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     1 %% $Id$
     2 \chapter{Substitution Tactics} \label{substitution}
     3 \index{substitution|(}
     4 Replacing equals by equals is a basic form of reasoning.  Isabelle supports
     5 several kinds of equality reasoning.  {\bf Substitution} means to replace
     6 free occurrences of~$t$ by~$u$ in a subgoal.  This is easily done, given an
     7 equality $t=u$, provided the logic possesses the appropriate rule ---
     8 unless you want to substitute even in the assumptions.  The tactic
     9 \ttindex{hyp_subst_tac} performs substitution in the assumptions, but it
    10 works via object-level implication, and therefore must be specially set up
    11 for each suitable object-logic.
    13 Substitution should not be confused with object-level {\bf rewriting}.
    14 Given equalities of the form $t=u$, rewriting replaces instances of~$t$ by
    15 corresponding instances of~$u$, and continues until it reaches a normal
    16 form.  Substitution handles `one-off' replacements by particular
    17 equalities, while rewriting handles general equalities.
    18 Chapter~\ref{simp-chap} discusses Isabelle's rewriting tactics.
    21 \section{Simple substitution}
    22 \index{substitution!simple}
    23 Many logics include a substitution rule of the form\indexbold{*subst}
    24 $$ \List{\Var{a}=\Var{b}; \Var{P}(\Var{a})} \Imp 
    25    \Var{P}(\Var{b})  \eqno(subst)$$
    26 In backward proof, this may seem difficult to use: the conclusion
    27 $\Var{P}(\Var{b})$ admits far too many unifiers.  But, if the theorem {\tt
    28 eqth} asserts $t=u$, then \hbox{\tt eqth RS subst} is the derived rule
    29 \[ \Var{P}(t) \Imp \Var{P}(u). \]
    30 Provided $u$ is not an unknown, resolution with this rule is
    31 well-behaved.\footnote{Unifying $\Var{P}(u)$ with a formula~$Q$
    32 expresses~$Q$ in terms of its dependence upon~$u$.  There are still $2^k$
    33 unifiers, if $Q$ has $k$ occurrences of~$u$, but Isabelle ensures that
    34 the first unifier includes all the occurrences.}  To replace $u$ by~$t$ in
    35 subgoal~$i$, use
    36 \begin{ttbox} 
    37 resolve_tac [eqth RS subst] \(i\) {\it.}
    38 \end{ttbox}
    39 To replace $t$ by~$u$ in
    40 subgoal~$i$, use
    41 \begin{ttbox} 
    42 resolve_tac [eqth RS ssubst] \(i\) {\it,}
    43 \end{ttbox}
    44 where \ttindexbold{ssubst} is the `swapped' substitution rule
    45 $$ \List{\Var{a}=\Var{b}; \Var{P}(\Var{b})} \Imp 
    46    \Var{P}(\Var{a}).  \eqno(ssubst)$$
    47 If \ttindex{sym} denotes the symmetry rule
    48 \(\Var{a}=\Var{b}\Imp\Var{b}=\Var{a}\), then {\tt ssubst} is just
    49 \hbox{\tt sym RS subst}.  Many logics with equality include the rules {\tt
    50 subst} and {\tt ssubst}, as well as {\tt refl}, {\tt sym} and {\tt trans}
    51 (for the usual equality laws).
    53 Elim-resolution is well-behaved with assumptions of the form $t=u$.
    54 To replace $u$ by~$t$ or $t$ by~$u$ in subgoal~$i$, use
    55 \begin{ttbox} 
    56 eresolve_tac [subst] \(i\)    {\it or}    eresolve_tac [ssubst] \(i\) {\it.}
    57 \end{ttbox}
    60 \section{Substitution in the hypotheses}
    61 \index{substitution!in hypotheses}
    62 Substitution rules, like other rules of natural deduction, do not affect
    63 the assumptions.  This can be inconvenient.  Consider proving the subgoal
    64 \[ \List{c=a; c=b} \Imp a=b. \]
    65 Calling \hbox{\tt eresolve_tac [ssubst] \(i\)} simply discards the
    66 assumption~$c=a$, since $c$ does not occur in~$a=b$.  Of course, we can
    67 work out a solution.  First apply \hbox{\tt eresolve_tac [subst] \(i\)},
    68 replacing~$a$ by~$c$:
    69 \[ \List{c=b} \Imp c=b \]
    70 Equality reasoning can be difficult, but this trivial proof requires
    71 nothing more sophisticated than substitution in the assumptions.
    72 Object-logics that include the rule~$(subst)$ provide a tactic for this
    73 purpose:
    74 \begin{ttbox} 
    75 hyp_subst_tac : int -> tactic
    76 \end{ttbox}
    77 \begin{description}
    78 \item[\ttindexbold{hyp_subst_tac} {\it i}] 
    79 selects an equality assumption of the form $t=u$ or $u=t$, where $t$ is a
    80 free variable or parameter.  Deleting this assumption, it replaces $t$
    81 by~$u$ throughout subgoal~$i$, including the other assumptions.
    82 \end{description}
    83 The term being replaced must be a free variable or parameter.  Substitution
    84 for constants is usually unhelpful, since they may appear in other
    85 theorems.  For instance, the best way to use the assumption $0=1$ is to
    86 contradict a theorem that states $0\not=1$, rather than to replace 0 by~1
    87 in the subgoal!
    89 Replacing a free variable causes similar problems if they appear in the
    90 premises of a rule being derived --- the substitution affects object-level
    91 assumptions, not meta-level assumptions.  For instance, replacing~$a$
    92 by~$b$ could make the premise~$P(a)$ worthless.  To avoid this problem, call
    93 \ttindex{cut_facts_tac} to insert the atomic premises as object-level
    94 assumptions.
    97 \section{Setting up {\tt hyp_subst_tac}} 
    98 Many Isabelle object-logics, such as {\tt FOL}, {\tt HOL} and their
    99 descendants, come with {\tt hyp_subst_tac} already defined.  A few others,
   100 such as {\tt CTT}, do not support this tactic because they lack the
   101 rule~$(subst)$.  When defining a new logic that includes a substitution
   102 rule and implication, you must set up {\tt hyp_subst_tac} yourself.  It
   103 is packaged as the \ML{} functor \ttindex{HypsubstFun}, which takes the
   104 argument signature~\ttindexbold{HYPSUBST_DATA}:
   105 \begin{ttbox} 
   106 signature HYPSUBST_DATA =
   107   sig
   108   val subst      : thm
   109   val sym        : thm
   110   val rev_cut_eq : thm
   111   val imp_intr   : thm
   112   val rev_mp     : thm
   113   val dest_eq    : term -> term*term
   114   end;
   115 \end{ttbox}
   116 Thus, the functor requires the following items:
   117 \begin{description}
   118 \item[\ttindexbold{subst}] should be the substitution rule
   119 $\List{\Var{a}=\Var{b};\; \Var{P}(\Var{a})} \Imp \Var{P}(\Var{b})$.
   121 \item[\ttindexbold{sym}] should be the symmetry rule
   122 $\Var{a}=\Var{b}\Imp\Var{b}=\Var{a}$.
   124 \item[\ttindexbold{rev_cut_eq}] should have the form
   125 $\List{\Var{a}=\Var{b};\; \Var{a}=\Var{b}\Imp\Var{R}} \Imp \Var{R}$.
   127 \item[\ttindexbold{imp_intr}] should be the implies introduction
   128 rule $(\Var{P}\Imp\Var{Q})\Imp \Var{P}\imp\Var{Q}$.
   130 \item[\ttindexbold{rev_mp}] should be the ``reversed'' implies elimination
   131 rule $\List{\Var{P};  \;\Var{P}\imp\Var{Q}} \Imp \Var{Q}$.
   133 \item[\ttindexbold{dest_eq}] should return the pair~$(t,u)$ when
   134 applied to the \ML{} term that represents~$t=u$.  For other terms, it
   135 should raise exception~\ttindex{Match}.
   136 \end{description}
   137 The functor resides in {\tt Provers/hypsubst.ML} on the Isabelle
   138 distribution directory.  It is not sensitive to the precise formalization
   139 of the object-logic.  It is not concerned with the names of the equality
   140 and implication symbols, or the types of formula and terms.  Coding the
   141 function {\tt dest_eq} requires knowledge of Isabelle's representation of
   142 terms.  For {\tt FOL} it is defined by
   143 \begin{ttbox} 
   144 fun dest_eq (Const("Trueprop",_) $ (Const("op =",_)$t$u)) = (t,u);
   145 \end{ttbox}
   146 Here {\tt Trueprop} is the coercion from type'~$o$ to type~$prop$, while
   147 \hbox{\tt op =} is the internal name of the infix operator~{\tt=}.
   148 Pattern-matching expresses the function concisely, using wildcards~({\tt_})
   149 to hide the types.
   151 Given a subgoal of the form
   152 \[ \List{P@1; \cdots ; t=u; \cdots ; P@n} \Imp Q, \]
   153 \ttindexbold{hyp_subst_tac} locates a suitable equality
   154 assumption and moves it to the last position using elim-resolution on {\tt
   155 rev_cut_eq} (possibly re-orienting it using~{\tt sym}):
   156 \[ \List{P@1; \cdots ; P@n; t=u} \Imp Q \]
   157 Using $n$ calls of \hbox{\tt eresolve_tac [rev_mp]}, it creates the subgoal
   158 \[ \List{t=u} \Imp P@1\imp \cdots \imp P@n \imp Q \]
   159 By \hbox{\tt eresolve_tac [ssubst]}, it replaces~$t$ by~$u$ throughout:
   160 \[ P'@1\imp \cdots \imp P'@n \imp Q' \]
   161 Finally, using $n$ calls of \hbox{\tt resolve_tac [imp_intr]}, it restores
   162 $P'@1$, \ldots, $P'@n$ as assumptions:
   163 \[ \List{P'@n; \cdots ; P'@1} \Imp Q' \]
   165 \index{substitution|)}