%% $Id$
\chapter{Tactics} \label{tactics}
\index{tactics|(}
Tactics have type \ttindexbold{tactic}. They are essentially
functions from theorems to theorem sequences, where the theorems represent
states of a backward proof. Tactics seldom need to be coded from scratch,
as functions; the basic tactics suffice for most proofs.
\section{Resolution and assumption tactics}
{\bf Resolution} is Isabelle's basic mechanism for refining a subgoal using
a rule. {\bf Elim-resolution} is particularly suited for elimination
rules, while {\bf destruct-resolution} is particularly suited for
destruction rules. The {\tt r}, {\tt e}, {\tt d} naming convention is
maintained for several different kinds of resolution tactics, as well as
the shortcuts in the subgoal module.
All the tactics in this section act on a subgoal designated by a positive
integer~$i$. They fail (by returning the empty sequence) if~$i$ is out of
range.
\subsection{Resolution tactics}
\index{tactics!resolution|bold}
\begin{ttbox}
resolve_tac : thm list -> int -> tactic
eresolve_tac : thm list -> int -> tactic
dresolve_tac : thm list -> int -> tactic
forward_tac : thm list -> int -> tactic
\end{ttbox}
These perform resolution on a list of theorems, $thms$, representing a list
of object-rules. When generating next states, they take each of the rules
in the order given. Each rule may yield several next states, or none:
higher-order resolution may yield multiple resolvents.
\begin{description}
\item[\ttindexbold{resolve_tac} {\it thms} {\it i}]
refines the proof state using the object-rules, which should normally be
introduction rules. It resolves an object-rule's conclusion with
subgoal~$i$ of the proof state.
\item[\ttindexbold{eresolve_tac} {\it thms} {\it i}]
refines the proof state by elim-resolution with the object-rules, which
should normally be elimination rules. It resolves with a rule, solves
its first premise by assumption, and finally {\bf deletes} that assumption
from any new subgoals.
\item[\ttindexbold{dresolve_tac} {\it thms} {\it i}]
performs destruct-resolution with the object-rules, which normally should
be destruction rules. This replaces an assumption by the result of
applying one of the rules.
\item[\ttindexbold{forward_tac}]
is like \ttindex{dresolve_tac} except that the selected assumption is not
deleted. It applies a rule to an assumption, adding the result as a new
assumption.
\end{description}
\subsection{Assumption tactics}
\index{tactics!assumption|bold}
\begin{ttbox}
assume_tac : int -> tactic
eq_assume_tac : int -> tactic
\end{ttbox}
\begin{description}
\item[\ttindexbold{assume_tac} {\it i}]
attempts to solve subgoal~$i$ by assumption.
\item[\ttindexbold{eq_assume_tac}]
is like {\tt assume_tac} but does not use unification. It succeeds (with a
{\bf unique} next state) if one of the assumptions is identical to the
subgoal's conclusion. Since it does not instantiate variables, it cannot
make other subgoals unprovable. It is intended to be called from proof
strategies, not interactively.
\end{description}
\subsection{Matching tactics} \label{match_tac}
\index{tactics!matching|bold}
\begin{ttbox}
match_tac : thm list -> int -> tactic
ematch_tac : thm list -> int -> tactic
dmatch_tac : thm list -> int -> tactic
\end{ttbox}
These are just like the resolution tactics except that they never
instantiate unknowns in the proof state. Flexible subgoals are not updated
willy-nilly, but are left alone. Matching --- strictly speaking --- means
treating the unknowns in the proof state as constants; these tactics merely
discard unifiers that would update the proof state.
\begin{description}
\item[\ttindexbold{match_tac} {\it thms} {\it i}]
refines the proof state using the object-rules, matching an object-rule's
conclusion with subgoal~$i$ of the proof state.
\item[\ttindexbold{ematch_tac}]
is like {\tt match_tac}, but performs elim-resolution.
\item[\ttindexbold{dmatch_tac}]
is like {\tt match_tac}, but performs destruct-resolution.
\end{description}
\subsection{Resolution with instantiation} \label{res_inst_tac}
\index{tactics!instantiation|bold}
\begin{ttbox}
res_inst_tac : (string*string)list -> thm -> int -> tactic
eres_inst_tac : (string*string)list -> thm -> int -> tactic
dres_inst_tac : (string*string)list -> thm -> int -> tactic
forw_inst_tac : (string*string)list -> thm -> int -> tactic
\end{ttbox}
These tactics are designed for applying rules such as substitution and
induction, which cause difficulties for higher-order unification. The
tactics accept explicit instantiations for schematic variables in the rule
--- typically, in the rule's conclusion. Each instantiation is a pair
{\tt($v$,$e$)}, where $v$ can be a type variable or ordinary variable:
\begin{itemize}
\item If $v$ is the {\bf type variable} {\tt'a}, then
the rule must contain a schematic type variable \verb$?'a$ of some
sort~$s$, and $e$ should be a type of sort $s$.
\item If $v$ is the {\bf variable} {\tt P}, then
the rule must contain a schematic variable \verb$?P$ of some type~$\tau$,
and $e$ should be a term of some type~$\sigma$ such that $\tau$ and
$\sigma$ are unifiable. If the unification of $\tau$ and $\sigma$
instantiates any schematic type variables in $\tau$, these instantiations
are recorded for application to the rule.
\end{itemize}
Types are instantiated before terms. Because type instantiations are
inferred from term instantiations, explicit type instantiations are seldom
necessary --- if \verb$?t$ has type \verb$?'a$, then the instantiation list
\verb$[("'a","bool"),("t","True")]$ may be simplified to
\verb$[("t","True")]$. Type unknowns in the proof state may cause
failure because the tactics cannot instantiate them.
The instantiation tactics act on a given subgoal. Terms in the
instantiations are type-checked in the context of that subgoal --- in
particular, they may refer to that subgoal's parameters. Any unknowns in
the terms receive subscripts and are lifted over the parameters; thus, you
may not refer to unknowns in the subgoal.
\begin{description}
\item[\ttindexbold{res_inst_tac} {\it insts} {\it thm} {\it i}]
instantiates the rule {\it thm} with the instantiations {\it insts}, as
described above, and then performs resolution on subgoal~$i$. Resolution
typically causes further instantiations; you need not give explicit
instantiations for every variable in the rule.
\item[\ttindexbold{eres_inst_tac}]
is like {\tt res_inst_tac}, but performs elim-resolution.
\item[\ttindexbold{dres_inst_tac}]
is like {\tt res_inst_tac}, but performs destruct-resolution.
\item[\ttindexbold{forw_inst_tac}]
is like {\tt dres_inst_tac} except that the selected assumption is not
deleted. It applies the instantiated rule to an assumption, adding the
result as a new assumption.
\end{description}
\section{Other basic tactics}
\subsection{Definitions and meta-level rewriting}
\index{tactics!meta-rewriting|bold}\index{rewriting!meta-level}
Definitions in Isabelle have the form $t\equiv u$, where typically $t$ is a
constant or a constant applied to a list of variables, for example $\it
sqr(n)\equiv n\times n$. (Conditional definitions, $\phi\Imp t\equiv u$,
are not supported.) {\bf Unfolding} the definition $t\equiv u$ means using
it as a rewrite rule, replacing~$t$ by~$u$ throughout a theorem. {\bf
Folding} $t\equiv u$ means replacing~$u$ by~$t$. Rewriting continues until
no rewrites are applicable to any subterm.
There are rules for unfolding and folding definitions; Isabelle does not do
this automatically. The corresponding tactics rewrite the proof state,
yielding a unique result. See also the {\tt goalw} command, which is the
easiest way of handling definitions.
\begin{ttbox}
rewrite_goals_tac : thm list -> tactic
rewrite_tac : thm list -> tactic
fold_goals_tac : thm list -> tactic
fold_tac : thm list -> tactic
\end{ttbox}
\begin{description}
\item[\ttindexbold{rewrite_goals_tac} {\it defs}]
unfolds the {\it defs} throughout the subgoals of the proof state, while
leaving the main goal unchanged. Use \ttindex{SELECT_GOAL} to restrict it to a
particular subgoal.
\item[\ttindexbold{rewrite_tac} {\it defs}]
unfolds the {\it defs} throughout the proof state, including the main goal
--- not normally desirable!
\item[\ttindexbold{fold_goals_tac} {\it defs}]
folds the {\it defs} throughout the subgoals of the proof state, while
leaving the main goal unchanged.
\item[\ttindexbold{fold_tac} {\it defs}]
folds the {\it defs} throughout the proof state.
\end{description}
\subsection{Tactic shortcuts}
\index{shortcuts!for tactics|bold}
\index{tactics!resolution}\index{tactics!assumption}
\index{tactics!meta-rewriting}
\begin{ttbox}
rtac : thm -> int ->tactic
etac : thm -> int ->tactic
dtac : thm -> int ->tactic
atac : int ->tactic
ares_tac : thm list -> int -> tactic
rewtac : thm -> tactic
\end{ttbox}
These abbreviate common uses of tactics.
\begin{description}
\item[\ttindexbold{rtac} {\it thm} {\it i}]
abbreviates \hbox{\tt resolve_tac [{\it thm}] {\it i}}, doing resolution.
\item[\ttindexbold{etac} {\it thm} {\it i}]
abbreviates \hbox{\tt eresolve_tac [{\it thm}] {\it i}}, doing elim-resolution.
\item[\ttindexbold{dtac} {\it thm} {\it i}]
abbreviates \hbox{\tt dresolve_tac [{\it thm}] {\it i}}, doing
destruct-resolution.
\item[\ttindexbold{atac} {\it i}]
is a synonym for \hbox{\tt assume_tac {\it i}}, doing proof by assumption.
\item[\ttindexbold{ares_tac} {\it thms} {\it i}]
tries proof by assumption and resolution; it abbreviates
\begin{ttbox}
assume_tac {\it i} ORELSE resolve_tac {\it thms} {\it i}
\end{ttbox}
\item[\ttindexbold{rewtac} {\it def}]
abbreviates \hbox{\tt rewrite_goals_tac [{\it def}]}, unfolding a definition.
\end{description}
\subsection{Inserting premises and facts}\label{cut_facts_tac}
\index{tactics!for inserting facts|bold}
\begin{ttbox}
cut_facts_tac : thm list -> int -> tactic
subgoal_tac : string -> int -> tactic
\end{ttbox}
These tactics add assumptions --- which must be proved, sooner or later ---
to a given subgoal.
\begin{description}
\item[\ttindexbold{cut_facts_tac} {\it thms} {\it i}]
adds the {\it thms} as new assumptions to subgoal~$i$. Once they have
been inserted as assumptions, they become subject to {\tt eresolve_tac}
and {\tt rewrite_goals_tac}. Only rules with no premises are inserted:
Isabelle cannot use assumptions that contain $\Imp$ or~$\Forall$. Sometimes
the theorems are premises of a rule being derived, returned by~{\tt
goal}; instead of calling this tactic, you could state the goal with an
outermost meta-quantifier.
\item[\ttindexbold{subgoal_tac} {\it formula} {\it i}]
adds the {\it formula} as a assumption to subgoal~$i$, and inserts the same
{\it formula} as a new subgoal, $i+1$.
\end{description}
\subsection{Theorems useful with tactics}
\index{theorems!of pure theory|bold}
\begin{ttbox}
asm_rl: thm
cut_rl: thm
\end{ttbox}
\begin{description}
\item[\ttindexbold{asm_rl}]
is $\psi\Imp\psi$. Under elim-resolution it does proof by assumption, and
\hbox{\tt eresolve_tac (asm_rl::{\it thms}) {\it i}} is equivalent to
\begin{ttbox}
assume_tac {\it i} ORELSE eresolve_tac {\it thms} {\it i}
\end{ttbox}
\item[\ttindexbold{cut_rl}]
is $\List{\psi\Imp\theta,\psi}\Imp\theta$. It is useful for inserting
assumptions; it underlies \ttindex{forward_tac}, \ttindex{cut_facts_tac}
and \ttindex{subgoal_tac}.
\end{description}
\section{Obscure tactics}
\subsection{Tidying the proof state}
\index{parameters!removing unused|bold}
\index{flex-flex constraints}
\begin{ttbox}
prune_params_tac : tactic
flexflex_tac : tactic
\end{ttbox}
\begin{description}
\item[\ttindexbold{prune_params_tac}]
removes unused parameters from all subgoals of the proof state. It works
by rewriting with the theorem $(\Forall x. V)\equiv V$. This tactic can
make the proof state more readable. It is used with
\ttindex{rule_by_tactic} to simplify the resulting theorem.
\item[\ttindexbold{flexflex_tac}]
removes all flex-flex pairs from the proof state by applying the trivial
unifier. This drastic step loses information, and should only be done as
the last step of a proof.
Flex-flex constraints arise from difficult cases of higher-order
unification. To prevent this, use \ttindex{res_inst_tac} to instantiate
some variables in a rule~(\S\ref{res_inst_tac}). Normally flex-flex
constraints can be ignored; they often disappear as unknowns get
instantiated.
\end{description}
\subsection{Renaming parameters in a goal} \index{parameters!renaming|bold}
\begin{ttbox}
rename_tac : string -> int -> tactic
rename_last_tac : string -> string list -> int -> tactic
Logic.set_rename_prefix : string -> unit
Logic.auto_rename : bool ref \hfill{\bf initially false}
\end{ttbox}
When creating a parameter, Isabelle chooses its name by matching variable
names via the object-rule. Given the rule $(\forall I)$ formalized as
$\left(\Forall x. P(x)\right) \Imp \forall x.P(x)$, Isabelle will note that
the $\Forall$-bound variable in the premise has the same name as the
$\forall$-bound variable in the conclusion.
Sometimes there is insufficient information and Isabelle chooses an
arbitrary name. The renaming tactics let you override Isabelle's choice.
Because renaming parameters has no logical effect on the proof state, the
{\tt by} command prints the needless message {\tt Warning:\ same as previous
level}.
Alternatively, you can suppress the naming mechanism described above and
have Isabelle generate uniform names for parameters. These names have the
form $p${\tt a}, $p${\tt b}, $p${\tt c},~\ldots, where $p$ is any desired
prefix. They are ugly but predictable.
\begin{description}
\item[\ttindexbold{rename_tac} {\it str} {\it i}]
interprets the string {\it str} as a series of blank-separated variable
names, and uses them to rename the parameters of subgoal~$i$. The names
must be distinct. If there are fewer names than parameters, then the
tactic renames the innermost parameters and may modify the remaining ones
to ensure that all the parameters are distinct.
\item[\ttindexbold{rename_last_tac} {\it prefix} {\it suffixes} {\it i}]
generates a list of names by attaching each of the {\it suffixes\/} to the
{\it prefix}. It is intended for coding structural induction tactics,
where several of the new parameters should have related names.
\item[\ttindexbold{Logic.set_rename_prefix} {\it prefix};]
sets the prefix for uniform renaming to~{\it prefix}. The default prefix
is {\tt"k"}.
\item[\ttindexbold{Logic.auto_rename} \tt:= true;]
makes Isabelle generate uniform names for parameters.
\end{description}
\subsection{Composition: resolution without lifting}
\index{tactics!for composition|bold}
\begin{ttbox}
compose_tac: (bool * thm * int) -> int -> tactic
\end{ttbox}
{\bf Composing} two rules means to resolve them without prior lifting or
renaming of unknowns. This low-level operation, which underlies the
resolution tactics, may occasionally be useful for special effects.
A typical application is \ttindex{res_inst_tac}, which lifts and instantiates a
rule, then passes the result to {\tt compose_tac}.
\begin{description}
\item[\ttindexbold{compose_tac} ($flag$, $rule$, $m$) $i$]
refines subgoal~$i$ using $rule$, without lifting. The $rule$ is taken to
have the form $\List{\psi@1; \ldots; \psi@m} \Imp \psi$, where $\psi$ need
{\bf not} be atomic; thus $m$ determines the number of new subgoals. If
$flag$ is {\tt true} then it performs elim-resolution --- it solves the
first premise of~$rule$ by assumption and deletes that assumption.
\end{description}
\section{Managing lots of rules}
These operations are not intended for interactive use. They are concerned
with the processing of large numbers of rules in automatic proof
strategies. Higher-order resolution involving a long list of rules is
slow. Filtering techniques can shorten the list of rules given to
resolution, and can also detect whether a given subgoal is too flexible,
with too many rules applicable.
\subsection{Combined resolution and elim-resolution} \label{biresolve_tac}
\index{tactics!resolution}
\begin{ttbox}
biresolve_tac : (bool*thm)list -> int -> tactic
bimatch_tac : (bool*thm)list -> int -> tactic
subgoals_of_brl : bool*thm -> int
lessb : (bool*thm) * (bool*thm) -> bool
\end{ttbox}
{\bf Bi-resolution} takes a list of $\it (flag,rule)$ pairs. For each
pair, it applies resolution if the flag is~{\tt false} and
elim-resolution if the flag is~{\tt true}. A single tactic call handles a
mixture of introduction and elimination rules.
\begin{description}
\item[\ttindexbold{biresolve_tac} {\it brls} {\it i}]
refines the proof state by resolution or elim-resolution on each rule, as
indicated by its flag. It affects subgoal~$i$ of the proof state.
\item[\ttindexbold{bimatch_tac}]
is like {\tt biresolve_tac}, but performs matching: unknowns in the
proof state are never updated (see~\S\ref{match_tac}).
\item[\ttindexbold{subgoals_of_brl}({\it flag},{\it rule})]
returns the number of new subgoals that bi-resolution would yield for the
pair (if applied to a suitable subgoal). This is $n$ if the flag is
{\tt false} and $n-1$ if the flag is {\tt true}, where $n$ is the number
of premises of the rule. Elim-resolution yields one fewer subgoal than
ordinary resolution because it solves the major premise by assumption.
\item[\ttindexbold{lessb} ({\it brl1},{\it brl2})]
returns the result of
\begin{ttbox}
subgoals_of_brl {\it brl1} < subgoals_of_brl {\it brl2}
\end{ttbox}
Note that \hbox{\tt sort lessb {\it brls}} sorts a list of $\it
(flag,rule)$ pairs by the number of new subgoals they will yield. Thus,
those that yield the fewest subgoals should be tried first.
\end{description}
\subsection{Discrimination nets for fast resolution}
\index{discrimination nets|bold}
\index{tactics!resolution}
\begin{ttbox}
net_resolve_tac : thm list -> int -> tactic
net_match_tac : thm list -> int -> tactic
net_biresolve_tac: (bool*thm) list -> int -> tactic
net_bimatch_tac : (bool*thm) list -> int -> tactic
filt_resolve_tac : thm list -> int -> int -> tactic
could_unify : term*term->bool
filter_thms : (term*term->bool) -> int*term*thm list -> thm list
\end{ttbox}
The module \ttindex{Net} implements a discrimination net data structure for
fast selection of rules \cite[Chapter 14]{charniak80}. A term is
classified by the symbol list obtained by flattening it in preorder.
The flattening takes account of function applications, constants, and free
and bound variables; it identifies all unknowns and also regards
$\lambda$-abstractions as unknowns, since they could $\eta$-contract to
anything.
A discrimination net serves as a polymorphic dictionary indexed by terms.
The module provides various functions for inserting and removing items from
nets. It provides functions for returning all items whose term could match
or unify with a target term. The matching and unification tests are
overly lax (due to the identifications mentioned above) but they serve as
useful filters.
A net can store introduction rules indexed by their conclusion, and
elimination rules indexed by their major premise. Isabelle provides
several functions for ``compiling'' long lists of rules into fast
resolution tactics. When supplied with a list of theorems, these functions
build a discrimination net; the net is used when the tactic is applied to a
goal. To avoid repreatedly constructing the nets, use currying: bind the
resulting tactics to \ML{} identifiers.
\begin{description}
\item[\ttindexbold{net_resolve_tac} {\it thms}]
builds a discrimination net to obtain the effect of a similar call to {\tt
resolve_tac}.
\item[\ttindexbold{net_match_tac} {\it thms}]
builds a discrimination net to obtain the effect of a similar call to {\tt
match_tac}.
\item[\ttindexbold{net_biresolve_tac} {\it brls}]
builds a discrimination net to obtain the effect of a similar call to {\tt
biresolve_tac}.
\item[\ttindexbold{net_bimatch_tac} {\it brls}]
builds a discrimination net to obtain the effect of a similar call to {\tt
bimatch_tac}.
\item[\ttindexbold{filt_resolve_tac} {\it thms} {\it maxr} {\it i}]
uses discrimination nets to extract the {\it thms} that are applicable to
subgoal~$i$. If more than {\it maxr\/} theorems are applicable then the
tactic fails. Otherwise it calls {\tt resolve_tac}.
This tactic helps avoid runaway instantiation of unknowns, for example in
type inference.
\item[\ttindexbold{could_unify} ({\it t},{\it u})]
returns {\tt false} if~$t$ and~$u$ are ``obviously'' non-unifiable, and
otherwise returns~{\tt true}. It assumes all variables are distinct,
reporting that {\tt ?a=?a} may unify with {\tt 0=1}.
\item[\ttindexbold{filter_thms} $could\; (limit,prem,thms)$]
returns the list of potentially resolvable rules (in {\it thms\/}) for the
subgoal {\it prem}, using the predicate {\it could\/} to compare the
conclusion of the subgoal with the conclusion of each rule. The resulting list
is no longer than {\it limit}.
\end{description}
\section{Programming tools for proof strategies}
Do not consider using the primitives discussed in this section unless you
really need to code tactics from scratch,
\subsection{Operations on type {\tt tactic}}
\index{tactics!primitives for coding|bold}
A tactic maps theorems to theorem sequences (lazy lists). The type
constructor for sequences is called \ttindex{Sequence.seq}. To simplify the
types of tactics and tacticals, Isabelle defines a type of tactics:
\begin{ttbox}
datatype tactic = Tactic of thm -> thm Sequence.seq
\end{ttbox}
{\tt Tactic} and {\tt tapply} convert between tactics and functions. The
other operations provide means for coding tactics in a clean style.
\begin{ttbox}
tapply : tactic * thm -> thm Sequence.seq
Tactic : (thm -> thm Sequence.seq) -> tactic
PRIMITIVE : (thm -> thm) -> tactic
STATE : (thm -> tactic) -> tactic
SUBGOAL : ((term*int) -> tactic) -> int -> tactic
\end{ttbox}
\begin{description}
\item[\ttindexbold{tapply} {\it tac} {\it thm}]
returns the result of applying the tactic, as a function, to {\it thm}.
\item[\ttindexbold{Tactic} {\it f}]
packages {\it f} as a tactic.
\item[\ttindexbold{PRIMITIVE} $f$]
applies $f$ to the proof state and returns the result as a
one-element sequence. This packages the meta-rule~$f$ as a tactic.
\item[\ttindexbold{STATE} $f$]
applies $f$ to the proof state and then applies the resulting tactic to the
same state. It supports the following style, where the tactic body is
expressed at a high level, but may peek at the proof state:
\begin{ttbox}
STATE (fn state => {\it some tactic})
\end{ttbox}
\item[\ttindexbold{SUBGOAL} $f$ $i$]
extracts subgoal~$i$ from the proof state as a term~$t$, and computes a
tactic by calling~$f(t,i)$. It applies the resulting tactic to the same
state. The tactic body is expressed at a high level, but may peek at a
particular subgoal:
\begin{ttbox}
SUBGOAL (fn (t,i) => {\it some tactic})
\end{ttbox}
\end{description}
\subsection{Tracing}
\index{tactics!tracing|bold}
\index{tracing!of tactics}
\begin{ttbox}
pause_tac: tactic
print_tac: tactic
\end{ttbox}
These violate the functional behaviour of tactics by printing information
when they are applied to a proof state. Their output may be difficult to
interpret. Note that certain of the searching tacticals, such as {\tt
REPEAT}, have built-in tracing options.
\begin{description}
\item[\ttindexbold{pause_tac}]
prints {\tt** Press RETURN to continue:} and then reads a line from the
terminal. If this line is blank then it returns the proof state unchanged;
otherwise it fails (which may terminate a repetition).
\item[\ttindexbold{print_tac}]
returns the proof state unchanged, with the side effect of printing it at
the terminal.
\end{description}
\subsection{Sequences}
\index{sequences (lazy lists)|bold}
The module \ttindex{Sequence} declares a type of lazy lists. It uses
Isabelle's type \ttindexbold{option} to represent the possible presence
(\ttindexbold{Some}) or absence (\ttindexbold{None}) of
a value:
\begin{ttbox}
datatype 'a option = None | Some of 'a;
\end{ttbox}
\subsubsection{Basic operations on sequences}
\begin{ttbox}
Sequence.null : 'a Sequence.seq
Sequence.seqof : (unit -> ('a * 'a Sequence.seq) option)
-> 'a Sequence.seq
Sequence.single : 'a -> 'a Sequence.seq
Sequence.pull : 'a Sequence.seq -> ('a * 'a Sequence.seq) option
\end{ttbox}
\begin{description}
\item[{\tt Sequence.null}]
is the empty sequence.
\item[\tt Sequence.seqof (fn()=> Some($x$,$s$))]
constructs the sequence with head~$x$ and tail~$s$, neither of which is
evaluated.
\item[{\tt Sequence.single} $x$]
constructs the sequence containing the single element~$x$.
\item[{\tt Sequence.pull} $s$]
returns {\tt None} if the sequence is empty and {\tt Some($x$,$s'$)} if the
sequence has head~$x$ and tail~$s'$. Warning: calling \hbox{Sequence.pull
$s$} again will {\bf recompute} the value of~$x$; it is not stored!
\end{description}
\subsubsection{Converting between sequences and lists}
\begin{ttbox}
Sequence.chop: int * 'a Sequence.seq -> 'a list * 'a Sequence.seq
Sequence.list_of_s : 'a Sequence.seq -> 'a list
Sequence.s_of_list : 'a list -> 'a Sequence.seq
\end{ttbox}
\begin{description}
\item[{\tt Sequence.chop} ($n$, $s$)]
returns the first~$n$ elements of~$s$ as a list, paired with the remaining
elements of~$s$. If $s$ has fewer than~$n$ elements, then so will the
list.
\item[{\tt Sequence.list_of_s} $s$]
returns the elements of~$s$, which must be finite, as a list.
\item[{\tt Sequence.s_of_list} $l$]
creates a sequence containing the elements of~$l$.
\end{description}
\subsubsection{Combining sequences}
\begin{ttbox}
Sequence.append: 'a Sequence.seq * 'a Sequence.seq -> 'a Sequence.seq
Sequence.interleave : 'a Sequence.seq * 'a Sequence.seq
-> 'a Sequence.seq
Sequence.flats : 'a Sequence.seq Sequence.seq -> 'a Sequence.seq
Sequence.maps : ('a -> 'b) -> 'a Sequence.seq -> 'b Sequence.seq
Sequence.filters : ('a -> bool) -> 'a Sequence.seq -> 'a Sequence.seq
\end{ttbox}
\begin{description}
\item[{\tt Sequence.append} ($s@1$, $s@2$)]
concatenates $s@1$ to $s@2$.
\item[{\tt Sequence.interleave} ($s@1$, $s@2$)]
joins $s@1$ with $s@2$ by interleaving their elements. The result contains
all the elements of the sequences, even if both are infinite.
\item[{\tt Sequence.flats} $ss$]
concatenates a sequence of sequences.
\item[{\tt Sequence.maps} $f$ $s$]
applies $f$ to every element of~$s=x@1,x@2,\ldots$, yielding the sequence
$f(x@1),f(x@2),\ldots$.
\item[{\tt Sequence.filters} $p$ $s$]
returns the sequence consisting of all elements~$x$ of~$s$ such that $p(x)$
is {\tt true}.
\end{description}
\index{tactics|)}