src/HOL/HOL.thy

   the_eq_trivial: "(THE x. x = a) = (a::'a)"
 
   impI:           "(P ==> Q) ==> P-->Q"
   mp:             "[| P-->Q;  P |] ==> Q"
 
-
-text{*Thanks to Stephan Merz*}
-theorem subst:
-  assumes eq: "s = t" and p: "P(s)"
-  shows "P(t::'a)"
-proof -
-  from eq have meta: "s \<equiv> t"
-    by (rule eq_reflection)
-  from p show ?thesis
-    by (unfold meta)
-qed
 
 defs
   True_def:     "True      == ((%x::bool. x) = (%x. x))"
   All_def:      "All(P)    == (P = (%x. True))"
   Ex_def:       "Ex(P)     == !Q. (!x. P x --> Q) --> Q"

   abs  ("\<bar>_\<bar>")
 const_syntax (HTML output)
   abs  ("\<bar>_\<bar>")
 
 
-subsection {*Equality*}
+subsection {* Fundamental rules *}
+
+subsubsection {*Equality*}
+
+text {* Thanks to Stephan Merz *}
+lemma subst:
+  assumes eq: "s = t" and p: "P s"
+  shows "P t"
+proof -
+  from eq have meta: "s \<equiv> t"
+    by (rule eq_reflection)
+  from p show ?thesis
+    by (unfold meta)
+qed
 
 lemma sym: "s = t ==> t = s"
   by (erule subst) (rule refl)
 
 lemma ssubst: "t = s ==> P s ==> P t"
   by (drule sym) (erule subst)
 
 lemma trans: "[| r=s; s=t |] ==> r=t"
   by (erule subst)
 
-lemma def_imp_eq: assumes meq: "A == B" shows "A = B"
+lemma def_imp_eq:
+  assumes meq: "A == B"
+  shows "A = B"
   by (unfold meq) (rule refl)
 
-
-(*Useful with eresolve_tac for proving equalties from known equalities.
-        a = b
+(*a mere copy*)
+lemma meta_eq_to_obj_eq: 
+  assumes meq: "A == B"
+  shows "A = B"
+  by (unfold meq) (rule refl)
+
+text {* Useful with eresolve\_tac for proving equalties from known equalities. *}
+     (* a = b
         |   |
         c = d   *)
 lemma box_equals: "[| a=b;  a=c;  b=d |] ==> c=d"
 apply (rule trans)
 apply (rule trans)

 
 lemma back_subst: "P a ==> a = b ==> P b"
   by (rule subst)
 
 
-subsection {*Congruence rules for application*}
+subsubsection {*Congruence rules for application*}
 
 (*similar to AP_THM in Gordon's HOL*)
 lemma fun_cong: "(f::'a=>'b) = g ==> f(x)=g(x)"
 apply (erule subst)
 apply (rule refl)

 lemma arg_cong2: "\<lbrakk> a = b; c = d \<rbrakk> \<Longrightarrow> f a c = f b d"
 apply (erule ssubst)+
 apply (rule refl)
 done
 
-
 lemma cong: "[| f = g; (x::'a) = y |] ==> f(x) = g(y)"
 apply (erule subst)+
 apply (rule refl)
 done
 
 
-subsection {*Equality of booleans -- iff*}
+subsubsection {*Equality of booleans -- iff*}
 
 lemma iffI: assumes prems: "P ==> Q" "Q ==> P" shows "P=Q"
   by (iprover intro: iff [THEN mp, THEN mp] impI prems)
 
 lemma iffD2: "[| P=Q; Q |] ==> P"

       and minor: "[| P --> Q; Q --> P |] ==> R"
   shows R
   by (iprover intro: minor impI major [THEN iffD2] major [THEN iffD1])
 
 
-subsection {*True*}
+subsubsection {*True*}
 
 lemma TrueI: "True"
   by (unfold True_def) (rule refl)
 
 lemma eqTrueI: "P ==> P=True"

 apply (erule iffD2)
 apply (rule TrueI)
 done
 
 
-subsection {*Universal quantifier*}
+subsubsection {*Universal quantifier*}
 
 lemma allI: assumes p: "!!x::'a. P(x)" shows "ALL x. P(x)"
 apply (unfold All_def)
 apply (iprover intro: ext eqTrueI p)
 done

       and minor: "[| P(x); ALL x. P(x) |] ==> R"
   shows "R"
 by (iprover intro: minor major major [THEN spec])
 
 
-subsection {*False*}
+subsubsection {*False*}
 (*Depends upon spec; it is impossible to do propositional logic before quantifiers!*)
 
 lemma FalseE: "False ==> P"
 apply (unfold False_def)
 apply (erule spec)

 
 lemma False_neq_True: "False=True ==> P"
 by (erule eqTrueE [THEN FalseE])
 
 
-subsection {*Negation*}
+subsubsection {*Negation*}
 
 lemma notI:
   assumes p: "P ==> False"
   shows "~P"
 apply (unfold not_def)

 
 (* Alternative ~ introduction rule: [| P ==> ~ Pa; P ==> Pa |] ==> ~ P *)
 lemmas notI2 = notE [THEN notI, standard]
 
 
-subsection {*Implication*}
+subsubsection {*Implication*}
 
 lemma impE:
   assumes "P-->Q" "P" "Q ==> R"
   shows "R"
 by (iprover intro: prems mp)

   shows "~P"
 apply (rule nq [THEN contrapos_nn])
 apply (erule pq)
 done
 
-subsection {*Existential quantifier*}
+subsubsection {*Existential quantifier*}
 
 lemma exI: "P x ==> EX x::'a. P x"
 apply (unfold Ex_def)
 apply (iprover intro: allI allE impI mp)
 done

 apply (rule major [unfolded Ex_def, THEN spec, THEN mp])
 apply (iprover intro: impI [THEN allI] minor)
 done
 
 
-subsection {*Conjunction*}
+subsubsection {*Conjunction*}
 
 lemma conjI: "[| P; Q |] ==> P&Q"
 apply (unfold and_def)
 apply (iprover intro: impI [THEN allI] mp)
 done

 lemma context_conjI:
   assumes prems: "P" "P ==> Q" shows "P & Q"
 by (iprover intro: conjI prems)
 
 
-subsection {*Disjunction*}
+subsubsection {*Disjunction*}
 
 lemma disjI1: "P ==> P|Q"
 apply (unfold or_def)
 apply (iprover intro: allI impI mp)
 done

   shows "R"
 by (iprover intro: minorP minorQ impI
                  major [unfolded or_def, THEN spec, THEN mp, THEN mp])
 
 
-subsection {*Classical logic*}
-
+subsubsection {*Classical logic*}
 
 lemma classical:
   assumes prem: "~P ==> P"
   shows "P"
 apply (rule True_or_False [THEN disjE, THEN eqTrueE])

       and p2: "~P ==> ~Q"
   shows "P"
 by (iprover intro: classical p1 p2 notE)
 
 
-subsection {*Unique existence*}
+subsubsection {*Unique existence*}
 
 lemma ex1I:
   assumes prems: "P a" "!!x. P(x) ==> x=a"
   shows "EX! x. P(x)"
 by (unfold Ex1_def, iprover intro: prems exI conjI allI impI)

 apply (rule exI)
 apply assumption
 done
 
 
-subsection {*THE: definite description operator*}
+subsubsection {*THE: definite description operator*}
 
 lemma the_equality:
   assumes prema: "P a"
       and premx: "!!x. P x ==> x=a"
   shows "(THE x. P x) = a"

 apply (rule refl)
 apply (erule sym)
 done
 
 
-subsection {*Classical intro rules for disjunction and existential quantifiers*}
+subsubsection {*Classical intro rules for disjunction and existential quantifiers*}
 
 lemma disjCI:
   assumes "~Q ==> P" shows "P|Q"
 apply (rule classical)
 apply (iprover intro: prems disjI1 disjI2 notI elim: notE)
 done
 
 lemma excluded_middle: "~P | P"
 by (iprover intro: disjCI)
 
-text{*case distinction as a natural deduction rule. Note that @{term "~P"}
-   is the second case, not the first.*}
+text {*
+  case distinction as a natural deduction rule.
+  Note that @{term "~P"} is the second case, not the first
+*}
 lemma case_split_thm:
   assumes prem1: "P ==> Q"
       and prem2: "~P ==> Q"
   shows "Q"
 apply (rule excluded_middle [THEN disjE])
 apply (erule prem2)
 apply (erule prem1)
 done
+lemmas case_split = case_split_thm [case_names True False]
 
 (*Classical implies (-->) elimination. *)
 lemma impCE:
   assumes major: "P-->Q"
       and minor: "~P ==> R" "Q ==> R"

   assumes "ALL x. ~P(x) ==> P(a)"
   shows "EX x. P(x)"
 apply (rule ccontr)
 apply (iprover intro: prems exI allI notI notE [of "\<exists>x. P x"])
 done
-
-
-
-subsection {* Theory and package setup *}
-
-ML
-{*
-val eq_reflection = thm "eq_reflection"
-val refl = thm "refl"
-val subst = thm "subst"
-val ext = thm "ext"
-val impI = thm "impI"
-val mp = thm "mp"
-val True_def = thm "True_def"
-val All_def = thm "All_def"
-val Ex_def = thm "Ex_def"
-val False_def = thm "False_def"
-val not_def = thm "not_def"
-val and_def = thm "and_def"
-val or_def = thm "or_def"
-val Ex1_def = thm "Ex1_def"
-val iff = thm "iff"
-val True_or_False = thm "True_or_False"
-val Let_def = thm "Let_def"
-val if_def = thm "if_def"
-val sym = thm "sym"
-val ssubst = thm "ssubst"
-val trans = thm "trans"
-val def_imp_eq = thm "def_imp_eq"
-val box_equals = thm "box_equals"
-val fun_cong = thm "fun_cong"
-val arg_cong = thm "arg_cong"
-val cong = thm "cong"
-val iffI = thm "iffI"
-val iffD2 = thm "iffD2"
-val rev_iffD2 = thm "rev_iffD2"
-val iffD1 = thm "iffD1"
-val rev_iffD1 = thm "rev_iffD1"
-val iffE = thm "iffE"
-val TrueI = thm "TrueI"
-val eqTrueI = thm "eqTrueI"
-val eqTrueE = thm "eqTrueE"
-val allI = thm "allI"
-val spec = thm "spec"
-val allE = thm "allE"
-val all_dupE = thm "all_dupE"
-val FalseE = thm "FalseE"
-val False_neq_True = thm "False_neq_True"
-val notI = thm "notI"
-val False_not_True = thm "False_not_True"
-val True_not_False = thm "True_not_False"
-val notE = thm "notE"
-val notI2 = thm "notI2"
-val impE = thm "impE"
-val rev_mp = thm "rev_mp"
-val contrapos_nn = thm "contrapos_nn"
-val contrapos_pn = thm "contrapos_pn"
-val not_sym = thm "not_sym"
-val rev_contrapos = thm "rev_contrapos"
-val exI = thm "exI"
-val exE = thm "exE"
-val conjI = thm "conjI"
-val conjunct1 = thm "conjunct1"
-val conjunct2 = thm "conjunct2"
-val conjE = thm "conjE"
-val context_conjI = thm "context_conjI"
-val disjI1 = thm "disjI1"
-val disjI2 = thm "disjI2"
-val disjE = thm "disjE"
-val classical = thm "classical"
-val ccontr = thm "ccontr"
-val rev_notE = thm "rev_notE"
-val notnotD = thm "notnotD"
-val contrapos_pp = thm "contrapos_pp"
-val ex1I = thm "ex1I"
-val ex_ex1I = thm "ex_ex1I"
-val ex1E = thm "ex1E"
-val ex1_implies_ex = thm "ex1_implies_ex"
-val the_equality = thm "the_equality"
-val theI = thm "theI"
-val theI' = thm "theI'"
-val theI2 = thm "theI2"
-val the1_equality = thm "the1_equality"
-val the_sym_eq_trivial = thm "the_sym_eq_trivial"
-val disjCI = thm "disjCI"
-val excluded_middle = thm "excluded_middle"
-val case_split_thm = thm "case_split_thm"
-val impCE = thm "impCE"
-val impCE = thm "impCE"
-val iffCE = thm "iffCE"
-val exCI = thm "exCI"
-
-(* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
-local
-  fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
-  |   wrong_prem (Bound _) = true
-  |   wrong_prem _ = false
-  val filter_right = List.filter (fn t => not (wrong_prem (HOLogic.dest_Trueprop (hd (Thm.prems_of t)))))
-in
-  fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp])
-  fun smp_tac j = EVERY'[dresolve_tac (smp j), atac]
-end
-
-
-fun strip_tac i = REPEAT(resolve_tac [impI,allI] i)
-
-(*Obsolete form of disjunctive case analysis*)
-fun excluded_middle_tac sP =
-    res_inst_tac [("Q",sP)] (excluded_middle RS disjE)
-
-fun case_tac a = res_inst_tac [("P",a)] case_split_thm
-*}
-
-theorems case_split = case_split_thm [case_names True False]
-
-ML {*
-structure ProjectRule = ProjectRuleFun
-(struct
-  val conjunct1 = thm "conjunct1";
-  val conjunct2 = thm "conjunct2";
-  val mp = thm "mp";
-end)
-*}
 
 
 subsubsection {* Intuitionistic Reasoning *}
 
 lemma impE':

 
 lemmas [symmetric, rulify] = atomize_all atomize_imp
   and [symmetric, defn] = atomize_all atomize_imp atomize_eq
 
 
+subsection {* Package setup *}
+
+subsubsection {* Fundamental ML bindings *}
+
+ML {*
+structure HOL =
+struct
+  (*FIXME reduce this to a minimum*)
+  val eq_reflection = thm "eq_reflection";
+  val def_imp_eq = thm "def_imp_eq";
+  val meta_eq_to_obj_eq = thm "meta_eq_to_obj_eq";
+  val ccontr = thm "ccontr";
+  val impI = thm "impI";
+  val impCE = thm "impCE";
+  val notI = thm "notI";
+  val notE = thm "notE";
+  val iffI = thm "iffI";
+  val iffCE = thm "iffCE";
+  val conjI = thm "conjI";
+  val conjE = thm "conjE";
+  val disjCI = thm "disjCI";
+  val disjE = thm "disjE";
+  val TrueI = thm "TrueI";
+  val FalseE = thm "FalseE";
+  val allI = thm "allI";
+  val allE = thm "allE";
+  val exI = thm "exI";
+  val exE = thm "exE";
+  val ex_ex1I = thm "ex_ex1I";
+  val the_equality = thm "the_equality";
+  val mp = thm "mp";
+  val rev_mp = thm "rev_mp"
+  val classical = thm "classical";
+  val subst = thm "subst";
+  val refl = thm "refl";
+  val sym = thm "sym";
+  val trans = thm "trans";
+  val arg_cong = thm "arg_cong";
+  val iffD1 = thm "iffD1";
+  val iffD2 = thm "iffD2";
+  val disjE = thm "disjE";
+  val conjE = thm "conjE";
+  val exE = thm "exE";
+  val contrapos_nn = thm "contrapos_nn";
+  val contrapos_pp = thm "contrapos_pp";
+  val notnotD = thm "notnotD";
+  val conjunct1 = thm "conjunct1";
+  val conjunct2 = thm "conjunct2";
+  val spec = thm "spec";
+  val imp_cong = thm "imp_cong";
+  val the_sym_eq_trivial = thm "the_sym_eq_trivial";
+  val triv_forall_equality = thm "triv_forall_equality";
+  val case_split = thm "case_split_thm";
+end
+*}
+
+
 subsubsection {* Classical Reasoner setup *}
 
+lemma thin_refl:
+  "\<And>X. \<lbrakk> x=x; PROP W \<rbrakk> \<Longrightarrow> PROP W" .
+
 use "cladata.ML"
-setup hypsubst_setup
-setup {* ContextRules.addSWrapper (fn tac => hyp_subst_tac' ORELSE' tac) *}
+ML {* val HOL_cs = HOL.cs *}
+setup Hypsubst.hypsubst_setup
+setup {* ContextRules.addSWrapper (fn tac => HOL.hyp_subst_tac' ORELSE' tac) *}
 setup Classical.setup
 setup ResAtpset.setup
-setup clasetup
+setup {* fn thy => (Classical.change_claset_of thy (K HOL.cs); thy) *}
 
 lemma contrapos_np: "~ Q ==> (~ P ==> Q) ==> P"
   apply (erule swap)
   apply (erule (1) meta_mp)
   done

   and ex1I [intro]
 
 lemmas [intro?] = ext
   and [elim?] = ex1_implies_ex
 
+(*Better then ex1E for classical reasoner: needs no quantifier duplication!*)
+lemma alt_ex1E:
+  assumes major: "\<exists>!x. P x"
+      and prem: "\<And>x. \<lbrakk> P x; \<forall>y y'. P y \<and> P y' \<longrightarrow> y = y' \<rbrakk> \<Longrightarrow> R"
+  shows R
+apply (rule ex1E [OF major])
+apply (rule prem)
+apply (tactic "ares_tac [HOL.allI] 1")+
+apply (tactic "etac (Classical.dup_elim HOL.allE) 1")
+by iprover
+
 use "blastdata.ML"
 setup Blast.setup
 
-
-subsubsection {* Simplifier setup *}
-
-lemma meta_eq_to_obj_eq: "x == y ==> x = y"
-proof -
-  assume r: "x == y"
-  show "x = y" by (unfold r) (rule refl)
-qed
+ML {*
+structure HOL =
+struct
+
+open HOL;
+
+fun case_tac a = res_inst_tac [("P", a)] case_split;
+
+end;
+*}
+
+
+subsubsection {* Simplifier *}
 
 lemma eta_contract_eq: "(%s. f s) = f" ..
 
 lemma simp_thms:
   shows not_not: "(~ ~ P) = P"
   and Not_eq_iff: "((~P) = (~Q)) = (P = Q)"
   and
     "(P ~= Q) = (P = (~Q))"
     "(P | ~P) = True"    "(~P | P) = True"
     "(x = x) = True"
-    "(~True) = False"  "(~False) = True"
+  and not_True_eq_False: "(\<not> True) = False"
+  and not_False_eq_True: "(\<not> False) = True"
+  and
     "(~P) ~= P"  "P ~= (~P)"
-    "(True=P) = P"  "(P=True) = P"  "(False=P) = (~P)"  "(P=False) = (~P)"
+    "(True=P) = P"
+  and eq_True: "(P = True) = P"
+  and "(False=P) = (~P)"
+  and eq_False: "(P = False) = (\<not> P)"
+  and
     "(True --> P) = P"  "(False --> P) = True"
     "(P --> True) = True"  "(P --> P) = True"
     "(P --> False) = (~P)"  "(P --> ~P) = (~P)"
     "(P & True) = P"  "(True & P) = P"
     "(P & False) = False"  "(False & P) = False"

 
 lemma disj_cong:
     "(P = P') ==> (~P' ==> (Q = Q')) ==> ((P | Q) = (P' | Q'))"
   by blast
 
-lemma eq_sym_conv: "(x = y) = (y = x)"
-  by iprover
-
 
 text {* \medskip if-then-else rules *}
 
 lemma if_True: "(if True then x else y) = x"
   by (unfold if_def) blast

 
 lemma split_if_asm: "P (if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
 by (simplesubst split_if, blast)
 
 lemmas if_splits = split_if split_if_asm
-
-lemma if_def2: "(if Q then x else y) = ((Q --> x) & (~ Q --> y))"
-  by (rule split_if)
 
 lemma if_cancel: "(if c then x else x) = x"
 by (simplesubst split_if, blast)
 
 lemma if_eq_cancel: "(if x = y then y else x) = x"

 qed
 
 
 text {* \medskip Actual Installation of the Simplifier. *}
 
+lemma True_implies_equals: "(True \<Longrightarrow> PROP P) \<equiv> PROP P"
+proof
+  assume prem: "True \<Longrightarrow> PROP P"
+  from prem [OF TrueI] show "PROP P" . 
+next
+  assume "PROP P"
+  show "PROP P" .
+qed
+
+lemma uncurry:
+  assumes "P \<longrightarrow> Q \<longrightarrow> R"
+  shows "P \<and> Q \<longrightarrow> R"
+  using prems by blast
+
+lemma iff_allI:
+  assumes "\<And>x. P x = Q x"
+  shows "(\<forall>x. P x) = (\<forall>x. Q x)"
+  using prems by blast
+
+lemma iff_exI:
+  assumes "\<And>x. P x = Q x"
+  shows "(\<exists>x. P x) = (\<exists>x. Q x)"
+  using prems by blast
+
+lemma all_comm:
+  "(\<forall>x y. P x y) = (\<forall>y x. P x y)"
+  by blast
+
+lemma ex_comm:
+  "(\<exists>x y. P x y) = (\<exists>y x. P x y)"
+  by blast
+
 use "simpdata.ML"
-setup "Simplifier.method_setup Splitter.split_modifiers" setup simpsetup
-setup Splitter.setup setup Clasimp.setup
+setup "Simplifier.method_setup Splitter.split_modifiers"
+setup simpsetup
+setup Splitter.setup
+setup Clasimp.setup
 setup EqSubst.setup
+
+text {* Simplifies x assuming c and y assuming ~c *}
+lemma if_cong:
+  assumes "b = c"
+      and "c \<Longrightarrow> x = u"
+      and "\<not> c \<Longrightarrow> y = v"
+  shows "(if b then x else y) = (if c then u else v)"
+  unfolding if_def using prems by simp
+
+text {* Prevents simplification of x and y:
+  faster and allows the execution of functional programs. *}
+lemma if_weak_cong [cong]:
+  assumes "b = c"
+  shows "(if b then x else y) = (if c then x else y)"
+  using prems by (rule arg_cong)
+
+text {* Prevents simplification of t: much faster *}
+lemma let_weak_cong:
+  assumes "a = b"
+  shows "(let x = a in t x) = (let x = b in t x)"
+  using prems by (rule arg_cong)
+
+text {* To tidy up the result of a simproc.  Only the RHS will be simplified. *}
+lemma eq_cong2:
+  assumes "u = u'"
+  shows "(t \<equiv> u) \<equiv> (t \<equiv> u')"
+  using prems by simp
+
+lemma if_distrib:
+  "f (if c then x else y) = (if c then f x else f y)"
+  by simp
+
+text {* For expand\_case\_tac *}
+lemma expand_case:
+  assumes "P \<Longrightarrow> Q True"
+      and "~P \<Longrightarrow> Q False"
+  shows "Q P"
+proof (tactic {* HOL.case_tac "P" 1 *})
+  assume P
+  then show "Q P" by simp
+next
+  assume "\<not> P"
+  then have "P = False" by simp
+  with prems show "Q P" by simp
+qed
+
+text {* This lemma restricts the effect of the rewrite rule u=v to the left-hand
+  side of an equality.  Used in {Integ,Real}/simproc.ML *}
+lemma restrict_to_left:
+  assumes "x = y"
+  shows "(x = z) = (y = z)"
+  using prems by simp
+
+
+subsubsection {* Generic cases and induction *}
+
+text {* Rule projections: *}
+
+ML {*
+structure ProjectRule = ProjectRuleFun
+(struct
+  val conjunct1 = thm "conjunct1";
+  val conjunct2 = thm "conjunct2";
+  val mp = thm "mp";
+end)
+*}
+
+constdefs
+  induct_forall where "induct_forall P == \<forall>x. P x"
+  induct_implies where "induct_implies A B == A \<longrightarrow> B"
+  induct_equal where "induct_equal x y == x = y"
+  induct_conj where "induct_conj A B == A \<and> B"
+
+lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
+  by (unfold atomize_all induct_forall_def)
+
+lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
+  by (unfold atomize_imp induct_implies_def)
+
+lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
+  by (unfold atomize_eq induct_equal_def)
+
+lemma induct_conj_eq:
+  includes meta_conjunction_syntax
+  shows "(A && B) == Trueprop (induct_conj A B)"
+  by (unfold atomize_conj induct_conj_def)
+
+lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
+lemmas induct_rulify [symmetric, standard] = induct_atomize
+lemmas induct_rulify_fallback =
+  induct_forall_def induct_implies_def induct_equal_def induct_conj_def
+
+
+lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
+    induct_conj (induct_forall A) (induct_forall B)"
+  by (unfold induct_forall_def induct_conj_def) iprover
+
+lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
+    induct_conj (induct_implies C A) (induct_implies C B)"
+  by (unfold induct_implies_def induct_conj_def) iprover
+
+lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
+proof
+  assume r: "induct_conj A B ==> PROP C" and A B
+  show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)
+next
+  assume r: "A ==> B ==> PROP C" and "induct_conj A B"
+  show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
+qed
+
+lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
+
+hide const induct_forall induct_implies induct_equal induct_conj
+
+text {* Method setup. *}
+
+ML {*
+  structure InductMethod = InductMethodFun
+  (struct
+    val cases_default = thm "case_split"
+    val atomize = thms "induct_atomize"
+    val rulify = thms "induct_rulify"
+    val rulify_fallback = thms "induct_rulify_fallback"
+  end);
+*}
+
+setup InductMethod.setup
 
 
 subsubsection {* Code generator setup *}
 
 types_code

 
 fun evaluation_tac i = Tactical.PRIMITIVE (Drule.fconv_rule
   (Drule.goals_conv (equal i) Codegen.evaluation_conv));
 
 val evaluation_meth =
-  Method.no_args (Method.METHOD (fn _ => evaluation_tac 1 THEN rtac TrueI 1));
+  Method.no_args (Method.METHOD (fn _ => evaluation_tac 1 THEN rtac HOL.TrueI 1));
 
 in
 
 Method.add_method ("evaluation", evaluation_meth, "solve goal by evaluation")
 
 end;
 *}
 
 
-subsection {* Other simple lemmas *}
-
-declare disj_absorb [simp] conj_absorb [simp]
-
-lemma ex1_eq[iff]: "EX! x. x = t" "EX! x. t = x"
-by blast+
-
-
-theorem choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
+text {* itself as a code generator datatype *}
+
+setup {*
+let fun add_itself thy =
+  let
+    val v = ("'a", []);
+    val t = Logic.mk_type (TFree v);
+    val Const (c, ty) = t;
+    val (_, Type (dtco, _)) = strip_type ty;
+  in
+    thy
+    |> CodegenData.add_datatype (dtco, (([v], [(c, [])]), CodegenData.lazy (fn () => [])))
+  end
+in add_itself end;
+*} 
+
+text {* code generation for arbitrary as exception *}
+
+setup {*
+  CodegenSerializer.add_undefined "SML" "arbitrary" "(raise Fail \"arbitrary\")"
+*}
+
+code_const arbitrary
+  (Haskell target_atom "(error \"arbitrary\")")
+
+
+subsection {* Other simple lemmas and lemma duplicates *}
+
+lemmas eq_sym_conv = eq_commute
+lemmas if_def2 = if_bool_eq_conj
+
+lemma ex1_eq [iff]: "EX! x. x = t" "EX! x. t = x"
+  by blast+
+
+lemma choice_eq: "(ALL x. EX! y. P x y) = (EX! f. ALL x. P x (f x))"
   apply (rule iffI)
   apply (rule_tac a = "%x. THE y. P x y" in ex1I)
   apply (fast dest!: theI')
   apply (fast intro: ext the1_equality [symmetric])
   apply (erule ex1E)

   apply (rule allI)
   apply (rule_tac P = "xa = x" in case_split_thm)
   apply (drule_tac [3] x = x in fun_cong, simp_all)
   done
 
-text{*Needs only HOL-lemmas:*}
+text {* Needs only HOL-lemmas *}
 lemma mk_left_commute:
   assumes a: "\<And>x y z. f (f x y) z = f x (f y z)" and
           c: "\<And>x y. f x y = f y x"
   shows "f x (f y z) = f y (f x z)"
-by(rule trans[OF trans[OF c a] arg_cong[OF c, of "f y"]])
-
-
-subsection {* Generic cases and induction *}
-
-constdefs
-  induct_forall where "induct_forall P == \<forall>x. P x"
-  induct_implies where "induct_implies A B == A \<longrightarrow> B"
-  induct_equal where "induct_equal x y == x = y"
-  induct_conj where "induct_conj A B == A \<and> B"
-
-lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
-  by (unfold atomize_all induct_forall_def)
-
-lemma induct_implies_eq: "(A ==> B) == Trueprop (induct_implies A B)"
-  by (unfold atomize_imp induct_implies_def)
-
-lemma induct_equal_eq: "(x == y) == Trueprop (induct_equal x y)"
-  by (unfold atomize_eq induct_equal_def)
-
-lemma induct_conj_eq:
-  includes meta_conjunction_syntax
-  shows "(A && B) == Trueprop (induct_conj A B)"
-  by (unfold atomize_conj induct_conj_def)
-
-lemmas induct_atomize = induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
-lemmas induct_rulify [symmetric, standard] = induct_atomize
-lemmas induct_rulify_fallback =
-  induct_forall_def induct_implies_def induct_equal_def induct_conj_def
-
-
-lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
-    induct_conj (induct_forall A) (induct_forall B)"
-  by (unfold induct_forall_def induct_conj_def) iprover
-
-lemma induct_implies_conj: "induct_implies C (induct_conj A B) =
-    induct_conj (induct_implies C A) (induct_implies C B)"
-  by (unfold induct_implies_def induct_conj_def) iprover
-
-lemma induct_conj_curry: "(induct_conj A B ==> PROP C) == (A ==> B ==> PROP C)"
-proof
-  assume r: "induct_conj A B ==> PROP C" and A B
-  show "PROP C" by (rule r) (simp add: induct_conj_def `A` `B`)
-next
-  assume r: "A ==> B ==> PROP C" and "induct_conj A B"
-  show "PROP C" by (rule r) (simp_all add: `induct_conj A B` [unfolded induct_conj_def])
-qed
-
-lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry
-
-hide const induct_forall induct_implies induct_equal induct_conj
-
-text {* Method setup. *}
+  by (rule trans [OF trans [OF c a] arg_cong [OF c, of "f y"]])
+
+
+subsection {* Conclude HOL structure *}
 
 ML {*
-  structure InductMethod = InductMethodFun
-  (struct
-    val cases_default = thm "case_split"
-    val atomize = thms "induct_atomize"
-    val rulify = thms "induct_rulify"
-    val rulify_fallback = thms "induct_rulify_fallback"
-  end);
+structure HOL =
+struct
+
+open HOL;
+
+(* combination of (spec RS spec RS ...(j times) ... spec RS mp) *)
+local
+  fun wrong_prem (Const ("All", _) $ (Abs (_, _, t))) = wrong_prem t
+    | wrong_prem (Bound _) = true
+    | wrong_prem _ = false;
+  val filter_right = filter (not o wrong_prem o HOLogic.dest_Trueprop o hd o Thm.prems_of);
+in
+  fun smp i = funpow i (fn m => filter_right ([spec] RL m)) ([mp]);
+  fun smp_tac j = EVERY'[dresolve_tac (smp j), atac];
+end;
+
+fun strip_tac i = REPEAT (resolve_tac [impI, allI] i);
+
+end;
 *}
 
-setup InductMethod.setup
-
-
-text {* itself as a code generator datatype *}
-
-setup {*
-let fun add_itself thy =
-  let
-    val v = ("'a", []);
-    val t = Logic.mk_type (TFree v);
-    val Const (c, ty) = t;
-    val (_, Type (dtco, _)) = strip_type ty;
-  in
-    thy
-    |> CodegenData.add_datatype (dtco, (([v], [(c, [])]), CodegenData.lazy (fn () => [])))
-  end
-in add_itself end;
-*} 
-
-text {* code generation for arbitrary as exception *}
-
-setup {*
-  CodegenSerializer.add_undefined "SML" "arbitrary" "(raise Fail \"arbitrary\")"
-*}
-code_const arbitrary
-  (Haskell target_atom "(error \"arbitrary\")")
-
 end