author paulson Thu, 05 Feb 1998 10:26:16 +0100 changeset 4596 0c32a609fcad parent 4595 fa8cee619732 child 4597 a0bdee64194c
Updated the description of how to set up hyp_subst_tac
--- a/doc-src/Ref/substitution.tex	Mon Feb 02 12:57:20 1998 +0100
+++ b/doc-src/Ref/substitution.tex	Thu Feb 05 10:26:16 1998 +0100
@@ -6,7 +6,7 @@
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 {\tt hyp_subst_tac} performs substitution even in the assumptions.
+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.

@@ -49,11 +49,11 @@
\Var{P}(\Var{a}).  \eqno(ssubst)

If \tdx{sym} denotes the symmetry rule
-$$\Var{a}=\Var{b}\Imp\Var{b}=\Var{a}$$, then {\tt ssubst} is just
+$$\Var{a}=\Var{b}\Imp\Var{b}=\Var{a}$$, then \texttt{ssubst} is just
\hbox{\tt sym RS subst}.  Many logics with equality include the rules {\tt
-subst} and {\tt ssubst}, as well as {\tt refl}, {\tt sym} and {\tt trans}
-(for the usual equality laws).  Examples include {\tt FOL} and {\tt HOL},
-but not {\tt CTT} (Constructive Type Theory).
+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
@@ -65,7 +65,7 @@
\begin{ttbox}
fun stac eqth = CHANGED o rtac (eqth RS ssubst);
\end{ttbox}
-Now {\tt stac~eqth} is like {\tt resolve_tac [eqth RS ssubst]} but with the
+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.

@@ -74,9 +74,9 @@
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 {\tt eresolve_tac\ts[ssubst]\ts$$i$$} simply discards the
+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 {\tt eresolve_tac\ts[subst]\ts$$i$$},
+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
@@ -109,38 +109,49 @@
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
-{\tt bound_hyp_subst_tac}; alternatively, call \ttindex{cut_facts_tac} to
+\texttt{bound_hyp_subst_tac}; alternatively, call \ttindex{cut_facts_tac} to
insert the atomic premises as object-level assumptions.

\section{Setting up {\tt hyp_subst_tac}}
-Many Isabelle object-logics, such as {\tt FOL}, {\tt HOL} and their
-descendants, come with {\tt hyp_subst_tac} already defined.  A few others,
-such as {\tt CTT}, do not support this tactic because they lack the
+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 {\tt hyp_subst_tac} yourself.  It
+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~{\tt HYPSUBST_DATA}:
+argument signature~\texttt{HYPSUBST_DATA}:
\begin{ttbox}
signature HYPSUBST_DATA =
sig
structure Simplifier : SIMPLIFIER
-  val dest_eq          : term -> term*term
+  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 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_eq}] should return the pair~$(t,u)$ when
-applied to the \ML{} term that represents~$t=u$.  For other terms, it
-should raise exception~\xdx{Match}.
+
+\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}.
@@ -156,28 +167,41 @@

\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 {\tt Provers/hypsubst.ML} in the Isabelle
+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
-function {\tt dest_eq} requires knowledge of Isabelle's representation of
-terms.  For {\tt FOL} it is defined by
+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_eq (Const("Trueprop",_) $(Const("op =",_)$t$u)) = (t,u) +fun dest_Trueprop (Const ("Trueprop", _)$ 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}
-Here {\tt Trueprop} is the coercion from type~$o$ to type~$prop$, while
-\hbox{\tt op =} is the internal name of the infix operator~{\tt=}.
-Pattern-matching expresses the function concisely, using wildcards~({\tt_})
-for the types.
+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 {\tt hyp_subst_tac} works as follows.  First, it identifies a
-suitable equality assumption, possibly re-orienting it using~{\tt sym}.  Then
-it moves other assumptions into the conclusion of the goal, by repeatedly
-caling {\tt eresolve_tac\ts[rev_mp]}.  Then, it uses {\tt asm_full_simp_tac}
-or {\tt ssubst} to substitute throughout the subgoal.  (If the equality
-involves unknowns then it must use {\tt ssubst}.)  Then, it deletes the
+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]}.