2 \chapter{Substitution Tactics} \label{substitution}
3 \index{tactics!substitution|(}\index{equality|(}
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.
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.
21 \section{Substitution rules}
22 \index{substitution!rules}\index{*subst theorem}
23 Many logics include a substitution rule of the form
25 \List{\Var{a}=\Var{b}; \Var{P}(\Var{a})} \Imp
26 \Var{P}(\Var{b}) \eqno(subst)
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
39 resolve_tac [eqth RS subst] \(i\){\it.}
41 To replace $t$ by~$u$ in
44 resolve_tac [eqth RS ssubst] \(i\){\it,}
46 where \tdxbold{ssubst} is the `swapped' substitution rule
48 \List{\Var{a}=\Var{b}; \Var{P}(\Var{b})} \Imp
49 \Var{P}(\Var{a}). \eqno(ssubst)
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).
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
61 eresolve_tac [subst] \(i\) {\rm or} eresolve_tac [ssubst] \(i\){\it.}
64 Logics HOL, FOL and ZF define the tactic \ttindexbold{stac} by
66 fun stac eqth = CHANGED o rtac (eqth RS ssubst);
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.
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\(i\)},
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
87 hyp_subst_tac : int -> tactic
88 bound_hyp_subst_tac : int -> tactic
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.
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.
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
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.
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}:
125 signature HYPSUBST_DATA =
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 *)
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}).
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).
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.
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.
157 \item[\tdxbold{eq_reflection}] is the theorem discussed
158 in~\S\ref{sec:setting-up-simp}.
160 \item[\tdxbold{rev_eq_reflection}] is the reverse of \texttt{eq_reflection}.
162 \item[\tdxbold{imp_intr}] should be the implies introduction
163 rule $(\Var{P}\Imp\Var{Q})\Imp \Var{P}\imp\Var{Q}$.
165 \item[\tdxbold{rev_mp}] should be the `reversed' implies elimination
166 rule $\List{\Var{P}; \;\Var{P}\imp\Var{Q}} \Imp \Var{Q}$.
168 \item[\tdxbold{subst}] should be the substitution rule
169 $\List{\Var{a}=\Var{b};\; \Var{P}(\Var{a})} \Imp \Var{P}(\Var{b})$.
171 \item[\tdxbold{sym}] should be the symmetry rule
172 $\Var{a}=\Var{b}\Imp\Var{b}=\Var{a}$.
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
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.
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
188 fun dest_Trueprop (Const ("Trueprop", _) $ P) = P
189 | dest_Trueprop t = raise TERM ("dest_Trueprop", [t]);
191 fun dest_eq (Const("op =",T) $ t $ u) = ((t, u), domain_type T)
193 fun dest_imp (Const("op -->",_) $ A $ B) = (A, B)
194 | dest_imp t = raise TERM ("dest_imp", [t]);
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.
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]}.
211 \index{equality|)}\index{tactics!substitution|)}
216 %%% TeX-master: "ref"