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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]}.