
diff -r 069d525f8f1d -r ca0ac2e75f6d ProgTutorial/Package/Ind_Code.thy
--- a/ProgTutorial/Package/Ind_Code.thy	Thu Mar 19 13:28:16 2009 +0100
+++ b/ProgTutorial/Package/Ind_Code.thy	Thu Mar 19 17:50:28 2009 +0100
@@ -33,24 +33,28 @@
   @{text [display]
   "\<forall>xs. As \<longrightarrow> (\<forall>ys. Bs \<longrightarrow> (\<forall>preds. orules \<longrightarrow> pred ss))\<^isup>* \<longrightarrow>  (\<forall>preds. orules \<longrightarrow> pred ts"}
   
-  applying as many allI and impI as possible
+  By applying as many allI and impI as possible, we have
   
-  so we have @{text "As"}, @{text "(\<forall>ys. Bs \<longrightarrow> (\<forall>preds. orules \<longrightarrow> pred ss))\<^isup>*"},
+  @{text "As"}, @{text "(\<forall>ys. Bs \<longrightarrow> (\<forall>preds. orules \<longrightarrow> pred ss))\<^isup>*"},
   @{text "orules"}; and have to show @{text "pred ts"}
 
   the $i$th @{text "orule"} is of the 
   form @{text "\<forall>xs. As \<longrightarrow> (\<forall>ys. Bs \<longrightarrow> pred ss)\<^isup>* \<longrightarrow> pred ts"}.
   
-  using the @{text "As"} we ????
+  So we apply the $i$th @{text "orule"}, but we have to show the @{text "As"} (by assumption)
+  and all @{text "(\<forall>ys. Bs \<longrightarrow> pred ss)\<^isup>*"}. For the latter we use the assumptions
+  @{text "(\<forall>ys. Bs \<longrightarrow> (\<forall>preds. orules \<longrightarrow> pred ss))\<^isup>*"} and @{text "orules"}.
+
 *}
 
 
 text {*
-  First we have to produce for each predicate its definitions of the form
+  First we have to produce for each predicate the definition of the form
 
   @{text [display] "pred \<equiv> \<lambda>zs. \<forall>preds. orules \<longrightarrow> pred zs"}
 
-  In order to make definitions, we use the following wrapper for 
+  and then ``register'' the definitions with Isabelle. 
+  To do the latter, we use the following wrapper for 
   @{ML LocalTheory.define}. The wrapper takes a predicate name, a syntax
   annotation and a term representing the right-hand side of the definition.
 *}
@@ -92,20 +96,18 @@
   @{text "> MyTrue \<equiv> True"}
   \end{isabelle}
 
-  The next two functions construct the terms we need for the definitions for
-  our \isacommand{simple\_inductive} command. These 
-  terms are of the form 
+  The next two functions construct the right-hand sides of the definitions, which
+  are of the form 
 
-  @{text [display] "\<lambda>\<^raw:$zs$>. \<forall>preds. orules \<longrightarrow> pred \<^raw:$zs$>"}
+  @{text [display] "\<lambda>zs. \<forall>preds. orules \<longrightarrow> pred zs"}
 
-  The variables @{text "\<^raw:$zs$>"} need to be chosen so that they do not occur
+  The variables @{text "zs"} need to be chosen so that they do not occur
   in the @{text orules} and also be distinct from the @{text "preds"}. 
 
   The first function constructs the term for one particular predicate, say
   @{text "pred"}; the number of arguments of this predicate is
-  determined by the number of argument types of @{text "arg_tys"}. 
-  So it takes these two parameters as arguments. The other arguments are
-  all the @{text "preds"} and the @{text "orules"}.
+  determined by the number of argument types given in @{text "arg_tys"}. 
+  The other arguments are all @{text "preds"} and the @{text "orules"}.
 *}
 
 ML %linenosgray{*fun defs_aux lthy orules preds (pred, arg_tys) =
@@ -127,19 +129,17 @@
 text {*
   The function in Line 3 is just a helper function for constructing universal
   quantifications. The code in Lines 5 to 9 produces the fresh @{text
-  "\<^raw:$zs$>"}. For this it pairs every argument type with the string
+  "zs"}. For this it pairs every argument type with the string
   @{text [quotes] "z"} (Line 7); then generates variants for all these strings
-  so that they are unique w.r.t.~to the @{text "orules"} and the predicates;
+  so that they are unique w.r.t.~to the predicates and @{text "orules"} (Line 8);
   in Line 9 it generates the corresponding variable terms for the unique
   strings.
 
   The unique free variables are applied to the predicate (Line 11) using the
   function @{ML list_comb}; then the @{text orules} are prefixed (Line 12); in
   Line 13 we quantify over all predicates; and in line 14 we just abstract
-  over all the @{text "\<^raw:$zs$>"}, i.e.~the fresh arguments of the
-  predicate.
-
-  A testcase for this function is
+  over all the @{text "zs"}, i.e.~the fresh arguments of the
+  predicate. A testcase for this function is
 *}
 
 local_setup %gray{* fn lthy =>
@@ -147,7 +147,7 @@
   val orules = [@{prop "even 0"},
                 @{prop "\<forall>n::nat. odd n \<longrightarrow> even (Suc n)"},
                 @{prop "\<forall>n::nat. even n \<longrightarrow> odd (Suc n)"}] 
-  val preds = [@{term "even::nat\<Rightarrow>bool"}, @{term "odd::nat\<Rightarrow>bool"}, @{term "z::nat"}]
+  val preds = [@{term "even::nat\<Rightarrow>bool"}, @{term "odd::nat\<Rightarrow>bool"}]
   val pred = @{term "even::nat\<Rightarrow>bool"}
   val arg_tys = [@{typ "nat"}]
   val def = defs_aux lthy orules preds (pred, arg_tys)
@@ -156,14 +156,14 @@
 end *}
 
 text {*
-  It constructs the left-hand side for the definition of @{text "even"}. So we obtain 
-  as printout the term
+  It calls @{ML defs_aux} for the definition of @{text "even"} and prints
+  out the definition. So we obtain as printout 
 
   @{text [display] 
 "\<lambda>z. \<forall>even odd. (even 0) \<longrightarrow> (\<forall>n. odd n \<longrightarrow> even (Suc n)) 
                          \<longrightarrow> (\<forall>n. even n \<longrightarrow> odd (Suc n)) \<longrightarrow> even z"}
 
-  The main function for the definitions now has to just iterate the function
+  The second function for the definitions has to just iterate the function
   @{ML defs_aux} over all predicates. The argument @{text "preds"} is again
   the the list of predicates as @{ML_type term}s; the argument @{text
   "prednames"} is the list of names of the predicates; @{text "arg_tyss"} is
@@ -188,10 +188,8 @@
   @{ML ObjectLogic.atomize_term} to make the transformation (Line 4). The call
   to @{ML defs_aux} in Line 5 produces all left-hand sides of the
   definitions. The actual definitions are then made in Line 7.  The result
-  of the function is a list of theorems and a local theory.
-
-
-  A testcase for this function is 
+  of the function is a list of theorems and a local theory. A testcase for 
+  this function is 
 *}
 
 local_setup %gray {* fn lthy =>
@@ -240,9 +238,9 @@
 done
 
 text {* 
-  The code for such induction principles has to accomplish two tasks: 
+  The code for automating such induction principles has to accomplish two tasks: 
   constructing the induction principles from the given introduction
-  rules and then automatically generating a proof of them using a tactic. 
+  rules and then automatically generating proofs for them using a tactic. 
   
   The tactic will use the following helper function for instantiating universal 
   quantifiers. 
@@ -260,7 +258,7 @@
   EVERY' (map (dtac o inst_spec) ctrms)*}
 
 text {*
-  we can use @{ML inst_spec} in the following proof to instantiate the 
+  we can use @{ML inst_spec_tac} in the following proof to instantiate the 
   three quantifiers in the assumption. 
 *}
 
@@ -290,11 +288,11 @@
           assume_tac]*}
 
 text {*
-  We only have to give it as arguments the definitions, the premise 
-  (like @{text "even n"}) 
-  and the instantiations. Compare this with the manual proof given for the
-  lemma @{thm [source] man_ind_principle}.  
-  A testcase for this tactic is the function
+  We only have to give it the definitions, the premise (like @{text "even n"})
+  and the instantiations as arguments. Compare this with the manual proof
+  given for the lemma @{thm [source] man_ind_principle}: there is almos a
+  one-to-one correspondence between the \isacommand{apply}-script and the
+  @{ML induction_tac}. A testcase for this tactic is the function
 *}
 
 ML{*fun test_tac prems = 
@@ -309,7 +307,7 @@
   which indeed proves the induction principle: 
 *}
 
-lemma 
+lemma auto_ind_principle:
 assumes prems: "even n"
 shows "P 0 \<Longrightarrow> (\<And>m. Q m \<Longrightarrow> P (Suc m)) \<Longrightarrow> (\<And>m. P m \<Longrightarrow> Q (Suc m)) \<Longrightarrow> P n"
 apply(tactic {* test_tac @{thms prems} *})
@@ -322,27 +320,37 @@
   @{text "pred"} a goal of the form
 
   @{text [display] 
-  "\<And>\<^raw:$zs$>. pred \<^raw:$zs$> \<Longrightarrow> rules[preds := \<^raw:$Ps$>] \<Longrightarrow> \<^raw:$P$> \<^raw:$zs$>"}
+  "pred ?zs \<Longrightarrow> rules[preds := ?Ps] \<Longrightarrow> ?P$ ?zs"}
 
-  where the given predicates @{text preds} are replaced in the introduction 
-  rules by new distinct variables written @{text "\<^raw:$Ps$>"}. 
+  where the predicates @{text preds} are replaced in the introduction 
+  rules by new distinct variables written @{text "Ps"}.
   We also need to generate fresh arguments for the predicate @{text "pred"} in
-  the premise and the @{text "\<^raw:$P$>"} in the conclusion. We achieve
+  the premise and the @{text "?P"} in the conclusion. Note 
+  that the  @{text "?Ps"}  and @{text "?zs"} need to be
+  schematic variables that can be instantiated. This corresponds to what the
+  @{thm [source] auto_ind_principle} looks like:
+
+  \begin{isabelle}
+  \isacommand{thm}~@{thm [source] auto_ind_principle}\\
+  @{text "> \<lbrakk>even ?n; ?P 0; \<And>m. ?Q m \<Longrightarrow> ?P (Suc m); \<And>m. ?P m \<Longrightarrow> ?Q (Suc m)\<rbrakk> \<Longrightarrow> ?P ?n"}
+  \end{isabelle}
+
+  We achieve
   that in two steps. 
 
   The function below expects that the introduction rules are already appropriately
   substituted. The argument @{text "srules"} stands for these substituted
    rules; @{text cnewpreds} are the certified terms coresponding
-  to the variables @{text "\<^raw:$Ps$>"}; @{text "pred"} is the predicate for
+  to the variables @{text "Ps"}; @{text "pred"} is the predicate for
   which we prove the introduction principle; @{text "newpred"} is its
   replacement and @{text "tys"} are the argument types of this predicate.
 *}
 
-ML %linenosgray{*fun prove_induction lthy defs srules cnewpreds ((pred, newpred), tys)  =
+ML %linenosgray{*fun prove_induction lthy defs srules cnewpreds ((pred, newpred), arg_tys)  =
 let
-  val zs = replicate (length tys) "z"
+  val zs = replicate (length arg_tys) "z"
   val (newargnames, lthy') = Variable.variant_fixes zs lthy;
-  val newargs = map Free (newargnames ~~ tys)
+  val newargs = map Free (newargnames ~~ arg_tys)
   
   val prem = HOLogic.mk_Trueprop (list_comb (pred, newargs))
   val goal = Logic.list_implies 
@@ -354,33 +362,58 @@
 end *}
 
 text {* 
-  In Line 3 we produce names @{text "\<^raw:$zs$>"} for each type in the 
+  In Line 3 we produce names @{text "zs"} for each type in the 
   argument type list. Line 4 makes these names unique and declares them as 
   \emph{free} (but fixed) variables in the local theory @{text "lthy'"}. In 
   Line 5 we just construct the terms corresponding to these variables. 
   The term variables are applied to the predicate in Line 7 (this corresponds
-  to the first premise @{text "pred \<^raw:$zs$>"} of the induction principle). 
-  In Line 8 and 9, we first construct the term  @{text "\<^raw:$P$>\<^raw:$zs$>"} 
+  to the first premise @{text "pred zs"} of the induction principle). 
+  In Line 8 and 9, we first construct the term  @{text "P zs"} 
   and then add the (substituded) introduction rules as premises. In case that
   no introduction rules are given, the conclusion of this implication needs
   to be wrapped inside a @{term Trueprop}, otherwise the Isabelle's goal
   mechanism will fail. 
 
-  In Line 11 we set up the goal to be proved; in the next line call the tactic
-  for proving the induction principle. This tactic expects definitions, the
+  In Line 11 we set up the goal to be proved; in the next line we call the tactic
+  for proving the induction principle. This tactic expects the definitions, the
   premise and the (certified) predicates with which the introduction rules
-  have been substituted. This will return a theorem. However, it is a theorem
+  have been substituted. The code in these two lines will return a theorem. 
+  However, it is a theorem
   proved inside the local theory @{text "lthy'"}, where the variables @{text
-  "\<^raw:$zs$>"} are fixed, but free. By exporting this theorem from @{text
-  "lthy'"} (which contains the @{text "\<^raw:$zs$>"} as free) to @{text
-  "lthy"} (which does not), we obtain the desired quantifications @{text
-  "\<And>\<^raw:$zs$>"}.
+  "zs"} are fixed, but free. By exporting this theorem from @{text
+  "lthy'"} (which contains the @{text "zs"} as free) to @{text
+  "lthy"} (which does not), we obtain the desired schematic variables.
+*}
 
-  (FIXME testcase)
-
+local_setup %gray{* fn lthy =>
+let
+  val defs = [@{thm even_def}, @{thm odd_def}]
+  val srules = [@{prop "P (0::nat)"},
+                @{prop "\<And>n::nat. Q n \<Longrightarrow> P (Suc n)"},
+                @{prop "\<And>n::nat. P n \<Longrightarrow> Q (Suc n)"}] 
+  val cnewpreds = [@{cterm "P::nat\<Rightarrow>bool"}, @{cterm "Q::nat\<Rightarrow>bool"}]
+  val pred = @{term "even::nat\<Rightarrow>bool"}
+  val newpred = @{term "P::nat\<Rightarrow>bool"}
+  val arg_tys = [@{typ "nat"}]
+  val intro = 
+    prove_induction lthy defs srules cnewpreds ((pred, newpred), arg_tys)
+in
+  warning (str_of_thm_raw lthy intro); lthy
+end *} 
 
-  Now it is left to produce the new predicates with which the introduction
-  rules are substituted. 
+text {*
+  This prints out:
+
+  @{text [display]
+  " \<lbrakk>even ?z; P 0; \<And>n. Q n \<Longrightarrow> P (Suc n); \<And>n. P n \<Longrightarrow> Q (Suc n)\<rbrakk> \<Longrightarrow> P ?z"}
+
+  Note that the export from @{text lthy'} to @{text lthy} in Line 13 above 
+  has turned the free, but fixed, @{text "z"} into a schematic 
+  variable @{text "?z"}.
+
+  We still have to produce the new predicates with which the introduction
+  rules are substituted and iterate @{ML prove_induction} over all
+  predicates. This is what the next function does. 
 *}
 
 ML %linenosgray{*fun inductions rules defs preds arg_tyss lthy  =
@@ -404,15 +437,16 @@
 text {*
   In Line 3 we generate a string @{text [quotes] "P"} for each predicate. 
   In Line 4, we use the same trick as in the previous function, that is making the 
-  @{text "\<^raw:$Ps$>"} fresh and declaring them as fixed but free in
+  @{text "Ps"} fresh and declaring them as fixed, but free, in
   the new local theory @{text "lthy'"}. From the local theory we extract
   the ambient theory in Line 6. We need this theory in order to certify 
   the new predicates. In Line 8 we calculate the types of these new predicates
-  using the argument types. Next we turn them into terms and subsequently
-  certify them. We can now produce the substituted introduction rules 
-  (Line 11). Line 14 and 15 just iterate the proofs for all predicates.
+  using the given argument types. Next we turn them into terms and subsequently
+  certify them (Line 9 and 10). We can now produce the substituted introduction rules 
+  (Line 11) using the function @{ML subst_free}. Line 14 and 15 just iterate 
+  the proofs for all predicates.
   From this we obtain a list of theorems. Finally we need to export the 
-  fixed variables @{text "\<^raw:$Ps$>"} to obtain the correct quantification 
+  fixed variables @{text "Ps"} to obtain the schematic variables 
   (Line 16).
 
   A testcase for this function is
@@ -428,32 +462,104 @@
   val tyss = [[@{typ "nat"}], [@{typ "nat"}]]
   val ind_thms = inductions rules defs preds tyss lthy
 in
-  warning (str_of_thms lthy ind_thms); lthy
-end  
-*}
+  warning (str_of_thms_raw lthy ind_thms); lthy
+end *}
 
 
 text {*
   which prints out
 
 @{text [display]
-"> even z \<Longrightarrow> 
->  P 0 \<Longrightarrow> (\<And>m. Pa m \<Longrightarrow> P (Suc m)) \<Longrightarrow> (\<And>m. P m \<Longrightarrow> Pa (Suc m)) \<Longrightarrow> P z,
-> odd z \<Longrightarrow> 
->  P 0 \<Longrightarrow> (\<And>m. Pa m \<Longrightarrow> P (Suc m)) \<Longrightarrow> (\<And>m. P m \<Longrightarrow> Pa (Suc m)) \<Longrightarrow> Pa z"}
+"> even ?z \<Longrightarrow> ?P1 0 \<Longrightarrow> 
+> (\<And>m. ?Pa1 m \<Longrightarrow> ?P1 (Suc m)) \<Longrightarrow> (\<And>m. ?P1 m \<Longrightarrow> ?Pa1 (Suc m)) \<Longrightarrow> ?P1 ?z,
+> odd ?z \<Longrightarrow> ?P1 0 
+> \<Longrightarrow> (\<And>m. ?Pa1 m \<Longrightarrow> ?P1 (Suc m)) \<Longrightarrow> (\<And>m. ?P1 m \<Longrightarrow> ?Pa1 (Suc m)) \<Longrightarrow> ?Pa1 ?z"}
 
 
-  This completes the code for the induction principles. Finally we can 
-  prove the introduction rules. 
+  Note that now both, the @{text "Ps"} and the @{text "zs"}, are schematic
+  variables. The numbers have been introduced by the pretty-printer and are 
+  not significant.
 
+  This completes the code for the induction principles.  Finally we can prove the 
+  introduction rules. Their proofs are quite a bit more involved. To ease them 
+  somewhat we  use the following two helper function.
 *}
 
-ML {* ObjectLogic.rulify  *}
-
-
 ML{*val all_elims = fold (fn ct => fn th => th RS inst_spec ct)
 val imp_elims = fold (fn th => fn th' => [th', th] MRS @{thm mp})*}
 
+text {* 
+  To see what they do, let us suppose whe have the follwoing three
+  theorems. 
+*}
+
+lemma all_elims_test:
+  fixes P::"nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"
+  shows "\<forall>x y z. P x y z" sorry
+
+lemma imp_elims_test:
+  fixes A B C::"bool"
+  shows "A \<longrightarrow> B \<longrightarrow> C" sorry
+
+lemma imp_elims_test':
+  fixes A::"bool"
+  shows "A" "B" sorry
+
+text {*
+  The function @{ML all_elims} takes a list of (certified) terms and instantiates
+  theorems of the form @{thm [source] all_elims_test}. For example we can instantiate
+  the quantifiers in this theorem with @{term a}, @{term b} and @{term c} as follows
+
+  @{ML_response_fake [display, gray]
+"let
+  val ctrms = [@{cterm \"a::nat\"}, @{cterm \"b::nat\"}, @{cterm \"c::nat\"}]
+  val new_thm = all_elims ctrms @{thm all_elims_test}
+in
+  warning (str_of_thm @{context} new_thm)
+end"
+  "P a b c"}
+
+  Similarly, the function @{ML imp_elims} eliminates preconditions from implications. 
+  For example 
+
+  @{ML_response_fake [display, gray]
+"warning (str_of_thm @{context} 
+            (imp_elims @{thms imp_elims_test'} @{thm imp_elims_test}))"
+  "C"}
+*}
+
+ML {* prems_of *}
+ML {* Logic.strip_params *}
+ML {* Logic.strip_assums_hyp *}
+
+ML {*
+fun chop_print_tac ctxt thm =
+let
+  val [trm] = prems_of thm
+  val params = map fst (Logic.strip_params trm)
+  val prems  = Logic.strip_assums_hyp trm
+  val (prems1, prems2) = chop (length prems - 3) prems;
+  val (params1, params2) = chop (length params - 2) params;
+  val _ = warning (Syntax.string_of_term ctxt trm)
+  val _ = warning (commas params)
+  val _ = warning (commas (map (Syntax.string_of_term ctxt) prems))
+  val _ = warning ((commas params1) ^ " | " ^ (commas params2))
+  val _ = warning ((commas (map (Syntax.string_of_term ctxt) prems1)) ^ " | " ^
+                   (commas (map (Syntax.string_of_term ctxt) prems2)))
+in
+   Seq.single thm
+end
+*}
+
+
+lemma intro1:
+  shows "\<And>m. odd m \<Longrightarrow> even (Suc m)"
+apply(tactic {* ObjectLogic.rulify_tac 1 *})
+apply(tactic {* rewrite_goals_tac [@{thm even_def}, @{thm odd_def}] *})
+apply(tactic {* REPEAT (resolve_tac [@{thm allI}, @{thm impI}] 1) *})
+apply(tactic {* chop_print_tac @{context} *})
+oops
+
 ML{*fun subproof2 prem params2 prems2 =  
  SUBPROOF (fn {prems, ...} =>
    let
@@ -468,43 +574,79 @@
      rtac prem'' 1 
    end)*}
 
-ML{*fun subproof1 rules preds i = 
+text {*
+
+*}
+
+
+ML %linenosgray{*fun subproof1 rules preds i = 
  SUBPROOF (fn {params, prems, context = ctxt', ...} =>
    let
      val (prems1, prems2) = chop (length prems - length rules) prems;
      val (params1, params2) = chop (length params - length preds) params;
    in
      rtac (ObjectLogic.rulify (all_elims params1 (nth prems2 i))) 1 
+     (* applicateion of the i-ith intro rule *)
      THEN
      EVERY1 (map (fn prem => subproof2 prem params2 prems2 ctxt') prems1)
    end)*}
 
+text {*
+  @{text "params1"} are the variables of the rules; @{text "params2"} is
+  the variables corresponding to the @{text "preds"}.
+
+  @{text "prems1"} are the assumption corresponding to the rules;
+  @{text "prems2"} are the assumptions coming from the allIs/impIs
+
+  you instantiate the parameters i-th introduction rule with the parameters
+  that come from the rule; and you apply it to the goal
+
+  this now generates subgoals corresponding to the premisses of this
+  intro rule 
+*}
+
 ML{*
-fun introductions_tac defs rules preds i ctxt =
+fun intros_tac defs rules preds i ctxt =
   EVERY1 [ObjectLogic.rulify_tac,
           K (rewrite_goals_tac defs),
           REPEAT o (resolve_tac [@{thm allI}, @{thm impI}]),
           subproof1 rules preds i ctxt]*}
 
-lemma evenS: 
-  shows "odd m \<Longrightarrow> even (Suc m)"
-apply(tactic {* 
+text {*
+  A test case
+*}
+
+ML{*fun intros_tac_test ctxt i =
 let
   val rules = [@{prop "even (0::nat)"},
-                 @{prop "\<And>n::nat. odd n \<Longrightarrow> even (Suc n)"},
-                 @{prop "\<And>n::nat. even n \<Longrightarrow> odd (Suc n)"}] 
+               @{prop "\<And>n::nat. odd n \<Longrightarrow> even (Suc n)"},
+               @{prop "\<And>n::nat. even n \<Longrightarrow> odd (Suc n)"}] 
   val defs = [@{thm even_def}, @{thm odd_def}]
   val preds = [@{term "even::nat\<Rightarrow>bool"}, @{term "odd::nat\<Rightarrow>bool"}]
 in
-  introductions_tac defs rules preds 1 @{context}
-end *})
+  intros_tac defs rules preds i ctxt
+end*}
+
+lemma intro0:
+  shows "even 0"
+apply(tactic {* intros_tac_test @{context} 0 *})
+done
+
+lemma intro1:
+  shows "\<And>m. odd m \<Longrightarrow> even (Suc m)"
+apply(tactic {* intros_tac_test @{context} 1 *})
+done
+
+lemma intro2:
+  shows "\<And>m. even m \<Longrightarrow> odd (Suc m)"
+apply(tactic {* intros_tac_test @{context} 2 *})
 done
 
 ML{*fun introductions rules preds defs lthy = 
 let
   fun prove_intro (i, goal) =
     Goal.prove lthy [] [] goal
-      (fn {context, ...} => introductions_tac defs rules preds i context)
+      (fn {context, ...} => intros_tac defs rules preds i context)
 in
   map_index prove_intro rules
 end*}




