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doc-src/Ref/substitution.tex

author | nipkow |

Wed, 04 Aug 2004 11:25:08 +0200 | |

changeset 15106 | e8cef6993701 |

parent 9695 | ec7d7f877712 |

child 20975 | 5bfa2e4ed789 |

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

aded comment

%% $Id$ \chapter{Substitution Tactics} \label{substitution} \index{tactics!substitution|(}\index{equality|(} Replacing equals by equals is a basic form of reasoning. Isabelle supports several kinds of equality reasoning. {\bf Substitution} means replacing free occurrences of~$t$ by~$u$ in a subgoal. This is easily done, given an equality $t=u$, provided the logic possesses the appropriate rule. The tactic \texttt{hyp_subst_tac} performs substitution even in the assumptions. But it works via object-level implication, and therefore must be specially set up for each suitable object-logic. Substitution should not be confused with object-level {\bf rewriting}. Given equalities of the form $t=u$, rewriting replaces instances of~$t$ by corresponding instances of~$u$, and continues until it reaches a normal form. Substitution handles `one-off' replacements by particular equalities while rewriting handles general equations. Chapter~\ref{chap:simplification} discusses Isabelle's rewriting tactics. \section{Substitution rules} \index{substitution!rules}\index{*subst theorem} Many logics include a substitution rule of the form $$ \List{\Var{a}=\Var{b}; \Var{P}(\Var{a})} \Imp \Var{P}(\Var{b}) \eqno(subst) $$ In backward proof, this may seem difficult to use: the conclusion $\Var{P}(\Var{b})$ admits far too many unifiers. But, if the theorem {\tt eqth} asserts $t=u$, then \hbox{\tt eqth RS subst} is the derived rule \[ \Var{P}(t) \Imp \Var{P}(u). \] Provided $u$ is not an unknown, resolution with this rule is well-behaved.\footnote{Unifying $\Var{P}(u)$ with a formula~$Q$ expresses~$Q$ in terms of its dependence upon~$u$. There are still $2^k$ unifiers, if $Q$ has $k$ occurrences of~$u$, but Isabelle ensures that the first unifier includes all the occurrences.} To replace $u$ by~$t$ in subgoal~$i$, use \begin{ttbox} resolve_tac [eqth RS subst] \(i\){\it.} \end{ttbox} To replace $t$ by~$u$ in subgoal~$i$, use \begin{ttbox} resolve_tac [eqth RS ssubst] \(i\){\it,} \end{ttbox} where \tdxbold{ssubst} is the `swapped' substitution rule $$ \List{\Var{a}=\Var{b}; \Var{P}(\Var{b})} \Imp \Var{P}(\Var{a}). \eqno(ssubst) $$ If \tdx{sym} denotes the symmetry rule \(\Var{a}=\Var{b}\Imp\Var{b}=\Var{a}\), then \texttt{ssubst} is just \hbox{\tt sym RS subst}. Many logics with equality include the rules {\tt subst} and \texttt{ssubst}, as well as \texttt{refl}, \texttt{sym} and \texttt{trans} (for the usual equality laws). Examples include \texttt{FOL} and \texttt{HOL}, but not \texttt{CTT} (Constructive Type Theory). Elim-resolution is well-behaved with assumptions of the form $t=u$. To replace $u$ by~$t$ or $t$ by~$u$ in subgoal~$i$, use \begin{ttbox} eresolve_tac [subst] \(i\) {\rm or} eresolve_tac [ssubst] \(i\){\it.} \end{ttbox} Logics HOL, FOL and ZF define the tactic \ttindexbold{stac} by \begin{ttbox} fun stac eqth = CHANGED o rtac (eqth RS ssubst); \end{ttbox} Now \texttt{stac~eqth} is like \texttt{resolve_tac [eqth RS ssubst]} but with the valuable property of failing if the substitution has no effect. \section{Substitution in the hypotheses} \index{assumptions!substitution in} Substitution rules, like other rules of natural deduction, do not affect the assumptions. This can be inconvenient. Consider proving the subgoal \[ \List{c=a; c=b} \Imp a=b. \] Calling \texttt{eresolve_tac\ts[ssubst]\ts\(i\)} simply discards the assumption~$c=a$, since $c$ does not occur in~$a=b$. Of course, we can work out a solution. First apply \texttt{eresolve_tac\ts[subst]\ts\(i\)}, replacing~$a$ by~$c$: \[ c=b \Imp c=b \] Equality reasoning can be difficult, but this trivial proof requires nothing more sophisticated than substitution in the assumptions. Object-logics that include the rule~$(subst)$ provide tactics for this purpose: \begin{ttbox} hyp_subst_tac : int -> tactic bound_hyp_subst_tac : int -> tactic \end{ttbox} \begin{ttdescription} \item[\ttindexbold{hyp_subst_tac} {\it i}] selects an equality assumption of the form $t=u$ or $u=t$, where $t$ is a free variable or parameter. Deleting this assumption, it replaces $t$ by~$u$ throughout subgoal~$i$, including the other assumptions. \item[\ttindexbold{bound_hyp_subst_tac} {\it i}] is similar but only substitutes for parameters (bound variables). Uses for this are discussed below. \end{ttdescription} The term being replaced must be a free variable or parameter. Substitution for constants is usually unhelpful, since they may appear in other theorems. For instance, the best way to use the assumption $0=1$ is to contradict a theorem that states $0\not=1$, rather than to replace 0 by~1 in the subgoal! Substitution for unknowns, such as $\Var{x}=0$, is a bad idea: we might prove the subgoal more easily by instantiating~$\Var{x}$ to~1. Substitution for free variables is unhelpful if they appear in the premises of a rule being derived: the substitution affects object-level assumptions, not meta-level assumptions. For instance, replacing~$a$ by~$b$ could make the premise~$P(a)$ worthless. To avoid this problem, use \texttt{bound_hyp_subst_tac}; alternatively, call \ttindex{cut_facts_tac} to insert the atomic premises as object-level assumptions. \section{Setting up the package} Many Isabelle object-logics, such as \texttt{FOL}, \texttt{HOL} and their descendants, come with \texttt{hyp_subst_tac} already defined. A few others, such as \texttt{CTT}, do not support this tactic because they lack the rule~$(subst)$. When defining a new logic that includes a substitution rule and implication, you must set up \texttt{hyp_subst_tac} yourself. It is packaged as the \ML{} functor \ttindex{HypsubstFun}, which takes the argument signature~\texttt{HYPSUBST_DATA}: \begin{ttbox} signature HYPSUBST_DATA = sig structure Simplifier : SIMPLIFIER val dest_Trueprop : term -> term val dest_eq : term -> term*term*typ val dest_imp : term -> term*term val eq_reflection : thm (* a=b ==> a==b *) val rev_eq_reflection: thm (* a==b ==> a=b *) val imp_intr : thm (*(P ==> Q) ==> P-->Q *) val rev_mp : thm (* [| P; P-->Q |] ==> Q *) val subst : thm (* [| a=b; P(a) |] ==> P(b) *) val sym : thm (* a=b ==> b=a *) val thin_refl : thm (* [|x=x; P|] ==> P *) end; \end{ttbox} Thus, the functor requires the following items: \begin{ttdescription} \item[Simplifier] should be an instance of the simplifier (see Chapter~\ref{chap:simplification}). \item[\ttindexbold{dest_Trueprop}] should coerce a meta-level formula to the corresponding object-level one. Typically, it should return $P$ when applied to the term $\texttt{Trueprop}\,P$ (see example below). \item[\ttindexbold{dest_eq}] should return the triple~$(t,u,T)$, where $T$ is the type of~$t$ and~$u$, when applied to the \ML{} term that represents~$t=u$. For other terms, it should raise an exception. \item[\ttindexbold{dest_imp}] should return the pair~$(P,Q)$ when applied to the \ML{} term that represents the implication $P\imp Q$. For other terms, it should raise an exception. \item[\tdxbold{eq_reflection}] is the theorem discussed in~\S\ref{sec:setting-up-simp}. \item[\tdxbold{rev_eq_reflection}] is the reverse of \texttt{eq_reflection}. \item[\tdxbold{imp_intr}] should be the implies introduction rule $(\Var{P}\Imp\Var{Q})\Imp \Var{P}\imp\Var{Q}$. \item[\tdxbold{rev_mp}] should be the `reversed' implies elimination rule $\List{\Var{P}; \;\Var{P}\imp\Var{Q}} \Imp \Var{Q}$. \item[\tdxbold{subst}] should be the substitution rule $\List{\Var{a}=\Var{b};\; \Var{P}(\Var{a})} \Imp \Var{P}(\Var{b})$. \item[\tdxbold{sym}] should be the symmetry rule $\Var{a}=\Var{b}\Imp\Var{b}=\Var{a}$. \item[\tdxbold{thin_refl}] should be the rule $\List{\Var{a}=\Var{a};\; \Var{P}} \Imp \Var{P}$, which is used to erase trivial equalities. \end{ttdescription} % The functor resides in file \texttt{Provers/hypsubst.ML} in the Isabelle distribution directory. It is not sensitive to the precise formalization of the object-logic. It is not concerned with the names of the equality and implication symbols, or the types of formula and terms. Coding the functions \texttt{dest_Trueprop}, \texttt{dest_eq} and \texttt{dest_imp} requires knowledge of Isabelle's representation of terms. For \texttt{FOL}, they are declared by \begin{ttbox} fun dest_Trueprop (Const ("Trueprop", _) $ P) = P | dest_Trueprop t = raise TERM ("dest_Trueprop", [t]); fun dest_eq (Const("op =",T) $ t $ u) = (t, u, domain_type T) fun dest_imp (Const("op -->",_) $ A $ B) = (A, B) | dest_imp t = raise TERM ("dest_imp", [t]); \end{ttbox} Recall that \texttt{Trueprop} is the coercion from type~$o$ to type~$prop$, while \hbox{\tt op =} is the internal name of the infix operator~\texttt{=}. Function \ttindexbold{domain_type}, given the function type $S\To T$, returns the type~$S$. Pattern-matching expresses the function concisely, using wildcards~({\tt_}) for the types. The tactic \texttt{hyp_subst_tac} works as follows. First, it identifies a suitable equality assumption, possibly re-orienting it using~\texttt{sym}. Then it moves other assumptions into the conclusion of the goal, by repeatedly calling \texttt{etac~rev_mp}. Then, it uses \texttt{asm_full_simp_tac} or \texttt{ssubst} to substitute throughout the subgoal. (If the equality involves unknowns then it must use \texttt{ssubst}.) Then, it deletes the equality. Finally, it moves the assumptions back to their original positions by calling \hbox{\tt resolve_tac\ts[imp_intr]}. \index{equality|)}\index{tactics!substitution|)} %%% Local Variables: %%% mode: latex %%% TeX-master: "ref" %%% End: