doc-src/Ref/substitution.tex
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     1 %% $Id$
     2 \chapter{Substitution Tactics} \label{substitution}
     3 \index{tactics!substitution|(}\index{equality|(}
     4 
     5 Replacing equals by equals is a basic form of reasoning.  Isabelle supports
     6 several kinds of equality reasoning.  {\bf Substitution} means replacing
     7 free occurrences of~$t$ by~$u$ in a subgoal.  This is easily done, given an
     8 equality $t=u$, provided the logic possesses the appropriate rule.  The
     9 tactic \texttt{hyp_subst_tac} performs substitution even in the assumptions.
    10 But it works via object-level implication, and therefore must be specially
    11 set up for each suitable object-logic.
    12 
    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 equations.
    18 Chapter~\ref{chap:simplification} discusses Isabelle's rewriting tactics.
    19 
    20 
    21 \section{Substitution rules}
    22 \index{substitution!rules}\index{*subst theorem}
    23 Many logics include a substitution rule of the form
    24 $$
    25 \List{\Var{a}=\Var{b}; \Var{P}(\Var{a})} \Imp 
    26 \Var{P}(\Var{b})  \eqno(subst)
    27 $$
    28 In backward proof, this may seem difficult to use: the conclusion
    29 $\Var{P}(\Var{b})$ admits far too many unifiers.  But, if the theorem {\tt
    30 eqth} asserts $t=u$, then \hbox{\tt eqth RS subst} is the derived rule
    31 \[ \Var{P}(t) \Imp \Var{P}(u). \]
    32 Provided $u$ is not an unknown, resolution with this rule is
    33 well-behaved.\footnote{Unifying $\Var{P}(u)$ with a formula~$Q$
    34 expresses~$Q$ in terms of its dependence upon~$u$.  There are still $2^k$
    35 unifiers, if $Q$ has $k$ occurrences of~$u$, but Isabelle ensures that
    36 the first unifier includes all the occurrences.}  To replace $u$ by~$t$ in
    37 subgoal~$i$, use
    38 \begin{ttbox} 
    39 resolve_tac [eqth RS subst] \(i\){\it.}
    40 \end{ttbox}
    41 To replace $t$ by~$u$ in
    42 subgoal~$i$, use
    43 \begin{ttbox} 
    44 resolve_tac [eqth RS ssubst] \(i\){\it,}
    45 \end{ttbox}
    46 where \tdxbold{ssubst} is the `swapped' substitution rule
    47 $$
    48 \List{\Var{a}=\Var{b}; \Var{P}(\Var{b})} \Imp 
    49 \Var{P}(\Var{a}).  \eqno(ssubst)
    50 $$
    51 If \tdx{sym} denotes the symmetry rule
    52 \(\Var{a}=\Var{b}\Imp\Var{b}=\Var{a}\), then \texttt{ssubst} is just
    53 \hbox{\tt sym RS subst}.  Many logics with equality include the rules {\tt
    54 subst} and \texttt{ssubst}, as well as \texttt{refl}, \texttt{sym} and \texttt{trans}
    55 (for the usual equality laws).  Examples include \texttt{FOL} and \texttt{HOL},
    56 but not \texttt{CTT} (Constructive Type Theory).
    57 
    58 Elim-resolution is well-behaved with assumptions of the form $t=u$.
    59 To replace $u$ by~$t$ or $t$ by~$u$ in subgoal~$i$, use
    60 \begin{ttbox} 
    61 eresolve_tac [subst] \(i\)    {\rm or}    eresolve_tac [ssubst] \(i\){\it.}
    62 \end{ttbox}
    63 
    64 Logics HOL, FOL and ZF define the tactic \ttindexbold{stac} by
    65 \begin{ttbox} 
    66 fun stac eqth = CHANGED o rtac (eqth RS ssubst);
    67 \end{ttbox}
    68 Now \texttt{stac~eqth} is like \texttt{resolve_tac [eqth RS ssubst]} but with the
    69 valuable property of failing if the substitution has no effect.
    70 
    71 
    72 \section{Substitution in the hypotheses}
    73 \index{assumptions!substitution in}
    74 Substitution rules, like other rules of natural deduction, do not affect
    75 the assumptions.  This can be inconvenient.  Consider proving the subgoal
    76 \[ \List{c=a; c=b} \Imp a=b. \]
    77 Calling \texttt{eresolve_tac\ts[ssubst]\ts\(i\)} simply discards the
    78 assumption~$c=a$, since $c$ does not occur in~$a=b$.  Of course, we can
    79 work out a solution.  First apply \texttt{eresolve_tac\ts[subst]\ts\(i\)},
    80 replacing~$a$ by~$c$:
    81 \[ c=b \Imp c=b \]
    82 Equality reasoning can be difficult, but this trivial proof requires
    83 nothing more sophisticated than substitution in the assumptions.
    84 Object-logics that include the rule~$(subst)$ provide tactics for this
    85 purpose:
    86 \begin{ttbox} 
    87 hyp_subst_tac       : int -> tactic
    88 bound_hyp_subst_tac : int -> tactic
    89 \end{ttbox}
    90 \begin{ttdescription}
    91 \item[\ttindexbold{hyp_subst_tac} {\it i}] 
    92   selects an equality assumption of the form $t=u$ or $u=t$, where $t$ is a
    93   free variable or parameter.  Deleting this assumption, it replaces $t$
    94   by~$u$ throughout subgoal~$i$, including the other assumptions.
    95 
    96 \item[\ttindexbold{bound_hyp_subst_tac} {\it i}] 
    97   is similar but only substitutes for parameters (bound variables).
    98   Uses for this are discussed below.
    99 \end{ttdescription}
   100 The term being replaced must be a free variable or parameter.  Substitution
   101 for constants is usually unhelpful, since they may appear in other
   102 theorems.  For instance, the best way to use the assumption $0=1$ is to
   103 contradict a theorem that states $0\not=1$, rather than to replace 0 by~1
   104 in the subgoal!
   105 
   106 Substitution for unknowns, such as $\Var{x}=0$, is a bad idea: we might prove
   107 the subgoal more easily by instantiating~$\Var{x}$ to~1.
   108 Substitution for free variables is unhelpful if they appear in the
   109 premises of a rule being derived: the substitution affects object-level
   110 assumptions, not meta-level assumptions.  For instance, replacing~$a$
   111 by~$b$ could make the premise~$P(a)$ worthless.  To avoid this problem, use
   112 \texttt{bound_hyp_subst_tac}; alternatively, call \ttindex{cut_facts_tac} to
   113 insert the atomic premises as object-level assumptions.
   114 
   115 
   116 \section{Setting up the package} 
   117 Many Isabelle object-logics, such as \texttt{FOL}, \texttt{HOL} and their
   118 descendants, come with \texttt{hyp_subst_tac} already defined.  A few others,
   119 such as \texttt{CTT}, do not support this tactic because they lack the
   120 rule~$(subst)$.  When defining a new logic that includes a substitution
   121 rule and implication, you must set up \texttt{hyp_subst_tac} yourself.  It
   122 is packaged as the \ML{} functor \ttindex{HypsubstFun}, which takes the
   123 argument signature~\texttt{HYPSUBST_DATA}:
   124 \begin{ttbox} 
   125 signature HYPSUBST_DATA =
   126   sig
   127   structure Simplifier : SIMPLIFIER
   128   val dest_Trueprop    : term -> term
   129   val dest_eq          : term -> term*term*typ
   130   val dest_imp         : term -> term*term
   131   val eq_reflection    : thm         (* a=b ==> a==b *)
   132   val rev_eq_reflection: thm         (* a==b ==> a=b *)
   133   val imp_intr         : thm         (*(P ==> Q) ==> P-->Q *)
   134   val rev_mp           : thm         (* [| P;  P-->Q |] ==> Q *)
   135   val subst            : thm         (* [| a=b;  P(a) |] ==> P(b) *)
   136   val sym              : thm         (* a=b ==> b=a *)
   137   val thin_refl        : thm         (* [|x=x; P|] ==> P *)
   138   end;
   139 \end{ttbox}
   140 Thus, the functor requires the following items:
   141 \begin{ttdescription}
   142 \item[Simplifier] should be an instance of the simplifier (see
   143   Chapter~\ref{chap:simplification}).
   144   
   145 \item[\ttindexbold{dest_Trueprop}] should coerce a meta-level formula to the
   146   corresponding object-level one.  Typically, it should return $P$ when
   147   applied to the term $\texttt{Trueprop}\,P$ (see example below).
   148   
   149 \item[\ttindexbold{dest_eq}] should return the triple~$(t,u,T)$, where $T$ is
   150   the type of~$t$ and~$u$, when applied to the \ML{} term that
   151   represents~$t=u$.  For other terms, it should raise an exception.
   152   
   153 \item[\ttindexbold{dest_imp}] should return the pair~$(P,Q)$ when applied to
   154   the \ML{} term that represents the implication $P\imp Q$.  For other terms,
   155   it should raise an exception.
   156 
   157 \item[\tdxbold{eq_reflection}] is the theorem discussed
   158   in~\S\ref{sec:setting-up-simp}.
   159   
   160 \item[\tdxbold{rev_eq_reflection}] is the reverse of \texttt{eq_reflection}.
   161 
   162 \item[\tdxbold{imp_intr}] should be the implies introduction
   163 rule $(\Var{P}\Imp\Var{Q})\Imp \Var{P}\imp\Var{Q}$.
   164 
   165 \item[\tdxbold{rev_mp}] should be the `reversed' implies elimination
   166 rule $\List{\Var{P};  \;\Var{P}\imp\Var{Q}} \Imp \Var{Q}$.
   167 
   168 \item[\tdxbold{subst}] should be the substitution rule
   169 $\List{\Var{a}=\Var{b};\; \Var{P}(\Var{a})} \Imp \Var{P}(\Var{b})$.
   170 
   171 \item[\tdxbold{sym}] should be the symmetry rule
   172 $\Var{a}=\Var{b}\Imp\Var{b}=\Var{a}$.
   173 
   174 \item[\tdxbold{thin_refl}] should be the rule
   175 $\List{\Var{a}=\Var{a};\; \Var{P}} \Imp \Var{P}$, which is used to erase
   176 trivial equalities.
   177 \end{ttdescription}
   178 %
   179 The functor resides in file \texttt{Provers/hypsubst.ML} in the Isabelle
   180 distribution directory.  It is not sensitive to the precise formalization
   181 of the object-logic.  It is not concerned with the names of the equality
   182 and implication symbols, or the types of formula and terms.  
   183 
   184 Coding the functions \texttt{dest_Trueprop}, \texttt{dest_eq} and
   185 \texttt{dest_imp} requires knowledge of Isabelle's representation of terms.
   186 For \texttt{FOL}, they are declared by
   187 \begin{ttbox} 
   188 fun dest_Trueprop (Const ("Trueprop", _) $ P) = P
   189   | dest_Trueprop t = raise TERM ("dest_Trueprop", [t]);
   190 
   191 fun dest_eq (Const("op =",T)  $ t $ u) = (t, u, domain_type T)
   192 
   193 fun dest_imp (Const("op -->",_) $ A $ B) = (A, B)
   194   | dest_imp  t = raise TERM ("dest_imp", [t]);
   195 \end{ttbox}
   196 Recall that \texttt{Trueprop} is the coercion from type~$o$ to type~$prop$,
   197 while \hbox{\tt op =} is the internal name of the infix operator~\texttt{=}.
   198 Function \ttindexbold{domain_type}, given the function type $S\To T$, returns
   199 the type~$S$.  Pattern-matching expresses the function concisely, using
   200 wildcards~({\tt_}) for the types.
   201 
   202 The tactic \texttt{hyp_subst_tac} works as follows.  First, it identifies a
   203 suitable equality assumption, possibly re-orienting it using~\texttt{sym}.
   204 Then it moves other assumptions into the conclusion of the goal, by repeatedly
   205 calling \texttt{etac~rev_mp}.  Then, it uses \texttt{asm_full_simp_tac} or
   206 \texttt{ssubst} to substitute throughout the subgoal.  (If the equality
   207 involves unknowns then it must use \texttt{ssubst}.)  Then, it deletes the
   208 equality.  Finally, it moves the assumptions back to their original positions
   209 by calling \hbox{\tt resolve_tac\ts[imp_intr]}.
   210 
   211 \index{equality|)}\index{tactics!substitution|)}
   212 
   213 
   214 %%% Local Variables: 
   215 %%% mode: latex
   216 %%% TeX-master: "ref"
   217 %%% End: