
diff -r a5e7ab090abf -r f7f1d8a98098 ProgTutorial/Tactical.thy
--- a/ProgTutorial/Tactical.thy	Thu Oct 29 16:02:15 2009 +0100
+++ b/ProgTutorial/Tactical.thy	Thu Oct 29 19:09:27 2009 +0100
@@ -20,7 +20,7 @@
   \isacommand{apply}-style reasoning at the user-level, where goals are
   modified in a sequence of proof steps until all of them are discharged.
   In this chapter we will explain simple tactics and how to combine them using
-  tactic combinators. We also explain briefly the simplifier and conversions.
+  tactic combinators. We also describe the simplifier, simprocs and conversions.
 *}
 
 
@@ -49,12 +49,11 @@
   val goal = @{prop \"P \<or> Q \<Longrightarrow> Q \<or> P\"}
 in
   Goal.prove ctxt [\"P\", \"Q\"] [] goal 
-   (fn _ => 
-      etac @{thm disjE} 1
-      THEN rtac @{thm disjI2} 1
-      THEN atac 1
-      THEN rtac @{thm disjI1} 1
-      THEN atac 1)
+   (fn _ =>  etac @{thm disjE} 1
+             THEN rtac @{thm disjI2} 1
+             THEN atac 1
+             THEN rtac @{thm disjI1} 1
+             THEN atac 1)
 end" "?P \<or> ?Q \<Longrightarrow> ?Q \<or> ?P"}
   
   To start the proof, the function @{ML_ind prove in Goal} sets up a goal
@@ -65,10 +64,10 @@
   into schematic variables once the goal is proved (in our case @{text P} and
   @{text Q}). The last argument is the tactic that proves the goal. This
   tactic can make use of the local assumptions (there are none in this
-  example). The tactics @{ML_ind etac}, @{ML_ind rtac} and @{ML_ind atac} in
-  the code above correspond roughly to @{text erule}, @{text rule} and @{text
-  assumption}, respectively. The combinator @{ML_ind THEN} strings the tactics
-  together.
+  example). The tactics @{ML_ind etac in Tactic}, @{ML_ind rtac in Tactic} and
+  @{ML_ind atac in Tactic} in the code above correspond roughly to @{text
+  erule}, @{text rule} and @{text assumption}, respectively. The combinator
+  @{ML THEN} strings the tactics together.
 
   \begin{readmore}
   To learn more about the function @{ML_ind prove in Goal} see
@@ -80,25 +79,25 @@
   \end{readmore}
 
   Note that in the code above we use antiquotations for referencing the
-  theorems. We  could also just have written @{ML "etac disjE 1"}. The reason 
-  is that many of the basic theorems have a corresponding ML-binding:
+  theorems. We  could also just have written @{ML "etac disjE 1"} because 
+  many of the basic theorems have a corresponding ML-binding:
 
   @{ML_response_fake [gray,display]
   "disjE"
   "\<lbrakk>?P \<or> ?Q; ?P \<Longrightarrow> ?R; ?Q \<Longrightarrow> ?R\<rbrakk> \<Longrightarrow> ?R"}
 
-  In case where there is no ML-binding theorems can also be obtained dynamically using 
-  the function @{ML thm} and the (string) name of the theorem; for example: 
+  In case where no ML-binding exists, theorems can also be looked up dynamically 
+  using the function @{ML thm} and the (string) name of the theorem. For example: 
 
   @{ML_response_fake [gray,display]
   "thm \"disjE\""
   "\<lbrakk>?P \<or> ?Q; ?P \<Longrightarrow> ?R; ?Q \<Longrightarrow> ?R\<rbrakk> \<Longrightarrow> ?R"}
 
-  Both ways however are considered bad style! The reason is that the binding
-  for @{ML disjE} can be re-assigned by the user and thus one does not have
+  Both ways however are considered \emph{bad} style! The reason is that the binding
+  for @{ML disjE} can be re-assigned and thus one does not have
   complete control over which theorem is actually applied. Similarly with the
-  string @{text [quotes] "disjE"}. Although theorems in the theorem database
-  must have a unique name, the string can stand for a dynamically updated
+  lookup of @{text [quotes] "disjE"}. Although theorems must have a unique name
+  in the theorem database, the string can stand for a dynamically updatable
   theorem list. Also in this case we cannot be sure which theorem is applied.
   These problems can be nicely prevented by using antiquotations, because then
   the theorems are fixed statically at compile-time.
@@ -146,11 +145,12 @@
        THEN' atac)*}text_raw{*\label{tac:footacprime}*}
 
 text {* 
-  where @{ML_ind THEN'} is used instead of @{ML THEN}. (For most combinators
-  that combine tactics---@{ML THEN} is only one such combinator---a ``primed''
-  version exists.)  With @{ML foo_tac'} you can give the number for the
-  subgoal explicitly when the tactic is called. So in the next proof you can
-  first discharge the second subgoal, and subsequently the first.
+  where @{ML_ind THEN' in Tactical} is used instead of @{ML THEN in
+  Tactical}. (For most combinators that combine tactics---@{ML THEN} is only
+  one such combinator---a ``primed'' version exists.)  With @{ML foo_tac'} you
+  can give the number for the subgoal explicitly when the tactic is called. So
+  in the next proof you can first discharge the second subgoal, and
+  subsequently the first.
 *}
 
 lemma 
@@ -167,8 +167,8 @@
   The tactics @{ML foo_tac} and @{ML foo_tac'} are very specific for analysing
   goals being only of the form @{prop "P \<or> Q \<Longrightarrow> Q \<or> P"}. If the goal is not of
   this form, then these tactics return the error message:\footnote{To be
-  precise, tactics do not produce this error message, it originates from the
-  \isacommand{apply} wrapper.}
+  precise, tactics do not produce this error message; the message originates from the
+  \isacommand{apply} wrapper that calls the tactic.}
 
 
   \begin{isabelle}
@@ -189,8 +189,9 @@
   willy-nilly; only in very grave failure situations should a tactic raise the 
   exception @{text THM}.
 
-  The simplest tactics are @{ML_ind  no_tac} and @{ML_ind  all_tac}. The first returns
-  the empty sequence and is defined as
+  The simplest tactics are @{ML_ind no_tac in Tactical} and 
+  @{ML_ind all_tac in Tactical}. The first returns the empty sequence and 
+  is defined as
 *}
 
 ML{*fun no_tac thm = Seq.empty*}
@@ -247,6 +248,29 @@
   Seq.single thm
 end*}
 
+text {*
+  which prints out the given theorem (using the string-function defined in
+  Section~\ref{sec:printing}) and then behaves like @{ML all_tac}. With this
+  tactic we are in the position to inspect every goal state in a proof. For
+  this consider the proof in Figure~\ref{fig:goalstates}: as can be seen,
+  internally every goal state is an implication of the form
+
+  @{text[display] "A\<^isub>1 \<Longrightarrow> \<dots> \<Longrightarrow> A\<^isub>n \<Longrightarrow> #C"}
+
+  where @{term C} is the goal to be proved and the @{term "A\<^isub>i"} are
+  the subgoals. So after setting up the lemma, the goal state is always of the
+  form @{text "C \<Longrightarrow> #C"}; when the proof is finished we are left with @{text
+  "#C"}. Since the goal @{term C} can potentially be an implication, there is a
+  ``protector'' wrapped around it (the wrapper is the outermost constant
+  @{text "Const (\"prop\", bool \<Rightarrow> bool)"}; in the figure we make it visible
+  as a @{text #}). This wrapper prevents that premises of @{text C} are
+  misinterpreted as open subgoals. While tactics can operate on the subgoals
+  (the @{text "A\<^isub>i"} above), they are expected to leave the conclusion
+  @{term C} intact, with the exception of possibly instantiating schematic
+  variables. This instantiations of schematic variables can be observed 
+  on the user level. Have a look at the following definition and proof.
+*}
+
 text_raw {*
   \begin{figure}[p]
   \begin{boxedminipage}{\textwidth}
@@ -333,29 +357,6 @@
   \end{figure}
 *}
 
-text {*
-  which prints out the given theorem (using the string-function defined in
-  Section~\ref{sec:printing}) and then behaves like @{ML all_tac}. With this
-  tactic we are in the position to inspect every goal state in a proof. For
-  this consider the proof in Figure~\ref{fig:goalstates}: as can be seen,
-  internally every goal state is an implication of the form
-
-  @{text[display] "A\<^isub>1 \<Longrightarrow> \<dots> \<Longrightarrow> A\<^isub>n \<Longrightarrow> #C"}
-
-  where @{term C} is the goal to be proved and the @{term "A\<^isub>i"} are
-  the subgoals. So after setting up the lemma, the goal state is always of the
-  form @{text "C \<Longrightarrow> #C"}; when the proof is finished we are left with @{text
-  "#C"}. Since the goal @{term C} can potentially be an implication, there is a
-  ``protector'' wrapped around it (the wrapper is the outermost constant
-  @{text "Const (\"prop\", bool \<Rightarrow> bool)"}; in the figure we make it visible
-  as an @{text #}). This wrapper prevents that premises of @{text C} are
-  misinterpreted as open subgoals. While tactics can operate on the subgoals
-  (the @{text "A\<^isub>i"} above), they are expected to leave the conclusion
-  @{term C} intact, with the exception of possibly instantiating schematic
-  variables. This instantiations of schematic variables can be observed 
-  on the user level. Have a look at the following definition and proof.
-*}
-
 definition 
   EQ_TRUE 
 where
@@ -363,8 +364,8 @@
 
 lemma test: 
   shows "EQ_TRUE ?X"
-  unfolding EQ_TRUE_def
-  by (rule refl)
+unfolding EQ_TRUE_def
+by (rule refl)
 
 text {*
   Although Isabelle issues a warning message when stating goals involving 
@@ -378,8 +379,8 @@
   \end{isabelle}
  
   The reason for this result is that the schematic variable @{text "?X"} 
-  is instantiated inside the proof and then the instantiation propagated 
-  to the ``outside''.
+  is instantiated inside the proof to be @{term True} and then the 
+  instantiation propagated to the ``outside''.
 
   \begin{readmore}
   For more information about the internals of goals see \isccite{sec:tactical-goals}.
@@ -390,8 +391,8 @@
 section {* Simple Tactics\label{sec:simpletacs} *}
 
 text {*
-  In this section we will describe more of the simple tactics in Isabelle. The 
-  first one is @{ML_ind  print_tac}, which is quite useful 
+  In this section we will introduce more simple tactics. The 
+  first one is @{ML_ind print_tac in Tactical}, which is quite useful 
   for low-level debugging of tactics. It just prints out a message and the current 
   goal state. Unlike @{ML my_print_tac} shown earlier, it prints the goal state 
   as the user would see it.  For example, processing the proof
@@ -411,8 +412,8 @@
 (*<*)oops(*>*)
 
 text {*
-  A simple tactic for easy discharge of any proof obligations is 
-  @{ML_ind  cheat_tac in Skip_Proof} in the structure @{ML_struct Skip_Proof}. 
+  A simple tactic for easy discharge of any proof obligations, even difficult ones, is 
+  @{ML_ind cheat_tac in Skip_Proof} in the structure @{ML_struct Skip_Proof}. 
   This tactic corresponds to the Isabelle command \isacommand{sorry} and is 
   sometimes useful during the development of tactics.
 *}
@@ -428,9 +429,14 @@
 text {*
   This tactic works however only if the quick-and-dirty mode of Isabelle 
   is switched on. This is done automatically in the ``interactive
-  mode'' of Isabelle, but must ne done manually in the ``batch mode''.
-
-  Another simple tactic is the function @{ML_ind  atac}, which, as shown 
+  mode'' of Isabelle, but must be done manually in the ``batch mode''
+  with for example the assignment
+*}
+
+ML{*quick_and_dirty := true*}
+
+text {*
+  Another simple tactic is the function @{ML_ind atac in Tactic}, which, as shown 
   earlier, corresponds to the assumption method.
 *}
 
@@ -443,11 +449,11 @@
 (*<*)oops(*>*)
 
 text {*
-  Similarly, @{ML_ind  rtac}, @{ML_ind  dtac}, @{ML_ind  etac} and 
-  @{ML_ind  ftac} correspond (roughly)
-  to @{text rule}, @{text drule}, @{text erule} and @{text frule}, 
-  respectively. Each of them takes a theorem as argument and attempts to 
-  apply it to a goal. Below are three self-explanatory examples.
+  Similarly, @{ML_ind rtac in Tactic}, @{ML_ind dtac in Tactic}, @{ML_ind etac
+  in Tactic} and @{ML_ind ftac in Tactic} correspond (roughly) to @{text
+  rule}, @{text drule}, @{text erule} and @{text frule}, respectively. Each of
+  them takes a theorem as argument and attempts to apply it to a goal. Below
+  are three self-explanatory examples.
 *}
 
 lemma 
@@ -475,8 +481,8 @@
 (*<*)oops(*>*)
 
 text {*
-  The function @{ML_ind  resolve_tac} is similar to @{ML_ind  rtac}, except that it
-  expects a list of theorems as arguments. From this list it will apply the
+  The function @{ML_ind  resolve_tac} is similar to @{ML rtac}, except that it
+  expects a list of theorems as argument. From this list it will apply the
   first applicable theorem (later theorems that are also applicable can be
   explored via the lazy sequences mechanism). Given the code
 *}
@@ -500,13 +506,13 @@
 
 text {* 
   Similar versions taking a list of theorems exist for the tactics 
-  @{ML dtac} (@{ML_ind  dresolve_tac}), @{ML etac} 
-  (@{ML_ind  eresolve_tac}) and so on.
-
-
-  Another simple tactic is @{ML_ind  cut_facts_tac}. It inserts a list of theorems
-  into the assumptions of the current goal state. Below we will insert the definitions
-  for the constants @{term True} and @{term False}. So
+  @{ML dtac} (@{ML_ind dresolve_tac in Tactic}), @{ML etac} 
+  (@{ML_ind eresolve_tac in Tactic}) and so on.
+
+  Another simple tactic is @{ML_ind cut_facts_tac in Tactic}. It inserts a
+  list of theorems into the assumptions of the current goal state. Below we
+  will insert the definitions for the constants @{term True} and @{term
+  False}. So
 *}
 
 lemma 
@@ -523,15 +529,14 @@
   Often proofs on the ML-level involve elaborate operations on assumptions and 
   @{text "\<And>"}-quantified variables. To do such operations using the basic tactics 
   shown so far is very unwieldy and brittle. Some convenience and
-  safety is provided by the functions @{ML_ind  FOCUS in Subgoal} and 
-  @{ML_ind  SUBPROOF}. These tactics fix the parameters 
+  safety is provided by the functions @{ML_ind FOCUS in Subgoal} and 
+  @{ML_ind SUBPROOF in Subgoal}. These tactics fix the parameters 
   and bind the various components of a goal state to a record. 
   To see what happens, suppose the function defined in Figure~\ref{fig:sptac}, which
-  takes a record and just prints out the contents of this record. Consider
-  now the proof:
+  takes a record and just prints out the contents of this record. Then consider
+  the proof:
 *}
 
-ML {* Subgoal.FOCUS *}
 
 text_raw{*
 \begin{figure}[t]
@@ -588,14 +593,14 @@
   \end{tabular}
   \end{quote}
 
-  The @{text params} and @{text schematics} stand or list of pairs where the left-hand
-  side of @{text ":="} is replaced by the right-hand side inside the tactic.
-  Notice that in the actual output the brown colour of the variables @{term x}, 
-  and @{term y}. Although they are parameters in the original goal, they are fixed inside
-  the tactic. By convention these fixed variables are printed in brown colour.
-  Similarly the schematic variable @{text ?z}. The assumption, or premise, 
-  @{prop "A x y"} is bound as @{ML_type cterm} to the record-variable 
-  @{text asms}, but also as @{ML_type thm} to @{text prems}.
+  The @{text params} and @{text schematics} stand or list of pairs where the
+  left-hand side of @{text ":="} is replaced by the right-hand side inside the
+  tactic.  Notice that in the actual output the variables @{term x} and @{term
+  y} have a brown colour. Although they are parameters in the original goal,
+  they are fixed inside the tactic. By convention these fixed variables are
+  printed in brown colour.  Similarly the schematic variable @{text ?z}. The
+  assumption, or premise, @{prop "A x y"} is bound as @{ML_type cterm} to the
+  record-variable @{text asms}, but also as @{ML_type thm} to @{text prems}.
 
   If we continue the proof script by applying the @{text impI}-rule
 *}
@@ -623,7 +628,7 @@
 
   One difference between the tactics @{ML SUBPROOF} and @{ML FOCUS in Subgoal}
   is that the former expects that the goal is solved completely, which the
-  latter does not. Another is that @{ML SUBPROOF} can not instantiate any schematic
+  latter does not. Another is that @{ML SUBPROOF} cannot instantiate any schematic
   variables.
 
   One convenience of both @{ML FOCUS in Subgoal} and @{ML SUBPROOF} is that we
@@ -676,7 +681,7 @@
   @{ML_ind  SUBGOAL} and @{ML_ind  CSUBGOAL}. They allow you to 
   inspect a given subgoal (the former
   presents the subgoal as a @{ML_type term}, while the latter as a @{ML_type
-  cterm}). With this you can implement a tactic that applies a rule according
+  cterm}). With them you can implement a tactic that applies a rule according
   to the topmost logic connective in the subgoal (to illustrate this we only
   analyse a few connectives). The code of the tactic is as
   follows.
@@ -701,8 +706,8 @@
   first five patterns we can use the @{text "@term"}-antiquotation to
   construct the patterns, the pattern in Line 8 cannot be constructed in this
   way. The reason is that an antiquotation would fix the type of the
-  quantified variable. So you really have to construct the pattern using the
-  basic term-constructors. This is not necessary in other cases, because their
+  quantified variable. So you really have to construct this pattern using the
+  basic term-constructors. This is not necessary in the other cases, because their
   type is always fixed to function types involving only the type @{typ
   bool}. (See Section \ref{sec:terms_types_manually} about constructing terms
   manually.) For the catch-all pattern, we chose to just return @{ML all_tac}. 
@@ -725,7 +730,7 @@
 (*<*)oops(*>*)
 
 text {*
-  where in all but the last the tactic applied an introduction rule. 
+  where in all but the last the tactic applies an introduction rule. 
   Note that we applied the tactic to the goals in ``reverse'' order. 
   This is a trick in order to be independent from the subgoals that are 
   produced by the rule. If we had applied it in the other order 
@@ -748,12 +753,12 @@
   contrived: there are much simpler methods available in Isabelle for
   implementing a tactic analysing a goal according to its topmost
   connective. These simpler methods use tactic combinators, which we will
-  explain in the next section, but before that we will show how
-  tactic application can be constraint.
+  explain in the next section. But before that we will show how
+  tactic application can be constrained.
 
   Since rules are applied using higher-order unification, an automatic proof
   procedure might become too fragile, if it just applies inference rules as 
-  shown above. The reason is that a number of rules introduce meta-variables 
+  shown above. The reason is that a number of rules introduce schematic variables 
   into the goal state. Consider for example the proof
 *}
 
@@ -767,13 +772,13 @@
 
 text {*
   where the application of rule @{text bspec} generates two subgoals involving the
-  meta-variable @{text "?x"}. Now, if you are not careful, tactics 
-  applied to the first subgoal might instantiate this meta-variable in such a 
+  schematic variable @{text "?x"}. Now, if you are not careful, tactics 
+  applied to the first subgoal might instantiate this schematic variable in such a 
   way that the second subgoal becomes unprovable. If it is clear what the @{text "?x"}
   should be, then this situation can be avoided by introducing a more
   constrained version of the @{text bspec}-rule. One way to give such 
   constraints is by pre-instantiating theorems with other theorems. 
-  The function @{ML_ind "RS"}, for example, does this.
+  The function @{ML_ind RS in Drule}, for example, does this.
   
   @{ML_response_fake [display,gray]
   "@{thm disjI1} RS @{thm conjI}" "\<lbrakk>?P1; ?Q\<rbrakk> \<Longrightarrow> (?P1 \<or> ?Q1) \<and> ?Q"}
@@ -787,9 +792,9 @@
 "*** Exception- THM (\"RSN: no unifiers\", 1, 
 [\"\<lbrakk>?P; ?Q\<rbrakk> \<Longrightarrow> ?P \<and> ?Q\", \"\<lbrakk>?P \<longrightarrow> ?Q; ?P\<rbrakk> \<Longrightarrow> ?Q\"]) raised"}
 
-  then the function raises an exception. The function @{ML_ind  RSN} is similar to @{ML RS}, but 
-  takes an additional number as argument that makes explicit which premise 
-  should be instantiated. 
+  then the function raises an exception. The function @{ML_ind  RSN in Drule} 
+  is similar to @{ML RS}, but takes an additional number as argument. This 
+  number makes explicit which premise should be instantiated. 
 
   To improve readability of the theorems we shall produce below, we will use the
   function @{ML no_vars} from Section~\ref{sec:printing}, which transforms
@@ -800,20 +805,24 @@
   "no_vars @{context} (@{thm disjI1} RS @{thm conjI})" "\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> (P \<or> Qa) \<and> Q"}
 
   If you want to instantiate more than one premise of a theorem, you can use 
-  the function @{ML_ind  MRS}:
+  the function @{ML_ind MRS in Drule}:
 
   @{ML_response_fake [display,gray]
   "no_vars @{context} ([@{thm disjI1}, @{thm disjI2}] MRS @{thm conjI})" 
   "\<lbrakk>P; Q\<rbrakk> \<Longrightarrow> (P \<or> Qa) \<and> (Pa \<or> Q)"}
 
   If you need to instantiate lists of theorems, you can use the
-  functions @{ML RL} and @{ML_ind  MRL}. For example in the code below,
-  every theorem in the second list is instantiated with every 
-  theorem in the first.
+  functions @{ML_ind RL in Drule} and @{ML_ind MRL in Drule}. For 
+  example in the code below, every theorem in the second list is 
+  instantiated with every theorem in the first.
 
   @{ML_response_fake [display,gray]
-"map (no_vars @{context}) 
-   ([@{thm impI}, @{thm disjI2}] RL [@{thm conjI}, @{thm disjI1}])" 
+"let
+  val list1 = [@{thm impI}, @{thm disjI2}]
+  val list2 = [@{thm conjI}, @{thm disjI1}]
+in
+  map (no_vars @{context}) (list1 RL list2)
+end" 
 "[\<lbrakk>P \<Longrightarrow> Q; Qa\<rbrakk> \<Longrightarrow> (P \<longrightarrow> Q) \<and> Qa,
  \<lbrakk>Q; Qa\<rbrakk> \<Longrightarrow> (P \<or> Q) \<and> Qa,
  (P \<Longrightarrow> Q) \<Longrightarrow> (P \<longrightarrow> Q) \<or> Qa,
@@ -846,12 +855,14 @@
 (*<*)oops(*>*)
 
 text {*
-  As you can see this is unfortunately \emph{not} the case. The problem is
+  As you can see this is unfortunately \emph{not} the case if we apply @{thm [source] 
+  cong} with the method @{text rule}. The problem is
   that higher-order unification produces an instantiation that is not the
   intended one. While we can use \isacommand{back} to interactively find the
   intended instantiation, this is not an option for an automatic proof
   procedure. Fortunately, the tactic @{ML_ind cong_tac in Cong_Tac} helps 
   with applying congruence rules and finding the intended instantiation.
+  For example
 *}
 
 lemma 
@@ -864,8 +875,8 @@
 (*<*)oops(*>*)
 
 text {*
-  However, sometimes it is necessary to expicitly match a theroem against a proof
-  state and in doing so finding manually an appropriate instantiation. Imagine 
+  However, frequently it is necessary to explicitly match a theorem against a proof
+  state and in doing so construct manually an appropriate instantiation. Imagine 
   you have the theorem
 *}
 
@@ -875,22 +886,21 @@
 
 text {* 
   and you want to apply it to the goal @{term "(t\<^isub>1 t\<^isub>2) \<le>
-  (s\<^isub>1 s\<^isub>2)"}. Since in this theorem, all variables are
+  (s\<^isub>1 s\<^isub>2)"}. Since in the theorem all variables are
   schematic, we have a nasty higher-order unification problem and @{text rtac}
-  will not be able to apply this rule as we want. However, in the proof below
-  we are only interested where @{term R} is instantiated to a constant and
-  similarly the instantiation for the other variables is ``obvious'' from the
-  proof state.  To use this rule we essentially match the conclusion of 
-  @{thm [source] rule} against the goal state reading of an instantiation for
+  will not be able to apply this rule in the way we want. For the goal at hand 
+  we want to use it so that @{term R} is instantiated to the constant @{text \<le>} and
+  similarly ``obvious'' instantiations for the other variables.  To achieve this 
+  we need to match the conclusion of 
+  @{thm [source] rule} against the goal reading off an instantiation for
   @{thm [source] rule}. For this the function @{ML_ind first_order_match in Thm} 
-  matches two @{ML_type cterm}s and produces, in the sucessful case, a matcher 
-  that can be used to instantiate the @{thm [source] rule}. The instantiation 
+  matches two @{ML_type cterm}s and produces, in the successful case, a matcher 
+  that can be used to instantiate the theorem. The instantiation 
   can be done with the function @{ML_ind instantiate in Drule}. To automate 
   this we implement the following function.
 *}
 
-ML{*fun fo_rtac thm = 
-  Subgoal.FOCUS (fn {concl, ...} =>
+ML %linenosgray{*fun fo_rtac thm = Subgoal.FOCUS (fn {concl, ...} =>
   let 
     val concl_pat = Drule.strip_imp_concl (cprop_of thm)
     val insts = Thm.first_order_match (concl_pat, concl)
@@ -899,10 +909,12 @@
   end)*}
 
 text {*
-  Note that we use @{ML FOCUS in Subgoal} because we have directly access
-  to the conclusion of the goal state and also because this function also
-  takes care about correctly handling parameters that might be present
-  in the goal state. An example for @{ML fo_rtac} is as follows.
+  Note that we use @{ML FOCUS in Subgoal} because it gives us directly access
+  to the conclusion of the goal state, but also because this function 
+  takes care of correctly handling parameters that might be present
+  in the goal state. In Line 3 we use the function @{ML_ind strip_imp_concl in Drule}
+  for calculating the conclusion of a theorem (produced as @{ML_type cterm}).
+  An example of @{ML fo_rtac} is as follows.
 *}
 
 lemma
@@ -916,17 +928,17 @@
 text {*
   We obtain the intended subgoals and also the parameters are correctly
   introduced in both of them. Such manual instantiations are quite frequently
-  necessary in order to contrain the application of inference rules. Otherwise
-  one ends up with ``wild'' higher-order unification problems that make
-  automatic proofs fails.
+  necessary in order to appropriately constrain the application of inference 
+  rules. Otherwise one would end up with ``wild'' higher-order unification 
+  problems that make automatic proofs fail.
 *}
 
 section {* Tactic Combinators *}
 
 text {* 
   The purpose of tactic combinators is to build compound tactics out of
-  smaller tactics. In the previous section we already used @{ML THEN}, which
-  just strings together two tactics in a sequence. For example:
+  smaller tactics. In the previous section we already used @{ML_ind THEN in Tactical}, 
+  which just strings together two tactics in a sequence. For example:
 *}
 
 lemma 
@@ -958,7 +970,7 @@
   in what follows we will usually prefer it over the ``unprimed'' one. 
 
   If there is a list of tactics that should all be tried out in 
-  sequence, you can use the combinator @{ML_ind  EVERY'}. For example
+  sequence, you can use the combinator @{ML_ind  EVERY' in Tactical}. For example
   the function @{ML foo_tac'} from page~\pageref{tac:footacprime} can also 
   be written as:
 *}
@@ -980,11 +992,11 @@
 text {*
   and call @{ML foo_tac1}.  
 
-  With the combinators @{ML THEN'}, @{ML EVERY'} and @{ML_ind  EVERY1} it must be
+  With the combinators @{ML THEN'}, @{ML EVERY'} and @{ML_ind  EVERY1 in Tactical} it must be
   guaranteed that all component tactics successfully apply; otherwise the
   whole tactic will fail. If you rather want to try out a number of tactics,
-  then you can use the combinator @{ML_ind  ORELSE'} for two tactics, and @{ML_ind
-   FIRST'} (or @{ML_ind  FIRST1}) for a list of tactics. For example, the tactic
+  then you can use the combinator @{ML_ind  ORELSE' in Tactical} for two tactics, and @{ML_ind
+   FIRST' in Tactical} (or @{ML_ind  FIRST1 in Tactical}) for a list of tactics. For example, the tactic
 
 *}
 
@@ -1038,7 +1050,7 @@
 text {* 
   Since such repeated applications of a tactic to the reverse order of 
   \emph{all} subgoals is quite common, there is the tactic combinator 
-  @{ML_ind  ALLGOALS} that simplifies this. Using this combinator you can simply 
+  @{ML_ind  ALLGOALS in Tactical} that simplifies this. Using this combinator you can simply 
   write: *}
 
 lemma 
@@ -1052,7 +1064,7 @@
 text {*
   Remember that we chose to implement @{ML select_tac'} so that it 
   always  succeeds by adding @{ML all_tac} at the end of the tactic
-  list. The same can be achieved with the tactic combinator @{ML_ind  TRY}.
+  list. The same can be achieved with the tactic combinator @{ML_ind  TRY in Tactical}.
   For example:
 *}
 
@@ -1086,7 +1098,7 @@
   from the end of the theorem list. As a result @{ML select_tac'} would only
   succeed on goals where it can make progress. But for the sake of argument,
   let us suppose that this deletion is \emph{not} an option. In such cases, you can
-  use the combinator @{ML_ind  CHANGED} to make sure the subgoal has been changed
+  use the combinator @{ML_ind  CHANGED in Tactical} to make sure the subgoal has been changed
   by the tactic. Because now
 
 *}
@@ -1105,7 +1117,7 @@
 text {*
   We can further extend @{ML select_tac'} so that it not just applies to the topmost
   connective, but also to the ones immediately ``underneath'', i.e.~analyse the goal 
-  completely. For this you can use the tactic combinator @{ML_ind  REPEAT}. As an example 
+  completely. For this you can use the tactic combinator @{ML_ind  REPEAT in Tactical}. As an example 
   suppose the following tactic
 *}
 
@@ -1127,7 +1139,7 @@
   Here it is crucial that @{ML select_tac'} is prefixed with @{ML CHANGED}, 
   because otherwise @{ML REPEAT} runs into an infinite loop (it applies the
   tactic as long as it succeeds). The function
-  @{ML_ind  REPEAT1} is similar, but runs the tactic at least once (failing if 
+  @{ML_ind  REPEAT1 in Tactical} is similar, but runs the tactic at least once (failing if 
   this is not possible).
 
   If you are after the ``primed'' version of @{ML repeat_xmp_tac}, then you 
@@ -1143,7 +1155,7 @@
   and @{ML repeat_xmp_tac'} are not yet quite what we are after: the problem is
   that goals 2 and 3 are not analysed. This is because the tactic
   is applied repeatedly only to the first subgoal. To analyse also all
-  resulting subgoals, you can use the tactic combinator @{ML_ind  REPEAT_ALL_NEW}. 
+  resulting subgoals, you can use the tactic combinator @{ML_ind  REPEAT_ALL_NEW in Tactical}. 
   Suppose the tactic
 *}
 
@@ -1199,7 +1211,7 @@
   Sometimes this leads to confusing behaviour of tactics and also has
   the potential to explode the search space for tactics.
   These problems can be avoided by prefixing the tactic with the tactic 
-  combinator @{ML_ind  DETERM}.
+  combinator @{ML_ind  DETERM in Tactical}.
 *}
 
 lemma 
@@ -1306,13 +1318,10 @@
   are discharged.  Note that in Isabelle the left-rules need to be implemented
   as elimination rules. You need to prove separate lemmas corresponding to
   $\longrightarrow_{L_{1..4}}$.  In order to explore all possibilities of applying 
-  the rules, use the tactic combinator @{ML_ind  DEPTH_SOLVE}, which searches 
+  the rules, use the tactic combinator @{ML_ind  DEPTH_SOLVE in Search}, which searches 
   for a state in which all subgoals are solved. Add also rules for equality and
   run your tactic on the de Bruijn formulae discussed in Exercise~\ref{ex:debruijn}. 
   \end{exercise}
-
-  \footnote{\bf FIXME: explain @{ML_ind Cong_Tac.cong_tac}}
-
 *}
 
 section {* Simplifier Tactics *}
@@ -2534,6 +2543,8 @@
   val this = Variable.export ctxt ctxt0 [this] 
 *}
 
-
+text {*
+  If a tactic should fail, return the empty sequence.
+*}
 
 end




