These are the basic tactics that are underlying any actual simplification work. The rewriting strategy is always strictly bottom up, except for congruence rules, which are applied while descending into a term. Conditions in conditional rewrite rules are solved recursively before the rewrite rule is applied. \begin{ttdescription} \item[\ttindexbold{simp_tac}, \ttindexbold{asm_simp_tac}, \ttindexbold{full_simp_tac}, \ttindexbold{asm_full_simp_tac}] are the basic simplification tactics that work exactly like their namesakes in \S\ref{sec:simp-for-dummies}, except that they are explicitly supplied with a simpset. \item[\ttindexbold{safe_asm_full_simp_tac}] is like \texttt{asm_full_simp_tac}, but uses the safe solver as explained in \S\ref{sec:simp-solver}. This tactic is mainly intended for building special tools, e.g.\ for combining the simplifier with the classical reasoner. It is rarely used directly. \end{ttdescription} \medskip Local modifications of simpsets within a proof are often much cleaner by using above tactics in conjunction with explicit simpsets, rather than their capitalized counterparts. For example \begin{ttbox} Addsimps \(thms\); by (Simp_tac \(i\)); Delsimps \(thms\); \end{ttbox} can be expressed more appropriately as \begin{ttbox} by (simp_tac (simpset() addsimps \(thms\)) \(i\)); \end{ttbox} \medskip Also note that functions depending implicitly on the current theory context (like capital \texttt{Simp_tac} and the other commands of \S\ref{sec:simp-for-dummies}) should be considered harmful outside of actual proof scripts. In particular, ML programs like theory definition packages or special tactics should refer to simpsets only explicitly, via the above tactics used in conjunction with \texttt{simpset_of} or the \texttt{SIMPSET} tacticals. \section{Forward rules and conversions} \index{simplification!forward rules}\index{simplification!conversions} \begin{ttbox}\index{*simplify}\index{*asm_simplify}\index{*full_simplify}\index{*asm_full_simplify}\index{*Simplifier.rewrite}\index{*Simplifier.asm_rewrite}\index{*Simplifier.full_rewrite}\index{*Simplifier.asm_full_rewrite} simplify : simpset -> thm -> thm asm_simplify : simpset -> thm -> thm full_simplify : simpset -> thm -> thm asm_full_simplify : simpset -> thm -> thm\medskip Simplifier.rewrite : simpset -> cterm -> thm Simplifier.asm_rewrite : simpset -> cterm -> thm Simplifier.full_rewrite : simpset -> cterm -> thm Simplifier.asm_full_rewrite : simpset -> cterm -> thm \end{ttbox} The first four of these functions provide \emph{forward} rules for simplification. Their effect is analogous to the corresponding tactics described in \S\ref{simp-tactics}, but affect the whole theorem instead of just a certain subgoal. Also note that the looper~/ solver process as described in \S\ref{sec:simp-looper} and \S\ref{sec:simp-solver} is omitted in forward simplification. The latter four are \emph{conversions}, establishing proven equations of the form $t \equiv u$ where the l.h.s.\ $t$ has been given as argument. \begin{warn} Forward simplification rules and conversions should be used rarely in ordinary proof scripts. The main intention is to provide an internal interface to the simplifier for special utilities. \end{warn} \section{Examples of using the simplifier} \index{examples!of simplification} Assume we are working within {\tt FOL} (see the file \texttt{FOL/ex/Nat}) and that \begin{ttdescription} \item[Nat.thy] is a theory including the constants $0$, $Suc$ and $+$, \item[add_0] is the rewrite rule $0+\Var{n} = \Var{n}$, \item[add_Suc] is the rewrite rule $Suc(\Var{m})+\Var{n} = Suc(\Var{m}+\Var{n})$, \item[induct] is the induction rule $\List{\Var{P}(0);\; \Forall x. \Var{P}(x)\Imp \Var{P}(Suc(x))} \Imp \Var{P}(\Var{n})$. \end{ttdescription} We augment the implicit simpset inherited from \texttt{Nat} with the basic rewrite rules for addition of natural numbers: \begin{ttbox} Addsimps [add_0, add_Suc]; \end{ttbox} \subsection{A trivial example} Proofs by induction typically involve simplification. Here is a proof that~0 is a right identity: \begin{ttbox} Goal "m+0 = m"; {\out Level 0} {\out m + 0 = m} {\out 1. m + 0 = m} \end{ttbox} The first step is to perform induction on the variable~$m$. This returns a base case and inductive step as two subgoals: \begin{ttbox} by (res_inst_tac [("n","m")] induct 1); {\out Level 1} {\out m + 0 = m} {\out 1. 0 + 0 = 0} {\out 2. !!x. x + 0 = x ==> Suc(x) + 0 = Suc(x)} \end{ttbox} Simplification solves the first subgoal trivially: \begin{ttbox} by (Simp_tac 1); {\out Level 2} {\out m + 0 = m} {\out 1. !!x. x + 0 = x ==> Suc(x) + 0 = Suc(x)} \end{ttbox} The remaining subgoal requires \ttindex{Asm_simp_tac} in order to use the induction hypothesis as a rewrite rule: \begin{ttbox} by (Asm_simp_tac 1); {\out Level 3} {\out m + 0 = m} {\out No subgoals!} \end{ttbox} \subsection{An example of tracing} \index{tracing!of simplification|(}\index{*trace_simp} Let us prove a similar result involving more complex terms. We prove that addition is commutative. \begin{ttbox} Goal "m+Suc(n) = Suc(m+n)"; {\out Level 0} {\out m + Suc(n) = Suc(m + n)} {\out 1. m + Suc(n) = Suc(m + n)} \end{ttbox} Performing induction on~$m$ yields two subgoals: \begin{ttbox} by (res_inst_tac [("n","m")] induct 1); {\out Level 1} {\out m + Suc(n) = Suc(m + n)} {\out 1. 0 + Suc(n) = Suc(0 + n)} {\out 2. !!x. x + Suc(n) = Suc(x + n) ==>} {\out Suc(x) + Suc(n) = Suc(Suc(x) + n)} \end{ttbox} Simplification solves the first subgoal, this time rewriting two occurrences of~0: \begin{ttbox} by (Simp_tac 1); {\out Level 2} {\out m + Suc(n) = Suc(m + n)} {\out 1. !!x. x + Suc(n) = Suc(x + n) ==>} {\out Suc(x) + Suc(n) = Suc(Suc(x) + n)} \end{ttbox} Switching tracing on illustrates how the simplifier solves the remaining subgoal: \begin{ttbox} set trace_simp; by (Asm_simp_tac 1); \ttbreak {\out Adding rewrite rule:} {\out .x + Suc n == Suc (.x + n)} \ttbreak {\out Applying instance of rewrite rule:} {\out ?m + Suc ?n == Suc (?m + ?n)} {\out Rewriting:} {\out Suc .x + Suc n == Suc (Suc .x + n)} \ttbreak {\out Applying instance of rewrite rule:} {\out Suc ?m + ?n == Suc (?m + ?n)} {\out Rewriting:} {\out Suc .x + n == Suc (.x + n)} \ttbreak {\out Applying instance of rewrite rule:} {\out Suc ?m + ?n == Suc (?m + ?n)} {\out Rewriting:} {\out Suc .x + n == Suc (.x + n)} \ttbreak {\out Applying instance of rewrite rule:} {\out ?x = ?x == True} {\out Rewriting:} {\out Suc (Suc (.x + n)) = Suc (Suc (.x + n)) == True} \ttbreak {\out Level 3} {\out m + Suc(n) = Suc(m + n)} {\out No subgoals!} \end{ttbox} Many variations are possible. At Level~1 (in either example) we could have solved both subgoals at once using the tactical \ttindex{ALLGOALS}: \begin{ttbox} by (ALLGOALS Asm_simp_tac); {\out Level 2} {\out m + Suc(n) = Suc(m + n)} {\out No subgoals!} \end{ttbox} \index{tracing!of simplification|)} \subsection{Free variables and simplification} Here is a conjecture to be proved for an arbitrary function~$f$ satisfying the law $f(Suc(\Var{n})) = Suc(f(\Var{n}))$: \begin{ttbox} val [prem] = Goal "(!!n. f(Suc(n)) = Suc(f(n))) ==> f(i+j) = i+f(j)"; {\out Level 0} {\out f(i + j) = i + f(j)} {\out 1. f(i + j) = i + f(j)} \ttbreak {\out val prem = "f(Suc(?n)) = Suc(f(?n))} {\out [!!n. f(Suc(n)) = Suc(f(n))]" : thm} \end{ttbox} In the theorem~\texttt{prem}, note that $f$ is a free variable while $\Var{n}$ is a schematic variable. \begin{ttbox} by (res_inst_tac [("n","i")] induct 1); {\out Level 1} {\out f(i + j) = i + f(j)} {\out 1. f(0 + j) = 0 + f(j)} {\out 2. !!x. f(x + j) = x + f(j) ==> f(Suc(x) + j) = Suc(x) + f(j)} \end{ttbox} We simplify each subgoal in turn. The first one is trivial: \begin{ttbox} by (Simp_tac 1); {\out Level 2} {\out f(i + j) = i + f(j)} {\out 1. !!x. f(x + j) = x + f(j) ==> f(Suc(x) + j) = Suc(x) + f(j)} \end{ttbox} The remaining subgoal requires rewriting by the premise, so we add it to the current simpset: \begin{ttbox} by (asm_simp_tac (simpset() addsimps [prem]) 1); {\out Level 3} {\out f(i + j) = i + f(j)} {\out No subgoals!} \end{ttbox} \subsection{Reordering assumptions} \label{sec:reordering-asms} \index{assumptions!reordering} As mentioned in \S\ref{sec:simp-for-dummies-tacs}, \ttindex{asm_full_simp_tac} may require the assumptions to be permuted to be more effective. Given the subgoal \begin{ttbox} {\out 1. [| ALL x. P x --> f x = g x; Q(f a); P a; R |] ==> S} \end{ttbox} we can rotate the assumptions two positions to the right \begin{ttbox} by (rotate_tac ~2 1); \end{ttbox} to obtain \begin{ttbox} {\out 1. [| P a; R; ALL x. P x --> f x = g x; Q(f a) |] ==> S} \end{ttbox} which enables \verb$asm_full_simp_tac$ to simplify \verb$Q(f a)$ to \verb$Q(g a)$ because now all required assumptions are to the left of \verb$Q(f a)$. Since rotation alone cannot produce arbitrary permutations, you can also pick out a particular assumption which needs to be rewritten and move it the the right end of the assumptions. In the above case rotation can be replaced by \begin{ttbox} by (dres_inst_tac [("psi","Q(f a)")] asm_rl 1); \end{ttbox} which is more directed and leads to \begin{ttbox} {\out 1. [| ALL x. P x --> f x = g x; P a; R; Q(f a) |] ==> S} \end{ttbox} \begin{warn} Reordering assumptions usually leads to brittle proofs and should be avoided. Future versions of \verb$asm_full_simp_tac$ will completely remove the need for such manipulations. \end{warn} \section{Permutative rewrite rules} \index{rewrite rules!permutative|(} A rewrite rule is {\bf permutative} if the left-hand side and right-hand side are the same up to renaming of variables. The most common permutative rule is commutativity: $x+y = y+x$. Other examples include $(x-y)-z = (x-z)-y$ in arithmetic and $insert(x,insert(y,A)) = insert(y,insert(x,A))$ for sets. Such rules are common enough to merit special attention. Because ordinary rewriting loops given such rules, the simplifier employs a special strategy, called {\bf ordered rewriting}\index{rewriting!ordered}. There is a standard lexicographic ordering on terms. This should be perfectly OK in most cases, but can be changed for special applications. \begin{ttbox} settermless : simpset * (term * term -> bool) -> simpset \hfill{\bf infix 4} \end{ttbox} \begin{ttdescription} \item[$ss$ \ttindexbold{settermless} $rel$] installs relation $rel$ as term order in simpset $ss$. \end{ttdescription} \medskip A permutative rewrite rule is applied only if it decreases the given term with respect to this ordering. For example, commutativity rewrites~$b+a$ to $a+b$, but then stops because $a+b$ is strictly less than $b+a$. The Boyer-Moore theorem prover~\cite{bm88book} also employs ordered rewriting. Permutative rewrite rules are added to simpsets just like other rewrite rules; the simplifier recognizes their special status automatically. They are most effective in the case of associative-commutative operators. (Associativity by itself is not permutative.) When dealing with an AC-operator~$f$, keep the following points in mind: \begin{itemize}\index{associative-commutative operators} \item The associative law must always be oriented from left to right, namely $f(f(x,y),z) = f(x,f(y,z))$. The opposite orientation, if used with commutativity, leads to looping in conjunction with the standard term order. \item To complete your set of rewrite rules, you must add not just associativity~(A) and commutativity~(C) but also a derived rule, {\bf left-com\-mut\-ativ\-ity} (LC): $f(x,f(y,z)) = f(y,f(x,z))$. \end{itemize} Ordered rewriting with the combination of A, C, and~LC sorts a term lexicographically: \[\def\maps#1{\stackrel{#1}{\longmapsto}} (b+c)+a \maps{A} b+(c+a) \maps{C} b+(a+c) \maps{LC} a+(b+c) \] Martin and Nipkow~\cite{martin-nipkow} discuss the theory and give many examples; other algebraic structures are amenable to ordered rewriting, such as boolean rings. \subsection{Example: sums of natural numbers} This example is again set in \HOL\ (see \texttt{HOL/ex/NatSum}). Theory \thydx{Arith} contains natural numbers arithmetic. Its associated simpset contains many arithmetic laws including distributivity of~$\times$ over~$+$, while \texttt{add_ac} is a list consisting of the A, C and LC laws for~$+$ on type \texttt{nat}. Let us prove the theorem \[ \sum@{i=1}^n i = n\times(n+1)/2. \] % A functional~\texttt{sum} represents the summation operator under the interpretation $\texttt{sum} \, f \, (n + 1) = \sum@{i=0}^n f\,i$. We extend \texttt{Arith} as follows: \begin{ttbox} NatSum = Arith + consts sum :: [nat=>nat, nat] => nat primrec "sum" nat "sum f 0 = 0" "sum f (Suc n) = f(n) + sum f n" end \end{ttbox} The \texttt{primrec} declaration automatically adds rewrite rules for \texttt{sum} to the default simpset. We now remove the \texttt{nat_cancel} simplification procedures (in order not to spoil the example) and insert the AC-rules for~$+$: \begin{ttbox} Delsimprocs nat_cancel; Addsimps add_ac; \end{ttbox} Our desired theorem now reads $\texttt{sum} \, (\lambda i.i) \, (n+1) = n\times(n+1)/2$. The Isabelle goal has both sides multiplied by~$2$: \begin{ttbox} Goal "2 * sum (\%i.i) (Suc n) = n * Suc n"; {\out Level 0} {\out 2 * sum (\%i. i) (Suc n) = n * Suc n} {\out 1. 2 * sum (\%i. i) (Suc n) = n * Suc n} \end{ttbox} Induction should not be applied until the goal is in the simplest form: \begin{ttbox} by (Simp_tac 1); {\out Level 1} {\out 2 * sum (\%i. i) (Suc n) = n * Suc n} {\out 1. n + (sum (\%i. i) n + sum (\%i. i) n) = n * n} \end{ttbox} Ordered rewriting has sorted the terms in the left-hand side. The subgoal is now ready for induction: \begin{ttbox} by (induct_tac "n" 1); {\out Level 2} {\out 2 * sum (\%i. i) (Suc n) = n * Suc n} {\out 1. 0 + (sum (\%i. i) 0 + sum (\%i. i) 0) = 0 * 0} \ttbreak {\out 2. !!n. n + (sum (\%i. i) n + sum (\%i. i) n) = n * n} {\out ==> Suc n + (sum (\%i. i) (Suc n) + sum (\%i. i) (Suc n)) =} {\out Suc n * Suc n} \end{ttbox} Simplification proves both subgoals immediately:\index{*ALLGOALS} \begin{ttbox} by (ALLGOALS Asm_simp_tac); {\out Level 3} {\out 2 * sum (\%i. i) (Suc n) = n * Suc n} {\out No subgoals!} \end{ttbox} Simplification cannot prove the induction step if we omit \texttt{add_ac} from the simpset. Observe that like terms have not been collected: \begin{ttbox} {\out Level 3} {\out 2 * sum (\%i. i) (Suc n) = n * Suc n} {\out 1. !!n. n + sum (\%i. i) n + (n + sum (\%i. i) n) = n + n * n} {\out ==> n + (n + sum (\%i. i) n) + (n + (n + sum (\%i. i) n)) =} {\out n + (n + (n + n * n))} \end{ttbox} Ordered rewriting proves this by sorting the left-hand side. Proving arithmetic theorems without ordered rewriting requires explicit use of commutativity. This is tedious; try it and see! Ordered rewriting is equally successful in proving $\sum@{i=1}^n i^3 = n^2\times(n+1)^2/4$. \subsection{Re-orienting equalities} Ordered rewriting with the derived rule \texttt{symmetry} can reverse equations: \begin{ttbox} val symmetry = prove_goal HOL.thy "(x=y) = (y=x)" (fn _ => [Blast_tac 1]); \end{ttbox} This is frequently useful. Assumptions of the form $s=t$, where $t$ occurs in the conclusion but not~$s$, can often be brought into the right form. For example, ordered rewriting with \texttt{symmetry} can prove the goal \[ f(a)=b \conj f(a)=c \imp b=c. \] Here \texttt{symmetry} reverses both $f(a)=b$ and $f(a)=c$ because $f(a)$ is lexicographically greater than $b$ and~$c$. These re-oriented equations, as rewrite rules, replace $b$ and~$c$ in the conclusion by~$f(a)$. Another example is the goal $\neg(t=u) \imp \neg(u=t)$. The differing orientations make this appear difficult to prove. Ordered rewriting with \texttt{symmetry} makes the equalities agree. (Without knowing more about~$t$ and~$u$ we cannot say whether they both go to $t=u$ or~$u=t$.) Then the simplifier can prove the goal outright. \index{rewrite rules!permutative|)} \section{*Coding simplification procedures} \begin{ttbox} mk_simproc: string -> cterm list -> (Sign.sg -> thm list -> term -> thm option) -> simproc \end{ttbox} \begin{ttdescription} \item[\ttindexbold{mk_simproc}~$name$~$lhss$~$proc$] makes $proc$ a simplification procedure for left-hand side patterns $lhss$. The name just serves as a comment. The function $proc$ may be invoked by the simplifier for redex positions matched by one of $lhss$ as described below. \end{ttdescription} Simplification procedures are applied in a two-stage process as follows: The simplifier tries to match the current redex position against any one of the $lhs$ patterns of any simplification procedure. If this succeeds, it invokes the corresponding {\ML} function, passing with the current signature, local assumptions and the (potential) redex. The result may be either \texttt{None} (indicating failure) or \texttt{Some~$thm$}. Any successful result is supposed to be a (possibly conditional) rewrite rule $t \equiv u$ that is applicable to the current redex. The rule will be applied just as any ordinary rewrite rule. It is expected to be already in \emph{internal form}, though, bypassing the automatic preprocessing of object-level equivalences. \medskip As an example of how to write your own simplification procedures, consider eta-expansion of pair abstraction (see also \texttt{HOL/Modelcheck/MCSyn} where this is used to provide external model checker syntax). The {\HOL} theory of tuples (see \texttt{HOL/Prod}) provides an operator \texttt{split} together with some concrete syntax supporting $\lambda\,(x,y).b$ abstractions. Assume that we would like to offer a tactic that rewrites any function $\lambda\,p.f\,p$ (where $p$ is of some pair type) to $\lambda\,(x,y).f\,(x,y)$. The corresponding rule is: \begin{ttbox} pair_eta_expand: (f::'a*'b=>'c) = (\%(x, y). f (x, y)) \end{ttbox} Unfortunately, term rewriting using this rule directly would not terminate! We now use the simplification procedure mechanism in order to stop the simplifier from applying this rule over and over again, making it rewrite only actual abstractions. The simplification procedure \texttt{pair_eta_expand_proc} is defined as follows: \begin{ttbox} local val lhss = [read_cterm (sign_of Prod.thy) ("f::'a*'b=>'c", TVar (("'z", 0), []))]; val rew = mk_meta_eq pair_eta_expand; \medskip fun proc _ _ (Abs _) = Some rew | proc _ _ _ = None; in val pair_eta_expand_proc = mk_simproc "pair_eta_expand" lhss proc; end; \end{ttbox} This is an example of using \texttt{pair_eta_expand_proc}: \begin{ttbox} {\out 1. P (\%p::'a * 'a. fst p + snd p + z)} by (simp_tac (simpset() addsimprocs [pair_eta_expand_proc]) 1); {\out 1. P (\%(x::'a,y::'a). x + y + z)} \end{ttbox} \medskip In the above example the simplification procedure just did fine grained control over rule application, beyond higher-order pattern matching. Usually, procedures would do some more work, in particular prove particular theorems depending on the current redex. \section{*Setting up the simplifier}\label{sec:setting-up-simp} \index{simplification!setting up} Setting up the simplifier for new logics is complicated. This section describes how the simplifier is installed for intuitionistic first-order logic; the code is largely taken from {\tt FOL/simpdata.ML} of the Isabelle sources. The simplifier and the case splitting tactic, which reside on separate files, are not part of Pure Isabelle. They must be loaded explicitly by the object-logic as follows (below \texttt{\~\relax\~\relax} refers to \texttt{\$ISABELLE_HOME}): \begin{ttbox} use "\~\relax\~\relax/src/Provers/simplifier.ML"; use "\~\relax\~\relax/src/Provers/splitter.ML"; \end{ttbox} Simplification requires converting object-equalities to meta-level rewrite rules. This demands rules stating that equal terms and equivalent formulae are also equal at the meta-level. The rule declaration part of the file \texttt{FOL/IFOL.thy} contains the two lines \begin{ttbox}\index{*eq_reflection theorem}\index{*iff_reflection theorem} eq_reflection "(x=y) ==> (x==y)" iff_reflection "(P<->Q) ==> (P==Q)" \end{ttbox} Of course, you should only assert such rules if they are true for your particular logic. In Constructive Type Theory, equality is a ternary relation of the form $a=b\in A$; the type~$A$ determines the meaning of the equality essentially as a partial equivalence relation. The present simplifier cannot be used. Rewriting in \texttt{CTT} uses another simplifier, which resides in the file {\tt Provers/typedsimp.ML} and is not documented. Even this does not work for later variants of Constructive Type Theory that use intensional equality~\cite{nordstrom90}. \subsection{A collection of standard rewrite rules} We first prove lots of standard rewrite rules about the logical connectives. These include cancellation and associative laws. We define a function that echoes the desired law and then supplies it the prover for intuitionistic \FOL: \begin{ttbox} fun int_prove_fun s = (writeln s; prove_goal IFOL.thy s (fn prems => [ (cut_facts_tac prems 1), (IntPr.fast_tac 1) ])); \end{ttbox} The following rewrite rules about conjunction are a selection of those proved on \texttt{FOL/simpdata.ML}. Later, these will be supplied to the standard simpset. \begin{ttbox} val conj_simps = map int_prove_fun ["P & True <-> P", "True & P <-> P", "P & False <-> False", "False & P <-> False", "P & P <-> P", "P & ~P <-> False", "~P & P <-> False", "(P & Q) & R <-> P & (Q & R)"]; \end{ttbox} The file also proves some distributive laws. As they can cause exponential blowup, they will not be included in the standard simpset. Instead they are merely bound to an \ML{} identifier, for user reference. \begin{ttbox} val distrib_simps = map int_prove_fun ["P & (Q | R) <-> P&Q | P&R", "(Q | R) & P <-> Q&P | R&P", "(P | Q --> R) <-> (P --> R) & (Q --> R)"]; \end{ttbox} \subsection{Functions for preprocessing the rewrite rules} \label{sec:setmksimps} \begin{ttbox}\indexbold{*setmksimps} setmksimps : simpset * (thm -> thm list) -> simpset \hfill{\bf infix 4} \end{ttbox} The next step is to define the function for preprocessing rewrite rules. This will be installed by calling \texttt{setmksimps} below. Preprocessing occurs whenever rewrite rules are added, whether by user command or automatically. Preprocessing involves extracting atomic rewrites at the object-level, then reflecting them to the meta-level. To start, the function \texttt{gen_all} strips any meta-level quantifiers from the front of the given theorem. Usually there are none anyway. \begin{ttbox} fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th; \end{ttbox} The function \texttt{atomize} analyses a theorem in order to extract atomic rewrite rules. The head of all the patterns, matched by the wildcard~\texttt{_}, is the coercion function \texttt{Trueprop}. \begin{ttbox} fun atomize th = case concl_of th of _ $ (Const("op &",_) $ _ $ _) => atomize(th RS conjunct1) \at atomize(th RS conjunct2) | _ $ (Const("op -->",_) $ _ $ _) => atomize(th RS mp) | _ $ (Const("All",_) $ _) => atomize(th RS spec) | _ $ (Const("True",_)) => [] | _ $ (Const("False",_)) => [] | _ => [th]; \end{ttbox} There are several cases, depending upon the form of the conclusion: \begin{itemize} \item Conjunction: extract rewrites from both conjuncts. \item Implication: convert $P\imp Q$ to the meta-implication $P\Imp Q$ and extract rewrites from~$Q$; these will be conditional rewrites with the condition~$P$. \item Universal quantification: remove the quantifier, replacing the bound variable by a schematic variable, and extract rewrites from the body. \item \texttt{True} and \texttt{False} contain no useful rewrites. \item Anything else: return the theorem in a singleton list. \end{itemize} The resulting theorems are not literally atomic --- they could be disjunctive, for example --- but are broken down as much as possible. See the file \texttt{ZF/simpdata.ML} for a sophisticated translation of set-theoretic formulae into rewrite rules. For standard situations like the above, there is a generic auxiliary function \ttindexbold{mk_atomize} that takes a list of pairs $(name, thms)$, where $name$ is an operator name and $thms$ is a list of theorems to resolve with in case the pattern matches, and returns a suitable \texttt{atomize} function. The simplified rewrites must now be converted into meta-equalities. The rule \texttt{eq_reflection} converts equality rewrites, while {\tt iff_reflection} converts if-and-only-if rewrites. The latter possibility can arise in two other ways: the negative theorem~$\neg P$ is converted to $P\equiv\texttt{False}$, and any other theorem~$P$ is converted to $P\equiv\texttt{True}$. The rules \texttt{iff_reflection_F} and {\tt iff_reflection_T} accomplish this conversion. \begin{ttbox} val P_iff_F = int_prove_fun "~P ==> (P <-> False)"; val iff_reflection_F = P_iff_F RS iff_reflection; \ttbreak val P_iff_T = int_prove_fun "P ==> (P <-> True)"; val iff_reflection_T = P_iff_T RS iff_reflection; \end{ttbox} The function \texttt{mk_eq} converts a theorem to a meta-equality using the case analysis described above. \begin{ttbox} fun mk_eq th = case concl_of th of _ $ (Const("op =",_)$_$_) => th RS eq_reflection | _ $ (Const("op <->",_)$_$_) => th RS iff_reflection | _ $ (Const("Not",_)$_) => th RS iff_reflection_F | _ => th RS iff_reflection_T; \end{ttbox} The three functions \texttt{gen_all}, \texttt{atomize} and \texttt{mk_eq} will be composed together and supplied below to \texttt{setmksimps}. \subsection{Making the initial simpset} It is time to assemble these items. We define the infix operator \ttindex{addcongs} to insert congruence rules; given a list of theorems, it converts their conclusions into meta-equalities and passes them to \ttindex{addeqcongs}. \begin{ttbox} infix 4 addcongs; fun ss addcongs congs = ss addeqcongs (congs RL [eq_reflection,iff_reflection]); \end{ttbox} The list \texttt{IFOL_simps} contains the default rewrite rules for intuitionistic first-order logic. The first of these is the reflexive law expressed as the equivalence $(a=a)\bimp\texttt{True}$; the rewrite rule $a=a$ is clearly useless. \begin{ttbox} val IFOL_simps = [refl RS P_iff_T] \at conj_simps \at disj_simps \at not_simps \at imp_simps \at iff_simps \at quant_simps; \end{ttbox} The list \texttt{triv_rls} contains trivial theorems for the solver. Any subgoal that is simplified to one of these will be removed. \begin{ttbox} val notFalseI = int_prove_fun "~False"; val triv_rls = [TrueI,refl,iff_refl,notFalseI]; \end{ttbox} % The basic simpset for intuitionistic \FOL{} is \ttindexbold{FOL_basic_ss}. It preprocess rewrites using {\tt gen_all}, \texttt{atomize} and \texttt{mk_eq}. It solves simplified subgoals using \texttt{triv_rls} and assumptions, and by detecting contradictions. It uses \ttindex{asm_simp_tac} to tackle subgoals of conditional rewrites. Other simpsets built from \texttt{FOL_basic_ss} will inherit these items. In particular, \ttindexbold{IFOL_ss}, which introduces {\tt IFOL_simps} as rewrite rules. \ttindexbold{FOL_ss} will later extend \texttt{IFOL_ss} with classical rewrite rules such as $\neg\neg P\bimp P$. \index{*setmksimps}\index{*setSSolver}\index{*setSolver}\index{*setsubgoaler} \index{*addsimps}\index{*addcongs} \begin{ttbox} fun unsafe_solver prems = FIRST'[resolve_tac (triv_rls {\at} prems), atac, etac FalseE]; fun safe_solver prems = FIRST'[match_tac (triv_rls {\at} prems), eq_assume_tac, ematch_tac [FalseE]]; val FOL_basic_ss = empty_ss setsubgoaler asm_simp_tac addsimprocs [defALL_regroup, defEX_regroup] setSSolver safe_solver setSolver unsafe_solver setmksimps (map mk_eq o atomize o gen_all); val IFOL_ss = FOL_basic_ss addsimps (IFOL_simps {\at} int_ex_simps {\at} int_all_simps) addcongs [imp_cong]; \end{ttbox} This simpset takes \texttt{imp_cong} as a congruence rule in order to use contextual information to simplify the conclusions of implications: \[ \List{\Var{P}\bimp\Var{P'};\; \Var{P'} \Imp \Var{Q}\bimp\Var{Q'}} \Imp (\Var{P}\imp\Var{Q}) \bimp (\Var{P'}\imp\Var{Q'}) \] By adding the congruence rule \texttt{conj_cong}, we could obtain a similar effect for conjunctions. \subsection{Splitter setup}\index{simplification!setting up the splitter} To set up case splitting, we have to call the \ML{} functor \ttindex{ SplitterFun}, which takes the argument signature \texttt{SPLITTER_DATA}. So we prove the theorem \texttt{meta_eq_to_iff} below and store it, together with the \texttt{mk_eq} function described above and several standard theorems, in the structure \texttt{SplitterData}. Calling the functor with this data yields a new instantiation of the splitter for our logic. \begin{ttbox} val meta_eq_to_iff = prove_goal IFOL.thy "x==y ==> x<->y" (fn [prem] => [rewtac prem, rtac iffI 1, atac 1, atac 1]); \ttbreak structure SplitterData = struct structure Simplifier = Simplifier val mk_eq = mk_eq val meta_eq_to_iff = meta_eq_to_iff val iffD = iffD2 val disjE = disjE val conjE = conjE val exE = exE val contrapos = contrapos val contrapos2 = contrapos2 val notnotD = notnotD end; \ttbreak structure Splitter = SplitterFun(SplitterData); \end{ttbox} \subsection{Theory setup}\index{simplification!setting up the theory} \begin{ttbox}\indexbold{*Simplifier.setup}\index{*setup!simplifier} Simplifier.setup: (theory -> theory) list \end{ttbox} Advanced theory related features of the simplifier (e.g.\ implicit simpset support) have to be set up explicitly. The simplifier already provides a suitable setup function definition. This has to be installed into the base theory of any new object-logic via a \texttt{setup} declaration. For example, this is done in \texttt{FOL/IFOL.thy} as follows: \begin{ttbox} setup Simplifier.setup \end{ttbox} \index{simplification|)} %%% Local Variables: %%% mode: latex %%% TeX-master: "ref" %%% End: