# HG changeset patch
# User haftmann
# Date 1274086711 -7200
# Node ID 5fb251d1c32fd3198690a5aa20b170820cb3bb73
# Parent  ef698bd61057c3ee67607b32ae8267fcc53ae1b6
dropped old Library/Word.thy and toy example ex/Adder.thy

diff -r ef698bd61057 -r 5fb251d1c32f src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Sun May 16 00:02:11 2010 +0200
+++ b/src/HOL/IsaMakefile	Mon May 17 10:58:31 2010 +0200
@@ -410,7 +410,7 @@
   Library/Library.thy Library/List_Prefix.thy Library/List_Set.thy	\
   Library/State_Monad.thy Library/Multiset.thy Library/Permutation.thy	\
   Library/Quotient_Type.thy Library/Quicksort.thy			\
-  Library/Nat_Infinity.thy Library/Word.thy Library/README.html		\
+  Library/Nat_Infinity.thy Library/README.html				\
   Library/Continuity.thy Library/Order_Relation.thy			\
   Library/Nested_Environment.thy Library/Ramsey.thy Library/Zorn.thy	\
   Library/Library/ROOT.ML Library/Library/document/root.tex		\
diff -r ef698bd61057 -r 5fb251d1c32f src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Sun May 16 00:02:11 2010 +0200
+++ b/src/HOL/Library/Library.thy	Mon May 17 10:58:31 2010 +0200
@@ -61,7 +61,6 @@
   Transitive_Closure_Table
   Univ_Poly
   While_Combinator
-  Word
   Zorn
 begin
 end
diff -r ef698bd61057 -r 5fb251d1c32f src/HOL/Library/Word.thy
--- a/src/HOL/Library/Word.thy	Sun May 16 00:02:11 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,2314 +0,0 @@
-(*  Title:      HOL/Library/Word.thy
-    Author:     Sebastian Skalberg, TU Muenchen
-*)
-
-header {* Binary Words *}
-
-theory Word
-imports Main
-begin
-
-subsection {* Auxilary Lemmas *}
-
-lemma max_le [intro!]: "[| x \<le> z; y \<le> z |] ==> max x y \<le> z"
-  by (simp add: max_def)
-
-lemma max_mono:
-  fixes x :: "'a::linorder"
-  assumes mf: "mono f"
-  shows       "max (f x) (f y) \<le> f (max x y)"
-proof -
-  from mf and le_maxI1 [of x y]
-  have fx: "f x \<le> f (max x y)" by (rule monoD)
-  from mf and le_maxI2 [of y x]
-  have fy: "f y \<le> f (max x y)" by (rule monoD)
-  from fx and fy
-  show "max (f x) (f y) \<le> f (max x y)" by auto
-qed
-
-declare zero_le_power [intro]
-  and zero_less_power [intro]
-
-lemma int_nat_two_exp: "2 ^ k = int (2 ^ k)"
-  by (simp add: zpower_int [symmetric])
-
-
-subsection {* Bits *}
-
-datatype bit =
-    Zero ("\<zero>")
-  | One ("\<one>")
-
-primrec bitval :: "bit => nat" where
-    "bitval \<zero> = 0"
-  | "bitval \<one> = 1"
-
-primrec bitnot :: "bit => bit" where
-    bitnot_zero: "(bitnot \<zero>) = \<one>"
-  | bitnot_one : "(bitnot \<one>)  = \<zero>"
-
-primrec bitand :: "bit => bit => bit" (infixr "bitand" 35) where
-    bitand_zero: "(\<zero> bitand y) = \<zero>"
-  | bitand_one:  "(\<one> bitand y) = y"
-
-primrec bitor  :: "bit => bit => bit" (infixr "bitor"  30) where
-    bitor_zero: "(\<zero> bitor y) = y"
-  | bitor_one:  "(\<one> bitor y) = \<one>"
-
-primrec bitxor :: "bit => bit => bit" (infixr "bitxor" 30) where
-    bitxor_zero: "(\<zero> bitxor y) = y"
-  | bitxor_one:  "(\<one> bitxor y) = (bitnot y)"
-
-notation (xsymbols)
-  bitnot ("\<not>\<^sub>b _" [40] 40) and
-  bitand (infixr "\<and>\<^sub>b" 35) and
-  bitor  (infixr "\<or>\<^sub>b" 30) and
-  bitxor (infixr "\<oplus>\<^sub>b" 30)
-
-notation (HTML output)
-  bitnot ("\<not>\<^sub>b _" [40] 40) and
-  bitand (infixr "\<and>\<^sub>b" 35) and
-  bitor  (infixr "\<or>\<^sub>b" 30) and
-  bitxor (infixr "\<oplus>\<^sub>b" 30)
-
-lemma bitnot_bitnot [simp]: "(bitnot (bitnot b)) = b"
-  by (cases b) simp_all
-
-lemma bitand_cancel [simp]: "(b bitand b) = b"
-  by (cases b) simp_all
-
-lemma bitor_cancel [simp]: "(b bitor b) = b"
-  by (cases b) simp_all
-
-lemma bitxor_cancel [simp]: "(b bitxor b) = \<zero>"
-  by (cases b) simp_all
-
-
-subsection {* Bit Vectors *}
-
-text {* First, a couple of theorems expressing case analysis and
-induction principles for bit vectors. *}
-
-lemma bit_list_cases:
-  assumes empty: "w = [] ==> P w"
-  and     zero:  "!!bs. w = \<zero> # bs ==> P w"
-  and     one:   "!!bs. w = \<one> # bs ==> P w"
-  shows   "P w"
-proof (cases w)
-  assume "w = []"
-  thus ?thesis by (rule empty)
-next
-  fix b bs
-  assume [simp]: "w = b # bs"
-  show "P w"
-  proof (cases b)
-    assume "b = \<zero>"
-    hence "w = \<zero> # bs" by simp
-    thus ?thesis by (rule zero)
-  next
-    assume "b = \<one>"
-    hence "w = \<one> # bs" by simp
-    thus ?thesis by (rule one)
-  qed
-qed
-
-lemma bit_list_induct:
-  assumes empty: "P []"
-  and     zero:  "!!bs. P bs ==> P (\<zero>#bs)"
-  and     one:   "!!bs. P bs ==> P (\<one>#bs)"
-  shows   "P w"
-proof (induct w, simp_all add: empty)
-  fix b bs
-  assume "P bs"
-  then show "P (b#bs)"
-    by (cases b) (auto intro!: zero one)
-qed
-
-definition
-  bv_msb :: "bit list => bit" where
-  "bv_msb w = (if w = [] then \<zero> else hd w)"
-
-definition
-  bv_extend :: "[nat,bit,bit list]=>bit list" where
-  "bv_extend i b w = (replicate (i - length w) b) @ w"
-
-definition
-  bv_not :: "bit list => bit list" where
-  "bv_not w = map bitnot w"
-
-lemma bv_length_extend [simp]: "length w \<le> i ==> length (bv_extend i b w) = i"
-  by (simp add: bv_extend_def)
-
-lemma bv_not_Nil [simp]: "bv_not [] = []"
-  by (simp add: bv_not_def)
-
-lemma bv_not_Cons [simp]: "bv_not (b#bs) = (bitnot b) # bv_not bs"
-  by (simp add: bv_not_def)
-
-lemma bv_not_bv_not [simp]: "bv_not (bv_not w) = w"
-  by (rule bit_list_induct [of _ w]) simp_all
-
-lemma bv_msb_Nil [simp]: "bv_msb [] = \<zero>"
-  by (simp add: bv_msb_def)
-
-lemma bv_msb_Cons [simp]: "bv_msb (b#bs) = b"
-  by (simp add: bv_msb_def)
-
-lemma bv_msb_bv_not [simp]: "0 < length w ==> bv_msb (bv_not w) = (bitnot (bv_msb w))"
-  by (cases w) simp_all
-
-lemma bv_msb_one_length [simp,intro]: "bv_msb w = \<one> ==> 0 < length w"
-  by (cases w) simp_all
-
-lemma length_bv_not [simp]: "length (bv_not w) = length w"
-  by (induct w) simp_all
-
-definition
-  bv_to_nat :: "bit list => nat" where
-  "bv_to_nat = foldl (%bn b. 2 * bn + bitval b) 0"
-
-lemma bv_to_nat_Nil [simp]: "bv_to_nat [] = 0"
-  by (simp add: bv_to_nat_def)
-
-lemma bv_to_nat_helper [simp]: "bv_to_nat (b # bs) = bitval b * 2 ^ length bs + bv_to_nat bs"
-proof -
-  let ?bv_to_nat' = "foldl (\<lambda>bn b. 2 * bn + bitval b)"
-  have helper: "\<And>base. ?bv_to_nat' base bs = base * 2 ^ length bs + ?bv_to_nat' 0 bs"
-  proof (induct bs)
-    case Nil
-    show ?case by simp
-  next
-    case (Cons x xs base)
-    show ?case
-      apply (simp only: foldl.simps)
-      apply (subst Cons [of "2 * base + bitval x"])
-      apply simp
-      apply (subst Cons [of "bitval x"])
-      apply (simp add: add_mult_distrib)
-      done
-  qed
-  show ?thesis by (simp add: bv_to_nat_def) (rule helper)
-qed
-
-lemma bv_to_nat0 [simp]: "bv_to_nat (\<zero>#bs) = bv_to_nat bs"
-  by simp
-
-lemma bv_to_nat1 [simp]: "bv_to_nat (\<one>#bs) = 2 ^ length bs + bv_to_nat bs"
-  by simp
-
-lemma bv_to_nat_upper_range: "bv_to_nat w < 2 ^ length w"
-proof (induct w, simp_all)
-  fix b bs
-  assume "bv_to_nat bs < 2 ^ length bs"
-  show "bitval b * 2 ^ length bs + bv_to_nat bs < 2 * 2 ^ length bs"
-  proof (cases b, simp_all)
-    have "bv_to_nat bs < 2 ^ length bs" by fact
-    also have "... < 2 * 2 ^ length bs" by auto
-    finally show "bv_to_nat bs < 2 * 2 ^ length bs" by simp
-  next
-    have "bv_to_nat bs < 2 ^ length bs" by fact
-    hence "2 ^ length bs + bv_to_nat bs < 2 ^ length bs + 2 ^ length bs" by arith
-    also have "... = 2 * (2 ^ length bs)" by simp
-    finally show "bv_to_nat bs < 2 ^ length bs" by simp
-  qed
-qed
-
-lemma bv_extend_longer [simp]:
-  assumes wn: "n \<le> length w"
-  shows       "bv_extend n b w = w"
-  by (simp add: bv_extend_def wn)
-
-lemma bv_extend_shorter [simp]:
-  assumes wn: "length w < n"
-  shows       "bv_extend n b w = bv_extend n b (b#w)"
-proof -
-  from wn
-  have s: "n - Suc (length w) + 1 = n - length w"
-    by arith
-  have "bv_extend n b w = replicate (n - length w) b @ w"
-    by (simp add: bv_extend_def)
-  also have "... = replicate (n - Suc (length w) + 1) b @ w"
-    by (subst s) rule
-  also have "... = (replicate (n - Suc (length w)) b @ replicate 1 b) @ w"
-    by (subst replicate_add) rule
-  also have "... = replicate (n - Suc (length w)) b @ b # w"
-    by simp
-  also have "... = bv_extend n b (b#w)"
-    by (simp add: bv_extend_def)
-  finally show "bv_extend n b w = bv_extend n b (b#w)" .
-qed
-
-primrec rem_initial :: "bit => bit list => bit list" where
-    "rem_initial b [] = []"
-  | "rem_initial b (x#xs) = (if b = x then rem_initial b xs else x#xs)"
-
-lemma rem_initial_length: "length (rem_initial b w) \<le> length w"
-  by (rule bit_list_induct [of _ w],simp_all (no_asm),safe,simp_all)
-
-lemma rem_initial_equal:
-  assumes p: "length (rem_initial b w) = length w"
-  shows      "rem_initial b w = w"
-proof -
-  have "length (rem_initial b w) = length w --> rem_initial b w = w"
-  proof (induct w, simp_all, clarify)
-    fix xs
-    assume "length (rem_initial b xs) = length xs --> rem_initial b xs = xs"
-    assume f: "length (rem_initial b xs) = Suc (length xs)"
-    with rem_initial_length [of b xs]
-    show "rem_initial b xs = b#xs"
-      by auto
-  qed
-  from this and p show ?thesis ..
-qed
-
-lemma bv_extend_rem_initial: "bv_extend (length w) b (rem_initial b w) = w"
-proof (induct w, simp_all, safe)
-  fix xs
-  assume ind: "bv_extend (length xs) b (rem_initial b xs) = xs"
-  from rem_initial_length [of b xs]
-  have [simp]: "Suc (length xs) - length (rem_initial b xs) =
-      1 + (length xs - length (rem_initial b xs))"
-    by arith
-  have "bv_extend (Suc (length xs)) b (rem_initial b xs) =
-      replicate (Suc (length xs) - length (rem_initial b xs)) b @ (rem_initial b xs)"
-    by (simp add: bv_extend_def)
-  also have "... =
-      replicate (1 + (length xs - length (rem_initial b xs))) b @ rem_initial b xs"
-    by simp
-  also have "... =
-      (replicate 1 b @ replicate (length xs - length (rem_initial b xs)) b) @ rem_initial b xs"
-    by (subst replicate_add) (rule refl)
-  also have "... = b # bv_extend (length xs) b (rem_initial b xs)"
-    by (auto simp add: bv_extend_def [symmetric])
-  also have "... = b # xs"
-    by (simp add: ind)
-  finally show "bv_extend (Suc (length xs)) b (rem_initial b xs) = b # xs" .
-qed
-
-lemma rem_initial_append1:
-  assumes "rem_initial b xs ~= []"
-  shows   "rem_initial b (xs @ ys) = rem_initial b xs @ ys"
-  using assms by (induct xs) auto
-
-lemma rem_initial_append2:
-  assumes "rem_initial b xs = []"
-  shows   "rem_initial b (xs @ ys) = rem_initial b ys"
-  using assms by (induct xs) auto
-
-definition
-  norm_unsigned :: "bit list => bit list" where
-  "norm_unsigned = rem_initial \<zero>"
-
-lemma norm_unsigned_Nil [simp]: "norm_unsigned [] = []"
-  by (simp add: norm_unsigned_def)
-
-lemma norm_unsigned_Cons0 [simp]: "norm_unsigned (\<zero>#bs) = norm_unsigned bs"
-  by (simp add: norm_unsigned_def)
-
-lemma norm_unsigned_Cons1 [simp]: "norm_unsigned (\<one>#bs) = \<one>#bs"
-  by (simp add: norm_unsigned_def)
-
-lemma norm_unsigned_idem [simp]: "norm_unsigned (norm_unsigned w) = norm_unsigned w"
-  by (rule bit_list_induct [of _ w],simp_all)
-
-fun
-  nat_to_bv_helper :: "nat => bit list => bit list"
-where
-  "nat_to_bv_helper n bs = (if n = 0 then bs
-                               else nat_to_bv_helper (n div 2) ((if n mod 2 = 0 then \<zero> else \<one>)#bs))"
-
-definition
-  nat_to_bv :: "nat => bit list" where
-  "nat_to_bv n = nat_to_bv_helper n []"
-
-lemma nat_to_bv0 [simp]: "nat_to_bv 0 = []"
-  by (simp add: nat_to_bv_def)
-
-lemmas [simp del] = nat_to_bv_helper.simps
-
-lemma n_div_2_cases:
-  assumes zero: "(n::nat) = 0 ==> R"
-  and     div : "[| n div 2 < n ; 0 < n |] ==> R"
-  shows         "R"
-proof (cases "n = 0")
-  assume "n = 0"
-  thus R by (rule zero)
-next
-  assume "n ~= 0"
-  hence "0 < n" by simp
-  hence "n div 2 < n" by arith
-  from this and `0 < n` show R by (rule div)
-qed
-
-lemma int_wf_ge_induct:
-  assumes ind :  "!!i::int. (!!j. [| k \<le> j ; j < i |] ==> P j) ==> P i"
-  shows          "P i"
-proof (rule wf_induct_rule [OF wf_int_ge_less_than])
-  fix x
-  assume ih: "(\<And>y\<Colon>int. (y, x) \<in> int_ge_less_than k \<Longrightarrow> P y)"
-  thus "P x"
-    by (rule ind) (simp add: int_ge_less_than_def)
-qed
-
-lemma unfold_nat_to_bv_helper:
-  "nat_to_bv_helper b l = nat_to_bv_helper b [] @ l"
-proof -
-  have "\<forall>l. nat_to_bv_helper b l = nat_to_bv_helper b [] @ l"
-  proof (induct b rule: less_induct)
-    fix n
-    assume ind: "!!j. j < n \<Longrightarrow> \<forall> l. nat_to_bv_helper j l = nat_to_bv_helper j [] @ l"
-    show "\<forall>l. nat_to_bv_helper n l = nat_to_bv_helper n [] @ l"
-    proof
-      fix l
-      show "nat_to_bv_helper n l = nat_to_bv_helper n [] @ l"
-      proof (cases "n < 0")
-        assume "n < 0"
-        thus ?thesis
-          by (simp add: nat_to_bv_helper.simps)
-      next
-        assume "~n < 0"
-        show ?thesis
-        proof (rule n_div_2_cases [of n])
-          assume [simp]: "n = 0"
-          show ?thesis
-            apply (simp only: nat_to_bv_helper.simps [of n])
-            apply simp
-            done
-        next
-          assume n2n: "n div 2 < n"
-          assume [simp]: "0 < n"
-          hence n20: "0 \<le> n div 2"
-            by arith
-          from ind [of "n div 2"] and n2n n20
-          have ind': "\<forall>l. nat_to_bv_helper (n div 2) l = nat_to_bv_helper (n div 2) [] @ l"
-            by blast
-          show ?thesis
-            apply (simp only: nat_to_bv_helper.simps [of n])
-            apply (cases "n=0")
-            apply simp
-            apply (simp only: if_False)
-            apply simp
-            apply (subst spec [OF ind',of "\<zero>#l"])
-            apply (subst spec [OF ind',of "\<one>#l"])
-            apply (subst spec [OF ind',of "[\<one>]"])
-            apply (subst spec [OF ind',of "[\<zero>]"])
-            apply simp
-            done
-        qed
-      qed
-    qed
-  qed
-  thus ?thesis ..
-qed
-
-lemma nat_to_bv_non0 [simp]: "n\<noteq>0 ==> nat_to_bv n = nat_to_bv (n div 2) @ [if n mod 2 = 0 then \<zero> else \<one>]"
-proof -
-  assume [simp]: "n\<noteq>0"
-  show ?thesis
-    apply (subst nat_to_bv_def [of n])
-    apply (simp only: nat_to_bv_helper.simps [of n])
-    apply (subst unfold_nat_to_bv_helper)
-    using prems
-    apply (simp)
-    apply (subst nat_to_bv_def [of "n div 2"])
-    apply auto
-    done
-qed
-
-lemma bv_to_nat_dist_append:
-  "bv_to_nat (l1 @ l2) = bv_to_nat l1 * 2 ^ length l2 + bv_to_nat l2"
-proof -
-  have "\<forall>l2. bv_to_nat (l1 @ l2) = bv_to_nat l1 * 2 ^ length l2 + bv_to_nat l2"
-  proof (induct l1, simp_all)
-    fix x xs
-    assume ind: "\<forall>l2. bv_to_nat (xs @ l2) = bv_to_nat xs * 2 ^ length l2 + bv_to_nat l2"
-    show "\<forall>l2::bit list. 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
-      fix l2
-      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
-        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
-        also have "... = (bitval x * 2 ^ length xs + bv_to_nat xs) * 2 ^ length l2"
-          by (simp add: ring_distribs)
-        finally show ?thesis by simp
-      qed
-    qed
-  qed
-  thus ?thesis ..
-qed
-
-lemma bv_nat_bv [simp]: "bv_to_nat (nat_to_bv n) = n"
-proof (induct n rule: less_induct)
-  fix n
-  assume ind: "!!j. j < n \<Longrightarrow> bv_to_nat (nat_to_bv j) = j"
-  show "bv_to_nat (nat_to_bv n) = n"
-  proof (rule n_div_2_cases [of n])
-    assume "n = 0" then show ?thesis by simp
-  next
-    assume nn: "n div 2 < n"
-    assume n0: "0 < n"
-    from ind and nn
-    have ind': "bv_to_nat (nat_to_bv (n div 2)) = n div 2" by blast
-    from n0 have n0': "n \<noteq> 0" by simp
-    show ?thesis
-      apply (subst nat_to_bv_def)
-      apply (simp only: nat_to_bv_helper.simps [of n])
-      apply (simp only: n0' if_False)
-      apply (subst unfold_nat_to_bv_helper)
-      apply (subst bv_to_nat_dist_append)
-      apply (fold nat_to_bv_def)
-      apply (simp add: ind' split del: split_if)
-      apply (cases "n mod 2 = 0")
-      proof (simp_all)
-        assume "n mod 2 = 0"
-        with mod_div_equality [of n 2]
-        show "n div 2 * 2 = n" by simp
-      next
-        assume "n mod 2 = Suc 0"
-        with mod_div_equality [of n 2]
-        show "Suc (n div 2 * 2) = n" by arith
-      qed
-  qed
-qed
-
-lemma bv_to_nat_type [simp]: "bv_to_nat (norm_unsigned w) = bv_to_nat w"
-  by (rule bit_list_induct) simp_all
-
-lemma length_norm_unsigned_le [simp]: "length (norm_unsigned w) \<le> length w"
-  by (rule bit_list_induct) simp_all
-
-lemma bv_to_nat_rew_msb: "bv_msb w = \<one> ==> bv_to_nat w = 2 ^ (length w - 1) + bv_to_nat (tl w)"
-  by (rule bit_list_cases [of w]) simp_all
-
-lemma norm_unsigned_result: "norm_unsigned xs = [] \<or> bv_msb (norm_unsigned xs) = \<one>"
-proof (rule length_induct [of _ xs])
-  fix xs :: "bit list"
-  assume ind: "\<forall>ys. length ys < length xs --> norm_unsigned ys = [] \<or> bv_msb (norm_unsigned ys) = \<one>"
-  show "norm_unsigned xs = [] \<or> bv_msb (norm_unsigned xs) = \<one>"
-  proof (rule bit_list_cases [of xs],simp_all)
-    fix bs
-    assume [simp]: "xs = \<zero>#bs"
-    from ind
-    have "length bs < length xs --> norm_unsigned bs = [] \<or> bv_msb (norm_unsigned bs) = \<one>" ..
-    thus "norm_unsigned bs = [] \<or> bv_msb (norm_unsigned bs) = \<one>" by simp
-  qed
-qed
-
-lemma norm_empty_bv_to_nat_zero:
-  assumes nw: "norm_unsigned w = []"
-  shows       "bv_to_nat w = 0"
-proof -
-  have "bv_to_nat w = bv_to_nat (norm_unsigned w)" by simp
-  also have "... = bv_to_nat []" by (subst nw) (rule refl)
-  also have "... = 0" by simp
-  finally show ?thesis .
-qed
-
-lemma bv_to_nat_lower_limit:
-  assumes w0: "0 < bv_to_nat w"
-  shows "2 ^ (length (norm_unsigned w) - 1) \<le> bv_to_nat w"
-proof -
-  from w0 and norm_unsigned_result [of w]
-  have msbw: "bv_msb (norm_unsigned w) = \<one>"
-    by (auto simp add: norm_empty_bv_to_nat_zero)
-  have "2 ^ (length (norm_unsigned w) - 1) \<le> bv_to_nat (norm_unsigned w)"
-    by (subst bv_to_nat_rew_msb [OF msbw],simp)
-  thus ?thesis by simp
-qed
-
-lemmas [simp del] = nat_to_bv_non0
-
-lemma norm_unsigned_length [intro!]: "length (norm_unsigned w) \<le> length w"
-by (subst norm_unsigned_def,rule rem_initial_length)
-
-lemma norm_unsigned_equal:
-  "length (norm_unsigned w) = length w ==> norm_unsigned w = w"
-by (simp add: norm_unsigned_def,rule rem_initial_equal)
-
-lemma bv_extend_norm_unsigned: "bv_extend (length w) \<zero> (norm_unsigned w) = w"
-by (simp add: norm_unsigned_def,rule bv_extend_rem_initial)
-
-lemma norm_unsigned_append1 [simp]:
-  "norm_unsigned xs \<noteq> [] ==> norm_unsigned (xs @ ys) = norm_unsigned xs @ ys"
-by (simp add: norm_unsigned_def,rule rem_initial_append1)
-
-lemma norm_unsigned_append2 [simp]:
-  "norm_unsigned xs = [] ==> norm_unsigned (xs @ ys) = norm_unsigned ys"
-by (simp add: norm_unsigned_def,rule rem_initial_append2)
-
-lemma bv_to_nat_zero_imp_empty:
-  "bv_to_nat w = 0 \<Longrightarrow> norm_unsigned w = []"
-by (atomize (full), induct w rule: bit_list_induct) simp_all
-
-lemma bv_to_nat_nzero_imp_nempty:
-  "bv_to_nat w \<noteq> 0 \<Longrightarrow> norm_unsigned w \<noteq> []"
-by (induct w rule: bit_list_induct) simp_all
-
-lemma nat_helper1:
-  assumes ass: "nat_to_bv (bv_to_nat w) = norm_unsigned w"
-  shows        "nat_to_bv (2 * bv_to_nat w + bitval x) = norm_unsigned (w @ [x])"
-proof (cases x)
-  assume [simp]: "x = \<one>"
-  have "nat_to_bv (Suc (2 * bv_to_nat w) div 2) @ [\<one>] =
-      nat_to_bv ((1 + 2 * bv_to_nat w) div 2) @ [\<one>]"
-    by (simp add: add_commute)
-  also have "... = nat_to_bv (bv_to_nat w) @ [\<one>]"
-    by (subst div_add1_eq) simp
-  also have "... = norm_unsigned w @ [\<one>]"
-    by (subst ass) (rule refl)
-  also have "... = norm_unsigned (w @ [\<one>])"
-    by (cases "norm_unsigned w") simp_all
-  finally have "nat_to_bv (Suc (2 * bv_to_nat w) div 2) @ [\<one>] = norm_unsigned (w @ [\<one>])" .
-  then show ?thesis by (simp add: nat_to_bv_non0)
-next
-  assume [simp]: "x = \<zero>"
-  show ?thesis
-  proof (cases "bv_to_nat w = 0")
-    assume "bv_to_nat w = 0"
-    thus ?thesis
-      by (simp add: bv_to_nat_zero_imp_empty)
-  next
-    assume "bv_to_nat w \<noteq> 0"
-    thus ?thesis
-      apply simp
-      apply (subst nat_to_bv_non0)
-      apply simp
-      apply auto
-      apply (subst ass)
-      apply (cases "norm_unsigned w")
-      apply (simp_all add: norm_empty_bv_to_nat_zero)
-      done
-  qed
-qed
-
-lemma nat_helper2: "nat_to_bv (2 ^ length xs + bv_to_nat xs) = \<one> # xs"
-proof -
-  have "\<forall>xs. nat_to_bv (2 ^ length (rev xs) + bv_to_nat (rev xs)) = \<one> # (rev xs)" (is "\<forall>xs. ?P xs")
-  proof
-    fix xs
-    show "?P xs"
-    proof (rule length_induct [of _ xs])
-      fix xs :: "bit list"
-      assume ind: "\<forall>ys. length ys < length xs --> ?P ys"
-      show "?P xs"
-      proof (cases xs)
-        assume "xs = []"
-        then show ?thesis by (simp add: nat_to_bv_non0)
-      next
-        fix y ys
-        assume [simp]: "xs = y # ys"
-        show ?thesis
-          apply simp
-          apply (subst bv_to_nat_dist_append)
-          apply simp
-        proof -
-          have "nat_to_bv (2 * 2 ^ length ys + (bv_to_nat (rev ys) * 2 + bitval y)) =
-            nat_to_bv (2 * (2 ^ length ys + bv_to_nat (rev ys)) + bitval y)"
-            by (simp add: add_ac mult_ac)
-          also have "... = nat_to_bv (2 * (bv_to_nat (\<one>#rev ys)) + bitval y)"
-            by simp
-          also have "... = norm_unsigned (\<one>#rev ys) @ [y]"
-          proof -
-            from ind
-            have "nat_to_bv (2 ^ length (rev ys) + bv_to_nat (rev ys)) = \<one> # rev ys"
-              by auto
-            hence [simp]: "nat_to_bv (2 ^ length ys + bv_to_nat (rev ys)) = \<one> # rev ys"
-              by simp
-            show ?thesis
-              apply (subst nat_helper1)
-              apply simp_all
-              done
-          qed
-          also have "... = (\<one>#rev ys) @ [y]"
-            by simp
-          also have "... = \<one> # rev ys @ [y]"
-            by simp
-          finally show "nat_to_bv (2 * 2 ^ length ys + (bv_to_nat (rev ys) * 2 + bitval y)) =
-              \<one> # rev ys @ [y]" .
-        qed
-      qed
-    qed
-  qed
-  hence "nat_to_bv (2 ^ length (rev (rev xs)) + bv_to_nat (rev (rev xs))) =
-      \<one> # rev (rev xs)" ..
-  thus ?thesis by simp
-qed
-
-lemma nat_bv_nat [simp]: "nat_to_bv (bv_to_nat w) = norm_unsigned w"
-proof (rule bit_list_induct [of _ w],simp_all)
-  fix xs
-  assume "nat_to_bv (bv_to_nat xs) = norm_unsigned xs"
-  have "bv_to_nat xs = bv_to_nat (norm_unsigned xs)" by simp
-  have "bv_to_nat xs < 2 ^ length xs"
-    by (rule bv_to_nat_upper_range)
-  show "nat_to_bv (2 ^ length xs + bv_to_nat xs) = \<one> # xs"
-    by (rule nat_helper2)
-qed
-
-lemma bv_to_nat_qinj:
-  assumes one: "bv_to_nat xs = bv_to_nat ys"
-  and     len: "length xs = length ys"
-  shows        "xs = ys"
-proof -
-  from one
-  have "nat_to_bv (bv_to_nat xs) = nat_to_bv (bv_to_nat ys)"
-    by simp
-  hence xsys: "norm_unsigned xs = norm_unsigned ys"
-    by simp
-  have "xs = bv_extend (length xs) \<zero> (norm_unsigned xs)"
-    by (simp add: bv_extend_norm_unsigned)
-  also have "... = bv_extend (length ys) \<zero> (norm_unsigned ys)"
-    by (simp add: xsys len)
-  also have "... = ys"
-    by (simp add: bv_extend_norm_unsigned)
-  finally show ?thesis .
-qed
-
-lemma norm_unsigned_nat_to_bv [simp]:
-  "norm_unsigned (nat_to_bv n) = nat_to_bv n"
-proof -
-  have "norm_unsigned (nat_to_bv n) = nat_to_bv (bv_to_nat (norm_unsigned (nat_to_bv n)))"
-    by (subst nat_bv_nat) simp
-  also have "... = nat_to_bv n" by simp
-  finally show ?thesis .
-qed
-
-lemma length_nat_to_bv_upper_limit:
-  assumes nk: "n \<le> 2 ^ k - 1"
-  shows       "length (nat_to_bv n) \<le> k"
-proof (cases "n = 0")
-  case True
-  thus ?thesis
-    by (simp add: nat_to_bv_def nat_to_bv_helper.simps)
-next
-  case False
-  hence n0: "0 < n" by simp
-  show ?thesis
-  proof (rule ccontr)
-    assume "~ length (nat_to_bv n) \<le> k"
-    hence "k < length (nat_to_bv n)" by simp
-    hence "k \<le> length (nat_to_bv n) - 1" by arith
-    hence "(2::nat) ^ k \<le> 2 ^ (length (nat_to_bv n) - 1)" by simp
-    also have "... = 2 ^ (length (norm_unsigned (nat_to_bv n)) - 1)" by simp
-    also have "... \<le> bv_to_nat (nat_to_bv n)"
-      by (rule bv_to_nat_lower_limit) (simp add: n0)
-    also have "... = n" by simp
-    finally have "2 ^ k \<le> n" .
-    with n0 have "2 ^ k - 1 < n" by arith
-    with nk show False by simp
-  qed
-qed
-
-lemma length_nat_to_bv_lower_limit:
-  assumes nk: "2 ^ k \<le> n"
-  shows       "k < length (nat_to_bv n)"
-proof (rule ccontr)
-  assume "~ k < length (nat_to_bv n)"
-  hence lnk: "length (nat_to_bv n) \<le> k" by simp
-  have "n = bv_to_nat (nat_to_bv n)" by simp
-  also have "... < 2 ^ length (nat_to_bv n)"
-    by (rule bv_to_nat_upper_range)
-  also from lnk have "... \<le> 2 ^ k" by simp
-  finally have "n < 2 ^ k" .
-  with nk show False by simp
-qed
-
-
-subsection {* Unsigned Arithmetic Operations *}
-
-definition
-  bv_add :: "[bit list, bit list ] => bit list" where
-  "bv_add w1 w2 = nat_to_bv (bv_to_nat w1 + bv_to_nat w2)"
-
-lemma bv_add_type1 [simp]: "bv_add (norm_unsigned w1) w2 = bv_add w1 w2"
-  by (simp add: bv_add_def)
-
-lemma bv_add_type2 [simp]: "bv_add w1 (norm_unsigned w2) = bv_add w1 w2"
-  by (simp add: bv_add_def)
-
-lemma bv_add_returntype [simp]: "norm_unsigned (bv_add w1 w2) = bv_add w1 w2"
-  by (simp add: bv_add_def)
-
-lemma bv_add_length: "length (bv_add w1 w2) \<le> Suc (max (length w1) (length w2))"
-proof (unfold bv_add_def,rule length_nat_to_bv_upper_limit)
-  from bv_to_nat_upper_range [of w1] and bv_to_nat_upper_range [of w2]
-  have "bv_to_nat w1 + bv_to_nat w2 \<le> (2 ^ length w1 - 1) + (2 ^ length w2 - 1)"
-    by arith
-  also have "... \<le>
-      max (2 ^ length w1 - 1) (2 ^ length w2 - 1) + max (2 ^ length w1 - 1) (2 ^ length w2 - 1)"
-    by (rule add_mono,safe intro!: le_maxI1 le_maxI2)
-  also have "... = 2 * max (2 ^ length w1 - 1) (2 ^ length w2 - 1)" by simp
-  also have "... \<le> 2 ^ Suc (max (length w1) (length w2)) - 2"
-  proof (cases "length w1 \<le> length w2")
-    assume w1w2: "length w1 \<le> length w2"
-    hence "(2::nat) ^ length w1 \<le> 2 ^ length w2" by simp
-    hence "(2::nat) ^ length w1 - 1 \<le> 2 ^ length w2 - 1" by arith
-    with w1w2 show ?thesis
-      by (simp add: diff_mult_distrib2 split: split_max)
-  next
-    assume [simp]: "~ (length w1 \<le> length w2)"
-    have "~ ((2::nat) ^ length w1 - 1 \<le> 2 ^ length w2 - 1)"
-    proof
-      assume "(2::nat) ^ length w1 - 1 \<le> 2 ^ length w2 - 1"
-      hence "((2::nat) ^ length w1 - 1) + 1 \<le> (2 ^ length w2 - 1) + 1"
-        by (rule add_right_mono)
-      hence "(2::nat) ^ length w1 \<le> 2 ^ length w2" by simp
-      hence "length w1 \<le> length w2" by simp
-      thus False by simp
-    qed
-    thus ?thesis
-      by (simp add: diff_mult_distrib2 split: split_max)
-  qed
-  finally show "bv_to_nat w1 + bv_to_nat w2 \<le> 2 ^ Suc (max (length w1) (length w2)) - 1"
-    by arith
-qed
-
-definition
-  bv_mult :: "[bit list, bit list ] => bit list" where
-  "bv_mult w1 w2 = nat_to_bv (bv_to_nat w1 * bv_to_nat w2)"
-
-lemma bv_mult_type1 [simp]: "bv_mult (norm_unsigned w1) w2 = bv_mult w1 w2"
-  by (simp add: bv_mult_def)
-
-lemma bv_mult_type2 [simp]: "bv_mult w1 (norm_unsigned w2) = bv_mult w1 w2"
-  by (simp add: bv_mult_def)
-
-lemma bv_mult_returntype [simp]: "norm_unsigned (bv_mult w1 w2) = bv_mult w1 w2"
-  by (simp add: bv_mult_def)
-
-lemma bv_mult_length: "length (bv_mult w1 w2) \<le> length w1 + length w2"
-proof (unfold bv_mult_def,rule length_nat_to_bv_upper_limit)
-  from bv_to_nat_upper_range [of w1] and bv_to_nat_upper_range [of w2]
-  have h: "bv_to_nat w1 \<le> 2 ^ length w1 - 1 \<and> bv_to_nat w2 \<le> 2 ^ length w2 - 1"
-    by arith
-  have "bv_to_nat w1 * bv_to_nat w2 \<le> (2 ^ length w1 - 1) * (2 ^ length w2 - 1)"
-    apply (cut_tac h)
-    apply (rule mult_mono)
-    apply auto
-    done
-  also have "... < 2 ^ length w1 * 2 ^ length w2"
-    by (rule mult_strict_mono,auto)
-  also have "... = 2 ^ (length w1 + length w2)"
-    by (simp add: power_add)
-  finally show "bv_to_nat w1 * bv_to_nat w2 \<le> 2 ^ (length w1 + length w2) - 1"
-    by arith
-qed
-
-subsection {* Signed Vectors *}
-
-primrec norm_signed :: "bit list => bit list" where
-    norm_signed_Nil: "norm_signed [] = []"
-  | norm_signed_Cons: "norm_signed (b#bs) =
-      (case b of
-        \<zero> => if norm_unsigned bs = [] then [] else b#norm_unsigned bs
-      | \<one> => b#rem_initial b bs)"
-
-lemma norm_signed0 [simp]: "norm_signed [\<zero>] = []"
-  by simp
-
-lemma norm_signed1 [simp]: "norm_signed [\<one>] = [\<one>]"
-  by simp
-
-lemma norm_signed01 [simp]: "norm_signed (\<zero>#\<one>#xs) = \<zero>#\<one>#xs"
-  by simp
-
-lemma norm_signed00 [simp]: "norm_signed (\<zero>#\<zero>#xs) = norm_signed (\<zero>#xs)"
-  by simp
-
-lemma norm_signed10 [simp]: "norm_signed (\<one>#\<zero>#xs) = \<one>#\<zero>#xs"
-  by simp
-
-lemma norm_signed11 [simp]: "norm_signed (\<one>#\<one>#xs) = norm_signed (\<one>#xs)"
-  by simp
-
-lemmas [simp del] = norm_signed_Cons
-
-definition
-  int_to_bv :: "int => bit list" where
-  "int_to_bv n = (if 0 \<le> n
-                 then norm_signed (\<zero>#nat_to_bv (nat n))
-                 else norm_signed (bv_not (\<zero>#nat_to_bv (nat (-n- 1)))))"
-
-lemma int_to_bv_ge0 [simp]: "0 \<le> n ==> int_to_bv n = norm_signed (\<zero> # nat_to_bv (nat n))"
-  by (simp add: int_to_bv_def)
-
-lemma int_to_bv_lt0 [simp]:
-    "n < 0 ==> int_to_bv n = norm_signed (bv_not (\<zero>#nat_to_bv (nat (-n- 1))))"
-  by (simp add: int_to_bv_def)
-
-lemma norm_signed_idem [simp]: "norm_signed (norm_signed w) = norm_signed w"
-proof (rule bit_list_induct [of _ w], simp_all)
-  fix xs
-  assume eq: "norm_signed (norm_signed xs) = norm_signed xs"
-  show "norm_signed (norm_signed (\<zero>#xs)) = norm_signed (\<zero>#xs)"
-  proof (rule bit_list_cases [of xs],simp_all)
-    fix ys
-    assume "xs = \<zero>#ys"
-    from this [symmetric] and eq
-    show "norm_signed (norm_signed (\<zero>#ys)) = norm_signed (\<zero>#ys)"
-      by simp
-  qed
-next
-  fix xs
-  assume eq: "norm_signed (norm_signed xs) = norm_signed xs"
-  show "norm_signed (norm_signed (\<one>#xs)) = norm_signed (\<one>#xs)"
-  proof (rule bit_list_cases [of xs],simp_all)
-    fix ys
-    assume "xs = \<one>#ys"
-    from this [symmetric] and eq
-    show "norm_signed (norm_signed (\<one>#ys)) = norm_signed (\<one>#ys)"
-      by simp
-  qed
-qed
-
-definition
-  bv_to_int :: "bit list => int" where
-  "bv_to_int w =
-    (case bv_msb w of \<zero> => int (bv_to_nat w)
-    | \<one> => - int (bv_to_nat (bv_not w) + 1))"
-
-lemma bv_to_int_Nil [simp]: "bv_to_int [] = 0"
-  by (simp add: bv_to_int_def)
-
-lemma bv_to_int_Cons0 [simp]: "bv_to_int (\<zero>#bs) = int (bv_to_nat bs)"
-  by (simp add: bv_to_int_def)
-
-lemma bv_to_int_Cons1 [simp]: "bv_to_int (\<one>#bs) = - int (bv_to_nat (bv_not bs) + 1)"
-  by (simp add: bv_to_int_def)
-
-lemma bv_to_int_type [simp]: "bv_to_int (norm_signed w) = bv_to_int w"
-proof (rule bit_list_induct [of _ w], simp_all)
-  fix xs
-  assume ind: "bv_to_int (norm_signed xs) = bv_to_int xs"
-  show "bv_to_int (norm_signed (\<zero>#xs)) = int (bv_to_nat xs)"
-  proof (rule bit_list_cases [of xs], simp_all)
-    fix ys
-    assume [simp]: "xs = \<zero>#ys"
-    from ind
-    show "bv_to_int (norm_signed (\<zero>#ys)) = int (bv_to_nat ys)"
-      by simp
-  qed
-next
-  fix xs
-  assume ind: "bv_to_int (norm_signed xs) = bv_to_int xs"
-  show "bv_to_int (norm_signed (\<one>#xs)) = -1 - int (bv_to_nat (bv_not xs))"
-  proof (rule bit_list_cases [of xs], simp_all)
-    fix ys
-    assume [simp]: "xs = \<one>#ys"
-    from ind
-    show "bv_to_int (norm_signed (\<one>#ys)) = -1 - int (bv_to_nat (bv_not ys))"
-      by simp
-  qed
-qed
-
-lemma bv_to_int_upper_range: "bv_to_int w < 2 ^ (length w - 1)"
-proof (rule bit_list_cases [of w],simp_all)
-  fix bs
-  from bv_to_nat_upper_range
-  show "int (bv_to_nat bs) < 2 ^ length bs"
-    by (simp add: int_nat_two_exp)
-next
-  fix bs
-  have "-1 - int (bv_to_nat (bv_not bs)) \<le> 0" by simp
-  also have "... < 2 ^ length bs" by (induct bs) simp_all
-  finally show "-1 - int (bv_to_nat (bv_not bs)) < 2 ^ length bs" .
-qed
-
-lemma bv_to_int_lower_range: "- (2 ^ (length w - 1)) \<le> bv_to_int w"
-proof (rule bit_list_cases [of w],simp_all)
-  fix bs :: "bit list"
-  have "- (2 ^ length bs) \<le> (0::int)" by (induct bs) simp_all
-  also have "... \<le> int (bv_to_nat bs)" by simp
-  finally show "- (2 ^ length bs) \<le> int (bv_to_nat bs)" .
-next
-  fix bs
-  from bv_to_nat_upper_range [of "bv_not bs"]
-  show "- (2 ^ length bs) \<le> -1 - int (bv_to_nat (bv_not bs))"
-    by (simp add: int_nat_two_exp)
-qed
-
-lemma int_bv_int [simp]: "int_to_bv (bv_to_int w) = norm_signed w"
-proof (rule bit_list_cases [of w],simp)
-  fix xs
-  assume [simp]: "w = \<zero>#xs"
-  show ?thesis
-    apply simp
-    apply (subst norm_signed_Cons [of "\<zero>" "xs"])
-    apply simp
-    using norm_unsigned_result [of xs]
-    apply safe
-    apply (rule bit_list_cases [of "norm_unsigned xs"])
-    apply simp_all
-    done
-next
-  fix xs
-  assume [simp]: "w = \<one>#xs"
-  show ?thesis
-    apply (simp del: int_to_bv_lt0)
-    apply (rule bit_list_induct [of _ xs])
-    apply simp
-    apply (subst int_to_bv_lt0)
-    apply (subgoal_tac "- int (bv_to_nat (bv_not (\<zero> # bs))) + -1 < 0 + 0")
-    apply simp
-    apply (rule add_le_less_mono)
-    apply simp
-    apply simp
-    apply (simp del: bv_to_nat1 bv_to_nat_helper)
-    apply simp
-    done
-qed
-
-lemma bv_int_bv [simp]: "bv_to_int (int_to_bv i) = i"
-  by (cases "0 \<le> i") simp_all
-
-lemma bv_msb_norm [simp]: "bv_msb (norm_signed w) = bv_msb w"
-  by (rule bit_list_cases [of w]) (simp_all add: norm_signed_Cons)
-
-lemma norm_signed_length: "length (norm_signed w) \<le> length w"
-  apply (cases w, simp_all)
-  apply (subst norm_signed_Cons)
-  apply (case_tac a, simp_all)
-  apply (rule rem_initial_length)
-  done
-
-lemma norm_signed_equal: "length (norm_signed w) = length w ==> norm_signed w = w"
-proof (rule bit_list_cases [of w], simp_all)
-  fix xs
-  assume "length (norm_signed (\<zero>#xs)) = Suc (length xs)"
-  thus "norm_signed (\<zero>#xs) = \<zero>#xs"
-    by (simp add: norm_signed_Cons norm_unsigned_equal [THEN eqTrueI]
-             split: split_if_asm)
-next
-  fix xs
-  assume "length (norm_signed (\<one>#xs)) = Suc (length xs)"
-  thus "norm_signed (\<one>#xs) = \<one>#xs"
-    apply (simp add: norm_signed_Cons)
-    apply (rule rem_initial_equal)
-    apply assumption
-    done
-qed
-
-lemma bv_extend_norm_signed: "bv_msb w = b ==> bv_extend (length w) b (norm_signed w) = w"
-proof (rule bit_list_cases [of w],simp_all)
-  fix xs
-  show "bv_extend (Suc (length xs)) \<zero> (norm_signed (\<zero>#xs)) = \<zero>#xs"
-  proof (simp add: norm_signed_def,auto)
-    assume "norm_unsigned xs = []"
-    hence xx: "rem_initial \<zero> xs = []"
-      by (simp add: norm_unsigned_def)
-    have "bv_extend (Suc (length xs)) \<zero> (\<zero>#rem_initial \<zero> xs) = \<zero>#xs"
-      apply (simp add: bv_extend_def replicate_app_Cons_same)
-      apply (fold bv_extend_def)
-      apply (rule bv_extend_rem_initial)
-      done
-    thus "bv_extend (Suc (length xs)) \<zero> [\<zero>] = \<zero>#xs"
-      by (simp add: xx)
-  next
-    show "bv_extend (Suc (length xs)) \<zero> (\<zero>#norm_unsigned xs) = \<zero>#xs"
-      apply (simp add: norm_unsigned_def)
-      apply (simp add: bv_extend_def replicate_app_Cons_same)
-      apply (fold bv_extend_def)
-      apply (rule bv_extend_rem_initial)
-      done
-  qed
-next
-  fix xs
-  show "bv_extend (Suc (length xs)) \<one> (norm_signed (\<one>#xs)) = \<one>#xs"
-    apply (simp add: norm_signed_Cons)
-    apply (simp add: bv_extend_def replicate_app_Cons_same)
-    apply (fold bv_extend_def)
-    apply (rule bv_extend_rem_initial)
-    done
-qed
-
-lemma bv_to_int_qinj:
-  assumes one: "bv_to_int xs = bv_to_int ys"
-  and     len: "length xs = length ys"
-  shows        "xs = ys"
-proof -
-  from one
-  have "int_to_bv (bv_to_int xs) = int_to_bv (bv_to_int ys)" by simp
-  hence xsys: "norm_signed xs = norm_signed ys" by simp
-  hence xsys': "bv_msb xs = bv_msb ys"
-  proof -
-    have "bv_msb xs = bv_msb (norm_signed xs)" by simp
-    also have "... = bv_msb (norm_signed ys)" by (simp add: xsys)
-    also have "... = bv_msb ys" by simp
-    finally show ?thesis .
-  qed
-  have "xs = bv_extend (length xs) (bv_msb xs) (norm_signed xs)"
-    by (simp add: bv_extend_norm_signed)
-  also have "... = bv_extend (length ys) (bv_msb ys) (norm_signed ys)"
-    by (simp add: xsys xsys' len)
-  also have "... = ys"
-    by (simp add: bv_extend_norm_signed)
-  finally show ?thesis .
-qed
-
-lemma int_to_bv_returntype [simp]: "norm_signed (int_to_bv w) = int_to_bv w"
-  by (simp add: int_to_bv_def)
-
-lemma bv_to_int_msb0: "0 \<le> bv_to_int w1 ==> bv_msb w1 = \<zero>"
-  by (rule bit_list_cases,simp_all)
-
-lemma bv_to_int_msb1: "bv_to_int w1 < 0 ==> bv_msb w1 = \<one>"
-  by (rule bit_list_cases,simp_all)
-
-lemma bv_to_int_lower_limit_gt0:
-  assumes w0: "0 < bv_to_int w"
-  shows       "2 ^ (length (norm_signed w) - 2) \<le> bv_to_int w"
-proof -
-  from w0
-  have "0 \<le> bv_to_int w" by simp
-  hence [simp]: "bv_msb w = \<zero>" by (rule bv_to_int_msb0)
-  have "2 ^ (length (norm_signed w) - 2) \<le> bv_to_int (norm_signed w)"
-  proof (rule bit_list_cases [of w])
-    assume "w = []"
-    with w0 show ?thesis by simp
-  next
-    fix w'
-    assume weq: "w = \<zero> # w'"
-    thus ?thesis
-    proof (simp add: norm_signed_Cons,safe)
-      assume "norm_unsigned w' = []"
-      with weq and w0 show False
-        by (simp add: norm_empty_bv_to_nat_zero)
-    next
-      assume w'0: "norm_unsigned w' \<noteq> []"
-      have "0 < bv_to_nat w'"
-      proof (rule ccontr)
-        assume "~ (0 < bv_to_nat w')"
-        hence "bv_to_nat w' = 0"
-          by arith
-        hence "norm_unsigned w' = []"
-          by (simp add: bv_to_nat_zero_imp_empty)
-        with w'0
-        show False by simp
-      qed
-      with bv_to_nat_lower_limit [of w']
-      show "2 ^ (length (norm_unsigned w') - Suc 0) \<le> int (bv_to_nat w')"
-        by (simp add: int_nat_two_exp)
-    qed
-  next
-    fix w'
-    assume "w = \<one> # w'"
-    from w0 have "bv_msb w = \<zero>" by simp
-    with prems show ?thesis by simp
-  qed
-  also have "...  = bv_to_int w" by simp
-  finally show ?thesis .
-qed
-
-lemma norm_signed_result: "norm_signed w = [] \<or> norm_signed w = [\<one>] \<or> bv_msb (norm_signed w) \<noteq> bv_msb (tl (norm_signed w))"
-  apply (rule bit_list_cases [of w],simp_all)
-  apply (case_tac "bs",simp_all)
-  apply (case_tac "a",simp_all)
-  apply (simp add: norm_signed_Cons)
-  apply safe
-  apply simp
-proof -
-  fix l
-  assume msb: "\<zero> = bv_msb (norm_unsigned l)"
-  assume "norm_unsigned l \<noteq> []"
-  with norm_unsigned_result [of l]
-  have "bv_msb (norm_unsigned l) = \<one>" by simp
-  with msb show False by simp
-next
-  fix xs
-  assume p: "\<one> = bv_msb (tl (norm_signed (\<one> # xs)))"
-  have "\<one> \<noteq> bv_msb (tl (norm_signed (\<one> # xs)))"
-    by (rule bit_list_induct [of _ xs],simp_all)
-  with p show False by simp
-qed
-
-lemma bv_to_int_upper_limit_lem1:
-  assumes w0: "bv_to_int w < -1"
-  shows       "bv_to_int w < - (2 ^ (length (norm_signed w) - 2))"
-proof -
-  from w0
-  have "bv_to_int w < 0" by simp
-  hence msbw [simp]: "bv_msb w = \<one>"
-    by (rule bv_to_int_msb1)
-  have "bv_to_int w = bv_to_int (norm_signed w)" by simp
-  also from norm_signed_result [of w]
-  have "... < - (2 ^ (length (norm_signed w) - 2))"
-  proof safe
-    assume "norm_signed w = []"
-    hence "bv_to_int (norm_signed w) = 0" by simp
-    with w0 show ?thesis by simp
-  next
-    assume "norm_signed w = [\<one>]"
-    hence "bv_to_int (norm_signed w) = -1" by simp
-    with w0 show ?thesis by simp
-  next
-    assume "bv_msb (norm_signed w) \<noteq> bv_msb (tl (norm_signed w))"
-    hence msb_tl: "\<one> \<noteq> bv_msb (tl (norm_signed w))" by simp
-    show "bv_to_int (norm_signed w) < - (2 ^ (length (norm_signed w) - 2))"
-    proof (rule bit_list_cases [of "norm_signed w"])
-      assume "norm_signed w = []"
-      hence "bv_to_int (norm_signed w) = 0" by simp
-      with w0 show ?thesis by simp
-    next
-      fix w'
-      assume nw: "norm_signed w = \<zero> # w'"
-      from msbw have "bv_msb (norm_signed w) = \<one>" by simp
-      with nw show ?thesis by simp
-    next
-      fix w'
-      assume weq: "norm_signed w = \<one> # w'"
-      show ?thesis
-      proof (rule bit_list_cases [of w'])
-        assume w'eq: "w' = []"
-        from w0 have "bv_to_int (norm_signed w) < -1" by simp
-        with w'eq and weq show ?thesis by simp
-      next
-        fix w''
-        assume w'eq: "w' = \<zero> # w''"
-        show ?thesis
-          apply (simp add: weq w'eq)
-          apply (subgoal_tac "- int (bv_to_nat (bv_not w'')) + -1 < 0 + 0")
-          apply (simp add: int_nat_two_exp)
-          apply (rule add_le_less_mono)
-          apply simp_all
-          done
-      next
-        fix w''
-        assume w'eq: "w' = \<one> # w''"
-        with weq and msb_tl show ?thesis by simp
-      qed
-    qed
-  qed
-  finally show ?thesis .
-qed
-
-lemma length_int_to_bv_upper_limit_gt0:
-  assumes w0: "0 < i"
-  and     wk: "i \<le> 2 ^ (k - 1) - 1"
-  shows       "length (int_to_bv i) \<le> k"
-proof (rule ccontr)
-  from w0 wk
-  have k1: "1 < k"
-    by (cases "k - 1",simp_all)
-  assume "~ length (int_to_bv i) \<le> k"
-  hence "k < length (int_to_bv i)" by simp
-  hence "k \<le> length (int_to_bv i) - 1" by arith
-  hence a: "k - 1 \<le> length (int_to_bv i) - 2" by arith
-  hence "(2::int) ^ (k - 1) \<le> 2 ^ (length (int_to_bv i) - 2)" by simp
-  also have "... \<le> i"
-  proof -
-    have "2 ^ (length (norm_signed (int_to_bv i)) - 2) \<le> bv_to_int (int_to_bv i)"
-    proof (rule bv_to_int_lower_limit_gt0)
-      from w0 show "0 < bv_to_int (int_to_bv i)" by simp
-    qed
-    thus ?thesis by simp
-  qed
-  finally have "2 ^ (k - 1) \<le> i" .
-  with wk show False by simp
-qed
-
-lemma pos_length_pos:
-  assumes i0: "0 < bv_to_int w"
-  shows       "0 < length w"
-proof -
-  from norm_signed_result [of w]
-  have "0 < length (norm_signed w)"
-  proof (auto)
-    assume ii: "norm_signed w = []"
-    have "bv_to_int (norm_signed w) = 0" by (subst ii) simp
-    hence "bv_to_int w = 0" by simp
-    with i0 show False by simp
-  next
-    assume ii: "norm_signed w = []"
-    assume jj: "bv_msb w \<noteq> \<zero>"
-    have "\<zero> = bv_msb (norm_signed w)"
-      by (subst ii) simp
-    also have "... \<noteq> \<zero>"
-      by (simp add: jj)
-    finally show False by simp
-  qed
-  also have "... \<le> length w"
-    by (rule norm_signed_length)
-  finally show ?thesis .
-qed
-
-lemma neg_length_pos:
-  assumes i0: "bv_to_int w < -1"
-  shows       "0 < length w"
-proof -
-  from norm_signed_result [of w]
-  have "0 < length (norm_signed w)"
-  proof (auto)
-    assume ii: "norm_signed w = []"
-    have "bv_to_int (norm_signed w) = 0"
-      by (subst ii) simp
-    hence "bv_to_int w = 0" by simp
-    with i0 show False by simp
-  next
-    assume ii: "norm_signed w = []"
-    assume jj: "bv_msb w \<noteq> \<zero>"
-    have "\<zero> = bv_msb (norm_signed w)" by (subst ii) simp
-    also have "... \<noteq> \<zero>" by (simp add: jj)
-    finally show False by simp
-  qed
-  also have "... \<le> length w"
-    by (rule norm_signed_length)
-  finally show ?thesis .
-qed
-
-lemma length_int_to_bv_lower_limit_gt0:
-  assumes wk: "2 ^ (k - 1) \<le> i"
-  shows       "k < length (int_to_bv i)"
-proof (rule ccontr)
-  have "0 < (2::int) ^ (k - 1)"
-    by (rule zero_less_power) simp
-  also have "... \<le> i" by (rule wk)
-  finally have i0: "0 < i" .
-  have lii0: "0 < length (int_to_bv i)"
-    apply (rule pos_length_pos)
-    apply (simp,rule i0)
-    done
-  assume "~ k < length (int_to_bv i)"
-  hence "length (int_to_bv i) \<le> k" by simp
-  with lii0
-  have a: "length (int_to_bv i) - 1 \<le> k - 1"
-    by arith
-  have "i < 2 ^ (length (int_to_bv i) - 1)"
-  proof -
-    have "i = bv_to_int (int_to_bv i)"
-      by simp
-    also have "... < 2 ^ (length (int_to_bv i) - 1)"
-      by (rule bv_to_int_upper_range)
-    finally show ?thesis .
-  qed
-  also have "(2::int) ^ (length (int_to_bv i) - 1) \<le> 2 ^ (k - 1)" using a
-    by simp
-  finally have "i < 2 ^ (k - 1)" .
-  with wk show False by simp
-qed
-
-lemma length_int_to_bv_upper_limit_lem1:
-  assumes w1: "i < -1"
-  and     wk: "- (2 ^ (k - 1)) \<le> i"
-  shows       "length (int_to_bv i) \<le> k"
-proof (rule ccontr)
-  from w1 wk
-  have k1: "1 < k" by (cases "k - 1") simp_all
-  assume "~ length (int_to_bv i) \<le> k"
-  hence "k < length (int_to_bv i)" by simp
-  hence "k \<le> length (int_to_bv i) - 1" by arith
-  hence a: "k - 1 \<le> length (int_to_bv i) - 2" by arith
-  have "i < - (2 ^ (length (int_to_bv i) - 2))"
-  proof -
-    have "i = bv_to_int (int_to_bv i)"
-      by simp
-    also have "... < - (2 ^ (length (norm_signed (int_to_bv i)) - 2))"
-      by (rule bv_to_int_upper_limit_lem1,simp,rule w1)
-    finally show ?thesis by simp
-  qed
-  also have "... \<le> -(2 ^ (k - 1))"
-  proof -
-    have "(2::int) ^ (k - 1) \<le> 2 ^ (length (int_to_bv i) - 2)" using a by simp
-    thus ?thesis by simp
-  qed
-  finally have "i < -(2 ^ (k - 1))" .
-  with wk show False by simp
-qed
-
-lemma length_int_to_bv_lower_limit_lem1:
-  assumes wk: "i < -(2 ^ (k - 1))"
-  shows       "k < length (int_to_bv i)"
-proof (rule ccontr)
-  from wk have "i \<le> -(2 ^ (k - 1)) - 1" by simp
-  also have "... < -1"
-  proof -
-    have "0 < (2::int) ^ (k - 1)"
-      by (rule zero_less_power) simp
-    hence "-((2::int) ^ (k - 1)) < 0" by simp
-    thus ?thesis by simp
-  qed
-  finally have i1: "i < -1" .
-  have lii0: "0 < length (int_to_bv i)"
-    apply (rule neg_length_pos)
-    apply (simp, rule i1)
-    done
-  assume "~ k < length (int_to_bv i)"
-  hence "length (int_to_bv i) \<le> k"
-    by simp
-  with lii0 have a: "length (int_to_bv i) - 1 \<le> k - 1" by arith
-  hence "(2::int) ^ (length (int_to_bv i) - 1) \<le> 2 ^ (k - 1)" by simp
-  hence "-((2::int) ^ (k - 1)) \<le> - (2 ^ (length (int_to_bv i) - 1))" by simp
-  also have "... \<le> i"
-  proof -
-    have "- (2 ^ (length (int_to_bv i) - 1)) \<le> bv_to_int (int_to_bv i)"
-      by (rule bv_to_int_lower_range)
-    also have "... = i"
-      by simp
-    finally show ?thesis .
-  qed
-  finally have "-(2 ^ (k - 1)) \<le> i" .
-  with wk show False by simp
-qed
-
-
-subsection {* Signed Arithmetic Operations *}
-
-subsubsection {* Conversion from unsigned to signed *}
-
-definition
-  utos :: "bit list => bit list" where
-  "utos w = norm_signed (\<zero> # w)"
-
-lemma utos_type [simp]: "utos (norm_unsigned w) = utos w"
-  by (simp add: utos_def norm_signed_Cons)
-
-lemma utos_returntype [simp]: "norm_signed (utos w) = utos w"
-  by (simp add: utos_def)
-
-lemma utos_length: "length (utos w) \<le> Suc (length w)"
-  by (simp add: utos_def norm_signed_Cons)
-
-lemma bv_to_int_utos: "bv_to_int (utos w) = int (bv_to_nat w)"
-proof (simp add: utos_def norm_signed_Cons, safe)
-  assume "norm_unsigned w = []"
-  hence "bv_to_nat (norm_unsigned w) = 0" by simp
-  thus "bv_to_nat w = 0" by simp
-qed
-
-
-subsubsection {* Unary minus *}
-
-definition
-  bv_uminus :: "bit list => bit list" where
-  "bv_uminus w = int_to_bv (- bv_to_int w)"
-
-lemma bv_uminus_type [simp]: "bv_uminus (norm_signed w) = bv_uminus w"
-  by (simp add: bv_uminus_def)
-
-lemma bv_uminus_returntype [simp]: "norm_signed (bv_uminus w) = bv_uminus w"
-  by (simp add: bv_uminus_def)
-
-lemma bv_uminus_length: "length (bv_uminus w) \<le> Suc (length w)"
-proof -
-  have "1 < -bv_to_int w \<or> -bv_to_int w = 1 \<or> -bv_to_int w = 0 \<or> -bv_to_int w = -1 \<or> -bv_to_int w < -1"
-    by arith
-  thus ?thesis
-  proof safe
-    assume p: "1 < - bv_to_int w"
-    have lw: "0 < length w"
-      apply (rule neg_length_pos)
-      using p
-      apply simp
-      done
-    show ?thesis
-    proof (simp add: bv_uminus_def,rule length_int_to_bv_upper_limit_gt0,simp_all)
-      from prems show "bv_to_int w < 0" by simp
-    next
-      have "-(2^(length w - 1)) \<le> bv_to_int w"
-        by (rule bv_to_int_lower_range)
-      hence "- bv_to_int w \<le> 2^(length w - 1)" by simp
-      also from lw have "... < 2 ^ length w" by simp
-      finally show "- bv_to_int w < 2 ^ length w" by simp
-    qed
-  next
-    assume p: "- bv_to_int w = 1"
-    hence lw: "0 < length w" by (cases w) simp_all
-    from p
-    show ?thesis
-      apply (simp add: bv_uminus_def)
-      using lw
-      apply (simp (no_asm) add: nat_to_bv_non0)
-      done
-  next
-    assume "- bv_to_int w = 0"
-    thus ?thesis by (simp add: bv_uminus_def)
-  next
-    assume p: "- bv_to_int w = -1"
-    thus ?thesis by (simp add: bv_uminus_def)
-  next
-    assume p: "- bv_to_int w < -1"
-    show ?thesis
-      apply (simp add: bv_uminus_def)
-      apply (rule length_int_to_bv_upper_limit_lem1)
-      apply (rule p)
-      apply simp
-    proof -
-      have "bv_to_int w < 2 ^ (length w - 1)"
-        by (rule bv_to_int_upper_range)
-      also have "... \<le> 2 ^ length w" by simp
-      finally show "bv_to_int w \<le> 2 ^ length w" by simp
-    qed
-  qed
-qed
-
-lemma bv_uminus_length_utos: "length (bv_uminus (utos w)) \<le> Suc (length w)"
-proof -
-  have "-bv_to_int (utos w) = 0 \<or> -bv_to_int (utos w) = -1 \<or> -bv_to_int (utos w) < -1"
-    by (simp add: bv_to_int_utos, arith)
-  thus ?thesis
-  proof safe
-    assume "-bv_to_int (utos w) = 0"
-    thus ?thesis by (simp add: bv_uminus_def)
-  next
-    assume "-bv_to_int (utos w) = -1"
-    thus ?thesis by (simp add: bv_uminus_def)
-  next
-    assume p: "-bv_to_int (utos w) < -1"
-    show ?thesis
-      apply (simp add: bv_uminus_def)
-      apply (rule length_int_to_bv_upper_limit_lem1)
-      apply (rule p)
-      apply (simp add: bv_to_int_utos)
-      using bv_to_nat_upper_range [of w]
-      apply (simp add: int_nat_two_exp)
-      done
-  qed
-qed
-
-definition
-  bv_sadd :: "[bit list, bit list ] => bit list" where
-  "bv_sadd w1 w2 = int_to_bv (bv_to_int w1 + bv_to_int w2)"
-
-lemma bv_sadd_type1 [simp]: "bv_sadd (norm_signed w1) w2 = bv_sadd w1 w2"
-  by (simp add: bv_sadd_def)
-
-lemma bv_sadd_type2 [simp]: "bv_sadd w1 (norm_signed w2) = bv_sadd w1 w2"
-  by (simp add: bv_sadd_def)
-
-lemma bv_sadd_returntype [simp]: "norm_signed (bv_sadd w1 w2) = bv_sadd w1 w2"
-  by (simp add: bv_sadd_def)
-
-lemma adder_helper:
-  assumes lw: "0 < max (length w1) (length w2)"
-  shows   "((2::int) ^ (length w1 - 1)) + (2 ^ (length w2 - 1)) \<le> 2 ^ max (length w1) (length w2)"
-proof -
-  have "((2::int) ^ (length w1 - 1)) + (2 ^ (length w2 - 1)) \<le>
-      2 ^ (max (length w1) (length w2) - 1) + 2 ^ (max (length w1) (length w2) - 1)"
-    by (auto simp:max_def)
-  also have "... = 2 ^ max (length w1) (length w2)"
-  proof -
-    from lw
-    show ?thesis
-      apply simp
-      apply (subst power_Suc [symmetric])
-      apply simp
-      done
-  qed
-  finally show ?thesis .
-qed
-
-lemma bv_sadd_length: "length (bv_sadd w1 w2) \<le> Suc (max (length w1) (length w2))"
-proof -
-  let ?Q = "bv_to_int w1 + bv_to_int w2"
-
-  have helper: "?Q \<noteq> 0 ==> 0 < max (length w1) (length w2)"
-  proof -
-    assume p: "?Q \<noteq> 0"
-    show "0 < max (length w1) (length w2)"
-    proof (simp add: less_max_iff_disj,rule)
-      assume [simp]: "w1 = []"
-      show "w2 \<noteq> []"
-      proof (rule ccontr,simp)
-        assume [simp]: "w2 = []"
-        from p show False by simp
-      qed
-    qed
-  qed
-
-  have "0 < ?Q \<or> ?Q = 0 \<or> ?Q = -1 \<or> ?Q < -1" by arith
-  thus ?thesis
-  proof safe
-    assume "?Q = 0"
-    thus ?thesis
-      by (simp add: bv_sadd_def)
-  next
-    assume "?Q = -1"
-    thus ?thesis
-      by (simp add: bv_sadd_def)
-  next
-    assume p: "0 < ?Q"
-    show ?thesis
-      apply (simp add: bv_sadd_def)
-      apply (rule length_int_to_bv_upper_limit_gt0)
-      apply (rule p)
-    proof simp
-      from bv_to_int_upper_range [of w2]
-      have "bv_to_int w2 \<le> 2 ^ (length w2 - 1)"
-        by simp
-      with bv_to_int_upper_range [of w1]
-      have "bv_to_int w1 + bv_to_int w2 < (2 ^ (length w1 - 1)) + (2 ^ (length w2 - 1))"
-        by (rule zadd_zless_mono)
-      also have "... \<le> 2 ^ max (length w1) (length w2)"
-        apply (rule adder_helper)
-        apply (rule helper)
-        using p
-        apply simp
-        done
-      finally show "?Q < 2 ^ max (length w1) (length w2)" .
-    qed
-  next
-    assume p: "?Q < -1"
-    show ?thesis
-      apply (simp add: bv_sadd_def)
-      apply (rule length_int_to_bv_upper_limit_lem1,simp_all)
-      apply (rule p)
-    proof -
-      have "(2 ^ (length w1 - 1)) + 2 ^ (length w2 - 1) \<le> (2::int) ^ max (length w1) (length w2)"
-        apply (rule adder_helper)
-        apply (rule helper)
-        using p
-        apply simp
-        done
-      hence "-((2::int) ^ max (length w1) (length w2)) \<le> - (2 ^ (length w1 - 1)) + -(2 ^ (length w2 - 1))"
-        by simp
-      also have "- (2 ^ (length w1 - 1)) + -(2 ^ (length w2 - 1)) \<le> ?Q"
-        apply (rule add_mono)
-        apply (rule bv_to_int_lower_range [of w1])
-        apply (rule bv_to_int_lower_range [of w2])
-        done
-      finally show "- (2^max (length w1) (length w2)) \<le> ?Q" .
-    qed
-  qed
-qed
-
-definition
-  bv_sub :: "[bit list, bit list] => bit list" where
-  "bv_sub w1 w2 = bv_sadd w1 (bv_uminus w2)"
-
-lemma bv_sub_type1 [simp]: "bv_sub (norm_signed w1) w2 = bv_sub w1 w2"
-  by (simp add: bv_sub_def)
-
-lemma bv_sub_type2 [simp]: "bv_sub w1 (norm_signed w2) = bv_sub w1 w2"
-  by (simp add: bv_sub_def)
-
-lemma bv_sub_returntype [simp]: "norm_signed (bv_sub w1 w2) = bv_sub w1 w2"
-  by (simp add: bv_sub_def)
-
-lemma bv_sub_length: "length (bv_sub w1 w2) \<le> Suc (max (length w1) (length w2))"
-proof (cases "bv_to_int w2 = 0")
-  assume p: "bv_to_int w2 = 0"
-  show ?thesis
-  proof (simp add: bv_sub_def bv_sadd_def bv_uminus_def p)
-    have "length (norm_signed w1) \<le> length w1"
-      by (rule norm_signed_length)
-    also have "... \<le> max (length w1) (length w2)"
-      by (rule le_maxI1)
-    also have "... \<le> Suc (max (length w1) (length w2))"
-      by arith
-    finally show "length (norm_signed w1) \<le> Suc (max (length w1) (length w2))" .
-  qed
-next
-  assume "bv_to_int w2 \<noteq> 0"
-  hence "0 < length w2" by (cases w2,simp_all)
-  hence lmw: "0 < max (length w1) (length w2)" by arith
-
-  let ?Q = "bv_to_int w1 - bv_to_int w2"
-
-  have "0 < ?Q \<or> ?Q = 0 \<or> ?Q = -1 \<or> ?Q < -1" by arith
-  thus ?thesis
-  proof safe
-    assume "?Q = 0"
-    thus ?thesis
-      by (simp add: bv_sub_def bv_sadd_def bv_uminus_def)
-  next
-    assume "?Q = -1"
-    thus ?thesis
-      by (simp add: bv_sub_def bv_sadd_def bv_uminus_def)
-  next
-    assume p: "0 < ?Q"
-    show ?thesis
-      apply (simp add: bv_sub_def bv_sadd_def bv_uminus_def)
-      apply (rule length_int_to_bv_upper_limit_gt0)
-      apply (rule p)
-    proof simp
-      from bv_to_int_lower_range [of w2]
-      have v2: "- bv_to_int w2 \<le> 2 ^ (length w2 - 1)" by simp
-      have "bv_to_int w1 + - bv_to_int w2 < (2 ^ (length w1 - 1)) + (2 ^ (length w2 - 1))"
-        apply (rule zadd_zless_mono)
-        apply (rule bv_to_int_upper_range [of w1])
-        apply (rule v2)
-        done
-      also have "... \<le> 2 ^ max (length w1) (length w2)"
-        apply (rule adder_helper)
-        apply (rule lmw)
-        done
-      finally show "?Q < 2 ^ max (length w1) (length w2)" by simp
-    qed
-  next
-    assume p: "?Q < -1"
-    show ?thesis
-      apply (simp add: bv_sub_def bv_sadd_def bv_uminus_def)
-      apply (rule length_int_to_bv_upper_limit_lem1)
-      apply (rule p)
-    proof simp
-      have "(2 ^ (length w1 - 1)) + 2 ^ (length w2 - 1) \<le> (2::int) ^ max (length w1) (length w2)"
-        apply (rule adder_helper)
-        apply (rule lmw)
-        done
-      hence "-((2::int) ^ max (length w1) (length w2)) \<le> - (2 ^ (length w1 - 1)) + -(2 ^ (length w2 - 1))"
-        by simp
-      also have "- (2 ^ (length w1 - 1)) + -(2 ^ (length w2 - 1)) \<le> bv_to_int w1 + -bv_to_int w2"
-        apply (rule add_mono)
-        apply (rule bv_to_int_lower_range [of w1])
-        using bv_to_int_upper_range [of w2]
-        apply simp
-        done
-      finally show "- (2^max (length w1) (length w2)) \<le> ?Q" by simp
-    qed
-  qed
-qed
-
-definition
-  bv_smult :: "[bit list, bit list] => bit list" where
-  "bv_smult w1 w2 = int_to_bv (bv_to_int w1 * bv_to_int w2)"
-
-lemma bv_smult_type1 [simp]: "bv_smult (norm_signed w1) w2 = bv_smult w1 w2"
-  by (simp add: bv_smult_def)
-
-lemma bv_smult_type2 [simp]: "bv_smult w1 (norm_signed w2) = bv_smult w1 w2"
-  by (simp add: bv_smult_def)
-
-lemma bv_smult_returntype [simp]: "norm_signed (bv_smult w1 w2) = bv_smult w1 w2"
-  by (simp add: bv_smult_def)
-
-lemma bv_smult_length: "length (bv_smult w1 w2) \<le> length w1 + length w2"
-proof -
-  let ?Q = "bv_to_int w1 * bv_to_int w2"
-
-  have lmw: "?Q \<noteq> 0 ==> 0 < length w1 \<and> 0 < length w2" by auto
-
-  have "0 < ?Q \<or> ?Q = 0 \<or> ?Q = -1 \<or> ?Q < -1" by arith
-  thus ?thesis
-  proof (safe dest!: iffD1 [OF mult_eq_0_iff])
-    assume "bv_to_int w1 = 0"
-    thus ?thesis by (simp add: bv_smult_def)
-  next
-    assume "bv_to_int w2 = 0"
-    thus ?thesis by (simp add: bv_smult_def)
-  next
-    assume p: "?Q = -1"
-    show ?thesis
-      apply (simp add: bv_smult_def p)
-      apply (cut_tac lmw)
-      apply arith
-      using p
-      apply simp
-      done
-  next
-    assume p: "0 < ?Q"
-    thus ?thesis
-    proof (simp add: zero_less_mult_iff,safe)
-      assume bi1: "0 < bv_to_int w1"
-      assume bi2: "0 < bv_to_int w2"
-      show ?thesis
-        apply (simp add: bv_smult_def)
-        apply (rule length_int_to_bv_upper_limit_gt0)
-        apply (rule p)
-      proof simp
-        have "?Q < 2 ^ (length w1 - 1) * 2 ^ (length w2 - 1)"
-          apply (rule mult_strict_mono)
-          apply (rule bv_to_int_upper_range)
-          apply (rule bv_to_int_upper_range)
-          apply (rule zero_less_power)
-          apply simp
-          using bi2
-          apply simp
-          done
-        also have "... \<le> 2 ^ (length w1 + length w2 - Suc 0)"
-          apply simp
-          apply (subst zpower_zadd_distrib [symmetric])
-          apply simp
-          done
-        finally show "?Q < 2 ^ (length w1 + length w2 - Suc 0)" .
-      qed
-    next
-      assume bi1: "bv_to_int w1 < 0"
-      assume bi2: "bv_to_int w2 < 0"
-      show ?thesis
-        apply (simp add: bv_smult_def)
-        apply (rule length_int_to_bv_upper_limit_gt0)
-        apply (rule p)
-      proof simp
-        have "-bv_to_int w1 * -bv_to_int w2 \<le> 2 ^ (length w1 - 1) * 2 ^(length w2 - 1)"
-          apply (rule mult_mono)
-          using bv_to_int_lower_range [of w1]
-          apply simp
-          using bv_to_int_lower_range [of w2]
-          apply simp
-          apply (rule zero_le_power,simp)
-          using bi2
-          apply simp
-          done
-        hence "?Q \<le> 2 ^ (length w1 - 1) * 2 ^(length w2 - 1)"
-          by simp
-        also have "... < 2 ^ (length w1 + length w2 - Suc 0)"
-          apply simp
-          apply (subst zpower_zadd_distrib [symmetric])
-          apply simp
-          apply (cut_tac lmw)
-          apply arith
-          apply (cut_tac p)
-          apply arith
-          done
-        finally show "?Q < 2 ^ (length w1 + length w2 - Suc 0)" .
-      qed
-    qed
-  next
-    assume p: "?Q < -1"
-    show ?thesis
-      apply (subst bv_smult_def)
-      apply (rule length_int_to_bv_upper_limit_lem1)
-      apply (rule p)
-    proof simp
-      have "(2::int) ^ (length w1 - 1) * 2 ^(length w2 - 1) \<le> 2 ^ (length w1 + length w2 - Suc 0)"
-        apply simp
-        apply (subst zpower_zadd_distrib [symmetric])
-        apply simp
-        done
-      hence "-((2::int) ^ (length w1 + length w2 - Suc 0)) \<le> -(2^(length w1 - 1) * 2 ^ (length w2 - 1))"
-        by simp
-      also have "... \<le> ?Q"
-      proof -
-        from p
-        have q: "bv_to_int w1 * bv_to_int w2 < 0"
-          by simp
-        thus ?thesis
-        proof (simp add: mult_less_0_iff,safe)
-          assume bi1: "0 < bv_to_int w1"
-          assume bi2: "bv_to_int w2 < 0"
-          have "-bv_to_int w2 * bv_to_int w1 \<le> ((2::int)^(length w2 - 1)) * (2 ^ (length w1 - 1))"
-            apply (rule mult_mono)
-            using bv_to_int_lower_range [of w2]
-            apply simp
-            using bv_to_int_upper_range [of w1]
-            apply simp
-            apply (rule zero_le_power,simp)
-            using bi1
-            apply simp
-            done
-          hence "-?Q \<le> ((2::int)^(length w1 - 1)) * (2 ^ (length w2 - 1))"
-            by (simp add: zmult_ac)
-          thus "-(((2::int)^(length w1 - Suc 0)) * (2 ^ (length w2 - Suc 0))) \<le> ?Q"
-            by simp
-        next
-          assume bi1: "bv_to_int w1 < 0"
-          assume bi2: "0 < bv_to_int w2"
-          have "-bv_to_int w1 * bv_to_int w2 \<le> ((2::int)^(length w1 - 1)) * (2 ^ (length w2 - 1))"
-            apply (rule mult_mono)
-            using bv_to_int_lower_range [of w1]
-            apply simp
-            using bv_to_int_upper_range [of w2]
-            apply simp
-            apply (rule zero_le_power,simp)
-            using bi2
-            apply simp
-            done
-          hence "-?Q \<le> ((2::int)^(length w1 - 1)) * (2 ^ (length w2 - 1))"
-            by (simp add: zmult_ac)
-          thus "-(((2::int)^(length w1 - Suc 0)) * (2 ^ (length w2 - Suc 0))) \<le> ?Q"
-            by simp
-        qed
-      qed
-      finally show "-(2 ^ (length w1 + length w2 - Suc 0)) \<le> ?Q" .
-    qed
-  qed
-qed
-
-lemma bv_msb_one: "bv_msb w = \<one> ==> bv_to_nat w \<noteq> 0"
-by (cases w) simp_all
-
-lemma bv_smult_length_utos: "length (bv_smult (utos w1) w2) \<le> length w1 + length w2"
-proof -
-  let ?Q = "bv_to_int (utos w1) * bv_to_int w2"
-
-  have lmw: "?Q \<noteq> 0 ==> 0 < length (utos w1) \<and> 0 < length w2" by auto
-
-  have "0 < ?Q \<or> ?Q = 0 \<or> ?Q = -1 \<or> ?Q < -1" by arith
-  thus ?thesis
-  proof (safe dest!: iffD1 [OF mult_eq_0_iff])
-    assume "bv_to_int (utos w1) = 0"
-    thus ?thesis by (simp add: bv_smult_def)
-  next
-    assume "bv_to_int w2 = 0"
-    thus ?thesis by (simp add: bv_smult_def)
-  next
-    assume p: "0 < ?Q"
-    thus ?thesis
-    proof (simp add: zero_less_mult_iff,safe)
-      assume biw2: "0 < bv_to_int w2"
-      show ?thesis
-        apply (simp add: bv_smult_def)
-        apply (rule length_int_to_bv_upper_limit_gt0)
-        apply (rule p)
-      proof simp
-        have "?Q < 2 ^ length w1 * 2 ^ (length w2 - 1)"
-          apply (rule mult_strict_mono)
-          apply (simp add: bv_to_int_utos int_nat_two_exp)
-          apply (rule bv_to_nat_upper_range)
-          apply (rule bv_to_int_upper_range)
-          apply (rule zero_less_power,simp)
-          using biw2
-          apply simp
-          done
-        also have "... \<le> 2 ^ (length w1 + length w2 - Suc 0)"
-          apply simp
-          apply (subst zpower_zadd_distrib [symmetric])
-          apply simp
-          apply (cut_tac lmw)
-          apply arith
-          using p
-          apply auto
-          done
-        finally show "?Q < 2 ^ (length w1 + length w2 - Suc 0)" .
-      qed
-    next
-      assume "bv_to_int (utos w1) < 0"
-      thus ?thesis by (simp add: bv_to_int_utos)
-    qed
-  next
-    assume p: "?Q = -1"
-    thus ?thesis
-      apply (simp add: bv_smult_def)
-      apply (cut_tac lmw)
-      apply arith
-      apply simp
-      done
-  next
-    assume p: "?Q < -1"
-    show ?thesis
-      apply (subst bv_smult_def)
-      apply (rule length_int_to_bv_upper_limit_lem1)
-      apply (rule p)
-    proof simp
-      have "(2::int) ^ length w1 * 2 ^(length w2 - 1) \<le> 2 ^ (length w1 + length w2 - Suc 0)"
-        apply simp
-        apply (subst zpower_zadd_distrib [symmetric])
-        apply simp
-        apply (cut_tac lmw)
-        apply arith
-        apply (cut_tac p)
-        apply arith
-        done
-      hence "-((2::int) ^ (length w1 + length w2 - Suc 0)) \<le> -(2^ length w1 * 2 ^ (length w2 - 1))"
-        by simp
-      also have "... \<le> ?Q"
-      proof -
-        from p
-        have q: "bv_to_int (utos w1) * bv_to_int w2 < 0"
-          by simp
-        thus ?thesis
-        proof (simp add: mult_less_0_iff,safe)
-          assume bi1: "0 < bv_to_int (utos w1)"
-          assume bi2: "bv_to_int w2 < 0"
-          have "-bv_to_int w2 * bv_to_int (utos w1) \<le> ((2::int)^(length w2 - 1)) * (2 ^ length w1)"
-            apply (rule mult_mono)
-            using bv_to_int_lower_range [of w2]
-            apply simp
-            apply (simp add: bv_to_int_utos)
-            using bv_to_nat_upper_range [of w1]
-            apply (simp add: int_nat_two_exp)
-            apply (rule zero_le_power,simp)
-            using bi1
-            apply simp
-            done
-          hence "-?Q \<le> ((2::int)^length w1) * (2 ^ (length w2 - 1))"
-            by (simp add: zmult_ac)
-          thus "-(((2::int)^length w1) * (2 ^ (length w2 - Suc 0))) \<le> ?Q"
-            by simp
-        next
-          assume bi1: "bv_to_int (utos w1) < 0"
-          thus "-(((2::int)^length w1) * (2 ^ (length w2 - Suc 0))) \<le> ?Q"
-            by (simp add: bv_to_int_utos)
-        qed
-      qed
-      finally show "-(2 ^ (length w1 + length w2 - Suc 0)) \<le> ?Q" .
-    qed
-  qed
-qed
-
-lemma bv_smult_sym: "bv_smult w1 w2 = bv_smult w2 w1"
-  by (simp add: bv_smult_def zmult_ac)
-
-subsection {* Structural operations *}
-
-definition
-  bv_select :: "[bit list,nat] => bit" where
-  "bv_select w i = w ! (length w - 1 - i)"
-
-definition
-  bv_chop :: "[bit list,nat] => bit list * bit list" where
-  "bv_chop w i = (let len = length w in (take (len - i) w,drop (len - i) w))"
-
-definition
-  bv_slice :: "[bit list,nat*nat] => bit list" where
-  "bv_slice w = (\<lambda>(b,e). fst (bv_chop (snd (bv_chop w (b+1))) e))"
-
-lemma bv_select_rev:
-  assumes notnull: "n < length w"
-  shows            "bv_select w n = rev w ! n"
-proof -
-  have "\<forall>n. n < length w --> bv_select w n = rev w ! n"
-  proof (rule length_induct [of _ w],auto simp add: bv_select_def)
-    fix xs :: "bit list"
-    fix n
-    assume ind: "\<forall>ys::bit list. length ys < length xs --> (\<forall>n. n < length ys --> ys ! (length ys - Suc n) = rev ys ! n)"
-    assume notx: "n < length xs"
-    show "xs ! (length xs - Suc n) = rev xs ! n"
-    proof (cases xs)
-      assume "xs = []"
-      with notx show ?thesis by simp
-    next
-      fix y ys
-      assume [simp]: "xs = y # ys"
-      show ?thesis
-      proof (auto simp add: nth_append)
-        assume noty: "n < length ys"
-        from spec [OF ind,of ys]
-        have "\<forall>n. n < length ys --> ys ! (length ys - Suc n) = rev ys ! n"
-          by simp
-        hence "n < length ys --> ys ! (length ys - Suc n) = rev ys ! n" ..
-        from this and noty
-        have "ys ! (length ys - Suc n) = rev ys ! n" ..
-        thus "(y # ys) ! (length ys - n) = rev ys ! n"
-          by (simp add: nth_Cons' noty linorder_not_less [symmetric])
-      next
-        assume "~ n < length ys"
-        hence x: "length ys \<le> n" by simp
-        from notx have "n < Suc (length ys)" by simp
-        hence "n \<le> length ys" by simp
-        with x have "length ys = n" by simp
-        thus "y = [y] ! (n - length ys)" by simp
-      qed
-    qed
-  qed
-  then have "n < length w --> bv_select w n = rev w ! n" ..
-  from this and notnull show ?thesis ..
-qed
-
-lemma bv_chop_append: "bv_chop (w1 @ w2) (length w2) = (w1,w2)"
-  by (simp add: bv_chop_def Let_def)
-
-lemma append_bv_chop_id: "fst (bv_chop w l) @ snd (bv_chop w l) = w"
-  by (simp add: bv_chop_def Let_def)
-
-lemma bv_chop_length_fst [simp]: "length (fst (bv_chop w i)) = length w - i"
-  by (simp add: bv_chop_def Let_def)
-
-lemma bv_chop_length_snd [simp]: "length (snd (bv_chop w i)) = min i (length w)"
-  by (simp add: bv_chop_def Let_def)
-
-lemma bv_slice_length [simp]: "[| j \<le> i; i < length w |] ==> length (bv_slice w (i,j)) = i - j + 1"
-  by (auto simp add: bv_slice_def)
-
-definition
-  length_nat :: "nat => nat" where
-  [code del]: "length_nat x = (LEAST n. x < 2 ^ n)"
-
-lemma length_nat: "length (nat_to_bv n) = length_nat n"
-  apply (simp add: length_nat_def)
-  apply (rule Least_equality [symmetric])
-  prefer 2
-  apply (rule length_nat_to_bv_upper_limit)
-  apply arith
-  apply (rule ccontr)
-proof -
-  assume "~ n < 2 ^ length (nat_to_bv n)"
-  hence "2 ^ length (nat_to_bv n) \<le> n" by simp
-  hence "length (nat_to_bv n) < length (nat_to_bv n)"
-    by (rule length_nat_to_bv_lower_limit)
-  thus False by simp
-qed
-
-lemma length_nat_0 [simp]: "length_nat 0 = 0"
-  by (simp add: length_nat_def Least_equality)
-
-lemma length_nat_non0:
-  assumes n0: "n \<noteq> 0"
-  shows       "length_nat n = Suc (length_nat (n div 2))"
-  apply (simp add: length_nat [symmetric])
-  apply (subst nat_to_bv_non0 [of n])
-  apply (simp_all add: n0)
-  done
-
-definition
-  length_int :: "int => nat" where
-  "length_int x =
-    (if 0 < x then Suc (length_nat (nat x))
-    else if x = 0 then 0
-    else Suc (length_nat (nat (-x - 1))))"
-
-lemma length_int: "length (int_to_bv i) = length_int i"
-proof (cases "0 < i")
-  assume i0: "0 < i"
-  hence "length (int_to_bv i) =
-      length (norm_signed (\<zero> # norm_unsigned (nat_to_bv (nat i))))" by simp
-  also from norm_unsigned_result [of "nat_to_bv (nat i)"]
-  have "... = Suc (length_nat (nat i))"
-    apply safe
-    apply (simp del: norm_unsigned_nat_to_bv)
-    apply (drule norm_empty_bv_to_nat_zero)
-    using prems
-    apply simp
-    apply (cases "norm_unsigned (nat_to_bv (nat i))")
-    apply (drule norm_empty_bv_to_nat_zero [of "nat_to_bv (nat i)"])
-    using prems
-    apply simp
-    apply simp
-    using prems
-    apply (simp add: length_nat [symmetric])
-    done
-  finally show ?thesis
-    using i0
-    by (simp add: length_int_def)
-next
-  assume "~ 0 < i"
-  hence i0: "i \<le> 0" by simp
-  show ?thesis
-  proof (cases "i = 0")
-    assume "i = 0"
-    thus ?thesis by (simp add: length_int_def)
-  next
-    assume "i \<noteq> 0"
-    with i0 have i0: "i < 0" by simp
-    hence "length (int_to_bv i) =
-        length (norm_signed (\<one> # bv_not (norm_unsigned (nat_to_bv (nat (- i) - 1)))))"
-      by (simp add: int_to_bv_def nat_diff_distrib)
-    also from norm_unsigned_result [of "nat_to_bv (nat (- i) - 1)"]
-    have "... = Suc (length_nat (nat (- i) - 1))"
-      apply safe
-      apply (simp del: norm_unsigned_nat_to_bv)
-      apply (drule norm_empty_bv_to_nat_zero [of "nat_to_bv (nat (-i) - Suc 0)"])
-      using prems
-      apply simp
-      apply (cases "- i - 1 = 0")
-      apply simp
-      apply (simp add: length_nat [symmetric])
-      apply (cases "norm_unsigned (nat_to_bv (nat (- i) - 1))")
-      apply simp
-      apply simp
-      done
-    finally
-    show ?thesis
-      using i0 by (simp add: length_int_def nat_diff_distrib del: int_to_bv_lt0)
-  qed
-qed
-
-lemma length_int_0 [simp]: "length_int 0 = 0"
-  by (simp add: length_int_def)
-
-lemma length_int_gt0: "0 < i ==> length_int i = Suc (length_nat (nat i))"
-  by (simp add: length_int_def)
-
-lemma length_int_lt0: "i < 0 ==> length_int i = Suc (length_nat (nat (- i) - 1))"
-  by (simp add: length_int_def nat_diff_distrib)
-
-lemma bv_chopI: "[| w = w1 @ w2 ; i = length w2 |] ==> bv_chop w i = (w1,w2)"
-  by (simp add: bv_chop_def Let_def)
-
-lemma bv_sliceI: "[| j \<le> i ; i < length w ; w = w1 @ w2 @ w3 ; Suc i = length w2 + j ; j = length w3  |] ==> bv_slice w (i,j) = w2"
-  apply (simp add: bv_slice_def)
-  apply (subst bv_chopI [of "w1 @ w2 @ w3" w1 "w2 @ w3"])
-  apply simp
-  apply simp
-  apply simp
-  apply (subst bv_chopI [of "w2 @ w3" w2 w3],simp_all)
-  done
-
-lemma bv_slice_bv_slice:
-  assumes ki: "k \<le> i"
-  and     ij: "i \<le> j"
-  and     jl: "j \<le> l"
-  and     lw: "l < length w"
-  shows       "bv_slice w (j,i) = bv_slice (bv_slice w (l,k)) (j-k,i-k)"
-proof -
-  def w1  == "fst (bv_chop w (Suc l))"
-  and w2  == "fst (bv_chop (snd (bv_chop w (Suc l))) (Suc j))"
-  and w3  == "fst (bv_chop (snd (bv_chop (snd (bv_chop w (Suc l))) (Suc j))) i)"
-  and w4  == "fst (bv_chop (snd (bv_chop (snd (bv_chop (snd (bv_chop w (Suc l))) (Suc j))) i)) k)"
-  and w5  == "snd (bv_chop (snd (bv_chop (snd (bv_chop (snd (bv_chop w (Suc l))) (Suc j))) i)) k)"
-  note w_defs = this
-
-  have w_def: "w = w1 @ w2 @ w3 @ w4 @ w5"
-    by (simp add: w_defs append_bv_chop_id)
-
-  from ki ij jl lw
-  show ?thesis
-    apply (subst bv_sliceI [where ?j = i and ?i = j and ?w = w and ?w1.0 = "w1 @ w2" and ?w2.0 = w3 and ?w3.0 = "w4 @ w5"])
-    apply simp_all
-    apply (rule w_def)
-    apply (simp add: w_defs)
-    apply (simp add: w_defs)
-    apply (subst bv_sliceI [where ?j = k and ?i = l and ?w = w and ?w1.0 = w1 and ?w2.0 = "w2 @ w3 @ w4" and ?w3.0 = w5])
-    apply simp_all
-    apply (rule w_def)
-    apply (simp add: w_defs)
-    apply (simp add: w_defs)
-    apply (subst bv_sliceI [where ?j = "i-k" and ?i = "j-k" and ?w = "w2 @ w3 @ w4" and ?w1.0 = w2 and ?w2.0 = w3 and ?w3.0 = w4])
-    apply simp_all
-    apply (simp_all add: w_defs)
-    done
-qed
-
-lemma bv_to_nat_extend [simp]: "bv_to_nat (bv_extend n \<zero> w) = bv_to_nat w"
-  apply (simp add: bv_extend_def)
-  apply (subst bv_to_nat_dist_append)
-  apply simp
-  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 (simp add: bv_extend_def)
-  apply (cases "n - length w")
-   apply simp_all
-  done
-
-lemma bv_to_int_extend [simp]:
-  assumes a: "bv_msb w = b"
-  shows      "bv_to_int (bv_extend n b w) = bv_to_int w"
-proof (cases "bv_msb w")
-  assume [simp]: "bv_msb w = \<zero>"
-  with a have [simp]: "b = \<zero>" by simp
-  show ?thesis by (simp add: bv_to_int_def)
-next
-  assume [simp]: "bv_msb w = \<one>"
-  with a have [simp]: "b = \<one>" by simp
-  show ?thesis
-    apply (simp add: bv_to_int_def)
-    apply (simp add: bv_extend_def)
-    apply (induct ("n - length w"), simp_all)
-    done
-qed
-
-lemma length_nat_mono [simp]: "x \<le> y ==> length_nat x \<le> length_nat y"
-proof (rule ccontr)
-  assume xy: "x \<le> y"
-  assume "~ length_nat x \<le> length_nat y"
-  hence lxly: "length_nat y < length_nat x"
-    by simp
-  hence "length_nat y < (LEAST n. x < 2 ^ n)"
-    by (simp add: length_nat_def)
-  hence "~ x < 2 ^ length_nat y"
-    by (rule not_less_Least)
-  hence xx: "2 ^ length_nat y \<le> x"
-    by simp
-  have yy: "y < 2 ^ length_nat y"
-    apply (simp add: length_nat_def)
-    apply (rule LeastI)
-    apply (subgoal_tac "y < 2 ^ y",assumption)
-    apply (cases "0 \<le> y")
-    apply (induct y,simp_all)
-    done
-  with xx have "y < x" by simp
-  with xy show False by simp
-qed
-
-lemma length_nat_mono_int [simp]: "x \<le> y ==> length_nat x \<le> length_nat y"
-  by (rule length_nat_mono) arith
-
-lemma length_nat_pos [simp,intro!]: "0 < x ==> 0 < length_nat x"
-  by (simp add: length_nat_non0)
-
-lemma length_int_mono_gt0: "[| 0 \<le> x ; x \<le> y |] ==> length_int x \<le> length_int y"
-  by (cases "x = 0") (simp_all add: length_int_gt0 nat_le_eq_zle)
-
-lemma length_int_mono_lt0: "[| x \<le> y ; y \<le> 0 |] ==> length_int y \<le> length_int x"
-  by (cases "y = 0") (simp_all add: length_int_lt0)
-
-lemmas [simp] = length_nat_non0
-
-lemma "nat_to_bv (number_of Int.Pls) = []"
-  by simp
-
-primrec fast_bv_to_nat_helper :: "[bit list, int] => int" where
-    fast_bv_to_nat_Nil: "fast_bv_to_nat_helper [] k = k"
-  | fast_bv_to_nat_Cons: "fast_bv_to_nat_helper (b#bs) k =
-      fast_bv_to_nat_helper bs ((bit_case Int.Bit0 Int.Bit1 b) k)"
-
-declare fast_bv_to_nat_helper.simps [code del]
-
-lemma fast_bv_to_nat_Cons0: "fast_bv_to_nat_helper (\<zero>#bs) bin =
-    fast_bv_to_nat_helper bs (Int.Bit0 bin)"
-  by simp
-
-lemma fast_bv_to_nat_Cons1: "fast_bv_to_nat_helper (\<one>#bs) bin =
-    fast_bv_to_nat_helper bs (Int.Bit1 bin)"
-  by simp
-
-lemma fast_bv_to_nat_def:
-  "bv_to_nat bs == number_of (fast_bv_to_nat_helper bs Int.Pls)"
-proof (simp add: bv_to_nat_def)
-  have "\<forall> bin. \<not> (neg (number_of bin :: int)) \<longrightarrow> (foldl (%bn b. 2 * bn + bitval b) (number_of bin) bs) = number_of (fast_bv_to_nat_helper bs bin)"
-    apply (induct bs,simp)
-    apply (case_tac a,simp_all)
-    done
-  thus "foldl (\<lambda>bn b. 2 * bn + bitval b) 0 bs \<equiv> number_of (fast_bv_to_nat_helper bs Int.Pls)"
-    by (simp del: nat_numeral_0_eq_0 add: nat_numeral_0_eq_0 [symmetric])
-qed
-
-declare fast_bv_to_nat_Cons [simp del]
-declare fast_bv_to_nat_Cons0 [simp]
-declare fast_bv_to_nat_Cons1 [simp]
-
-setup {*
-(*comments containing lcp are the removal of fast_bv_of_nat*)
-let
-  fun is_const_bool (Const("True",_)) = true
-    | is_const_bool (Const("False",_)) = true
-    | is_const_bool _ = false
-  fun is_const_bit (Const("Word.bit.Zero",_)) = true
-    | is_const_bit (Const("Word.bit.One",_)) = true
-    | is_const_bit _ = false
-  fun num_is_usable (Const(@{const_name Int.Pls},_)) = true
-    | num_is_usable (Const(@{const_name Int.Min},_)) = false
-    | num_is_usable (Const(@{const_name Int.Bit0},_) $ w) =
-        num_is_usable w
-    | num_is_usable (Const(@{const_name Int.Bit1},_) $ w) =
-        num_is_usable w
-    | num_is_usable _ = false
-  fun vec_is_usable (Const("List.list.Nil",_)) = true
-    | vec_is_usable (Const("List.list.Cons",_) $ b $ bs) =
-        vec_is_usable bs andalso is_const_bit b
-    | vec_is_usable _ = false
-  (*lcp** val fast1_th = PureThy.get_thm thy "Word.fast_nat_to_bv_def"*)
-  val fast2_th = @{thm "Word.fast_bv_to_nat_def"};
-  (*lcp** fun f sg thms (Const("Word.nat_to_bv",_) $ (Const(@{const_name Int.number_of},_) $ t)) =
-    if num_is_usable t
-      then SOME (Drule.cterm_instantiate [(cterm_of sg (Var (("w", 0), @{typ int})), cterm_of sg t)] fast1_th)
-      else NONE
-    | f _ _ _ = NONE *)
-  fun g sg thms (Const("Word.bv_to_nat",_) $ (t as (Const("List.list.Cons",_) $ _ $ _))) =
-        if vec_is_usable t then
-          SOME (Drule.cterm_instantiate [(cterm_of sg (Var(("bs",0),Type("List.list",[Type("Word.bit",[])]))),cterm_of sg t)] fast2_th)
-        else NONE
-    | g _ _ _ = NONE
-  (*lcp** val simproc1 = Simplifier.simproc thy "nat_to_bv" ["Word.nat_to_bv (number_of w)"] f *)
-  val simproc2 = Simplifier.simproc @{theory} "bv_to_nat" ["Word.bv_to_nat (x # xs)"] g
-in
-  Simplifier.map_simpset (fn ss => ss addsimprocs [(*lcp*simproc1,*)simproc2])
-end*}
-
-declare bv_to_nat1 [simp del]
-declare bv_to_nat_helper [simp del]
-
-definition
-  bv_mapzip :: "[bit => bit => bit,bit list, bit list] => bit list" where
-  "bv_mapzip f w1 w2 =
-    (let g = bv_extend (max (length w1) (length w2)) \<zero>
-     in map (split f) (zip (g w1) (g w2)))"
-
-lemma bv_length_bv_mapzip [simp]:
-    "length (bv_mapzip f w1 w2) = max (length w1) (length w2)"
-  by (simp add: bv_mapzip_def Let_def split: split_max)
-
-lemma bv_mapzip_Nil [simp]: "bv_mapzip f [] [] = []"
-  by (simp add: bv_mapzip_def Let_def)
-
-lemma bv_mapzip_Cons [simp]: "length w1 = length w2 ==>
-    bv_mapzip f (x#w1) (y#w2) = f x y # bv_mapzip f w1 w2"
-  by (simp add: bv_mapzip_def Let_def)
-
-end
diff -r ef698bd61057 -r 5fb251d1c32f src/HOL/ex/Codegenerator_Candidates.thy
--- a/src/HOL/ex/Codegenerator_Candidates.thy	Sun May 16 00:02:11 2010 +0200
+++ b/src/HOL/ex/Codegenerator_Candidates.thy	Mon May 17 10:58:31 2010 +0200
@@ -23,7 +23,6 @@
   RBT
   SetsAndFunctions
   While_Combinator
-  Word
 begin
 
 inductive sublist :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" where
diff -r ef698bd61057 -r 5fb251d1c32f src/HOL/ex/ROOT.ML
--- a/src/HOL/ex/ROOT.ML	Sun May 16 00:02:11 2010 +0200
+++ b/src/HOL/ex/ROOT.ML	Mon May 17 10:58:31 2010 +0200
@@ -7,7 +7,6 @@
   "State_Monad",
   "Efficient_Nat_examples",
   "FuncSet",
-  "Word",
   "Eval_Examples",
   "Codegenerator_Test",
   "Codegenerator_Pretty_Test",
@@ -46,7 +45,6 @@
   "Unification",
   "Primrec",
   "Tarski",
-  "Adder",
   "Classical",
   "set",
   "Meson_Test",