# HG changeset patch
# User haftmann
# Date 1232226494 -3600
# Node ID aa8a1ed95a575a478656be3c65f76cfb5d9aa609
# Parent  64cc846d4a63c6fb34ee99f2f1555f9b68843c4b# Parent  4be5e49c74b1c887a4d5000ac72cc0360128679d
merged

diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOL/Fundamental_Theorem_Algebra.thy
--- a/src/HOL/Fundamental_Theorem_Algebra.thy	Sat Jan 17 22:07:29 2009 +0100
+++ b/src/HOL/Fundamental_Theorem_Algebra.thy	Sat Jan 17 22:08:14 2009 +0100
@@ -135,15 +135,15 @@
 done
 
 definition
-  "plength p = (if p = 0 then 0 else Suc (degree p))"
+  "psize p = (if p = 0 then 0 else Suc (degree p))"
 
-lemma plength_eq_0_iff [simp]: "plength p = 0 \<longleftrightarrow> p = 0"
-  unfolding plength_def by simp
+lemma psize_eq_0_iff [simp]: "psize p = 0 \<longleftrightarrow> p = 0"
+  unfolding psize_def by simp
 
-lemma poly_offset: "\<exists> q. plength q = plength p \<and> (\<forall>x. poly q (x::complex) = poly p (a + x))"
+lemma poly_offset: "\<exists> q. psize q = psize p \<and> (\<forall>x. poly q (x::complex) = poly p (a + x))"
 proof (intro exI conjI)
-  show "plength (offset_poly p a) = plength p"
-    unfolding plength_def
+  show "psize (offset_poly p a) = psize p"
+    unfolding psize_def
     by (simp add: offset_poly_eq_0_iff degree_offset_poly)
   show "\<forall>x. poly (offset_poly p a) x = poly p (a + x)"
     by (simp add: poly_offset_poly)
@@ -719,8 +719,8 @@
 text{* Constant function (non-syntactic characterization). *}
 definition "constant f = (\<forall>x y. f x = f y)"
 
-lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> plength p \<ge> 2"
-  unfolding constant_def plength_def
+lemma nonconstant_length: "\<not> (constant (poly p)) \<Longrightarrow> psize p \<ge> 2"
+  unfolding constant_def psize_def
   apply (induct p, auto)
   done
  
@@ -733,9 +733,9 @@
 
 lemma poly_decompose_lemma:
  assumes nz: "\<not>(\<forall>z. z\<noteq>0 \<longrightarrow> poly p z = (0::'a::{recpower,idom}))"
-  shows "\<exists>k a q. a\<noteq>0 \<and> Suc (plength q + k) = plength p \<and> 
+  shows "\<exists>k a q. a\<noteq>0 \<and> Suc (psize q + k) = psize p \<and> 
                  (\<forall>z. poly p z = z^k * poly (pCons a q) z)"
-unfolding plength_def
+unfolding psize_def
 using nz
 proof(induct p)
   case 0 thus ?case by simp
@@ -761,7 +761,7 @@
 lemma poly_decompose:
   assumes nc: "~constant(poly p)"
   shows "\<exists>k a q. a\<noteq>(0::'a::{recpower,idom}) \<and> k\<noteq>0 \<and>
-               plength q + k + 1 = plength p \<and> 
+               psize q + k + 1 = psize p \<and> 
               (\<forall>z. poly p z = poly p 0 + z^k * poly (pCons a q) z)"
 using nc 
 proof(induct p)
@@ -781,7 +781,7 @@
     apply simp
     apply (rule_tac x="q" in exI)
     apply (auto simp add: power_Suc)
-    apply (auto simp add: plength_def split: if_splits)
+    apply (auto simp add: psize_def split: if_splits)
     done
 qed
 
@@ -791,10 +791,10 @@
   assumes nc: "~constant(poly p)"
   shows "\<exists>z::complex. poly p z = 0"
 using nc
-proof(induct n\<equiv> "plength p" arbitrary: p rule: nat_less_induct)
+proof(induct n\<equiv> "psize p" arbitrary: p rule: nat_less_induct)
   fix n fix p :: "complex poly"
   let ?p = "poly p"
-  assume H: "\<forall>m<n. \<forall>p. \<not> constant (poly p) \<longrightarrow> m = plength p \<longrightarrow> (\<exists>(z::complex). poly p z = 0)" and nc: "\<not> constant ?p" and n: "n = plength 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)
@@ -804,7 +804,7 @@
   moreover
   {assume pc0: "?p c \<noteq> 0"
     from poly_offset[of p c] obtain q where
-      q: "plength q = plength p" "\<forall>x. poly q x = ?p (c+x)" by blast
+      q: "psize q = psize p" "\<forall>x. poly q x = ?p (c+x)" by blast
     {assume h: "constant (poly q)"
       from q(2) have th: "\<forall>x. poly q (x - c) = ?p x" by auto
       {fix x y
@@ -823,8 +823,8 @@
     have qr: "\<forall>z. poly q z = poly (smult (inverse ?a0) q) z * ?a0"
       by simp
     let ?r = "smult (inverse ?a0) q"
-    have lgqr: "plength q = plength ?r"
-      using a00 unfolding plength_def degree_def
+    have lgqr: "psize q = psize ?r"
+      using a00 unfolding psize_def degree_def
       by (simp add: expand_poly_eq)
     {assume h: "\<And>x y. poly ?r x = poly ?r y"
       {fix x y
@@ -844,7 +844,7 @@
       finally have "cmod (poly ?r w) < 1 \<longleftrightarrow> cmod (poly q w) < cmod ?a0" .}
     note mrmq_eq = this
     from poly_decompose[OF rnc] obtain k a s where 
-      kas: "a\<noteq>0" "k\<noteq>0" "plength s + k + 1 = plength ?r" 
+      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
@@ -861,8 +861,8 @@
 	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 = plength (pCons 1 (monom a (k - 1)))"
-	by (simp add: plength_def degree_monom_eq)
+      from kas(1) kas(2) have th02: "k+1 = psize (pCons 1 (monom a (k - 1)))"
+	by (simp add: psize_def degree_monom_eq)
       from H[rule_format, OF k1n th01 th02]
       obtain w where w: "1 + w^k * a = 0"
 	unfolding poly_pCons poly_monom
@@ -1353,11 +1353,11 @@
 lemma basic_cqe_conv3:
   fixes p q :: "complex poly"
   assumes l: "p \<noteq> 0" 
-  shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (plength p)))"
+  shows "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
 proof-
-  from l have dp:"degree (pCons a p) = plength p" by (simp add: plength_def)
+  from l have dp:"degree (pCons a p) = psize p" by (simp add: psize_def)
   from nullstellensatz_univariate[of "pCons a p" q] l
-  show "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (plength p)))"
+  show "(\<exists>x. poly (pCons a p) x = 0 \<and> poly q x \<noteq> 0) \<equiv> \<not> ((pCons a p) dvd (q ^ (psize p)))"
     unfolding dp
     by - (atomize (full), auto)
 qed
diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOL/Polynomial.thy
--- a/src/HOL/Polynomial.thy	Sat Jan 17 22:07:29 2009 +0100
+++ b/src/HOL/Polynomial.thy	Sat Jan 17 22:08:14 2009 +0100
@@ -293,6 +293,14 @@
 
 end
 
+instance poly ::
+  ("{cancel_ab_semigroup_add,comm_monoid_add}") cancel_ab_semigroup_add
+proof
+  fix p q r :: "'a poly"
+  assume "p + q = p + r" thus "q = r"
+    by (simp add: expand_poly_eq)
+qed
+
 instantiation poly :: (ab_group_add) ab_group_add
 begin
 
@@ -345,19 +353,22 @@
   "pCons a p - pCons b q = pCons (a - b) (p - q)"
   by (rule poly_ext, simp add: coeff_pCons split: nat.split)
 
-lemma degree_add_le: "degree (p + q) \<le> max (degree p) (degree q)"
+lemma degree_add_le_max: "degree (p + q) \<le> max (degree p) (degree q)"
   by (rule degree_le, auto simp add: coeff_eq_0)
 
+lemma degree_add_le:
+  "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p + q) \<le> n"
+  by (auto intro: order_trans degree_add_le_max)
+
 lemma degree_add_less:
   "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p + q) < n"
-  by (auto intro: le_less_trans degree_add_le)
+  by (auto intro: le_less_trans degree_add_le_max)
 
 lemma degree_add_eq_right:
   "degree p < degree q \<Longrightarrow> degree (p + q) = degree q"
   apply (cases "q = 0", simp)
   apply (rule order_antisym)
-  apply (rule ord_le_eq_trans [OF degree_add_le])
-  apply simp
+  apply (simp add: degree_add_le)
   apply (rule le_degree)
   apply (simp add: coeff_eq_0)
   done
@@ -370,13 +381,17 @@
 lemma degree_minus [simp]: "degree (- p) = degree p"
   unfolding degree_def by simp
 
-lemma degree_diff_le: "degree (p - q) \<le> max (degree p) (degree q)"
+lemma degree_diff_le_max: "degree (p - q) \<le> max (degree p) (degree q)"
   using degree_add_le [where p=p and q="-q"]
   by (simp add: diff_minus)
 
+lemma degree_diff_le:
+  "\<lbrakk>degree p \<le> n; degree q \<le> n\<rbrakk> \<Longrightarrow> degree (p - q) \<le> n"
+  by (simp add: diff_minus degree_add_le)
+
 lemma degree_diff_less:
   "\<lbrakk>degree p < n; degree q < n\<rbrakk> \<Longrightarrow> degree (p - q) < n"
-  by (auto intro: le_less_trans degree_diff_le)
+  by (simp add: diff_minus degree_add_less)
 
 lemma add_monom: "monom a n + monom b n = monom (a + b) n"
   by (rule poly_ext) simp
@@ -534,6 +549,8 @@
 
 end
 
+instance poly :: (comm_semiring_0_cancel) comm_semiring_0_cancel ..
+
 lemma coeff_mult:
   "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n-i))"
 proof (induct p arbitrary: n)
@@ -575,6 +592,8 @@
 
 end
 
+instance poly :: (comm_semiring_1_cancel) comm_semiring_1_cancel ..
+
 lemma coeff_1 [simp]: "coeff 1 n = (if n = 0 then 1 else 0)"
   unfolding one_poly_def
   by (simp add: coeff_pCons split: nat.split)
@@ -647,19 +666,19 @@
 subsection {* Long division of polynomials *}
 
 definition
-  divmod_poly_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
+  pdivmod_rel :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"
 where
   [code del]:
-  "divmod_poly_rel x y q r \<longleftrightarrow>
+  "pdivmod_rel x y q r \<longleftrightarrow>
     x = q * y + r \<and> (if y = 0 then q = 0 else r = 0 \<or> degree r < degree y)"
 
-lemma divmod_poly_rel_0:
-  "divmod_poly_rel 0 y 0 0"
-  unfolding divmod_poly_rel_def by simp
+lemma pdivmod_rel_0:
+  "pdivmod_rel 0 y 0 0"
+  unfolding pdivmod_rel_def by simp
 
-lemma divmod_poly_rel_by_0:
-  "divmod_poly_rel x 0 0 x"
-  unfolding divmod_poly_rel_def by simp
+lemma pdivmod_rel_by_0:
+  "pdivmod_rel x 0 0 x"
+  unfolding pdivmod_rel_def by simp
 
 lemma eq_zero_or_degree_less:
   assumes "degree p \<le> n" and "coeff p n = 0"
@@ -685,62 +704,59 @@
   then show ?thesis ..
 qed
 
-lemma divmod_poly_rel_pCons:
-  assumes rel: "divmod_poly_rel x y q r"
+lemma pdivmod_rel_pCons:
+  assumes rel: "pdivmod_rel x y q r"
   assumes y: "y \<noteq> 0"
   assumes b: "b = coeff (pCons a r) (degree y) / coeff y (degree y)"
-  shows "divmod_poly_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
-    (is "divmod_poly_rel ?x y ?q ?r")
+  shows "pdivmod_rel (pCons a x) y (pCons b q) (pCons a r - smult b y)"
+    (is "pdivmod_rel ?x y ?q ?r")
 proof -
   have x: "x = q * y + r" and r: "r = 0 \<or> degree r < degree y"
-    using assms unfolding divmod_poly_rel_def by simp_all
+    using assms unfolding pdivmod_rel_def by simp_all
 
   have 1: "?x = ?q * y + ?r"
     using b x by simp
 
   have 2: "?r = 0 \<or> degree ?r < degree y"
   proof (rule eq_zero_or_degree_less)
-    have "degree ?r \<le> max (degree (pCons a r)) (degree (smult b y))"
-      by (rule degree_diff_le)
-    also have "\<dots> \<le> degree y"
-    proof (rule min_max.le_supI)
+    show "degree ?r \<le> degree y"
+    proof (rule degree_diff_le)
       show "degree (pCons a r) \<le> degree y"
         using r by auto
       show "degree (smult b y) \<le> degree y"
         by (rule degree_smult_le)
     qed
-    finally show "degree ?r \<le> degree y" .
   next
     show "coeff ?r (degree y) = 0"
       using `y \<noteq> 0` unfolding b by simp
   qed
 
   from 1 2 show ?thesis
-    unfolding divmod_poly_rel_def
+    unfolding pdivmod_rel_def
     using `y \<noteq> 0` by simp
 qed
 
-lemma divmod_poly_rel_exists: "\<exists>q r. divmod_poly_rel x y q r"
+lemma pdivmod_rel_exists: "\<exists>q r. pdivmod_rel x y q r"
 apply (cases "y = 0")
-apply (fast intro!: divmod_poly_rel_by_0)
+apply (fast intro!: pdivmod_rel_by_0)
 apply (induct x)
-apply (fast intro!: divmod_poly_rel_0)
-apply (fast intro!: divmod_poly_rel_pCons)
+apply (fast intro!: pdivmod_rel_0)
+apply (fast intro!: pdivmod_rel_pCons)
 done
 
-lemma divmod_poly_rel_unique:
-  assumes 1: "divmod_poly_rel x y q1 r1"
-  assumes 2: "divmod_poly_rel x y q2 r2"
+lemma pdivmod_rel_unique:
+  assumes 1: "pdivmod_rel x y q1 r1"
+  assumes 2: "pdivmod_rel x y q2 r2"
   shows "q1 = q2 \<and> r1 = r2"
 proof (cases "y = 0")
   assume "y = 0" with assms show ?thesis
-    by (simp add: divmod_poly_rel_def)
+    by (simp add: pdivmod_rel_def)
 next
   assume [simp]: "y \<noteq> 0"
   from 1 have q1: "x = q1 * y + r1" and r1: "r1 = 0 \<or> degree r1 < degree y"
-    unfolding divmod_poly_rel_def by simp_all
+    unfolding pdivmod_rel_def by simp_all
   from 2 have q2: "x = q2 * y + r2" and r2: "r2 = 0 \<or> degree r2 < degree y"
-    unfolding divmod_poly_rel_def by simp_all
+    unfolding pdivmod_rel_def by simp_all
   from q1 q2 have q3: "(q1 - q2) * y = r2 - r1"
     by (simp add: ring_simps)
   from r1 r2 have r3: "(r2 - r1) = 0 \<or> degree (r2 - r1) < degree y"
@@ -761,36 +777,36 @@
   qed
 qed
 
-lemmas divmod_poly_rel_unique_div =
-  divmod_poly_rel_unique [THEN conjunct1, standard]
+lemmas pdivmod_rel_unique_div =
+  pdivmod_rel_unique [THEN conjunct1, standard]
 
-lemmas divmod_poly_rel_unique_mod =
-  divmod_poly_rel_unique [THEN conjunct2, standard]
+lemmas pdivmod_rel_unique_mod =
+  pdivmod_rel_unique [THEN conjunct2, standard]
 
 instantiation poly :: (field) ring_div
 begin
 
 definition div_poly where
-  [code del]: "x div y = (THE q. \<exists>r. divmod_poly_rel x y q r)"
+  [code del]: "x div y = (THE q. \<exists>r. pdivmod_rel x y q r)"
 
 definition mod_poly where
-  [code del]: "x mod y = (THE r. \<exists>q. divmod_poly_rel x y q r)"
+  [code del]: "x mod y = (THE r. \<exists>q. pdivmod_rel x y q r)"
 
 lemma div_poly_eq:
-  "divmod_poly_rel x y q r \<Longrightarrow> x div y = q"
+  "pdivmod_rel x y q r \<Longrightarrow> x div y = q"
 unfolding div_poly_def
-by (fast elim: divmod_poly_rel_unique_div)
+by (fast elim: pdivmod_rel_unique_div)
 
 lemma mod_poly_eq:
-  "divmod_poly_rel x y q r \<Longrightarrow> x mod y = r"
+  "pdivmod_rel x y q r \<Longrightarrow> x mod y = r"
 unfolding mod_poly_def
-by (fast elim: divmod_poly_rel_unique_mod)
+by (fast elim: pdivmod_rel_unique_mod)
 
-lemma divmod_poly_rel:
-  "divmod_poly_rel x y (x div y) (x mod y)"
+lemma pdivmod_rel:
+  "pdivmod_rel x y (x div y) (x mod y)"
 proof -
-  from divmod_poly_rel_exists
-    obtain q r where "divmod_poly_rel x y q r" by fast
+  from pdivmod_rel_exists
+    obtain q r where "pdivmod_rel x y q r" by fast
   thus ?thesis
     by (simp add: div_poly_eq mod_poly_eq)
 qed
@@ -798,26 +814,26 @@
 instance proof
   fix x y :: "'a poly"
   show "x div y * y + x mod y = x"
-    using divmod_poly_rel [of x y]
-    by (simp add: divmod_poly_rel_def)
+    using pdivmod_rel [of x y]
+    by (simp add: pdivmod_rel_def)
 next
   fix x :: "'a poly"
-  have "divmod_poly_rel x 0 0 x"
-    by (rule divmod_poly_rel_by_0)
+  have "pdivmod_rel x 0 0 x"
+    by (rule pdivmod_rel_by_0)
   thus "x div 0 = 0"
     by (rule div_poly_eq)
 next
   fix y :: "'a poly"
-  have "divmod_poly_rel 0 y 0 0"
-    by (rule divmod_poly_rel_0)
+  have "pdivmod_rel 0 y 0 0"
+    by (rule pdivmod_rel_0)
   thus "0 div y = 0"
     by (rule div_poly_eq)
 next
   fix x y z :: "'a poly"
   assume "y \<noteq> 0"
-  hence "divmod_poly_rel (x + z * y) y (z + x div y) (x mod y)"
-    using divmod_poly_rel [of x y]
-    by (simp add: divmod_poly_rel_def left_distrib)
+  hence "pdivmod_rel (x + z * y) y (z + x div y) (x mod y)"
+    using pdivmod_rel [of x y]
+    by (simp add: pdivmod_rel_def left_distrib)
   thus "(x + z * y) div y = z + x div y"
     by (rule div_poly_eq)
 qed
@@ -826,22 +842,22 @@
 
 lemma degree_mod_less:
   "y \<noteq> 0 \<Longrightarrow> x mod y = 0 \<or> degree (x mod y) < degree y"
-  using divmod_poly_rel [of x y]
-  unfolding divmod_poly_rel_def by simp
+  using pdivmod_rel [of x y]
+  unfolding pdivmod_rel_def by simp
 
 lemma div_poly_less: "degree x < degree y \<Longrightarrow> x div y = 0"
 proof -
   assume "degree x < degree y"
-  hence "divmod_poly_rel x y 0 x"
-    by (simp add: divmod_poly_rel_def)
+  hence "pdivmod_rel x y 0 x"
+    by (simp add: pdivmod_rel_def)
   thus "x div y = 0" by (rule div_poly_eq)
 qed
 
 lemma mod_poly_less: "degree x < degree y \<Longrightarrow> x mod y = x"
 proof -
   assume "degree x < degree y"
-  hence "divmod_poly_rel x y 0 x"
-    by (simp add: divmod_poly_rel_def)
+  hence "pdivmod_rel x y 0 x"
+    by (simp add: pdivmod_rel_def)
   thus "x mod y = x" by (rule mod_poly_eq)
 qed
 
@@ -852,7 +868,7 @@
   shows "(pCons a x) mod y = (pCons a (x mod y) - smult b y)"
 unfolding b
 apply (rule mod_poly_eq)
-apply (rule divmod_poly_rel_pCons [OF divmod_poly_rel y refl])
+apply (rule pdivmod_rel_pCons [OF pdivmod_rel y refl])
 done
 
 
@@ -1087,30 +1103,30 @@
 text {* code generator setup for div and mod *}
 
 definition
-  divmod_poly :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
+  pdivmod :: "'a::field poly \<Rightarrow> 'a poly \<Rightarrow> 'a poly \<times> 'a poly"
 where
-  [code del]: "divmod_poly x y = (x div y, x mod y)"
+  [code del]: "pdivmod x y = (x div y, x mod y)"
 
-lemma div_poly_code [code]: "x div y = fst (divmod_poly x y)"
-  unfolding divmod_poly_def by simp
+lemma div_poly_code [code]: "x div y = fst (pdivmod x y)"
+  unfolding pdivmod_def by simp
 
-lemma mod_poly_code [code]: "x mod y = snd (divmod_poly x y)"
-  unfolding divmod_poly_def by simp
+lemma mod_poly_code [code]: "x mod y = snd (pdivmod x y)"
+  unfolding pdivmod_def by simp
 
-lemma divmod_poly_0 [code]: "divmod_poly 0 y = (0, 0)"
-  unfolding divmod_poly_def by simp
+lemma pdivmod_0 [code]: "pdivmod 0 y = (0, 0)"
+  unfolding pdivmod_def by simp
 
-lemma divmod_poly_pCons [code]:
-  "divmod_poly (pCons a x) y =
+lemma pdivmod_pCons [code]:
+  "pdivmod (pCons a x) y =
     (if y = 0 then (0, pCons a x) else
-      (let (q, r) = divmod_poly x y;
+      (let (q, r) = pdivmod x y;
            b = coeff (pCons a r) (degree y) / coeff y (degree y)
         in (pCons b q, pCons a r - smult b y)))"
-apply (simp add: divmod_poly_def Let_def, safe)
+apply (simp add: pdivmod_def Let_def, safe)
 apply (rule div_poly_eq)
-apply (erule divmod_poly_rel_pCons [OF divmod_poly_rel _ refl])
+apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
 apply (rule mod_poly_eq)
-apply (erule divmod_poly_rel_pCons [OF divmod_poly_rel _ refl])
+apply (erule pdivmod_rel_pCons [OF pdivmod_rel _ refl])
 done
 
 end
diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOL/Tools/lin_arith.ML
--- a/src/HOL/Tools/lin_arith.ML	Sat Jan 17 22:07:29 2009 +0100
+++ b/src/HOL/Tools/lin_arith.ML	Sat Jan 17 22:08:14 2009 +0100
@@ -340,7 +340,7 @@
 
 fun subst_term ([] : (term * term) list) (t : term) = t
   | subst_term pairs                     t          =
-      (case AList.lookup (op aconv) pairs t of
+      (case AList.lookup Pattern.aeconv pairs t of
         SOME new =>
           new
       | NONE     =>
@@ -682,7 +682,7 @@
 
 fun negated_term_occurs_positively (terms : term list) : bool =
   List.exists
-    (fn (Trueprop $ (Const ("Not", _) $ t)) => member (op aconv) terms (Trueprop $ t)
+    (fn (Trueprop $ (Const ("Not", _) $ t)) => member Pattern.aeconv terms (Trueprop $ t)
       | _                                   => false)
     terms;
 
diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOLCF/Cfun.thy
--- a/src/HOLCF/Cfun.thy	Sat Jan 17 22:07:29 2009 +0100
+++ b/src/HOLCF/Cfun.thy	Sat Jan 17 22:08:14 2009 +0100
@@ -7,8 +7,7 @@
 header {* The type of continuous functions *}
 
 theory Cfun
-imports Pcpodef Ffun
-uses ("Tools/cont_proc.ML")
+imports Pcpodef Ffun Product_Cpo
 begin
 
 defaultsort cpo
@@ -301,7 +300,7 @@
 
 text {* cont2cont lemma for @{term Rep_CFun} *}
 
-lemma cont2cont_Rep_CFun:
+lemma cont2cont_Rep_CFun [cont2cont]:
   assumes f: "cont (\<lambda>x. f x)"
   assumes t: "cont (\<lambda>x. t x)"
   shows "cont (\<lambda>x. (f x)\<cdot>(t x))"
@@ -321,6 +320,11 @@
 
 text {* cont2cont Lemma for @{term "%x. LAM y. f x y"} *}
 
+text {*
+  Not suitable as a cont2cont rule, because on nested lambdas
+  it causes exponential blow-up in the number of subgoals.
+*}
+
 lemma cont2cont_LAM:
   assumes f1: "\<And>x. cont (\<lambda>y. f x y)"
   assumes f2: "\<And>y. cont (\<lambda>x. f x y)"
@@ -331,17 +335,40 @@
   from f2 show "cont f" by (rule cont2cont_lambda)
 qed
 
-text {* continuity simplification procedure *}
+text {*
+  This version does work as a cont2cont rule, since it
+  has only a single subgoal.
+*}
+
+lemma cont2cont_LAM' [cont2cont]:
+  fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo"
+  assumes f: "cont (\<lambda>p. f (fst p) (snd p))"
+  shows "cont (\<lambda>x. \<Lambda> y. f x y)"
+proof (rule cont2cont_LAM)
+  fix x :: 'a
+  have "cont (\<lambda>y. (x, y))"
+    by (rule cont_pair2)
+  with f have "cont (\<lambda>y. f (fst (x, y)) (snd (x, y)))"
+    by (rule cont2cont_app3)
+  thus "cont (\<lambda>y. f x y)"
+    by (simp only: fst_conv snd_conv)
+next
+  fix y :: 'b
+  have "cont (\<lambda>x. (x, y))"
+    by (rule cont_pair1)
+  with f have "cont (\<lambda>x. f (fst (x, y)) (snd (x, y)))"
+    by (rule cont2cont_app3)
+  thus "cont (\<lambda>x. f x y)"
+    by (simp only: fst_conv snd_conv)
+qed
+
+lemma cont2cont_LAM_discrete [cont2cont]:
+  "(\<And>y::'a::discrete_cpo. cont (\<lambda>x. f x y)) \<Longrightarrow> cont (\<lambda>x. \<Lambda> y. f x y)"
+by (simp add: cont2cont_LAM)
 
 lemmas cont_lemmas1 =
   cont_const cont_id cont_Rep_CFun2 cont2cont_Rep_CFun cont2cont_LAM
 
-use "Tools/cont_proc.ML";
-setup ContProc.setup;
-
-(*val cont_tac = (fn i => (resolve_tac cont_lemmas i));*)
-(*val cont_tacR = (fn i => (REPEAT (cont_tac i)));*)
-
 subsection {* Miscellaneous *}
 
 text {* Monotonicity of @{term Abs_CFun} *}
@@ -530,7 +557,8 @@
   monocontlub2cont [OF monofun_strictify2 contlub_strictify2, standard]
 
 lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)"
-by (unfold strictify_def, simp add: cont_strictify1 cont_strictify2)
+  unfolding strictify_def
+  by (simp add: cont_strictify1 cont_strictify2 cont2cont_LAM)
 
 lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>"
 by (simp add: strictify_conv_if)
diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOLCF/Cont.thy
--- a/src/HOLCF/Cont.thy	Sat Jan 17 22:07:29 2009 +0100
+++ b/src/HOLCF/Cont.thy	Sat Jan 17 22:08:14 2009 +0100
@@ -104,6 +104,8 @@
 apply simp
 done
 
+lemmas cont2monofunE = cont2mono [THEN monofunE]
+
 lemmas ch2ch_cont = cont2mono [THEN ch2ch_monofun]
 
 text {* right to left: @{prop "cont f \<Longrightarrow> monofun f \<and> contlub f"} *}
@@ -135,22 +137,82 @@
 apply (erule cpo_lubI)
 done
 
+subsection {* Continuity simproc *}
+
+ML {*
+structure Cont2ContData = NamedThmsFun
+  ( val name = "cont2cont" val description = "continuity intro rule" )
+*}
+
+setup {* Cont2ContData.setup *}
+
+text {*
+  Given the term @{term "cont f"}, the procedure tries to construct the
+  theorem @{term "cont f == True"}. If this theorem cannot be completely
+  solved by the introduction rules, then the procedure returns a
+  conditional rewrite rule with the unsolved subgoals as premises.
+*}
+
+setup {*
+let
+  fun solve_cont thy ss t =
+    let
+      val tr = instantiate' [] [SOME (cterm_of thy t)] Eq_TrueI;
+      val rules = Cont2ContData.get (Simplifier.the_context ss);
+      val tac = REPEAT_ALL_NEW (match_tac rules);
+    in Option.map fst (Seq.pull (tac 1 tr)) end
+
+  val proc =
+    Simplifier.simproc @{theory} "cont_proc" ["cont f"] solve_cont;
+in
+  Simplifier.map_simpset (fn ss => ss addsimprocs [proc])
+end
+*}
+
 subsection {* Continuity of basic functions *}
 
 text {* The identity function is continuous *}
 
-lemma cont_id: "cont (\<lambda>x. x)"
+lemma cont_id [cont2cont]: "cont (\<lambda>x. x)"
 apply (rule contI)
 apply (erule cpo_lubI)
 done
 
 text {* constant functions are continuous *}
 
-lemma cont_const: "cont (\<lambda>x. c)"
+lemma cont_const [cont2cont]: "cont (\<lambda>x. c)"
 apply (rule contI)
 apply (rule lub_const)
 done
 
+text {* application of functions is continuous *}
+
+lemma cont2cont_apply:
+  fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo" and t :: "'a \<Rightarrow> 'b"
+  assumes f1: "\<And>y. cont (\<lambda>x. f x y)"
+  assumes f2: "\<And>x. cont (\<lambda>y. f x y)"
+  assumes t: "cont (\<lambda>x. t x)"
+  shows "cont (\<lambda>x. (f x) (t x))"
+proof (rule monocontlub2cont [OF monofunI contlubI])
+  fix x y :: "'a" assume "x \<sqsubseteq> y"
+  then show "f x (t x) \<sqsubseteq> f y (t y)"
+    by (auto intro: cont2monofunE [OF f1]
+                    cont2monofunE [OF f2]
+                    cont2monofunE [OF t]
+                    trans_less)
+next
+  fix Y :: "nat \<Rightarrow> 'a" assume "chain Y"
+  then show "f (\<Squnion>i. Y i) (t (\<Squnion>i. Y i)) = (\<Squnion>i. f (Y i) (t (Y i)))"
+    by (simp only: cont2contlubE [OF t]  ch2ch_cont [OF t]
+                   cont2contlubE [OF f1] ch2ch_cont [OF f1]
+                   cont2contlubE [OF f2] ch2ch_cont [OF f2]
+                   diag_lub)
+qed
+
+lemma cont2cont_compose:
+  "\<lbrakk>cont c; cont (\<lambda>x. f x)\<rbrakk> \<Longrightarrow> cont (\<lambda>x. c (f x))"
+by (rule cont2cont_apply [OF cont_const])
+
 text {* if-then-else is continuous *}
 
 lemma cont_if [simp]:
diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOLCF/Cprod.thy
--- a/src/HOLCF/Cprod.thy	Sat Jan 17 22:07:29 2009 +0100
+++ b/src/HOLCF/Cprod.thy	Sat Jan 17 22:08:14 2009 +0100
@@ -12,23 +12,6 @@
 
 subsection {* Type @{typ unit} is a pcpo *}
 
-instantiation unit :: sq_ord
-begin
-
-definition
-  less_unit_def [simp]: "x \<sqsubseteq> (y::unit) \<equiv> True"
-
-instance ..
-end
-
-instance unit :: discrete_cpo
-by intro_classes simp
-
-instance unit :: finite_po ..
-
-instance unit :: pcpo
-by intro_classes simp
-
 definition
   unit_when :: "'a \<rightarrow> unit \<rightarrow> 'a" where
   "unit_when = (\<Lambda> a _. a)"
@@ -39,165 +22,6 @@
 lemma unit_when [simp]: "unit_when\<cdot>a\<cdot>u = a"
 by (simp add: unit_when_def)
 
-
-subsection {* Product type is a partial order *}
-
-instantiation "*" :: (sq_ord, sq_ord) sq_ord
-begin
-
-definition
-  less_cprod_def: "(op \<sqsubseteq>) \<equiv> \<lambda>p1 p2. (fst p1 \<sqsubseteq> fst p2 \<and> snd p1 \<sqsubseteq> snd p2)"
-
-instance ..
-end
-
-instance "*" :: (po, po) po
-proof
-  fix x :: "'a \<times> 'b"
-  show "x \<sqsubseteq> x"
-    unfolding less_cprod_def by simp
-next
-  fix x y :: "'a \<times> 'b"
-  assume "x \<sqsubseteq> y" "y \<sqsubseteq> x" thus "x = y"
-    unfolding less_cprod_def Pair_fst_snd_eq
-    by (fast intro: antisym_less)
-next
-  fix x y z :: "'a \<times> 'b"
-  assume "x \<sqsubseteq> y" "y \<sqsubseteq> z" thus "x \<sqsubseteq> z"
-    unfolding less_cprod_def
-    by (fast intro: trans_less)
-qed
-
-subsection {* Monotonicity of @{text "(_,_)"}, @{term fst}, @{term snd} *}
-
-lemma prod_lessI: "\<lbrakk>fst p \<sqsubseteq> fst q; snd p \<sqsubseteq> snd q\<rbrakk> \<Longrightarrow> p \<sqsubseteq> q"
-unfolding less_cprod_def by simp
-
-lemma Pair_less_iff [simp]: "(a, b) \<sqsubseteq> (c, d) = (a \<sqsubseteq> c \<and> b \<sqsubseteq> d)"
-unfolding less_cprod_def by simp
-
-text {* Pair @{text "(_,_)"}  is monotone in both arguments *}
-
-lemma monofun_pair1: "monofun (\<lambda>x. (x, y))"
-by (simp add: monofun_def)
-
-lemma monofun_pair2: "monofun (\<lambda>y. (x, y))"
-by (simp add: monofun_def)
-
-lemma monofun_pair:
-  "\<lbrakk>x1 \<sqsubseteq> x2; y1 \<sqsubseteq> y2\<rbrakk> \<Longrightarrow> (x1, y1) \<sqsubseteq> (x2, y2)"
-by simp
-
-text {* @{term fst} and @{term snd} are monotone *}
-
-lemma monofun_fst: "monofun fst"
-by (simp add: monofun_def less_cprod_def)
-
-lemma monofun_snd: "monofun snd"
-by (simp add: monofun_def less_cprod_def)
-
-subsection {* Product type is a cpo *}
-
-lemma is_lub_Pair:
-  "\<lbrakk>range X <<| x; range Y <<| y\<rbrakk> \<Longrightarrow> range (\<lambda>i. (X i, Y i)) <<| (x, y)"
-apply (rule is_lubI [OF ub_rangeI])
-apply (simp add: less_cprod_def is_ub_lub)
-apply (frule ub2ub_monofun [OF monofun_fst])
-apply (drule ub2ub_monofun [OF monofun_snd])
-apply (simp add: less_cprod_def is_lub_lub)
-done
-
-lemma lub_cprod:
-  fixes S :: "nat \<Rightarrow> ('a::cpo \<times> 'b::cpo)"
-  assumes S: "chain S"
-  shows "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
-proof -
-  have "chain (\<lambda>i. fst (S i))"
-    using monofun_fst S by (rule ch2ch_monofun)
-  hence 1: "range (\<lambda>i. fst (S i)) <<| (\<Squnion>i. fst (S i))"
-    by (rule cpo_lubI)
-  have "chain (\<lambda>i. snd (S i))"
-    using monofun_snd S by (rule ch2ch_monofun)
-  hence 2: "range (\<lambda>i. snd (S i)) <<| (\<Squnion>i. snd (S i))"
-    by (rule cpo_lubI)
-  show "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
-    using is_lub_Pair [OF 1 2] by simp
-qed
-
-lemma thelub_cprod:
-  "chain (S::nat \<Rightarrow> 'a::cpo \<times> 'b::cpo)
-    \<Longrightarrow> (\<Squnion>i. S i) = (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
-by (rule lub_cprod [THEN thelubI])
-
-instance "*" :: (cpo, cpo) cpo
-proof
-  fix S :: "nat \<Rightarrow> ('a \<times> 'b)"
-  assume "chain S"
-  hence "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
-    by (rule lub_cprod)
-  thus "\<exists>x. range S <<| x" ..
-qed
-
-instance "*" :: (finite_po, finite_po) finite_po ..
-
-instance "*" :: (discrete_cpo, discrete_cpo) discrete_cpo
-proof
-  fix x y :: "'a \<times> 'b"
-  show "x \<sqsubseteq> y \<longleftrightarrow> x = y"
-    unfolding less_cprod_def Pair_fst_snd_eq
-    by simp
-qed
-
-subsection {* Product type is pointed *}
-
-lemma minimal_cprod: "(\<bottom>, \<bottom>) \<sqsubseteq> p"
-by (simp add: less_cprod_def)
-
-instance "*" :: (pcpo, pcpo) pcpo
-by intro_classes (fast intro: minimal_cprod)
-
-lemma inst_cprod_pcpo: "\<bottom> = (\<bottom>, \<bottom>)"
-by (rule minimal_cprod [THEN UU_I, symmetric])
-
-
-subsection {* Continuity of @{text "(_,_)"}, @{term fst}, @{term snd} *}
-
-lemma cont_pair1: "cont (\<lambda>x. (x, y))"
-apply (rule contI)
-apply (rule is_lub_Pair)
-apply (erule cpo_lubI)
-apply (rule lub_const)
-done
-
-lemma cont_pair2: "cont (\<lambda>y. (x, y))"
-apply (rule contI)
-apply (rule is_lub_Pair)
-apply (rule lub_const)
-apply (erule cpo_lubI)
-done
-
-lemma contlub_fst: "contlub fst"
-apply (rule contlubI)
-apply (simp add: thelub_cprod)
-done
-
-lemma contlub_snd: "contlub snd"
-apply (rule contlubI)
-apply (simp add: thelub_cprod)
-done
-
-lemma cont_fst: "cont fst"
-apply (rule monocontlub2cont)
-apply (rule monofun_fst)
-apply (rule contlub_fst)
-done
-
-lemma cont_snd: "cont snd"
-apply (rule monocontlub2cont)
-apply (rule monofun_snd)
-apply (rule contlub_snd)
-done
-
 subsection {* Continuous versions of constants *}
 
 definition
@@ -245,10 +69,10 @@
 by (simp add: cpair_eq_pair)
 
 lemma cpair_less [iff]: "(<a, b> \<sqsubseteq> <a', b'>) = (a \<sqsubseteq> a' \<and> b \<sqsubseteq> b')"
-by (simp add: cpair_eq_pair less_cprod_def)
+by (simp add: cpair_eq_pair)
 
 lemma cpair_defined_iff [iff]: "(<x, y> = \<bottom>) = (x = \<bottom> \<and> y = \<bottom>)"
-by (simp add: inst_cprod_pcpo cpair_eq_pair)
+by (simp add: cpair_eq_pair)
 
 lemma cpair_strict [simp]: "\<langle>\<bottom>, \<bottom>\<rangle> = \<bottom>"
 by simp
@@ -273,10 +97,10 @@
 by (simp add: cpair_eq_pair csnd_def cont_snd)
 
 lemma cfst_strict [simp]: "cfst\<cdot>\<bottom> = \<bottom>"
-unfolding inst_cprod_pcpo2 by (rule cfst_cpair)
+by (simp add: cfst_def)
 
 lemma csnd_strict [simp]: "csnd\<cdot>\<bottom> = \<bottom>"
-unfolding inst_cprod_pcpo2 by (rule csnd_cpair)
+by (simp add: csnd_def)
 
 lemma cpair_cfst_csnd: "\<langle>cfst\<cdot>p, csnd\<cdot>p\<rangle> = p"
 by (cases p rule: cprodE, simp)
@@ -302,19 +126,10 @@
 by (rule compactI, simp add: csnd_less_iff)
 
 lemma compact_cpair: "\<lbrakk>compact x; compact y\<rbrakk> \<Longrightarrow> compact <x, y>"
-by (rule compactI, simp add: less_cprod)
+by (simp add: cpair_eq_pair)
 
 lemma compact_cpair_iff [simp]: "compact <x, y> = (compact x \<and> compact y)"
-apply (safe intro!: compact_cpair)
-apply (drule compact_cfst, simp)
-apply (drule compact_csnd, simp)
-done
-
-instance "*" :: (chfin, chfin) chfin
-apply intro_classes
-apply (erule compact_imp_max_in_chain)
-apply (rule_tac p="\<Squnion>i. Y i" in cprodE, simp)
-done
+by (simp add: cpair_eq_pair)
 
 lemma lub_cprod2: 
   "chain S \<Longrightarrow> range S <<| <\<Squnion>i. cfst\<cdot>(S i), \<Squnion>i. csnd\<cdot>(S i)>"
diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOLCF/Dsum.thy
--- a/src/HOLCF/Dsum.thy	Sat Jan 17 22:07:29 2009 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,251 +0,0 @@
-(*  Title:      HOLCF/Dsum.thy
-    Author:     Brian Huffman
-*)
-
-header {* The cpo of disjoint sums *}
-
-theory Dsum
-imports Bifinite
-begin
-
-subsection {* Ordering on type @{typ "'a + 'b"} *}
-
-instantiation "+" :: (sq_ord, sq_ord) sq_ord
-begin
-
-definition
-  less_sum_def: "x \<sqsubseteq> y \<equiv> case x of
-         Inl a \<Rightarrow> (case y of Inl b \<Rightarrow> a \<sqsubseteq> b | Inr b \<Rightarrow> False) |
-         Inr a \<Rightarrow> (case y of Inl b \<Rightarrow> False | Inr b \<Rightarrow> a \<sqsubseteq> b)"
-
-instance ..
-end
-
-lemma Inl_less_iff [simp]: "Inl x \<sqsubseteq> Inl y = x \<sqsubseteq> y"
-unfolding less_sum_def by simp
-
-lemma Inr_less_iff [simp]: "Inr x \<sqsubseteq> Inr y = x \<sqsubseteq> y"
-unfolding less_sum_def by simp
-
-lemma Inl_less_Inr [simp]: "\<not> Inl x \<sqsubseteq> Inr y"
-unfolding less_sum_def by simp
-
-lemma Inr_less_Inl [simp]: "\<not> Inr x \<sqsubseteq> Inl y"
-unfolding less_sum_def by simp
-
-lemma Inl_mono: "x \<sqsubseteq> y \<Longrightarrow> Inl x \<sqsubseteq> Inl y"
-by simp
-
-lemma Inr_mono: "x \<sqsubseteq> y \<Longrightarrow> Inr x \<sqsubseteq> Inr y"
-by simp
-
-lemma Inl_lessE: "\<lbrakk>Inl a \<sqsubseteq> x; \<And>b. \<lbrakk>x = Inl b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
-by (cases x, simp_all)
-
-lemma Inr_lessE: "\<lbrakk>Inr a \<sqsubseteq> x; \<And>b. \<lbrakk>x = Inr b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
-by (cases x, simp_all)
-
-lemmas sum_less_elims = Inl_lessE Inr_lessE
-
-lemma sum_less_cases:
-  "\<lbrakk>x \<sqsubseteq> y;
-    \<And>a b. \<lbrakk>x = Inl a; y = Inl b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R;
-    \<And>a b. \<lbrakk>x = Inr a; y = Inr b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R\<rbrakk>
-      \<Longrightarrow> R"
-by (cases x, safe elim!: sum_less_elims, auto)
-
-subsection {* Sum type is a complete partial order *}
-
-instance "+" :: (po, po) po
-proof
-  fix x :: "'a + 'b"
-  show "x \<sqsubseteq> x"
-    by (induct x, simp_all)
-next
-  fix x y :: "'a + 'b"
-  assume "x \<sqsubseteq> y" and "y \<sqsubseteq> x" thus "x = y"
-    by (induct x, auto elim!: sum_less_elims intro: antisym_less)
-next
-  fix x y z :: "'a + 'b"
-  assume "x \<sqsubseteq> y" and "y \<sqsubseteq> z" thus "x \<sqsubseteq> z"
-    by (induct x, auto elim!: sum_less_elims intro: trans_less)
-qed
-
-lemma monofun_inv_Inl: "monofun (\<lambda>p. THE a. p = Inl a)"
-by (rule monofunI, erule sum_less_cases, simp_all)
-
-lemma monofun_inv_Inr: "monofun (\<lambda>p. THE b. p = Inr b)"
-by (rule monofunI, erule sum_less_cases, simp_all)
-
-lemma sum_chain_cases:
-  assumes Y: "chain Y"
-  assumes A: "\<And>A. \<lbrakk>chain A; Y = (\<lambda>i. Inl (A i))\<rbrakk> \<Longrightarrow> R"
-  assumes B: "\<And>B. \<lbrakk>chain B; Y = (\<lambda>i. Inr (B i))\<rbrakk> \<Longrightarrow> R"
-  shows "R"
- apply (cases "Y 0")
-  apply (rule A)
-   apply (rule ch2ch_monofun [OF monofun_inv_Inl Y])
-  apply (rule ext)
-  apply (cut_tac j=i in chain_mono [OF Y le0], simp)
-  apply (erule Inl_lessE, simp)
- apply (rule B)
-  apply (rule ch2ch_monofun [OF monofun_inv_Inr Y])
- apply (rule ext)
- apply (cut_tac j=i in chain_mono [OF Y le0], simp)
- apply (erule Inr_lessE, simp)
-done
-
-lemma is_lub_Inl: "range S <<| x \<Longrightarrow> range (\<lambda>i. Inl (S i)) <<| Inl x"
- apply (rule is_lubI)
-  apply (rule ub_rangeI)
-  apply (simp add: is_ub_lub)
- apply (frule ub_rangeD [where i=arbitrary])
- apply (erule Inl_lessE, simp)
- apply (erule is_lub_lub)
- apply (rule ub_rangeI)
- apply (drule ub_rangeD, simp)
-done
-
-lemma is_lub_Inr: "range S <<| x \<Longrightarrow> range (\<lambda>i. Inr (S i)) <<| Inr x"
- apply (rule is_lubI)
-  apply (rule ub_rangeI)
-  apply (simp add: is_ub_lub)
- apply (frule ub_rangeD [where i=arbitrary])
- apply (erule Inr_lessE, simp)
- apply (erule is_lub_lub)
- apply (rule ub_rangeI)
- apply (drule ub_rangeD, simp)
-done
-
-instance "+" :: (cpo, cpo) cpo
- apply intro_classes
- apply (erule sum_chain_cases, safe)
-  apply (rule exI)
-  apply (rule is_lub_Inl)
-  apply (erule cpo_lubI)
- apply (rule exI)
- apply (rule is_lub_Inr)
- apply (erule cpo_lubI)
-done
-
-subsection {* Continuity of @{term Inl}, @{term Inr}, @{term sum_case} *}
-
-lemma cont2cont_Inl [simp]: "cont f \<Longrightarrow> cont (\<lambda>x. Inl (f x))"
-by (fast intro: contI is_lub_Inl elim: contE)
-
-lemma cont2cont_Inr [simp]: "cont f \<Longrightarrow> cont (\<lambda>x. Inr (f x))"
-by (fast intro: contI is_lub_Inr elim: contE)
-
-lemma cont_Inl: "cont Inl"
-by (rule cont2cont_Inl [OF cont_id])
-
-lemma cont_Inr: "cont Inr"
-by (rule cont2cont_Inr [OF cont_id])
-
-lemmas ch2ch_Inl [simp] = ch2ch_cont [OF cont_Inl]
-lemmas ch2ch_Inr [simp] = ch2ch_cont [OF cont_Inr]
-
-lemmas lub_Inl = cont2contlubE [OF cont_Inl, symmetric]
-lemmas lub_Inr = cont2contlubE [OF cont_Inr, symmetric]
-
-lemma cont_sum_case1:
-  assumes f: "\<And>a. cont (\<lambda>x. f x a)"
-  assumes g: "\<And>b. cont (\<lambda>x. g x b)"
-  shows "cont (\<lambda>x. case y of Inl a \<Rightarrow> f x a | Inr b \<Rightarrow> g x b)"
-by (induct y, simp add: f, simp add: g)
-
-lemma cont_sum_case2: "\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (sum_case f g)"
-apply (rule contI)
-apply (erule sum_chain_cases)
-apply (simp add: cont2contlubE [OF cont_Inl, symmetric] contE)
-apply (simp add: cont2contlubE [OF cont_Inr, symmetric] contE)
-done
-
-lemma cont2cont_sum_case [simp]:
-  assumes f1: "\<And>a. cont (\<lambda>x. f x a)" and f2: "\<And>x. cont (\<lambda>a. f x a)"
-  assumes g1: "\<And>b. cont (\<lambda>x. g x b)" and g2: "\<And>x. cont (\<lambda>b. g x b)"
-  assumes h: "cont (\<lambda>x. h x)"
-  shows "cont (\<lambda>x. case h x of Inl a \<Rightarrow> f x a | Inr b \<Rightarrow> g x b)"
-apply (rule cont2cont_app2 [OF cont2cont_lambda _ h])
-apply (rule cont_sum_case1 [OF f1 g1])
-apply (rule cont_sum_case2 [OF f2 g2])
-done
-
-subsection {* Compactness and chain-finiteness *}
-
-lemma compact_Inl: "compact a \<Longrightarrow> compact (Inl a)"
-apply (rule compactI2)
-apply (erule sum_chain_cases, safe)
-apply (simp add: lub_Inl)
-apply (erule (2) compactD2)
-apply (simp add: lub_Inr)
-done
-
-lemma compact_Inr: "compact a \<Longrightarrow> compact (Inr a)"
-apply (rule compactI2)
-apply (erule sum_chain_cases, safe)
-apply (simp add: lub_Inl)
-apply (simp add: lub_Inr)
-apply (erule (2) compactD2)
-done
-
-lemma compact_Inl_rev: "compact (Inl a) \<Longrightarrow> compact a"
-unfolding compact_def
-by (drule adm_subst [OF cont_Inl], simp)
-
-lemma compact_Inr_rev: "compact (Inr a) \<Longrightarrow> compact a"
-unfolding compact_def
-by (drule adm_subst [OF cont_Inr], simp)
-
-lemma compact_Inl_iff [simp]: "compact (Inl a) = compact a"
-by (safe elim!: compact_Inl compact_Inl_rev)
-
-lemma compact_Inr_iff [simp]: "compact (Inr a) = compact a"
-by (safe elim!: compact_Inr compact_Inr_rev)
-
-instance "+" :: (chfin, chfin) chfin
-apply intro_classes
-apply (erule compact_imp_max_in_chain)
-apply (case_tac "\<Squnion>i. Y i", simp_all)
-done
-
-instance "+" :: (finite_po, finite_po) finite_po ..
-
-instance "+" :: (discrete_cpo, discrete_cpo) discrete_cpo
-by intro_classes (simp add: less_sum_def split: sum.split)
-
-subsection {* Sum type is a bifinite domain *}
-
-instantiation "+" :: (profinite, profinite) profinite
-begin
-
-definition
-  approx_sum_def: "approx =
-    (\<lambda>n. \<Lambda> x. case x of Inl a \<Rightarrow> Inl (approx n\<cdot>a) | Inr b \<Rightarrow> Inr (approx n\<cdot>b))"
-
-lemma approx_Inl [simp]: "approx n\<cdot>(Inl x) = Inl (approx n\<cdot>x)"
-  unfolding approx_sum_def by simp
-
-lemma approx_Inr [simp]: "approx n\<cdot>(Inr x) = Inr (approx n\<cdot>x)"
-  unfolding approx_sum_def by simp
-
-instance proof
-  fix i :: nat and x :: "'a + 'b"
-  show "chain (approx :: nat \<Rightarrow> 'a + 'b \<rightarrow> 'a + 'b)"
-    unfolding approx_sum_def
-    by (rule ch2ch_LAM, case_tac x, simp_all)
-  show "(\<Squnion>i. approx i\<cdot>x) = x"
-    by (induct x, simp_all add: lub_Inl lub_Inr)
-  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
-    by (induct x, simp_all)
-  have "{x::'a + 'b. approx i\<cdot>x = x} \<subseteq>
-        {x::'a. approx i\<cdot>x = x} <+> {x::'b. approx i\<cdot>x = x}"
-    by (rule subsetI, case_tac x, simp_all add: InlI InrI)
-  thus "finite {x::'a + 'b. approx i\<cdot>x = x}"
-    by (rule finite_subset,
-        intro finite_Plus finite_fixes_approx)
-qed
-
-end
-
-end
diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOLCF/Fixrec.thy
--- a/src/HOLCF/Fixrec.thy	Sat Jan 17 22:07:29 2009 +0100
+++ b/src/HOLCF/Fixrec.thy	Sat Jan 17 22:08:14 2009 +0100
@@ -55,6 +55,7 @@
   "maybe_when\<cdot>f\<cdot>r\<cdot>fail = f"
   "maybe_when\<cdot>f\<cdot>r\<cdot>(return\<cdot>x) = r\<cdot>x"
 by (simp_all add: return_def fail_def maybe_when_def cont_Rep_maybe
+                  cont2cont_LAM
                   cont_Abs_maybe Abs_maybe_inverse Rep_maybe_strict)
 
 translations
diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOLCF/HOLCF.thy
--- a/src/HOLCF/HOLCF.thy	Sat Jan 17 22:07:29 2009 +0100
+++ b/src/HOLCF/HOLCF.thy	Sat Jan 17 22:08:14 2009 +0100
@@ -6,10 +6,11 @@
 
 theory HOLCF
 imports
-  Domain ConvexPD Algebraic Universal Dsum Main
+  Domain ConvexPD Algebraic Universal Sum_Cpo Main
 uses
   "holcf_logic.ML"
   "Tools/cont_consts.ML"
+  "Tools/cont_proc.ML"
   "Tools/domain/domain_library.ML"
   "Tools/domain/domain_syntax.ML"
   "Tools/domain/domain_axioms.ML"
diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOLCF/IsaMakefile
--- a/src/HOLCF/IsaMakefile	Sat Jan 17 22:07:29 2009 +0100
+++ b/src/HOLCF/IsaMakefile	Sat Jan 17 22:08:14 2009 +0100
@@ -1,5 +1,3 @@
-#
-# $Id$
 #
 # IsaMakefile for HOLCF
 #
@@ -41,7 +39,6 @@
   Discrete.thy \
   Deflation.thy \
   Domain.thy \
-  Dsum.thy \
   Eventual.thy \
   Ffun.thy \
   Fixrec.thy \
@@ -54,8 +51,10 @@
   Pcpodef.thy \
   Pcpo.thy \
   Porder.thy \
+  Product_Cpo.thy \
   Sprod.thy \
   Ssum.thy \
+  Sum_Cpo.thy \
   Tr.thy \
   Universal.thy \
   UpperPD.thy \
diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOLCF/Product_Cpo.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Product_Cpo.thy	Sat Jan 17 22:08:14 2009 +0100
@@ -0,0 +1,247 @@
+(*  Title:      HOLCF/Product_Cpo.thy
+    Author:     Franz Regensburger
+*)
+
+header {* The cpo of cartesian products *}
+
+theory Product_Cpo
+imports Adm
+begin
+
+defaultsort cpo
+
+subsection {* Type @{typ unit} is a pcpo *}
+
+instantiation unit :: sq_ord
+begin
+
+definition
+  less_unit_def [simp]: "x \<sqsubseteq> (y::unit) \<equiv> True"
+
+instance ..
+end
+
+instance unit :: discrete_cpo
+by intro_classes simp
+
+instance unit :: finite_po ..
+
+instance unit :: pcpo
+by intro_classes simp
+
+
+subsection {* Product type is a partial order *}
+
+instantiation "*" :: (sq_ord, sq_ord) sq_ord
+begin
+
+definition
+  less_cprod_def: "(op \<sqsubseteq>) \<equiv> \<lambda>p1 p2. (fst p1 \<sqsubseteq> fst p2 \<and> snd p1 \<sqsubseteq> snd p2)"
+
+instance ..
+end
+
+instance "*" :: (po, po) po
+proof
+  fix x :: "'a \<times> 'b"
+  show "x \<sqsubseteq> x"
+    unfolding less_cprod_def by simp
+next
+  fix x y :: "'a \<times> 'b"
+  assume "x \<sqsubseteq> y" "y \<sqsubseteq> x" thus "x = y"
+    unfolding less_cprod_def Pair_fst_snd_eq
+    by (fast intro: antisym_less)
+next
+  fix x y z :: "'a \<times> 'b"
+  assume "x \<sqsubseteq> y" "y \<sqsubseteq> z" thus "x \<sqsubseteq> z"
+    unfolding less_cprod_def
+    by (fast intro: trans_less)
+qed
+
+subsection {* Monotonicity of @{text "(_,_)"}, @{term fst}, @{term snd} *}
+
+lemma prod_lessI: "\<lbrakk>fst p \<sqsubseteq> fst q; snd p \<sqsubseteq> snd q\<rbrakk> \<Longrightarrow> p \<sqsubseteq> q"
+unfolding less_cprod_def by simp
+
+lemma Pair_less_iff [simp]: "(a, b) \<sqsubseteq> (c, d) \<longleftrightarrow> a \<sqsubseteq> c \<and> b \<sqsubseteq> d"
+unfolding less_cprod_def by simp
+
+text {* Pair @{text "(_,_)"}  is monotone in both arguments *}
+
+lemma monofun_pair1: "monofun (\<lambda>x. (x, y))"
+by (simp add: monofun_def)
+
+lemma monofun_pair2: "monofun (\<lambda>y. (x, y))"
+by (simp add: monofun_def)
+
+lemma monofun_pair:
+  "\<lbrakk>x1 \<sqsubseteq> x2; y1 \<sqsubseteq> y2\<rbrakk> \<Longrightarrow> (x1, y1) \<sqsubseteq> (x2, y2)"
+by simp
+
+text {* @{term fst} and @{term snd} are monotone *}
+
+lemma monofun_fst: "monofun fst"
+by (simp add: monofun_def less_cprod_def)
+
+lemma monofun_snd: "monofun snd"
+by (simp add: monofun_def less_cprod_def)
+
+subsection {* Product type is a cpo *}
+
+lemma is_lub_Pair:
+  "\<lbrakk>range X <<| x; range Y <<| y\<rbrakk> \<Longrightarrow> range (\<lambda>i. (X i, Y i)) <<| (x, y)"
+apply (rule is_lubI [OF ub_rangeI])
+apply (simp add: less_cprod_def is_ub_lub)
+apply (frule ub2ub_monofun [OF monofun_fst])
+apply (drule ub2ub_monofun [OF monofun_snd])
+apply (simp add: less_cprod_def is_lub_lub)
+done
+
+lemma lub_cprod:
+  fixes S :: "nat \<Rightarrow> ('a::cpo \<times> 'b::cpo)"
+  assumes S: "chain S"
+  shows "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
+proof -
+  have "chain (\<lambda>i. fst (S i))"
+    using monofun_fst S by (rule ch2ch_monofun)
+  hence 1: "range (\<lambda>i. fst (S i)) <<| (\<Squnion>i. fst (S i))"
+    by (rule cpo_lubI)
+  have "chain (\<lambda>i. snd (S i))"
+    using monofun_snd S by (rule ch2ch_monofun)
+  hence 2: "range (\<lambda>i. snd (S i)) <<| (\<Squnion>i. snd (S i))"
+    by (rule cpo_lubI)
+  show "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
+    using is_lub_Pair [OF 1 2] by simp
+qed
+
+lemma thelub_cprod:
+  "chain (S::nat \<Rightarrow> 'a::cpo \<times> 'b::cpo)
+    \<Longrightarrow> (\<Squnion>i. S i) = (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
+by (rule lub_cprod [THEN thelubI])
+
+instance "*" :: (cpo, cpo) cpo
+proof
+  fix S :: "nat \<Rightarrow> ('a \<times> 'b)"
+  assume "chain S"
+  hence "range S <<| (\<Squnion>i. fst (S i), \<Squnion>i. snd (S i))"
+    by (rule lub_cprod)
+  thus "\<exists>x. range S <<| x" ..
+qed
+
+instance "*" :: (finite_po, finite_po) finite_po ..
+
+instance "*" :: (discrete_cpo, discrete_cpo) discrete_cpo
+proof
+  fix x y :: "'a \<times> 'b"
+  show "x \<sqsubseteq> y \<longleftrightarrow> x = y"
+    unfolding less_cprod_def Pair_fst_snd_eq
+    by simp
+qed
+
+subsection {* Product type is pointed *}
+
+lemma minimal_cprod: "(\<bottom>, \<bottom>) \<sqsubseteq> p"
+by (simp add: less_cprod_def)
+
+instance "*" :: (pcpo, pcpo) pcpo
+by intro_classes (fast intro: minimal_cprod)
+
+lemma inst_cprod_pcpo: "\<bottom> = (\<bottom>, \<bottom>)"
+by (rule minimal_cprod [THEN UU_I, symmetric])
+
+lemma Pair_defined_iff [simp]: "(x, y) = \<bottom> \<longleftrightarrow> x = \<bottom> \<and> y = \<bottom>"
+unfolding inst_cprod_pcpo by simp
+
+lemma fst_strict [simp]: "fst \<bottom> = \<bottom>"
+unfolding inst_cprod_pcpo by (rule fst_conv)
+
+lemma csnd_strict [simp]: "snd \<bottom> = \<bottom>"
+unfolding inst_cprod_pcpo by (rule snd_conv)
+
+lemma Pair_strict [simp]: "(\<bottom>, \<bottom>) = \<bottom>"
+by simp
+
+lemma split_strict [simp]: "split f \<bottom> = f \<bottom> \<bottom>"
+unfolding split_def by simp
+
+subsection {* Continuity of @{text "(_,_)"}, @{term fst}, @{term snd} *}
+
+lemma cont_pair1: "cont (\<lambda>x. (x, y))"
+apply (rule contI)
+apply (rule is_lub_Pair)
+apply (erule cpo_lubI)
+apply (rule lub_const)
+done
+
+lemma cont_pair2: "cont (\<lambda>y. (x, y))"
+apply (rule contI)
+apply (rule is_lub_Pair)
+apply (rule lub_const)
+apply (erule cpo_lubI)
+done
+
+lemma contlub_fst: "contlub fst"
+apply (rule contlubI)
+apply (simp add: thelub_cprod)
+done
+
+lemma contlub_snd: "contlub snd"
+apply (rule contlubI)
+apply (simp add: thelub_cprod)
+done
+
+lemma cont_fst: "cont fst"
+apply (rule monocontlub2cont)
+apply (rule monofun_fst)
+apply (rule contlub_fst)
+done
+
+lemma cont_snd: "cont snd"
+apply (rule monocontlub2cont)
+apply (rule monofun_snd)
+apply (rule contlub_snd)
+done
+
+lemma cont2cont_Pair [cont2cont]:
+  assumes f: "cont (\<lambda>x. f x)"
+  assumes g: "cont (\<lambda>x. g x)"
+  shows "cont (\<lambda>x. (f x, g x))"
+apply (rule cont2cont_apply [OF _ cont_pair1 f])
+apply (rule cont2cont_apply [OF _ cont_pair2 g])
+apply (rule cont_const)
+done
+
+lemmas cont2cont_fst [cont2cont] = cont2cont_compose [OF cont_fst]
+
+lemmas cont2cont_snd [cont2cont] = cont2cont_compose [OF cont_snd]
+
+subsection {* Compactness and chain-finiteness *}
+
+lemma fst_less_iff: "fst (x::'a \<times> 'b) \<sqsubseteq> y \<longleftrightarrow> x \<sqsubseteq> (y, snd x)"
+unfolding less_cprod_def by simp
+
+lemma snd_less_iff: "snd (x::'a \<times> 'b) \<sqsubseteq> y = x \<sqsubseteq> (fst x, y)"
+unfolding less_cprod_def by simp
+
+lemma compact_fst: "compact x \<Longrightarrow> compact (fst x)"
+by (rule compactI, simp add: fst_less_iff)
+
+lemma compact_snd: "compact x \<Longrightarrow> compact (snd x)"
+by (rule compactI, simp add: snd_less_iff)
+
+lemma compact_Pair: "\<lbrakk>compact x; compact y\<rbrakk> \<Longrightarrow> compact (x, y)"
+by (rule compactI, simp add: less_cprod_def)
+
+lemma compact_Pair_iff [simp]: "compact (x, y) \<longleftrightarrow> compact x \<and> compact y"
+apply (safe intro!: compact_Pair)
+apply (drule compact_fst, simp)
+apply (drule compact_snd, simp)
+done
+
+instance "*" :: (chfin, chfin) chfin
+apply intro_classes
+apply (erule compact_imp_max_in_chain)
+apply (case_tac "\<Squnion>i. Y i", simp)
+done
+
+end
diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOLCF/Ssum.thy
--- a/src/HOLCF/Ssum.thy	Sat Jan 17 22:07:29 2009 +0100
+++ b/src/HOLCF/Ssum.thy	Sat Jan 17 22:08:14 2009 +0100
@@ -188,7 +188,7 @@
 
 lemma beta_sscase:
   "sscase\<cdot>f\<cdot>g\<cdot>s = (\<Lambda><t, x, y>. If t then f\<cdot>x else g\<cdot>y fi)\<cdot>(Rep_Ssum s)"
-unfolding sscase_def by (simp add: cont_Rep_Ssum)
+unfolding sscase_def by (simp add: cont_Rep_Ssum cont2cont_LAM)
 
 lemma sscase1 [simp]: "sscase\<cdot>f\<cdot>g\<cdot>\<bottom> = \<bottom>"
 unfolding beta_sscase by (simp add: Rep_Ssum_strict)
diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOLCF/Sum_Cpo.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Sum_Cpo.thy	Sat Jan 17 22:08:14 2009 +0100
@@ -0,0 +1,251 @@
+(*  Title:      HOLCF/Sum_Cpo.thy
+    Author:     Brian Huffman
+*)
+
+header {* The cpo of disjoint sums *}
+
+theory Sum_Cpo
+imports Bifinite
+begin
+
+subsection {* Ordering on type @{typ "'a + 'b"} *}
+
+instantiation "+" :: (sq_ord, sq_ord) sq_ord
+begin
+
+definition
+  less_sum_def: "x \<sqsubseteq> y \<equiv> case x of
+         Inl a \<Rightarrow> (case y of Inl b \<Rightarrow> a \<sqsubseteq> b | Inr b \<Rightarrow> False) |
+         Inr a \<Rightarrow> (case y of Inl b \<Rightarrow> False | Inr b \<Rightarrow> a \<sqsubseteq> b)"
+
+instance ..
+end
+
+lemma Inl_less_iff [simp]: "Inl x \<sqsubseteq> Inl y = x \<sqsubseteq> y"
+unfolding less_sum_def by simp
+
+lemma Inr_less_iff [simp]: "Inr x \<sqsubseteq> Inr y = x \<sqsubseteq> y"
+unfolding less_sum_def by simp
+
+lemma Inl_less_Inr [simp]: "\<not> Inl x \<sqsubseteq> Inr y"
+unfolding less_sum_def by simp
+
+lemma Inr_less_Inl [simp]: "\<not> Inr x \<sqsubseteq> Inl y"
+unfolding less_sum_def by simp
+
+lemma Inl_mono: "x \<sqsubseteq> y \<Longrightarrow> Inl x \<sqsubseteq> Inl y"
+by simp
+
+lemma Inr_mono: "x \<sqsubseteq> y \<Longrightarrow> Inr x \<sqsubseteq> Inr y"
+by simp
+
+lemma Inl_lessE: "\<lbrakk>Inl a \<sqsubseteq> x; \<And>b. \<lbrakk>x = Inl b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
+by (cases x, simp_all)
+
+lemma Inr_lessE: "\<lbrakk>Inr a \<sqsubseteq> x; \<And>b. \<lbrakk>x = Inr b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
+by (cases x, simp_all)
+
+lemmas sum_less_elims = Inl_lessE Inr_lessE
+
+lemma sum_less_cases:
+  "\<lbrakk>x \<sqsubseteq> y;
+    \<And>a b. \<lbrakk>x = Inl a; y = Inl b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R;
+    \<And>a b. \<lbrakk>x = Inr a; y = Inr b; a \<sqsubseteq> b\<rbrakk> \<Longrightarrow> R\<rbrakk>
+      \<Longrightarrow> R"
+by (cases x, safe elim!: sum_less_elims, auto)
+
+subsection {* Sum type is a complete partial order *}
+
+instance "+" :: (po, po) po
+proof
+  fix x :: "'a + 'b"
+  show "x \<sqsubseteq> x"
+    by (induct x, simp_all)
+next
+  fix x y :: "'a + 'b"
+  assume "x \<sqsubseteq> y" and "y \<sqsubseteq> x" thus "x = y"
+    by (induct x, auto elim!: sum_less_elims intro: antisym_less)
+next
+  fix x y z :: "'a + 'b"
+  assume "x \<sqsubseteq> y" and "y \<sqsubseteq> z" thus "x \<sqsubseteq> z"
+    by (induct x, auto elim!: sum_less_elims intro: trans_less)
+qed
+
+lemma monofun_inv_Inl: "monofun (\<lambda>p. THE a. p = Inl a)"
+by (rule monofunI, erule sum_less_cases, simp_all)
+
+lemma monofun_inv_Inr: "monofun (\<lambda>p. THE b. p = Inr b)"
+by (rule monofunI, erule sum_less_cases, simp_all)
+
+lemma sum_chain_cases:
+  assumes Y: "chain Y"
+  assumes A: "\<And>A. \<lbrakk>chain A; Y = (\<lambda>i. Inl (A i))\<rbrakk> \<Longrightarrow> R"
+  assumes B: "\<And>B. \<lbrakk>chain B; Y = (\<lambda>i. Inr (B i))\<rbrakk> \<Longrightarrow> R"
+  shows "R"
+ apply (cases "Y 0")
+  apply (rule A)
+   apply (rule ch2ch_monofun [OF monofun_inv_Inl Y])
+  apply (rule ext)
+  apply (cut_tac j=i in chain_mono [OF Y le0], simp)
+  apply (erule Inl_lessE, simp)
+ apply (rule B)
+  apply (rule ch2ch_monofun [OF monofun_inv_Inr Y])
+ apply (rule ext)
+ apply (cut_tac j=i in chain_mono [OF Y le0], simp)
+ apply (erule Inr_lessE, simp)
+done
+
+lemma is_lub_Inl: "range S <<| x \<Longrightarrow> range (\<lambda>i. Inl (S i)) <<| Inl x"
+ apply (rule is_lubI)
+  apply (rule ub_rangeI)
+  apply (simp add: is_ub_lub)
+ apply (frule ub_rangeD [where i=arbitrary])
+ apply (erule Inl_lessE, simp)
+ apply (erule is_lub_lub)
+ apply (rule ub_rangeI)
+ apply (drule ub_rangeD, simp)
+done
+
+lemma is_lub_Inr: "range S <<| x \<Longrightarrow> range (\<lambda>i. Inr (S i)) <<| Inr x"
+ apply (rule is_lubI)
+  apply (rule ub_rangeI)
+  apply (simp add: is_ub_lub)
+ apply (frule ub_rangeD [where i=arbitrary])
+ apply (erule Inr_lessE, simp)
+ apply (erule is_lub_lub)
+ apply (rule ub_rangeI)
+ apply (drule ub_rangeD, simp)
+done
+
+instance "+" :: (cpo, cpo) cpo
+ apply intro_classes
+ apply (erule sum_chain_cases, safe)
+  apply (rule exI)
+  apply (rule is_lub_Inl)
+  apply (erule cpo_lubI)
+ apply (rule exI)
+ apply (rule is_lub_Inr)
+ apply (erule cpo_lubI)
+done
+
+subsection {* Continuity of @{term Inl}, @{term Inr}, @{term sum_case} *}
+
+lemma cont2cont_Inl [simp]: "cont f \<Longrightarrow> cont (\<lambda>x. Inl (f x))"
+by (fast intro: contI is_lub_Inl elim: contE)
+
+lemma cont2cont_Inr [simp]: "cont f \<Longrightarrow> cont (\<lambda>x. Inr (f x))"
+by (fast intro: contI is_lub_Inr elim: contE)
+
+lemma cont_Inl: "cont Inl"
+by (rule cont2cont_Inl [OF cont_id])
+
+lemma cont_Inr: "cont Inr"
+by (rule cont2cont_Inr [OF cont_id])
+
+lemmas ch2ch_Inl [simp] = ch2ch_cont [OF cont_Inl]
+lemmas ch2ch_Inr [simp] = ch2ch_cont [OF cont_Inr]
+
+lemmas lub_Inl = cont2contlubE [OF cont_Inl, symmetric]
+lemmas lub_Inr = cont2contlubE [OF cont_Inr, symmetric]
+
+lemma cont_sum_case1:
+  assumes f: "\<And>a. cont (\<lambda>x. f x a)"
+  assumes g: "\<And>b. cont (\<lambda>x. g x b)"
+  shows "cont (\<lambda>x. case y of Inl a \<Rightarrow> f x a | Inr b \<Rightarrow> g x b)"
+by (induct y, simp add: f, simp add: g)
+
+lemma cont_sum_case2: "\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (sum_case f g)"
+apply (rule contI)
+apply (erule sum_chain_cases)
+apply (simp add: cont2contlubE [OF cont_Inl, symmetric] contE)
+apply (simp add: cont2contlubE [OF cont_Inr, symmetric] contE)
+done
+
+lemma cont2cont_sum_case [simp]:
+  assumes f1: "\<And>a. cont (\<lambda>x. f x a)" and f2: "\<And>x. cont (\<lambda>a. f x a)"
+  assumes g1: "\<And>b. cont (\<lambda>x. g x b)" and g2: "\<And>x. cont (\<lambda>b. g x b)"
+  assumes h: "cont (\<lambda>x. h x)"
+  shows "cont (\<lambda>x. case h x of Inl a \<Rightarrow> f x a | Inr b \<Rightarrow> g x b)"
+apply (rule cont2cont_app2 [OF cont2cont_lambda _ h])
+apply (rule cont_sum_case1 [OF f1 g1])
+apply (rule cont_sum_case2 [OF f2 g2])
+done
+
+subsection {* Compactness and chain-finiteness *}
+
+lemma compact_Inl: "compact a \<Longrightarrow> compact (Inl a)"
+apply (rule compactI2)
+apply (erule sum_chain_cases, safe)
+apply (simp add: lub_Inl)
+apply (erule (2) compactD2)
+apply (simp add: lub_Inr)
+done
+
+lemma compact_Inr: "compact a \<Longrightarrow> compact (Inr a)"
+apply (rule compactI2)
+apply (erule sum_chain_cases, safe)
+apply (simp add: lub_Inl)
+apply (simp add: lub_Inr)
+apply (erule (2) compactD2)
+done
+
+lemma compact_Inl_rev: "compact (Inl a) \<Longrightarrow> compact a"
+unfolding compact_def
+by (drule adm_subst [OF cont_Inl], simp)
+
+lemma compact_Inr_rev: "compact (Inr a) \<Longrightarrow> compact a"
+unfolding compact_def
+by (drule adm_subst [OF cont_Inr], simp)
+
+lemma compact_Inl_iff [simp]: "compact (Inl a) = compact a"
+by (safe elim!: compact_Inl compact_Inl_rev)
+
+lemma compact_Inr_iff [simp]: "compact (Inr a) = compact a"
+by (safe elim!: compact_Inr compact_Inr_rev)
+
+instance "+" :: (chfin, chfin) chfin
+apply intro_classes
+apply (erule compact_imp_max_in_chain)
+apply (case_tac "\<Squnion>i. Y i", simp_all)
+done
+
+instance "+" :: (finite_po, finite_po) finite_po ..
+
+instance "+" :: (discrete_cpo, discrete_cpo) discrete_cpo
+by intro_classes (simp add: less_sum_def split: sum.split)
+
+subsection {* Sum type is a bifinite domain *}
+
+instantiation "+" :: (profinite, profinite) profinite
+begin
+
+definition
+  approx_sum_def: "approx =
+    (\<lambda>n. \<Lambda> x. case x of Inl a \<Rightarrow> Inl (approx n\<cdot>a) | Inr b \<Rightarrow> Inr (approx n\<cdot>b))"
+
+lemma approx_Inl [simp]: "approx n\<cdot>(Inl x) = Inl (approx n\<cdot>x)"
+  unfolding approx_sum_def by simp
+
+lemma approx_Inr [simp]: "approx n\<cdot>(Inr x) = Inr (approx n\<cdot>x)"
+  unfolding approx_sum_def by simp
+
+instance proof
+  fix i :: nat and x :: "'a + 'b"
+  show "chain (approx :: nat \<Rightarrow> 'a + 'b \<rightarrow> 'a + 'b)"
+    unfolding approx_sum_def
+    by (rule ch2ch_LAM, case_tac x, simp_all)
+  show "(\<Squnion>i. approx i\<cdot>x) = x"
+    by (induct x, simp_all add: lub_Inl lub_Inr)
+  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
+    by (induct x, simp_all)
+  have "{x::'a + 'b. approx i\<cdot>x = x} \<subseteq>
+        {x::'a. approx i\<cdot>x = x} <+> {x::'b. approx i\<cdot>x = x}"
+    by (rule subsetI, case_tac x, simp_all add: InlI InrI)
+  thus "finite {x::'a + 'b. approx i\<cdot>x = x}"
+    by (rule finite_subset,
+        intro finite_Plus finite_fixes_approx)
+qed
+
+end
+
+end
diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOLCF/Up.thy
--- a/src/HOLCF/Up.thy	Sat Jan 17 22:07:29 2009 +0100
+++ b/src/HOLCF/Up.thy	Sat Jan 17 22:08:14 2009 +0100
@@ -282,10 +282,10 @@
 text {* properties of fup *}
 
 lemma fup1 [simp]: "fup\<cdot>f\<cdot>\<bottom> = \<bottom>"
-by (simp add: fup_def cont_Ifup1 cont_Ifup2 inst_up_pcpo)
+by (simp add: fup_def cont_Ifup1 cont_Ifup2 inst_up_pcpo cont2cont_LAM)
 
 lemma fup2 [simp]: "fup\<cdot>f\<cdot>(up\<cdot>x) = f\<cdot>x"
-by (simp add: up_def fup_def cont_Iup cont_Ifup1 cont_Ifup2)
+by (simp add: up_def fup_def cont_Iup cont_Ifup1 cont_Ifup2 cont2cont_LAM)
 
 lemma fup3 [simp]: "fup\<cdot>up\<cdot>x = x"
 by (cases x, simp_all)
diff -r 4be5e49c74b1 -r aa8a1ed95a57 src/HOLCF/ex/Stream.thy
--- a/src/HOLCF/ex/Stream.thy	Sat Jan 17 22:07:29 2009 +0100
+++ b/src/HOLCF/ex/Stream.thy	Sat Jan 17 22:08:14 2009 +0100
@@ -521,7 +521,7 @@
 section "smap"
 
 lemma smap_unfold: "smap = (\<Lambda> f t. case t of x&&xs \<Rightarrow> f$x && smap$f$xs)"
-by (insert smap_def [THEN eq_reflection, THEN fix_eq2], auto)
+by (insert smap_def [where 'a='a and 'b='b, THEN eq_reflection, THEN fix_eq2], auto)
 
 lemma smap_empty [simp]: "smap\<cdot>f\<cdot>\<bottom> = \<bottom>"
 by (subst smap_unfold, simp)
@@ -540,7 +540,7 @@
 lemma sfilter_unfold:
  "sfilter = (\<Lambda> p s. case s of x && xs \<Rightarrow>
   If p\<cdot>x then x && sfilter\<cdot>p\<cdot>xs else sfilter\<cdot>p\<cdot>xs fi)"
-by (insert sfilter_def [THEN eq_reflection, THEN fix_eq2], auto)
+by (insert sfilter_def [where 'a='a, THEN eq_reflection, THEN fix_eq2], auto)
 
 lemma strict_sfilter: "sfilter\<cdot>\<bottom> = \<bottom>"
 apply (rule ext_cfun)