doc-src/Ref/document/substitution.tex

+
+\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: