# HG changeset patch
# User haftmann
# Date 1256907762 -3600
# Node ID 85adf83af7ce1c93b82984ee31f17eae0694110d
# Parent  ee5b5ef5e970abcdf3d14dc0ea6130499824b97c# Parent  1f18de40b43f8166a2ceeb85ff9f43aa7cac14fb
merged

diff -r ee5b5ef5e970 -r 85adf83af7ce NEWS
--- a/NEWS	Fri Oct 30 11:31:34 2009 +0100
+++ b/NEWS	Fri Oct 30 14:02:42 2009 +0100
@@ -37,6 +37,11 @@
 
 *** HOL ***
 
+* Combined former theories Divides and IntDiv to one theory Divides
+in the spirit of other number theory theories in HOL;  some constants
+(and to a lesser extent also facts) have been suffixed with _nat und _int
+respectively.  INCOMPATIBILITY.
+
 * Most rules produced by inductive and datatype package
 have mandatory prefixes.
 INCOMPATIBILITY.
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Decision_Procs/Commutative_Ring.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/Commutative_Ring.thy	Fri Oct 30 14:02:42 2009 +0100
@@ -0,0 +1,319 @@
+(*  Author:     Bernhard Haeupler
+
+Proving equalities in commutative rings done "right" in Isabelle/HOL.
+*)
+
+header {* Proving equalities in commutative rings *}
+
+theory Commutative_Ring
+imports Main Parity
+uses ("commutative_ring_tac.ML")
+begin
+
+text {* Syntax of multivariate polynomials (pol) and polynomial expressions. *}
+
+datatype 'a pol =
+    Pc 'a
+  | Pinj nat "'a pol"
+  | PX "'a pol" nat "'a pol"
+
+datatype 'a polex =
+    Pol "'a pol"
+  | Add "'a polex" "'a polex"
+  | Sub "'a polex" "'a polex"
+  | Mul "'a polex" "'a polex"
+  | Pow "'a polex" nat
+  | Neg "'a polex"
+
+text {* Interpretation functions for the shadow syntax. *}
+
+primrec
+  Ipol :: "'a::{comm_ring_1} list \<Rightarrow> 'a pol \<Rightarrow> 'a"
+where
+    "Ipol l (Pc c) = c"
+  | "Ipol l (Pinj i P) = Ipol (drop i l) P"
+  | "Ipol l (PX P x Q) = Ipol l P * (hd l)^x + Ipol (drop 1 l) Q"
+
+primrec
+  Ipolex :: "'a::{comm_ring_1} list \<Rightarrow> 'a polex \<Rightarrow> 'a"
+where
+    "Ipolex l (Pol P) = Ipol l P"
+  | "Ipolex l (Add P Q) = Ipolex l P + Ipolex l Q"
+  | "Ipolex l (Sub P Q) = Ipolex l P - Ipolex l Q"
+  | "Ipolex l (Mul P Q) = Ipolex l P * Ipolex l Q"
+  | "Ipolex l (Pow p n) = Ipolex l p ^ n"
+  | "Ipolex l (Neg P) = - Ipolex l P"
+
+text {* Create polynomial normalized polynomials given normalized inputs. *}
+
+definition
+  mkPinj :: "nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol" where
+  "mkPinj x P = (case P of
+    Pc c \<Rightarrow> Pc c |
+    Pinj y P \<Rightarrow> Pinj (x + y) P |
+    PX p1 y p2 \<Rightarrow> Pinj x P)"
+
+definition
+  mkPX :: "'a::{comm_ring} pol \<Rightarrow> nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol" where
+  "mkPX P i Q = (case P of
+    Pc c \<Rightarrow> (if (c = 0) then (mkPinj 1 Q) else (PX P i Q)) |
+    Pinj j R \<Rightarrow> PX P i Q |
+    PX P2 i2 Q2 \<Rightarrow> (if (Q2 = (Pc 0)) then (PX P2 (i+i2) Q) else (PX P i Q)) )"
+
+text {* Defining the basic ring operations on normalized polynomials *}
+
+function
+  add :: "'a::{comm_ring} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<oplus>" 65)
+where
+    "Pc a \<oplus> Pc b = Pc (a + b)"
+  | "Pc c \<oplus> Pinj i P = Pinj i (P \<oplus> Pc c)"
+  | "Pinj i P \<oplus> Pc c = Pinj i (P \<oplus> Pc c)"
+  | "Pc c \<oplus> PX P i Q = PX P i (Q \<oplus> Pc c)"
+  | "PX P i Q \<oplus> Pc c = PX P i (Q \<oplus> Pc c)"
+  | "Pinj x P \<oplus> Pinj y Q =
+      (if x = y then mkPinj x (P \<oplus> Q)
+       else (if x > y then mkPinj y (Pinj (x - y) P \<oplus> Q)
+         else mkPinj x (Pinj (y - x) Q \<oplus> P)))"
+  | "Pinj x P \<oplus> PX Q y R =
+      (if x = 0 then P \<oplus> PX Q y R
+       else (if x = 1 then PX Q y (R \<oplus> P)
+         else PX Q y (R \<oplus> Pinj (x - 1) P)))"
+  | "PX P x R \<oplus> Pinj y Q =
+      (if y = 0 then PX P x R \<oplus> Q
+       else (if y = 1 then PX P x (R \<oplus> Q)
+         else PX P x (R \<oplus> Pinj (y - 1) Q)))"
+  | "PX P1 x P2 \<oplus> PX Q1 y Q2 =
+      (if x = y then mkPX (P1 \<oplus> Q1) x (P2 \<oplus> Q2)
+       else (if x > y then mkPX (PX P1 (x - y) (Pc 0) \<oplus> Q1) y (P2 \<oplus> Q2)
+         else mkPX (PX Q1 (y-x) (Pc 0) \<oplus> P1) x (P2 \<oplus> Q2)))"
+by pat_completeness auto
+termination by (relation "measure (\<lambda>(x, y). size x + size y)") auto
+
+function
+  mul :: "'a::{comm_ring} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<otimes>" 70)
+where
+    "Pc a \<otimes> Pc b = Pc (a * b)"
+  | "Pc c \<otimes> Pinj i P =
+      (if c = 0 then Pc 0 else mkPinj i (P \<otimes> Pc c))"
+  | "Pinj i P \<otimes> Pc c =
+      (if c = 0 then Pc 0 else mkPinj i (P \<otimes> Pc c))"
+  | "Pc c \<otimes> PX P i Q =
+      (if c = 0 then Pc 0 else mkPX (P \<otimes> Pc c) i (Q \<otimes> Pc c))"
+  | "PX P i Q \<otimes> Pc c =
+      (if c = 0 then Pc 0 else mkPX (P \<otimes> Pc c) i (Q \<otimes> Pc c))"
+  | "Pinj x P \<otimes> Pinj y Q =
+      (if x = y then mkPinj x (P \<otimes> Q) else
+         (if x > y then mkPinj y (Pinj (x-y) P \<otimes> Q)
+           else mkPinj x (Pinj (y - x) Q \<otimes> P)))"
+  | "Pinj x P \<otimes> PX Q y R =
+      (if x = 0 then P \<otimes> PX Q y R else
+         (if x = 1 then mkPX (Pinj x P \<otimes> Q) y (R \<otimes> P)
+           else mkPX (Pinj x P \<otimes> Q) y (R \<otimes> Pinj (x - 1) P)))"
+  | "PX P x R \<otimes> Pinj y Q =
+      (if y = 0 then PX P x R \<otimes> Q else
+         (if y = 1 then mkPX (Pinj y Q \<otimes> P) x (R \<otimes> Q)
+           else mkPX (Pinj y Q \<otimes> P) x (R \<otimes> Pinj (y - 1) Q)))"
+  | "PX P1 x P2 \<otimes> PX Q1 y Q2 =
+      mkPX (P1 \<otimes> Q1) (x + y) (P2 \<otimes> Q2) \<oplus>
+        (mkPX (P1 \<otimes> mkPinj 1 Q2) x (Pc 0) \<oplus>
+          (mkPX (Q1 \<otimes> mkPinj 1 P2) y (Pc 0)))"
+by pat_completeness auto
+termination by (relation "measure (\<lambda>(x, y). size x + size y)")
+  (auto simp add: mkPinj_def split: pol.split)
+
+text {* Negation*}
+primrec
+  neg :: "'a::{comm_ring} pol \<Rightarrow> 'a pol"
+where
+    "neg (Pc c) = Pc (-c)"
+  | "neg (Pinj i P) = Pinj i (neg P)"
+  | "neg (PX P x Q) = PX (neg P) x (neg Q)"
+
+text {* Substraction *}
+definition
+  sub :: "'a::{comm_ring} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<ominus>" 65)
+where
+  "sub P Q = P \<oplus> neg Q"
+
+text {* Square for Fast Exponentation *}
+primrec
+  sqr :: "'a::{comm_ring_1} pol \<Rightarrow> 'a pol"
+where
+    "sqr (Pc c) = Pc (c * c)"
+  | "sqr (Pinj i P) = mkPinj i (sqr P)"
+  | "sqr (PX A x B) = mkPX (sqr A) (x + x) (sqr B) \<oplus>
+      mkPX (Pc (1 + 1) \<otimes> A \<otimes> mkPinj 1 B) x (Pc 0)"
+
+text {* Fast Exponentation *}
+fun
+  pow :: "nat \<Rightarrow> 'a::{comm_ring_1} pol \<Rightarrow> 'a pol"
+where
+    "pow 0 P = Pc 1"
+  | "pow n P = (if even n then pow (n div 2) (sqr P)
+       else P \<otimes> pow (n div 2) (sqr P))"
+  
+lemma pow_if:
+  "pow n P =
+   (if n = 0 then Pc 1 else if even n then pow (n div 2) (sqr P)
+    else P \<otimes> pow (n div 2) (sqr P))"
+  by (cases n) simp_all
+
+
+text {* Normalization of polynomial expressions *}
+
+primrec
+  norm :: "'a::{comm_ring_1} polex \<Rightarrow> 'a pol"
+where
+    "norm (Pol P) = P"
+  | "norm (Add P Q) = norm P \<oplus> norm Q"
+  | "norm (Sub P Q) = norm P \<ominus> norm Q"
+  | "norm (Mul P Q) = norm P \<otimes> norm Q"
+  | "norm (Pow P n) = pow n (norm P)"
+  | "norm (Neg P) = neg (norm P)"
+
+text {* mkPinj preserve semantics *}
+lemma mkPinj_ci: "Ipol l (mkPinj a B) = Ipol l (Pinj a B)"
+  by (induct B) (auto simp add: mkPinj_def algebra_simps)
+
+text {* mkPX preserves semantics *}
+lemma mkPX_ci: "Ipol l (mkPX A b C) = Ipol l (PX A b C)"
+  by (cases A) (auto simp add: mkPX_def mkPinj_ci power_add algebra_simps)
+
+text {* Correctness theorems for the implemented operations *}
+
+text {* Negation *}
+lemma neg_ci: "Ipol l (neg P) = -(Ipol l P)"
+  by (induct P arbitrary: l) auto
+
+text {* Addition *}
+lemma add_ci: "Ipol l (P \<oplus> Q) = Ipol l P + Ipol l Q"
+proof (induct P Q arbitrary: l rule: add.induct)
+  case (6 x P y Q)
+  show ?case
+  proof (rule linorder_cases)
+    assume "x < y"
+    with 6 show ?case by (simp add: mkPinj_ci algebra_simps)
+  next
+    assume "x = y"
+    with 6 show ?case by (simp add: mkPinj_ci)
+  next
+    assume "x > y"
+    with 6 show ?case by (simp add: mkPinj_ci algebra_simps)
+  qed
+next
+  case (7 x P Q y R)
+  have "x = 0 \<or> x = 1 \<or> x > 1" by arith
+  moreover
+  { assume "x = 0" with 7 have ?case by simp }
+  moreover
+  { assume "x = 1" with 7 have ?case by (simp add: algebra_simps) }
+  moreover
+  { assume "x > 1" from 7 have ?case by (cases x) simp_all }
+  ultimately show ?case by blast
+next
+  case (8 P x R y Q)
+  have "y = 0 \<or> y = 1 \<or> y > 1" by arith
+  moreover
+  { assume "y = 0" with 8 have ?case by simp }
+  moreover
+  { assume "y = 1" with 8 have ?case by simp }
+  moreover
+  { assume "y > 1" with 8 have ?case by simp }
+  ultimately show ?case by blast
+next
+  case (9 P1 x P2 Q1 y Q2)
+  show ?case
+  proof (rule linorder_cases)
+    assume a: "x < y" hence "EX d. d + x = y" by arith
+    with 9 a show ?case by (auto simp add: mkPX_ci power_add algebra_simps)
+  next
+    assume a: "y < x" hence "EX d. d + y = x" by arith
+    with 9 a show ?case by (auto simp add: power_add mkPX_ci algebra_simps)
+  next
+    assume "x = y"
+    with 9 show ?case by (simp add: mkPX_ci algebra_simps)
+  qed
+qed (auto simp add: algebra_simps)
+
+text {* Multiplication *}
+lemma mul_ci: "Ipol l (P \<otimes> Q) = Ipol l P * Ipol l Q"
+  by (induct P Q arbitrary: l rule: mul.induct)
+    (simp_all add: mkPX_ci mkPinj_ci algebra_simps add_ci power_add)
+
+text {* Substraction *}
+lemma sub_ci: "Ipol l (P \<ominus> Q) = Ipol l P - Ipol l Q"
+  by (simp add: add_ci neg_ci sub_def)
+
+text {* Square *}
+lemma sqr_ci: "Ipol ls (sqr P) = Ipol ls P * Ipol ls P"
+  by (induct P arbitrary: ls)
+    (simp_all add: add_ci mkPinj_ci mkPX_ci mul_ci algebra_simps power_add)
+
+text {* Power *}
+lemma even_pow:"even n \<Longrightarrow> pow n P = pow (n div 2) (sqr P)"
+  by (induct n) simp_all
+
+lemma pow_ci: "Ipol ls (pow n P) = Ipol ls P ^ n"
+proof (induct n arbitrary: P rule: nat_less_induct)
+  case (1 k)
+  show ?case
+  proof (cases k)
+    case 0
+    then show ?thesis by simp
+  next
+    case (Suc l)
+    show ?thesis
+    proof cases
+      assume "even l"
+      then have "Suc l div 2 = l div 2"
+        by (simp add: nat_number even_nat_plus_one_div_two)
+      moreover
+      from Suc have "l < k" by simp
+      with 1 have "\<And>P. Ipol ls (pow l P) = Ipol ls P ^ l" by simp
+      moreover
+      note Suc `even l` even_nat_plus_one_div_two
+      ultimately show ?thesis by (auto simp add: mul_ci power_Suc even_pow)
+    next
+      assume "odd l"
+      {
+        fix p
+        have "Ipol ls (sqr P) ^ (Suc l div 2) = Ipol ls P ^ Suc l"
+        proof (cases l)
+          case 0
+          with `odd l` show ?thesis by simp
+        next
+          case (Suc w)
+          with `odd l` have "even w" by simp
+          have two_times: "2 * (w div 2) = w"
+            by (simp only: numerals even_nat_div_two_times_two [OF `even w`])
+          have "Ipol ls P * Ipol ls P = Ipol ls P ^ Suc (Suc 0)"
+            by (simp add: power_Suc)
+          then have "Ipol ls P * Ipol ls P = Ipol ls P ^ 2"
+            by (simp add: numerals)
+          with Suc show ?thesis
+            by (auto simp add: power_mult [symmetric, of _ 2 _] two_times mul_ci sqr_ci
+                     simp del: power_Suc)
+        qed
+      } with 1 Suc `odd l` show ?thesis by simp
+    qed
+  qed
+qed
+
+text {* Normalization preserves semantics  *}
+lemma norm_ci: "Ipolex l Pe = Ipol l (norm Pe)"
+  by (induct Pe) (simp_all add: add_ci sub_ci mul_ci neg_ci pow_ci)
+
+text {* Reflection lemma: Key to the (incomplete) decision procedure *}
+lemma norm_eq:
+  assumes "norm P1 = norm P2"
+  shows "Ipolex l P1 = Ipolex l P2"
+proof -
+  from prems have "Ipol l (norm P1) = Ipol l (norm P2)" by simp
+  then show ?thesis by (simp only: norm_ci)
+qed
+
+
+use "commutative_ring_tac.ML"
+setup Commutative_Ring_Tac.setup
+
+end
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Decision_Procs/Commutative_Ring_Complete.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/Commutative_Ring_Complete.thy	Fri Oct 30 14:02:42 2009 +0100
@@ -0,0 +1,391 @@
+(*  Author:     Bernhard Haeupler
+
+This theory is about of the relative completeness of method comm-ring
+method.  As long as the reified atomic polynomials of type 'a pol are
+in normal form, the cring method is complete.
+*)
+
+header {* Proof of the relative completeness of method comm-ring *}
+
+theory Commutative_Ring_Complete
+imports Commutative_Ring
+begin
+
+text {* Formalization of normal form *}
+fun
+  isnorm :: "('a::{comm_ring}) pol \<Rightarrow> bool"
+where
+    "isnorm (Pc c) \<longleftrightarrow> True"
+  | "isnorm (Pinj i (Pc c)) \<longleftrightarrow> False"
+  | "isnorm (Pinj i (Pinj j Q)) \<longleftrightarrow> False"
+  | "isnorm (Pinj 0 P) \<longleftrightarrow> False"
+  | "isnorm (Pinj i (PX Q1 j Q2)) \<longleftrightarrow> isnorm (PX Q1 j Q2)"
+  | "isnorm (PX P 0 Q) \<longleftrightarrow> False"
+  | "isnorm (PX (Pc c) i Q) \<longleftrightarrow> c \<noteq> 0 \<and> isnorm Q"
+  | "isnorm (PX (PX P1 j (Pc c)) i Q) \<longleftrightarrow> c \<noteq> 0 \<and> isnorm (PX P1 j (Pc c)) \<and> isnorm Q"
+  | "isnorm (PX P i Q) \<longleftrightarrow> isnorm P \<and> isnorm Q"
+
+(* Some helpful lemmas *)
+lemma norm_Pinj_0_False:"isnorm (Pinj 0 P) = False"
+by(cases P, auto)
+
+lemma norm_PX_0_False:"isnorm (PX (Pc 0) i Q) = False"
+by(cases i, auto)
+
+lemma norm_Pinj:"isnorm (Pinj i Q) \<Longrightarrow> isnorm Q"
+by(cases i,simp add: norm_Pinj_0_False norm_PX_0_False,cases Q) auto
+
+lemma norm_PX2:"isnorm (PX P i Q) \<Longrightarrow> isnorm Q"
+by(cases i, auto, cases P, auto, case_tac pol2, auto)
+
+lemma norm_PX1:"isnorm (PX P i Q) \<Longrightarrow> isnorm P"
+by(cases i, auto, cases P, auto, case_tac pol2, auto)
+
+lemma mkPinj_cn:"\<lbrakk>y~=0; isnorm Q\<rbrakk> \<Longrightarrow> isnorm (mkPinj y Q)" 
+apply(auto simp add: mkPinj_def norm_Pinj_0_False split: pol.split)
+apply(case_tac nat, auto simp add: norm_Pinj_0_False)
+by(case_tac pol, auto) (case_tac y, auto)
+
+lemma norm_PXtrans: 
+  assumes A:"isnorm (PX P x Q)" and "isnorm Q2" 
+  shows "isnorm (PX P x Q2)"
+proof(cases P)
+  case (PX p1 y p2) from prems show ?thesis by(cases x, auto, cases p2, auto)
+next
+  case Pc from prems show ?thesis by(cases x, auto)
+next
+  case Pinj from prems show ?thesis by(cases x, auto)
+qed
+ 
+lemma norm_PXtrans2: assumes A:"isnorm (PX P x Q)" and "isnorm Q2" shows "isnorm (PX P (Suc (n+x)) Q2)"
+proof(cases P)
+  case (PX p1 y p2)
+  from prems show ?thesis by(cases x, auto, cases p2, auto)
+next
+  case Pc
+  from prems show ?thesis by(cases x, auto)
+next
+  case Pinj
+  from prems show ?thesis by(cases x, auto)
+qed
+
+text {* mkPX conserves normalizedness (@{text "_cn"}) *}
+lemma mkPX_cn: 
+  assumes "x \<noteq> 0" and "isnorm P" and "isnorm Q" 
+  shows "isnorm (mkPX P x Q)"
+proof(cases P)
+  case (Pc c)
+  from prems show ?thesis by (cases x) (auto simp add: mkPinj_cn mkPX_def)
+next
+  case (Pinj i Q)
+  from prems show ?thesis by (cases x) (auto simp add: mkPinj_cn mkPX_def)
+next
+  case (PX P1 y P2)
+  from prems have Y0:"y>0" by(cases y, auto)
+  from prems have "isnorm P1" "isnorm P2" by (auto simp add: norm_PX1[of P1 y P2] norm_PX2[of P1 y P2])
+  with prems Y0 show ?thesis by (cases x, auto simp add: mkPX_def norm_PXtrans2[of P1 y _ Q _], cases P2, auto)
+qed
+
+text {* add conserves normalizedness *}
+lemma add_cn:"isnorm P \<Longrightarrow> isnorm Q \<Longrightarrow> isnorm (P \<oplus> Q)"
+proof(induct P Q rule: add.induct)
+  case (2 c i P2) thus ?case by (cases P2, simp_all, cases i, simp_all)
+next
+  case (3 i P2 c) thus ?case by (cases P2, simp_all, cases i, simp_all)
+next
+  case (4 c P2 i Q2)
+  from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
+  with prems show ?case by(cases i, simp, cases P2, auto, case_tac pol2, auto)
+next
+  case (5 P2 i Q2 c)
+  from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
+  with prems show ?case by(cases i, simp, cases P2, auto, case_tac pol2, auto)
+next
+  case (6 x P2 y Q2)
+  from prems have Y0:"y>0" by (cases y, auto simp add: norm_Pinj_0_False) 
+  from prems have X0:"x>0" by (cases x, auto simp add: norm_Pinj_0_False) 
+  have "x < y \<or> x = y \<or> x > y" by arith
+  moreover
+  { assume "x<y" hence "EX d. y=d+x" by arith
+    then obtain d where "y=d+x"..
+    moreover
+    note prems X0
+    moreover
+    from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+    moreover
+    with prems have "isnorm (Pinj d Q2)" by (cases d, simp, cases Q2, auto)
+    ultimately have ?case by (simp add: mkPinj_cn)}
+  moreover
+  { assume "x=y"
+    moreover
+    from prems have "isnorm P2" "isnorm Q2" by(auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+    moreover
+    note prems Y0
+    moreover
+    ultimately have ?case by (simp add: mkPinj_cn) }
+  moreover
+  { assume "x>y" hence "EX d. x=d+y" by arith
+    then obtain d where "x=d+y"..
+    moreover
+    note prems Y0
+    moreover
+    from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+    moreover
+    with prems have "isnorm (Pinj d P2)" by (cases d, simp, cases P2, auto)
+    ultimately have ?case by (simp add: mkPinj_cn)}
+  ultimately show ?case by blast
+next
+  case (7 x P2 Q2 y R)
+  have "x=0 \<or> (x = 1) \<or> (x > 1)" by arith
+  moreover
+  { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
+  moreover
+  { assume "x=1"
+    from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
+    with prems have "isnorm (R \<oplus> P2)" by simp
+    with prems have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
+  moreover
+  { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
+    then obtain d where X:"x=Suc (Suc d)" ..
+    from prems have NR:"isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
+    with prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
+    with prems NR have "isnorm (R \<oplus> Pinj (x - 1) P2)" "isnorm (PX Q2 y R)" by simp fact
+    with X have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
+  ultimately show ?case by blast
+next
+  case (8 Q2 y R x P2)
+  have "x = 0 \<or> x = 1 \<or> x > 1" by arith
+  moreover
+  { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
+  moreover
+  { assume "x=1"
+    from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
+    with prems have "isnorm (R \<oplus> P2)" by simp
+    with prems have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
+  moreover
+  { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
+    then obtain d where X:"x=Suc (Suc d)" ..
+    from prems have NR:"isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
+    with prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
+    with prems NR have "isnorm (R \<oplus> Pinj (x - 1) P2)" "isnorm (PX Q2 y R)" by simp fact
+    with X have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
+  ultimately show ?case by blast
+next
+  case (9 P1 x P2 Q1 y Q2)
+  from prems have Y0:"y>0" by(cases y, auto)
+  from prems have X0:"x>0" by(cases x, auto)
+  from prems have NP1:"isnorm P1" and NP2:"isnorm P2" by (auto simp add: norm_PX1[of P1 _ P2] norm_PX2[of P1 _ P2])
+  from prems have NQ1:"isnorm Q1" and NQ2:"isnorm Q2" by (auto simp add: norm_PX1[of Q1 _ Q2] norm_PX2[of Q1 _ Q2])
+  have "y < x \<or> x = y \<or> x < y" by arith
+  moreover
+  {assume sm1:"y < x" hence "EX d. x=d+y" by arith
+    then obtain d where sm2:"x=d+y"..
+    note prems NQ1 NP1 NP2 NQ2 sm1 sm2
+    moreover
+    have "isnorm (PX P1 d (Pc 0))" 
+    proof(cases P1)
+      case (PX p1 y p2)
+      with prems show ?thesis by(cases d, simp,cases p2, auto)
+    next case Pc   from prems show ?thesis by(cases d, auto)
+    next case Pinj from prems show ?thesis by(cases d, auto)
+    qed
+    ultimately have "isnorm (P2 \<oplus> Q2)" "isnorm (PX P1 (x - y) (Pc 0) \<oplus> Q1)" by auto
+    with Y0 sm1 sm2 have ?case by (simp add: mkPX_cn)}
+  moreover
+  {assume "x=y"
+    from prems NP1 NP2 NQ1 NQ2 have "isnorm (P2 \<oplus> Q2)" "isnorm (P1 \<oplus> Q1)" by auto
+    with Y0 prems have ?case by (simp add: mkPX_cn) }
+  moreover
+  {assume sm1:"x<y" hence "EX d. y=d+x" by arith
+    then obtain d where sm2:"y=d+x"..
+    note prems NQ1 NP1 NP2 NQ2 sm1 sm2
+    moreover
+    have "isnorm (PX Q1 d (Pc 0))" 
+    proof(cases Q1)
+      case (PX p1 y p2)
+      with prems show ?thesis by(cases d, simp,cases p2, auto)
+    next case Pc   from prems show ?thesis by(cases d, auto)
+    next case Pinj from prems show ?thesis by(cases d, auto)
+    qed
+    ultimately have "isnorm (P2 \<oplus> Q2)" "isnorm (PX Q1 (y - x) (Pc 0) \<oplus> P1)" by auto
+    with X0 sm1 sm2 have ?case by (simp add: mkPX_cn)}
+  ultimately show ?case by blast
+qed simp
+
+text {* mul concerves normalizedness *}
+lemma mul_cn :"isnorm P \<Longrightarrow> isnorm Q \<Longrightarrow> isnorm (P \<otimes> Q)"
+proof(induct P Q rule: mul.induct)
+  case (2 c i P2) thus ?case 
+    by (cases P2, simp_all) (cases "i",simp_all add: mkPinj_cn)
+next
+  case (3 i P2 c) thus ?case 
+    by (cases P2, simp_all) (cases "i",simp_all add: mkPinj_cn)
+next
+  case (4 c P2 i Q2)
+  from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
+  with prems show ?case 
+    by - (case_tac "c=0",simp_all,case_tac "i=0",simp_all add: mkPX_cn)
+next
+  case (5 P2 i Q2 c)
+  from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
+  with prems show ?case
+    by - (case_tac "c=0",simp_all,case_tac "i=0",simp_all add: mkPX_cn)
+next
+  case (6 x P2 y Q2)
+  have "x < y \<or> x = y \<or> x > y" by arith
+  moreover
+  { assume "x<y" hence "EX d. y=d+x" by arith
+    then obtain d where "y=d+x"..
+    moreover
+    note prems
+    moreover
+    from prems have "x>0" by (cases x, auto simp add: norm_Pinj_0_False) 
+    moreover
+    from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+    moreover
+    with prems have "isnorm (Pinj d Q2)" by (cases d, simp, cases Q2, auto) 
+    ultimately have ?case by (simp add: mkPinj_cn)}
+  moreover
+  { assume "x=y"
+    moreover
+    from prems have "isnorm P2" "isnorm Q2" by(auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+    moreover
+    with prems have "y>0" by (cases y, auto simp add: norm_Pinj_0_False)
+    moreover
+    note prems
+    moreover
+    ultimately have ?case by (simp add: mkPinj_cn) }
+  moreover
+  { assume "x>y" hence "EX d. x=d+y" by arith
+    then obtain d where "x=d+y"..
+    moreover
+    note prems
+    moreover
+    from prems have "y>0" by (cases y, auto simp add: norm_Pinj_0_False) 
+    moreover
+    from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
+    moreover
+    with prems have "isnorm (Pinj d P2)"  by (cases d, simp, cases P2, auto)
+    ultimately have ?case by (simp add: mkPinj_cn) }
+  ultimately show ?case by blast
+next
+  case (7 x P2 Q2 y R)
+  from prems have Y0:"y>0" by(cases y, auto)
+  have "x=0 \<or> (x = 1) \<or> (x > 1)" by arith
+  moreover
+  { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
+  moreover
+  { assume "x=1"
+    from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
+    with prems have "isnorm (R \<otimes> P2)" "isnorm Q2" by (auto simp add: norm_PX1[of Q2 y R])
+    with Y0 prems have ?case by (simp add: mkPX_cn)}
+  moreover
+  { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
+    then obtain d where X:"x=Suc (Suc d)" ..
+    from prems have NR:"isnorm R" "isnorm Q2" by (auto simp add: norm_PX2[of Q2 y R] norm_PX1[of Q2 y R])
+    moreover
+    from prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
+    moreover
+    from prems have "isnorm (Pinj x P2)" by(cases P2, auto)
+    moreover
+    note prems
+    ultimately have "isnorm (R \<otimes> Pinj (x - 1) P2)" "isnorm (Pinj x P2 \<otimes> Q2)" by auto
+    with Y0 X have ?case by (simp add: mkPX_cn)}
+  ultimately show ?case by blast
+next
+  case (8 Q2 y R x P2)
+  from prems have Y0:"y>0" by(cases y, auto)
+  have "x=0 \<or> (x = 1) \<or> (x > 1)" by arith
+  moreover
+  { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
+  moreover
+  { assume "x=1"
+    from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
+    with prems have "isnorm (R \<otimes> P2)" "isnorm Q2" by (auto simp add: norm_PX1[of Q2 y R])
+    with Y0 prems have ?case by (simp add: mkPX_cn) }
+  moreover
+  { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
+    then obtain d where X:"x=Suc (Suc d)" ..
+    from prems have NR:"isnorm R" "isnorm Q2" by (auto simp add: norm_PX2[of Q2 y R] norm_PX1[of Q2 y R])
+    moreover
+    from prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
+    moreover
+    from prems have "isnorm (Pinj x P2)" by(cases P2, auto)
+    moreover
+    note prems
+    ultimately have "isnorm (R \<otimes> Pinj (x - 1) P2)" "isnorm (Pinj x P2 \<otimes> Q2)" by auto
+    with Y0 X have ?case by (simp add: mkPX_cn) }
+  ultimately show ?case by blast
+next
+  case (9 P1 x P2 Q1 y Q2)
+  from prems have X0:"x>0" by(cases x, auto)
+  from prems have Y0:"y>0" by(cases y, auto)
+  note prems
+  moreover
+  from prems have "isnorm P1" "isnorm P2" by (auto simp add: norm_PX1[of P1 x P2] norm_PX2[of P1 x P2])
+  moreover 
+  from prems have "isnorm Q1" "isnorm Q2" by (auto simp add: norm_PX1[of Q1 y Q2] norm_PX2[of Q1 y Q2])
+  ultimately have "isnorm (P1 \<otimes> Q1)" "isnorm (P2 \<otimes> Q2)"
+    "isnorm (P1 \<otimes> mkPinj 1 Q2)" "isnorm (Q1 \<otimes> mkPinj 1 P2)" 
+    by (auto simp add: mkPinj_cn)
+  with prems X0 Y0 have
+    "isnorm (mkPX (P1 \<otimes> Q1) (x + y) (P2 \<otimes> Q2))"
+    "isnorm (mkPX (P1 \<otimes> mkPinj (Suc 0) Q2) x (Pc 0))"  
+    "isnorm (mkPX (Q1 \<otimes> mkPinj (Suc 0) P2) y (Pc 0))" 
+    by (auto simp add: mkPX_cn)
+  thus ?case by (simp add: add_cn)
+qed(simp)
+
+text {* neg conserves normalizedness *}
+lemma neg_cn: "isnorm P \<Longrightarrow> isnorm (neg P)"
+proof (induct P)
+  case (Pinj i P2)
+  from prems have "isnorm P2" by (simp add: norm_Pinj[of i P2])
+  with prems show ?case by(cases P2, auto, cases i, auto)
+next
+  case (PX P1 x P2)
+  from prems have "isnorm P2" "isnorm P1" by (auto simp add: norm_PX1[of P1 x P2] norm_PX2[of P1 x P2])
+  with prems show ?case
+  proof(cases P1)
+    case (PX p1 y p2)
+    with prems show ?thesis by(cases x, auto, cases p2, auto)
+  next
+    case Pinj
+    with prems show ?thesis by(cases x, auto)
+  qed(cases x, auto)
+qed(simp)
+
+text {* sub conserves normalizedness *}
+lemma sub_cn:"isnorm p \<Longrightarrow> isnorm q \<Longrightarrow> isnorm (p \<ominus> q)"
+by (simp add: sub_def add_cn neg_cn)
+
+text {* sqr conserves normalizizedness *}
+lemma sqr_cn:"isnorm P \<Longrightarrow> isnorm (sqr P)"
+proof(induct P)
+  case (Pinj i Q)
+  from prems show ?case by(cases Q, auto simp add: mkPX_cn mkPinj_cn, cases i, auto simp add: mkPX_cn mkPinj_cn)
+next 
+  case (PX P1 x P2)
+  from prems have "x+x~=0" "isnorm P2" "isnorm P1" by (cases x,  auto simp add: norm_PX1[of P1 x P2] norm_PX2[of P1 x P2])
+  with prems have
+    "isnorm (mkPX (Pc (1 + 1) \<otimes> P1 \<otimes> mkPinj (Suc 0) P2) x (Pc 0))"
+    and "isnorm (mkPX (sqr P1) (x + x) (sqr P2))"
+   by (auto simp add: add_cn mkPX_cn mkPinj_cn mul_cn)
+  thus ?case by (auto simp add: add_cn mkPX_cn mkPinj_cn mul_cn)
+qed simp
+
+text {* pow conserves normalizedness *}
+lemma pow_cn:"isnorm P \<Longrightarrow> isnorm (pow n P)"
+proof (induct n arbitrary: P rule: nat_less_induct)
+  case (1 k)
+  show ?case 
+  proof (cases "k=0")
+    case False
+    then have K2:"k div 2 < k" by (cases k, auto)
+    from prems have "isnorm (sqr P)" by (simp add: sqr_cn)
+    with prems K2 show ?thesis
+    by (simp add: allE[of _ "(k div 2)" _] allE[of _ "(sqr P)" _], cases k, auto simp add: mul_cn)
+  qed simp
+qed
+
+end
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Decision_Procs/Decision_Procs.thy
--- a/src/HOL/Decision_Procs/Decision_Procs.thy	Fri Oct 30 11:31:34 2009 +0100
+++ b/src/HOL/Decision_Procs/Decision_Procs.thy	Fri Oct 30 14:02:42 2009 +0100
@@ -1,7 +1,10 @@
-header {* Various decision procedures. typically involving reflection *}
+header {* Various decision procedures, typically involving reflection *}
 
 theory Decision_Procs
-imports Cooper Ferrack MIR Approximation Dense_Linear_Order "ex/Approximation_Ex" "ex/Dense_Linear_Order_Ex" Parametric_Ferrante_Rackoff
+imports
+  Commutative_Ring Cooper Ferrack MIR Approximation Dense_Linear_Order Parametric_Ferrante_Rackoff
+  Commutative_Ring_Complete
+  "ex/Commutative_Ring_Ex" "ex/Approximation_Ex" "ex/Dense_Linear_Order_Ex"
 begin
 
 end
\ No newline at end of file
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Decision_Procs/commutative_ring_tac.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/commutative_ring_tac.ML	Fri Oct 30 14:02:42 2009 +0100
@@ -0,0 +1,104 @@
+(*  Author:     Amine Chaieb
+
+Tactic for solving equalities over commutative rings.
+*)
+
+signature COMMUTATIVE_RING_TAC =
+sig
+  val tac: Proof.context -> int -> tactic
+  val setup: theory -> theory
+end
+
+structure Commutative_Ring_Tac: COMMUTATIVE_RING_TAC =
+struct
+
+(* Zero and One of the commutative ring *)
+fun cring_zero T = Const (@{const_name HOL.zero}, T);
+fun cring_one T = Const (@{const_name HOL.one}, T);
+
+(* reification functions *)
+(* add two polynom expressions *)
+fun polT t = Type (@{type_name Commutative_Ring.pol}, [t]);
+fun polexT t = Type (@{type_name Commutative_Ring.polex}, [t]);
+
+(* pol *)
+fun pol_Pc t = Const (@{const_name Commutative_Ring.pol.Pc}, t --> polT t);
+fun pol_Pinj t = Const (@{const_name Commutative_Ring.pol.Pinj}, HOLogic.natT --> polT t --> polT t);
+fun pol_PX t = Const (@{const_name Commutative_Ring.pol.PX}, polT t --> HOLogic.natT --> polT t --> polT t);
+
+(* polex *)
+fun polex_add t = Const (@{const_name Commutative_Ring.polex.Add}, polexT t --> polexT t --> polexT t);
+fun polex_sub t = Const (@{const_name Commutative_Ring.polex.Sub}, polexT t --> polexT t --> polexT t);
+fun polex_mul t = Const (@{const_name Commutative_Ring.polex.Mul}, polexT t --> polexT t --> polexT t);
+fun polex_neg t = Const (@{const_name Commutative_Ring.polex.Neg}, polexT t --> polexT t);
+fun polex_pol t = Const (@{const_name Commutative_Ring.polex.Pol}, polT t --> polexT t);
+fun polex_pow t = Const (@{const_name Commutative_Ring.polex.Pow}, polexT t --> HOLogic.natT --> polexT t);
+
+(* reification of polynoms : primitive cring expressions *)
+fun reif_pol T vs (t as Free _) =
+      let
+        val one = @{term "1::nat"};
+        val i = find_index (fn t' => t' = t) vs
+      in if i = 0
+        then pol_PX T $ (pol_Pc T $ cring_one T)
+          $ one $ (pol_Pc T $ cring_zero T)
+        else pol_Pinj T $ HOLogic.mk_nat i
+          $ (pol_PX T $ (pol_Pc T $ cring_one T)
+            $ one $ (pol_Pc T $ cring_zero T))
+        end
+  | reif_pol T vs t = pol_Pc T $ t;
+
+(* reification of polynom expressions *)
+fun reif_polex T vs (Const (@{const_name HOL.plus}, _) $ a $ b) =
+      polex_add T $ reif_polex T vs a $ reif_polex T vs b
+  | reif_polex T vs (Const (@{const_name HOL.minus}, _) $ a $ b) =
+      polex_sub T $ reif_polex T vs a $ reif_polex T vs b
+  | reif_polex T vs (Const (@{const_name HOL.times}, _) $ a $ b) =
+      polex_mul T $ reif_polex T vs a $ reif_polex T vs b
+  | reif_polex T vs (Const (@{const_name HOL.uminus}, _) $ a) =
+      polex_neg T $ reif_polex T vs a
+  | reif_polex T vs (Const (@{const_name Power.power}, _) $ a $ n) =
+      polex_pow T $ reif_polex T vs a $ n
+  | reif_polex T vs t = polex_pol T $ reif_pol T vs t;
+
+(* reification of the equation *)
+val cr_sort = @{sort "comm_ring_1"};
+
+fun reif_eq thy (eq as Const(@{const_name "op ="}, Type("fun", [T, _])) $ lhs $ rhs) =
+      if Sign.of_sort thy (T, cr_sort) then
+        let
+          val fs = OldTerm.term_frees eq;
+          val cvs = cterm_of thy (HOLogic.mk_list T fs);
+          val clhs = cterm_of thy (reif_polex T fs lhs);
+          val crhs = cterm_of thy (reif_polex T fs rhs);
+          val ca = ctyp_of thy T;
+        in (ca, cvs, clhs, crhs) end
+      else error ("reif_eq: not an equation over " ^ Syntax.string_of_sort_global thy cr_sort)
+  | reif_eq _ _ = error "reif_eq: not an equation";
+
+(* The cring tactic *)
+(* Attention: You have to make sure that no t^0 is in the goal!! *)
+(* Use simply rewriting t^0 = 1 *)
+val cring_simps =
+  [@{thm mkPX_def}, @{thm mkPinj_def}, @{thm sub_def}, @{thm power_add},
+    @{thm even_def}, @{thm pow_if}, sym OF [@{thm power_add}]];
+
+fun tac ctxt = SUBGOAL (fn (g, i) =>
+  let
+    val thy = ProofContext.theory_of ctxt;
+    val cring_ss = Simplifier.simpset_of ctxt  (*FIXME really the full simpset!?*)
+      addsimps cring_simps;
+    val (ca, cvs, clhs, crhs) = reif_eq thy (HOLogic.dest_Trueprop g)
+    val norm_eq_th =
+      simplify cring_ss (instantiate' [SOME ca] [SOME clhs, SOME crhs, SOME cvs] @{thm norm_eq})
+  in
+    cut_rules_tac [norm_eq_th] i
+    THEN (simp_tac cring_ss i)
+    THEN (simp_tac cring_ss i)
+  end);
+
+val setup =
+  Method.setup @{binding comm_ring} (Scan.succeed (SIMPLE_METHOD' o tac))
+    "reflective decision procedure for equalities over commutative rings";
+
+end;
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Decision_Procs/ex/Commutative_Ring_Ex.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Decision_Procs/ex/Commutative_Ring_Ex.thy	Fri Oct 30 14:02:42 2009 +0100
@@ -0,0 +1,48 @@
+(*  Author:     Bernhard Haeupler *)
+
+header {* Some examples demonstrating the comm-ring method *}
+
+theory Commutative_Ring_Ex
+imports Commutative_Ring
+begin
+
+lemma "4*(x::int)^5*y^3*x^2*3 + x*z + 3^5 = 12*x^7*y^3 + z*x + 243"
+by comm_ring
+
+lemma "((x::int) + y)^2  = x^2 + y^2 + 2*x*y"
+by comm_ring
+
+lemma "((x::int) + y)^3  = x^3 + y^3 + 3*x^2*y + 3*y^2*x"
+by comm_ring
+
+lemma "((x::int) - y)^3  = x^3 + 3*x*y^2 + (-3)*y*x^2 - y^3"
+by comm_ring
+
+lemma "((x::int) - y)^2  = x^2 + y^2 - 2*x*y"
+by comm_ring
+
+lemma " ((a::int) + b + c)^2 = a^2 + b^2 + c^2 + 2*a*b + 2*b*c + 2*a*c"
+by comm_ring
+
+lemma "((a::int) - b - c)^2 = a^2 + b^2 + c^2 - 2*a*b + 2*b*c - 2*a*c"
+by comm_ring
+
+lemma "(a::int)*b + a*c = a*(b+c)"
+by comm_ring
+
+lemma "(a::int)^2 - b^2 = (a - b) * (a + b)"
+by comm_ring
+
+lemma "(a::int)^3 - b^3 = (a - b) * (a^2 + a*b + b^2)"
+by comm_ring
+
+lemma "(a::int)^3 + b^3 = (a + b) * (a^2 - a*b + b^2)"
+by comm_ring
+
+lemma "(a::int)^4 - b^4 = (a - b) * (a + b)*(a^2 + b^2)"
+by comm_ring
+
+lemma "(a::int)^10 - b^10 = (a - b) * (a^9 + a^8*b + a^7*b^2 + a^6*b^3 + a^5*b^4 + a^4*b^5 + a^3*b^6 + a^2*b^7 + a*b^8 + b^9 )"
+by comm_ring
+
+end
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Divides.thy
--- a/src/HOL/Divides.thy	Fri Oct 30 11:31:34 2009 +0100
+++ b/src/HOL/Divides.thy	Fri Oct 30 14:02:42 2009 +0100
@@ -634,6 +634,11 @@
 
 end
 
+lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
+  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
+by (simp add: divmod_nat_zero divmod_nat_base divmod_nat_step)
+    (simp add: divmod_nat_div_mod)
+
 text {* Simproc for cancelling @{const div} and @{const mod} *}
 
 ML {*
@@ -666,22 +671,6 @@
 end
 *}
 
-text {* code generator setup *}
-
-lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
-  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
-by (simp add: divmod_nat_zero divmod_nat_base divmod_nat_step)
-    (simp add: divmod_nat_div_mod)
-
-code_modulename SML
-  Divides Nat
-
-code_modulename OCaml
-  Divides Nat
-
-code_modulename Haskell
-  Divides Nat
-
 
 subsubsection {* Quotient *}
 
@@ -1136,4 +1125,1339 @@
     Suc_mod_eq_add3_mod [of _ "number_of v", standard]
 declare Suc_mod_eq_add3_mod_number_of [simp]
 
+
+lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" 
+apply (induct "m")
+apply (simp_all add: mod_Suc)
+done
+
+declare Suc_times_mod_eq [of "number_of w", standard, simp]
+
+lemma [simp]: "n div k \<le> (Suc n) div k"
+by (simp add: div_le_mono) 
+
+lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
+by (cases n) simp_all
+
+lemma div_2_gt_zero [simp]: "(1::nat) < n ==> 0 < n div 2" 
+using Suc_n_div_2_gt_zero [of "n - 1"] by simp
+
+  (* Potential use of algebra : Equality modulo n*)
+lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
+by (simp add: mult_ac add_ac)
+
+lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
+proof -
+  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
+  also have "... = Suc m mod n" by (rule mod_mult_self3) 
+  finally show ?thesis .
+qed
+
+lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
+apply (subst mod_Suc [of m]) 
+apply (subst mod_Suc [of "m mod n"], simp) 
+done
+
+
+subsection {* Division on @{typ int} *}
+
+definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
+    --{*definition of quotient and remainder*}
+    [code]: "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
+               (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"
+
+definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
+    --{*for the division algorithm*}
+    [code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
+                         else (2 * q, r))"
+
+text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
+function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
+  "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
+     else adjust b (posDivAlg a (2 * b)))"
+by auto
+termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
+  (auto simp add: mult_2)
+
+text{*algorithm for the case @{text "a<0, b>0"}*}
+function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
+  "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
+     else adjust b (negDivAlg a (2 * b)))"
+by auto
+termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
+  (auto simp add: mult_2)
+
+text{*algorithm for the general case @{term "b\<noteq>0"}*}
+definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where
+  [code_unfold]: "negateSnd = apsnd uminus"
+
+definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
+    --{*The full division algorithm considers all possible signs for a, b
+       including the special case @{text "a=0, b<0"} because 
+       @{term negDivAlg} requires @{term "a<0"}.*}
+  "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
+                  else if a = 0 then (0, 0)
+                       else negateSnd (negDivAlg (-a) (-b))
+               else 
+                  if 0 < b then negDivAlg a b
+                  else negateSnd (posDivAlg (-a) (-b)))"
+
+instantiation int :: Divides.div
+begin
+
+definition
+  "a div b = fst (divmod_int a b)"
+
+definition
+ "a mod b = snd (divmod_int a b)"
+
+instance ..
+
 end
+
+lemma divmod_int_mod_div:
+  "divmod_int p q = (p div q, p mod q)"
+  by (auto simp add: div_int_def mod_int_def)
+
+text{*
+Here is the division algorithm in ML:
+
+\begin{verbatim}
+    fun posDivAlg (a,b) =
+      if a<b then (0,a)
+      else let val (q,r) = posDivAlg(a, 2*b)
+               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
+           end
+
+    fun negDivAlg (a,b) =
+      if 0\<le>a+b then (~1,a+b)
+      else let val (q,r) = negDivAlg(a, 2*b)
+               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
+           end;
+
+    fun negateSnd (q,r:int) = (q,~r);
+
+    fun divmod (a,b) = if 0\<le>a then 
+                          if b>0 then posDivAlg (a,b) 
+                           else if a=0 then (0,0)
+                                else negateSnd (negDivAlg (~a,~b))
+                       else 
+                          if 0<b then negDivAlg (a,b)
+                          else        negateSnd (posDivAlg (~a,~b));
+\end{verbatim}
+*}
+
+
+
+subsubsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
+
+lemma unique_quotient_lemma:
+     "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
+      ==> q' \<le> (q::int)"
+apply (subgoal_tac "r' + b * (q'-q) \<le> r")
+ prefer 2 apply (simp add: right_diff_distrib)
+apply (subgoal_tac "0 < b * (1 + q - q') ")
+apply (erule_tac [2] order_le_less_trans)
+ prefer 2 apply (simp add: right_diff_distrib right_distrib)
+apply (subgoal_tac "b * q' < b * (1 + q) ")
+ prefer 2 apply (simp add: right_diff_distrib right_distrib)
+apply (simp add: mult_less_cancel_left)
+done
+
+lemma unique_quotient_lemma_neg:
+     "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
+      ==> q \<le> (q'::int)"
+by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
+    auto)
+
+lemma unique_quotient:
+     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]  
+      ==> q = q'"
+apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
+apply (blast intro: order_antisym
+             dest: order_eq_refl [THEN unique_quotient_lemma] 
+             order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
+done
+
+
+lemma unique_remainder:
+     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]  
+      ==> r = r'"
+apply (subgoal_tac "q = q'")
+ apply (simp add: divmod_int_rel_def)
+apply (blast intro: unique_quotient)
+done
+
+
+subsubsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}
+
+text{*And positive divisors*}
+
+lemma adjust_eq [simp]:
+     "adjust b (q,r) = 
+      (let diff = r-b in  
+        if 0 \<le> diff then (2*q + 1, diff)   
+                     else (2*q, r))"
+by (simp add: Let_def adjust_def)
+
+declare posDivAlg.simps [simp del]
+
+text{*use with a simproc to avoid repeatedly proving the premise*}
+lemma posDivAlg_eqn:
+     "0 < b ==>  
+      posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
+by (rule posDivAlg.simps [THEN trans], simp)
+
+text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
+theorem posDivAlg_correct:
+  assumes "0 \<le> a" and "0 < b"
+  shows "divmod_int_rel a b (posDivAlg a b)"
+using prems apply (induct a b rule: posDivAlg.induct)
+apply auto
+apply (simp add: divmod_int_rel_def)
+apply (subst posDivAlg_eqn, simp add: right_distrib)
+apply (case_tac "a < b")
+apply simp_all
+apply (erule splitE)
+apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
+done
+
+
+subsubsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
+
+text{*And positive divisors*}
+
+declare negDivAlg.simps [simp del]
+
+text{*use with a simproc to avoid repeatedly proving the premise*}
+lemma negDivAlg_eqn:
+     "0 < b ==>  
+      negDivAlg a b =       
+       (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
+by (rule negDivAlg.simps [THEN trans], simp)
+
+(*Correctness of negDivAlg: it computes quotients correctly
+  It doesn't work if a=0 because the 0/b equals 0, not -1*)
+lemma negDivAlg_correct:
+  assumes "a < 0" and "b > 0"
+  shows "divmod_int_rel a b (negDivAlg a b)"
+using prems apply (induct a b rule: negDivAlg.induct)
+apply (auto simp add: linorder_not_le)
+apply (simp add: divmod_int_rel_def)
+apply (subst negDivAlg_eqn, assumption)
+apply (case_tac "a + b < (0\<Colon>int)")
+apply simp_all
+apply (erule splitE)
+apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
+done
+
+
+subsubsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
+
+(*the case a=0*)
+lemma divmod_int_rel_0: "b \<noteq> 0 ==> divmod_int_rel 0 b (0, 0)"
+by (auto simp add: divmod_int_rel_def linorder_neq_iff)
+
+lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
+by (subst posDivAlg.simps, auto)
+
+lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
+by (subst negDivAlg.simps, auto)
+
+lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
+by (simp add: negateSnd_def)
+
+lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (negateSnd qr)"
+by (auto simp add: split_ifs divmod_int_rel_def)
+
+lemma divmod_int_correct: "b \<noteq> 0 ==> divmod_int_rel a b (divmod_int a b)"
+by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
+                    posDivAlg_correct negDivAlg_correct)
+
+text{*Arbitrary definitions for division by zero.  Useful to simplify 
+    certain equations.*}
+
+lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
+by (simp add: div_int_def mod_int_def divmod_int_def posDivAlg.simps)  
+
+
+text{*Basic laws about division and remainder*}
+
+lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
+apply (case_tac "b = 0", simp)
+apply (cut_tac a = a and b = b in divmod_int_correct)
+apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)
+done
+
+lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
+by(simp add: zmod_zdiv_equality[symmetric])
+
+lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
+by(simp add: mult_commute zmod_zdiv_equality[symmetric])
+
+text {* Tool setup *}
+
+ML {*
+local
+
+structure CancelDivMod = CancelDivModFun(struct
+
+  val div_name = @{const_name div};
+  val mod_name = @{const_name mod};
+  val mk_binop = HOLogic.mk_binop;
+  val mk_sum = Arith_Data.mk_sum HOLogic.intT;
+  val dest_sum = Arith_Data.dest_sum;
+
+  val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];
+
+  val trans = trans;
+
+  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
+    (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))
+
+end)
+
+in
+
+val cancel_div_mod_int_proc = Simplifier.simproc @{theory}
+  "cancel_zdiv_zmod" ["(k::int) + l"] (K CancelDivMod.proc);
+
+val _ = Addsimprocs [cancel_div_mod_int_proc];
+
+end
+*}
+
+lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
+apply (cut_tac a = a and b = b in divmod_int_correct)
+apply (auto simp add: divmod_int_rel_def mod_int_def)
+done
+
+lemmas pos_mod_sign  [simp] = pos_mod_conj [THEN conjunct1, standard]
+   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]
+
+lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
+apply (cut_tac a = a and b = b in divmod_int_correct)
+apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)
+done
+
+lemmas neg_mod_sign  [simp] = neg_mod_conj [THEN conjunct1, standard]
+   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]
+
+
+
+subsubsection{*General Properties of div and mod*}
+
+lemma divmod_int_rel_div_mod: "b \<noteq> 0 ==> divmod_int_rel a b (a div b, a mod b)"
+apply (cut_tac a = a and b = b in zmod_zdiv_equality)
+apply (force simp add: divmod_int_rel_def linorder_neq_iff)
+done
+
+lemma divmod_int_rel_div: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"
+by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
+
+lemma divmod_int_rel_mod: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"
+by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
+
+lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
+apply (rule divmod_int_rel_div)
+apply (auto simp add: divmod_int_rel_def)
+done
+
+lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
+apply (rule divmod_int_rel_div)
+apply (auto simp add: divmod_int_rel_def)
+done
+
+lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
+apply (rule divmod_int_rel_div)
+apply (auto simp add: divmod_int_rel_def)
+done
+
+(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
+
+lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
+apply (rule_tac q = 0 in divmod_int_rel_mod)
+apply (auto simp add: divmod_int_rel_def)
+done
+
+lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
+apply (rule_tac q = 0 in divmod_int_rel_mod)
+apply (auto simp add: divmod_int_rel_def)
+done
+
+lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
+apply (rule_tac q = "-1" in divmod_int_rel_mod)
+apply (auto simp add: divmod_int_rel_def)
+done
+
+text{*There is no @{text mod_neg_pos_trivial}.*}
+
+
+(*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
+lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
+apply (case_tac "b = 0", simp)
+apply (simp add: divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, 
+                                 THEN divmod_int_rel_div, THEN sym])
+
+done
+
+(*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
+lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
+apply (case_tac "b = 0", simp)
+apply (subst divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, THEN divmod_int_rel_mod],
+       auto)
+done
+
+
+subsubsection{*Laws for div and mod with Unary Minus*}
+
+lemma zminus1_lemma:
+     "divmod_int_rel a b (q, r)
+      ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  
+                          if r=0 then 0 else b-r)"
+by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
+
+
+lemma zdiv_zminus1_eq_if:
+     "b \<noteq> (0::int)  
+      ==> (-a) div b =  
+          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
+by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_div])
+
+lemma zmod_zminus1_eq_if:
+     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
+apply (case_tac "b = 0", simp)
+apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_mod])
+done
+
+lemma zmod_zminus1_not_zero:
+  fixes k l :: int
+  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
+  unfolding zmod_zminus1_eq_if by auto
+
+lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
+by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
+
+lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
+by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
+
+lemma zdiv_zminus2_eq_if:
+     "b \<noteq> (0::int)  
+      ==> a div (-b) =  
+          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
+by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
+
+lemma zmod_zminus2_eq_if:
+     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
+by (simp add: zmod_zminus1_eq_if zmod_zminus2)
+
+lemma zmod_zminus2_not_zero:
+  fixes k l :: int
+  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
+  unfolding zmod_zminus2_eq_if by auto 
+
+
+subsubsection{*Division of a Number by Itself*}
+
+lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
+apply (subgoal_tac "0 < a*q")
+ apply (simp add: zero_less_mult_iff, arith)
+done
+
+lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
+apply (subgoal_tac "0 \<le> a* (1-q) ")
+ apply (simp add: zero_le_mult_iff)
+apply (simp add: right_diff_distrib)
+done
+
+lemma self_quotient: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"
+apply (simp add: split_ifs divmod_int_rel_def linorder_neq_iff)
+apply (rule order_antisym, safe, simp_all)
+apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
+apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
+apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
+done
+
+lemma self_remainder: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"
+apply (frule self_quotient, assumption)
+apply (simp add: divmod_int_rel_def)
+done
+
+lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
+by (simp add: divmod_int_rel_div_mod [THEN self_quotient])
+
+(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
+lemma zmod_self [simp]: "a mod a = (0::int)"
+apply (case_tac "a = 0", simp)
+apply (simp add: divmod_int_rel_div_mod [THEN self_remainder])
+done
+
+
+subsubsection{*Computation of Division and Remainder*}
+
+lemma zdiv_zero [simp]: "(0::int) div b = 0"
+by (simp add: div_int_def divmod_int_def)
+
+lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
+by (simp add: div_int_def divmod_int_def)
+
+lemma zmod_zero [simp]: "(0::int) mod b = 0"
+by (simp add: mod_int_def divmod_int_def)
+
+lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
+by (simp add: mod_int_def divmod_int_def)
+
+text{*a positive, b positive *}
+
+lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
+by (simp add: div_int_def divmod_int_def)
+
+lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
+by (simp add: mod_int_def divmod_int_def)
+
+text{*a negative, b positive *}
+
+lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
+by (simp add: div_int_def divmod_int_def)
+
+lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
+by (simp add: mod_int_def divmod_int_def)
+
+text{*a positive, b negative *}
+
+lemma div_pos_neg:
+     "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
+by (simp add: div_int_def divmod_int_def)
+
+lemma mod_pos_neg:
+     "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
+by (simp add: mod_int_def divmod_int_def)
+
+text{*a negative, b negative *}
+
+lemma div_neg_neg:
+     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
+by (simp add: div_int_def divmod_int_def)
+
+lemma mod_neg_neg:
+     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
+by (simp add: mod_int_def divmod_int_def)
+
+text {*Simplify expresions in which div and mod combine numerical constants*}
+
+lemma divmod_int_relI:
+  "\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk>
+    \<Longrightarrow> divmod_int_rel a b (q, r)"
+  unfolding divmod_int_rel_def by simp
+
+lemmas divmod_int_rel_div_eq = divmod_int_relI [THEN divmod_int_rel_div, THEN eq_reflection]
+lemmas divmod_int_rel_mod_eq = divmod_int_relI [THEN divmod_int_rel_mod, THEN eq_reflection]
+lemmas arithmetic_simps =
+  arith_simps
+  add_special
+  OrderedGroup.add_0_left
+  OrderedGroup.add_0_right
+  mult_zero_left
+  mult_zero_right
+  mult_1_left
+  mult_1_right
+
+(* simprocs adapted from HOL/ex/Binary.thy *)
+ML {*
+local
+  val mk_number = HOLogic.mk_number HOLogic.intT;
+  fun mk_cert u k l = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $
+    (@{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $ u $ mk_number k) $
+      mk_number l;
+  fun prove ctxt prop = Goal.prove ctxt [] [] prop
+    (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps @{thms arithmetic_simps}))));
+  fun binary_proc proc ss ct =
+    (case Thm.term_of ct of
+      _ $ t $ u =>
+      (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
+        SOME args => proc (Simplifier.the_context ss) args
+      | NONE => NONE)
+    | _ => NONE);
+in
+  fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
+    if n = 0 then NONE
+    else let val (k, l) = Integer.div_mod m n;
+    in SOME (rule OF [prove ctxt (Logic.mk_equals (t, mk_cert u k l))]) end);
+end
+*}
+
+simproc_setup binary_int_div ("number_of m div number_of n :: int") =
+  {* K (divmod_proc (@{thm divmod_int_rel_div_eq})) *}
+
+simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =
+  {* K (divmod_proc (@{thm divmod_int_rel_mod_eq})) *}
+
+lemmas posDivAlg_eqn_number_of [simp] =
+    posDivAlg_eqn [of "number_of v" "number_of w", standard]
+
+lemmas negDivAlg_eqn_number_of [simp] =
+    negDivAlg_eqn [of "number_of v" "number_of w", standard]
+
+
+text{*Special-case simplification *}
+
+lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
+apply (cut_tac a = a and b = "-1" in neg_mod_sign)
+apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
+apply (auto simp del: neg_mod_sign neg_mod_bound)
+done
+
+lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
+by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
+
+(** The last remaining special cases for constant arithmetic:
+    1 div z and 1 mod z **)
+
+lemmas div_pos_pos_1_number_of [simp] =
+    div_pos_pos [OF int_0_less_1, of "number_of w", standard]
+
+lemmas div_pos_neg_1_number_of [simp] =
+    div_pos_neg [OF int_0_less_1, of "number_of w", standard]
+
+lemmas mod_pos_pos_1_number_of [simp] =
+    mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
+
+lemmas mod_pos_neg_1_number_of [simp] =
+    mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
+
+
+lemmas posDivAlg_eqn_1_number_of [simp] =
+    posDivAlg_eqn [of concl: 1 "number_of w", standard]
+
+lemmas negDivAlg_eqn_1_number_of [simp] =
+    negDivAlg_eqn [of concl: 1 "number_of w", standard]
+
+
+
+subsubsection{*Monotonicity in the First Argument (Dividend)*}
+
+lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
+apply (cut_tac a = a and b = b in zmod_zdiv_equality)
+apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
+apply (rule unique_quotient_lemma)
+apply (erule subst)
+apply (erule subst, simp_all)
+done
+
+lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
+apply (cut_tac a = a and b = b in zmod_zdiv_equality)
+apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
+apply (rule unique_quotient_lemma_neg)
+apply (erule subst)
+apply (erule subst, simp_all)
+done
+
+
+subsubsection{*Monotonicity in the Second Argument (Divisor)*}
+
+lemma q_pos_lemma:
+     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
+apply (subgoal_tac "0 < b'* (q' + 1) ")
+ apply (simp add: zero_less_mult_iff)
+apply (simp add: right_distrib)
+done
+
+lemma zdiv_mono2_lemma:
+     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
+         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
+      ==> q \<le> (q'::int)"
+apply (frule q_pos_lemma, assumption+) 
+apply (subgoal_tac "b*q < b* (q' + 1) ")
+ apply (simp add: mult_less_cancel_left)
+apply (subgoal_tac "b*q = r' - r + b'*q'")
+ prefer 2 apply simp
+apply (simp (no_asm_simp) add: right_distrib)
+apply (subst add_commute, rule zadd_zless_mono, arith)
+apply (rule mult_right_mono, auto)
+done
+
+lemma zdiv_mono2:
+     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
+apply (subgoal_tac "b \<noteq> 0")
+ prefer 2 apply arith
+apply (cut_tac a = a and b = b in zmod_zdiv_equality)
+apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
+apply (rule zdiv_mono2_lemma)
+apply (erule subst)
+apply (erule subst, simp_all)
+done
+
+lemma q_neg_lemma:
+     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
+apply (subgoal_tac "b'*q' < 0")
+ apply (simp add: mult_less_0_iff, arith)
+done
+
+lemma zdiv_mono2_neg_lemma:
+     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
+         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
+      ==> q' \<le> (q::int)"
+apply (frule q_neg_lemma, assumption+) 
+apply (subgoal_tac "b*q' < b* (q + 1) ")
+ apply (simp add: mult_less_cancel_left)
+apply (simp add: right_distrib)
+apply (subgoal_tac "b*q' \<le> b'*q'")
+ prefer 2 apply (simp add: mult_right_mono_neg, arith)
+done
+
+lemma zdiv_mono2_neg:
+     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
+apply (cut_tac a = a and b = b in zmod_zdiv_equality)
+apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
+apply (rule zdiv_mono2_neg_lemma)
+apply (erule subst)
+apply (erule subst, simp_all)
+done
+
+
+subsubsection{*More Algebraic Laws for div and mod*}
+
+text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
+
+lemma zmult1_lemma:
+     "[| divmod_int_rel b c (q, r);  c \<noteq> 0 |]  
+      ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
+by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)
+
+lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
+apply (case_tac "c = 0", simp)
+apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_div])
+done
+
+lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
+apply (case_tac "c = 0", simp)
+apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_mod])
+done
+
+lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
+apply (case_tac "b = 0", simp)
+apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
+done
+
+text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
+
+lemma zadd1_lemma:
+     "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br);  c \<noteq> 0 |]  
+      ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
+by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)
+
+(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
+lemma zdiv_zadd1_eq:
+     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
+apply (case_tac "c = 0", simp)
+apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] divmod_int_rel_div)
+done
+
+instance int :: ring_div
+proof
+  fix a b c :: int
+  assume not0: "b \<noteq> 0"
+  show "(a + c * b) div b = c + a div b"
+    unfolding zdiv_zadd1_eq [of a "c * b"] using not0 
+      by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
+next
+  fix a b c :: int
+  assume "a \<noteq> 0"
+  then show "(a * b) div (a * c) = b div c"
+  proof (cases "b \<noteq> 0 \<and> c \<noteq> 0")
+    case False then show ?thesis by auto
+  next
+    case True then have "b \<noteq> 0" and "c \<noteq> 0" by auto
+    with `a \<noteq> 0`
+    have "\<And>q r. divmod_int_rel b c (q, r) \<Longrightarrow> divmod_int_rel (a * b) (a * c) (q, a * r)"
+      apply (auto simp add: divmod_int_rel_def) 
+      apply (auto simp add: algebra_simps)
+      apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff mult_commute [of a] mult_less_cancel_right)
+      done
+    moreover with `c \<noteq> 0` divmod_int_rel_div_mod have "divmod_int_rel b c (b div c, b mod c)" by auto
+    ultimately have "divmod_int_rel (a * b) (a * c) (b div c, a * (b mod c))" .
+    moreover from  `a \<noteq> 0` `c \<noteq> 0` have "a * c \<noteq> 0" by simp
+    ultimately show ?thesis by (rule divmod_int_rel_div)
+  qed
+qed auto
+
+lemma posDivAlg_div_mod:
+  assumes "k \<ge> 0"
+  and "l \<ge> 0"
+  shows "posDivAlg k l = (k div l, k mod l)"
+proof (cases "l = 0")
+  case True then show ?thesis by (simp add: posDivAlg.simps)
+next
+  case False with assms posDivAlg_correct
+    have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
+    by simp
+  from divmod_int_rel_div [OF this `l \<noteq> 0`] divmod_int_rel_mod [OF this `l \<noteq> 0`]
+  show ?thesis by simp
+qed
+
+lemma negDivAlg_div_mod:
+  assumes "k < 0"
+  and "l > 0"
+  shows "negDivAlg k l = (k div l, k mod l)"
+proof -
+  from assms have "l \<noteq> 0" by simp
+  from assms negDivAlg_correct
+    have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
+    by simp
+  from divmod_int_rel_div [OF this `l \<noteq> 0`] divmod_int_rel_mod [OF this `l \<noteq> 0`]
+  show ?thesis by simp
+qed
+
+lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
+by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
+
+(* REVISIT: should this be generalized to all semiring_div types? *)
+lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
+
+
+subsubsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
+
+(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
+  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
+  to cause particular problems.*)
+
+text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
+
+lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
+apply (subgoal_tac "b * (c - q mod c) < r * 1")
+ apply (simp add: algebra_simps)
+apply (rule order_le_less_trans)
+ apply (erule_tac [2] mult_strict_right_mono)
+ apply (rule mult_left_mono_neg)
+  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)
+ apply (simp)
+apply (simp)
+done
+
+lemma zmult2_lemma_aux2:
+     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
+apply (subgoal_tac "b * (q mod c) \<le> 0")
+ apply arith
+apply (simp add: mult_le_0_iff)
+done
+
+lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
+apply (subgoal_tac "0 \<le> b * (q mod c) ")
+apply arith
+apply (simp add: zero_le_mult_iff)
+done
+
+lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
+apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
+ apply (simp add: right_diff_distrib)
+apply (rule order_less_le_trans)
+ apply (erule mult_strict_right_mono)
+ apply (rule_tac [2] mult_left_mono)
+  apply simp
+ using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)
+apply simp
+done
+
+lemma zmult2_lemma: "[| divmod_int_rel a b (q, r);  b \<noteq> 0;  0 < c |]  
+      ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
+by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff
+                   zero_less_mult_iff right_distrib [symmetric] 
+                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
+
+lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
+apply (case_tac "b = 0", simp)
+apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_div])
+done
+
+lemma zmod_zmult2_eq:
+     "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
+apply (case_tac "b = 0", simp)
+apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_mod])
+done
+
+
+subsubsection {*Splitting Rules for div and mod*}
+
+text{*The proofs of the two lemmas below are essentially identical*}
+
+lemma split_pos_lemma:
+ "0<k ==> 
+    P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
+apply (rule iffI, clarify)
+ apply (erule_tac P="P ?x ?y" in rev_mp)  
+ apply (subst mod_add_eq) 
+ apply (subst zdiv_zadd1_eq) 
+ apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
+txt{*converse direction*}
+apply (drule_tac x = "n div k" in spec) 
+apply (drule_tac x = "n mod k" in spec, simp)
+done
+
+lemma split_neg_lemma:
+ "k<0 ==>
+    P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
+apply (rule iffI, clarify)
+ apply (erule_tac P="P ?x ?y" in rev_mp)  
+ apply (subst mod_add_eq) 
+ apply (subst zdiv_zadd1_eq) 
+ apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
+txt{*converse direction*}
+apply (drule_tac x = "n div k" in spec) 
+apply (drule_tac x = "n mod k" in spec, simp)
+done
+
+lemma split_zdiv:
+ "P(n div k :: int) =
+  ((k = 0 --> P 0) & 
+   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
+   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
+apply (case_tac "k=0", simp)
+apply (simp only: linorder_neq_iff)
+apply (erule disjE) 
+ apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
+                      split_neg_lemma [of concl: "%x y. P x"])
+done
+
+lemma split_zmod:
+ "P(n mod k :: int) =
+  ((k = 0 --> P n) & 
+   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
+   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
+apply (case_tac "k=0", simp)
+apply (simp only: linorder_neq_iff)
+apply (erule disjE) 
+ apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
+                      split_neg_lemma [of concl: "%x y. P y"])
+done
+
+(* Enable arith to deal with div 2 and mod 2: *)
+declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
+declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
+
+
+subsubsection{*Speeding up the Division Algorithm with Shifting*}
+
+text{*computing div by shifting *}
+
+lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
+proof cases
+  assume "a=0"
+    thus ?thesis by simp
+next
+  assume "a\<noteq>0" and le_a: "0\<le>a"   
+  hence a_pos: "1 \<le> a" by arith
+  hence one_less_a2: "1 < 2 * a" by arith
+  hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
+    unfolding mult_le_cancel_left
+    by (simp add: add1_zle_eq add_commute [of 1])
+  with a_pos have "0 \<le> b mod a" by simp
+  hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
+    by (simp add: mod_pos_pos_trivial one_less_a2)
+  with  le_2a
+  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
+    by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
+                  right_distrib) 
+  thus ?thesis
+    by (subst zdiv_zadd1_eq,
+        simp add: mod_mult_mult1 one_less_a2
+                  div_pos_pos_trivial)
+qed
+
+lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
+apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
+apply (rule_tac [2] pos_zdiv_mult_2)
+apply (auto simp add: right_diff_distrib)
+apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
+apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric])
+apply (simp_all add: algebra_simps)
+apply (simp only: ab_diff_minus minus_add_distrib [symmetric] number_of_Min zdiv_zminus_zminus)
+done
+
+lemma zdiv_number_of_Bit0 [simp]:
+     "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  
+          number_of v div (number_of w :: int)"
+by (simp only: number_of_eq numeral_simps) (simp add: mult_2 [symmetric])
+
+lemma zdiv_number_of_Bit1 [simp]:
+     "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  
+          (if (0::int) \<le> number_of w                    
+           then number_of v div (number_of w)     
+           else (number_of v + (1::int)) div (number_of w))"
+apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) 
+apply (simp add: pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac mult_2 [symmetric])
+done
+
+
+subsubsection{*Computing mod by Shifting (proofs resemble those for div)*}
+
+lemma pos_zmod_mult_2:
+  fixes a b :: int
+  assumes "0 \<le> a"
+  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
+proof (cases "0 < a")
+  case False with assms show ?thesis by simp
+next
+  case True
+  then have "b mod a < a" by (rule pos_mod_bound)
+  then have "1 + b mod a \<le> a" by simp
+  then have A: "2 * (1 + b mod a) \<le> 2 * a" by simp
+  from `0 < a` have "0 \<le> b mod a" by (rule pos_mod_sign)
+  then have B: "0 \<le> 1 + 2 * (b mod a)" by simp
+  have "((1\<Colon>int) mod ((2\<Colon>int) * a) + (2\<Colon>int) * b mod ((2\<Colon>int) * a)) mod ((2\<Colon>int) * a) = (1\<Colon>int) + (2\<Colon>int) * (b mod a)"
+    using `0 < a` and A
+    by (auto simp add: mod_mult_mult1 mod_pos_pos_trivial ring_distribs intro!: mod_pos_pos_trivial B)
+  then show ?thesis by (subst mod_add_eq)
+qed
+
+lemma neg_zmod_mult_2:
+  fixes a b :: int
+  assumes "a \<le> 0"
+  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
+proof -
+  from assms have "0 \<le> - a" by auto
+  then have "(1 + 2 * (- b - 1)) mod (2 * (- a)) = 1 + 2 * ((- b - 1) mod (- a))"
+    by (rule pos_zmod_mult_2)
+  then show ?thesis by (simp add: zmod_zminus2 algebra_simps)
+     (simp add: diff_minus add_ac)
+qed
+
+lemma zmod_number_of_Bit0 [simp]:
+     "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  
+      (2::int) * (number_of v mod number_of w)"
+apply (simp only: number_of_eq numeral_simps) 
+apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
+                 neg_zmod_mult_2 add_ac mult_2 [symmetric])
+done
+
+lemma zmod_number_of_Bit1 [simp]:
+     "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  
+      (if (0::int) \<le> number_of w  
+                then 2 * (number_of v mod number_of w) + 1     
+                else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
+apply (simp only: number_of_eq numeral_simps) 
+apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
+                 neg_zmod_mult_2 add_ac mult_2 [symmetric])
+done
+
+
+subsubsection{*Quotients of Signs*}
+
+lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
+apply (subgoal_tac "a div b \<le> -1", force)
+apply (rule order_trans)
+apply (rule_tac a' = "-1" in zdiv_mono1)
+apply (auto simp add: div_eq_minus1)
+done
+
+lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
+by (drule zdiv_mono1_neg, auto)
+
+lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
+by (drule zdiv_mono1, auto)
+
+lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
+apply auto
+apply (drule_tac [2] zdiv_mono1)
+apply (auto simp add: linorder_neq_iff)
+apply (simp (no_asm_use) add: linorder_not_less [symmetric])
+apply (blast intro: div_neg_pos_less0)
+done
+
+lemma neg_imp_zdiv_nonneg_iff:
+     "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
+apply (subst zdiv_zminus_zminus [symmetric])
+apply (subst pos_imp_zdiv_nonneg_iff, auto)
+done
+
+(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
+lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
+by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
+
+(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
+lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
+by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
+
+
+subsubsection {* The Divides Relation *}
+
+lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
+  dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]
+
+lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
+  by (rule dvd_mod) (* TODO: remove *)
+
+lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
+  by (rule dvd_mod_imp_dvd) (* TODO: remove *)
+
+lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
+  using zmod_zdiv_equality[where a="m" and b="n"]
+  by (simp add: algebra_simps)
+
+lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
+apply (induct "y", auto)
+apply (rule zmod_zmult1_eq [THEN trans])
+apply (simp (no_asm_simp))
+apply (rule mod_mult_eq [symmetric])
+done
+
+lemma zdiv_int: "int (a div b) = (int a) div (int b)"
+apply (subst split_div, auto)
+apply (subst split_zdiv, auto)
+apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)
+apply (auto simp add: divmod_int_rel_def of_nat_mult)
+done
+
+lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
+apply (subst split_mod, auto)
+apply (subst split_zmod, auto)
+apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
+       in unique_remainder)
+apply (auto simp add: divmod_int_rel_def of_nat_mult)
+done
+
+lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
+by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
+
+lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
+apply (subgoal_tac "m mod n = 0")
+ apply (simp add: zmult_div_cancel)
+apply (simp only: dvd_eq_mod_eq_0)
+done
+
+text{*Suggested by Matthias Daum*}
+lemma int_power_div_base:
+     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
+apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
+ apply (erule ssubst)
+ apply (simp only: power_add)
+ apply simp_all
+done
+
+text {* by Brian Huffman *}
+lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
+by (rule mod_minus_eq [symmetric])
+
+lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
+by (rule mod_diff_left_eq [symmetric])
+
+lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
+by (rule mod_diff_right_eq [symmetric])
+
+lemmas zmod_simps =
+  mod_add_left_eq  [symmetric]
+  mod_add_right_eq [symmetric]
+  zmod_zmult1_eq   [symmetric]
+  mod_mult_left_eq [symmetric]
+  zpower_zmod
+  zminus_zmod zdiff_zmod_left zdiff_zmod_right
+
+text {* Distributive laws for function @{text nat}. *}
+
+lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
+apply (rule linorder_cases [of y 0])
+apply (simp add: div_nonneg_neg_le0)
+apply simp
+apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
+done
+
+(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
+lemma nat_mod_distrib:
+  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
+apply (case_tac "y = 0", simp)
+apply (simp add: nat_eq_iff zmod_int)
+done
+
+text  {* transfer setup *}
+
+lemma transfer_nat_int_functions:
+    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
+    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
+  by (auto simp add: nat_div_distrib nat_mod_distrib)
+
+lemma transfer_nat_int_function_closures:
+    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
+    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
+  apply (cases "y = 0")
+  apply (auto simp add: pos_imp_zdiv_nonneg_iff)
+  apply (cases "y = 0")
+  apply auto
+done
+
+declare TransferMorphism_nat_int [transfer add return:
+  transfer_nat_int_functions
+  transfer_nat_int_function_closures
+]
+
+lemma transfer_int_nat_functions:
+    "(int x) div (int y) = int (x div y)"
+    "(int x) mod (int y) = int (x mod y)"
+  by (auto simp add: zdiv_int zmod_int)
+
+lemma transfer_int_nat_function_closures:
+    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
+    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
+  by (simp_all only: is_nat_def transfer_nat_int_function_closures)
+
+declare TransferMorphism_int_nat [transfer add return:
+  transfer_int_nat_functions
+  transfer_int_nat_function_closures
+]
+
+text{*Suggested by Matthias Daum*}
+lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
+apply (subgoal_tac "nat x div nat k < nat x")
+ apply (simp (asm_lr) add: nat_div_distrib [symmetric])
+apply (rule Divides.div_less_dividend, simp_all)
+done
+
+text {* code generator setup *}
+
+context ring_1
+begin
+
+lemma of_int_num [code]:
+  "of_int k = (if k = 0 then 0 else if k < 0 then
+     - of_int (- k) else let
+       (l, m) = divmod_int k 2;
+       l' = of_int l
+     in if m = 0 then l' + l' else l' + l' + 1)"
+proof -
+  have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow> 
+    of_int k = of_int (k div 2 * 2 + 1)"
+  proof -
+    have "k mod 2 < 2" by (auto intro: pos_mod_bound)
+    moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)
+    moreover assume "k mod 2 \<noteq> 0"
+    ultimately have "k mod 2 = 1" by arith
+    moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
+    ultimately show ?thesis by auto
+  qed
+  have aux2: "\<And>x. of_int 2 * x = x + x"
+  proof -
+    fix x
+    have int2: "(2::int) = 1 + 1" by arith
+    show "of_int 2 * x = x + x"
+    unfolding int2 of_int_add left_distrib by simp
+  qed
+  have aux3: "\<And>x. x * of_int 2 = x + x"
+  proof -
+    fix x
+    have int2: "(2::int) = 1 + 1" by arith
+    show "x * of_int 2 = x + x" 
+    unfolding int2 of_int_add right_distrib by simp
+  qed
+  from aux1 show ?thesis by (auto simp add: divmod_int_mod_div Let_def aux2 aux3)
+qed
+
+end
+
+lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
+proof
+  assume H: "x mod n = y mod n"
+  hence "x mod n - y mod n = 0" by simp
+  hence "(x mod n - y mod n) mod n = 0" by simp 
+  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
+  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
+next
+  assume H: "n dvd x - y"
+  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
+  hence "x = n*k + y" by simp
+  hence "x mod n = (n*k + y) mod n" by simp
+  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
+qed
+
+lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
+  shows "\<exists>q. x = y + n * q"
+proof-
+  from xy have th: "int x - int y = int (x - y)" by simp 
+  from xyn have "int x mod int n = int y mod int n" 
+    by (simp add: zmod_int[symmetric])
+  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
+  hence "n dvd x - y" by (simp add: th zdvd_int)
+  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
+qed
+
+lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
+  (is "?lhs = ?rhs")
+proof
+  assume H: "x mod n = y mod n"
+  {assume xy: "x \<le> y"
+    from H have th: "y mod n = x mod n" by simp
+    from nat_mod_eq_lemma[OF th xy] have ?rhs 
+      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
+  moreover
+  {assume xy: "y \<le> x"
+    from nat_mod_eq_lemma[OF H xy] have ?rhs 
+      apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
+  ultimately  show ?rhs using linear[of x y] by blast  
+next
+  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
+  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
+  thus  ?lhs by simp
+qed
+
+lemma div_nat_number_of [simp]:
+     "(number_of v :: nat)  div  number_of v' =  
+          (if neg (number_of v :: int) then 0  
+           else nat (number_of v div number_of v'))"
+  unfolding nat_number_of_def number_of_is_id neg_def
+  by (simp add: nat_div_distrib)
+
+lemma one_div_nat_number_of [simp]:
+     "Suc 0 div number_of v' = nat (1 div number_of v')" 
+by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
+
+lemma mod_nat_number_of [simp]:
+     "(number_of v :: nat)  mod  number_of v' =  
+        (if neg (number_of v :: int) then 0  
+         else if neg (number_of v' :: int) then number_of v  
+         else nat (number_of v mod number_of v'))"
+  unfolding nat_number_of_def number_of_is_id neg_def
+  by (simp add: nat_mod_distrib)
+
+lemma one_mod_nat_number_of [simp]:
+     "Suc 0 mod number_of v' =  
+        (if neg (number_of v' :: int) then Suc 0
+         else nat (1 mod number_of v'))"
+by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
+
+lemmas dvd_eq_mod_eq_0_number_of =
+  dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
+
+declare dvd_eq_mod_eq_0_number_of [simp]
+
+
+subsubsection {* Code generation *}
+
+definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
+  "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
+
+lemma pdivmod_posDivAlg [code]:
+  "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
+by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
+
+lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
+  apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
+    then pdivmod k l
+    else (let (r, s) = pdivmod k l in
+      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
+proof -
+  have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
+  show ?thesis
+    by (simp add: divmod_int_mod_div pdivmod_def)
+      (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
+      zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
+qed
+
+lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
+  apsnd ((op *) (sgn l)) (if sgn k = sgn l
+    then pdivmod k l
+    else (let (r, s) = pdivmod k l in
+      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
+proof -
+  have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"
+    by (auto simp add: not_less sgn_if)
+  then show ?thesis by (simp add: divmod_int_pdivmod)
+qed
+
+end
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Groebner_Basis.thy
--- a/src/HOL/Groebner_Basis.thy	Fri Oct 30 11:31:34 2009 +0100
+++ b/src/HOL/Groebner_Basis.thy	Fri Oct 30 14:02:42 2009 +0100
@@ -5,7 +5,7 @@
 header {* Semiring normalization and Groebner Bases *}
 
 theory Groebner_Basis
-imports IntDiv
+imports Numeral_Simprocs
 uses
   "Tools/Groebner_Basis/misc.ML"
   "Tools/Groebner_Basis/normalizer_data.ML"
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/IntDiv.thy
--- a/src/HOL/IntDiv.thy	Fri Oct 30 11:31:34 2009 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,1474 +0,0 @@
-(*  Title:      HOL/IntDiv.thy
-    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   1999  University of Cambridge
-*)
-
-header{* The Division Operators div and mod *}
-
-theory IntDiv
-imports Int Divides FunDef
-uses
-  "~~/src/Provers/Arith/assoc_fold.ML"
-  "~~/src/Provers/Arith/cancel_numerals.ML"
-  "~~/src/Provers/Arith/combine_numerals.ML"
-  "~~/src/Provers/Arith/cancel_numeral_factor.ML"
-  "~~/src/Provers/Arith/extract_common_term.ML"
-  ("Tools/numeral_simprocs.ML")
-  ("Tools/nat_numeral_simprocs.ML")
-begin
-
-definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
-    --{*definition of quotient and remainder*}
-    [code]: "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
-               (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"
-
-definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
-    --{*for the division algorithm*}
-    [code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
-                         else (2 * q, r))"
-
-text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
-function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
-  "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
-     else adjust b (posDivAlg a (2 * b)))"
-by auto
-termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
-  (auto simp add: mult_2)
-
-text{*algorithm for the case @{text "a<0, b>0"}*}
-function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
-  "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
-     else adjust b (negDivAlg a (2 * b)))"
-by auto
-termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
-  (auto simp add: mult_2)
-
-text{*algorithm for the general case @{term "b\<noteq>0"}*}
-definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where
-  [code_unfold]: "negateSnd = apsnd uminus"
-
-definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
-    --{*The full division algorithm considers all possible signs for a, b
-       including the special case @{text "a=0, b<0"} because 
-       @{term negDivAlg} requires @{term "a<0"}.*}
-  "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
-                  else if a = 0 then (0, 0)
-                       else negateSnd (negDivAlg (-a) (-b))
-               else 
-                  if 0 < b then negDivAlg a b
-                  else negateSnd (posDivAlg (-a) (-b)))"
-
-instantiation int :: Divides.div
-begin
-
-definition
-  "a div b = fst (divmod_int a b)"
-
-definition
- "a mod b = snd (divmod_int a b)"
-
-instance ..
-
-end
-
-lemma divmod_int_mod_div:
-  "divmod_int p q = (p div q, p mod q)"
-  by (auto simp add: div_int_def mod_int_def)
-
-text{*
-Here is the division algorithm in ML:
-
-\begin{verbatim}
-    fun posDivAlg (a,b) =
-      if a<b then (0,a)
-      else let val (q,r) = posDivAlg(a, 2*b)
-               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
-           end
-
-    fun negDivAlg (a,b) =
-      if 0\<le>a+b then (~1,a+b)
-      else let val (q,r) = negDivAlg(a, 2*b)
-               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
-           end;
-
-    fun negateSnd (q,r:int) = (q,~r);
-
-    fun divmod (a,b) = if 0\<le>a then 
-                          if b>0 then posDivAlg (a,b) 
-                           else if a=0 then (0,0)
-                                else negateSnd (negDivAlg (~a,~b))
-                       else 
-                          if 0<b then negDivAlg (a,b)
-                          else        negateSnd (posDivAlg (~a,~b));
-\end{verbatim}
-*}
-
-
-
-subsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
-
-lemma unique_quotient_lemma:
-     "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
-      ==> q' \<le> (q::int)"
-apply (subgoal_tac "r' + b * (q'-q) \<le> r")
- prefer 2 apply (simp add: right_diff_distrib)
-apply (subgoal_tac "0 < b * (1 + q - q') ")
-apply (erule_tac [2] order_le_less_trans)
- prefer 2 apply (simp add: right_diff_distrib right_distrib)
-apply (subgoal_tac "b * q' < b * (1 + q) ")
- prefer 2 apply (simp add: right_diff_distrib right_distrib)
-apply (simp add: mult_less_cancel_left)
-done
-
-lemma unique_quotient_lemma_neg:
-     "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
-      ==> q \<le> (q'::int)"
-by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
-    auto)
-
-lemma unique_quotient:
-     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]  
-      ==> q = q'"
-apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
-apply (blast intro: order_antisym
-             dest: order_eq_refl [THEN unique_quotient_lemma] 
-             order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
-done
-
-
-lemma unique_remainder:
-     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]  
-      ==> r = r'"
-apply (subgoal_tac "q = q'")
- apply (simp add: divmod_int_rel_def)
-apply (blast intro: unique_quotient)
-done
-
-
-subsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}
-
-text{*And positive divisors*}
-
-lemma adjust_eq [simp]:
-     "adjust b (q,r) = 
-      (let diff = r-b in  
-        if 0 \<le> diff then (2*q + 1, diff)   
-                     else (2*q, r))"
-by (simp add: Let_def adjust_def)
-
-declare posDivAlg.simps [simp del]
-
-text{*use with a simproc to avoid repeatedly proving the premise*}
-lemma posDivAlg_eqn:
-     "0 < b ==>  
-      posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
-by (rule posDivAlg.simps [THEN trans], simp)
-
-text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
-theorem posDivAlg_correct:
-  assumes "0 \<le> a" and "0 < b"
-  shows "divmod_int_rel a b (posDivAlg a b)"
-using prems apply (induct a b rule: posDivAlg.induct)
-apply auto
-apply (simp add: divmod_int_rel_def)
-apply (subst posDivAlg_eqn, simp add: right_distrib)
-apply (case_tac "a < b")
-apply simp_all
-apply (erule splitE)
-apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
-done
-
-
-subsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
-
-text{*And positive divisors*}
-
-declare negDivAlg.simps [simp del]
-
-text{*use with a simproc to avoid repeatedly proving the premise*}
-lemma negDivAlg_eqn:
-     "0 < b ==>  
-      negDivAlg a b =       
-       (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
-by (rule negDivAlg.simps [THEN trans], simp)
-
-(*Correctness of negDivAlg: it computes quotients correctly
-  It doesn't work if a=0 because the 0/b equals 0, not -1*)
-lemma negDivAlg_correct:
-  assumes "a < 0" and "b > 0"
-  shows "divmod_int_rel a b (negDivAlg a b)"
-using prems apply (induct a b rule: negDivAlg.induct)
-apply (auto simp add: linorder_not_le)
-apply (simp add: divmod_int_rel_def)
-apply (subst negDivAlg_eqn, assumption)
-apply (case_tac "a + b < (0\<Colon>int)")
-apply simp_all
-apply (erule splitE)
-apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
-done
-
-
-subsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
-
-(*the case a=0*)
-lemma divmod_int_rel_0: "b \<noteq> 0 ==> divmod_int_rel 0 b (0, 0)"
-by (auto simp add: divmod_int_rel_def linorder_neq_iff)
-
-lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
-by (subst posDivAlg.simps, auto)
-
-lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
-by (subst negDivAlg.simps, auto)
-
-lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
-by (simp add: negateSnd_def)
-
-lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (negateSnd qr)"
-by (auto simp add: split_ifs divmod_int_rel_def)
-
-lemma divmod_int_correct: "b \<noteq> 0 ==> divmod_int_rel a b (divmod_int a b)"
-by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
-                    posDivAlg_correct negDivAlg_correct)
-
-text{*Arbitrary definitions for division by zero.  Useful to simplify 
-    certain equations.*}
-
-lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
-by (simp add: div_int_def mod_int_def divmod_int_def posDivAlg.simps)  
-
-
-text{*Basic laws about division and remainder*}
-
-lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
-apply (case_tac "b = 0", simp)
-apply (cut_tac a = a and b = b in divmod_int_correct)
-apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)
-done
-
-lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
-by(simp add: zmod_zdiv_equality[symmetric])
-
-lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
-by(simp add: mult_commute zmod_zdiv_equality[symmetric])
-
-text {* Tool setup *}
-
-ML {*
-local
-
-fun mk_number T n = HOLogic.number_of_const T $ HOLogic.mk_numeral n;
-
-fun find_first_numeral past (t::terms) =
-        ((snd (HOLogic.dest_number t), rev past @ terms)
-         handle TERM _ => find_first_numeral (t::past) terms)
-  | find_first_numeral past [] = raise TERM("find_first_numeral", []);
-
-val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
-
-fun mk_minus t = 
-  let val T = Term.fastype_of t
-  in Const (@{const_name HOL.uminus}, T --> T) $ t end;
-
-(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
-fun mk_sum T []        = mk_number T 0
-  | mk_sum T [t,u]     = mk_plus (t, u)
-  | mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
-
-(*this version ALWAYS includes a trailing zero*)
-fun long_mk_sum T []        = mk_number T 0
-  | long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
-
-val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} Term.dummyT;
-
-(*decompose additions AND subtractions as a sum*)
-fun dest_summing (pos, Const (@{const_name HOL.plus}, _) $ t $ u, ts) =
-        dest_summing (pos, t, dest_summing (pos, u, ts))
-  | dest_summing (pos, Const (@{const_name HOL.minus}, _) $ t $ u, ts) =
-        dest_summing (pos, t, dest_summing (not pos, u, ts))
-  | dest_summing (pos, t, ts) =
-        if pos then t::ts else mk_minus t :: ts;
-
-fun dest_sum t = dest_summing (true, t, []);
-
-structure CancelDivMod = CancelDivModFun(struct
-
-  val div_name = @{const_name div};
-  val mod_name = @{const_name mod};
-  val mk_binop = HOLogic.mk_binop;
-  val mk_sum = mk_sum HOLogic.intT;
-  val dest_sum = dest_sum;
-
-  val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];
-
-  val trans = trans;
-
-  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
-    (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))
-
-end)
-
-in
-
-val cancel_div_mod_int_proc = Simplifier.simproc @{theory}
-  "cancel_zdiv_zmod" ["(k::int) + l"] (K CancelDivMod.proc);
-
-val _ = Addsimprocs [cancel_div_mod_int_proc];
-
-end
-*}
-
-lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
-apply (cut_tac a = a and b = b in divmod_int_correct)
-apply (auto simp add: divmod_int_rel_def mod_int_def)
-done
-
-lemmas pos_mod_sign  [simp] = pos_mod_conj [THEN conjunct1, standard]
-   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]
-
-lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
-apply (cut_tac a = a and b = b in divmod_int_correct)
-apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)
-done
-
-lemmas neg_mod_sign  [simp] = neg_mod_conj [THEN conjunct1, standard]
-   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]
-
-
-
-subsection{*General Properties of div and mod*}
-
-lemma divmod_int_rel_div_mod: "b \<noteq> 0 ==> divmod_int_rel a b (a div b, a mod b)"
-apply (cut_tac a = a and b = b in zmod_zdiv_equality)
-apply (force simp add: divmod_int_rel_def linorder_neq_iff)
-done
-
-lemma divmod_int_rel_div: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"
-by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
-
-lemma divmod_int_rel_mod: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"
-by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
-
-lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
-apply (rule divmod_int_rel_div)
-apply (auto simp add: divmod_int_rel_def)
-done
-
-lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
-apply (rule divmod_int_rel_div)
-apply (auto simp add: divmod_int_rel_def)
-done
-
-lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
-apply (rule divmod_int_rel_div)
-apply (auto simp add: divmod_int_rel_def)
-done
-
-(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
-
-lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
-apply (rule_tac q = 0 in divmod_int_rel_mod)
-apply (auto simp add: divmod_int_rel_def)
-done
-
-lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
-apply (rule_tac q = 0 in divmod_int_rel_mod)
-apply (auto simp add: divmod_int_rel_def)
-done
-
-lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
-apply (rule_tac q = "-1" in divmod_int_rel_mod)
-apply (auto simp add: divmod_int_rel_def)
-done
-
-text{*There is no @{text mod_neg_pos_trivial}.*}
-
-
-(*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
-lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
-apply (case_tac "b = 0", simp)
-apply (simp add: divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, 
-                                 THEN divmod_int_rel_div, THEN sym])
-
-done
-
-(*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
-lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
-apply (case_tac "b = 0", simp)
-apply (subst divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, THEN divmod_int_rel_mod],
-       auto)
-done
-
-
-subsection{*Laws for div and mod with Unary Minus*}
-
-lemma zminus1_lemma:
-     "divmod_int_rel a b (q, r)
-      ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  
-                          if r=0 then 0 else b-r)"
-by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
-
-
-lemma zdiv_zminus1_eq_if:
-     "b \<noteq> (0::int)  
-      ==> (-a) div b =  
-          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
-by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_div])
-
-lemma zmod_zminus1_eq_if:
-     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
-apply (case_tac "b = 0", simp)
-apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_mod])
-done
-
-lemma zmod_zminus1_not_zero:
-  fixes k l :: int
-  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
-  unfolding zmod_zminus1_eq_if by auto
-
-lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
-by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
-
-lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
-by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
-
-lemma zdiv_zminus2_eq_if:
-     "b \<noteq> (0::int)  
-      ==> a div (-b) =  
-          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
-by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
-
-lemma zmod_zminus2_eq_if:
-     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
-by (simp add: zmod_zminus1_eq_if zmod_zminus2)
-
-lemma zmod_zminus2_not_zero:
-  fixes k l :: int
-  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
-  unfolding zmod_zminus2_eq_if by auto 
-
-
-subsection{*Division of a Number by Itself*}
-
-lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
-apply (subgoal_tac "0 < a*q")
- apply (simp add: zero_less_mult_iff, arith)
-done
-
-lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
-apply (subgoal_tac "0 \<le> a* (1-q) ")
- apply (simp add: zero_le_mult_iff)
-apply (simp add: right_diff_distrib)
-done
-
-lemma self_quotient: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"
-apply (simp add: split_ifs divmod_int_rel_def linorder_neq_iff)
-apply (rule order_antisym, safe, simp_all)
-apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
-apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
-apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
-done
-
-lemma self_remainder: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"
-apply (frule self_quotient, assumption)
-apply (simp add: divmod_int_rel_def)
-done
-
-lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
-by (simp add: divmod_int_rel_div_mod [THEN self_quotient])
-
-(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
-lemma zmod_self [simp]: "a mod a = (0::int)"
-apply (case_tac "a = 0", simp)
-apply (simp add: divmod_int_rel_div_mod [THEN self_remainder])
-done
-
-
-subsection{*Computation of Division and Remainder*}
-
-lemma zdiv_zero [simp]: "(0::int) div b = 0"
-by (simp add: div_int_def divmod_int_def)
-
-lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
-by (simp add: div_int_def divmod_int_def)
-
-lemma zmod_zero [simp]: "(0::int) mod b = 0"
-by (simp add: mod_int_def divmod_int_def)
-
-lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
-by (simp add: mod_int_def divmod_int_def)
-
-text{*a positive, b positive *}
-
-lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
-by (simp add: div_int_def divmod_int_def)
-
-lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
-by (simp add: mod_int_def divmod_int_def)
-
-text{*a negative, b positive *}
-
-lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
-by (simp add: div_int_def divmod_int_def)
-
-lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
-by (simp add: mod_int_def divmod_int_def)
-
-text{*a positive, b negative *}
-
-lemma div_pos_neg:
-     "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
-by (simp add: div_int_def divmod_int_def)
-
-lemma mod_pos_neg:
-     "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
-by (simp add: mod_int_def divmod_int_def)
-
-text{*a negative, b negative *}
-
-lemma div_neg_neg:
-     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
-by (simp add: div_int_def divmod_int_def)
-
-lemma mod_neg_neg:
-     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
-by (simp add: mod_int_def divmod_int_def)
-
-text {*Simplify expresions in which div and mod combine numerical constants*}
-
-lemma divmod_int_relI:
-  "\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk>
-    \<Longrightarrow> divmod_int_rel a b (q, r)"
-  unfolding divmod_int_rel_def by simp
-
-lemmas divmod_int_rel_div_eq = divmod_int_relI [THEN divmod_int_rel_div, THEN eq_reflection]
-lemmas divmod_int_rel_mod_eq = divmod_int_relI [THEN divmod_int_rel_mod, THEN eq_reflection]
-lemmas arithmetic_simps =
-  arith_simps
-  add_special
-  OrderedGroup.add_0_left
-  OrderedGroup.add_0_right
-  mult_zero_left
-  mult_zero_right
-  mult_1_left
-  mult_1_right
-
-(* simprocs adapted from HOL/ex/Binary.thy *)
-ML {*
-local
-  val mk_number = HOLogic.mk_number HOLogic.intT;
-  fun mk_cert u k l = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $
-    (@{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $ u $ mk_number k) $
-      mk_number l;
-  fun prove ctxt prop = Goal.prove ctxt [] [] prop
-    (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps @{thms arithmetic_simps}))));
-  fun binary_proc proc ss ct =
-    (case Thm.term_of ct of
-      _ $ t $ u =>
-      (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
-        SOME args => proc (Simplifier.the_context ss) args
-      | NONE => NONE)
-    | _ => NONE);
-in
-  fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
-    if n = 0 then NONE
-    else let val (k, l) = Integer.div_mod m n;
-    in SOME (rule OF [prove ctxt (Logic.mk_equals (t, mk_cert u k l))]) end);
-end
-*}
-
-simproc_setup binary_int_div ("number_of m div number_of n :: int") =
-  {* K (divmod_proc (@{thm divmod_int_rel_div_eq})) *}
-
-simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =
-  {* K (divmod_proc (@{thm divmod_int_rel_mod_eq})) *}
-
-lemmas posDivAlg_eqn_number_of [simp] =
-    posDivAlg_eqn [of "number_of v" "number_of w", standard]
-
-lemmas negDivAlg_eqn_number_of [simp] =
-    negDivAlg_eqn [of "number_of v" "number_of w", standard]
-
-
-text{*Special-case simplification *}
-
-lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
-apply (cut_tac a = a and b = "-1" in neg_mod_sign)
-apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
-apply (auto simp del: neg_mod_sign neg_mod_bound)
-done
-
-lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
-by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
-
-(** The last remaining special cases for constant arithmetic:
-    1 div z and 1 mod z **)
-
-lemmas div_pos_pos_1_number_of [simp] =
-    div_pos_pos [OF int_0_less_1, of "number_of w", standard]
-
-lemmas div_pos_neg_1_number_of [simp] =
-    div_pos_neg [OF int_0_less_1, of "number_of w", standard]
-
-lemmas mod_pos_pos_1_number_of [simp] =
-    mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
-
-lemmas mod_pos_neg_1_number_of [simp] =
-    mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
-
-
-lemmas posDivAlg_eqn_1_number_of [simp] =
-    posDivAlg_eqn [of concl: 1 "number_of w", standard]
-
-lemmas negDivAlg_eqn_1_number_of [simp] =
-    negDivAlg_eqn [of concl: 1 "number_of w", standard]
-
-
-
-subsection{*Monotonicity in the First Argument (Dividend)*}
-
-lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
-apply (cut_tac a = a and b = b in zmod_zdiv_equality)
-apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
-apply (rule unique_quotient_lemma)
-apply (erule subst)
-apply (erule subst, simp_all)
-done
-
-lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
-apply (cut_tac a = a and b = b in zmod_zdiv_equality)
-apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
-apply (rule unique_quotient_lemma_neg)
-apply (erule subst)
-apply (erule subst, simp_all)
-done
-
-
-subsection{*Monotonicity in the Second Argument (Divisor)*}
-
-lemma q_pos_lemma:
-     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
-apply (subgoal_tac "0 < b'* (q' + 1) ")
- apply (simp add: zero_less_mult_iff)
-apply (simp add: right_distrib)
-done
-
-lemma zdiv_mono2_lemma:
-     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
-         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
-      ==> q \<le> (q'::int)"
-apply (frule q_pos_lemma, assumption+) 
-apply (subgoal_tac "b*q < b* (q' + 1) ")
- apply (simp add: mult_less_cancel_left)
-apply (subgoal_tac "b*q = r' - r + b'*q'")
- prefer 2 apply simp
-apply (simp (no_asm_simp) add: right_distrib)
-apply (subst add_commute, rule zadd_zless_mono, arith)
-apply (rule mult_right_mono, auto)
-done
-
-lemma zdiv_mono2:
-     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
-apply (subgoal_tac "b \<noteq> 0")
- prefer 2 apply arith
-apply (cut_tac a = a and b = b in zmod_zdiv_equality)
-apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
-apply (rule zdiv_mono2_lemma)
-apply (erule subst)
-apply (erule subst, simp_all)
-done
-
-lemma q_neg_lemma:
-     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
-apply (subgoal_tac "b'*q' < 0")
- apply (simp add: mult_less_0_iff, arith)
-done
-
-lemma zdiv_mono2_neg_lemma:
-     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
-         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
-      ==> q' \<le> (q::int)"
-apply (frule q_neg_lemma, assumption+) 
-apply (subgoal_tac "b*q' < b* (q + 1) ")
- apply (simp add: mult_less_cancel_left)
-apply (simp add: right_distrib)
-apply (subgoal_tac "b*q' \<le> b'*q'")
- prefer 2 apply (simp add: mult_right_mono_neg, arith)
-done
-
-lemma zdiv_mono2_neg:
-     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
-apply (cut_tac a = a and b = b in zmod_zdiv_equality)
-apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
-apply (rule zdiv_mono2_neg_lemma)
-apply (erule subst)
-apply (erule subst, simp_all)
-done
-
-
-subsection{*More Algebraic Laws for div and mod*}
-
-text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
-
-lemma zmult1_lemma:
-     "[| divmod_int_rel b c (q, r);  c \<noteq> 0 |]  
-      ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
-by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)
-
-lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
-apply (case_tac "c = 0", simp)
-apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_div])
-done
-
-lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
-apply (case_tac "c = 0", simp)
-apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_mod])
-done
-
-lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
-apply (case_tac "b = 0", simp)
-apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
-done
-
-text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
-
-lemma zadd1_lemma:
-     "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br);  c \<noteq> 0 |]  
-      ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
-by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)
-
-(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
-lemma zdiv_zadd1_eq:
-     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
-apply (case_tac "c = 0", simp)
-apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] divmod_int_rel_div)
-done
-
-instance int :: ring_div
-proof
-  fix a b c :: int
-  assume not0: "b \<noteq> 0"
-  show "(a + c * b) div b = c + a div b"
-    unfolding zdiv_zadd1_eq [of a "c * b"] using not0 
-      by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
-next
-  fix a b c :: int
-  assume "a \<noteq> 0"
-  then show "(a * b) div (a * c) = b div c"
-  proof (cases "b \<noteq> 0 \<and> c \<noteq> 0")
-    case False then show ?thesis by auto
-  next
-    case True then have "b \<noteq> 0" and "c \<noteq> 0" by auto
-    with `a \<noteq> 0`
-    have "\<And>q r. divmod_int_rel b c (q, r) \<Longrightarrow> divmod_int_rel (a * b) (a * c) (q, a * r)"
-      apply (auto simp add: divmod_int_rel_def) 
-      apply (auto simp add: algebra_simps)
-      apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff mult_commute [of a] mult_less_cancel_right)
-      done
-    moreover with `c \<noteq> 0` divmod_int_rel_div_mod have "divmod_int_rel b c (b div c, b mod c)" by auto
-    ultimately have "divmod_int_rel (a * b) (a * c) (b div c, a * (b mod c))" .
-    moreover from  `a \<noteq> 0` `c \<noteq> 0` have "a * c \<noteq> 0" by simp
-    ultimately show ?thesis by (rule divmod_int_rel_div)
-  qed
-qed auto
-
-lemma posDivAlg_div_mod:
-  assumes "k \<ge> 0"
-  and "l \<ge> 0"
-  shows "posDivAlg k l = (k div l, k mod l)"
-proof (cases "l = 0")
-  case True then show ?thesis by (simp add: posDivAlg.simps)
-next
-  case False with assms posDivAlg_correct
-    have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
-    by simp
-  from divmod_int_rel_div [OF this `l \<noteq> 0`] divmod_int_rel_mod [OF this `l \<noteq> 0`]
-  show ?thesis by simp
-qed
-
-lemma negDivAlg_div_mod:
-  assumes "k < 0"
-  and "l > 0"
-  shows "negDivAlg k l = (k div l, k mod l)"
-proof -
-  from assms have "l \<noteq> 0" by simp
-  from assms negDivAlg_correct
-    have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
-    by simp
-  from divmod_int_rel_div [OF this `l \<noteq> 0`] divmod_int_rel_mod [OF this `l \<noteq> 0`]
-  show ?thesis by simp
-qed
-
-lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
-by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
-
-(* REVISIT: should this be generalized to all semiring_div types? *)
-lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
-
-
-subsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
-
-(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
-  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
-  to cause particular problems.*)
-
-text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
-
-lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
-apply (subgoal_tac "b * (c - q mod c) < r * 1")
- apply (simp add: algebra_simps)
-apply (rule order_le_less_trans)
- apply (erule_tac [2] mult_strict_right_mono)
- apply (rule mult_left_mono_neg)
-  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps pos_mod_bound)
- apply (simp)
-apply (simp)
-done
-
-lemma zmult2_lemma_aux2:
-     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
-apply (subgoal_tac "b * (q mod c) \<le> 0")
- apply arith
-apply (simp add: mult_le_0_iff)
-done
-
-lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
-apply (subgoal_tac "0 \<le> b * (q mod c) ")
-apply arith
-apply (simp add: zero_le_mult_iff)
-done
-
-lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
-apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
- apply (simp add: right_diff_distrib)
-apply (rule order_less_le_trans)
- apply (erule mult_strict_right_mono)
- apply (rule_tac [2] mult_left_mono)
-  apply simp
- using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps pos_mod_bound)
-apply simp
-done
-
-lemma zmult2_lemma: "[| divmod_int_rel a b (q, r);  b \<noteq> 0;  0 < c |]  
-      ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
-by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff
-                   zero_less_mult_iff right_distrib [symmetric] 
-                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
-
-lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
-apply (case_tac "b = 0", simp)
-apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_div])
-done
-
-lemma zmod_zmult2_eq:
-     "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
-apply (case_tac "b = 0", simp)
-apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_mod])
-done
-
-
-subsection {*Splitting Rules for div and mod*}
-
-text{*The proofs of the two lemmas below are essentially identical*}
-
-lemma split_pos_lemma:
- "0<k ==> 
-    P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
-apply (rule iffI, clarify)
- apply (erule_tac P="P ?x ?y" in rev_mp)  
- apply (subst mod_add_eq) 
- apply (subst zdiv_zadd1_eq) 
- apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
-txt{*converse direction*}
-apply (drule_tac x = "n div k" in spec) 
-apply (drule_tac x = "n mod k" in spec, simp)
-done
-
-lemma split_neg_lemma:
- "k<0 ==>
-    P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
-apply (rule iffI, clarify)
- apply (erule_tac P="P ?x ?y" in rev_mp)  
- apply (subst mod_add_eq) 
- apply (subst zdiv_zadd1_eq) 
- apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
-txt{*converse direction*}
-apply (drule_tac x = "n div k" in spec) 
-apply (drule_tac x = "n mod k" in spec, simp)
-done
-
-lemma split_zdiv:
- "P(n div k :: int) =
-  ((k = 0 --> P 0) & 
-   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
-   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
-apply (case_tac "k=0", simp)
-apply (simp only: linorder_neq_iff)
-apply (erule disjE) 
- apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
-                      split_neg_lemma [of concl: "%x y. P x"])
-done
-
-lemma split_zmod:
- "P(n mod k :: int) =
-  ((k = 0 --> P n) & 
-   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
-   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
-apply (case_tac "k=0", simp)
-apply (simp only: linorder_neq_iff)
-apply (erule disjE) 
- apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
-                      split_neg_lemma [of concl: "%x y. P y"])
-done
-
-(* Enable arith to deal with div 2 and mod 2: *)
-declare split_zdiv [of _ _ "number_of k", simplified, standard, arith_split]
-declare split_zmod [of _ _ "number_of k", simplified, standard, arith_split]
-
-
-subsection{*Speeding up the Division Algorithm with Shifting*}
-
-text{*computing div by shifting *}
-
-lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
-proof cases
-  assume "a=0"
-    thus ?thesis by simp
-next
-  assume "a\<noteq>0" and le_a: "0\<le>a"   
-  hence a_pos: "1 \<le> a" by arith
-  hence one_less_a2: "1 < 2 * a" by arith
-  hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
-    unfolding mult_le_cancel_left
-    by (simp add: add1_zle_eq add_commute [of 1])
-  with a_pos have "0 \<le> b mod a" by simp
-  hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
-    by (simp add: mod_pos_pos_trivial one_less_a2)
-  with  le_2a
-  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
-    by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
-                  right_distrib) 
-  thus ?thesis
-    by (subst zdiv_zadd1_eq,
-        simp add: mod_mult_mult1 one_less_a2
-                  div_pos_pos_trivial)
-qed
-
-lemma neg_zdiv_mult_2: "a \<le> (0::int) ==> (1 + 2*b) div (2*a) = (b+1) div a"
-apply (subgoal_tac " (1 + 2* (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a) ")
-apply (rule_tac [2] pos_zdiv_mult_2)
-apply (auto simp add: right_diff_distrib)
-apply (subgoal_tac " (-1 - (2 * b)) = - (1 + (2 * b))")
-apply (simp only: zdiv_zminus_zminus diff_minus minus_add_distrib [symmetric])
-apply (simp_all add: algebra_simps)
-apply (simp only: ab_diff_minus minus_add_distrib [symmetric] number_of_Min zdiv_zminus_zminus)
-done
-
-lemma zdiv_number_of_Bit0 [simp]:
-     "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  
-          number_of v div (number_of w :: int)"
-by (simp only: number_of_eq numeral_simps) (simp add: mult_2 [symmetric])
-
-lemma zdiv_number_of_Bit1 [simp]:
-     "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  
-          (if (0::int) \<le> number_of w                    
-           then number_of v div (number_of w)     
-           else (number_of v + (1::int)) div (number_of w))"
-apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) 
-apply (simp add: pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac mult_2 [symmetric])
-done
-
-
-subsection{*Computing mod by Shifting (proofs resemble those for div)*}
-
-lemma pos_zmod_mult_2:
-  fixes a b :: int
-  assumes "0 \<le> a"
-  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
-proof (cases "0 < a")
-  case False with assms show ?thesis by simp
-next
-  case True
-  then have "b mod a < a" by (rule pos_mod_bound)
-  then have "1 + b mod a \<le> a" by simp
-  then have A: "2 * (1 + b mod a) \<le> 2 * a" by simp
-  from `0 < a` have "0 \<le> b mod a" by (rule pos_mod_sign)
-  then have B: "0 \<le> 1 + 2 * (b mod a)" by simp
-  have "((1\<Colon>int) mod ((2\<Colon>int) * a) + (2\<Colon>int) * b mod ((2\<Colon>int) * a)) mod ((2\<Colon>int) * a) = (1\<Colon>int) + (2\<Colon>int) * (b mod a)"
-    using `0 < a` and A
-    by (auto simp add: mod_mult_mult1 mod_pos_pos_trivial ring_distribs intro!: mod_pos_pos_trivial B)
-  then show ?thesis by (subst mod_add_eq)
-qed
-
-lemma neg_zmod_mult_2:
-  fixes a b :: int
-  assumes "a \<le> 0"
-  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
-proof -
-  from assms have "0 \<le> - a" by auto
-  then have "(1 + 2 * (- b - 1)) mod (2 * (- a)) = 1 + 2 * ((- b - 1) mod (- a))"
-    by (rule pos_zmod_mult_2)
-  then show ?thesis by (simp add: zmod_zminus2 algebra_simps)
-     (simp add: diff_minus add_ac)
-qed
-
-lemma zmod_number_of_Bit0 [simp]:
-     "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  
-      (2::int) * (number_of v mod number_of w)"
-apply (simp only: number_of_eq numeral_simps) 
-apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
-                 neg_zmod_mult_2 add_ac mult_2 [symmetric])
-done
-
-lemma zmod_number_of_Bit1 [simp]:
-     "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  
-      (if (0::int) \<le> number_of w  
-                then 2 * (number_of v mod number_of w) + 1     
-                else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
-apply (simp only: number_of_eq numeral_simps) 
-apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
-                 neg_zmod_mult_2 add_ac mult_2 [symmetric])
-done
-
-
-subsection{*Quotients of Signs*}
-
-lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
-apply (subgoal_tac "a div b \<le> -1", force)
-apply (rule order_trans)
-apply (rule_tac a' = "-1" in zdiv_mono1)
-apply (auto simp add: div_eq_minus1)
-done
-
-lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
-by (drule zdiv_mono1_neg, auto)
-
-lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
-by (drule zdiv_mono1, auto)
-
-lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
-apply auto
-apply (drule_tac [2] zdiv_mono1)
-apply (auto simp add: linorder_neq_iff)
-apply (simp (no_asm_use) add: linorder_not_less [symmetric])
-apply (blast intro: div_neg_pos_less0)
-done
-
-lemma neg_imp_zdiv_nonneg_iff:
-     "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
-apply (subst zdiv_zminus_zminus [symmetric])
-apply (subst pos_imp_zdiv_nonneg_iff, auto)
-done
-
-(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
-lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
-by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
-
-(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
-lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
-by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
-
-
-subsection {* The Divides Relation *}
-
-lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
-  dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]
-
-lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
-  by (rule dvd_mod) (* TODO: remove *)
-
-lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
-  by (rule dvd_mod_imp_dvd) (* TODO: remove *)
-
-lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
-  using zmod_zdiv_equality[where a="m" and b="n"]
-  by (simp add: algebra_simps)
-
-lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
-apply (induct "y", auto)
-apply (rule zmod_zmult1_eq [THEN trans])
-apply (simp (no_asm_simp))
-apply (rule mod_mult_eq [symmetric])
-done
-
-lemma zdiv_int: "int (a div b) = (int a) div (int b)"
-apply (subst split_div, auto)
-apply (subst split_zdiv, auto)
-apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
-apply (auto simp add: IntDiv.divmod_int_rel_def of_nat_mult)
-done
-
-lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
-apply (subst split_mod, auto)
-apply (subst split_zmod, auto)
-apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
-       in unique_remainder)
-apply (auto simp add: IntDiv.divmod_int_rel_def of_nat_mult)
-done
-
-lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
-by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
-
-lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
-apply (subgoal_tac "m mod n = 0")
- apply (simp add: zmult_div_cancel)
-apply (simp only: dvd_eq_mod_eq_0)
-done
-
-text{*Suggested by Matthias Daum*}
-lemma int_power_div_base:
-     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
-apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
- apply (erule ssubst)
- apply (simp only: power_add)
- apply simp_all
-done
-
-text {* by Brian Huffman *}
-lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
-by (rule mod_minus_eq [symmetric])
-
-lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
-by (rule mod_diff_left_eq [symmetric])
-
-lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
-by (rule mod_diff_right_eq [symmetric])
-
-lemmas zmod_simps =
-  mod_add_left_eq  [symmetric]
-  mod_add_right_eq [symmetric]
-  zmod_zmult1_eq   [symmetric]
-  mod_mult_left_eq [symmetric]
-  zpower_zmod
-  zminus_zmod zdiff_zmod_left zdiff_zmod_right
-
-text {* Distributive laws for function @{text nat}. *}
-
-lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
-apply (rule linorder_cases [of y 0])
-apply (simp add: div_nonneg_neg_le0)
-apply simp
-apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
-done
-
-(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
-lemma nat_mod_distrib:
-  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
-apply (case_tac "y = 0", simp add: DIVISION_BY_ZERO)
-apply (simp add: nat_eq_iff zmod_int)
-done
-
-text{*Suggested by Matthias Daum*}
-lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
-apply (subgoal_tac "nat x div nat k < nat x")
- apply (simp (asm_lr) add: nat_div_distrib [symmetric])
-apply (rule Divides.div_less_dividend, simp_all)
-done
-
-text {* code generator setup *}
-
-context ring_1
-begin
-
-lemma of_int_num [code]:
-  "of_int k = (if k = 0 then 0 else if k < 0 then
-     - of_int (- k) else let
-       (l, m) = divmod_int k 2;
-       l' = of_int l
-     in if m = 0 then l' + l' else l' + l' + 1)"
-proof -
-  have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow> 
-    of_int k = of_int (k div 2 * 2 + 1)"
-  proof -
-    have "k mod 2 < 2" by (auto intro: pos_mod_bound)
-    moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)
-    moreover assume "k mod 2 \<noteq> 0"
-    ultimately have "k mod 2 = 1" by arith
-    moreover have "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
-    ultimately show ?thesis by auto
-  qed
-  have aux2: "\<And>x. of_int 2 * x = x + x"
-  proof -
-    fix x
-    have int2: "(2::int) = 1 + 1" by arith
-    show "of_int 2 * x = x + x"
-    unfolding int2 of_int_add left_distrib by simp
-  qed
-  have aux3: "\<And>x. x * of_int 2 = x + x"
-  proof -
-    fix x
-    have int2: "(2::int) = 1 + 1" by arith
-    show "x * of_int 2 = x + x" 
-    unfolding int2 of_int_add right_distrib by simp
-  qed
-  from aux1 show ?thesis by (auto simp add: divmod_int_mod_div Let_def aux2 aux3)
-qed
-
-end
-
-lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
-proof
-  assume H: "x mod n = y mod n"
-  hence "x mod n - y mod n = 0" by simp
-  hence "(x mod n - y mod n) mod n = 0" by simp 
-  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
-  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
-next
-  assume H: "n dvd x - y"
-  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
-  hence "x = n*k + y" by simp
-  hence "x mod n = (n*k + y) mod n" by simp
-  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
-qed
-
-lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
-  shows "\<exists>q. x = y + n * q"
-proof-
-  from xy have th: "int x - int y = int (x - y)" by simp 
-  from xyn have "int x mod int n = int y mod int n" 
-    by (simp add: zmod_int[symmetric])
-  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
-  hence "n dvd x - y" by (simp add: th zdvd_int)
-  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
-qed
-
-lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
-  (is "?lhs = ?rhs")
-proof
-  assume H: "x mod n = y mod n"
-  {assume xy: "x \<le> y"
-    from H have th: "y mod n = x mod n" by simp
-    from nat_mod_eq_lemma[OF th xy] have ?rhs 
-      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
-  moreover
-  {assume xy: "y \<le> x"
-    from nat_mod_eq_lemma[OF H xy] have ?rhs 
-      apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
-  ultimately  show ?rhs using linear[of x y] by blast  
-next
-  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
-  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
-  thus  ?lhs by simp
-qed
-
-lemma div_nat_number_of [simp]:
-     "(number_of v :: nat)  div  number_of v' =  
-          (if neg (number_of v :: int) then 0  
-           else nat (number_of v div number_of v'))"
-  unfolding nat_number_of_def number_of_is_id neg_def
-  by (simp add: nat_div_distrib)
-
-lemma one_div_nat_number_of [simp]:
-     "Suc 0 div number_of v' = nat (1 div number_of v')" 
-by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
-
-lemma mod_nat_number_of [simp]:
-     "(number_of v :: nat)  mod  number_of v' =  
-        (if neg (number_of v :: int) then 0  
-         else if neg (number_of v' :: int) then number_of v  
-         else nat (number_of v mod number_of v'))"
-  unfolding nat_number_of_def number_of_is_id neg_def
-  by (simp add: nat_mod_distrib)
-
-lemma one_mod_nat_number_of [simp]:
-     "Suc 0 mod number_of v' =  
-        (if neg (number_of v' :: int) then Suc 0
-         else nat (1 mod number_of v'))"
-by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
-
-lemmas dvd_eq_mod_eq_0_number_of =
-  dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
-
-declare dvd_eq_mod_eq_0_number_of [simp]
-
-
-subsection {* Transfer setup *}
-
-lemma transfer_nat_int_functions:
-    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
-    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
-  by (auto simp add: nat_div_distrib nat_mod_distrib)
-
-lemma transfer_nat_int_function_closures:
-    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
-    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
-  apply (cases "y = 0")
-  apply (auto simp add: pos_imp_zdiv_nonneg_iff)
-  apply (cases "y = 0")
-  apply auto
-done
-
-declare TransferMorphism_nat_int[transfer add return:
-  transfer_nat_int_functions
-  transfer_nat_int_function_closures
-]
-
-lemma transfer_int_nat_functions:
-    "(int x) div (int y) = int (x div y)"
-    "(int x) mod (int y) = int (x mod y)"
-  by (auto simp add: zdiv_int zmod_int)
-
-lemma transfer_int_nat_function_closures:
-    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
-    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
-  by (simp_all only: is_nat_def transfer_nat_int_function_closures)
-
-declare TransferMorphism_int_nat[transfer add return:
-  transfer_int_nat_functions
-  transfer_int_nat_function_closures
-]
-
-
-subsection {* Code generation *}
-
-definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
-  "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
-
-lemma pdivmod_posDivAlg [code]:
-  "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
-by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
-
-lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
-  apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
-    then pdivmod k l
-    else (let (r, s) = pdivmod k l in
-      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
-proof -
-  have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
-  show ?thesis
-    by (simp add: divmod_int_mod_div pdivmod_def)
-      (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
-      zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
-qed
-
-lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
-  apsnd ((op *) (sgn l)) (if sgn k = sgn l
-    then pdivmod k l
-    else (let (r, s) = pdivmod k l in
-      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
-proof -
-  have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"
-    by (auto simp add: not_less sgn_if)
-  then show ?thesis by (simp add: divmod_int_pdivmod)
-qed
-
-code_modulename SML
-  IntDiv Integer
-
-code_modulename OCaml
-  IntDiv Integer
-
-code_modulename Haskell
-  IntDiv Integer
-
-
-
-subsection {* Proof Tools setup; Combination and Cancellation Simprocs *}
-
-declare split_div[of _ _ "number_of k", standard, arith_split]
-declare split_mod[of _ _ "number_of k", standard, arith_split]
-
-
-subsubsection{*For @{text combine_numerals}*}
-
-lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
-by (simp add: add_mult_distrib)
-
-
-subsubsection{*For @{text cancel_numerals}*}
-
-lemma nat_diff_add_eq1:
-     "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
-by (simp split add: nat_diff_split add: add_mult_distrib)
-
-lemma nat_diff_add_eq2:
-     "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"
-by (simp split add: nat_diff_split add: add_mult_distrib)
-
-lemma nat_eq_add_iff1:
-     "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_eq_add_iff2:
-     "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_less_add_iff1:
-     "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_less_add_iff2:
-     "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_le_add_iff1:
-     "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-lemma nat_le_add_iff2:
-     "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"
-by (auto split add: nat_diff_split simp add: add_mult_distrib)
-
-
-subsubsection{*For @{text cancel_numeral_factors} *}
-
-lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
-by auto
-
-lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"
-by auto
-
-lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
-by auto
-
-lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
-by auto
-
-lemma nat_mult_dvd_cancel_disj[simp]:
-  "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"
-by(auto simp: dvd_eq_mod_eq_0 mod_mult_distrib2[symmetric])
-
-lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)"
-by(auto)
-
-
-subsubsection{*For @{text cancel_factor} *}
-
-lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"
-by auto
-
-lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"
-by auto
-
-lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"
-by auto
-
-lemma nat_mult_div_cancel_disj[simp]:
-     "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"
-by (simp add: nat_mult_div_cancel1)
-
-
-use "Tools/numeral_simprocs.ML"
-
-use "Tools/nat_numeral_simprocs.ML"
-
-declaration {* 
-  K (Lin_Arith.add_simps (@{thms neg_simps} @ [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}])
-  #> Lin_Arith.add_simps (@{thms ring_distribs} @ [@{thm Let_number_of}, @{thm Let_0}, @{thm Let_1},
-     @{thm nat_0}, @{thm nat_1},
-     @{thm add_nat_number_of}, @{thm diff_nat_number_of}, @{thm mult_nat_number_of},
-     @{thm eq_nat_number_of}, @{thm less_nat_number_of}, @{thm le_number_of_eq_not_less},
-     @{thm le_Suc_number_of}, @{thm le_number_of_Suc},
-     @{thm less_Suc_number_of}, @{thm less_number_of_Suc},
-     @{thm Suc_eq_number_of}, @{thm eq_number_of_Suc},
-     @{thm mult_Suc}, @{thm mult_Suc_right},
-     @{thm add_Suc}, @{thm add_Suc_right},
-     @{thm eq_number_of_0}, @{thm eq_0_number_of}, @{thm less_0_number_of},
-     @{thm of_int_number_of_eq}, @{thm of_nat_number_of_eq}, @{thm nat_number_of},
-     @{thm if_True}, @{thm if_False}])
-  #> Lin_Arith.add_simprocs (Numeral_Simprocs.assoc_fold_simproc
-      :: Numeral_Simprocs.combine_numerals
-      :: Numeral_Simprocs.cancel_numerals)
-  #> Lin_Arith.add_simprocs (Nat_Numeral_Simprocs.combine_numerals :: Nat_Numeral_Simprocs.cancel_numerals))
-*}
-
-end
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Fri Oct 30 11:31:34 2009 +0100
+++ b/src/HOL/IsaMakefile	Fri Oct 30 14:02:42 2009 +0100
@@ -248,12 +248,12 @@
   Groebner_Basis.thy \
   Hilbert_Choice.thy \
   Int.thy \
-  IntDiv.thy \
   List.thy \
   Main.thy \
   Map.thy \
   Nat_Numeral.thy \
   Nat_Transfer.thy \
+  Numeral_Simprocs.thy \
   Presburger.thy \
   Predicate_Compile.thy \
   Quickcheck.thy \
@@ -382,7 +382,6 @@
   Library/While_Combinator.thy Library/Product_ord.thy			\
   Library/Char_nat.thy Library/Char_ord.thy Library/Option_ord.thy	\
   Library/Sublist_Order.thy Library/List_lexord.thy			\
-  Library/Commutative_Ring.thy Library/comm_ring.ML			\
   Library/Coinductive_List.thy Library/AssocList.thy			\
   Library/Formal_Power_Series.thy Library/Binomial.thy			\
   Library/Eval_Witness.thy Library/Code_Char.thy			\
@@ -785,6 +784,9 @@
 
 $(LOG)/HOL-Decision_Procs.gz: $(OUT)/HOL \
   Decision_Procs/Approximation.thy \
+  Decision_Procs/Commutative_Ring.thy \
+  Decision_Procs/Commutative_Ring_Complete.thy \
+  Decision_Procs/commutative_ring_tac.ML \
   Decision_Procs/Cooper.thy \
   Decision_Procs/cooper_tac.ML \
   Decision_Procs/Dense_Linear_Order.thy \
@@ -795,6 +797,7 @@
   Decision_Procs/Decision_Procs.thy \
   Decision_Procs/ex/Dense_Linear_Order_Ex.thy \
   Decision_Procs/ex/Approximation_Ex.thy \
+  Decision_Procs/ex/Commutative_Ring_Ex.thy \
   Decision_Procs/ROOT.ML
 	@$(ISABELLE_TOOL) usedir $(OUT)/HOL Decision_Procs
 
@@ -937,7 +940,7 @@
 
 HOL-ex: HOL $(LOG)/HOL-ex.gz
 
-$(LOG)/HOL-ex.gz: $(OUT)/HOL Library/Commutative_Ring.thy		\
+$(LOG)/HOL-ex.gz: $(OUT)/HOL Decision_Procs/Commutative_Ring.thy		\
   Number_Theory/Primes.thy						\
   Tools/Predicate_Compile/predicate_compile_core.ML			\
   ex/Abstract_NAT.thy ex/Antiquote.thy ex/Arith_Examples.thy		\
@@ -945,8 +948,8 @@
   ex/Binary.thy ex/CTL.thy ex/Chinese.thy ex/Classical.thy		\
   ex/CodegenSML_Test.thy ex/Codegenerator_Candidates.thy		\
   ex/Codegenerator_Pretty.thy ex/Codegenerator_Pretty_Test.thy		\
-  ex/Codegenerator_Test.thy ex/Coherent.thy ex/Commutative_RingEx.thy	\
-  ex/Commutative_Ring_Complete.thy ex/Efficient_Nat_examples.thy	\
+  ex/Codegenerator_Test.thy ex/Coherent.thy	\
+  ex/Efficient_Nat_examples.thy	\
   ex/Eval_Examples.thy ex/Fundefs.thy ex/Groebner_Examples.thy		\
   ex/Guess.thy ex/HarmonicSeries.thy ex/Hebrew.thy			\
   ex/Hex_Bin_Examples.thy ex/Higher_Order_Logic.thy			\
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Library/Commutative_Ring.thy
--- a/src/HOL/Library/Commutative_Ring.thy	Fri Oct 30 11:31:34 2009 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,319 +0,0 @@
-(*  Author:     Bernhard Haeupler
-
-Proving equalities in commutative rings done "right" in Isabelle/HOL.
-*)
-
-header {* Proving equalities in commutative rings *}
-
-theory Commutative_Ring
-imports List Parity Main
-uses ("comm_ring.ML")
-begin
-
-text {* Syntax of multivariate polynomials (pol) and polynomial expressions. *}
-
-datatype 'a pol =
-    Pc 'a
-  | Pinj nat "'a pol"
-  | PX "'a pol" nat "'a pol"
-
-datatype 'a polex =
-    Pol "'a pol"
-  | Add "'a polex" "'a polex"
-  | Sub "'a polex" "'a polex"
-  | Mul "'a polex" "'a polex"
-  | Pow "'a polex" nat
-  | Neg "'a polex"
-
-text {* Interpretation functions for the shadow syntax. *}
-
-primrec
-  Ipol :: "'a::{comm_ring_1} list \<Rightarrow> 'a pol \<Rightarrow> 'a"
-where
-    "Ipol l (Pc c) = c"
-  | "Ipol l (Pinj i P) = Ipol (drop i l) P"
-  | "Ipol l (PX P x Q) = Ipol l P * (hd l)^x + Ipol (drop 1 l) Q"
-
-primrec
-  Ipolex :: "'a::{comm_ring_1} list \<Rightarrow> 'a polex \<Rightarrow> 'a"
-where
-    "Ipolex l (Pol P) = Ipol l P"
-  | "Ipolex l (Add P Q) = Ipolex l P + Ipolex l Q"
-  | "Ipolex l (Sub P Q) = Ipolex l P - Ipolex l Q"
-  | "Ipolex l (Mul P Q) = Ipolex l P * Ipolex l Q"
-  | "Ipolex l (Pow p n) = Ipolex l p ^ n"
-  | "Ipolex l (Neg P) = - Ipolex l P"
-
-text {* Create polynomial normalized polynomials given normalized inputs. *}
-
-definition
-  mkPinj :: "nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol" where
-  "mkPinj x P = (case P of
-    Pc c \<Rightarrow> Pc c |
-    Pinj y P \<Rightarrow> Pinj (x + y) P |
-    PX p1 y p2 \<Rightarrow> Pinj x P)"
-
-definition
-  mkPX :: "'a::{comm_ring} pol \<Rightarrow> nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol" where
-  "mkPX P i Q = (case P of
-    Pc c \<Rightarrow> (if (c = 0) then (mkPinj 1 Q) else (PX P i Q)) |
-    Pinj j R \<Rightarrow> PX P i Q |
-    PX P2 i2 Q2 \<Rightarrow> (if (Q2 = (Pc 0)) then (PX P2 (i+i2) Q) else (PX P i Q)) )"
-
-text {* Defining the basic ring operations on normalized polynomials *}
-
-function
-  add :: "'a::{comm_ring} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<oplus>" 65)
-where
-    "Pc a \<oplus> Pc b = Pc (a + b)"
-  | "Pc c \<oplus> Pinj i P = Pinj i (P \<oplus> Pc c)"
-  | "Pinj i P \<oplus> Pc c = Pinj i (P \<oplus> Pc c)"
-  | "Pc c \<oplus> PX P i Q = PX P i (Q \<oplus> Pc c)"
-  | "PX P i Q \<oplus> Pc c = PX P i (Q \<oplus> Pc c)"
-  | "Pinj x P \<oplus> Pinj y Q =
-      (if x = y then mkPinj x (P \<oplus> Q)
-       else (if x > y then mkPinj y (Pinj (x - y) P \<oplus> Q)
-         else mkPinj x (Pinj (y - x) Q \<oplus> P)))"
-  | "Pinj x P \<oplus> PX Q y R =
-      (if x = 0 then P \<oplus> PX Q y R
-       else (if x = 1 then PX Q y (R \<oplus> P)
-         else PX Q y (R \<oplus> Pinj (x - 1) P)))"
-  | "PX P x R \<oplus> Pinj y Q =
-      (if y = 0 then PX P x R \<oplus> Q
-       else (if y = 1 then PX P x (R \<oplus> Q)
-         else PX P x (R \<oplus> Pinj (y - 1) Q)))"
-  | "PX P1 x P2 \<oplus> PX Q1 y Q2 =
-      (if x = y then mkPX (P1 \<oplus> Q1) x (P2 \<oplus> Q2)
-       else (if x > y then mkPX (PX P1 (x - y) (Pc 0) \<oplus> Q1) y (P2 \<oplus> Q2)
-         else mkPX (PX Q1 (y-x) (Pc 0) \<oplus> P1) x (P2 \<oplus> Q2)))"
-by pat_completeness auto
-termination by (relation "measure (\<lambda>(x, y). size x + size y)") auto
-
-function
-  mul :: "'a::{comm_ring} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<otimes>" 70)
-where
-    "Pc a \<otimes> Pc b = Pc (a * b)"
-  | "Pc c \<otimes> Pinj i P =
-      (if c = 0 then Pc 0 else mkPinj i (P \<otimes> Pc c))"
-  | "Pinj i P \<otimes> Pc c =
-      (if c = 0 then Pc 0 else mkPinj i (P \<otimes> Pc c))"
-  | "Pc c \<otimes> PX P i Q =
-      (if c = 0 then Pc 0 else mkPX (P \<otimes> Pc c) i (Q \<otimes> Pc c))"
-  | "PX P i Q \<otimes> Pc c =
-      (if c = 0 then Pc 0 else mkPX (P \<otimes> Pc c) i (Q \<otimes> Pc c))"
-  | "Pinj x P \<otimes> Pinj y Q =
-      (if x = y then mkPinj x (P \<otimes> Q) else
-         (if x > y then mkPinj y (Pinj (x-y) P \<otimes> Q)
-           else mkPinj x (Pinj (y - x) Q \<otimes> P)))"
-  | "Pinj x P \<otimes> PX Q y R =
-      (if x = 0 then P \<otimes> PX Q y R else
-         (if x = 1 then mkPX (Pinj x P \<otimes> Q) y (R \<otimes> P)
-           else mkPX (Pinj x P \<otimes> Q) y (R \<otimes> Pinj (x - 1) P)))"
-  | "PX P x R \<otimes> Pinj y Q =
-      (if y = 0 then PX P x R \<otimes> Q else
-         (if y = 1 then mkPX (Pinj y Q \<otimes> P) x (R \<otimes> Q)
-           else mkPX (Pinj y Q \<otimes> P) x (R \<otimes> Pinj (y - 1) Q)))"
-  | "PX P1 x P2 \<otimes> PX Q1 y Q2 =
-      mkPX (P1 \<otimes> Q1) (x + y) (P2 \<otimes> Q2) \<oplus>
-        (mkPX (P1 \<otimes> mkPinj 1 Q2) x (Pc 0) \<oplus>
-          (mkPX (Q1 \<otimes> mkPinj 1 P2) y (Pc 0)))"
-by pat_completeness auto
-termination by (relation "measure (\<lambda>(x, y). size x + size y)")
-  (auto simp add: mkPinj_def split: pol.split)
-
-text {* Negation*}
-primrec
-  neg :: "'a::{comm_ring} pol \<Rightarrow> 'a pol"
-where
-    "neg (Pc c) = Pc (-c)"
-  | "neg (Pinj i P) = Pinj i (neg P)"
-  | "neg (PX P x Q) = PX (neg P) x (neg Q)"
-
-text {* Substraction *}
-definition
-  sub :: "'a::{comm_ring} pol \<Rightarrow> 'a pol \<Rightarrow> 'a pol" (infixl "\<ominus>" 65)
-where
-  "sub P Q = P \<oplus> neg Q"
-
-text {* Square for Fast Exponentation *}
-primrec
-  sqr :: "'a::{comm_ring_1} pol \<Rightarrow> 'a pol"
-where
-    "sqr (Pc c) = Pc (c * c)"
-  | "sqr (Pinj i P) = mkPinj i (sqr P)"
-  | "sqr (PX A x B) = mkPX (sqr A) (x + x) (sqr B) \<oplus>
-      mkPX (Pc (1 + 1) \<otimes> A \<otimes> mkPinj 1 B) x (Pc 0)"
-
-text {* Fast Exponentation *}
-fun
-  pow :: "nat \<Rightarrow> 'a::{comm_ring_1} pol \<Rightarrow> 'a pol"
-where
-    "pow 0 P = Pc 1"
-  | "pow n P = (if even n then pow (n div 2) (sqr P)
-       else P \<otimes> pow (n div 2) (sqr P))"
-  
-lemma pow_if:
-  "pow n P =
-   (if n = 0 then Pc 1 else if even n then pow (n div 2) (sqr P)
-    else P \<otimes> pow (n div 2) (sqr P))"
-  by (cases n) simp_all
-
-
-text {* Normalization of polynomial expressions *}
-
-primrec
-  norm :: "'a::{comm_ring_1} polex \<Rightarrow> 'a pol"
-where
-    "norm (Pol P) = P"
-  | "norm (Add P Q) = norm P \<oplus> norm Q"
-  | "norm (Sub P Q) = norm P \<ominus> norm Q"
-  | "norm (Mul P Q) = norm P \<otimes> norm Q"
-  | "norm (Pow P n) = pow n (norm P)"
-  | "norm (Neg P) = neg (norm P)"
-
-text {* mkPinj preserve semantics *}
-lemma mkPinj_ci: "Ipol l (mkPinj a B) = Ipol l (Pinj a B)"
-  by (induct B) (auto simp add: mkPinj_def algebra_simps)
-
-text {* mkPX preserves semantics *}
-lemma mkPX_ci: "Ipol l (mkPX A b C) = Ipol l (PX A b C)"
-  by (cases A) (auto simp add: mkPX_def mkPinj_ci power_add algebra_simps)
-
-text {* Correctness theorems for the implemented operations *}
-
-text {* Negation *}
-lemma neg_ci: "Ipol l (neg P) = -(Ipol l P)"
-  by (induct P arbitrary: l) auto
-
-text {* Addition *}
-lemma add_ci: "Ipol l (P \<oplus> Q) = Ipol l P + Ipol l Q"
-proof (induct P Q arbitrary: l rule: add.induct)
-  case (6 x P y Q)
-  show ?case
-  proof (rule linorder_cases)
-    assume "x < y"
-    with 6 show ?case by (simp add: mkPinj_ci algebra_simps)
-  next
-    assume "x = y"
-    with 6 show ?case by (simp add: mkPinj_ci)
-  next
-    assume "x > y"
-    with 6 show ?case by (simp add: mkPinj_ci algebra_simps)
-  qed
-next
-  case (7 x P Q y R)
-  have "x = 0 \<or> x = 1 \<or> x > 1" by arith
-  moreover
-  { assume "x = 0" with 7 have ?case by simp }
-  moreover
-  { assume "x = 1" with 7 have ?case by (simp add: algebra_simps) }
-  moreover
-  { assume "x > 1" from 7 have ?case by (cases x) simp_all }
-  ultimately show ?case by blast
-next
-  case (8 P x R y Q)
-  have "y = 0 \<or> y = 1 \<or> y > 1" by arith
-  moreover
-  { assume "y = 0" with 8 have ?case by simp }
-  moreover
-  { assume "y = 1" with 8 have ?case by simp }
-  moreover
-  { assume "y > 1" with 8 have ?case by simp }
-  ultimately show ?case by blast
-next
-  case (9 P1 x P2 Q1 y Q2)
-  show ?case
-  proof (rule linorder_cases)
-    assume a: "x < y" hence "EX d. d + x = y" by arith
-    with 9 a show ?case by (auto simp add: mkPX_ci power_add algebra_simps)
-  next
-    assume a: "y < x" hence "EX d. d + y = x" by arith
-    with 9 a show ?case by (auto simp add: power_add mkPX_ci algebra_simps)
-  next
-    assume "x = y"
-    with 9 show ?case by (simp add: mkPX_ci algebra_simps)
-  qed
-qed (auto simp add: algebra_simps)
-
-text {* Multiplication *}
-lemma mul_ci: "Ipol l (P \<otimes> Q) = Ipol l P * Ipol l Q"
-  by (induct P Q arbitrary: l rule: mul.induct)
-    (simp_all add: mkPX_ci mkPinj_ci algebra_simps add_ci power_add)
-
-text {* Substraction *}
-lemma sub_ci: "Ipol l (P \<ominus> Q) = Ipol l P - Ipol l Q"
-  by (simp add: add_ci neg_ci sub_def)
-
-text {* Square *}
-lemma sqr_ci: "Ipol ls (sqr P) = Ipol ls P * Ipol ls P"
-  by (induct P arbitrary: ls)
-    (simp_all add: add_ci mkPinj_ci mkPX_ci mul_ci algebra_simps power_add)
-
-text {* Power *}
-lemma even_pow:"even n \<Longrightarrow> pow n P = pow (n div 2) (sqr P)"
-  by (induct n) simp_all
-
-lemma pow_ci: "Ipol ls (pow n P) = Ipol ls P ^ n"
-proof (induct n arbitrary: P rule: nat_less_induct)
-  case (1 k)
-  show ?case
-  proof (cases k)
-    case 0
-    then show ?thesis by simp
-  next
-    case (Suc l)
-    show ?thesis
-    proof cases
-      assume "even l"
-      then have "Suc l div 2 = l div 2"
-        by (simp add: nat_number even_nat_plus_one_div_two)
-      moreover
-      from Suc have "l < k" by simp
-      with 1 have "\<And>P. Ipol ls (pow l P) = Ipol ls P ^ l" by simp
-      moreover
-      note Suc `even l` even_nat_plus_one_div_two
-      ultimately show ?thesis by (auto simp add: mul_ci power_Suc even_pow)
-    next
-      assume "odd l"
-      {
-        fix p
-        have "Ipol ls (sqr P) ^ (Suc l div 2) = Ipol ls P ^ Suc l"
-        proof (cases l)
-          case 0
-          with `odd l` show ?thesis by simp
-        next
-          case (Suc w)
-          with `odd l` have "even w" by simp
-          have two_times: "2 * (w div 2) = w"
-            by (simp only: numerals even_nat_div_two_times_two [OF `even w`])
-          have "Ipol ls P * Ipol ls P = Ipol ls P ^ Suc (Suc 0)"
-            by (simp add: power_Suc)
-          then have "Ipol ls P * Ipol ls P = Ipol ls P ^ 2"
-            by (simp add: numerals)
-          with Suc show ?thesis
-            by (auto simp add: power_mult [symmetric, of _ 2 _] two_times mul_ci sqr_ci
-                     simp del: power_Suc)
-        qed
-      } with 1 Suc `odd l` show ?thesis by simp
-    qed
-  qed
-qed
-
-text {* Normalization preserves semantics  *}
-lemma norm_ci: "Ipolex l Pe = Ipol l (norm Pe)"
-  by (induct Pe) (simp_all add: add_ci sub_ci mul_ci neg_ci pow_ci)
-
-text {* Reflection lemma: Key to the (incomplete) decision procedure *}
-lemma norm_eq:
-  assumes "norm P1 = norm P2"
-  shows "Ipolex l P1 = Ipolex l P2"
-proof -
-  from prems have "Ipol l (norm P1) = Ipol l (norm P2)" by simp
-  then show ?thesis by (simp only: norm_ci)
-qed
-
-
-use "comm_ring.ML"
-setup CommRing.setup
-
-end
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Library/Library.thy
--- a/src/HOL/Library/Library.thy	Fri Oct 30 11:31:34 2009 +0100
+++ b/src/HOL/Library/Library.thy	Fri Oct 30 14:02:42 2009 +0100
@@ -11,7 +11,6 @@
   Code_Char_chr
   Code_Integer
   Coinductive_List
-  Commutative_Ring
   Continuity
   ContNotDenum
   Countable
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Library/Numeral_Type.thy
--- a/src/HOL/Library/Numeral_Type.thy	Fri Oct 30 11:31:34 2009 +0100
+++ b/src/HOL/Library/Numeral_Type.thy	Fri Oct 30 14:02:42 2009 +0100
@@ -188,7 +188,7 @@
 by (rule type_definition.Abs_inverse [OF type])
 
 lemma Rep_Abs_mod: "Rep (Abs (m mod n)) = m mod n"
-by (simp add: Abs_inverse IntDiv.pos_mod_conj [OF size0])
+by (simp add: Abs_inverse pos_mod_conj [OF size0])
 
 lemma Rep_Abs_0: "Rep (Abs 0) = 0"
 by (simp add: Abs_inverse size0)
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Library/Word.thy
--- a/src/HOL/Library/Word.thy	Fri Oct 30 11:31:34 2009 +0100
+++ b/src/HOL/Library/Word.thy	Fri Oct 30 14:02:42 2009 +0100
@@ -5,7 +5,7 @@
 header {* Binary Words *}
 
 theory Word
-imports "~~/src/HOL/Main"
+imports Main
 begin
 
 subsection {* Auxilary Lemmas *}
@@ -561,35 +561,17 @@
   shows        "nat_to_bv (2 * bv_to_nat w + bitval x) = norm_unsigned (w @ [x])"
 proof (cases x)
   assume [simp]: "x = \<one>"
-  show ?thesis
-    apply (simp add: nat_to_bv_non0)
-    apply safe
-  proof -
-    fix q
-    assume "Suc (2 * bv_to_nat w) = 2 * q"
-    hence orig: "(2 * bv_to_nat w + 1) mod 2 = 2 * q mod 2" (is "?lhs = ?rhs")
-      by simp
-    have "?lhs = (1 + 2 * bv_to_nat w) mod 2"
-      by (simp add: add_commute)
-    also have "... = 1"
-      by (subst mod_add_eq) simp
-    finally have eq1: "?lhs = 1" .
-    have "?rhs  = 0" by simp
-    with orig and eq1
-    show "nat_to_bv (Suc (2 * bv_to_nat w) div 2) @ [\<zero>] = norm_unsigned (w @ [\<one>])"
-      by simp
-  next
-    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 show "nat_to_bv (Suc (2 * bv_to_nat w) div 2) @ [\<one>] = norm_unsigned (w @ [\<one>])" .
-  qed
+  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
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Library/comm_ring.ML
--- a/src/HOL/Library/comm_ring.ML	Fri Oct 30 11:31:34 2009 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,109 +0,0 @@
-(*  Author:     Amine Chaieb
-
-Tactic for solving equalities over commutative rings.
-*)
-
-signature COMM_RING =
-sig
-  val comm_ring_tac : Proof.context -> int -> tactic
-  val setup : theory -> theory
-end
-
-structure CommRing: COMM_RING =
-struct
-
-(* The Cring exception for erronous uses of cring_tac *)
-exception CRing of string;
-
-(* Zero and One of the commutative ring *)
-fun cring_zero T = Const (@{const_name HOL.zero}, T);
-fun cring_one T = Const (@{const_name HOL.one}, T);
-
-(* reification functions *)
-(* add two polynom expressions *)
-fun polT t = Type ("Commutative_Ring.pol", [t]);
-fun polexT t = Type ("Commutative_Ring.polex", [t]);
-
-(* pol *)
-fun pol_Pc t = Const ("Commutative_Ring.pol.Pc", t --> polT t);
-fun pol_Pinj t = Const ("Commutative_Ring.pol.Pinj", HOLogic.natT --> polT t --> polT t);
-fun pol_PX t = Const ("Commutative_Ring.pol.PX", polT t --> HOLogic.natT --> polT t --> polT t);
-
-(* polex *)
-fun polex_add t = Const ("Commutative_Ring.polex.Add", polexT t --> polexT t --> polexT t);
-fun polex_sub t = Const ("Commutative_Ring.polex.Sub", polexT t --> polexT t --> polexT t);
-fun polex_mul t = Const ("Commutative_Ring.polex.Mul", polexT t --> polexT t --> polexT t);
-fun polex_neg t = Const ("Commutative_Ring.polex.Neg", polexT t --> polexT t);
-fun polex_pol t = Const ("Commutative_Ring.polex.Pol", polT t --> polexT t);
-fun polex_pow t = Const ("Commutative_Ring.polex.Pow", polexT t --> HOLogic.natT --> polexT t);
-
-(* reification of polynoms : primitive cring expressions *)
-fun reif_pol T vs (t as Free _) =
-      let
-        val one = @{term "1::nat"};
-        val i = find_index (fn t' => t' = t) vs
-      in if i = 0
-        then pol_PX T $ (pol_Pc T $ cring_one T)
-          $ one $ (pol_Pc T $ cring_zero T)
-        else pol_Pinj T $ HOLogic.mk_nat i
-          $ (pol_PX T $ (pol_Pc T $ cring_one T)
-            $ one $ (pol_Pc T $ cring_zero T))
-        end
-  | reif_pol T vs t = pol_Pc T $ t;
-
-(* reification of polynom expressions *)
-fun reif_polex T vs (Const (@{const_name HOL.plus}, _) $ a $ b) =
-      polex_add T $ reif_polex T vs a $ reif_polex T vs b
-  | reif_polex T vs (Const (@{const_name HOL.minus}, _) $ a $ b) =
-      polex_sub T $ reif_polex T vs a $ reif_polex T vs b
-  | reif_polex T vs (Const (@{const_name HOL.times}, _) $ a $ b) =
-      polex_mul T $ reif_polex T vs a $ reif_polex T vs b
-  | reif_polex T vs (Const (@{const_name HOL.uminus}, _) $ a) =
-      polex_neg T $ reif_polex T vs a
-  | reif_polex T vs (Const (@{const_name Power.power}, _) $ a $ n) =
-      polex_pow T $ reif_polex T vs a $ n
-  | reif_polex T vs t = polex_pol T $ reif_pol T vs t;
-
-(* reification of the equation *)
-val cr_sort = @{sort "comm_ring_1"};
-
-fun reif_eq thy (eq as Const("op =", Type("fun", [T, _])) $ lhs $ rhs) =
-      if Sign.of_sort thy (T, cr_sort) then
-        let
-          val fs = OldTerm.term_frees eq;
-          val cvs = cterm_of thy (HOLogic.mk_list T fs);
-          val clhs = cterm_of thy (reif_polex T fs lhs);
-          val crhs = cterm_of thy (reif_polex T fs rhs);
-          val ca = ctyp_of thy T;
-        in (ca, cvs, clhs, crhs) end
-      else raise CRing ("reif_eq: not an equation over " ^ Syntax.string_of_sort_global thy cr_sort)
-  | reif_eq _ _ = raise CRing "reif_eq: not an equation";
-
-(* The cring tactic *)
-(* Attention: You have to make sure that no t^0 is in the goal!! *)
-(* Use simply rewriting t^0 = 1 *)
-val cring_simps =
-  [@{thm mkPX_def}, @{thm mkPinj_def}, @{thm sub_def}, @{thm power_add},
-    @{thm even_def}, @{thm pow_if}, sym OF [@{thm power_add}]];
-
-fun comm_ring_tac ctxt = SUBGOAL (fn (g, i) =>
-  let
-    val thy = ProofContext.theory_of ctxt;
-    val cring_ss = Simplifier.simpset_of ctxt  (*FIXME really the full simpset!?*)
-      addsimps cring_simps;
-    val (ca, cvs, clhs, crhs) = reif_eq thy (HOLogic.dest_Trueprop g)
-    val norm_eq_th =
-      simplify cring_ss (instantiate' [SOME ca] [SOME clhs, SOME crhs, SOME cvs] @{thm norm_eq})
-  in
-    cut_rules_tac [norm_eq_th] i
-    THEN (simp_tac cring_ss i)
-    THEN (simp_tac cring_ss i)
-  end);
-
-val setup =
-  Method.setup @{binding comm_ring} (Scan.succeed (SIMPLE_METHOD' o comm_ring_tac))
-    "reflective decision procedure for equalities over commutative rings" #>
-  Method.setup @{binding algebra} (Scan.succeed (SIMPLE_METHOD' o comm_ring_tac))
-    "method for proving algebraic properties (same as comm_ring)";
-
-end;
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Parity.thy
--- a/src/HOL/Parity.thy	Fri Oct 30 11:31:34 2009 +0100
+++ b/src/HOL/Parity.thy	Fri Oct 30 14:02:42 2009 +0100
@@ -315,42 +315,6 @@
   qed
 qed
 
-subsection {* General Lemmas About Division *}
-
-(*FIXME move to Divides.thy*)
-
-lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" 
-apply (induct "m")
-apply (simp_all add: mod_Suc)
-done
-
-declare Suc_times_mod_eq [of "number_of w", standard, simp]
-
-lemma [simp]: "n div k \<le> (Suc n) div k"
-by (simp add: div_le_mono) 
-
-lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
-by arith
-
-lemma div_2_gt_zero [simp]: "(1::nat) < n ==> 0 < n div 2" 
-by arith
-
-  (* Potential use of algebra : Equality modulo n*)
-lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
-by (simp add: mult_ac add_ac)
-
-lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
-proof -
-  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
-  also have "... = Suc m mod n" by (rule mod_mult_self3) 
-  finally show ?thesis .
-qed
-
-lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
-apply (subst mod_Suc [of m]) 
-apply (subst mod_Suc [of "m mod n"], simp) 
-done
-
 
 subsection {* More Even/Odd Results *}
  
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Tools/arith_data.ML
--- a/src/HOL/Tools/arith_data.ML	Fri Oct 30 11:31:34 2009 +0100
+++ b/src/HOL/Tools/arith_data.ML	Fri Oct 30 14:02:42 2009 +0100
@@ -1,7 +1,7 @@
 (*  Title:      HOL/Tools/arith_data.ML
     Author:     Markus Wenzel, Stefan Berghofer, and Tobias Nipkow
 
-Common arithmetic proof auxiliary.
+Common arithmetic proof auxiliary and legacy.
 *)
 
 signature ARITH_DATA =
@@ -11,6 +11,11 @@
   val add_tactic: string -> (bool -> Proof.context -> int -> tactic) -> theory -> theory
   val get_arith_facts: Proof.context -> thm list
 
+  val mk_number: typ -> int -> term
+  val mk_sum: typ -> term list -> term
+  val long_mk_sum: typ -> term list -> term
+  val dest_sum: term -> term list
+
   val prove_conv_nohyps: tactic list -> Proof.context -> term * term -> thm option
   val prove_conv: tactic list -> Proof.context -> thm list -> term * term -> thm option
   val prove_conv2: tactic -> (simpset -> tactic) -> simpset -> term * term -> thm
@@ -67,6 +72,36 @@
     "various arithmetic decision procedures";
 
 
+(* some specialized syntactic operations *)
+
+fun mk_number T n = HOLogic.number_of_const T $ HOLogic.mk_numeral n;
+
+val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
+
+fun mk_minus t = 
+  let val T = Term.fastype_of t
+  in Const (@{const_name HOL.uminus}, T --> T) $ t end;
+
+(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
+fun mk_sum T []        = mk_number T 0
+  | mk_sum T [t,u]     = mk_plus (t, u)
+  | mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
+
+(*this version ALWAYS includes a trailing zero*)
+fun long_mk_sum T []        = mk_number T 0
+  | long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
+
+(*decompose additions AND subtractions as a sum*)
+fun dest_summing (pos, Const (@{const_name HOL.plus}, _) $ t $ u, ts) =
+        dest_summing (pos, t, dest_summing (pos, u, ts))
+  | dest_summing (pos, Const (@{const_name HOL.minus}, _) $ t $ u, ts) =
+        dest_summing (pos, t, dest_summing (not pos, u, ts))
+  | dest_summing (pos, t, ts) =
+        if pos then t::ts else mk_minus t :: ts;
+
+fun dest_sum t = dest_summing (true, t, []);
+
+
 (* various auxiliary and legacy *)
 
 fun prove_conv_nohyps tacs ctxt (t, u) =
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/Tools/numeral_simprocs.ML
--- a/src/HOL/Tools/numeral_simprocs.ML	Fri Oct 30 11:31:34 2009 +0100
+++ b/src/HOL/Tools/numeral_simprocs.ML	Fri Oct 30 14:02:42 2009 +0100
@@ -16,9 +16,6 @@
 
 signature NUMERAL_SIMPROCS =
 sig
-  val mk_sum: typ -> term list -> term
-  val dest_sum: term -> term list
-
   val assoc_fold_simproc: simproc
   val combine_numerals: simproc
   val cancel_numerals: simproc list
@@ -32,39 +29,10 @@
 structure Numeral_Simprocs : NUMERAL_SIMPROCS =
 struct
 
-fun mk_number T n = HOLogic.number_of_const T $ HOLogic.mk_numeral n;
-
-fun find_first_numeral past (t::terms) =
-        ((snd (HOLogic.dest_number t), rev past @ terms)
-         handle TERM _ => find_first_numeral (t::past) terms)
-  | find_first_numeral past [] = raise TERM("find_first_numeral", []);
-
-val mk_plus = HOLogic.mk_binop @{const_name HOL.plus};
-
-fun mk_minus t = 
-  let val T = Term.fastype_of t
-  in Const (@{const_name HOL.uminus}, T --> T) $ t end;
-
-(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
-fun mk_sum T []        = mk_number T 0
-  | mk_sum T [t,u]     = mk_plus (t, u)
-  | mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
-
-(*this version ALWAYS includes a trailing zero*)
-fun long_mk_sum T []        = mk_number T 0
-  | long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
-
-val dest_plus = HOLogic.dest_bin @{const_name HOL.plus} Term.dummyT;
-
-(*decompose additions AND subtractions as a sum*)
-fun dest_summing (pos, Const (@{const_name HOL.plus}, _) $ t $ u, ts) =
-        dest_summing (pos, t, dest_summing (pos, u, ts))
-  | dest_summing (pos, Const (@{const_name HOL.minus}, _) $ t $ u, ts) =
-        dest_summing (pos, t, dest_summing (not pos, u, ts))
-  | dest_summing (pos, t, ts) =
-        if pos then t::ts else mk_minus t :: ts;
-
-fun dest_sum t = dest_summing (true, t, []);
+val mk_number = Arith_Data.mk_number;
+val mk_sum = Arith_Data.mk_sum;
+val long_mk_sum = Arith_Data.long_mk_sum;
+val dest_sum = Arith_Data.dest_sum;
 
 val mk_diff = HOLogic.mk_binop @{const_name HOL.minus};
 val dest_diff = HOLogic.dest_bin @{const_name HOL.minus} Term.dummyT;
@@ -95,6 +63,11 @@
       in dest_prod t @ dest_prod u end
       handle TERM _ => [t];
 
+fun find_first_numeral past (t::terms) =
+        ((snd (HOLogic.dest_number t), rev past @ terms)
+         handle TERM _ => find_first_numeral (t::past) terms)
+  | find_first_numeral past [] = raise TERM("find_first_numeral", []);
+
 (*DON'T do the obvious simplifications; that would create special cases*)
 fun mk_coeff (k, t) = mk_times (mk_number (Term.fastype_of t) k, t);
 
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/ex/Codegenerator_Candidates.thy
--- a/src/HOL/ex/Codegenerator_Candidates.thy	Fri Oct 30 11:31:34 2009 +0100
+++ b/src/HOL/ex/Codegenerator_Candidates.thy	Fri Oct 30 14:02:42 2009 +0100
@@ -9,7 +9,6 @@
   AssocList
   Binomial
   Fset
-  Commutative_Ring
   Enum
   List_Prefix
   Nat_Infinity
@@ -22,7 +21,7 @@
   Tree
   While_Combinator
   Word
-  "~~/src/HOL/ex/Commutative_Ring_Complete"
+  "~~/src/HOL/Decision_Procs/Commutative_Ring_Complete"
   "~~/src/HOL/ex/Records"
 begin
 
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/ex/Commutative_RingEx.thy
--- a/src/HOL/ex/Commutative_RingEx.thy	Fri Oct 30 11:31:34 2009 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,50 +0,0 @@
-(*  ID:         $Id$
-    Author:     Bernhard Haeupler
-*)
-
-header {* Some examples demonstrating the comm-ring method *}
-
-theory Commutative_RingEx
-imports Commutative_Ring
-begin
-
-lemma "4*(x::int)^5*y^3*x^2*3 + x*z + 3^5 = 12*x^7*y^3 + z*x + 243"
-by comm_ring
-
-lemma "((x::int) + y)^2  = x^2 + y^2 + 2*x*y"
-by comm_ring
-
-lemma "((x::int) + y)^3  = x^3 + y^3 + 3*x^2*y + 3*y^2*x"
-by comm_ring
-
-lemma "((x::int) - y)^3  = x^3 + 3*x*y^2 + (-3)*y*x^2 - y^3"
-by comm_ring
-
-lemma "((x::int) - y)^2  = x^2 + y^2 - 2*x*y"
-by comm_ring
-
-lemma " ((a::int) + b + c)^2 = a^2 + b^2 + c^2 + 2*a*b + 2*b*c + 2*a*c"
-by comm_ring
-
-lemma "((a::int) - b - c)^2 = a^2 + b^2 + c^2 - 2*a*b + 2*b*c - 2*a*c"
-by comm_ring
-
-lemma "(a::int)*b + a*c = a*(b+c)"
-by comm_ring
-
-lemma "(a::int)^2 - b^2 = (a - b) * (a + b)"
-by comm_ring
-
-lemma "(a::int)^3 - b^3 = (a - b) * (a^2 + a*b + b^2)"
-by comm_ring
-
-lemma "(a::int)^3 + b^3 = (a + b) * (a^2 - a*b + b^2)"
-by comm_ring
-
-lemma "(a::int)^4 - b^4 = (a - b) * (a + b)*(a^2 + b^2)"
-by comm_ring
-
-lemma "(a::int)^10 - b^10 = (a - b) * (a^9 + a^8*b + a^7*b^2 + a^6*b^3 + a^5*b^4 + a^4*b^5 + a^3*b^6 + a^2*b^7 + a*b^8 + b^9 )"
-by comm_ring
-
-end
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/ex/Commutative_Ring_Complete.thy
--- a/src/HOL/ex/Commutative_Ring_Complete.thy	Fri Oct 30 11:31:34 2009 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,391 +0,0 @@
-(*  Author:     Bernhard Haeupler
-
-This theory is about of the relative completeness of method comm-ring
-method.  As long as the reified atomic polynomials of type 'a pol are
-in normal form, the cring method is complete.
-*)
-
-header {* Proof of the relative completeness of method comm-ring *}
-
-theory Commutative_Ring_Complete
-imports Commutative_Ring
-begin
-
-text {* Formalization of normal form *}
-fun
-  isnorm :: "('a::{comm_ring}) pol \<Rightarrow> bool"
-where
-    "isnorm (Pc c) \<longleftrightarrow> True"
-  | "isnorm (Pinj i (Pc c)) \<longleftrightarrow> False"
-  | "isnorm (Pinj i (Pinj j Q)) \<longleftrightarrow> False"
-  | "isnorm (Pinj 0 P) \<longleftrightarrow> False"
-  | "isnorm (Pinj i (PX Q1 j Q2)) \<longleftrightarrow> isnorm (PX Q1 j Q2)"
-  | "isnorm (PX P 0 Q) \<longleftrightarrow> False"
-  | "isnorm (PX (Pc c) i Q) \<longleftrightarrow> c \<noteq> 0 \<and> isnorm Q"
-  | "isnorm (PX (PX P1 j (Pc c)) i Q) \<longleftrightarrow> c \<noteq> 0 \<and> isnorm (PX P1 j (Pc c)) \<and> isnorm Q"
-  | "isnorm (PX P i Q) \<longleftrightarrow> isnorm P \<and> isnorm Q"
-
-(* Some helpful lemmas *)
-lemma norm_Pinj_0_False:"isnorm (Pinj 0 P) = False"
-by(cases P, auto)
-
-lemma norm_PX_0_False:"isnorm (PX (Pc 0) i Q) = False"
-by(cases i, auto)
-
-lemma norm_Pinj:"isnorm (Pinj i Q) \<Longrightarrow> isnorm Q"
-by(cases i,simp add: norm_Pinj_0_False norm_PX_0_False,cases Q) auto
-
-lemma norm_PX2:"isnorm (PX P i Q) \<Longrightarrow> isnorm Q"
-by(cases i, auto, cases P, auto, case_tac pol2, auto)
-
-lemma norm_PX1:"isnorm (PX P i Q) \<Longrightarrow> isnorm P"
-by(cases i, auto, cases P, auto, case_tac pol2, auto)
-
-lemma mkPinj_cn:"\<lbrakk>y~=0; isnorm Q\<rbrakk> \<Longrightarrow> isnorm (mkPinj y Q)" 
-apply(auto simp add: mkPinj_def norm_Pinj_0_False split: pol.split)
-apply(case_tac nat, auto simp add: norm_Pinj_0_False)
-by(case_tac pol, auto) (case_tac y, auto)
-
-lemma norm_PXtrans: 
-  assumes A:"isnorm (PX P x Q)" and "isnorm Q2" 
-  shows "isnorm (PX P x Q2)"
-proof(cases P)
-  case (PX p1 y p2) from prems show ?thesis by(cases x, auto, cases p2, auto)
-next
-  case Pc from prems show ?thesis by(cases x, auto)
-next
-  case Pinj from prems show ?thesis by(cases x, auto)
-qed
- 
-lemma norm_PXtrans2: assumes A:"isnorm (PX P x Q)" and "isnorm Q2" shows "isnorm (PX P (Suc (n+x)) Q2)"
-proof(cases P)
-  case (PX p1 y p2)
-  from prems show ?thesis by(cases x, auto, cases p2, auto)
-next
-  case Pc
-  from prems show ?thesis by(cases x, auto)
-next
-  case Pinj
-  from prems show ?thesis by(cases x, auto)
-qed
-
-text {* mkPX conserves normalizedness (@{text "_cn"}) *}
-lemma mkPX_cn: 
-  assumes "x \<noteq> 0" and "isnorm P" and "isnorm Q" 
-  shows "isnorm (mkPX P x Q)"
-proof(cases P)
-  case (Pc c)
-  from prems show ?thesis by (cases x) (auto simp add: mkPinj_cn mkPX_def)
-next
-  case (Pinj i Q)
-  from prems show ?thesis by (cases x) (auto simp add: mkPinj_cn mkPX_def)
-next
-  case (PX P1 y P2)
-  from prems have Y0:"y>0" by(cases y, auto)
-  from prems have "isnorm P1" "isnorm P2" by (auto simp add: norm_PX1[of P1 y P2] norm_PX2[of P1 y P2])
-  with prems Y0 show ?thesis by (cases x, auto simp add: mkPX_def norm_PXtrans2[of P1 y _ Q _], cases P2, auto)
-qed
-
-text {* add conserves normalizedness *}
-lemma add_cn:"isnorm P \<Longrightarrow> isnorm Q \<Longrightarrow> isnorm (P \<oplus> Q)"
-proof(induct P Q rule: add.induct)
-  case (2 c i P2) thus ?case by (cases P2, simp_all, cases i, simp_all)
-next
-  case (3 i P2 c) thus ?case by (cases P2, simp_all, cases i, simp_all)
-next
-  case (4 c P2 i Q2)
-  from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
-  with prems show ?case by(cases i, simp, cases P2, auto, case_tac pol2, auto)
-next
-  case (5 P2 i Q2 c)
-  from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
-  with prems show ?case by(cases i, simp, cases P2, auto, case_tac pol2, auto)
-next
-  case (6 x P2 y Q2)
-  from prems have Y0:"y>0" by (cases y, auto simp add: norm_Pinj_0_False) 
-  from prems have X0:"x>0" by (cases x, auto simp add: norm_Pinj_0_False) 
-  have "x < y \<or> x = y \<or> x > y" by arith
-  moreover
-  { assume "x<y" hence "EX d. y=d+x" by arith
-    then obtain d where "y=d+x"..
-    moreover
-    note prems X0
-    moreover
-    from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
-    moreover
-    with prems have "isnorm (Pinj d Q2)" by (cases d, simp, cases Q2, auto)
-    ultimately have ?case by (simp add: mkPinj_cn)}
-  moreover
-  { assume "x=y"
-    moreover
-    from prems have "isnorm P2" "isnorm Q2" by(auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
-    moreover
-    note prems Y0
-    moreover
-    ultimately have ?case by (simp add: mkPinj_cn) }
-  moreover
-  { assume "x>y" hence "EX d. x=d+y" by arith
-    then obtain d where "x=d+y"..
-    moreover
-    note prems Y0
-    moreover
-    from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
-    moreover
-    with prems have "isnorm (Pinj d P2)" by (cases d, simp, cases P2, auto)
-    ultimately have ?case by (simp add: mkPinj_cn)}
-  ultimately show ?case by blast
-next
-  case (7 x P2 Q2 y R)
-  have "x=0 \<or> (x = 1) \<or> (x > 1)" by arith
-  moreover
-  { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
-  moreover
-  { assume "x=1"
-    from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
-    with prems have "isnorm (R \<oplus> P2)" by simp
-    with prems have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
-  moreover
-  { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
-    then obtain d where X:"x=Suc (Suc d)" ..
-    from prems have NR:"isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
-    with prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
-    with prems NR have "isnorm (R \<oplus> Pinj (x - 1) P2)" "isnorm (PX Q2 y R)" by simp fact
-    with X have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
-  ultimately show ?case by blast
-next
-  case (8 Q2 y R x P2)
-  have "x = 0 \<or> x = 1 \<or> x > 1" by arith
-  moreover
-  { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
-  moreover
-  { assume "x=1"
-    from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
-    with prems have "isnorm (R \<oplus> P2)" by simp
-    with prems have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
-  moreover
-  { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
-    then obtain d where X:"x=Suc (Suc d)" ..
-    from prems have NR:"isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
-    with prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
-    with prems NR have "isnorm (R \<oplus> Pinj (x - 1) P2)" "isnorm (PX Q2 y R)" by simp fact
-    with X have ?case by (simp add: norm_PXtrans[of Q2 y _]) }
-  ultimately show ?case by blast
-next
-  case (9 P1 x P2 Q1 y Q2)
-  from prems have Y0:"y>0" by(cases y, auto)
-  from prems have X0:"x>0" by(cases x, auto)
-  from prems have NP1:"isnorm P1" and NP2:"isnorm P2" by (auto simp add: norm_PX1[of P1 _ P2] norm_PX2[of P1 _ P2])
-  from prems have NQ1:"isnorm Q1" and NQ2:"isnorm Q2" by (auto simp add: norm_PX1[of Q1 _ Q2] norm_PX2[of Q1 _ Q2])
-  have "y < x \<or> x = y \<or> x < y" by arith
-  moreover
-  {assume sm1:"y < x" hence "EX d. x=d+y" by arith
-    then obtain d where sm2:"x=d+y"..
-    note prems NQ1 NP1 NP2 NQ2 sm1 sm2
-    moreover
-    have "isnorm (PX P1 d (Pc 0))" 
-    proof(cases P1)
-      case (PX p1 y p2)
-      with prems show ?thesis by(cases d, simp,cases p2, auto)
-    next case Pc   from prems show ?thesis by(cases d, auto)
-    next case Pinj from prems show ?thesis by(cases d, auto)
-    qed
-    ultimately have "isnorm (P2 \<oplus> Q2)" "isnorm (PX P1 (x - y) (Pc 0) \<oplus> Q1)" by auto
-    with Y0 sm1 sm2 have ?case by (simp add: mkPX_cn)}
-  moreover
-  {assume "x=y"
-    from prems NP1 NP2 NQ1 NQ2 have "isnorm (P2 \<oplus> Q2)" "isnorm (P1 \<oplus> Q1)" by auto
-    with Y0 prems have ?case by (simp add: mkPX_cn) }
-  moreover
-  {assume sm1:"x<y" hence "EX d. y=d+x" by arith
-    then obtain d where sm2:"y=d+x"..
-    note prems NQ1 NP1 NP2 NQ2 sm1 sm2
-    moreover
-    have "isnorm (PX Q1 d (Pc 0))" 
-    proof(cases Q1)
-      case (PX p1 y p2)
-      with prems show ?thesis by(cases d, simp,cases p2, auto)
-    next case Pc   from prems show ?thesis by(cases d, auto)
-    next case Pinj from prems show ?thesis by(cases d, auto)
-    qed
-    ultimately have "isnorm (P2 \<oplus> Q2)" "isnorm (PX Q1 (y - x) (Pc 0) \<oplus> P1)" by auto
-    with X0 sm1 sm2 have ?case by (simp add: mkPX_cn)}
-  ultimately show ?case by blast
-qed simp
-
-text {* mul concerves normalizedness *}
-lemma mul_cn :"isnorm P \<Longrightarrow> isnorm Q \<Longrightarrow> isnorm (P \<otimes> Q)"
-proof(induct P Q rule: mul.induct)
-  case (2 c i P2) thus ?case 
-    by (cases P2, simp_all) (cases "i",simp_all add: mkPinj_cn)
-next
-  case (3 i P2 c) thus ?case 
-    by (cases P2, simp_all) (cases "i",simp_all add: mkPinj_cn)
-next
-  case (4 c P2 i Q2)
-  from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
-  with prems show ?case 
-    by - (case_tac "c=0",simp_all,case_tac "i=0",simp_all add: mkPX_cn)
-next
-  case (5 P2 i Q2 c)
-  from prems have "isnorm P2" "isnorm Q2" by (auto simp only: norm_PX1[of P2 i Q2] norm_PX2[of P2 i Q2])
-  with prems show ?case
-    by - (case_tac "c=0",simp_all,case_tac "i=0",simp_all add: mkPX_cn)
-next
-  case (6 x P2 y Q2)
-  have "x < y \<or> x = y \<or> x > y" by arith
-  moreover
-  { assume "x<y" hence "EX d. y=d+x" by arith
-    then obtain d where "y=d+x"..
-    moreover
-    note prems
-    moreover
-    from prems have "x>0" by (cases x, auto simp add: norm_Pinj_0_False) 
-    moreover
-    from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
-    moreover
-    with prems have "isnorm (Pinj d Q2)" by (cases d, simp, cases Q2, auto) 
-    ultimately have ?case by (simp add: mkPinj_cn)}
-  moreover
-  { assume "x=y"
-    moreover
-    from prems have "isnorm P2" "isnorm Q2" by(auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
-    moreover
-    with prems have "y>0" by (cases y, auto simp add: norm_Pinj_0_False)
-    moreover
-    note prems
-    moreover
-    ultimately have ?case by (simp add: mkPinj_cn) }
-  moreover
-  { assume "x>y" hence "EX d. x=d+y" by arith
-    then obtain d where "x=d+y"..
-    moreover
-    note prems
-    moreover
-    from prems have "y>0" by (cases y, auto simp add: norm_Pinj_0_False) 
-    moreover
-    from prems have "isnorm P2" "isnorm Q2" by (auto simp add: norm_Pinj[of _ P2] norm_Pinj[of _ Q2])
-    moreover
-    with prems have "isnorm (Pinj d P2)"  by (cases d, simp, cases P2, auto)
-    ultimately have ?case by (simp add: mkPinj_cn) }
-  ultimately show ?case by blast
-next
-  case (7 x P2 Q2 y R)
-  from prems have Y0:"y>0" by(cases y, auto)
-  have "x=0 \<or> (x = 1) \<or> (x > 1)" by arith
-  moreover
-  { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
-  moreover
-  { assume "x=1"
-    from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
-    with prems have "isnorm (R \<otimes> P2)" "isnorm Q2" by (auto simp add: norm_PX1[of Q2 y R])
-    with Y0 prems have ?case by (simp add: mkPX_cn)}
-  moreover
-  { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
-    then obtain d where X:"x=Suc (Suc d)" ..
-    from prems have NR:"isnorm R" "isnorm Q2" by (auto simp add: norm_PX2[of Q2 y R] norm_PX1[of Q2 y R])
-    moreover
-    from prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
-    moreover
-    from prems have "isnorm (Pinj x P2)" by(cases P2, auto)
-    moreover
-    note prems
-    ultimately have "isnorm (R \<otimes> Pinj (x - 1) P2)" "isnorm (Pinj x P2 \<otimes> Q2)" by auto
-    with Y0 X have ?case by (simp add: mkPX_cn)}
-  ultimately show ?case by blast
-next
-  case (8 Q2 y R x P2)
-  from prems have Y0:"y>0" by(cases y, auto)
-  have "x=0 \<or> (x = 1) \<or> (x > 1)" by arith
-  moreover
-  { assume "x=0" with prems have ?case by (auto simp add: norm_Pinj_0_False)}
-  moreover
-  { assume "x=1"
-    from prems have "isnorm R" "isnorm P2" by (auto simp add: norm_Pinj[of _ P2] norm_PX2[of Q2 y R])
-    with prems have "isnorm (R \<otimes> P2)" "isnorm Q2" by (auto simp add: norm_PX1[of Q2 y R])
-    with Y0 prems have ?case by (simp add: mkPX_cn) }
-  moreover
-  { assume "x > 1" hence "EX d. x=Suc (Suc d)" by arith
-    then obtain d where X:"x=Suc (Suc d)" ..
-    from prems have NR:"isnorm R" "isnorm Q2" by (auto simp add: norm_PX2[of Q2 y R] norm_PX1[of Q2 y R])
-    moreover
-    from prems have "isnorm (Pinj (x - 1) P2)" by(cases P2, auto)
-    moreover
-    from prems have "isnorm (Pinj x P2)" by(cases P2, auto)
-    moreover
-    note prems
-    ultimately have "isnorm (R \<otimes> Pinj (x - 1) P2)" "isnorm (Pinj x P2 \<otimes> Q2)" by auto
-    with Y0 X have ?case by (simp add: mkPX_cn) }
-  ultimately show ?case by blast
-next
-  case (9 P1 x P2 Q1 y Q2)
-  from prems have X0:"x>0" by(cases x, auto)
-  from prems have Y0:"y>0" by(cases y, auto)
-  note prems
-  moreover
-  from prems have "isnorm P1" "isnorm P2" by (auto simp add: norm_PX1[of P1 x P2] norm_PX2[of P1 x P2])
-  moreover 
-  from prems have "isnorm Q1" "isnorm Q2" by (auto simp add: norm_PX1[of Q1 y Q2] norm_PX2[of Q1 y Q2])
-  ultimately have "isnorm (P1 \<otimes> Q1)" "isnorm (P2 \<otimes> Q2)"
-    "isnorm (P1 \<otimes> mkPinj 1 Q2)" "isnorm (Q1 \<otimes> mkPinj 1 P2)" 
-    by (auto simp add: mkPinj_cn)
-  with prems X0 Y0 have
-    "isnorm (mkPX (P1 \<otimes> Q1) (x + y) (P2 \<otimes> Q2))"
-    "isnorm (mkPX (P1 \<otimes> mkPinj (Suc 0) Q2) x (Pc 0))"  
-    "isnorm (mkPX (Q1 \<otimes> mkPinj (Suc 0) P2) y (Pc 0))" 
-    by (auto simp add: mkPX_cn)
-  thus ?case by (simp add: add_cn)
-qed(simp)
-
-text {* neg conserves normalizedness *}
-lemma neg_cn: "isnorm P \<Longrightarrow> isnorm (neg P)"
-proof (induct P)
-  case (Pinj i P2)
-  from prems have "isnorm P2" by (simp add: norm_Pinj[of i P2])
-  with prems show ?case by(cases P2, auto, cases i, auto)
-next
-  case (PX P1 x P2)
-  from prems have "isnorm P2" "isnorm P1" by (auto simp add: norm_PX1[of P1 x P2] norm_PX2[of P1 x P2])
-  with prems show ?case
-  proof(cases P1)
-    case (PX p1 y p2)
-    with prems show ?thesis by(cases x, auto, cases p2, auto)
-  next
-    case Pinj
-    with prems show ?thesis by(cases x, auto)
-  qed(cases x, auto)
-qed(simp)
-
-text {* sub conserves normalizedness *}
-lemma sub_cn:"isnorm p \<Longrightarrow> isnorm q \<Longrightarrow> isnorm (p \<ominus> q)"
-by (simp add: sub_def add_cn neg_cn)
-
-text {* sqr conserves normalizizedness *}
-lemma sqr_cn:"isnorm P \<Longrightarrow> isnorm (sqr P)"
-proof(induct P)
-  case (Pinj i Q)
-  from prems show ?case by(cases Q, auto simp add: mkPX_cn mkPinj_cn, cases i, auto simp add: mkPX_cn mkPinj_cn)
-next 
-  case (PX P1 x P2)
-  from prems have "x+x~=0" "isnorm P2" "isnorm P1" by (cases x,  auto simp add: norm_PX1[of P1 x P2] norm_PX2[of P1 x P2])
-  with prems have
-    "isnorm (mkPX (Pc (1 + 1) \<otimes> P1 \<otimes> mkPinj (Suc 0) P2) x (Pc 0))"
-    and "isnorm (mkPX (sqr P1) (x + x) (sqr P2))"
-   by (auto simp add: add_cn mkPX_cn mkPinj_cn mul_cn)
-  thus ?case by (auto simp add: add_cn mkPX_cn mkPinj_cn mul_cn)
-qed simp
-
-text {* pow conserves normalizedness *}
-lemma pow_cn:"isnorm P \<Longrightarrow> isnorm (pow n P)"
-proof (induct n arbitrary: P rule: nat_less_induct)
-  case (1 k)
-  show ?case 
-  proof (cases "k=0")
-    case False
-    then have K2:"k div 2 < k" by (cases k, auto)
-    from prems have "isnorm (sqr P)" by (simp add: sqr_cn)
-    with prems K2 show ?thesis
-    by (simp add: allE[of _ "(k div 2)" _] allE[of _ "(sqr P)" _], cases k, auto simp add: mul_cn)
-  qed simp
-qed
-
-end
diff -r ee5b5ef5e970 -r 85adf83af7ce src/HOL/ex/ROOT.ML
--- a/src/HOL/ex/ROOT.ML	Fri Oct 30 11:31:34 2009 +0100
+++ b/src/HOL/ex/ROOT.ML	Fri Oct 30 14:02:42 2009 +0100
@@ -45,8 +45,6 @@
   "Groebner_Examples",
   "MT",
   "Unification",
-  "Commutative_RingEx",
-  "Commutative_Ring_Complete",
   "Primrec",
   "Tarski",
   "Adder",
diff -r ee5b5ef5e970 -r 85adf83af7ce src/Pure/Isar/expression.ML
--- a/src/Pure/Isar/expression.ML	Fri Oct 30 11:31:34 2009 +0100
+++ b/src/Pure/Isar/expression.ML	Fri Oct 30 14:02:42 2009 +0100
@@ -611,7 +611,7 @@
   else raise Match);
 
 (* define one predicate including its intro rule and axioms
-   - bname: predicate name
+   - binding: predicate name
    - parms: locale parameters
    - defs: thms representing substitutions from defines elements
    - ts: terms representing locale assumptions (not normalised wrt. defs)
@@ -619,9 +619,9 @@
    - thy: the theory
 *)
 
-fun def_pred bname parms defs ts norm_ts thy =
+fun def_pred binding parms defs ts norm_ts thy =
   let
-    val name = Sign.full_name thy bname;
+    val name = Sign.full_name thy binding;
 
     val (body, bodyT, body_eq) = atomize_spec thy norm_ts;
     val env = Term.add_free_names body [];
@@ -639,9 +639,9 @@
     val ([pred_def], defs_thy) =
       thy
       |> bodyT = propT ? Sign.add_advanced_trfuns ([], [], [aprop_tr' (length args) name], [])
-      |> Sign.declare_const ((bname, predT), NoSyn) |> snd
+      |> Sign.declare_const ((Binding.conceal binding, predT), NoSyn) |> snd
       |> PureThy.add_defs false
-        [((Binding.conceal (Binding.map_name Thm.def_name bname),
+        [((Binding.conceal (Binding.map_name Thm.def_name binding),
             Logic.mk_equals (head, body)), [])];
     val defs_ctxt = ProofContext.init defs_thy |> Variable.declare_term head;
 
@@ -667,7 +667,7 @@
 
 (* main predicate definition function *)
 
-fun define_preds pname parms (((exts, exts'), (ints, ints')), defs) thy =
+fun define_preds binding parms (((exts, exts'), (ints, ints')), defs) thy =
   let
     val defs' = map (cterm_of thy #> Assumption.assume #> Drule.abs_def) defs;
 
@@ -675,13 +675,13 @@
       if null exts then (NONE, NONE, [], thy)
       else
         let
-          val aname = if null ints then pname else Binding.suffix_name ("_" ^ axiomsN) pname;
+          val abinding = if null ints then binding else Binding.suffix_name ("_" ^ axiomsN) binding;
           val ((statement, intro, axioms), thy') =
             thy
-            |> def_pred aname parms defs' exts exts';
+            |> def_pred abinding parms defs' exts exts';
           val (_, thy'') =
             thy'
-            |> Sign.mandatory_path (Binding.name_of aname)
+            |> Sign.mandatory_path (Binding.name_of abinding)
             |> PureThy.note_thmss Thm.internalK
               [((Binding.conceal (Binding.name introN), []), [([intro], [Locale.unfold_add])])]
             ||> Sign.restore_naming thy';
@@ -692,10 +692,10 @@
         let
           val ((statement, intro, axioms), thy''') =
             thy''
-            |> def_pred pname parms defs' (ints @ the_list a_pred) (ints' @ the_list a_pred);
+            |> def_pred binding parms defs' (ints @ the_list a_pred) (ints' @ the_list a_pred);
           val (_, thy'''') =
             thy'''
-            |> Sign.mandatory_path (Binding.name_of pname)
+            |> Sign.mandatory_path (Binding.name_of binding)
             |> PureThy.note_thmss Thm.internalK
                  [((Binding.conceal (Binding.name introN), []), [([intro], [Locale.intro_add])]),
                   ((Binding.conceal (Binding.name axiomsN), []),
@@ -723,9 +723,9 @@
   | defines_to_notes _ e = e;
 
 fun gen_add_locale prep_decl
-    bname raw_predicate_bname raw_import raw_body thy =
+    binding raw_predicate_binding raw_import raw_body thy =
   let
-    val name = Sign.full_name thy bname;
+    val name = Sign.full_name thy binding;
     val _ = Locale.defined thy name andalso
       error ("Duplicate definition of locale " ^ quote name);
 
@@ -733,17 +733,17 @@
       prep_decl raw_import I raw_body (ProofContext.init thy);
     val text as (((_, exts'), _), defs) = eval ctxt' deps body_elems;
 
-    val predicate_bname =
-      if Binding.is_empty raw_predicate_bname then bname
-      else raw_predicate_bname;
+    val predicate_binding =
+      if Binding.is_empty raw_predicate_binding then binding
+      else raw_predicate_binding;
     val ((a_statement, a_intro, a_axioms), (b_statement, b_intro, b_axioms), thy') =
-      define_preds predicate_bname parms text thy;
+      define_preds predicate_binding parms text thy;
 
     val extraTs = subtract (op =) (fold Term.add_tfreesT (map snd parms) []) (fold Term.add_tfrees exts' []);
     val _ =
       if null extraTs then ()
       else warning ("Additional type variable(s) in locale specification " ^
-        quote (Binding.str_of bname));
+        quote (Binding.str_of binding));
 
     val a_satisfy = Element.satisfy_morphism a_axioms;
     val b_satisfy = Element.satisfy_morphism b_axioms;
@@ -755,7 +755,7 @@
 
     val notes =
       if is_some asm then
-        [(Thm.internalK, [((Binding.conceal (Binding.suffix_name ("_" ^ axiomsN) bname), []),
+        [(Thm.internalK, [((Binding.conceal (Binding.suffix_name ("_" ^ axiomsN) binding), []),
           [([Assumption.assume (cterm_of thy' (the asm))],
             [(Attrib.internal o K) Locale.witness_add])])])]
       else [];
@@ -772,7 +772,7 @@
     val axioms = map Element.conclude_witness b_axioms;
 
     val loc_ctxt = thy'
-      |> Locale.register_locale bname (extraTs, params)
+      |> Locale.register_locale binding (extraTs, params)
           (asm, rev defs) (a_intro, b_intro) axioms ([], []) (rev notes) (rev deps')
       |> TheoryTarget.init (SOME name)
       |> fold (fn (kind, facts) => LocalTheory.notes kind facts #> snd) notes';