author blanchet Wed, 20 Jan 2010 10:38:19 +0100 changeset 34937 fb90752a9271 parent 34936 c4f04bee79f3 (current diff) parent 34934 440605046777 (diff) child 34938 f4d3daddac42
merged
```--- a/NEWS	Wed Jan 20 10:38:06 2010 +0100
+++ b/NEWS	Wed Jan 20 10:38:19 2010 +0100
@@ -38,6 +38,7 @@
INTER_fold_inter              ~> INFI_fold_inf
UNION_fold_union              ~> SUPR_fold_sup

*** ML ***
```
```--- a/src/FOL/FOL.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/FOL/FOL.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -383,6 +383,8 @@
val atomize = @{thms induct_atomize}
val rulify = @{thms induct_rulify}
val rulify_fallback = @{thms induct_rulify_fallback}
+    fun dest_def _ = NONE
+    fun trivial_tac _ = no_tac
);
*}
```
```--- a/src/HOL/Algebra/UnivPoly.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Algebra/UnivPoly.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -1581,14 +1581,10 @@
{
(*JE: we now apply the induction hypothesis with some additional facts required*)
from f_in_P deg_g_le_deg_f show ?thesis
-      proof (induct n \<equiv> "deg R f" arbitrary: "f" rule: nat_less_induct)
-        fix n f
-        assume hypo: "\<forall>m<n. \<forall>x. x \<in> carrier P \<longrightarrow>
-          deg R g \<le> deg R x \<longrightarrow>
-          m = deg R x \<longrightarrow>
-          (\<exists>q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and> lcoeff g (^) k \<odot>\<^bsub>P\<^esub> x = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r & (r = \<zero>\<^bsub>P\<^esub> | deg R r < deg R g))"
-          and prem: "n = deg R f" and f_in_P [simp]: "f \<in> carrier P"
-          and deg_g_le_deg_f: "deg R g \<le> deg R f"
+      proof (induct "deg R f" arbitrary: "f" rule: less_induct)
+        case less
+        note f_in_P [simp] = `f \<in> carrier P`
+          and deg_g_le_deg_f = `deg R g \<le> deg R f`
let ?k = "1::nat" and ?r = "(g \<otimes>\<^bsub>P\<^esub> (monom P (lcoeff f) (deg R f - deg R g))) \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> (lcoeff g \<odot>\<^bsub>P\<^esub> f)"
and ?q = "monom P (lcoeff f) (deg R f - deg R g)"
show "\<exists> q r (k::nat). q \<in> carrier P \<and> r \<in> carrier P \<and> lcoeff g (^) k \<odot>\<^bsub>P\<^esub> f = g \<otimes>\<^bsub>P\<^esub> q \<oplus>\<^bsub>P\<^esub> r & (r = \<zero>\<^bsub>P\<^esub> | deg R r < deg R g)"
@@ -1631,13 +1627,13 @@
{
(*JE: now it only remains the case where the induction hypothesis can be used.*)
(*JE: we first prove that the degree of the remainder is smaller than the one of f*)
-                  have deg_remainder_l_f: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < n"
+                  have deg_remainder_l_f: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R f"
proof -
have "deg R (\<ominus>\<^bsub>P\<^esub> ?r) = deg R ?r" using deg_uminus [of ?r] by simp
also have "\<dots> < deg R f"
proof (rule deg_lcoeff_cancel)
show "deg R (\<ominus>\<^bsub>P\<^esub> (lcoeff g \<odot>\<^bsub>P\<^esub> f)) \<le> deg R f"
-                        using deg_smult_ring [of "lcoeff g" f] using prem
+                        using deg_smult_ring [of "lcoeff g" f]
using lcoeff_nonzero2 [OF g_in_P g_not_zero] by simp
show "deg R (g \<otimes>\<^bsub>P\<^esub> ?q) \<le> deg R f"
using monom_deg_mult [OF _ g_in_P, of f "lcoeff f"] and deg_g_le_deg_f
@@ -1651,7 +1647,7 @@
using R.finsum_singleton [of "deg R g" "{.. deg R f}" "(\<lambda>i. coeff P g i \<otimes> lcoeff f)"]
unfolding Pi_def using deg_g_le_deg_f by force
-                    finally show "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < n" unfolding prem .
+                    finally show "deg R (\<ominus>\<^bsub>P\<^esub> ?r) < deg R f" .
qed
moreover have "\<ominus>\<^bsub>P\<^esub> ?r \<in> carrier P" by simp
moreover obtain m where deg_rem_eq_m: "deg R (\<ominus>\<^bsub>P\<^esub> ?r) = m" by auto
@@ -1660,7 +1656,7 @@
ultimately obtain q' r' k'
where rem_desc: "lcoeff g (^) (k'::nat) \<odot>\<^bsub>P\<^esub> (\<ominus>\<^bsub>P\<^esub> ?r) = g \<otimes>\<^bsub>P\<^esub> q' \<oplus>\<^bsub>P\<^esub> r'"and rem_deg: "(r' = \<zero>\<^bsub>P\<^esub> \<or> deg R r' < deg R g)"
and q'_in_carrier: "q' \<in> carrier P" and r'_in_carrier: "r' \<in> carrier P"
-                    using hypo by blast
+                    using less by blast
(*JE: we now prove that the new quotient, remainder and exponent can be used to get
the quotient, remainder and exponent of the long division theorem*)
show ?thesis```
```--- a/src/HOL/Bali/Basis.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Bali/Basis.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -1,7 +1,5 @@
(*  Title:      HOL/Bali/Basis.thy
-    ID:         \$Id\$
Author:     David von Oheimb
-
*)
header {* Definitions extending HOL as logical basis of Bali *}

@@ -66,8 +64,6 @@
"\<lbrakk> \<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c; (a,x)\<in>r\<^sup>*; (a,y)\<in>r\<^sup>*\<rbrakk>
\<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
proof -
-  note converse_rtrancl_induct = converse_rtrancl_induct [consumes 1]
-  note converse_rtranclE = converse_rtranclE [consumes 1]
assume unique: "\<And> a b c. \<lbrakk>(a,b)\<in>r; (a,c)\<in>r\<rbrakk> \<Longrightarrow> b=c"
assume "(a,x)\<in>r\<^sup>*"
then show "(a,y)\<in>r\<^sup>* \<Longrightarrow> (x,y)\<in>r\<^sup>* \<or> (y,x)\<in>r\<^sup>*"
@@ -110,13 +106,6 @@
apply (auto dest: rtrancl_into_trancl1)
done

-(* ### To Transitive_Closure *)
-theorems converse_rtrancl_induct
- = converse_rtrancl_induct [consumes 1,case_names Id Step]
-
-theorems converse_trancl_induct
-         = converse_trancl_induct [consumes 1,case_names Single Step]
-
(* context (theory "Set") *)
lemma Ball_weaken:"\<lbrakk>Ball s P;\<And> x. P x\<longrightarrow>Q x\<rbrakk>\<Longrightarrow>Ball s Q"
by auto```
```--- a/src/HOL/Bali/DeclConcepts.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Bali/DeclConcepts.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -1,5 +1,4 @@
(*  Title:      HOL/Bali/DeclConcepts.thy
-    ID:         \$Id\$
Author:     Norbert Schirmer
*)
@@ -2282,74 +2281,47 @@
done

lemma superclasses_mono:
-"\<lbrakk>G\<turnstile>C \<prec>\<^sub>C D;ws_prog G; class G C = Some c;
-  \<And> C c. \<lbrakk>class G C = Some c;C\<noteq>Object\<rbrakk> \<Longrightarrow> \<exists> sc. class G (super c) = Some sc;
-  x\<in>superclasses G D
-\<rbrakk> \<Longrightarrow> x\<in>superclasses G C"
-proof -
-
-  assume     ws: "ws_prog G"          and
-          cls_C: "class G C = Some c" and
-             wf: "\<And>C c. \<lbrakk>class G C = Some c; C \<noteq> Object\<rbrakk>
-                  \<Longrightarrow> \<exists>sc. class G (super c) = Some sc"
-  assume clsrel: "G\<turnstile>C\<prec>\<^sub>C D"
-  thus "\<And> c. \<lbrakk>class G C = Some c; x\<in>superclasses G D\<rbrakk>\<Longrightarrow>
-        x\<in>superclasses G C" (is "PROP ?P C"
-                             is "\<And> c. ?CLS C c \<Longrightarrow> ?SUP D \<Longrightarrow> ?SUP C")
-  proof (induct ?P C  rule: converse_trancl_induct)
-    fix C c
-    assume "G\<turnstile>C\<prec>\<^sub>C\<^sub>1D" "class G C = Some c" "x \<in> superclasses G D"
-    with wf ws show "?SUP C"
-      by (auto    intro: no_subcls1_Object
-  next
-    fix C S c
-    assume clsrel': "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 S" "G\<turnstile>S \<prec>\<^sub>C D"
-       and    hyp : "\<And> s. \<lbrakk>class G S = Some s; x \<in> superclasses G D\<rbrakk>
-                           \<Longrightarrow> x \<in> superclasses G S"
-       and  cls_C': "class G C = Some c"
-       and       x: "x \<in> superclasses G D"
-    moreover note wf ws
-    moreover from calculation
-    have "?SUP S"
-      by (force intro: no_subcls1_Object simp add: subcls1_def)
-    moreover from calculation
-    have "super c = S"
-      by (auto intro: no_subcls1_Object simp add: subcls1_def)
-    ultimately show "?SUP C"
-      by (auto intro: no_subcls1_Object simp add: superclasses_rec)
-  qed
+  assumes clsrel: "G\<turnstile>C\<prec>\<^sub>C D"
+  and ws: "ws_prog G"
+  and cls_C: "class G C = Some c"
+  and wf: "\<And>C c. \<lbrakk>class G C = Some c; C \<noteq> Object\<rbrakk>
+    \<Longrightarrow> \<exists>sc. class G (super c) = Some sc"
+  and x: "x\<in>superclasses G D"
+  shows "x\<in>superclasses G C" using clsrel cls_C x
+proof (induct arbitrary: c rule: converse_trancl_induct)
+  case (base C)
+  with wf ws show ?case
+    by (auto    intro: no_subcls1_Object
+next
+  case (step C S)
+  moreover note wf ws
+  moreover from calculation
+  have "x\<in>superclasses G S"
+    by (force intro: no_subcls1_Object simp add: subcls1_def)
+  moreover from calculation
+  have "super c = S"
+    by (auto intro: no_subcls1_Object simp add: subcls1_def)
+  ultimately show ?case
+    by (auto intro: no_subcls1_Object simp add: superclasses_rec)
qed

lemma subclsEval:
-"\<lbrakk>G\<turnstile>C \<prec>\<^sub>C D;ws_prog G; class G C = Some c;
-  \<And> C c. \<lbrakk>class G C = Some c;C\<noteq>Object\<rbrakk> \<Longrightarrow> \<exists> sc. class G (super c) = Some sc
- \<rbrakk> \<Longrightarrow> D\<in>superclasses G C"
-proof -
-  note converse_trancl_induct
-       = converse_trancl_induct [consumes 1,case_names Single Step]
-  assume
-             ws: "ws_prog G"          and
-          cls_C: "class G C = Some c" and
-             wf: "\<And>C c. \<lbrakk>class G C = Some c; C \<noteq> Object\<rbrakk>
-                  \<Longrightarrow> \<exists>sc. class G (super c) = Some sc"
-  assume clsrel: "G\<turnstile>C\<prec>\<^sub>C D"
-  thus "\<And> c. class G C = Some c\<Longrightarrow> D\<in>superclasses G C"
-    (is "PROP ?P C"  is "\<And> c. ?CLS C c  \<Longrightarrow> ?SUP C")
-  proof (induct ?P C  rule: converse_trancl_induct)
-    fix C c
-    assume "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 D" "class G C = Some c"
-    with ws wf show "?SUP C"
-      by (auto intro: no_subcls1_Object simp add: superclasses_rec subcls1_def)
-  next
-    fix C S c
-    assume "G\<turnstile>C \<prec>\<^sub>C\<^sub>1 S" "G\<turnstile>S\<prec>\<^sub>C D"
-           "\<And>s. class G S = Some s \<Longrightarrow> D \<in> superclasses G S"
-           "class G C = Some c"
-    with ws wf show "?SUP C"
-      by - (rule superclasses_mono,
-            auto dest: no_subcls1_Object simp add: subcls1_def )
-  qed
+  assumes clsrel: "G\<turnstile>C\<prec>\<^sub>C D"
+  and ws: "ws_prog G"
+  and cls_C: "class G C = Some c"
+  and wf: "\<And>C c. \<lbrakk>class G C = Some c; C \<noteq> Object\<rbrakk>
+    \<Longrightarrow> \<exists>sc. class G (super c) = Some sc"
+  shows "D\<in>superclasses G C" using clsrel cls_C
+proof (induct arbitrary: c rule: converse_trancl_induct)
+  case (base C)
+  with ws wf show ?case
+    by (auto intro: no_subcls1_Object simp add: superclasses_rec subcls1_def)
+next
+  case (step C S)
+  with ws wf show ?case
+    by - (rule superclasses_mono,
+          auto dest: no_subcls1_Object simp add: subcls1_def )
qed

end```
```--- a/src/HOL/Bali/WellForm.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Bali/WellForm.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -1,5 +1,4 @@
(*  Title:      HOL/Bali/WellForm.thy
-    ID:         \$Id\$
Author:     David von Oheimb and Norbert Schirmer
*)

@@ -1409,8 +1408,7 @@
from clsC ws
show "methd G C sig = Some m
\<Longrightarrow> G\<turnstile>(mdecl (sig,mthd m)) declared_in (declclass m)"
-    (is "PROP ?P C")
-  proof (induct ?P C rule: ws_class_induct')
+  proof (induct C rule: ws_class_induct')
case Object
assume "methd G Object sig = Some m"
with wf show ?thesis
@@ -1755,28 +1753,20 @@
lemma ballE': "\<forall>x\<in>A. P x \<Longrightarrow> (x \<notin> A \<Longrightarrow> Q) \<Longrightarrow> (P x \<Longrightarrow> Q) \<Longrightarrow> Q" by blast

lemma subint_widen_imethds:
- "\<lbrakk>G\<turnstile>I\<preceq>I J; wf_prog G; is_iface G J; jm \<in> imethds G J sig\<rbrakk> \<Longrightarrow>
-  \<exists> im \<in> imethds G I sig. is_static im = is_static jm \<and>
+  assumes irel: "G\<turnstile>I\<preceq>I J"
+  and wf: "wf_prog G"
+  and is_iface: "is_iface G J"
+  and jm: "jm \<in> imethds G J sig"
+  shows "\<exists>im \<in> imethds G I sig. is_static im = is_static jm \<and>
accmodi im = accmodi jm \<and>
G\<turnstile>resTy im\<preceq>resTy jm"
-proof -
-  assume irel: "G\<turnstile>I\<preceq>I J" and
-           wf: "wf_prog G" and
-     is_iface: "is_iface G J"
-  from irel show "jm \<in> imethds G J sig \<Longrightarrow> ?thesis"
-               (is "PROP ?P I" is "PROP ?Prem J \<Longrightarrow> ?Concl I")
-  proof (induct ?P I rule: converse_rtrancl_induct)
-    case Id
-    assume "jm \<in> imethds G J sig"
-    then show "?Concl J" by  (blast elim: bexI')
+  using irel jm
+proof (induct rule: converse_rtrancl_induct)
+    case base
+    then show ?case by  (blast elim: bexI')
next
-    case Step
-    fix I SI
-    assume subint1_I_SI: "G\<turnstile>I \<prec>I1 SI" and
-            subint_SI_J: "G\<turnstile>SI \<preceq>I J" and
-                    hyp: "PROP ?P SI" and
-                     jm: "jm \<in> imethds G J sig"
-    from subint1_I_SI
+    case (step I SI)
+    from `G\<turnstile>I \<prec>I1 SI`
obtain i where
ifI: "iface G I = Some i" and
SI: "SI \<in> set (isuperIfs i)"
@@ -1784,10 +1774,10 @@

let ?newMethods
= "(Option.set \<circ> table_of (map (\<lambda>(sig, mh). (sig, I, mh)) (imethods i)))"
-    show "?Concl I"
+    show ?case
proof (cases "?newMethods sig = {}")
case True
-      with ifI SI hyp wf jm
+      with ifI SI step wf
show "?thesis"
next
@@ -1816,7 +1806,7 @@
wf_SI: "wf_idecl G (SI,si)"
by (auto dest!: wf_idecl_supD is_acc_ifaceD
dest: wf_prog_idecl)
-      from jm hyp
+      from step
sim: "sim \<in> imethds G SI sig" and
eq_static_sim_jm: "is_static sim = is_static jm" and
@@ -1841,7 +1831,6 @@
show ?thesis
by auto
qed
-  qed
qed

(* Tactical version *)
@@ -1974,30 +1963,20 @@
from clsC ws
show "\<And> m d. \<lbrakk>methd G C sig = Some m; class G (declclass m) = Some d\<rbrakk>
\<Longrightarrow> table_of (methods d) sig  = Some (mthd m)"
-         (is "PROP ?P C")
-  proof (induct ?P C rule: ws_class_induct)
+  proof (induct rule: ws_class_induct)
case Object
-    fix m d
-    assume "methd G Object sig = Some m"
-           "class G (declclass m) = Some d"
with wf show "?thesis m d" by auto
next
-    case Subcls
-    fix C c m d
-    assume hyp: "PROP ?P (super c)"
-    and      m: "methd G C sig = Some m"
-    and  declC: "class G (declclass m) = Some d"
-    and   clsC: "class G C = Some c"
-    and   nObj: "C \<noteq> Object"
+    case (Subcls C c)
let ?newMethods = "table_of (map (\<lambda>(s, m). (s, C, m)) (methods c)) sig"
show "?thesis m d"
proof (cases "?newMethods")
case None
-      from None clsC nObj ws m declC
-      show "?thesis" by (auto simp add: methd_rec) (rule hyp)
+      from None ws Subcls
+      show "?thesis" by (auto simp add: methd_rec) (rule Subcls)
next
case Some
-      from Some clsC nObj ws m declC
+      from Some ws Subcls
show "?thesis"
dest: wf_prog_cdecl wf_cdecl_supD is_acc_class_is_class)```
```--- a/src/HOL/Boogie/Tools/boogie_tactics.ML	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Boogie/Tools/boogie_tactics.ML	Wed Jan 20 10:38:19 2010 +0100
@@ -69,8 +69,9 @@
|> Seq.map (pair (map (rpair [] o case_name_of) (Thm.prems_of st)))) #>
Seq.maps (fn (names, st) =>
CASES
-        (Rule_Cases.make_common false
-          (ProofContext.theory_of ctxt, Thm.prop_of st) names)
+        (Rule_Cases.make_common
+          (ProofContext.theory_of ctxt,
+           Thm.prop_of (Rule_Cases.internalize_params st)) names)
Tactical.all_tac st))
in
val setup_boogie_cases = Method.setup @{binding boogie_cases}```
```--- a/src/HOL/Code_Numeral.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Code_Numeral.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -83,7 +83,7 @@
then have "\<And>n. P (of_nat n) \<Longrightarrow> P (Suc_code_numeral (of_nat n))" .
then have step: "\<And>n. P (of_nat n) \<Longrightarrow> P (of_nat (Suc n))" by simp
from init step have "P (of_nat (nat_of k))"
-    by (induct "nat_of k") simp_all
+    by (induct ("nat_of k")) simp_all
then show "P k" by simp
qed simp_all

@@ -100,7 +100,7 @@
fix k
have "code_numeral_size k = nat_size (nat_of k)"
by (induct k rule: code_numeral.induct) (simp_all del: zero_code_numeral_def Suc_code_numeral_def, simp_all)
-  also have "nat_size (nat_of k) = nat_of k" by (induct "nat_of k") simp_all
+  also have "nat_size (nat_of k) = nat_of k" by (induct ("nat_of k")) simp_all
finally show "code_numeral_size k = nat_of k" .
qed

@@ -296,8 +296,11 @@

setup {*
-    false false) ["SML", "Haskell", "Scala"]
-  #> Numeral.add_code @{const_name number_code_numeral_inst.number_of_code_numeral} false true "OCaml"
+    false false Code_Printer.str) ["SML", "Haskell"]
+    false true Code_Printer.str "OCaml"
+    false false Code_Printer.str "Scala"
*}

code_reserved SML Int int
@@ -323,9 +326,10 @@
(Scala infixl 8 "*")

code_const div_mod_code_numeral
-  (SML "(fn n => fn m =>/ if m = 0/ then (0, n) else/ (n div m, n mod m))")
-  (OCaml "(fun k -> fun l ->/ Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int k)/ (Big'_int.abs'_big'_int l))")
+  (SML "!(fn n => fn m =>/ if m = 0/ then (0, n) else/ (n div m, n mod m))")
+  (OCaml "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)")
+  (Scala "!((n: Int) => (m: Int) =>/ if (m == 0)/ (0, n) else/ (n '/ m, n % m))")

code_const "eq_class.eq \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
(SML "!((_ : Int.int) = _)")
@@ -337,12 +341,12 @@
(SML "Int.<=/ ((_),/ (_))")
(OCaml "Big'_int.le'_big'_int")
-  (Scala infix 4 "<=")
+  (Scala infixl 4 "<=")

code_const "op < \<Colon> code_numeral \<Rightarrow> code_numeral \<Rightarrow> bool"
(SML "Int.</ ((_),/ (_))")
(OCaml "Big'_int.lt'_big'_int")
-  (Scala infix 4 "<")
+  (Scala infixl 4 "<")

end```
```--- a/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Decision_Procs/Reflected_Multivariate_Polynomial.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -987,16 +987,14 @@
assumes nq: "isnpolyh p n0" and eq :"\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a::{ring_char_0,power,division_by_zero,field})"
shows "p = 0\<^sub>p"
using nq eq
-proof (induct n\<equiv>"maxindex p" arbitrary: p n0 rule: nat_less_induct)
-  fix n p n0
-  assume H: "\<forall>m<n. \<forall>p n0. isnpolyh p n0 \<longrightarrow>
-    (\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)) \<longrightarrow> m = maxindex p \<longrightarrow> p = 0\<^sub>p"
-    and np: "isnpolyh p n0" and zp: "\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" and n: "n = maxindex p"
-  {assume nz: "n = 0"
-    then obtain c where "p = C c" using n np by (cases p, auto)
+proof (induct "maxindex p" arbitrary: p n0 rule: less_induct)
+  case less
+  note np = `isnpolyh p n0` and zp = `\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)`
+  {assume nz: "maxindex p = 0"
+    then obtain c where "p = C c" using np by (cases p, auto)
with zp np have "p = 0\<^sub>p" unfolding wf_bs_def by simp}
moreover
-  {assume nz: "n \<noteq> 0"
+  {assume nz: "maxindex p \<noteq> 0"
let ?hd = "decrpoly ?h"
let ?ihd = "maxindex ?hd"
@@ -1005,24 +1003,23 @@
hence nhd: "isnpolyh ?hd (n0 - 1)" using decrpoly_normh by blast

from maxindex_coefficients[of p] coefficients_head[of p, symmetric]
-    have mihn: "maxindex ?h \<le> n" unfolding n by auto
-    with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < n" by auto
+    have mihn: "maxindex ?h \<le> maxindex p" by auto
+    with decr_maxindex[OF h(2)] nz  have ihd_lt_n: "?ihd < maxindex p" by auto
{fix bs:: "'a list"  assume bs: "wf_bs bs ?hd"
let ?ts = "take ?ihd bs"
let ?rs = "drop ?ihd bs"
have ts: "wf_bs ?ts ?hd" using bs unfolding wf_bs_def by simp
have bs_ts_eq: "?ts@ ?rs = bs" by simp
from wf_bs_decrpoly[OF ts] have tsh: " \<forall>x. wf_bs (x#?ts) ?h" by simp
-      from ihd_lt_n have "ALL x. length (x#?ts) \<le> n" by simp
-      with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = n" by blast
-      hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" using n unfolding wf_bs_def by simp
+      from ihd_lt_n have "ALL x. length (x#?ts) \<le> maxindex p" by simp
+      with length_le_list_ex obtain xs where xs:"length ((x#?ts) @ xs) = maxindex p" by blast
+      hence "\<forall> x. wf_bs ((x#?ts) @ xs) p" unfolding wf_bs_def by simp
with zp have "\<forall> x. Ipoly ((x#?ts) @ xs) p = 0" by blast
hence "\<forall> x. Ipoly (x#(?ts @ xs)) p = 0" by simp
with polypoly_poly[OF np, where ?'a = 'a] polypoly_polypoly'[OF np, where ?'a = 'a]
have "\<forall>x. poly (polypoly' (?ts @ xs) p) x = poly [] x"  by simp
hence "poly (polypoly' (?ts @ xs) p) = poly []" by (auto intro: ext)
hence "\<forall> c \<in> set (coefficients p). Ipoly (?ts @ xs) (decrpoly c) = 0"
-        thm poly_zero
using poly_zero[where ?'a='a] by (simp add: polypoly'_def list_all_iff)
have th0: "Ipoly (?ts @ xs) ?hd = 0" by simp
@@ -1031,7 +1028,7 @@
with wf'' wf_bs_I[of ?ts ?hd ?rs] bs_ts_eq have "\<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = 0" by simp }
then have hdz: "\<forall>bs. wf_bs bs ?hd \<longrightarrow> \<lparr>?hd\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" by blast

-    from H[rule_format, OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
+    from less(1)[OF ihd_lt_n nhd] hdz have "?hd = 0\<^sub>p" by blast
hence "?h = 0\<^sub>p" by simp
with head_nz[OF np] have "p = 0\<^sub>p" by simp}
ultimately show "p = 0\<^sub>p" by blast
@@ -1357,8 +1354,8 @@
(polydivide_aux (a,n,p,k,s) = (k',r) \<longrightarrow> (k' \<ge> k) \<and> (degree r = 0 \<or> degree r < degree p)
\<and> (\<exists>nr. isnpolyh r nr) \<and> (\<exists>q n1. isnpolyh q n1 \<and> ((polypow (k' - k) a) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)))"
using ns
-proof(induct d\<equiv>"degree s" arbitrary: s k k' r n1 rule: nat_less_induct)
-  fix d s k k' r n1
+proof(induct "degree s" arbitrary: s k k' r n1 rule: less_induct)
+  case less
let ?D = "polydivide_aux_dom"
let ?dths = "?D (a, n, p, k, s)"
let ?qths = "\<exists>q n1. isnpolyh q n1 \<and> (a ^\<^sub>p (k' - k) *\<^sub>p s = p *\<^sub>p q +\<^sub>p r)"
@@ -1366,20 +1363,13 @@
\<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
let ?ths = "?dths \<and> ?qrths"
-  let ?p' = "funpow (d - n) shift1 p"
-  let ?xdn = "funpow (d - n) shift1 1\<^sub>p"
+  let ?p' = "funpow (degree s - n) shift1 p"
+  let ?xdn = "funpow (degree s - n) shift1 1\<^sub>p"
let ?akk' = "a ^\<^sub>p (k' - k)"
-  assume H: "\<forall>m<d. \<forall>x xa xb xc xd.
-    isnpolyh x xd \<longrightarrow>
-    m = degree x \<longrightarrow> ?D (a, n, p, xa, x) \<and>
-    (polydivide_aux (a, n, p, xa, x) = (xb, xc) \<longrightarrow>
-    xa \<le> xb \<and> (degree xc = 0 \<or> degree xc < degree p) \<and>
-    (\<exists> nr. isnpolyh xc nr) \<and>
-    (\<exists>q n1. isnpolyh q n1 \<and> a ^\<^sub>p xb - xa *\<^sub>p x = p *\<^sub>p q +\<^sub>p xc))"
-    and ns: "isnpolyh s n1" and ds: "d = degree s"
+  note ns = `isnpolyh s n1`
from np have np0: "isnpolyh p 0"
using isnpolyh_mono[where n="n0" and n'="0" and p="p"]  by simp
-  have np': "isnpolyh ?p' 0" using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="d -n"] isnpoly_def by simp
+  have np': "isnpolyh ?p' 0" using funpow_shift1_isnpoly[OF np0[simplified isnpoly_def[symmetric]] pnz, where n="degree s - n"] isnpoly_def by simp
from funpow_shift1_isnpoly[where p="1\<^sub>p"] have nxdn: "isnpolyh ?xdn 0" by (simp add: isnpoly_def)
from polypow_normh [OF head_isnpolyh[OF np0], where k="k' - k"] ap
@@ -1391,31 +1381,31 @@
hence ?ths using dom by blast}
moreover
{assume sz: "s \<noteq> 0\<^sub>p"
-    {assume dn: "d < n"
-      with sz ds  have dom:"?dths" by - (rule polydivide_aux_real_domintros,simp_all)
-      from polydivide_aux.psimps[OF dom] sz dn ds
+    {assume dn: "degree s < n"
+      with sz have dom:"?dths" by - (rule polydivide_aux_real_domintros,simp_all)
+      from polydivide_aux.psimps[OF dom] sz dn
have "?qrths" using ns ndp np by auto (rule exI[where x="0\<^sub>p"],simp)
with dom have ?ths by blast}
moreover
-    {assume dn': "\<not> d < n" hence dn: "d \<ge> n" by arith
+    {assume dn': "\<not> degree s < n" hence dn: "degree s \<ge> n" by arith
have degsp': "degree s = degree ?p'"
-        using ds dn ndp funpow_shift1_degree[where k = "d - n" and p="p"] by simp
+        using dn ndp funpow_shift1_degree[where k = "degree s - n" and p="p"] by simp
{assume ba: "?b = a"
have nr: "isnpolyh (s -\<^sub>p ?p') 0" using polysub_normh[OF ns np'] by simp
-        have "degree (s -\<^sub>p ?p') < d \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
+        have "degree (s -\<^sub>p ?p') < degree s \<or> s -\<^sub>p ?p' = 0\<^sub>p" by simp
moreover
-        {assume deglt:"degree (s -\<^sub>p ?p') < d"
-          from  H[rule_format, OF deglt nr,simplified]
+        {assume deglt:"degree (s -\<^sub>p ?p') < degree s"
+          from  less(1)[OF deglt nr]
have domsp: "?D (a, n, p, k, s -\<^sub>p ?p')" by blast
have dom: ?dths apply (rule polydivide_aux_real_domintros)
-            using ba ds dn' domsp by simp_all
-          from polydivide_aux.psimps[OF dom] sz dn' ba ds
+            using ba dn' domsp by simp_all
+          from polydivide_aux.psimps[OF dom] sz dn' ba
have eq: "polydivide_aux (a,n,p,k,s) = polydivide_aux (a,n,p,k, s -\<^sub>p ?p')"
{assume h1: "polydivide_aux (a, n, p, k, s) = (k', r)"
-            from H[rule_format, OF deglt nr, where xa = "k" and xb="k'" and xc="r", simplified]
+            from less(1)[OF deglt nr, of k k' r]
trans[OF eq[symmetric] h1]
have kk': "k \<le> k'" and nr:"\<exists>nr. isnpolyh r nr" and dr: "degree r = 0 \<or> degree r < degree p"
and q1:"\<exists>q nq. isnpolyh q nq \<and> (a ^\<^sub>p k' - k *\<^sub>p (s -\<^sub>p ?p') = p *\<^sub>p q +\<^sub>p r)" by auto
@@ -1434,19 +1424,19 @@
Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs ?p' + Ipoly bs p * Ipoly bs q + Ipoly bs r"
hence " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
-              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (d - n) shift1 1\<^sub>p *\<^sub>p p)
+              Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p *\<^sub>p p)
+ Ipoly bs p * Ipoly bs q + Ipoly bs r"
by (auto simp only: funpow_shift1_1)
hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
-              Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (d - n) shift1 1\<^sub>p)
+              Ipoly bs p * (Ipoly bs (a^\<^sub>p (k' - k)) * Ipoly bs (funpow (degree s - n) shift1 1\<^sub>p)
+ Ipoly bs q) + Ipoly bs r" by (simp add: ring_simps)
hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
-              Ipoly bs (p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (d - n) shift1 1\<^sub>p) +\<^sub>p q) +\<^sub>p r)" by simp
+              Ipoly bs (p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q) +\<^sub>p r)" by simp
with isnpolyh_unique[OF nakks' nqr']
have "a ^\<^sub>p (k' - k) *\<^sub>p s =
-              p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (d - n) shift1 1\<^sub>p) +\<^sub>p q) +\<^sub>p r" by blast
+              p *\<^sub>p ((a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q) +\<^sub>p r" by blast
hence ?qths using nq'
-              apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (d - n) shift1 1\<^sub>p) +\<^sub>p q" in exI)
+              apply (rule_tac x="(a^\<^sub>p (k' - k)) *\<^sub>p (funpow (degree s - n) shift1 1\<^sub>p) +\<^sub>p q" in exI)
apply (rule_tac x="0" in exI) by simp
with kk' nr dr have "k \<le> k' \<and> (degree r = 0 \<or> degree r < degree p) \<and> (\<exists>nr. isnpolyh r nr) \<and> ?qths"
by blast } hence ?qrths by blast
@@ -1456,25 +1446,23 @@
hence domsp: "?D (a, n, p, k, s -\<^sub>p ?p')"
apply (simp) by (rule polydivide_aux_real_domintros, simp_all)
have dom: ?dths apply (rule polydivide_aux_real_domintros)
-            using ba ds dn' domsp by simp_all
+            using ba dn' domsp by simp_all
from spz isnpolyh_unique[OF polysub_normh[OF ns np'], where q="0\<^sub>p", symmetric, where ?'a = "'a::{ring_char_0,division_by_zero,field}"]
have " \<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs s = Ipoly bs ?p'" by simp
hence "\<forall>(bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs s = Ipoly bs (?xdn *\<^sub>p p)" using np nxdn apply simp
by (simp only: funpow_shift1_1) simp
hence sp': "s = ?xdn *\<^sub>p p" using isnpolyh_unique[OF ns polymul_normh[OF nxdn np]] by blast
{assume h1: "polydivide_aux (a,n,p,k,s) = (k',r)"
-            from polydivide_aux.psimps[OF dom] sz dn' ba ds
+            from polydivide_aux.psimps[OF dom] sz dn' ba
have eq: "polydivide_aux (a,n,p,k,s) = polydivide_aux (a,n,p,k, s -\<^sub>p ?p')"
also have "\<dots> = (k,0\<^sub>p)" using polydivide_aux.psimps[OF domsp] spz by simp
finally have "(k',r) = (k,0\<^sub>p)" using h1 by simp
-            with sp' ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
+            with sp'[symmetric] ns np nxdn polyadd_0(1)[OF polymul_normh[OF np nxdn]]
polyadd_0(2)[OF polymul_normh[OF np nxdn]] have ?qrths
apply auto
apply (rule exI[where x="?xdn"])
-              apply auto
-              apply (rule polymul_commute)
-              apply simp_all
+              apply (auto simp add: polymul_commute[of p])
done}
with dom have ?ths by blast}
ultimately have ?ths by blast }
@@ -1488,31 +1476,30 @@
funpow_shift1_nz[OF pnz] by simp_all
from polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
-          funpow_shift1_nz[OF pnz, where n="d - n"]
+          funpow_shift1_nz[OF pnz, where n="degree s - n"]
-          ndp ds[symmetric] dn
+          ndp dn
have degth: "degree (a *\<^sub>p s) = degree (?b *\<^sub>p ?p') "
-        {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < d"
+        {assume dth: "degree ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) < degree s"
have th: "?D (a, n, p, Suc k, (a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p'))"
-            using H[rule_format, OF dth nth, simplified] by blast
-          have dom: ?dths
-            using ba ds dn' th apply simp apply (rule polydivide_aux_real_domintros)
-            using ba ds dn'  by simp_all
+            using less(1)[OF dth nth] by blast
+          have dom: ?dths using ba dn' th
+            by - (rule polydivide_aux_real_domintros, simp_all)
ap have nasbp': "isnpolyh ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) 0" by simp
{assume h1:"polydivide_aux (a,n,p,k,s) = (k', r)"
-            from h1  polydivide_aux.psimps[OF dom] sz dn' ba ds
+            from h1  polydivide_aux.psimps[OF dom] sz dn' ba
have eq:"polydivide_aux (a,n,p,Suc k,(a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = (k',r)"
-            with H[rule_format, OF dth nasbp', simplified, where xa="Suc k" and xb="k'" and xc="r"]
+            with less(1)[OF dth nasbp', of "Suc k" k' r]
obtain q nq nr where kk': "Suc k \<le> k'" and nr: "isnpolyh r nr" and nq: "isnpolyh q nq"
and dr: "degree r = 0 \<or> degree r < degree p"
and qr: "a ^\<^sub>p (k' - Suc k) *\<^sub>p ((a *\<^sub>p s) -\<^sub>p (?b *\<^sub>p ?p')) = p *\<^sub>p q +\<^sub>p r" by auto
@@ -1524,7 +1511,7 @@
hence "Ipoly bs a ^ (Suc (k' - Suc k)) * Ipoly bs s = Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?p' + Ipoly bs r"
hence "Ipoly bs a ^ (k' - k)  * Ipoly bs s = Ipoly bs p * Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn * Ipoly bs p + Ipoly bs r"
-              by (simp add:kk'' funpow_shift1_1[where n="d - n" and p="p"])
+              by (simp add:kk'' funpow_shift1_1[where n="degree s - n" and p="p"])
hence "Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) = Ipoly bs p * (Ipoly bs q + Ipoly bs a ^ (k' - Suc k) * Ipoly bs ?b * Ipoly bs ?xdn) + Ipoly bs r"
hence ieq:"\<forall>(bs :: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a ^\<^sub>p (k' - k) *\<^sub>p s) =
@@ -1546,13 +1533,13 @@
{assume spz: "a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p') = 0\<^sub>p"
hence domsp: "?D (a, n, p, Suc k, a *\<^sub>p s -\<^sub>p (?b *\<^sub>p ?p'))"
apply (simp) by (rule polydivide_aux_real_domintros, simp_all)
-          have dom: ?dths using sz ba dn' ds domsp
+          have dom: ?dths using sz ba dn' domsp
by - (rule polydivide_aux_real_domintros, simp_all)
{fix bs :: "'a::{ring_char_0,division_by_zero,field} list"
from isnpolyh_unique[OF nth, where ?'a="'a" and q="0\<^sub>p",simplified,symmetric] spz
have "Ipoly bs (a*\<^sub>p s) = Ipoly bs ?b * Ipoly bs ?p'" by simp
hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (?b *\<^sub>p ?xdn) * Ipoly bs p"
-            by (simp add: funpow_shift1_1[where n="d - n" and p="p"])
+            by (simp add: funpow_shift1_1[where n="degree s - n" and p="p"])
hence "Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" by simp
}
hence hth: "\<forall> (bs:: 'a::{ring_char_0,division_by_zero,field} list). Ipoly bs (a*\<^sub>p s) = Ipoly bs (p *\<^sub>p (?b *\<^sub>p ?xdn))" ..
@@ -1562,7 +1549,7 @@
polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]],
simplified ap] by simp
{assume h1: "polydivide_aux (a,n,p,k,s) = (k', r)"
-          from h1 sz ds ba dn' spz polydivide_aux.psimps[OF dom] polydivide_aux.psimps[OF domsp]
+          from h1 sz ba dn' spz polydivide_aux.psimps[OF dom] polydivide_aux.psimps[OF domsp]
have "(k',r) = (Suc k, 0\<^sub>p)" by (simp add: Let_def)
with h1 np head_isnpolyh[OF np, simplified ap] ns polymul_normh[OF head_isnpolyh[OF ns] nxdn]
polymul_normh[OF np polymul_normh[OF head_isnpolyh[OF ns] nxdn]] asq
@@ -1573,7 +1560,7 @@
hence ?qrths by blast
with dom have ?ths by blast}
ultimately have ?ths using  degree_polysub_samehead[OF polymul_normh[OF head_isnpolyh[OF np0, simplified ap] ns] polymul_normh[OF head_isnpolyh[OF ns] np'] hdth degth] polymul_degreen[OF head_isnpolyh[OF np] ns, where m="0"]
-          head_nz[OF np] pnz sz ap[symmetric] ds[symmetric]
+          head_nz[OF np] pnz sz ap[symmetric]
by (simp add: degree_eq_degreen0[symmetric]) blast }
ultimately have ?ths by blast
}```
```--- a/src/HOL/Extraction.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Extraction.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -1,5 +1,4 @@
(*  Title:      HOL/Extraction.thy
-    ID:         \$Id\$
Author:     Stefan Berghofer, TU Muenchen
*)

@@ -28,11 +27,13 @@
allE rev_mp conjE Eq_TrueI Eq_FalseI eqTrueI eqTrueE eq_cong2
notE' impE' impE iffE imp_cong simp_thms eq_True eq_False
induct_forall_eq induct_implies_eq induct_equal_eq induct_conj_eq
-  induct_atomize induct_rulify induct_rulify_fallback
+  induct_atomize induct_atomize' induct_rulify induct_rulify'
+  induct_rulify_fallback induct_trueI
True_implies_equals TrueE

lemmas [extraction_expand_def] =
induct_forall_def induct_implies_def induct_equal_def induct_conj_def
+  induct_true_def induct_false_def

datatype sumbool = Left | Right
```
```--- a/src/HOL/GCD.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/GCD.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -16,7 +16,7 @@
another extension of the notions to the integers, and added a number
of results to "Primes" and "GCD". IntPrimes also defined and developed
the congruence relations on the integers. The notion was extended to
-the natural numbers by Chiaeb.
+the natural numbers by Chaieb.

natural numbers and the integers, and added a number of new theorems.
@@ -25,7 +25,7 @@
*)

-header {* Greates common divisor and least common multiple *}
+header {* Greatest common divisor and least common multiple *}

theory GCD
imports Fact Parity
@@ -1074,34 +1074,35 @@
assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
shows "P a b"
-proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
-  fix n a b
-  assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
+proof(induct "a + b" arbitrary: a b rule: less_induct)
+  case less
have "a = b \<or> a < b \<or> b < a" by arith
moreover {assume eq: "a= b"
from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
by simp}
moreover
{assume lt: "a < b"
-    hence "a + b - a < n \<or> a = 0"  using H(2) by arith
+    hence "a + b - a < a + b \<or> a = 0" by arith
moreover
{assume "a =0" with z c have "P a b" by blast }
moreover
-    {assume ab: "a + b - a < n"
-      have th0: "a + b - a = a + (b - a)" using lt by arith
-      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
-      have "P a b" by (simp add: th0[symmetric])}
+    {assume "a + b - a < a + b"
+      also have th0: "a + b - a = a + (b - a)" using lt by arith
+      finally have "a + (b - a) < a + b" .
+      then have "P a (a + (b - a))" by (rule add[rule_format, OF less])
+      then have "P a b" by (simp add: th0[symmetric])}
ultimately have "P a b" by blast}
moreover
{assume lt: "a > b"
-    hence "b + a - b < n \<or> b = 0"  using H(2) by arith
+    hence "b + a - b < a + b \<or> b = 0" by arith
moreover
{assume "b =0" with z c have "P a b" by blast }
moreover
-    {assume ab: "b + a - b < n"
-      have th0: "b + a - b = b + (a - b)" using lt by arith
-      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
-      have "P b a" by (simp add: th0[symmetric])
+    {assume "b + a - b < a + b"
+      also have th0: "b + a - b = b + (a - b)" using lt by arith
+      finally have "b + (a - b) < a + b" .
+      then have "P b (b + (a - b))" by (rule add[rule_format, OF less])
+      then have "P b a" by (simp add: th0[symmetric])
hence "P a b" using c by blast }
ultimately have "P a b" by blast}
ultimately  show "P a b" by blast```
```--- a/src/HOL/HOL.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/HOL.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -1398,6 +1398,8 @@
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"
+  induct_true where "induct_true == True"
+  induct_false where "induct_false == False"

lemma induct_forall_eq: "(!!x. P x) == Trueprop (induct_forall (\<lambda>x. P x))"
by (unfold atomize_all induct_forall_def)
@@ -1411,10 +1413,13 @@
lemma induct_conj_eq: "(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_atomize' = induct_forall_eq induct_implies_eq induct_conj_eq
+lemmas induct_atomize = induct_atomize' induct_equal_eq
+lemmas induct_rulify' [symmetric, standard] = induct_atomize'
lemmas induct_rulify [symmetric, standard] = induct_atomize
lemmas induct_rulify_fallback =
induct_forall_def induct_implies_def induct_equal_def induct_conj_def
+  induct_true_def induct_false_def

lemma induct_forall_conj: "induct_forall (\<lambda>x. induct_conj (A x) (B x)) =
@@ -1436,7 +1441,8 @@

lemmas induct_conj = induct_forall_conj induct_implies_conj induct_conj_curry

-hide const induct_forall induct_implies induct_equal induct_conj
+lemma induct_trueI: "induct_true"

text {* Method setup. *}

@@ -1445,12 +1451,93 @@
(
val cases_default = @{thm case_split}
val atomize = @{thms induct_atomize}
-  val rulify = @{thms induct_rulify}
+  val rulify = @{thms induct_rulify'}
val rulify_fallback = @{thms induct_rulify_fallback}
+  fun dest_def (Const (@{const_name induct_equal}, _) \$ t \$ u) = SOME (t, u)
+    | dest_def _ = NONE
+  val trivial_tac = match_tac @{thms induct_trueI}
)
*}

-setup Induct.setup
+setup {*
+  Induct.setup #>
+  Context.theory_map (Induct.map_simpset (fn ss => ss
+    setmksimps (Simpdata.mksimps Simpdata.mksimps_pairs #>
+      map (Simplifier.rewrite_rule (map Thm.symmetric
+        @{thms induct_rulify_fallback induct_true_def induct_false_def})))
+      [Simplifier.simproc @{theory} "swap_induct_false"
+         ["induct_false ==> PROP P ==> PROP Q"]
+         (fn _ => fn _ =>
+            (fn _ \$ (P as _ \$ @{const induct_false}) \$ (_ \$ Q \$ _) =>
+                  if P <> Q then SOME Drule.swap_prems_eq else NONE
+              | _ => NONE)),
+       Simplifier.simproc @{theory} "induct_equal_conj_curry"
+         ["induct_conj P Q ==> PROP R"]
+         (fn _ => fn _ =>
+            (fn _ \$ (_ \$ P) \$ _ =>
+                let
+                  fun is_conj (@{const induct_conj} \$ P \$ Q) =
+                        is_conj P andalso is_conj Q
+                    | is_conj (Const (@{const_name induct_equal}, _) \$ _ \$ _) = true
+                    | is_conj @{const induct_true} = true
+                    | is_conj @{const induct_false} = true
+                    | is_conj _ = false
+                in if is_conj P then SOME @{thm induct_conj_curry} else NONE end
+              | _ => NONE))]))
+*}
+
+text {* Pre-simplification of induction and cases rules *}
+
+lemma [induct_simp]: "(!!x. induct_equal x t ==> PROP P x) == PROP P t"
+  unfolding induct_equal_def
+proof
+  assume R: "!!x. x = t ==> PROP P x"
+  show "PROP P t" by (rule R [OF refl])
+next
+  fix x assume "PROP P t" "x = t"
+  then show "PROP P x" by simp
+qed
+
+lemma [induct_simp]: "(!!x. induct_equal t x ==> PROP P x) == PROP P t"
+  unfolding induct_equal_def
+proof
+  assume R: "!!x. t = x ==> PROP P x"
+  show "PROP P t" by (rule R [OF refl])
+next
+  fix x assume "PROP P t" "t = x"
+  then show "PROP P x" by simp
+qed
+
+lemma [induct_simp]: "(induct_false ==> P) == Trueprop induct_true"
+  unfolding induct_false_def induct_true_def
+  by (iprover intro: equal_intr_rule)
+
+lemma [induct_simp]: "(induct_true ==> PROP P) == PROP P"
+  unfolding induct_true_def
+proof
+  assume R: "True \<Longrightarrow> PROP P"
+  from TrueI show "PROP P" by (rule R)
+next
+  assume "PROP P"
+  then show "PROP P" .
+qed
+
+lemma [induct_simp]: "(PROP P ==> induct_true) == Trueprop induct_true"
+  unfolding induct_true_def
+  by (iprover intro: equal_intr_rule)
+
+lemma [induct_simp]: "(!!x. induct_true) == Trueprop induct_true"
+  unfolding induct_true_def
+  by (iprover intro: equal_intr_rule)
+
+lemma [induct_simp]: "induct_implies induct_true P == P"
+  by (simp add: induct_implies_def induct_true_def)
+
+lemma [induct_simp]: "(x = x) = True"
+  by (rule simp_thms)
+
+hide const induct_forall induct_implies induct_equal induct_conj induct_true induct_false

use "~~/src/Tools/induct_tacs.ML"
setup InductTacs.setup```
```--- a/src/HOL/Induct/Common_Patterns.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Induct/Common_Patterns.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -73,7 +73,7 @@
show "P (a x)" sorry
next
case (Suc n)
-  note hyp = `\<And>x. A (a x) \<Longrightarrow> n = a x \<Longrightarrow> P (a x)`
+  note hyp = `\<And>x. n = a x \<Longrightarrow> A (a x) \<Longrightarrow> P (a x)`
and prem = `A (a x)`
and defn = `Suc n = a x`
show "P (a x)" sorry```
```--- a/src/HOL/Isar_Examples/Puzzle.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Isar_Examples/Puzzle.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -22,17 +22,16 @@
proof (rule order_antisym)
{
fix n show "n \<le> f n"
-    proof (induct k \<equiv> "f n" arbitrary: n rule: less_induct)
-      case (less k n)
-      then have hyp: "\<And>m. f m < f n \<Longrightarrow> m \<le> f m" by (simp only:)
+    proof (induct "f n" arbitrary: n rule: less_induct)
+      case less
show "n \<le> f n"
proof (cases n)
case (Suc m)
from f_ax have "f (f m) < f n" by (simp only: Suc)
-        with hyp have "f m \<le> f (f m)" .
+        with less have "f m \<le> f (f m)" .
also from f_ax have "\<dots> < f n" by (simp only: Suc)
finally have "f m < f n" .
-        with hyp have "m \<le> f m" .
+        with less have "m \<le> f m" .
also note `\<dots> < f n`
finally have "m < f n" .
then have "n \<le> f n" by (simp only: Suc)```
```--- a/src/HOL/Library/Code_Integer.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Library/Code_Integer.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -18,13 +18,16 @@
(SML "IntInf.int")
(OCaml "Big'_int.big'_int")
+  (Scala "BigInt")

code_instance int :: eq

setup {*
-    true true) ["SML", "OCaml", "Haskell"]
+    true true Code_Printer.str) ["SML", "OCaml", "Haskell", "Scala"]
+    true true (fn s => (Pretty.block o map Code_Printer.str) ["BigInt", s]) "Scala"
*}

code_const "Int.Pls" and "Int.Min" and "Int.Bit0" and "Int.Bit1"
@@ -40,63 +43,80 @@
and "error/ \"Min\""
and "error/ \"Bit0\""
and "error/ \"Bit1\"")
+  (Scala "!error(\"Pls\")"
+     and "!error(\"Min\")"
+     and "!error(\"Bit0\")"
+     and "!error(\"Bit1\")")
+

code_const Int.pred
(SML "IntInf.- ((_), 1)")
(OCaml "Big'_int.pred'_big'_int")
+  (Scala "!(_/ -/ 1)")

code_const Int.succ
(SML "IntInf.+ ((_), 1)")
(OCaml "Big'_int.succ'_big'_int")
+  (Scala "!(_/ +/ 1)")

code_const "op + \<Colon> int \<Rightarrow> int \<Rightarrow> int"
(SML "IntInf.+ ((_), (_))")
+  (Scala infixl 7 "+")

code_const "uminus \<Colon> int \<Rightarrow> int"
(SML "IntInf.~")
(OCaml "Big'_int.minus'_big'_int")
+  (Scala "!(- _)")

code_const "op - \<Colon> int \<Rightarrow> int \<Rightarrow> int"
(SML "IntInf.- ((_), (_))")
(OCaml "Big'_int.sub'_big'_int")
+  (Scala infixl 7 "-")

code_const "op * \<Colon> int \<Rightarrow> int \<Rightarrow> int"
(SML "IntInf.* ((_), (_))")
(OCaml "Big'_int.mult'_big'_int")
+  (Scala infixl 8 "*")

code_const pdivmod
-  (SML "(fn k => fn l =>/ IntInf.divMod/ (IntInf.abs k,/ IntInf.abs l))")
-  (OCaml "(fun k -> fun l ->/ Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int k)/ (Big'_int.abs'_big'_int l))")
-  (Haskell "(\\k l ->/ divMod/ (abs k)/ (abs l))")
+  (SML "IntInf.divMod/ (IntInf.abs _,/ IntInf.abs _)")
+  (OCaml "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)")
+  (Haskell "divMod/ (abs _)/ (abs _))")
+  (Scala "!(_.abs '/% _.abs)")

code_const "eq_class.eq \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
(SML "!((_ : IntInf.int) = _)")
(OCaml "Big'_int.eq'_big'_int")
+  (Scala infixl 5 "==")

code_const "op \<le> \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
(SML "IntInf.<= ((_), (_))")
(OCaml "Big'_int.le'_big'_int")
+  (Scala infixl 4 "<=")

code_const "op < \<Colon> int \<Rightarrow> int \<Rightarrow> bool"
(SML "IntInf.< ((_), (_))")
(OCaml "Big'_int.lt'_big'_int")
+  (Scala infixl 4 "<=")

code_const Code_Numeral.int_of
(SML "IntInf.fromInt")
(OCaml "_")
+  (Scala "!BigInt(_)")

code_reserved SML IntInf
+code_reserved Scala BigInt

text {* Evaluation *}
```
```--- a/src/HOL/Library/Efficient_Nat.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Library/Efficient_Nat.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -226,7 +226,7 @@

text {*
For ML, we map @{typ nat} to target language integers, where we
-  assert that values are always non-negative.
+  ensure that values are always non-negative.
*}

code_type nat
@@ -245,9 +245,9 @@
*}

text {*
-  For Haskell we define our own @{typ nat} type.  The reason
-  is that we have to distinguish type class instances
-  for @{typ nat} and @{typ int}.
+  For Haskell ans Scala we define our own @{typ nat} type.  The reason
+  is that we have to distinguish type class instances for @{typ nat}
+  and @{typ int}.
*}

@@ -286,8 +286,53 @@

+code_include Scala "Nat" {*
+object Nat {
+
+  def apply(numeral: BigInt): Nat = new Nat(numeral max 0)
+  def apply(numeral: Int): Nat = Nat(BigInt(numeral))
+  def apply(numeral: String): Nat = Nat(BigInt(numeral))
+
+}
+
+class Nat private(private val value: BigInt) {
+
+  override def hashCode(): Int = this.value.hashCode()
+
+  override def equals(that: Any): Boolean = that match {
+    case that: Nat => this equals that
+    case _ => false
+  }
+
+  override def toString(): String = this.value.toString
+
+  def equals(that: Nat): Boolean = this.value == that.value
+
+  def as_BigInt: BigInt = this.value
+  def as_Int: Int = this.value
+
+  def +(that: Nat): Nat = new Nat(this.value + that.value)
+  def -(that: Nat): Nat = Nat(this.value + that.value)
+  def *(that: Nat): Nat = new Nat(this.value * that.value)
+
+  def /%(that: Nat): (Nat, Nat) = if (that.value == 0) (new Nat(0), this)
+    else {
+      val (k, l) = this.value /% that.value
+      (new Nat(k), new Nat(l))
+    }
+
+  def <=(that: Nat): Boolean = this.value <= that.value
+
+  def <(that: Nat): Boolean = this.value < that.value
+
+}
+*}
+
+code_reserved Scala Nat
+
code_type nat
+  (Scala "Nat.Nat")

code_instance nat :: eq
@@ -303,7 +348,9 @@

setup {*
-    false true) ["SML", "OCaml", "Haskell"]
+    false true Code_Printer.str) ["SML", "OCaml", "Haskell"]
+    false true (fn s => (Pretty.block o map Code_Printer.str) ["Nat.Nat", s]) "Scala"
*}

text {*
@@ -349,10 +396,11 @@
(SML "IntInf.max/ (/0,/ _)")
(OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int")

-text {* For Haskell, things are slightly different again. *}
+text {* For Haskell ans Scala, things are slightly different again. *}

code_const int and nat
+  (Scala "!_.as'_BigInt" and "!Nat.Nat((_))")

text {* Conversion from and to indices. *}

@@ -360,11 +408,13 @@
(SML "IntInf.toInt")
(OCaml "_")
+  (Scala "!_.as'_Int")

code_const Code_Numeral.nat_of
(SML "IntInf.fromInt")
(OCaml "_")
+  (Scala "!Nat.Nat((_))")

text {* Using target language arithmetic operations whenever appropriate *}

@@ -372,31 +422,45 @@
(SML "IntInf.+ ((_), (_))")
+  (Scala infixl 7 "+")
+
+code_const "op - \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
+  (Scala infixl 7 "-")

code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
(SML "IntInf.* ((_), (_))")
(OCaml "Big'_int.mult'_big'_int")
+  (Scala infixl 8 "*")

code_const divmod_aux
(SML "IntInf.divMod/ ((_),/ (_))")
(OCaml "Big'_int.quomod'_big'_int")
+  (Scala infixl 8 "/%")
+
+code_const divmod_nat
+  (Scala infixl 8 "/%")

code_const "eq_class.eq \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
(SML "!((_ : IntInf.int) = _)")
(OCaml "Big'_int.eq'_big'_int")
+  (Scala infixl 5 "==")

code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
(SML "IntInf.<= ((_), (_))")
(OCaml "Big'_int.le'_big'_int")
+  (Scala infixl 4 "<=")

code_const "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
(SML "IntInf.< ((_), (_))")
(OCaml "Big'_int.lt'_big'_int")
+  (Scala infixl 4 "<")

consts_code
"0::nat"                     ("0")```
```--- a/src/HOL/Library/Fundamental_Theorem_Algebra.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Library/Fundamental_Theorem_Algebra.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -621,19 +621,18 @@
done
qed

-text{* Fundamental theorem of algebral *}
+text{* Fundamental theorem of algebra *}

lemma fundamental_theorem_of_algebra:
assumes nc: "~constant(poly p)"
shows "\<exists>z::complex. poly p z = 0"
using nc
-proof(induct n\<equiv> "psize p" arbitrary: p rule: nat_less_induct)
-  fix n fix p :: "complex poly"
+proof(induct "psize p" arbitrary: p rule: less_induct)
+  case less
let ?p = "poly p"
-  assume H: "\<forall>m<n. \<forall>p. \<not> constant (poly p) \<longrightarrow> m = psize p \<longrightarrow> (\<exists>(z::complex). poly p z = 0)" and nc: "\<not> constant ?p" and n: "n = psize p"
let ?ths = "\<exists>z. ?p z = 0"

-  from nonconstant_length[OF nc] have n2: "n\<ge> 2" by (simp add: n)
+  from nonconstant_length[OF less(2)] have n2: "psize p \<ge> 2" .
from poly_minimum_modulus obtain c where
c: "\<forall>w. cmod (?p c) \<le> cmod (?p w)" by blast
{assume pc: "?p c = 0" hence ?ths by blast}
@@ -649,7 +648,7 @@
using h unfolding constant_def by blast
also have "\<dots> = ?p y" using th by auto
finally have "?p x = ?p y" .}
-      with nc have False unfolding constant_def by blast }
+      with less(2) have False unfolding constant_def by blast }
hence qnc: "\<not> constant (poly q)" by blast
from q(2) have pqc0: "?p c = poly q 0" by simp
from c pqc0 have cq0: "\<forall>w. cmod (poly q 0) \<le> cmod (?p w)" by simp
@@ -682,8 +681,8 @@
from poly_decompose[OF rnc] obtain k a s where
kas: "a\<noteq>0" "k\<noteq>0" "psize s + k + 1 = psize ?r"
"\<forall>z. poly ?r z = poly ?r 0 + z^k* poly (pCons a s) z" by blast
-    {assume "k + 1 = n"
-      with kas(3) lgqr[symmetric] q(1) n[symmetric] have s0:"s=0" by auto
+    {assume "psize p = k + 1"
+      with kas(3) lgqr[symmetric] q(1) have s0:"s=0" by auto
{fix w
have "cmod (poly ?r w) = cmod (1 + a * w ^ k)"
using kas(4)[rule_format, of w] s0 r01 by (simp add: algebra_simps)}
@@ -691,15 +690,15 @@
from reduce_poly_simple[OF kas(1,2)]
have "\<exists>w. cmod (poly ?r w) < 1" unfolding hth by blast}
moreover
-    {assume kn: "k+1 \<noteq> n"
-      from kn kas(3) q(1) n[symmetric] lgqr have k1n: "k + 1 < n" by simp
+    {assume kn: "psize p \<noteq> k+1"
+      from kn kas(3) q(1) lgqr have k1n: "k + 1 < psize p" by simp
have th01: "\<not> constant (poly (pCons 1 (monom a (k - 1))))"
unfolding constant_def poly_pCons poly_monom
using kas(1) apply simp
by (rule exI[where x=0], rule exI[where x=1], simp)
from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
-      from H[rule_format, OF k1n th01 th02]
+      from less(1) [OF k1n [simplified th02] th01]
obtain w where w: "1 + w^k * a = 0"
unfolding poly_pCons poly_monom
using kas(2) by (cases k, auto simp add: algebra_simps)```
```--- a/src/HOL/Library/Polynomial.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Library/Polynomial.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -1384,7 +1384,7 @@
with k have "degree p = Suc (degree k)"
by (simp add: degree_mult_eq del: mult_pCons_left)
with `Suc n = degree p` have "n = degree k" by simp
-    with `k \<noteq> 0` have "finite {x. poly k x = 0}" by (rule Suc.hyps)
+    then have "finite {x. poly k x = 0}" using `k \<noteq> 0` by (rule Suc.hyps)
then have "finite (insert a {x. poly k x = 0})" by simp
then show "finite {x. poly p x = 0}"
```--- a/src/HOL/Library/Transitive_Closure_Table.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Library/Transitive_Closure_Table.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -17,11 +17,11 @@
assume "r\<^sup>*\<^sup>* x y"
then show "\<exists>xs. rtrancl_path r x xs y"
proof (induct rule: converse_rtranclp_induct)
-    case 1
+    case base
have "rtrancl_path r y [] y" by (rule rtrancl_path.base)
then show ?case ..
next
-    case (2 x z)
+    case (step x z)
from `\<exists>xs. rtrancl_path r z xs y`
obtain xs where "rtrancl_path r z xs y" ..
with `r x z` have "rtrancl_path r x (z # xs) y"```
```--- a/src/HOL/Library/Word.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Library/Word.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -436,7 +436,7 @@
show "bitval x * 2 ^ (length xs + length l2) + bv_to_nat xs * 2 ^ length l2 = (bitval x * 2 ^ length xs + bv_to_nat xs) * 2 ^ length l2"
proof -
have "(2::nat) ^ (length xs + length l2) = 2 ^ length xs * 2 ^ length l2"
-          by (induct "length xs",simp_all)
+          by (induct ("length xs")) simp_all
hence "bitval x * 2 ^ (length xs + length l2) + bv_to_nat xs * 2 ^ length l2 =
bitval x * 2 ^ length xs * 2 ^ length l2 + bv_to_nat xs * 2 ^ length l2"
by simp
@@ -2165,13 +2165,13 @@
apply (subst bv_to_nat_dist_append)
apply simp
-  apply (induct "n - length w")
+  apply (induct ("n - length w"))
apply simp_all
done

lemma bv_msb_extend_same [simp]: "bv_msb w = b ==> bv_msb (bv_extend n b w) = b"
-  apply (induct "n - length w")
+  apply (cases "n - length w")
apply simp_all
done

@@ -2188,7 +2188,7 @@
show ?thesis
-    apply (induct "n - length w",simp_all)
+    apply (induct ("n - length w"), simp_all)
done
qed
```
```--- a/src/HOL/List.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/List.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -578,13 +578,13 @@
apply fastsimp
done

-lemma same_append_eq [iff]: "(xs @ ys = xs @ zs) = (ys = zs)"
+lemma same_append_eq [iff, induct_simp]: "(xs @ ys = xs @ zs) = (ys = zs)"
by simp

lemma append1_eq_conv [iff]: "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
by simp

-lemma append_same_eq [iff]: "(ys @ xs = zs @ xs) = (ys = zs)"
+lemma append_same_eq [iff, induct_simp]: "(ys @ xs = zs @ xs) = (ys = zs)"
by simp

lemma append_self_conv2 [iff]: "(xs @ ys = ys) = (xs = [])"
@@ -2383,7 +2383,6 @@
unfolding SUPR_def set_map [symmetric] Sup_set_fold foldl_map

-
subsubsection {* List summation: @{const listsum} and @{text"\<Sum>"}*}

lemma listsum_append [simp]: "listsum (xs @ ys) = listsum xs + listsum ys"
@@ -3195,6 +3194,117 @@
apply auto
done

+subsubsection{*Transpose*}
+
+function transpose where
+"transpose []             = []" |
+"transpose ([]     # xss) = transpose xss" |
+"transpose ((x#xs) # xss) =
+  (x # [h. (h#t) \<leftarrow> xss]) # transpose (xs # [t. (h#t) \<leftarrow> xss])"
+by pat_completeness auto
+
+  "concat (map (list_case [] (\<lambda>h t. [h])) xss) =
+  map (\<lambda>xs. hd xs) [ys\<leftarrow>xss . ys \<noteq> []]"
+  by (induct xss) (auto split: list.split)
+
+lemma transpose_aux_filter_tail:
+  "concat (map (list_case [] (\<lambda>h t. [t])) xss) =
+  map (\<lambda>xs. tl xs) [ys\<leftarrow>xss . ys \<noteq> []]"
+  by (induct xss) (auto split: list.split)
+
+lemma transpose_aux_max:
+  "max (Suc (length xs)) (foldr (\<lambda>xs. max (length xs)) xss 0) =
+  Suc (max (length xs) (foldr (\<lambda>x. max (length x - Suc 0)) [ys\<leftarrow>xss . ys\<noteq>[]] 0))"
+  (is "max _ ?foldB = Suc (max _ ?foldA)")
+proof (cases "[ys\<leftarrow>xss . ys\<noteq>[]] = []")
+  case True
+  hence "foldr (\<lambda>xs. max (length xs)) xss 0 = 0"
+  proof (induct xss)
+    case (Cons x xs)
+    moreover hence "x = []" by (cases x) auto
+    ultimately show ?case by auto
+  qed simp
+  thus ?thesis using True by simp
+next
+  case False
+
+  have foldA: "?foldA = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0 - 1"
+    by (induct xss) auto
+  have foldB: "?foldB = foldr (\<lambda>x. max (length x)) [ys\<leftarrow>xss . ys \<noteq> []] 0"
+    by (induct xss) auto
+
+  have "0 < ?foldB"
+  proof -
+    from False
+    obtain z zs where zs: "[ys\<leftarrow>xss . ys \<noteq> []] = z#zs" by (auto simp: neq_Nil_conv)
+    hence "z \<in> set ([ys\<leftarrow>xss . ys \<noteq> []])" by auto
+    hence "z \<noteq> []" by auto
+    thus ?thesis
+      unfolding foldB zs
+      by (auto simp: max_def intro: less_le_trans)
+  qed
+  thus ?thesis
+    unfolding foldA foldB max_Suc_Suc[symmetric]
+    by simp
+qed
+
+termination transpose
+  by (relation "measure (\<lambda>xs. foldr (\<lambda>xs. max (length xs)) xs 0 + length xs)")
+     (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max less_Suc_eq_le)
+
+lemma transpose_empty: "(transpose xs = []) \<longleftrightarrow> (\<forall>x \<in> set xs. x = [])"
+  by (induct rule: transpose.induct) simp_all
+
+lemma length_transpose:
+  fixes xs :: "'a list list"
+  shows "length (transpose xs) = foldr (\<lambda>xs. max (length xs)) xs 0"
+  by (induct rule: transpose.induct)
+    (auto simp: transpose_aux_filter_tail foldr_map comp_def transpose_aux_max
+                max_Suc_Suc[symmetric] simp del: max_Suc_Suc)
+
+lemma nth_transpose:
+  fixes xs :: "'a list list"
+  assumes "i < length (transpose xs)"
+  shows "transpose xs ! i = map (\<lambda>xs. xs ! i) [ys \<leftarrow> xs. i < length ys]"
+using assms proof (induct arbitrary: i rule: transpose.induct)
+  case (3 x xs xss)
+  def XS == "(x # xs) # xss"
+  hence [simp]: "XS \<noteq> []" by auto
+  thus ?case
+  proof (cases i)
+    case 0
+  next
+    case (Suc j)
+    have *: "\<And>xss. xs # map tl xss = map tl ((x#xs)#xss)" by simp
+    have **: "\<And>xss. (x#xs) # filter (\<lambda>ys. ys \<noteq> []) xss = filter (\<lambda>ys. ys \<noteq> []) ((x#xs)#xss)" by simp
+    { fix x have "Suc j < length x \<longleftrightarrow> x \<noteq> [] \<and> j < length x - Suc 0"
+      by (cases x) simp_all
+    } note *** = this
+
+    have j_less: "j < length (transpose (xs # concat (map (list_case [] (\<lambda>h t. [t])) xss)))"
+      using "3.prems" by (simp add: transpose_aux_filter_tail length_transpose Suc)
+
+    show ?thesis
+      unfolding transpose.simps `i = Suc j` nth_Cons_Suc "3.hyps"[OF j_less]
+      apply (auto simp: transpose_aux_filter_tail filter_map comp_def length_transpose * ** *** XS_def[symmetric])
+      apply (rule_tac y=x in list.exhaust)
+      by auto
+  qed
+qed simp_all
+
+lemma transpose_map_map:
+  "transpose (map (map f) xs) = map (map f) (transpose xs)"
+proof (rule nth_equalityI, safe)
+  have [simp]: "length (transpose (map (map f) xs)) = length (transpose xs)"
+    by (simp add: length_transpose foldr_map comp_def)
+  show "length (transpose (map (map f) xs)) = length (map (map f) (transpose xs))" by simp
+
+  fix i assume "i < length (transpose (map (map f) xs))"
+  thus "transpose (map (map f) xs) ! i = map (map f) (transpose xs) ! i"
+    by (simp add: nth_transpose filter_map comp_def)
+qed

subsubsection {* (In)finiteness *}

@@ -3406,6 +3516,45 @@
lemma sorted_takeWhile: "sorted xs \<Longrightarrow> sorted (takeWhile P xs)"
apply (subst takeWhile_eq_take) by (rule sorted_take)

+lemma sorted_filter:
+  "sorted (map f xs) \<Longrightarrow> sorted (map f (filter P xs))"
+  by (induct xs) (simp_all add: sorted_Cons)
+
+lemma foldr_max_sorted:
+  assumes "sorted (rev xs)"
+  shows "foldr max xs y = (if xs = [] then y else max (xs ! 0) y)"
+using assms proof (induct xs)
+  case (Cons x xs)
+  moreover hence "sorted (rev xs)" using sorted_append by auto
+  ultimately show ?case
+    by (cases xs, auto simp add: sorted_append max_def)
+qed simp
+
+lemma filter_equals_takeWhile_sorted_rev:
+  assumes sorted: "sorted (rev (map f xs))"
+  shows "[x \<leftarrow> xs. t < f x] = takeWhile (\<lambda> x. t < f x) xs"
+    (is "filter ?P xs = ?tW")
+proof (rule takeWhile_eq_filter[symmetric])
+  let "?dW" = "dropWhile ?P xs"
+  fix x assume "x \<in> set ?dW"
+  then obtain i where i: "i < length ?dW" and nth_i: "x = ?dW ! i"
+    unfolding in_set_conv_nth by auto
+  hence "length ?tW + i < length (?tW @ ?dW)"
+    unfolding length_append by simp
+  hence i': "length (map f ?tW) + i < length (map f xs)" by simp
+  have "(map f ?tW @ map f ?dW) ! (length (map f ?tW) + i) \<le>
+        (map f ?tW @ map f ?dW) ! (length (map f ?tW) + 0)"
+    using sorted_rev_nth_mono[OF sorted _ i', of "length ?tW"]
+    unfolding map_append[symmetric] by simp
+  hence "f x \<le> f (?dW ! 0)"
+    unfolding nth_append_length_plus nth_i
+    using i preorder_class.le_less_trans[OF le0 i] by simp
+  also have "... \<le> t"
+    using hd_dropWhile[of "?P" xs] le0[THEN preorder_class.le_less_trans, OF i]
+    using hd_conv_nth[of "?dW"] by simp
+  finally show "\<not> t < f x" by simp
+qed
+
end

lemma sorted_upt[simp]: "sorted[i..<j]"
@@ -3417,6 +3566,153 @@
done

+subsubsection {*@{const transpose} on sorted lists*}
+
+lemma sorted_transpose[simp]:
+  shows "sorted (rev (map length (transpose xs)))"
+  by (auto simp: sorted_equals_nth_mono rev_nth nth_transpose
+    length_filter_conv_card intro: card_mono)
+
+lemma transpose_max_length:
+  "foldr (\<lambda>xs. max (length xs)) (transpose xs) 0 = length [x \<leftarrow> xs. x \<noteq> []]"
+  (is "?L = ?R")
+proof (cases "transpose xs = []")
+  case False
+  have "?L = foldr max (map length (transpose xs)) 0"
+    by (simp add: foldr_map comp_def)
+  also have "... = length (transpose xs ! 0)"
+    using False sorted_transpose by (simp add: foldr_max_sorted)
+  finally show ?thesis
+    using False by (simp add: nth_transpose)
+next
+  case True
+  hence "[x \<leftarrow> xs. x \<noteq> []] = []"
+    by (auto intro!: filter_False simp: transpose_empty)
+  thus ?thesis by (simp add: transpose_empty True)
+qed
+
+lemma length_transpose_sorted:
+  fixes xs :: "'a list list"
+  assumes sorted: "sorted (rev (map length xs))"
+  shows "length (transpose xs) = (if xs = [] then 0 else length (xs ! 0))"
+proof (cases "xs = []")
+  case False
+  thus ?thesis
+    using foldr_max_sorted[OF sorted] False
+    unfolding length_transpose foldr_map comp_def
+    by simp
+qed simp
+
+lemma nth_nth_transpose_sorted[simp]:
+  fixes xs :: "'a list list"
+  assumes sorted: "sorted (rev (map length xs))"
+  and i: "i < length (transpose xs)"
+  and j: "j < length [ys \<leftarrow> xs. i < length ys]"
+  shows "transpose xs ! i ! j = xs ! j  ! i"
+  using j filter_equals_takeWhile_sorted_rev[OF sorted, of i]
+    nth_transpose[OF i] nth_map[OF j]
+
+lemma transpose_column_length:
+  fixes xs :: "'a list list"
+  assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
+  shows "length (filter (\<lambda>ys. i < length ys) (transpose xs)) = length (xs ! i)"
+proof -
+  have "xs \<noteq> []" using `i < length xs` by auto
+  note filter_equals_takeWhile_sorted_rev[OF sorted, simp]
+  { fix j assume "j \<le> i"
+    note sorted_rev_nth_mono[OF sorted, of j i, simplified, OF this `i < length xs`]
+  } note sortedE = this[consumes 1]
+
+  have "{j. j < length (transpose xs) \<and> i < length (transpose xs ! j)}
+    = {..< length (xs ! i)}"
+  proof safe
+    fix j
+    assume "j < length (transpose xs)" and "i < length (transpose xs ! j)"
+    with this(2) nth_transpose[OF this(1)]
+    have "i < length (takeWhile (\<lambda>ys. j < length ys) xs)" by simp
+    from nth_mem[OF this] takeWhile_nth[OF this]
+    show "j < length (xs ! i)" by (auto dest: set_takeWhileD)
+  next
+    fix j assume "j < length (xs ! i)"
+    thus "j < length (transpose xs)"
+      using foldr_max_sorted[OF sorted] `xs \<noteq> []` sortedE[OF le0]
+      by (auto simp: length_transpose comp_def foldr_map)
+
+    have "Suc i \<le> length (takeWhile (\<lambda>ys. j < length ys) xs)"
+      using `i < length xs` `j < length (xs ! i)` less_Suc_eq_le
+      by (auto intro!: length_takeWhile_less_P_nth dest!: sortedE)
+    with nth_transpose[OF `j < length (transpose xs)`]
+    show "i < length (transpose xs ! j)" by simp
+  qed
+  thus ?thesis by (simp add: length_filter_conv_card)
+qed
+
+lemma transpose_column:
+  fixes xs :: "'a list list"
+  assumes sorted: "sorted (rev (map length xs))" and "i < length xs"
+  shows "map (\<lambda>ys. ys ! i) (filter (\<lambda>ys. i < length ys) (transpose xs))
+    = xs ! i" (is "?R = _")
+proof (rule nth_equalityI, safe)
+  show length: "length ?R = length (xs ! i)"
+    using transpose_column_length[OF assms] by simp
+
+  fix j assume j: "j < length ?R"
+  note * = less_le_trans[OF this, unfolded length_map, OF length_filter_le]
+  from j have j_less: "j < length (xs ! i)" using length by simp
+  have i_less_tW: "Suc i \<le> length (takeWhile (\<lambda>ys. Suc j \<le> length ys) xs)"
+  proof (rule length_takeWhile_less_P_nth)
+    show "Suc i \<le> length xs" using `i < length xs` by simp
+    fix k assume "k < Suc i"
+    hence "k \<le> i" by auto
+    with sorted_rev_nth_mono[OF sorted this] `i < length xs`
+    have "length (xs ! i) \<le> length (xs ! k)" by simp
+    thus "Suc j \<le> length (xs ! k)" using j_less by simp
+  qed
+  have i_less_filter: "i < length [ys\<leftarrow>xs . j < length ys]"
+    unfolding filter_equals_takeWhile_sorted_rev[OF sorted, of j]
+    using i_less_tW by (simp_all add: Suc_le_eq)
+  from j show "?R ! j = xs ! i ! j"
+    unfolding filter_equals_takeWhile_sorted_rev[OF sorted_transpose, of i]
+    by (simp add: takeWhile_nth nth_nth_transpose_sorted[OF sorted * i_less_filter])
+qed
+
+lemma transpose_transpose:
+  fixes xs :: "'a list list"
+  assumes sorted: "sorted (rev (map length xs))"
+  shows "transpose (transpose xs) = takeWhile (\<lambda>x. x \<noteq> []) xs" (is "?L = ?R")
+proof -
+  have len: "length ?L = length ?R"
+    unfolding length_transpose transpose_max_length
+    using filter_equals_takeWhile_sorted_rev[OF sorted, of 0]
+    by simp
+
+  { fix i assume "i < length ?R"
+    with less_le_trans[OF _ length_takeWhile_le[of _ xs]]
+    have "i < length xs" by simp
+  } note * = this
+  show ?thesis
+    by (rule nth_equalityI)
+       (simp_all add: len nth_transpose transpose_column[OF sorted] * takeWhile_nth)
+qed
+
+theorem transpose_rectangle:
+  assumes "xs = [] \<Longrightarrow> n = 0"
+  assumes rect: "\<And> i. i < length xs \<Longrightarrow> length (xs ! i) = n"
+  shows "transpose xs = map (\<lambda> i. map (\<lambda> j. xs ! j ! i) [0..<length xs]) [0..<n]"
+    (is "?trans = ?map")
+proof (rule nth_equalityI)
+  have "sorted (rev (map length xs))"
+    by (auto simp: rev_nth rect intro!: sorted_nth_monoI)
+  from foldr_max_sorted[OF this] assms
+  show len: "length ?trans = length ?map"
+    by (simp_all add: length_transpose foldr_map comp_def)
+  moreover
+  { fix i assume "i < n" hence "[ys\<leftarrow>xs . i < length ys] = xs"
+      using rect by (auto simp: in_set_conv_nth intro!: filter_True) }
+  ultimately show "\<forall>i < length ?trans. ?trans ! i = ?map ! i"
+    by (auto simp: nth_transpose intro: nth_equalityI)
+qed

subsubsection {* @{text sorted_list_of_set} *}

@@ -3449,7 +3745,6 @@

end

-
subsubsection {* @{text lists}: the list-forming operator over sets *}

inductive_set
@@ -3548,7 +3843,6 @@
"listset []    = {[]}"
"listset(A#As) = set_Cons A (listset As)"

-
subsection{*Relations on Lists*}

subsubsection {* Length Lexicographic Ordering *}```
```--- a/src/HOL/MicroJava/BV/EffectMono.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/MicroJava/BV/EffectMono.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -15,12 +15,13 @@

lemma sup_loc_some [rule_format]:
"\<forall>y n. (G \<turnstile> b <=l y) \<longrightarrow> n < length y \<longrightarrow> y!n = OK t \<longrightarrow>
-  (\<exists>t. b!n = OK t \<and> (G \<turnstile> (b!n) <=o (y!n)))" (is "?P b")
-proof (induct ?P b)
-  show "?P []" by simp
+  (\<exists>t. b!n = OK t \<and> (G \<turnstile> (b!n) <=o (y!n)))"
+proof (induct b)
+  case Nil
+  show ?case by simp
next
case (Cons a list)
-  show "?P (a#list)"
+  show ?case
proof (clarsimp simp add: list_all2_Cons1 sup_loc_def Listn.le_def lesub_def)
fix z zs n
assume *:
@@ -60,13 +61,14 @@

lemma append_length_n [rule_format]:
-"\<forall>n. n \<le> length x \<longrightarrow> (\<exists>a b. x = a@b \<and> length a = n)" (is "?P x")
-proof (induct ?P x)
-  show "?P []" by simp
+"\<forall>n. n \<le> length x \<longrightarrow> (\<exists>a b. x = a@b \<and> length a = n)"
+proof (induct x)
+  case Nil
+  show ?case by simp
next
-  fix l ls assume Cons: "?P ls"
+  case (Cons l ls)

-  show "?P (l#ls)"
+  show ?case
proof (intro allI impI)
fix n
assume l: "n \<le> length (l # ls)"```
```--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -170,8 +170,8 @@
next assume "card s > 2" thus ?thesis using as and n_def proof(induct n arbitrary: u s)
case (Suc n) fix s::"'a set" and u::"'a \<Rightarrow> real"
assume IA:"\<And>u s.  \<lbrakk>2 < card s; \<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V; finite s;
-               s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n \<equiv> card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" and
-        as:"Suc n \<equiv> card s" "2 < card s" "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
+               s \<noteq> {}; s \<subseteq> V; setsum u s = 1; n = card s \<rbrakk> \<Longrightarrow> (\<Sum>x\<in>s. u x *\<^sub>R x) \<in> V" and
+        as:"Suc n = card s" "2 < card s" "\<forall>x\<in>V. \<forall>y\<in>V. \<forall>u v. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> V"
"finite s" "s \<noteq> {}" "s \<subseteq> V" "setsum u s = 1"
have "\<exists>x\<in>s. u x \<noteq> 1" proof(rule_tac ccontr)
assume " \<not> (\<exists>x\<in>s. u x \<noteq> 1)" hence "setsum u s = real_of_nat (card s)" unfolding card_eq_setsum by auto
@@ -1345,7 +1345,7 @@
next
case False then obtain w where "w\<in>s" by auto
show ?thesis unfolding caratheodory[of s]
-  proof(induct "CARD('n) + 1")
+  proof(induct ("CARD('n) + 1"))
have *:"{x.\<exists>sa. finite sa \<and> sa \<subseteq> s \<and> card sa \<le> 0 \<and> x \<in> convex hull sa} = {}"
using compact_empty by (auto simp add: convex_hull_empty)
case 0 thus ?case unfolding * by simp```
```--- a/src/HOL/Multivariate_Analysis/Derivative.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -607,7 +607,8 @@
have ***:"\<And>y y1 y2 d dx::real. (y1\<le>y\<and>y2\<le>y) \<or> (y\<le>y1\<and>y\<le>y2) \<Longrightarrow> d < abs dx \<Longrightarrow> abs(y1 - y - - dx) \<le> d \<Longrightarrow> (abs (y2 - y - dx) \<le> d) \<Longrightarrow> False" by arith
show False apply(rule ***[OF **, where dx="d * D \$ k \$ j" and d="\<bar>D \$ k \$ j\<bar> / 2 * \<bar>d\<bar>"])
using *[of "-d"] and *[of d] and d[THEN conjunct1] and j unfolding mult_minus_left
-    unfolding abs_mult diff_minus_eq_add scaleR.minus_left unfolding  group_simps by auto qed
+    unfolding abs_mult diff_minus_eq_add scaleR.minus_left unfolding group_simps by (auto intro: mult_pos_pos)
+qed

subsection {* In particular if we have a mapping into R^1. *}
```
```--- a/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Multivariate_Analysis/Euclidean_Space.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -3542,17 +3542,9 @@
and sp:"s \<subseteq> span t"
shows "\<exists>t'. (card t' = card t) \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
using f i sp
-proof(induct c\<equiv>"card(t - s)" arbitrary: s t rule: nat_less_induct)
-  fix n:: nat and s t :: "('a ^'n) set"
-  assume H: " \<forall>m<n. \<forall>(x:: ('a ^'n) set) xa.
-                finite xa \<longrightarrow>
-                independent x \<longrightarrow>
-                x \<subseteq> span xa \<longrightarrow>
-                m = card (xa - x) \<longrightarrow>
-                (\<exists>t'. (card t' = card xa) \<and> finite t' \<and>
-                      x \<subseteq> t' \<and> t' \<subseteq> x \<union> xa \<and> x \<subseteq> span t')"
-    and ft: "finite t" and s: "independent s" and sp: "s \<subseteq> span t"
-    and n: "n = card (t - s)"
+proof(induct "card (t - s)" arbitrary: s t rule: less_induct)
+  case less
+  note ft = `finite t` and s = `independent s` and sp = `s \<subseteq> span t`
let ?P = "\<lambda>t'. (card t' = card t) \<and> finite t' \<and> s \<subseteq> t' \<and> t' \<subseteq> s \<union> t \<and> s \<subseteq> span t'"
let ?ths = "\<exists>t'. ?P t'"
{assume st: "s \<subseteq> t"
@@ -3568,12 +3560,12 @@
{assume st: "\<not> s \<subseteq> t" "\<not> t \<subseteq> s"
from st(2) obtain b where b: "b \<in> t" "b \<notin> s" by blast
from b have "t - {b} - s \<subset> t - s" by blast
-      then have cardlt: "card (t - {b} - s) < n" using n ft
+      then have cardlt: "card (t - {b} - s) < card (t - s)" using ft
by (auto intro: psubset_card_mono)
from b ft have ct0: "card t \<noteq> 0" by auto
{assume stb: "s \<subseteq> span(t -{b})"
from ft have ftb: "finite (t -{b})" by auto
-      from H[rule_format, OF cardlt ftb s stb]
+      from less(1)[OF cardlt ftb s stb]
obtain u where u: "card u = card (t-{b})" "s \<subseteq> u" "u \<subseteq> s \<union> (t - {b})" "s \<subseteq> span u" and fu: "finite u" by blast
let ?w = "insert b u"
have th0: "s \<subseteq> insert b u" using u by blast
@@ -3594,8 +3586,8 @@
from stb obtain a where a: "a \<in> s" "a \<notin> span (t - {b})" by blast
have ab: "a \<noteq> b" using a b by blast
have at: "a \<notin> t" using a ab span_superset[of a "t- {b}"] by auto
-      have mlt: "card ((insert a (t - {b})) - s) < n"
-        using cardlt ft n  a b by auto
+      have mlt: "card ((insert a (t - {b})) - s) < card (t - s)"
+        using cardlt ft a b by auto
have ft': "finite (insert a (t - {b}))" using ft by auto
{fix x assume xs: "x \<in> s"
have t: "t \<subseteq> (insert b (insert a (t -{b})))" using b by auto
@@ -3608,7 +3600,7 @@
from span_trans[OF bs x] have "x \<in> span (insert a (t - {b}))"  .}
then have sp': "s \<subseteq> span (insert a (t - {b}))" by blast

-      from H[rule_format, OF mlt ft' s sp' refl] obtain u where
+      from less(1)[OF mlt ft' s sp'] obtain u where
u: "card u = card (insert a (t -{b}))" "finite u" "s \<subseteq> u" "u \<subseteq> s \<union> insert a (t -{b})"
"s \<subseteq> span u" by blast
from u a b ft at ct0 have "?P u" by auto
@@ -3657,11 +3649,9 @@
assumes sv: "(S::(real^'n) set) \<subseteq> V" and iS: "independent S"
shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
using sv iS
-proof(induct d\<equiv> "CARD('n) - card S" arbitrary: S rule: nat_less_induct)
-  fix n and S:: "(real^'n) set"
-  assume H: "\<forall>m<n. \<forall>S \<subseteq> V. independent S \<longrightarrow> m = CARD('n) - card S \<longrightarrow>
-              (\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B)"
-    and sv: "S \<subseteq> V" and i: "independent S" and n: "n = CARD('n) - card S"
+proof(induct "CARD('n) - card S" arbitrary: S rule: less_induct)
+  case less
+  note sv = `S \<subseteq> V` and i = `independent S`
let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
let ?ths = "\<exists>x. ?P x"
let ?d = "CARD('n)"
@@ -3674,11 +3664,11 @@
have th0: "insert a S \<subseteq> V" using a sv by blast
from independent_insert[of a S]  i a
have th1: "independent (insert a S)" by auto
-    have mlt: "?d - card (insert a S) < n"
-      using aS a n independent_bound[OF th1]
+    have mlt: "?d - card (insert a S) < ?d - card S"
+      using aS a independent_bound[OF th1]
by auto

-    from H[rule_format, OF mlt th0 th1 refl]
+    from less(1)[OF mlt th0 th1]
obtain B where B: "insert a S \<subseteq> B" "B \<subseteq> V" "independent B" " V \<subseteq> span B"
by blast
from B have "?P B" by auto```
```--- a/src/HOL/Nominal/Examples/Class.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Nominal/Examples/Class.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -15069,11 +15069,9 @@
assumes a: "(M[a\<turnstile>c>b]) \<longrightarrow>\<^isub>a M'" "a\<noteq>b"
shows "\<exists>M0. M0[a\<turnstile>c>b]=M' \<and> M \<longrightarrow>\<^isub>a M0"
using a
-apply(nominal_induct M\<equiv>"M[a\<turnstile>c>b]" M' avoiding: M a b rule: a_redu.strong_induct)
-apply(simp)
+apply(nominal_induct "M[a\<turnstile>c>b]" M' avoiding: M a b rule: a_redu.strong_induct)
apply(drule  crename_lredu)
apply(blast)
-apply(simp)
apply(drule  crename_credu)
apply(blast)
(* Cut *)
@@ -16132,11 +16130,9 @@
assumes a: "(M[x\<turnstile>n>y]) \<longrightarrow>\<^isub>a M'" "x\<noteq>y"
shows "\<exists>M0. M0[x\<turnstile>n>y]=M' \<and> M \<longrightarrow>\<^isub>a M0"
using a
-apply(nominal_induct M\<equiv>"M[x\<turnstile>n>y]" M' avoiding: M x y rule: a_redu.strong_induct)
-apply(simp)
+apply(nominal_induct "M[x\<turnstile>n>y]" M' avoiding: M x y rule: a_redu.strong_induct)
apply(drule  nrename_lredu)
apply(blast)
-apply(simp)
apply(drule  nrename_credu)
apply(blast)
(* Cut *)```
```--- a/src/HOL/Nominal/Examples/Fsub.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Nominal/Examples/Fsub.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -982,19 +982,18 @@
from `(\<Delta>@[(TVarB X Q)]@\<Gamma>) \<turnstile> M <: N`
and `\<Gamma> \<turnstile> P<:Q`
show "(\<Delta>@[(TVarB X P)]@\<Gamma>) \<turnstile> M <: N"
-    proof (induct \<Gamma>\<equiv>"\<Delta>@[(TVarB X Q)]@\<Gamma>" M N arbitrary: \<Gamma> X \<Delta> rule: subtype_of.induct)
-      case (SA_Top _ S \<Gamma> X \<Delta>)
-      then have lh_drv_prm\<^isub>1: "\<turnstile> (\<Delta>@[(TVarB X Q)]@\<Gamma>) ok"
-        and lh_drv_prm\<^isub>2: "S closed_in (\<Delta>@[(TVarB X Q)]@\<Gamma>)" by simp_all
-      have rh_drv: "\<Gamma> \<turnstile> P <: Q" by fact
-      hence "P closed_in \<Gamma>" by (simp add: subtype_implies_closed)
-      with lh_drv_prm\<^isub>1 have "\<turnstile> (\<Delta>@[(TVarB X P)]@\<Gamma>) ok" by (simp add: replace_type)
+    proof (induct "\<Delta>@[(TVarB X Q)]@\<Gamma>" M N arbitrary: \<Gamma> X \<Delta> rule: subtype_of.induct)
+      case (SA_Top S \<Gamma> X \<Delta>)
+      from `\<Gamma> \<turnstile> P <: Q`
+      have "P closed_in \<Gamma>" by (simp add: subtype_implies_closed)
+      with `\<turnstile> (\<Delta>@[(TVarB X Q)]@\<Gamma>) ok` have "\<turnstile> (\<Delta>@[(TVarB X P)]@\<Gamma>) ok"
moreover
-      from lh_drv_prm\<^isub>2 have "S closed_in (\<Delta>@[(TVarB X P)]@\<Gamma>)"
+      from `S closed_in (\<Delta>@[(TVarB X Q)]@\<Gamma>)` have "S closed_in (\<Delta>@[(TVarB X P)]@\<Gamma>)"
ultimately show "(\<Delta>@[(TVarB X P)]@\<Gamma>) \<turnstile> S <: Top" by (simp add: subtype_of.SA_Top)
next
-      case (SA_trans_TVar Y S _ N \<Gamma> X \<Delta>)
+      case (SA_trans_TVar Y S N \<Gamma> X \<Delta>)
then have IH_inner: "(\<Delta>@[(TVarB X P)]@\<Gamma>) \<turnstile> S <: N"
and lh_drv_prm: "(TVarB Y S) \<in> set (\<Delta>@[(TVarB X Q)]@\<Gamma>)"
and rh_drv: "\<Gamma> \<turnstile> P<:Q"
@@ -1020,23 +1019,23 @@
then show "(\<Delta>@[(TVarB X P)]@\<Gamma>) \<turnstile> Tvar Y <: N" using memb\<^isub>X\<^isub>P eq by auto
qed
next
-      case (SA_refl_TVar _ Y \<Gamma> X \<Delta>)
-      then have lh_drv_prm\<^isub>1: "\<turnstile> (\<Delta>@[(TVarB X Q)]@\<Gamma>) ok"
-        and lh_drv_prm\<^isub>2: "Y \<in> ty_dom (\<Delta>@[(TVarB X Q)]@\<Gamma>)" by simp_all
-      have "\<Gamma> \<turnstile> P <: Q" by fact
-      hence "P closed_in \<Gamma>" by (simp add: subtype_implies_closed)
-      with lh_drv_prm\<^isub>1 have "\<turnstile> (\<Delta>@[(TVarB X P)]@\<Gamma>) ok" by (simp add: replace_type)
+      case (SA_refl_TVar Y \<Gamma> X \<Delta>)
+      from `\<Gamma> \<turnstile> P <: Q`
+      have "P closed_in \<Gamma>" by (simp add: subtype_implies_closed)
+      with `\<turnstile> (\<Delta>@[(TVarB X Q)]@\<Gamma>) ok` have "\<turnstile> (\<Delta>@[(TVarB X P)]@\<Gamma>) ok"
moreover
-      from lh_drv_prm\<^isub>2 have "Y \<in> ty_dom (\<Delta>@[(TVarB X P)]@\<Gamma>)" by (simp add: doms_append)
+      from `Y \<in> ty_dom (\<Delta>@[(TVarB X Q)]@\<Gamma>)` have "Y \<in> ty_dom (\<Delta>@[(TVarB X P)]@\<Gamma>)"
ultimately show "(\<Delta>@[(TVarB X P)]@\<Gamma>) \<turnstile> Tvar Y <: Tvar Y" by (simp add: subtype_of.SA_refl_TVar)
next
-      case (SA_arrow _ S\<^isub>1 Q\<^isub>1 Q\<^isub>2 S\<^isub>2 \<Gamma> X \<Delta>)
+      case (SA_arrow S\<^isub>1 Q\<^isub>1 Q\<^isub>2 S\<^isub>2 \<Gamma> X \<Delta>)
then show "(\<Delta>@[(TVarB X P)]@\<Gamma>) \<turnstile> Q\<^isub>1 \<rightarrow> Q\<^isub>2 <: S\<^isub>1 \<rightarrow> S\<^isub>2" by blast
next
-      case (SA_all _ T\<^isub>1 S\<^isub>1 Y S\<^isub>2 T\<^isub>2 \<Gamma> X \<Delta>)
-      from SA_all(2,4,5,6)
+      case (SA_all T\<^isub>1 S\<^isub>1 Y S\<^isub>2 T\<^isub>2 \<Gamma> X \<Delta>)
have IH_inner\<^isub>1: "(\<Delta>@[(TVarB X P)]@\<Gamma>) \<turnstile> T\<^isub>1 <: S\<^isub>1"
-        and IH_inner\<^isub>2: "(((TVarB Y T\<^isub>1)#\<Delta>)@[(TVarB X P)]@\<Gamma>) \<turnstile> S\<^isub>2 <: T\<^isub>2" by force+
+        and IH_inner\<^isub>2: "(((TVarB Y T\<^isub>1)#\<Delta>)@[(TVarB X P)]@\<Gamma>) \<turnstile> S\<^isub>2 <: T\<^isub>2"
+        by (fastsimp intro: SA_all)+
then show "(\<Delta>@[(TVarB X P)]@\<Gamma>) \<turnstile> (\<forall>Y<:S\<^isub>1. S\<^isub>2) <: (\<forall>Y<:T\<^isub>1. T\<^isub>2)" by auto
qed
}
@@ -1263,7 +1262,7 @@
assumes "\<turnstile> (\<Gamma> @ VarB x Q # \<Delta>) ok"
shows "\<turnstile> (\<Gamma> @ \<Delta>) ok"
using assms
-proof (induct  \<Gamma>' \<equiv> "\<Gamma> @ VarB x Q # \<Delta>" arbitrary: \<Gamma> \<Delta>)
+proof (induct "\<Gamma> @ VarB x Q # \<Delta>" arbitrary: \<Gamma> \<Delta>)
case valid_nil
have "[] = \<Gamma> @ VarB x Q # \<Delta>" by fact
then have "False" by auto
@@ -1314,14 +1313,14 @@
and     "\<turnstile> (\<Delta> @ B # \<Gamma>) ok"
shows   "(\<Delta> @ B # \<Gamma>) \<turnstile> t : T"
using assms
-proof(nominal_induct \<Gamma>'\<equiv> "\<Delta> @ \<Gamma>" t T avoiding: \<Delta> \<Gamma> B rule: typing.strong_induct)
-  case (T_Var x' T \<Gamma>' \<Gamma>'' \<Delta>')
+proof(nominal_induct "\<Delta> @ \<Gamma>" t T avoiding: \<Delta> \<Gamma> B rule: typing.strong_induct)
+  case (T_Var x T)
then show ?case by auto
next
-  case (T_App \<Gamma> t\<^isub>1 T\<^isub>1 T\<^isub>2 t\<^isub>2 \<Gamma> \<Delta>)
+  case (T_App X t\<^isub>1 T\<^isub>2 T\<^isub>1\<^isub>1 T\<^isub>1\<^isub>2)
then show ?case by force
next
-  case (T_Abs y T\<^isub>1 \<Gamma>' t\<^isub>2 T\<^isub>2 \<Delta> \<Gamma>)
+  case (T_Abs y T\<^isub>1 t\<^isub>2 T\<^isub>2 \<Delta> \<Gamma>)
then have "VarB y T\<^isub>1 # \<Delta> @ \<Gamma> \<turnstile> t\<^isub>2 : T\<^isub>2" by simp
then have closed: "T\<^isub>1 closed_in (\<Delta> @ \<Gamma>)"
by (auto dest: typing_ok)
@@ -1336,22 +1335,22 @@
apply (rule closed)
done
then have "\<turnstile> ((VarB y T\<^isub>1 # \<Delta>) @ B # \<Gamma>) ok" by simp
-  then have "(VarB y T\<^isub>1 # \<Delta>) @ B # \<Gamma> \<turnstile> t\<^isub>2 : T\<^isub>2"
-    by (rule T_Abs) (simp add: T_Abs)
+  with _ have "(VarB y T\<^isub>1 # \<Delta>) @ B # \<Gamma> \<turnstile> t\<^isub>2 : T\<^isub>2"
+    by (rule T_Abs) simp
then have "VarB y T\<^isub>1 # \<Delta> @ B # \<Gamma> \<turnstile> t\<^isub>2 : T\<^isub>2" by simp
then show ?case by (rule typing.T_Abs)
next
-  case (T_Sub \<Gamma>' t S T \<Delta> \<Gamma>)
-  from `\<turnstile> (\<Delta> @ B # \<Gamma>) ok` and `\<Gamma>' = \<Delta> @ \<Gamma>`
+  case (T_Sub t S T \<Delta> \<Gamma>)
+  from refl and `\<turnstile> (\<Delta> @ B # \<Gamma>) ok`
have "\<Delta> @ B # \<Gamma> \<turnstile> t : S" by (rule T_Sub)
-  moreover from  `\<Gamma>'\<turnstile>S<:T` and `\<turnstile> (\<Delta> @ B # \<Gamma>) ok`
+  moreover from  `(\<Delta> @ \<Gamma>)\<turnstile>S<:T` and `\<turnstile> (\<Delta> @ B # \<Gamma>) ok`
have "(\<Delta> @ B # \<Gamma>)\<turnstile>S<:T"
by (rule weakening) (simp add: extends_def T_Sub)
ultimately show ?case by (rule typing.T_Sub)
next
-  case (T_TAbs X T\<^isub>1 \<Gamma>' t\<^isub>2 T\<^isub>2 \<Delta> \<Gamma>)
-  then have "TVarB X T\<^isub>1 # \<Delta> @ \<Gamma> \<turnstile> t\<^isub>2 : T\<^isub>2" by simp
-  then have closed: "T\<^isub>1 closed_in (\<Delta> @ \<Gamma>)"
+  case (T_TAbs X T\<^isub>1 t\<^isub>2 T\<^isub>2 \<Delta> \<Gamma>)
+  from `TVarB X T\<^isub>1 # \<Delta> @ \<Gamma> \<turnstile> t\<^isub>2 : T\<^isub>2`
+  have closed: "T\<^isub>1 closed_in (\<Delta> @ \<Gamma>)"
by (auto dest: typing_ok)
have "\<turnstile> (TVarB X T\<^isub>1 # \<Delta> @ B # \<Gamma>) ok"
apply (rule valid_consT)
@@ -1364,15 +1363,15 @@
apply (rule closed)
done
then have "\<turnstile> ((TVarB X T\<^isub>1 # \<Delta>) @ B # \<Gamma>) ok" by simp
-  then have "(TVarB X T\<^isub>1 # \<Delta>) @ B # \<Gamma> \<turnstile> t\<^isub>2 : T\<^isub>2"
-    by (rule T_TAbs) (simp add: T_TAbs)
+  with _ have "(TVarB X T\<^isub>1 # \<Delta>) @ B # \<Gamma> \<turnstile> t\<^isub>2 : T\<^isub>2"
+    by (rule T_TAbs) simp
then have "TVarB X T\<^isub>1 # \<Delta> @ B # \<Gamma> \<turnstile> t\<^isub>2 : T\<^isub>2" by simp
then show ?case by (rule typing.T_TAbs)
next
-  case (T_TApp X \<Gamma>' t\<^isub>1 T2 T11 T12 \<Delta> \<Gamma>)
+  case (T_TApp X t\<^isub>1 T2 T11 T12 \<Delta> \<Gamma>)
have "\<Delta> @ B # \<Gamma> \<turnstile> t\<^isub>1 : (\<forall>X<:T11. T12)"
-    by (rule T_TApp)+
-  moreover from `\<Gamma>'\<turnstile>T2<:T11` and `\<turnstile> (\<Delta> @ B # \<Gamma>) ok`
+    by (rule T_TApp refl)+
+  moreover from `(\<Delta> @ \<Gamma>)\<turnstile>T2<:T11` and `\<turnstile> (\<Delta> @ B # \<Gamma>) ok`
have "(\<Delta> @ B # \<Gamma>)\<turnstile>T2<:T11"
by (rule weakening) (simp add: extends_def T_TApp)
ultimately show ?case by (rule better_T_TApp)
@@ -1393,24 +1392,22 @@
assumes "(\<Gamma> @ VarB x Q # \<Delta>) \<turnstile> S <: T"
shows  "(\<Gamma>@\<Delta>) \<turnstile> S <: T"
using assms
-proof (induct  \<Gamma>' \<equiv> "\<Gamma> @ VarB x Q # \<Delta>" S T arbitrary: \<Gamma>)
-  case (SA_Top G' S G)
-  then have "\<turnstile> (G @ \<Delta>) ok" by (auto dest: valid_cons')
-  moreover have "S closed_in (G @ \<Delta>)" using SA_Top by (auto dest: closed_in_cons)
+proof (induct "\<Gamma> @ VarB x Q # \<Delta>" S T arbitrary: \<Gamma>)
+  case (SA_Top S)
+  then have "\<turnstile> (\<Gamma> @ \<Delta>) ok" by (auto dest: valid_cons')
+  moreover have "S closed_in (\<Gamma> @ \<Delta>)" using SA_Top by (auto dest: closed_in_cons)
ultimately show ?case using subtype_of.SA_Top by auto
next
-  case (SA_refl_TVar G X' G')
-  then have "\<turnstile> (G' @ VarB x Q # \<Delta>) ok" by simp
-  then have h1:"\<turnstile> (G' @ \<Delta>) ok" by (auto dest: valid_cons')
-  have "X' \<in> ty_dom (G' @ VarB x Q # \<Delta>)" using SA_refl_TVar by auto
-  then have h2:"X' \<in> ty_dom (G' @ \<Delta>)" using ty_dom_vrs by auto
+  case (SA_refl_TVar X)
+  from `\<turnstile> (\<Gamma> @ VarB x Q # \<Delta>) ok`
+  have h1:"\<turnstile> (\<Gamma> @ \<Delta>) ok" by (auto dest: valid_cons')
+  have "X \<in> ty_dom (\<Gamma> @ VarB x Q # \<Delta>)" using SA_refl_TVar by auto
+  then have h2:"X \<in> ty_dom (\<Gamma> @ \<Delta>)" using ty_dom_vrs by auto
show ?case using h1 h2 by auto
next
-  case (SA_all G T1 S1 X S2 T2 G')
-  have ih1:"TVarB X T1 # G = (TVarB X T1 # G') @ VarB x Q # \<Delta> \<Longrightarrow> ((TVarB X T1 # G') @ \<Delta>)\<turnstile>S2<:T2" by fact
-  then have h1:"(TVarB X T1 # (G' @ \<Delta>))\<turnstile>S2<:T2" using SA_all by auto
-  have ih2:"G = G' @ VarB x Q # \<Delta> \<Longrightarrow> (G' @ \<Delta>)\<turnstile>T1<:S1" by fact
-  then have h2:"(G' @ \<Delta>)\<turnstile>T1<:S1" using SA_all by auto
+  case (SA_all T1 S1 X S2 T2)
+  have h1:"((TVarB X T1 # \<Gamma>) @ \<Delta>)\<turnstile>S2<:T2" by (fastsimp intro: SA_all)
+  have h2:"(\<Gamma> @ \<Delta>)\<turnstile>T1<:S1" using SA_all by auto
then show ?case using h1 h2 by auto
qed (auto)

@@ -1418,26 +1415,26 @@
assumes H: "\<Delta> @ (TVarB X Q) # \<Gamma> \<turnstile> t : T"
shows "\<Gamma> \<turnstile> P <: Q \<Longrightarrow> \<Delta> @ (TVarB X P) # \<Gamma> \<turnstile> t : T"
using H
-  proof (nominal_induct \<Gamma>' \<equiv> "\<Delta> @ (TVarB X Q) # \<Gamma>" t T avoiding: P arbitrary: \<Delta> rule: typing.strong_induct)
-    case (T_Var x T G P D)
+  proof (nominal_induct "\<Delta> @ (TVarB X Q) # \<Gamma>" t T avoiding: P arbitrary: \<Delta> rule: typing.strong_induct)
+    case (T_Var x T P D)
then have "VarB x T \<in> set (D @ TVarB X P # \<Gamma>)"
and "\<turnstile>  (D @ TVarB X P # \<Gamma>) ok"
by (auto intro: replace_type dest!: subtype_implies_closed)
then show ?case by auto
next
-    case (T_App G t1 T1 T2 t2 P D)
+    case (T_App t1 T1 T2 t2 P D)
then show ?case by force
next
-    case (T_Abs x T1 G t2 T2 P D)
+    case (T_Abs x T1 t2 T2 P D)
then show ?case by (fastsimp dest: typing_ok)
next
-    case (T_Sub G t S T D)
+    case (T_Sub t S T P D)
then show ?case using subtype_narrow by fastsimp
next
-    case (T_TAbs X' T1 G t2 T2 P D)
+    case (T_TAbs X' T1 t2 T2 P D)
then show ?case by (fastsimp dest: typing_ok)
next
-    case (T_TApp X' G t1 T2 T11 T12 P D)
+    case (T_TApp X' t1 T2 T11 T12 P D)
then have "D @ TVarB X P # \<Gamma> \<turnstile> t1 : Forall X' T12 T11" by fastsimp
moreover have "(D @ [TVarB X Q] @ \<Gamma>) \<turnstile> T2<:T11" using T_TApp by auto
then have "(D @ [TVarB X P] @ \<Gamma>) \<turnstile> T2<:T11" using `\<Gamma>\<turnstile>P<:Q`
@@ -1454,8 +1451,8 @@
theorem subst_type: -- {* A.8 *}
assumes H: "(\<Delta> @ (VarB x U) # \<Gamma>) \<turnstile> t : T"
shows "\<Gamma> \<turnstile> u : U \<Longrightarrow> \<Delta> @ \<Gamma> \<turnstile> t[x \<mapsto> u] : T" using H
- proof (nominal_induct \<Gamma>' \<equiv> "\<Delta> @ (VarB x U) # \<Gamma>" t T avoiding: x u arbitrary: \<Delta> rule: typing.strong_induct)
-   case (T_Var y T G x u D)
+ proof (nominal_induct "\<Delta> @ (VarB x U) # \<Gamma>" t T avoiding: x u arbitrary: \<Delta> rule: typing.strong_induct)
+   case (T_Var y T x u D)
show ?case
proof (cases "x = y")
assume eq:"x=y"
@@ -1468,23 +1465,23 @@
by (auto simp add:binding.inject dest: valid_cons')
qed
next
-   case (T_App G t1 T1 T2 t2 x u D)
+   case (T_App t1 T1 T2 t2 x u D)
then show ?case by force
next
-   case (T_Abs y T1 G t2 T2 x u D)
+   case (T_Abs y T1 t2 T2 x u D)
then show ?case by force
next
-   case (T_Sub G t S T x u D)
+   case (T_Sub t S T x u D)
then have "D @ \<Gamma> \<turnstile> t[x \<mapsto> u] : S" by auto
moreover have "(D @ \<Gamma>) \<turnstile> S<:T" using T_Sub by (auto dest: strengthening)
ultimately show ?case by auto
next
-   case (T_TAbs X T1 G t2 T2 x u D)
-   from `TVarB X T1 # G \<turnstile> t2 : T2` have "X \<sharp> T1"
+   case (T_TAbs X T1 t2 T2 x u D)
+   from `TVarB X T1 # D @ VarB x U # \<Gamma> \<turnstile> t2 : T2` have "X \<sharp> T1"
by (auto simp add: valid_ty_dom_fresh dest: typing_ok intro!: closed_in_fresh)
with `X \<sharp> u` and T_TAbs show ?case by fastsimp
next
-   case (T_TApp X G t1 T2 T11 T12 x u D)
+   case (T_TApp X t1 T2 T11 T12 x u D)
then have "(D@\<Gamma>) \<turnstile>T2<:T11" using T_TApp by (auto dest: strengthening)
then show "((D @ \<Gamma>) \<turnstile> ((t1 \<cdot>\<^sub>\<tau> T2)[x \<mapsto> u]) : (T12[X \<mapsto> T2]\<^sub>\<tau>))" using T_TApp
by (force simp add: fresh_prod fresh_list_append fresh_list_cons subst_trm_fresh_tyvar)
@@ -1496,8 +1493,8 @@
assumes H: "(\<Delta> @ ((TVarB X Q) # \<Gamma>)) \<turnstile> S <: T"
shows "\<Gamma> \<turnstile> P <: Q \<Longrightarrow> (\<Delta>[X \<mapsto> P]\<^sub>e @ \<Gamma>) \<turnstile> S[X \<mapsto> P]\<^sub>\<tau> <: T[X \<mapsto> P]\<^sub>\<tau>"
using H
-proof (nominal_induct \<Gamma>' \<equiv> "\<Delta> @ TVarB X Q # \<Gamma>" S T avoiding: X P arbitrary: \<Delta> rule: subtype_of.strong_induct)
-  case (SA_Top G S X P D)
+proof (nominal_induct "\<Delta> @ TVarB X Q # \<Gamma>" S T avoiding: X P arbitrary: \<Delta> rule: subtype_of.strong_induct)
+  case (SA_Top S X P D)
then have "\<turnstile> (D @ TVarB X Q # \<Gamma>) ok" by simp
moreover have closed: "P closed_in \<Gamma>" using SA_Top subtype_implies_closed by auto
ultimately have "\<turnstile> (D[X \<mapsto> P]\<^sub>e @ \<Gamma>) ok" by (rule valid_subst)
@@ -1505,17 +1502,18 @@
then have "S[X \<mapsto> P]\<^sub>\<tau> closed_in  (D[X \<mapsto> P]\<^sub>e @ \<Gamma>)" using closed by (rule subst_closed_in)
ultimately show ?case by auto
next
-  case (SA_trans_TVar Y S G T X P D)
-  have h:"G\<turnstile>S<:T" by fact
+  case (SA_trans_TVar Y S T X P D)
+  have h:"(D @ TVarB X Q # \<Gamma>)\<turnstile>S<:T" by fact
then have ST: "(D[X \<mapsto> P]\<^sub>e @ \<Gamma>) \<turnstile> S[X \<mapsto> P]\<^sub>\<tau> <: T[X \<mapsto> P]\<^sub>\<tau>" using SA_trans_TVar by auto
-  from `G\<turnstile>S<:T` have G_ok: "\<turnstile> G ok" by (rule subtype_implies_ok)
+  from h have G_ok: "\<turnstile> (D @ TVarB X Q # \<Gamma>) ok" by (rule subtype_implies_ok)
from G_ok and SA_trans_TVar have X\<Gamma>_ok: "\<turnstile> (TVarB X Q # \<Gamma>) ok"
by (auto intro: validE_append)
show "(D[X \<mapsto> P]\<^sub>e @ \<Gamma>) \<turnstile> Tvar Y[X \<mapsto> P]\<^sub>\<tau><:T[X \<mapsto> P]\<^sub>\<tau>"
proof (cases "X = Y")
assume eq: "X = Y"
-    from eq and SA_trans_TVar have "TVarB Y Q \<in> set G" by simp
-    with G_ok have QS: "Q = S" using `TVarB Y S \<in> set G` by (rule uniqueness_of_ctxt)
+    from eq and SA_trans_TVar have "TVarB Y Q \<in> set (D @ TVarB X Q # \<Gamma>)" by simp
+    with G_ok have QS: "Q = S" using `TVarB Y S \<in> set (D @ TVarB X Q # \<Gamma>)`
+      by (rule uniqueness_of_ctxt)
from X\<Gamma>_ok have "X \<sharp> ty_dom \<Gamma>" and "Q closed_in \<Gamma>" by auto
then have XQ: "X \<sharp> Q" by (rule closed_in_fresh)
note `\<Gamma>\<turnstile>P<:Q`
@@ -1552,8 +1550,8 @@
qed
qed
next
-  case (SA_refl_TVar G Y X P D)
-  then have "\<turnstile> (D @ TVarB X Q # \<Gamma>) ok" by simp
+  case (SA_refl_TVar Y X P D)
+  note `\<turnstile> (D @ TVarB X Q # \<Gamma>) ok`
moreover from SA_refl_TVar have closed: "P closed_in \<Gamma>"
by (auto dest: subtype_implies_closed)
ultimately have ok: "\<turnstile> (D[X \<mapsto> P]\<^sub>e @ \<Gamma>) ok" using valid_subst by auto
@@ -1571,12 +1569,12 @@
with neq and ok show ?thesis by auto
qed
next
-  case (SA_arrow G T1 S1 S2 T2 X P D)
+  case (SA_arrow T1 S1 S2 T2 X P D)
then have h1:"(D[X \<mapsto> P]\<^sub>e @ \<Gamma>)\<turnstile>T1[X \<mapsto> P]\<^sub>\<tau><:S1[X \<mapsto> P]\<^sub>\<tau>" using SA_arrow by auto
from SA_arrow have h2:"(D[X \<mapsto> P]\<^sub>e @ \<Gamma>)\<turnstile>S2[X \<mapsto> P]\<^sub>\<tau><:T2[X \<mapsto> P]\<^sub>\<tau>" using SA_arrow by auto
show ?case using subtype_of.SA_arrow h1 h2 by auto
next
-  case (SA_all G T1 S1 Y S2 T2 X P D)
+  case (SA_all T1 S1 Y S2 T2 X P D)
then have Y: "Y \<sharp> ty_dom (D @ TVarB X Q # \<Gamma>)"
by (auto dest: subtype_implies_ok intro: fresh_dom)
moreover from SA_all have "S1 closed_in (D @ TVarB X Q # \<Gamma>)"
@@ -1594,13 +1592,13 @@
assumes H: "(D @ TVarB X Q # G) \<turnstile> t : T"
shows "G \<turnstile> P <: Q \<Longrightarrow>
(D[X \<mapsto> P]\<^sub>e @ G) \<turnstile> t[X \<mapsto>\<^sub>\<tau> P] : T[X \<mapsto> P]\<^sub>\<tau>" using H
-proof (nominal_induct \<Gamma>'\<equiv>"(D @ TVarB X Q # G)" t T avoiding: X P arbitrary: D rule: typing.strong_induct)
-  case (T_Var x T G' X P D')
+proof (nominal_induct "D @ TVarB X Q # G" t T avoiding: X P arbitrary: D rule: typing.strong_induct)
+  case (T_Var x T X P D')
have "G\<turnstile>P<:Q" by fact
then have "P closed_in G" using subtype_implies_closed by auto
-  moreover have "\<turnstile> (D' @ TVarB X Q # G) ok" using T_Var by auto
+  moreover note `\<turnstile> (D' @ TVarB X Q # G) ok`
ultimately have "\<turnstile> (D'[X \<mapsto> P]\<^sub>e @ G) ok" using valid_subst by auto
-  moreover have "VarB x T \<in> set (D' @ TVarB X Q # G)" using T_Var by auto
+  moreover note `VarB x T \<in> set (D' @ TVarB X Q # G)`
then have "VarB x T \<in> set D' \<or> VarB x T \<in> set G" by simp
then have "(VarB x (T[X \<mapsto> P]\<^sub>\<tau>)) \<in> set (D'[X \<mapsto> P]\<^sub>e @ G)"
proof
@@ -1621,25 +1619,25 @@
qed
ultimately show ?case by auto
next
-  case (T_App G' t1 T1 T2 t2 X P D')
+  case (T_App t1 T1 T2 t2 X P D')
then have "D'[X \<mapsto> P]\<^sub>e @ G \<turnstile> t1[X \<mapsto>\<^sub>\<tau> P] : (T1 \<rightarrow> T2)[X \<mapsto> P]\<^sub>\<tau>" by auto
moreover from T_App have "D'[X \<mapsto> P]\<^sub>e @ G \<turnstile> t2[X \<mapsto>\<^sub>\<tau> P] : T1[X \<mapsto> P]\<^sub>\<tau>" by auto
ultimately show ?case by auto
next
-  case (T_Abs x T1 G' t2 T2 X P D')
+  case (T_Abs x T1 t2 T2 X P D')
then show ?case by force
next
-  case (T_Sub G' t S T X P D')
+  case (T_Sub t S T X P D')
then show ?case using substT_subtype by force
next
-  case (T_TAbs X' G' T1 t2 T2 X P D')
+  case (T_TAbs X' T1 t2 T2 X P D')
then have "X' \<sharp> ty_dom (D' @ TVarB X Q # G)"
-  and "G' closed_in (D' @ TVarB X Q # G)"
+  and "T1 closed_in (D' @ TVarB X Q # G)"
by (auto dest: typing_ok)
-  then have "X' \<sharp> G'" by (rule closed_in_fresh)
+  then have "X' \<sharp> T1" by (rule closed_in_fresh)
with T_TAbs show ?case by force
next
-  case (T_TApp X' G' t1 T2 T11 T12 X P D')
+  case (T_TApp X' t1 T2 T11 T12 X P D')
then have "X' \<sharp> ty_dom (D' @ TVarB X Q # G)"
moreover from T_TApp have "T11 closed_in (D' @ TVarB X Q # G)"
@@ -1824,22 +1822,22 @@
lemma Fun_canonical: -- {* A.14(1) *}
assumes ty: "[] \<turnstile> v : T\<^isub>1 \<rightarrow> T\<^isub>2"
shows "val v \<Longrightarrow> \<exists>x t S. v = (\<lambda>x:S. t)" using ty
-proof (induct \<Gamma>\<equiv>"[]::env" v T\<equiv>"T\<^isub>1 \<rightarrow> T\<^isub>2" arbitrary: T\<^isub>1 T\<^isub>2)
-  case (T_Sub \<Gamma> t S T)
-  hence "\<Gamma> \<turnstile> S <: T\<^isub>1 \<rightarrow> T\<^isub>2" by simp
-  then obtain S\<^isub>1 S\<^isub>2 where S: "S = S\<^isub>1 \<rightarrow> S\<^isub>2"
+proof (induct "[]::env" v "T\<^isub>1 \<rightarrow> T\<^isub>2" arbitrary: T\<^isub>1 T\<^isub>2)
+  case (T_Sub t S)
+  from `[] \<turnstile> S <: T\<^isub>1 \<rightarrow> T\<^isub>2`
+  obtain S\<^isub>1 S\<^isub>2 where S: "S = S\<^isub>1 \<rightarrow> S\<^isub>2"
by cases (auto simp add: T_Sub)
-  with `val t` and `\<Gamma> = []` show ?case by (rule T_Sub)
+  then show ?case using `val t` by (rule T_Sub)
qed (auto)

lemma TyAll_canonical: -- {* A.14(3) *}
fixes X::tyvrs
assumes ty: "[] \<turnstile> v : (\<forall>X<:T\<^isub>1. T\<^isub>2)"
shows "val v \<Longrightarrow> \<exists>X t S. v = (\<lambda>X<:S. t)" using ty
-proof (induct \<Gamma>\<equiv>"[]::env" v T\<equiv>"\<forall>X<:T\<^isub>1. T\<^isub>2" arbitrary: X T\<^isub>1 T\<^isub>2)
-  case (T_Sub  \<Gamma> t S T)
-  hence "\<Gamma> \<turnstile> S <: (\<forall>X<:T\<^isub>1. T\<^isub>2)" by simp
-  then obtain X S\<^isub>1 S\<^isub>2 where S: "S = (\<forall>X<:S\<^isub>1. S\<^isub>2)"
+proof (induct "[]::env" v "\<forall>X<:T\<^isub>1. T\<^isub>2" arbitrary: X T\<^isub>1 T\<^isub>2)
+  case (T_Sub t S)
+  from `[] \<turnstile> S <: (\<forall>X<:T\<^isub>1. T\<^isub>2)`
+  obtain X S\<^isub>1 S\<^isub>2 where S: "S = (\<forall>X<:S\<^isub>1. S\<^isub>2)"
by cases (auto simp add: T_Sub)
then show ?case using T_Sub by auto
qed (auto)
@@ -1848,8 +1846,8 @@
assumes "[] \<turnstile> t : T"
shows "val t \<or> (\<exists>t'. t \<longmapsto> t')"
using assms
-proof (induct \<Gamma> \<equiv> "[]::env" t T)
-  case (T_App \<Gamma> t\<^isub>1 T\<^isub>1\<^isub>1  T\<^isub>1\<^isub>2 t\<^isub>2)
+proof (induct "[]::env" t T)
+  case (T_App t\<^isub>1 T\<^isub>1\<^isub>1  T\<^isub>1\<^isub>2 t\<^isub>2)
hence "val t\<^isub>1 \<or> (\<exists>t'. t\<^isub>1 \<longmapsto> t')" by simp
thus ?case
proof
@@ -1875,7 +1873,7 @@
thus ?case by auto
qed
next
-  case (T_TApp X \<Gamma> t\<^isub>1 T\<^isub>2 T\<^isub>1\<^isub>1 T\<^isub>1\<^isub>2)
+  case (T_TApp X t\<^isub>1 T\<^isub>2 T\<^isub>1\<^isub>1 T\<^isub>1\<^isub>2)
hence "val t\<^isub>1 \<or> (\<exists>t'. t\<^isub>1 \<longmapsto> t')" by simp
thus ?case
proof```
```--- a/src/HOL/Nominal/Examples/Pattern.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Nominal/Examples/Pattern.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -410,37 +410,34 @@
and b: "\<Gamma> \<turnstile> u : U"
shows "\<Delta> @ \<Gamma> \<turnstile> t[x\<mapsto>u] : T" using a b
proof (nominal_induct \<Gamma>'\<equiv>"\<Delta> @ [(x, U)] @ \<Gamma>" t T avoiding: x u \<Delta> rule: typing.strong_induct)
-  case (Var \<Gamma>' y T x u \<Delta>)
-  then have a1: "valid (\<Delta> @ [(x, U)] @ \<Gamma>)"
-       and  a2: "(y, T) \<in> set (\<Delta> @ [(x, U)] @ \<Gamma>)"
-       and  a3: "\<Gamma> \<turnstile> u : U" by simp_all
-  from a1 have a4: "valid (\<Delta> @ \<Gamma>)" by (rule valid_insert)
+  case (Var y T x u \<Delta>)
+  from `valid (\<Delta> @ [(x, U)] @ \<Gamma>)`
+  have valid: "valid (\<Delta> @ \<Gamma>)" by (rule valid_insert)
show "\<Delta> @ \<Gamma> \<turnstile> Var y[x\<mapsto>u] : T"
proof cases
assume eq: "x = y"
-    from a1 a2 have "T = U" using eq by (auto intro: context_unique)
-    with a3 show "\<Delta> @ \<Gamma> \<turnstile> Var y[x\<mapsto>u] : T" using eq a4 by (auto intro: weakening)
+    from Var eq have "T = U" by (auto intro: context_unique)
+    with Var eq valid show "\<Delta> @ \<Gamma> \<turnstile> Var y[x\<mapsto>u] : T" by (auto intro: weakening)
next
assume ineq: "x \<noteq> y"
-    from a2 have "(y, T) \<in> set (\<Delta> @ \<Gamma>)" using ineq by simp
-    then show "\<Delta> @ \<Gamma> \<turnstile> Var y[x\<mapsto>u] : T" using ineq a4 by auto
+    from Var ineq have "(y, T) \<in> set (\<Delta> @ \<Gamma>)" by simp
+    then show "\<Delta> @ \<Gamma> \<turnstile> Var y[x\<mapsto>u] : T" using ineq valid by auto
qed
next
-  case (Tuple \<Gamma>' t\<^isub>1 T\<^isub>1 t\<^isub>2 T\<^isub>2)
-  from `\<Gamma> \<turnstile> u : U` `\<Gamma>' = \<Delta> @ [(x, U)] @ \<Gamma>`
+  case (Tuple t\<^isub>1 T\<^isub>1 t\<^isub>2 T\<^isub>2)
+  from refl `\<Gamma> \<turnstile> u : U`
have "\<Delta> @ \<Gamma> \<turnstile> t\<^isub>1[x\<mapsto>u] : T\<^isub>1" by (rule Tuple)
-  moreover from `\<Gamma> \<turnstile> u : U` `\<Gamma>' = \<Delta> @ [(x, U)] @ \<Gamma>`
+  moreover from refl `\<Gamma> \<turnstile> u : U`
have "\<Delta> @ \<Gamma> \<turnstile> t\<^isub>2[x\<mapsto>u] : T\<^isub>2" by (rule Tuple)
ultimately have "\<Delta> @ \<Gamma> \<turnstile> \<langle>t\<^isub>1[x\<mapsto>u], t\<^isub>2[x\<mapsto>u]\<rangle> : T\<^isub>1 \<otimes> T\<^isub>2" ..
then show ?case by simp
next
-  case (Let p t \<Gamma>' T \<Delta>' s S)
-  from `\<Gamma> \<turnstile> u : U` `\<Gamma>' = \<Delta> @ [(x, U)] @ \<Gamma>`
+  case (Let p t T \<Delta>' s S)
+  from refl `\<Gamma> \<turnstile> u : U`
have "\<Delta> @ \<Gamma> \<turnstile> t[x\<mapsto>u] : T" by (rule Let)
moreover note `\<turnstile> p : T \<Rightarrow> \<Delta>'`
-  moreover from `\<Gamma>' = \<Delta> @ [(x, U)] @ \<Gamma>`
-  have "\<Delta>' @ \<Gamma>' = (\<Delta>' @ \<Delta>) @ [(x, U)] @ \<Gamma>" by simp
-  with `\<Gamma> \<turnstile> u : U` have "(\<Delta>' @ \<Delta>) @ \<Gamma> \<turnstile> s[x\<mapsto>u] : S" by (rule Let)
+  moreover have "\<Delta>' @ (\<Delta> @ [(x, U)] @ \<Gamma>) = (\<Delta>' @ \<Delta>) @ [(x, U)] @ \<Gamma>" by simp
+  then have "(\<Delta>' @ \<Delta>) @ \<Gamma> \<turnstile> s[x\<mapsto>u] : S" using `\<Gamma> \<turnstile> u : U` by (rule Let)
then have "\<Delta>' @ \<Delta> @ \<Gamma> \<turnstile> s[x\<mapsto>u] : S" by simp
ultimately have "\<Delta> @ \<Gamma> \<turnstile> (LET p = t[x\<mapsto>u] IN s[x\<mapsto>u]) : S"
by (rule better_T_Let)
@@ -448,10 +445,10 @@
by (simp add: fresh_star_def fresh_list_nil fresh_list_cons)
ultimately show ?case by simp
next
-  case (Abs y T \<Gamma>' t S)
-  from `\<Gamma>' = \<Delta> @ [(x, U)] @ \<Gamma>` have "(y, T) # \<Gamma>' = ((y, T) # \<Delta>) @ [(x, U)] @ \<Gamma>"
+  case (Abs y T t S)
+  have "(y, T) # \<Delta> @ [(x, U)] @ \<Gamma> = ((y, T) # \<Delta>) @ [(x, U)] @ \<Gamma>"
by simp
-  with `\<Gamma> \<turnstile> u : U` have "((y, T) # \<Delta>) @ \<Gamma> \<turnstile> t[x\<mapsto>u] : S" by (rule Abs)
+  then have "((y, T) # \<Delta>) @ \<Gamma> \<turnstile> t[x\<mapsto>u] : S" using `\<Gamma> \<turnstile> u : U` by (rule Abs)
then have "(y, T) # \<Delta> @ \<Gamma> \<turnstile> t[x\<mapsto>u] : S" by simp
then have "\<Delta> @ \<Gamma> \<turnstile> (\<lambda>y:T. t[x\<mapsto>u]) : T \<rightarrow> S"
by (rule typing.Abs)
@@ -459,10 +456,10 @@
ultimately show ?case by simp
next
-  case (App \<Gamma>' t\<^isub>1 T S t\<^isub>2)
-  from `\<Gamma> \<turnstile> u : U` `\<Gamma>' = \<Delta> @ [(x, U)] @ \<Gamma>`
+  case (App t\<^isub>1 T S t\<^isub>2)
+  from refl `\<Gamma> \<turnstile> u : U`
have "\<Delta> @ \<Gamma> \<turnstile> t\<^isub>1[x\<mapsto>u] : T \<rightarrow> S" by (rule App)
-  moreover from `\<Gamma> \<turnstile> u : U` `\<Gamma>' = \<Delta> @ [(x, U)] @ \<Gamma>`
+  moreover from refl `\<Gamma> \<turnstile> u : U`
have "\<Delta> @ \<Gamma> \<turnstile> t\<^isub>2[x\<mapsto>u] : T" by (rule App)
ultimately have "\<Delta> @ \<Gamma> \<turnstile> (t\<^isub>1[x\<mapsto>u]) \<cdot> (t\<^isub>2[x\<mapsto>u]) : S"
by (rule typing.App)```
```--- a/src/HOL/Nominal/Examples/SOS.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Nominal/Examples/SOS.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -220,10 +220,10 @@
shows "(\<Delta>@\<Gamma>) \<turnstile> e[x::=e'] : T"
using a b
proof (nominal_induct \<Gamma>\<equiv>"\<Delta>@[(x,T')]@\<Gamma>" e T avoiding: e' \<Delta> rule: typing.strong_induct)
-  case (t_Var \<Gamma>' y T e' \<Delta>)
+  case (t_Var y T e' \<Delta>)
then have a1: "valid (\<Delta>@[(x,T')]@\<Gamma>)"
and  a2: "(y,T) \<in> set (\<Delta>@[(x,T')]@\<Gamma>)"
-       and  a3: "\<Gamma> \<turnstile> e' : T'" by simp_all
+       and  a3: "\<Gamma> \<turnstile> e' : T'" .
from a1 have a4: "valid (\<Delta>@\<Gamma>)" by (rule valid_insert)
{ assume eq: "x=y"
from a1 a2 have "T=T'" using eq by (auto intro: context_unique)```
```--- a/src/HOL/Nominal/nominal_induct.ML	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Nominal/nominal_induct.ML	Wed Jan 20 10:38:19 2010 +0100
@@ -5,7 +5,7 @@

structure NominalInduct:
sig
-  val nominal_induct_tac: Proof.context -> (binding option * term) option list list ->
+  val nominal_induct_tac: Proof.context -> bool -> (binding option * (term * bool)) option list list ->
(string * typ) list -> (string * typ) list list -> thm list ->
thm list -> int -> Rule_Cases.cases_tactic
val nominal_induct_method: (Proof.context -> Proof.method) context_parser
@@ -74,26 +74,29 @@
else map (tune o #1) (take (p - n) ps) @ xs;
in Logic.list_rename_params (ys, prem) end;
fun rename_prems prop =
-      let val (As, C) = Logic.strip_horn (Thm.prop_of rule)
+      let val (As, C) = Logic.strip_horn prop
in Logic.list_implies (map rename As, C) end;
in Thm.equal_elim (Thm.reflexive (Drule.cterm_fun rename_prems (Thm.cprop_of rule))) rule end;

(* nominal_induct_tac *)

-fun nominal_induct_tac ctxt def_insts avoiding fixings rules facts =
+fun nominal_induct_tac ctxt simp def_insts avoiding fixings rules facts =
let
val thy = ProofContext.theory_of ctxt;
val cert = Thm.cterm_of thy;

val ((insts, defs), defs_ctxt) = fold_map Induct.add_defs def_insts ctxt |>> split_list;
-    val atomized_defs = map (map (Conv.fconv_rule ObjectLogic.atomize)) defs;
+    val atomized_defs = map (map (Conv.fconv_rule Induct.atomize_cterm)) defs;

val finish_rule =
split_all_tuples
#> rename_params_rule true
(map (Name.clean o ProofContext.revert_skolem defs_ctxt o fst) avoiding);
-    fun rule_cases r = Rule_Cases.make_nested true (Thm.prop_of r) (Induct.rulified_term r);
+
+    fun rule_cases ctxt r =
+      let val r' = if simp then Induct.simplified_rule ctxt r else r
+      in Rule_Cases.make_nested (Thm.prop_of r') (Induct.rulified_term r') end;
in
(fn i => fn st =>
rules
@@ -102,19 +105,32 @@
|> Seq.maps (fn (((cases, concls), (more_consumes, more_facts)), rule) =>
(PRECISE_CONJUNCTS (length concls) (ALLGOALS (fn j =>
(CONJUNCTS (ALLGOALS
-            (Method.insert_tac (more_facts @ nth_list atomized_defs (j - 1))
-              THEN' Induct.fix_tac defs_ctxt
-                (nth concls (j - 1) + more_consumes)
-                (nth_list fixings (j - 1))))
+            let
+              val adefs = nth_list atomized_defs (j - 1);
+              val xs = nth_list fixings (j - 1);
+              val k = nth concls (j - 1) + more_consumes
+            in
+              Method.insert_tac (more_facts @ adefs) THEN'
+                (if simp then
+                   Induct.rotate_tac k (length adefs) THEN'
+                   Induct.fix_tac defs_ctxt k
+                     (List.partition (member op = frees) xs |> op @)
+                 else
+                   Induct.fix_tac defs_ctxt k xs)
+            end)
THEN' Induct.inner_atomize_tac) j))
THEN' Induct.atomize_tac) i st |> Seq.maps (fn st' =>
Induct.guess_instance ctxt
(finish_rule (Induct.internalize more_consumes rule)) i st'
|> Seq.maps (fn rule' =>
-              CASES (rule_cases rule' cases)
+              CASES (rule_cases ctxt rule' cases)
(Tactic.rtac (rename_params_rule false [] rule') i THEN
PRIMITIVE (singleton (ProofContext.export defs_ctxt ctxt))) st'))))
-    THEN_ALL_NEW_CASES Induct.rulify_tac
+    THEN_ALL_NEW_CASES
+      ((if simp then Induct.simplify_tac ctxt THEN' (TRY o Induct.trivial_tac)
+        else K all_tac)
+       THEN_ALL_NEW Induct.rulify_tac)
end;

@@ -128,11 +144,14 @@
val fixingN = "arbitrary";  (* to be consistent with induct; hopefully this changes again *)
val ruleN = "rule";

-val inst = Scan.lift (Args.\$\$\$ "_") >> K NONE || Args.term >> SOME;
+val inst = Scan.lift (Args.\$\$\$ "_") >> K NONE ||
+  Args.term >> (SOME o rpair false) ||
+  Scan.lift (Args.\$\$\$ "(") |-- (Args.term >> (SOME o rpair true)) --|
+    Scan.lift (Args.\$\$\$ ")");

val def_inst =
((Scan.lift (Args.binding --| (Args.\$\$\$ "\<equiv>" || Args.\$\$\$ "==")) >> SOME)
-      -- Args.term) >> SOME ||
+      -- (Args.term >> rpair false)) >> SOME ||
inst >> Option.map (pair NONE);

val free = Args.context -- Args.term >> (fn (_, Free v) => v | (ctxt, t) =>
@@ -153,11 +172,11 @@
in

val nominal_induct_method =
-  P.and_list' (Scan.repeat (unless_more_args def_inst)) --
-  avoiding -- fixing -- rule_spec >>
-  (fn (((x, y), z), w) => fn ctxt =>
+  Args.mode Induct.no_simpN -- (P.and_list' (Scan.repeat (unless_more_args def_inst)) --
+  avoiding -- fixing -- rule_spec) >>
+  (fn (no_simp, (((x, y), z), w)) => fn ctxt =>
RAW_METHOD_CASES (fn facts =>
-      HEADGOAL (nominal_induct_tac ctxt x y z w facts)));
+      HEADGOAL (nominal_induct_tac ctxt (not no_simp) x y z w facts)));

end;
```
```--- a/src/HOL/Nominal/nominal_inductive2.ML	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Nominal/nominal_inductive2.ML	Wed Jan 20 10:38:19 2010 +0100
@@ -196,7 +196,7 @@
| add_set (t, T) ((u, U) :: ps) =
if T = U then
let val S = HOLogic.mk_setT T
-                in (Const (@{const_name union}, S --> S --> S) \$ u \$ t, T) :: ps
+                in (Const (@{const_name sup}, S --> S --> S) \$ u \$ t, T) :: ps
end
else (u, U) :: add_set (t, T) ps
in```
```--- a/src/HOL/Old_Number_Theory/Legacy_GCD.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Old_Number_Theory/Legacy_GCD.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -233,37 +233,39 @@
with gcd_unique[of "gcd u v" x y]  show ?thesis by auto
qed

-lemma ind_euclid:
-  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
-  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
+lemma ind_euclid:
+  assumes c: " \<forall>a b. P (a::nat) b \<longleftrightarrow> P b a" and z: "\<forall>a. P a 0"
+  and add: "\<forall>a b. P a b \<longrightarrow> P a (a + b)"
shows "P a b"
-proof(induct n\<equiv>"a+b" arbitrary: a b rule: nat_less_induct)
-  fix n a b
-  assume H: "\<forall>m < n. \<forall>a b. m = a + b \<longrightarrow> P a b" "n = a + b"
+proof(induct "a + b" arbitrary: a b rule: less_induct)
+  case less
have "a = b \<or> a < b \<or> b < a" by arith
moreover {assume eq: "a= b"
-    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq by simp}
+    from add[rule_format, OF z[rule_format, of a]] have "P a b" using eq
+    by simp}
moreover
{assume lt: "a < b"
-    hence "a + b - a < n \<or> a = 0"  using H(2) by arith
+    hence "a + b - a < a + b \<or> a = 0" by arith
moreover
{assume "a =0" with z c have "P a b" by blast }
moreover
-    {assume ab: "a + b - a < n"
-      have th0: "a + b - a = a + (b - a)" using lt by arith
-      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
-      have "P a b" by (simp add: th0[symmetric])}
+    {assume "a + b - a < a + b"
+      also have th0: "a + b - a = a + (b - a)" using lt by arith
+      finally have "a + (b - a) < a + b" .
+      then have "P a (a + (b - a))" by (rule add[rule_format, OF less])
+      then have "P a b" by (simp add: th0[symmetric])}
ultimately have "P a b" by blast}
moreover
{assume lt: "a > b"
-    hence "b + a - b < n \<or> b = 0"  using H(2) by arith
+    hence "b + a - b < a + b \<or> b = 0" by arith
moreover
{assume "b =0" with z c have "P a b" by blast }
moreover
-    {assume ab: "b + a - b < n"
-      have th0: "b + a - b = b + (a - b)" using lt by arith
-      from add[rule_format, OF H(1)[rule_format, OF ab th0]]
-      have "P b a" by (simp add: th0[symmetric])
+    {assume "b + a - b < a + b"
+      also have th0: "b + a - b = b + (a - b)" using lt by arith
+      finally have "b + (a - b) < a + b" .
+      then have "P b (b + (a - b))" by (rule add[rule_format, OF less])
+      then have "P b a" by (simp add: th0[symmetric])
hence "P a b" using c by blast }
ultimately have "P a b" by blast}
ultimately  show "P a b" by blast```
```--- a/src/HOL/Product_Type.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Product_Type.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -1000,7 +1000,7 @@
(SML infix 2 "*")
(OCaml infix 2 "*")
-  (Scala "!((_),/ (_))")
+  (Scala "((_),/ (_))")

code_instance * :: eq
```--- a/src/HOL/Tools/Datatype/datatype_data.ML	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Tools/Datatype/datatype_data.ML	Wed Jan 20 10:38:19 2010 +0100
@@ -341,7 +341,8 @@
((Binding.empty, map (fn th => th RS notE) (flat distinct)),
[Classical.safe_elim NONE]),
-        ((Binding.empty, weak_case_congs), [Simplifier.attrib (op addcongs)])]
+        ((Binding.empty, weak_case_congs), [Simplifier.attrib (op addcongs)]),
+        ((Binding.empty, flat (distinct @ inject)), [Induct.add_simp_rule])]
@ named_rules @ unnamed_rules)
|> snd
```--- a/src/HOL/Tools/inductive_set.ML	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Tools/inductive_set.ML	Wed Jan 20 10:38:19 2010 +0100
@@ -34,7 +34,7 @@
val collect_mem_simproc =
Simplifier.simproc (theory "Set") "Collect_mem" ["Collect t"] (fn thy => fn ss =>
fn S as Const ("Collect", Type ("fun", [_, T])) \$ t =>
-         let val (u, Ts, ps) = HOLogic.strip_psplits t
+         let val (u, _, ps) = HOLogic.strip_psplits t
in case u of
(c as Const ("op :", _)) \$ q \$ S' =>
(case try (HOLogic.strip_ptuple ps) q of```
```--- a/src/HOL/Tools/numeral.ML	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Tools/numeral.ML	Wed Jan 20 10:38:19 2010 +0100
@@ -8,7 +8,7 @@
sig
val mk_cnumeral: int -> cterm
val mk_cnumber: ctyp -> int -> cterm
-  val add_code: string -> bool -> bool -> string -> theory -> theory
+  val add_code: string -> bool -> bool -> (string -> Pretty.T) -> string -> theory -> theory
end;

structure Numeral: NUMERAL =
@@ -56,7 +56,7 @@

local open Basic_Code_Thingol in

-fun add_code number_of negative unbounded target thy =
+fun add_code number_of negative unbounded print target thy =
let
fun dest_numeral pls' min' bit0' bit1' thm =
let
@@ -74,11 +74,12 @@
| dest_num _ = Code_Printer.eqn_error thm "Illegal numeral expression: illegal term";
in dest_num end;
fun pretty literals [pls', min', bit0', bit1'] _ thm _ _ [(t, _)] =
-      (Code_Printer.str o Code_Printer.literal_numeral literals unbounded
+      (print o Code_Printer.literal_numeral literals unbounded
o the_default 0 o dest_numeral pls' min' bit0' bit1' thm) t;
in
-      (SOME (1, ([@{const_name Int.Pls}, @{const_name Int.Min}, @{const_name Int.Bit0}, @{const_name Int.Bit1}], pretty)))
+      (SOME (1, ([@{const_name Int.Pls}, @{const_name Int.Min},
+        @{const_name Int.Bit0}, @{const_name Int.Bit1}], pretty)))
end;

end; (*local*)```
```--- a/src/HOL/Transitive_Closure.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/Transitive_Closure.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -106,12 +106,8 @@
theorem rtranclp_induct [consumes 1, case_names base step, induct set: rtranclp]:
assumes a: "r^** a b"
and cases: "P a" "!!y z. [| r^** a y; r y z; P y |] ==> P z"
-  shows "P b"
-proof -
-  from a have "a = a --> P b"
-    by (induct "%x y. x = a --> P y" a b) (iprover intro: cases)+
-  then show ?thesis by iprover
-qed
+  shows "P b" using a
+  by (induct x\<equiv>a b) (rule cases)+

lemmas rtrancl_induct [induct set: rtrancl] = rtranclp_induct [to_set]

@@ -257,7 +253,7 @@
lemma sym_rtrancl: "sym r ==> sym (r^*)"
by (simp only: sym_conv_converse_eq rtrancl_converse [symmetric])

-theorem converse_rtranclp_induct[consumes 1]:
+theorem converse_rtranclp_induct [consumes 1, case_names base step]:
assumes major: "r^** a b"
and cases: "P b" "!!y z. [| r y z; r^** z b; P z |] ==> P y"
shows "P a"
@@ -274,7 +270,7 @@
converse_rtrancl_induct [of "(ax,ay)" "(bx,by)", split_format (complete),
consumes 1, case_names refl step]

-lemma converse_rtranclpE:
+lemma converse_rtranclpE [consumes 1, case_names base step]:
assumes major: "r^** x z"
and cases: "x=z ==> P"
"!!y. [| r x y; r^** y z |] ==> P"
@@ -352,15 +348,11 @@

text {* Nice induction rule for @{text trancl} *}
lemma tranclp_induct [consumes 1, case_names base step, induct pred: tranclp]:
-  assumes "r^++ a b"
+  assumes a: "r^++ a b"
and cases: "!!y. r a y ==> P y"
"!!y z. r^++ a y ==> r y z ==> P y ==> P z"
-  shows "P b"
-proof -
-  from `r^++ a b` have "a = a --> P b"
-    by (induct "%x y. x = a --> P y" a b) (iprover intro: cases)+
-  then show ?thesis by iprover
-qed
+  shows "P b" using a
+  by (induct x\<equiv>a b) (iprover intro: cases)+

lemmas trancl_induct [induct set: trancl] = tranclp_induct [to_set]

@@ -484,7 +476,7 @@
lemma sym_trancl: "sym r ==> sym (r^+)"
by (simp only: sym_conv_converse_eq trancl_converse [symmetric])

-lemma converse_tranclp_induct:
+lemma converse_tranclp_induct [consumes 1, case_names base step]:
assumes major: "r^++ a b"
and cases: "!!y. r y b ==> P(y)"
"!!y z.[| r y z;  r^++ z b;  P(z) |] ==> P(y)"```
```--- a/src/HOL/ex/ThreeDivides.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOL/ex/ThreeDivides.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -178,21 +178,17 @@

lemma exp_exists:
"m = (\<Sum>x<nlen m. (m div (10::nat)^x mod 10) * 10^x)"
-proof (induct nd \<equiv> "nlen m" arbitrary: m)
+proof (induct "nlen m" arbitrary: m)
case 0 thus ?case by (simp add: nlen_zero)
next
case (Suc nd)
-  hence IH:
-    "nd = nlen (m div 10) \<Longrightarrow>
-    m div 10 = (\<Sum>x<nd. m div 10 div 10^x mod 10 * 10^x)"
-    by blast
obtain c where mexp: "m = 10*(m div 10) + c \<and> c < 10"
and cdef: "c = m mod 10" by simp
show "m = (\<Sum>x<nlen m. m div 10^x mod 10 * 10^x)"
proof -
from `Suc nd = nlen m`
have "nd = nlen (m div 10)" by (rule nlen_suc)
-    with IH have
+    with Suc have
"m div 10 = (\<Sum>x<nd. m div 10 div 10^x mod 10 * 10^x)" by simp
with mexp have
"m = 10*(\<Sum>x<nd. m div 10 div 10^x mod 10 * 10^x) + c" by simp```
```--- a/src/HOLCF/Universal.thy	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/HOLCF/Universal.thy	Wed Jan 20 10:38:19 2010 +0100
@@ -694,13 +694,8 @@

lemma basis_emb_mono:
"x \<sqsubseteq> y \<Longrightarrow> ubasis_le (basis_emb x) (basis_emb y)"
-proof (induct n \<equiv> "max (place x) (place y)" arbitrary: x y rule: less_induct)
-  case (less n)
-  hence IH:
-    "\<And>(a::'a compact_basis) b.
-     \<lbrakk>max (place a) (place b) < max (place x) (place y); a \<sqsubseteq> b\<rbrakk>
-        \<Longrightarrow> ubasis_le (basis_emb a) (basis_emb b)"
-    by simp
+proof (induct "max (place x) (place y)" arbitrary: x y rule: less_induct)
+  case less
show ?case proof (rule linorder_cases)
assume "place x < place y"
then have "rank x < rank y"
@@ -709,7 +704,7 @@
apply (case_tac "y = compact_bot", simp)
apply (simp add: basis_emb.simps [of y])
apply (rule ubasis_le_trans [OF _ ubasis_le_lower [OF fin2]])
-      apply (rule IH)
+      apply (rule less)
apply (erule place_sub_less)
apply (erule rank_less_imp_below_sub [OF `x \<sqsubseteq> y`])
@@ -724,7 +719,7 @@
apply (case_tac "x = compact_bot", simp add: ubasis_le_minimal)
apply (simp add: basis_emb.simps [of x])
apply (rule ubasis_le_upper [OF fin2], simp)
-      apply (rule IH)
+      apply (rule less)
apply (erule place_sub_less)
apply (erule rev_below_trans)```
```--- a/src/Pure/Isar/code.ML	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/Pure/Isar/code.ML	Wed Jan 20 10:38:19 2010 +0100
@@ -746,6 +746,10 @@
:: Pretty.str "of"
:: map (Pretty.quote o Syntax.pretty_typ_global thy) tys)) cos)
);
+    fun pretty_case (const, (_, (_, []))) = Pretty.str (string_of_const thy const)
+      | pretty_case (const, (_, (_, cos))) = (Pretty.block o Pretty.breaks) [
+          Pretty.str (string_of_const thy const), Pretty.str "with",
+          (Pretty.block o Pretty.commas o map (Pretty.str o string_of_const thy)) cos];
val eqns = the_eqns exec
|> Symtab.dest
|> (map o apfst) (string_of_const thy)
@@ -755,18 +759,26 @@
|> Symtab.dest
|> map (fn (dtco, (_, (vs, cos)) :: _) =>
(string_of_typ thy (Type (dtco, map TFree vs)), cos))
-      |> sort (string_ord o pairself fst)
+      |> sort (string_ord o pairself fst);
+    val cases = Symtab.dest ((fst o the_cases o the_exec) thy);
+    val undefineds = Symtab.keys ((snd o the_cases o the_exec) thy);
in
(Pretty.writeln o Pretty.chunks) [
Pretty.block (
-        Pretty.str "code equations:"
-        :: Pretty.fbrk
+        Pretty.str "code equations:" :: Pretty.fbrk
:: (Pretty.fbreaks o map pretty_eqns) eqns
),
Pretty.block (
-        Pretty.str "datatypes:"
-        :: Pretty.fbrk
+        Pretty.str "datatypes:" :: Pretty.fbrk
:: (Pretty.fbreaks o map pretty_dtyp) dtyps
+      ),
+      Pretty.block (
+        Pretty.str "cases:" :: Pretty.fbrk
+        :: (Pretty.fbreaks o map pretty_case) cases
+      ),
+      Pretty.block (
+        Pretty.str "undefined:" :: Pretty.fbrk
+        :: (Pretty.commas o map (Pretty.str o string_of_const thy)) undefineds
)
]
end;```
```--- a/src/Pure/Isar/proof.ML	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/Pure/Isar/proof.ML	Wed Jan 20 10:38:19 2010 +0100
@@ -387,7 +387,7 @@
fun no_goal_cases st = map (rpair NONE) (goals st);

fun goal_cases st =
-  Rule_Cases.make_common true (Thm.theory_of_thm st, Thm.prop_of st) (map (rpair []) (goals st));
+  Rule_Cases.make_common (Thm.theory_of_thm st, Thm.prop_of st) (map (rpair []) (goals st));

fun apply_method current_context meth state =
let```
```--- a/src/Pure/Isar/rule_cases.ML	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/Pure/Isar/rule_cases.ML	Wed Jan 20 10:38:19 2010 +0100
@@ -25,8 +25,8 @@
binds: (indexname * term option) list,
cases: (string * T) list}
val strip_params: term -> (string * typ) list
-  val make_common: bool -> theory * term -> (string * string list) list -> cases
-  val make_nested: bool -> term -> theory * term -> (string * string list) list -> cases
+  val make_common: theory * term -> (string * string list) list -> cases
+  val make_nested: term -> theory * term -> (string * string list) list -> cases
val apply: term list -> T -> T
val consume: thm list -> thm list -> ('a * int) * thm ->
(('a * (int * thm list)) * thm) Seq.seq
@@ -43,6 +43,7 @@
val get: thm -> (string * string list) list * int
val rename_params: string list list -> thm -> thm
val params: string list list -> attribute
+  val internalize_params: thm -> thm
val mutual_rule: Proof.context -> thm list -> (int list * thm) option
val strict_mutual_rule: Proof.context -> thm list -> int list * thm
end;
@@ -90,18 +91,15 @@
chop (length (Logic.strip_assums_hyp outline)) (Logic.strip_assums_hyp prop)
in ([(qual case_hypsN, hyps)], [(qual case_premsN, prems)]) end;

-fun extract_case is_open thy (case_outline, raw_prop) name concls =
+fun extract_case thy (case_outline, raw_prop) name concls =
let
-    val rename = if is_open then I else apfst (Name.internal o Name.clean);
-
val props = Logic.dest_conjunctions (Drule.norm_hhf thy raw_prop);
val len = length props;
val nested = is_some case_outline andalso len > 1;

fun extract prop =
let
-        val (fixes1, fixes2) = extract_fixes case_outline prop
-          |> apfst (map rename);
+        val (fixes1, fixes2) = extract_fixes case_outline prop;
val abs_fixes = abs (fixes1 @ fixes2);
fun abs_fixes1 t =
if not nested then abs_fixes t
@@ -135,7 +133,7 @@
else SOME (nested_case (hd cases))
end;

-fun make is_open rule_struct (thy, prop) cases =
+fun make rule_struct (thy, prop) cases =
let
val n = length cases;
val nprems = Logic.count_prems prop;
@@ -143,13 +141,13 @@
((case try (fn () =>
(Option.map (curry Logic.nth_prem i) rule_struct, Logic.nth_prem (i, prop))) () of
NONE => (name, NONE)
-      | SOME p => (name, extract_case is_open thy p name concls)) :: cs, i - 1);
+      | SOME p => (name, extract_case thy p name concls)) :: cs, i - 1);
in fold_rev add_case (drop (Int.max (n - nprems, 0)) cases) ([], n) |> #1 end;

in

-fun make_common is_open = make is_open NONE;
-fun make_nested is_open rule_struct = make is_open (SOME rule_struct);
+val make_common = make NONE;
+fun make_nested rule_struct = make (SOME rule_struct);

fun apply args =
let
@@ -334,6 +332,20 @@
fun params xss = Thm.rule_attribute (K (rename_params xss));

+(* internalize parameter names *)
+
+fun internalize_params rule =
+  let
+    fun rename prem =
+      let val xs =
+        map (Name.internal o Name.clean o fst) (Logic.strip_params prem)
+      in Logic.list_rename_params (xs, prem) end;
+    fun rename_prems prop =
+      let val (As, C) = Logic.strip_horn (Thm.prop_of rule)
+      in Logic.list_implies (map rename As, C) end;
+  in Thm.equal_elim (Thm.reflexive (Drule.cterm_fun rename_prems (Thm.cprop_of rule))) rule end;
+
+

(** mutual_rule **)
```
```--- a/src/Tools/Code/code_scala.ML	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/Tools/Code/code_scala.ML	Wed Jan 20 10:38:19 2010 +0100
@@ -71,7 +71,7 @@
(applify "[" "]" NOBR ((str o deresolve) c) (map (print_typ tyvars NOBR) tys'))
(map (print_term tyvars is_pat thm vars NOBR) ts))
| SOME (_, print) => (false, fn ts =>
-              print (print_term tyvars is_pat thm) thm vars fxy (ts ~~ take k tys_args));
+              print (print_term tyvars is_pat thm) thm vars fxy (ts ~~ take l tys_args));
in if k = l then print' ts
else if k < l then
print_term tyvars is_pat thm vars fxy (Code_Thingol.eta_expand l app)
@@ -404,11 +404,17 @@
let
val s = ML_Syntax.print_char c;
in if s = "'" then "\\'" else s end;
+  fun bigint_scala k = "(" ^ (if k <= 2147483647
+    then string_of_int k else quote (string_of_int k)) ^ ")"
in Literals {
literal_char = Library.enclose "'" "'" o char_scala,
literal_string = quote o translate_string char_scala,
-  literal_numeral = fn unbounded => fn k => if k >= 0 then string_of_int k
-    else Library.enclose "(" ")" (signed_string_of_int k),
+  literal_numeral = fn unbounded => fn k => if k >= 0 then
+      if unbounded then bigint_scala k
+      else Library.enclose "(" ")" (string_of_int k)
+    else
+      if unbounded then "(- " ^ bigint_scala (~ k) ^ ")"
+      else Library.enclose "(" ")" (signed_string_of_int k),
literal_list = fn [] => str "Nil" | ps => Pretty.block [str "List", enum "," "(" ")" ps],
infix_cons = (6, "::")
} end;
@@ -424,7 +430,7 @@
#> Code_Target.add_syntax_tyco target "fun" (SOME (2, fn print_typ => fn fxy => fn [ty1, ty2] =>
brackify_infix (1, R) fxy [
-        print_typ (INFX (1, X)) ty1,
+        print_typ BR ty1 (*product type vs. tupled arguments!*),
str "=>",
print_typ (INFX (1, R)) ty2
]))```
```--- a/src/Tools/induct.ML	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/Tools/induct.ML	Wed Jan 20 10:38:19 2010 +0100
@@ -10,6 +10,8 @@
val atomize: thm list
val rulify: thm list
val rulify_fallback: thm list
+  val dest_def: term -> (term * term) option
+  val trivial_tac: int -> tactic
end;

signature INDUCT =
@@ -42,6 +44,9 @@
val coinduct_type: string -> attribute
val coinduct_pred: string -> attribute
val coinduct_del: attribute
+  val map_simpset: (simpset -> simpset) -> Context.generic -> Context.generic
+  val no_simpN: string
val casesN: string
val inductN: string
val coinductN: string
@@ -50,19 +55,24 @@
val setN: string
(*proof methods*)
val fix_tac: Proof.context -> int -> (string * typ) list -> int -> tactic
-  val add_defs: (binding option * term) option list -> Proof.context ->
+  val add_defs: (binding option * (term * bool)) option list -> Proof.context ->
(term option list * thm list) * Proof.context
val atomize_term: theory -> term -> term
+  val atomize_cterm: conv
val atomize_tac: int -> tactic
val inner_atomize_tac: int -> tactic
val rulified_term: thm -> theory * term
val rulify_tac: int -> tactic
+  val simplified_rule: Proof.context -> thm -> thm
+  val simplify_tac: Proof.context -> int -> tactic
+  val trivial_tac: int -> tactic
+  val rotate_tac: int -> int -> int -> tactic
val internalize: int -> thm -> thm
val guess_instance: Proof.context -> thm -> int -> thm -> thm Seq.seq
val cases_tac: Proof.context -> term option list list -> thm option ->
thm list -> int -> cases_tactic
val get_inductT: Proof.context -> term option list list -> thm list list
-  val induct_tac: Proof.context -> (binding option * term) option list list ->
+  val induct_tac: Proof.context -> bool -> (binding option * (term * bool)) option list list ->
(string * typ) list list -> term option list -> thm list option ->
thm list -> int -> cases_tactic
val coinduct_tac: Proof.context -> term option list -> term option list -> thm option ->
@@ -107,6 +117,77 @@

+(** constraint simplification **)
+
+(* rearrange parameters and premises to allow application of one-point-rules *)
+
+fun swap_params_conv ctxt i j cv =
+  let
+    fun conv1 0 ctxt = Conv.forall_conv (cv o snd) ctxt
+      | conv1 k ctxt =
+          Conv.rewr_conv @{thm swap_params} then_conv
+          Conv.forall_conv (conv1 (k-1) o snd) ctxt
+    fun conv2 0 ctxt = conv1 j ctxt
+      | conv2 k ctxt = Conv.forall_conv (conv2 (k-1) o snd) ctxt
+  in conv2 i ctxt end;
+
+fun swap_prems_conv 0 = Conv.all_conv
+  | swap_prems_conv i =
+      Conv.implies_concl_conv (swap_prems_conv (i-1)) then_conv
+      Conv.rewr_conv Drule.swap_prems_eq
+
+fun drop_judgment ctxt = ObjectLogic.drop_judgment (ProofContext.theory_of ctxt);
+
+fun find_eq ctxt t =
+  let
+    val l = length (Logic.strip_params t);
+    val Hs = Logic.strip_assums_hyp t;
+    fun find (i, t) =
+      case Data.dest_def (drop_judgment ctxt t) of
+        SOME (Bound j, _) => SOME (i, j)
+      | SOME (_, Bound j) => SOME (i, j)
+      | _ => NONE
+  in
+    case get_first find (map_index I Hs) of
+      NONE => NONE
+    | SOME (0, 0) => NONE
+    | SOME (i, j) => SOME (i, l-j-1, j)
+  end;
+
+fun mk_swap_rrule ctxt ct = case find_eq ctxt (term_of ct) of
+    NONE => NONE
+  | SOME (i, k, j) => SOME (swap_params_conv ctxt k j (K (swap_prems_conv i)) ct);
+
+val rearrange_eqs_simproc = Simplifier.simproc
+  (Thm.theory_of_thm Drule.swap_prems_eq) "rearrange_eqs" ["all t"]
+  (fn thy => fn ss => fn t =>
+     mk_swap_rrule (Simplifier.the_context ss) (cterm_of thy t))
+
+(* rotate k premises to the left by j, skipping over first j premises *)
+
+fun rotate_conv 0 j 0 = Conv.all_conv
+  | rotate_conv 0 j k = swap_prems_conv j then_conv rotate_conv 1 j (k-1)
+  | rotate_conv i j k = Conv.implies_concl_conv (rotate_conv (i-1) j k);
+
+fun rotate_tac j 0 = K all_tac
+  | rotate_tac j k = SUBGOAL (fn (goal, i) => CONVERSION (rotate_conv
+      j (length (Logic.strip_assums_hyp goal) - j - k) k) i);
+
+(* rulify operators around definition *)
+
+fun rulify_defs_conv ctxt ct =
+  if exists_subterm (is_some o Data.dest_def) (term_of ct) andalso
+    not (is_some (Data.dest_def (drop_judgment ctxt (term_of ct))))
+  then
+    (Conv.forall_conv (rulify_defs_conv o snd) ctxt else_conv
+     Conv.implies_conv (Conv.try_conv (rulify_defs_conv ctxt))
+       (Conv.try_conv (rulify_defs_conv ctxt)) else_conv
+     Conv.first_conv (map Conv.rewr_conv Data.rulify) then_conv
+       Conv.try_conv (rulify_defs_conv ctxt)) ct
+  else Conv.no_conv ct;
+
+
+
(** induct data **)

(* rules *)
@@ -132,23 +213,25 @@

structure InductData = Generic_Data
(
-  type T = (rules * rules) * (rules * rules) * (rules * rules);
+  type T = (rules * rules) * (rules * rules) * (rules * rules) * simpset;
val empty =
((init_rules (left_var_prem o #2), init_rules (Thm.major_prem_of o #2)),
(init_rules (right_var_concl o #2), init_rules (Thm.major_prem_of o #2)),
-     (init_rules (left_var_concl o #2), init_rules (Thm.concl_of o #2)));
+     (init_rules (left_var_concl o #2), init_rules (Thm.concl_of o #2)),
val extend = I;
-  fun merge (((casesT1, casesP1), (inductT1, inductP1), (coinductT1, coinductP1)),
-      ((casesT2, casesP2), (inductT2, inductP2), (coinductT2, coinductP2))) =
+  fun merge (((casesT1, casesP1), (inductT1, inductP1), (coinductT1, coinductP1), simpset1),
+      ((casesT2, casesP2), (inductT2, inductP2), (coinductT2, coinductP2), simpset2)) =
((Item_Net.merge (casesT1, casesT2), Item_Net.merge (casesP1, casesP2)),
-      (Item_Net.merge (inductT1, inductT2), Item_Net.merge (inductP1, inductP2)),
-      (Item_Net.merge (coinductT1, coinductT2), Item_Net.merge (coinductP1, coinductP2)));
+     (Item_Net.merge (inductT1, inductT2), Item_Net.merge (inductP1, inductP2)),
+     (Item_Net.merge (coinductT1, coinductT2), Item_Net.merge (coinductP1, coinductP2)),
+     merge_ss (simpset1, simpset2));
);

val get_local = InductData.get o Context.Proof;

fun dest_rules ctxt =
-  let val ((casesT, casesP), (inductT, inductP), (coinductT, coinductP)) = get_local ctxt in
+  let val ((casesT, casesP), (inductT, inductP), (coinductT, coinductP), _) = get_local ctxt in
{type_cases = Item_Net.content casesT,
pred_cases = Item_Net.content casesP,
type_induct = Item_Net.content inductT,
@@ -158,7 +241,7 @@
end;

fun print_rules ctxt =
-  let val ((casesT, casesP), (inductT, inductP), (coinductT, coinductP)) = get_local ctxt in
+  let val ((casesT, casesP), (inductT, inductP), (coinductT, coinductP), _) = get_local ctxt in
[pretty_rules ctxt "coinduct type:" coinductT,
pretty_rules ctxt "coinduct pred:" coinductP,
pretty_rules ctxt "induct type:" inductT,
@@ -206,9 +289,10 @@
fun del_att which = Thm.declaration_attribute (fn th => InductData.map (which (pairself (fn rs =>
fold Item_Net.remove (filter_rules rs th) rs))));

-fun map1 f (x, y, z) = (f x, y, z);
-fun map2 f (x, y, z) = (x, f y, z);
-fun map3 f (x, y, z) = (x, y, f z);
+fun map1 f (x, y, z, s) = (f x, y, z, s);
+fun map2 f (x, y, z, s) = (x, f y, z, s);
+fun map3 f (x, y, z, s) = (x, y, f z, s);
+fun map4 f (x, y, z, s) = (x, y, z, f s);

fun add_casesT rule x = map1 (apfst (Item_Net.update rule)) x;
fun add_casesP rule x = map1 (apsnd (Item_Net.update rule)) x;
@@ -234,12 +318,17 @@
val coinduct_pred = mk_att add_coinductP consumes1;
val coinduct_del = del_att map3;

+fun map_simpset f = InductData.map (map4 f);
+  (map_simpset (fn ss => ss addsimps [thm]) ctxt, thm);
+
end;

(** attribute syntax **)

+val no_simpN = "no_simp";
val casesN = "cases";
val inductN = "induct";
val coinductN = "coinduct";
@@ -268,7 +357,9 @@
Attrib.setup @{binding induct} (attrib induct_type induct_pred induct_del)
"declaration of induction rule" #>
Attrib.setup @{binding coinduct} (attrib coinduct_type coinduct_pred coinduct_del)
-    "declaration of coinduction rule";
+    "declaration of coinduction rule" #>
+  Attrib.setup @{binding induct_simp} (Scan.succeed add_simp_rule)
+    "declaration of rules for simplifying induction or cases rules";

end;

@@ -362,7 +453,8 @@
ruleq
|> Seq.maps (Rule_Cases.consume [] facts)
|> Seq.maps (fn ((cases, (_, more_facts)), rule) =>
-        CASES (Rule_Cases.make_common false (thy, Thm.prop_of rule) cases)
+        CASES (Rule_Cases.make_common (thy,
+            Thm.prop_of (Rule_Cases.internalize_params rule)) cases)
(Method.insert_tac more_facts i THEN Tactic.rtac rule i) st)
end;

@@ -409,6 +501,22 @@
(Simplifier.rewrite_goal_tac [@{thm Pure.conjunction_imp}] THEN' Goal.norm_hhf_tac);

+(* simplify *)
+
+fun simplify_conv ctxt ct =
+  if exists_subterm (is_some o Data.dest_def) (term_of ct) then
+    (Conv.try_conv (rulify_defs_conv ctxt) then_conv
+       Simplifier.full_rewrite (Simplifier.context ctxt (#4 (get_local ctxt)))) ct
+  else Conv.all_conv ct;
+
+fun simplified_rule ctxt thm =
+  Conv.fconv_rule (Conv.prems_conv ~1 (simplify_conv ctxt)) thm;
+
+fun simplify_tac ctxt = CONVERSION (simplify_conv ctxt);
+
+val trivial_tac = Data.trivial_tac;
+
+
(* prepare rule *)

fun rule_instance ctxt inst rule =
@@ -548,11 +656,19 @@

let
-    fun add (SOME (SOME x, t)) ctxt =
+    fun add (SOME (_, (t, true))) ctxt = ((SOME t, []), ctxt)
+      | add (SOME (SOME x, (t, _))) ctxt =
let val ([(lhs, (_, th))], ctxt') =
LocalDefs.add_defs [((x, NoSyn), (Thm.empty_binding, t))] ctxt
in ((SOME lhs, [th]), ctxt') end
-      | add (SOME (NONE, t)) ctxt = ((SOME t, []), ctxt)
+      | add (SOME (NONE, (t as Free _, _))) ctxt = ((SOME t, []), ctxt)
+      | add (SOME (NONE, (t, _))) ctxt =
+          let
+            val ([s], _) = Name.variants ["x"] (Variable.names_of ctxt);
+            val ([(lhs, (_, th))], ctxt') =
+                (Thm.empty_binding, t))] ctxt
+          in ((SOME lhs, [th]), ctxt') end
| add NONE ctxt = ((NONE, []), ctxt);
in fold_map add def_insts #> apfst (split_list #> apsnd flat) end;

@@ -576,12 +692,12 @@
fun get_inductP ctxt (fact :: _) = map single (find_inductP ctxt (Thm.concl_of fact))
| get_inductP _ _ = [];

-fun induct_tac ctxt def_insts arbitrary taking opt_rule facts =
+fun induct_tac ctxt simp def_insts arbitrary taking opt_rule facts =
let
val thy = ProofContext.theory_of ctxt;

val ((insts, defs), defs_ctxt) = fold_map add_defs def_insts ctxt |>> split_list;
-    val atomized_defs = map (map (Conv.fconv_rule ObjectLogic.atomize)) defs;
+    val atomized_defs = map (map (Conv.fconv_rule atomize_cterm)) defs;

fun inst_rule (concls, r) =
(if null insts then `Rule_Cases.get r
@@ -601,8 +717,10 @@
|> tap (trace_rules ctxt inductN o map #2)
|> Seq.of_list |> Seq.maps (Seq.try inst_rule));

-    fun rule_cases rule =
-      Rule_Cases.make_nested false (Thm.prop_of rule) (rulified_term rule);
+    fun rule_cases ctxt rule =
+      let val rule' = (if simp then simplified_rule ctxt else I)
+        (Rule_Cases.internalize_params rule);
+      in Rule_Cases.make_nested (Thm.prop_of rule') (rulified_term rule') end;
in
(fn i => fn st =>
ruleq
@@ -610,19 +728,32 @@
|> Seq.maps (fn (((cases, concls), (more_consumes, more_facts)), rule) =>
(PRECISE_CONJUNCTS (length concls) (ALLGOALS (fn j =>
(CONJUNCTS (ALLGOALS
-            (Method.insert_tac (more_facts @ nth_list atomized_defs (j - 1))
-              THEN' fix_tac defs_ctxt
-                (nth concls (j - 1) + more_consumes)
-                (nth_list arbitrary (j - 1))))
+            let
+              val adefs = nth_list atomized_defs (j - 1);
+              val xs = nth_list arbitrary (j - 1);
+              val k = nth concls (j - 1) + more_consumes
+            in
+              Method.insert_tac (more_facts @ adefs) THEN'
+                (if simp then
+                   rotate_tac k (length adefs) THEN'
+                   fix_tac defs_ctxt k
+                     (List.partition (member op = frees) xs |> op @)
+                 else
+                   fix_tac defs_ctxt k xs)
+             end)
THEN' inner_atomize_tac) j))
THEN' atomize_tac) i st |> Seq.maps (fn st' =>
guess_instance ctxt (internalize more_consumes rule) i st'
|> Seq.map (rule_instance ctxt (burrow_options (Variable.polymorphic ctxt) taking))
|> Seq.maps (fn rule' =>
-              CASES (rule_cases rule' cases)
+              CASES (rule_cases ctxt rule' cases)
(Tactic.rtac rule' i THEN
PRIMITIVE (singleton (ProofContext.export defs_ctxt ctxt))) st'))))
-    THEN_ALL_NEW_CASES rulify_tac
+    THEN_ALL_NEW_CASES
+      ((if simp then simplify_tac ctxt THEN' (TRY o trivial_tac)
+        else K all_tac)
+       THEN_ALL_NEW rulify_tac)
end;

@@ -672,7 +803,8 @@
guess_instance ctxt rule i st
|> Seq.map (rule_instance ctxt (burrow_options (Variable.polymorphic ctxt) taking))
|> Seq.maps (fn rule' =>
-          CASES (Rule_Cases.make_common false (thy, Thm.prop_of rule') cases)
+          CASES (Rule_Cases.make_common (thy,
+              Thm.prop_of (Rule_Cases.internalize_params rule')) cases)
(Method.insert_tac more_facts i THEN Tactic.rtac rule' i) st)))
end;

@@ -711,10 +843,15 @@

val inst = Scan.lift (Args.\$\$\$ "_") >> K NONE || Args.term >> SOME;

+val inst' = Scan.lift (Args.\$\$\$ "_") >> K NONE ||
+  Args.term >> (SOME o rpair false) ||
+  Scan.lift (Args.\$\$\$ "(") |-- (Args.term >> (SOME o rpair true)) --|
+    Scan.lift (Args.\$\$\$ ")");
+
val def_inst =
((Scan.lift (Args.binding --| (Args.\$\$\$ "\<equiv>" || Args.\$\$\$ "==")) >> SOME)
-      -- Args.term) >> SOME ||
-    inst >> Option.map (pair NONE);
+      -- (Args.term >> rpair false)) >> SOME ||
+    inst' >> Option.map (pair NONE);

val free = Args.context -- Args.term >> (fn (_, Free v) => v | (ctxt, t) =>
error ("Bad free variable: " ^ Syntax.string_of_term ctxt t));
@@ -740,11 +877,11 @@

val induct_setup =
Method.setup @{binding induct}
-    (P.and_list' (Scan.repeat (unless_more_args def_inst)) --
-      (arbitrary -- taking -- Scan.option induct_rule) >>
-      (fn (insts, ((arbitrary, taking), opt_rule)) => fn ctxt =>
+    (Args.mode no_simpN -- (P.and_list' (Scan.repeat (unless_more_args def_inst)) --
+      (arbitrary -- taking -- Scan.option induct_rule)) >>
+      (fn (no_simp, (insts, ((arbitrary, taking), opt_rule))) => fn ctxt =>
RAW_METHOD_CASES (fn facts =>
-          Seq.DETERM (HEADGOAL (induct_tac ctxt insts arbitrary taking opt_rule facts)))))
+          Seq.DETERM (HEADGOAL (induct_tac ctxt (not no_simp) insts arbitrary taking opt_rule facts)))))
"induction on types or predicates/sets";

val coinduct_setup =```
```--- a/src/Tools/jEdit/README_BUILD	Wed Jan 20 10:38:06 2010 +0100
+++ b/src/Tools/jEdit/README_BUILD	Wed Jan 20 10:38:19 2010 +0100
@@ -31,6 +31,9 @@
* Isabelle/Pure Scala components
Netbeans Library "Isabelle-Pure" = ~~/lib/classes/Pure.jar

+* Scala Compiler
+  Netbeans Library "Scala-compiler" = \$SCALA_HOME/lib/scala-compiler.jar
+

Running the application within Netbeans
=======================================```