# HG changeset patch
# User chaieb
# Date 1126711552 -7200
# Node ID 105519771c67db045124ef02eedeb6cb0467586f
# Parent  afaa031ed4dad2592873d4fbbbd9cec55d33b2cc
The oracle for Presburger has been changer: It is automatically generated form a verified formaliztion of Cooper's Algorithm ex/Reflected_Presburger.thy
comm_ring : a reflected Method for proving equalities in a commutative ring

diff -r afaa031ed4da -r 105519771c67 src/HOL/Commutative_Ring.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Commutative_Ring.thy	Wed Sep 14 17:25:52 2005 +0200
@@ -0,0 +1,310 @@
+
+(*  ID:         $Id$
+    Author:     Bernhard Haeupler
+
+  Proving equalities in a commutative Ring done "right" in Isabelle/HOL
+*)
+
+theory Commutative_Ring
+imports List
+uses ("comm_ring.ML")
+begin
+
+  (* 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"
+
+  (* Interpretation functions for the shadow syntax *)
+consts 
+  Ipol :: "('a::{comm_ring,recpower}) list \<Rightarrow> 'a pol \<Rightarrow> 'a"
+  Ipolex :: "('a::{comm_ring,recpower}) list \<Rightarrow> 'a polex \<Rightarrow> 'a"
+
+primrec
+  "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 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)"
+
+  (* Create polynomial normalized polynomials given normalized inputs *)
+constdefs mkPinj :: "nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol"
+  mkPinj_def: "mkPinj x P \<equiv> (case P of   
+  (Pc c) \<Rightarrow> (Pc c) |
+  (Pinj y P) \<Rightarrow> Pinj (x+y) P |
+  (PX p1 y p2) \<Rightarrow> Pinj x P )"
+
+constdefs mkPX :: "('a::{comm_ring,recpower}) pol \<Rightarrow> nat \<Rightarrow> 'a pol \<Rightarrow> 'a pol"
+  mkPX_def: "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)) )"
+
+  (* Defining the basic ring operations on normalized polynomials *)
+consts
+add :: "(('a::{comm_ring,recpower}) pol) \<times> ('a pol) \<Rightarrow> 'a pol"
+mul :: "(('a::{comm_ring,recpower}) pol) \<times> ('a pol) \<Rightarrow> 'a pol"
+neg :: "('a::{comm_ring,recpower}) pol \<Rightarrow> 'a pol"
+sqr :: "('a::{comm_ring,recpower}) pol  \<Rightarrow> 'a pol"
+pow :: "('a::{comm_ring,recpower}) pol \<times> nat \<Rightarrow> 'a pol"
+
+
+  (* Addition *)
+recdef add "measure (\<lambda>(x, y). size x + size y)"
+  "add (Pc a, Pc b) = Pc (a+b)"
+  "add (Pc c, Pinj i P) = Pinj i (add (P, Pc c))"
+  "add (Pinj i P, Pc c) = Pinj i (add (P, Pc c))"
+  "add (Pc c, PX P i Q) = PX P i (add (Q, Pc c))"
+  "add (PX P i Q, Pc c) = PX P i (add (Q, Pc c))"
+  "add (Pinj x P, Pinj y Q) = 
+  (if x=y then mkPinj x (add (P, Q)) 
+   else (if x>y then mkPinj y (add (Pinj (x-y) P, Q))
+         else mkPinj x (add (Pinj (y-x) Q, P)) ))"
+  "add (Pinj x P, PX Q y R) = 
+  (if x=0 then add(P, PX Q y R) 
+   else (if x=1 then PX Q y (add (R, P)) 
+         else PX Q y (add (R, Pinj (x - 1) P))))"
+  "add (PX P x R, Pinj y Q) = 
+  (if y=0 then add(PX P x R, Q) 
+   else (if y=1 then PX P x (add (R, Q)) 
+         else PX P x (add (R, Pinj (y - 1) Q))))"
+  "add (PX P1 x P2, PX Q1 y Q2) = 
+  (if x=y then mkPX (add (P1, Q1)) x (add (P2, Q2)) 
+  else (if x>y then mkPX (add (PX P1 (x-y) (Pc 0), Q1)) y (add (P2,Q2)) 
+        else mkPX (add (PX Q1 (y-x) (Pc 0), P1)) x (add (P2,Q2)) ))"
+
+  (* Multiplication *)
+recdef mul "measure (\<lambda>(x, y). size x + size y)"
+  "mul (Pc a, Pc b) = Pc (a*b)"
+  "mul (Pc c, Pinj i P) = (if c=0 then Pc 0 else mkPinj i (mul (P, Pc c)))"
+  "mul (Pinj i P, Pc c) = (if c=0 then Pc 0 else mkPinj i (mul (P, Pc c)))"
+  "mul (Pc c, PX P i Q) = 
+  (if c=0 then Pc 0 else mkPX (mul (P, Pc c)) i (mul (Q, Pc c)))"
+  "mul (PX P i Q, Pc c) = 
+  (if c=0 then Pc 0 else mkPX (mul (P, Pc c)) i (mul (Q, Pc c)))"
+  "mul (Pinj x P, Pinj y Q) = 
+  (if x=y then mkPinj x (mul (P, Q)) 
+   else (if x>y then mkPinj y (mul (Pinj (x-y) P, Q))
+         else mkPinj x (mul (Pinj (y-x) Q, P)) ))"
+  "mul (Pinj x P, PX Q y R) = 
+  (if x=0 then mul(P, PX Q y R) 
+   else (if x=1 then mkPX (mul (Pinj x P, Q)) y (mul (R, P)) 
+         else mkPX (mul (Pinj x P, Q)) y (mul (R, Pinj (x - 1) P))))"
+  "mul (PX P x R, Pinj y Q) = 
+  (if y=0 then mul(PX P x R, Q) 
+   else (if y=1 then mkPX (mul (Pinj y Q, P)) x (mul (R, Q)) 
+         else mkPX (mul (Pinj y Q, P)) x (mul (R, Pinj (y - 1) Q))))"
+  "mul (PX P1 x P2, PX Q1 y Q2) =  
+  add (mkPX (mul (P1, Q1)) (x+y) (mul (P2, Q2)), 
+  add (mkPX (mul (P1, mkPinj 1 Q2)) x (Pc 0), mkPX (mul (Q1, mkPinj 1 P2)) y (Pc 0)) )"
+(hints simp add: mkPinj_def split: pol.split)
+
+  (* Negation*)
+primrec
+  "neg (Pc c) = (Pc (-c))"
+  "neg (Pinj i P) = Pinj i (neg P)"
+  "neg (PX P x Q) = PX (neg P) x (neg Q)"
+
+  (* Substraction*)
+constdefs sub :: "(('a::{comm_ring,recpower}) pol) \<Rightarrow> ('a pol) \<Rightarrow> 'a pol"
+  "sub p q \<equiv> add (p,neg q)"
+
+  (* Square for Fast Exponentation *)
+primrec
+  "sqr (Pc c) = Pc (c*c)"
+  "sqr (Pinj i P) = mkPinj i (sqr P)"
+  "sqr (PX A x B) = add (mkPX (sqr A) (x+x) (sqr B), mkPX (mul (mul (Pc (1+1), A), mkPinj 1 B)) x (Pc 0))"
+
+  (* Fast Exponentation *)
+lemma pow_wf:"odd n \<longrightarrow> (n::nat) div 2 < n" by(cases n, auto)
+recdef pow "measure (\<lambda>(x, y). y)"
+  "pow (p, 0) = Pc 1"
+  "pow (p, n) = (if even n then (pow (sqr p, n div 2)) else mul (p, pow (sqr p, n div 2)))"
+(hints simp add: pow_wf)
+
+lemma pow_if: "pow (p,n) = (if n=0 then Pc 1 else (if even n then (pow (sqr p, n div 2)) else mul (p, pow (sqr p, n div 2))))"
+by (cases n) simp_all
+
+(*
+lemma number_of_nat_B0: "(number_of (w BIT bit.B0) ::nat) = 2* (number_of w)"
+by simp
+
+lemma number_of_nat_even: "even (number_of (w BIT bit.B0)::nat)"
+by simp
+  
+lemma pow_even : "pow (p, number_of(w BIT bit.B0)) = pow (sqr p, number_of w)"
+  ( is "pow(?p,?n) = pow (_,?n2)")
+proof-
+  have "even ?n" by simp
+  hence "pow (p, ?n) = pow (sqr p, ?n div 2)" 
+    apply simp
+    apply (cases "IntDef.neg (number_of w)")
+    apply simp
+    done  
+*)
+  (* Normalization of polynomial expressions *)
+
+consts norm :: "('a::{comm_ring,recpower}) polex \<Rightarrow> 'a pol"
+primrec
+  "norm (Pol P) = P"
+  "norm (Add P Q) = add (norm P, norm Q)"
+  "norm (Sub p q) = sub (norm p) (norm q)"
+  "norm (Mul P Q) = mul (norm P, norm Q)"
+  "norm (Pow p n) = pow (norm p, n)"
+  "norm (Neg P) = neg (norm P)"
+
+  (* mkPinj preserve semantics *)
+lemma mkPinj_ci: "ALL a l. Ipol l (mkPinj a B) = Ipol l (Pinj a B)"
+  by(induct B, auto simp add: mkPinj_def ring_eq_simps)
+
+  (* mkPX preserves semantics *)
+lemma mkPX_ci: "ALL b l. Ipol l (mkPX A b C) = Ipol l (PX A b C)"
+  by(case_tac A, auto simp add: mkPX_def mkPinj_ci power_add ring_eq_simps)
+
+  (* Correctness theorems for the implemented operations *)
+  (* Negation *)
+lemma neg_ci: "ALL l. Ipol l (neg P) = -(Ipol l P)"
+  by(induct P, auto)
+
+  (* Addition *)
+lemma add_ci: "ALL l. Ipol l (add (P, Q)) = Ipol l P + Ipol l Q"
+proof(induct P Q rule: add.induct)
+  case (6 x P y Q)
+  have "x < y \<or> x = y \<or> x > y" by arith
+  moreover
+  { assume "x<y" hence "EX d. d+x=y" by arith
+    then obtain d where "d+x=y"..
+    with prems have ?case by (auto simp add: mkPinj_ci ring_eq_simps) }
+  moreover
+  { assume "x=y" with prems have ?case by (auto simp add: mkPinj_ci)}
+  moreover
+  { assume "x>y" hence "EX d. d+y=x" by arith
+    then obtain d where "d+y=x"..
+    with prems have ?case by (auto simp add: mkPinj_ci ring_eq_simps) }
+  ultimately show ?case by blast
+next 
+  case (7 x P Q y R)
+  have "x=0 \<or> (x = 1) \<or> (x > 1)" by arith
+  moreover
+  { assume "x=0" with prems have ?case by auto }
+  moreover
+  { assume "x=1" with prems have ?case by (auto simp add: ring_eq_simps) }
+  moreover
+  { assume "x > 1" from prems have ?case by(cases x, auto) }
+  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 prems have ?case by simp}
+  moreover
+  {assume "y=1" with prems have ?case by simp}
+  moreover
+  {assume "y>1" hence "EX d. d+1=y" by arith
+    then obtain d where "d+1=y" ..
+    with prems have ?case by auto }
+  ultimately show ?case by blast
+next 
+  case (9 P1 x P2 Q1 y Q2)
+  have "y < x \<or> x = y \<or> x < y" by arith
+  moreover
+  {assume "y < x" hence "EX d. d+y=x" by arith
+    then obtain d where "d+y=x"..
+    with prems have ?case by (auto simp add: power_add mkPX_ci ring_eq_simps) }
+  moreover
+  {assume "x=y" with prems have ?case by(auto simp add: mkPX_ci ring_eq_simps) }
+  moreover
+  {assume "x<y" hence "EX d. d+x=y" by arith
+    then obtain d where "d+x=y" ..
+    with prems have ?case by (auto simp add: mkPX_ci power_add ring_eq_simps) }
+  ultimately show ?case by blast
+qed (auto simp add: ring_eq_simps)
+
+    (* Multiplication *)
+lemma mul_ci: "ALL l. Ipol l (mul (P, Q)) = Ipol l P * Ipol l Q" 
+  by (induct P Q rule: mul.induct, auto simp add: mkPX_ci mkPinj_ci ring_eq_simps add_ci power_add) 
+
+  (* Substraction *)
+lemma sub_ci: "\<forall> l. Ipol l (sub p q) = (Ipol l p) - (Ipol l q)"
+  by (auto simp add: add_ci neg_ci sub_def)
+
+  (* Square *)
+lemma sqr_ci:"ALL ls. Ipol ls (sqr p) = Ipol ls p * Ipol ls p"
+by(induct p, auto simp add: add_ci mkPinj_ci mkPX_ci mul_ci ring_eq_simps power_add)
+
+
+  (* Power *)
+lemma even_pow:"even n \<longrightarrow> pow (p, n) = pow (sqr p, n div 2)" by(induct n,auto)
+lemma pow_ci: "ALL p. Ipol ls (pow (p, n)) = (Ipol ls p) ^ n"
+proof(induct n rule: nat_less_induct)
+  case (1 k)
+  have two:"2=Suc( Suc 0)" by simp
+  from prems show ?case 
+  proof(cases k)
+    case (Suc l)
+    hence KL:"k=Suc l" by simp
+    have "even l \<or> odd l" by (simp)
+    moreover
+    {assume EL:"even l"
+      have "Suc l div 2 = l div 2" by (simp add: nat_number even_nat_plus_one_div_two[OF EL])
+      moreover
+      from KL have"l<k" by simp
+      with prems have "ALL p. Ipol ls (pow (p, l)) = Ipol ls p ^ l" by simp
+      moreover
+      note prems even_nat_plus_one_div_two[OF EL]
+      ultimately have ?thesis by (simp add: mul_ci power_Suc even_pow) }
+    moreover
+    {assume OL:"odd l"
+      with prems have "\<lbrakk>\<forall>m<Suc l. \<forall>p. Ipol ls (pow (p, m)) = Ipol ls p ^ m; k = Suc l; odd l\<rbrakk> \<Longrightarrow> \<forall>p. Ipol ls (sqr p) ^ (Suc l div 2) = Ipol ls p ^ Suc l"
+      proof(cases l)
+	case (Suc w)
+	from prems have EW:"even w" by simp
+	from two have two_times:"(2 * (w div 2))= w" by (simp only: even_nat_div_two_times_two[OF EW])
+	have A:"ALL p. (Ipol ls p * Ipol ls p) = (Ipol ls p) ^ (Suc (Suc 0))" by (simp add: power_Suc)
+	from A two[symmetric] have "ALL p.(Ipol ls p * Ipol ls p) = (Ipol ls p) ^ 2" by simp
+	with prems show ?thesis by (auto simp add: power_mult[symmetric, of _ 2 _] two_times mul_ci sqr_ci)
+      qed(simp)
+      with prems have ?thesis by simp }
+    ultimately show ?thesis by blast
+  qed(simp)
+qed
+
+  (* 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)
+
+(* Reflection lemma: Key to the (incomplete) decision procedure *)
+lemma norm_eq:
+  assumes A:"norm P1  = norm P2" 
+  shows "Ipolex l P1 = Ipolex l P2"
+proof -
+  from A have "Ipol l (norm P1) = Ipol l (norm P2)" by simp
+  thus ?thesis by(simp only: norm_ci)
+qed
+
+
+    (* Code generation *)
+(*
+Does not work, since no generic ring operations implementation is there
+generate_code ("ring.ML") test = "norm"*)
+
+
+ use "comm_ring.ML"
+setup "CommRing.setup"
+
+end
\ No newline at end of file
diff -r afaa031ed4da -r 105519771c67 src/HOL/Integ/Presburger.thy
--- a/src/HOL/Integ/Presburger.thy	Wed Sep 14 10:24:39 2005 +0200
+++ b/src/HOL/Integ/Presburger.thy	Wed Sep 14 17:25:52 2005 +0200
@@ -10,7 +10,8 @@
 
 theory Presburger
 imports NatSimprocs SetInterval
-uses ("cooper_dec.ML") ("cooper_proof.ML") ("qelim.ML") ("presburger.ML")
+uses ("cooper_dec.ML") ("cooper_proof.ML") ("qelim.ML") 
+	("reflected_presburger.ML") ("reflected_cooper.ML") ("presburger.ML")
 begin
 
 text {* Theorem for unitifying the coeffitients of @{text x} in an existential formula*}
@@ -982,8 +983,10 @@
   by (simp cong: conj_cong)
 
 use "cooper_dec.ML"
+use "reflected_presburger.ML" 
+use "reflected_cooper.ML"
 oracle
-  presburger_oracle ("term") = CooperDec.presburger_oracle
+  presburger_oracle ("term") = ReflectedCooper.presburger_oracle
 
 use "cooper_proof.ML"
 use "qelim.ML"
diff -r afaa031ed4da -r 105519771c67 src/HOL/Integ/cooper_dec.ML
--- a/src/HOL/Integ/cooper_dec.ML	Wed Sep 14 10:24:39 2005 +0200
+++ b/src/HOL/Integ/cooper_dec.ML	Wed Sep 14 17:25:52 2005 +0200
@@ -44,7 +44,6 @@
   val evalc : term -> term
   val cooper_w : string list -> term -> (term option * term)
   val integer_qelim : Term.term -> Term.term
-  val presburger_oracle : theory -> term -> term
 end;
 
 structure  CooperDec : COOPER_DEC =
@@ -938,8 +937,4 @@
 
 val integer_qelim = simpl o evalc o (lift_qelim linform (cnnf posineq o evalc) cooper is_arith_rel) ; 
 
-fun presburger_oracle thy t =
-    if (!quick_and_dirty) 
-    then HOLogic.mk_Trueprop (HOLogic.mk_eq(t,integer_qelim t))
-    else error "Presburger oracle: not in quick_and_dirty mode"
 end;
diff -r afaa031ed4da -r 105519771c67 src/HOL/Integ/reflected_cooper.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Integ/reflected_cooper.ML	Wed Sep 14 17:25:52 2005 +0200
@@ -0,0 +1,124 @@
+(* $Id$ *)
+(* The oracle for Presburger arithmetic based on the verified Code *)
+    (* in HOL/ex/Reflected_Presburger.thy *)
+
+structure ReflectedCooper =
+struct
+
+open Generated;
+(* pseudo reification : term -> intterm *)
+
+fun i_of_term vs t = 
+    case t of
+	Free(xn,xT) => (case assoc(vs,t) of 
+			   NONE   => error "Variable not found in the list!!"
+			 | SOME n => Var n)
+      | Const("0",iT) => Cst 0
+      | Const("1",iT) => Cst 1
+      | Bound i => Var (nat i)
+      | Const("uminus",_)$t' => Neg (i_of_term vs t')
+      | Const ("op +",_)$t1$t2 => Add (i_of_term vs t1,i_of_term vs t2)
+      | Const ("op -",_)$t1$t2 => Sub (i_of_term vs t1,i_of_term vs t2)
+      | Const ("op *",_)$t1$t2 => Mult (i_of_term vs t1,i_of_term vs t2)
+      | Const ("Numeral.number_of",_)$t' => Cst (HOLogic.dest_binum t')
+      | _ => error "i_of_term: unknown term";
+	
+
+(* pseudo reification : term -> QF *)
+fun qf_of_term vs t = 
+    case t of 
+	Const("True",_) => T
+      | Const("False",_) => F
+      | Const("op <",_)$t1$t2 => Lt (i_of_term vs t1,i_of_term vs t2)
+      | Const("op <=",_)$t1$t2 => Le (i_of_term vs t1,i_of_term vs t2)
+      | Const ("Divides.op dvd",_)$t1$t2 => 
+	Divides(i_of_term vs t1,i_of_term vs t2)
+      | Const("op =",eqT)$t1$t2 => 
+	if (domain_type eqT = HOLogic.intT)
+	then let val i1 = i_of_term vs t1
+		 val i2 = i_of_term vs t2
+	     in	Eq (i1,i2)
+	     end 
+	else Equ(qf_of_term vs t1,qf_of_term vs t2)
+      | Const("op &",_)$t1$t2 => And(qf_of_term vs t1,qf_of_term vs t2)
+      | Const("op |",_)$t1$t2 => Or(qf_of_term vs t1,qf_of_term vs t2)
+      | Const("op -->",_)$t1$t2 => Imp(qf_of_term vs t1,qf_of_term vs t2)
+      | Const("Not",_)$t' => NOT(qf_of_term vs t')
+      | Const("Ex",_)$Abs(xn,xT,p) => 
+	QEx(qf_of_term (map (fn(v,n) => (v,n + 1)) vs) p)
+      | Const("All",_)$Abs(xn,xT,p) => 
+	QAll(qf_of_term (map (fn(v,n) => (v,n + 1)) vs) p)
+      | _ => error "qf_of_term : unknown term!";
+
+(*
+fun parse s = term_of (read_cterm (sign_of Main.thy) (s,HOLogic.boolT));
+
+val t = parse "ALL (i::int) (j::int). i < 8* j --> (i - 1 = j + 3 + 2*j) & (j <= -i + k ) | 4 = i | 5 dvd i";
+*)
+fun zip [] [] = []
+  | zip (x::xs) (y::ys) = (x,y)::(zip xs ys);
+
+
+fun start_vs t =
+    let val fs = term_frees t
+    in zip fs (map nat (0 upto (length fs - 1)))
+    end ;
+
+(* transform intterm and QF back to terms *)
+val iT = HOLogic.intT;
+val bT = HOLogic.boolT;
+fun myassoc2 l v =
+    case l of
+	[] => NONE
+      | (x,v')::xs => if v = v' then SOME x
+		      else myassoc2 xs v;
+
+fun term_of_i vs t =
+    case t of 
+	Cst i => CooperDec.mk_numeral i
+      | Var n => valOf (myassoc2 vs n)
+      | Neg t' => Const("uminus",iT --> iT)$(term_of_i vs t')
+      | Add(t1,t2) => Const("op +",[iT,iT] ---> iT)$
+			   (term_of_i vs t1)$(term_of_i vs t2)
+      | Sub(t1,t2) => Const("op -",[iT,iT] ---> iT)$
+			   (term_of_i vs t1)$(term_of_i vs t2)
+      | Mult(t1,t2) => Const("op *",[iT,iT] ---> iT)$
+			   (term_of_i vs t1)$(term_of_i vs t2);
+
+fun term_of_qf vs t = 
+    case t of 
+	T => HOLogic.true_const 
+      | F => HOLogic.false_const
+      | Lt(t1,t2) => Const("op <",[iT,iT] ---> bT)$
+			   (term_of_i vs t1)$(term_of_i vs t2)
+      | Le(t1,t2) => Const("op <=",[iT,iT] ---> bT)$
+			  (term_of_i vs t1)$(term_of_i vs t2)
+      | Gt(t1,t2) => Const("op <",[iT,iT] ---> bT)$
+			   (term_of_i vs t2)$(term_of_i vs t1)
+      | Ge(t1,t2) => Const("op <=",[iT,iT] ---> bT)$
+			  (term_of_i vs t2)$(term_of_i vs t1)
+      | Eq(t1,t2) => Const("op =",[iT,iT] ---> bT)$
+			   (term_of_i vs t1)$(term_of_i vs t2)
+      | Divides(t1,t2) => Const("Divides.op dvd",[iT,iT] ---> bT)$
+			       (term_of_i vs t1)$(term_of_i vs t2)
+      | NOT t' => HOLogic.Not$(term_of_qf vs t')
+      | And(t1,t2) => HOLogic.conj$(term_of_qf vs t1)$(term_of_qf vs t2)
+      | Or(t1,t2) => HOLogic.disj$(term_of_qf vs t1)$(term_of_qf vs t2)
+      | Imp(t1,t2) => HOLogic.imp$(term_of_qf vs t1)$(term_of_qf vs t2)
+      | Equ(t1,t2) => (HOLogic.eq_const bT)$(term_of_qf vs t1)$
+					   (term_of_qf vs t2)
+      | _ => error "If this is raised, Isabelle/HOL or generate_code is inconsistent!";
+
+(* The oracle *)
+ exception COOPER; 
+
+fun presburger_oracle thy t =
+    let val vs = start_vs t
+	val result = lift_un (term_of_qf vs) (pa (qf_of_term vs t))
+    in 
+    case result of 
+	None => raise COOPER
+      | Some t' => HOLogic.mk_Trueprop (HOLogic.mk_eq(t,t'))
+    end ;
+ 
+end;
\ No newline at end of file
diff -r afaa031ed4da -r 105519771c67 src/HOL/Integ/reflected_presburger.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Integ/reflected_presburger.ML	Wed Sep 14 17:25:52 2005 +0200
@@ -0,0 +1,2172 @@
+(* $Id$ *)
+
+    (* Caution: This file should not be modified. *)
+    (* It is autmatically generated from HOL/ex/Reflected_Presburger.thy *)
+fun nat i = if i < 0 then 0 else i;
+structure Generated =
+struct
+
+datatype intterm = Cst of int | Var of int | Neg of intterm
+  | Add of intterm * intterm | Sub of intterm * intterm
+  | Mult of intterm * intterm;
+
+datatype QF = Lt of intterm * intterm | Gt of intterm * intterm
+  | Le of intterm * intterm | Ge of intterm * intterm | Eq of intterm * intterm
+  | Divides of intterm * intterm | T | F | NOT of QF | And of QF * QF
+  | Or of QF * QF | Imp of QF * QF | Equ of QF * QF | QAll of QF | QEx of QF;
+
+datatype 'a option = None | Some of 'a;
+
+fun lift_un c None = None
+  | lift_un c (Some p) = Some (c p);
+
+fun lift_bin (c, (Some a, Some b)) = Some (c a b)
+  | lift_bin (c, (None, y)) = None
+  | lift_bin (c, (Some y, None)) = None;
+
+fun lift_qe qe None = None
+  | lift_qe qe (Some p) = qe p;
+
+fun qelim (qe, QAll p) = lift_un NOT (lift_qe qe (lift_un NOT (qelim (qe, p))))
+  | qelim (qe, QEx p) = lift_qe qe (qelim (qe, p))
+  | qelim (qe, And (p, q)) =
+    lift_bin ((fn x => fn xa => And (x, xa)), (qelim (qe, p), qelim (qe, q)))
+  | qelim (qe, Or (p, q)) =
+    lift_bin ((fn x => fn xa => Or (x, xa)), (qelim (qe, p), qelim (qe, q)))
+  | qelim (qe, Imp (p, q)) =
+    lift_bin ((fn x => fn xa => Imp (x, xa)), (qelim (qe, p), qelim (qe, q)))
+  | qelim (qe, Equ (p, q)) =
+    lift_bin ((fn x => fn xa => Equ (x, xa)), (qelim (qe, p), qelim (qe, q)))
+  | qelim (qe, NOT p) = lift_un NOT (qelim (qe, p))
+  | qelim (qe, Lt (w, x)) = Some (Lt (w, x))
+  | qelim (qe, Gt (y, z)) = Some (Gt (y, z))
+  | qelim (qe, Le (aa, ab)) = Some (Le (aa, ab))
+  | qelim (qe, Ge (ac, ad)) = Some (Ge (ac, ad))
+  | qelim (qe, Eq (ae, af)) = Some (Eq (ae, af))
+  | qelim (qe, Divides (ag, ah)) = Some (Divides (ag, ah))
+  | qelim (qe, T) = Some T
+  | qelim (qe, F) = Some F;
+
+fun lin_mul (c, Cst i) = Cst (c * i)
+  | lin_mul (c, Add (Mult (Cst c', Var n), r)) =
+    (if (c = 0) then Cst 0
+      else Add (Mult (Cst (c * c'), Var n), lin_mul (c, r)));
+
+fun op_60_def0 m n = ((m) < (n));
+
+fun op_60_61_def0 m n = not (op_60_def0 n m);
+
+fun lin_add (Add (Mult (Cst c1, Var n1), r1), Add (Mult (Cst c2, Var n2), r2)) =
+    (if (n1 = n2)
+      then let val c = Cst (c1 + c2)
+           in (if ((c1 + c2) = 0) then lin_add (r1, r2)
+                else Add (Mult (c, Var n1), lin_add (r1, r2)))
+           end
+      else (if op_60_61_def0 n1 n2
+             then Add (Mult (Cst c1, Var n1),
+                        lin_add (r1, Add (Mult (Cst c2, Var n2), r2)))
+             else Add (Mult (Cst c2, Var n2),
+                        lin_add (Add (Mult (Cst c1, Var n1), r1), r2))))
+  | lin_add (Add (Mult (Cst c1, Var n1), r1), Cst b) =
+    Add (Mult (Cst c1, Var n1), lin_add (r1, Cst b))
+  | lin_add (Cst x, Add (Mult (Cst c2, Var n2), r2)) =
+    Add (Mult (Cst c2, Var n2), lin_add (Cst x, r2))
+  | lin_add (Cst b1, Cst b2) = Cst (b1 + b2);
+
+fun lin_neg i = lin_mul (~1, i);
+
+fun linearize (Cst b) = Some (Cst b)
+  | linearize (Var n) = Some (Add (Mult (Cst 1, Var n), Cst 0))
+  | linearize (Neg i) = lift_un lin_neg (linearize i)
+  | linearize (Add (i, j)) =
+    lift_bin ((fn x => fn y => lin_add (x, y)), (linearize i, linearize j))
+  | linearize (Sub (i, j)) =
+    lift_bin
+      ((fn x => fn y => lin_add (x, lin_neg y)), (linearize i, linearize j))
+  | linearize (Mult (i, j)) =
+    (case linearize i of None => None
+      | Some x =>
+          (case x of
+            Cst xa =>
+              (case linearize j of None => None
+                | Some x => Some (lin_mul (xa, x)))
+            | Var xa =>
+                (case linearize j of None => None
+                  | Some xa =>
+                      (case xa of Cst xa => Some (lin_mul (xa, x))
+                        | Var xa => None | Neg xa => None | Add (xa, xb) => None
+                        | Sub (xa, xb) => None | Mult (xa, xb) => None))
+            | Neg xa =>
+                (case linearize j of None => None
+                  | Some xa =>
+                      (case xa of Cst xa => Some (lin_mul (xa, x))
+                        | Var xa => None | Neg xa => None | Add (xa, xb) => None
+                        | Sub (xa, xb) => None | Mult (xa, xb) => None))
+            | Add (xa, xb) =>
+                (case linearize j of None => None
+                  | Some xa =>
+                      (case xa of Cst xa => Some (lin_mul (xa, x))
+                        | Var xa => None | Neg xa => None | Add (xa, xb) => None
+                        | Sub (xa, xb) => None | Mult (xa, xb) => None))
+            | Sub (xa, xb) =>
+                (case linearize j of None => None
+                  | Some xa =>
+                      (case xa of Cst xa => Some (lin_mul (xa, x))
+                        | Var xa => None | Neg xa => None | Add (xa, xb) => None
+                        | Sub (xa, xb) => None | Mult (xa, xb) => None))
+            | Mult (xa, xb) =>
+                (case linearize j of None => None
+                  | Some xa =>
+                      (case xa of Cst xa => Some (lin_mul (xa, x))
+                        | Var xa => None | Neg xa => None | Add (xa, xb) => None
+                        | Sub (xa, xb) => None | Mult (xa, xb) => None))));
+
+fun linform (Le (it1, it2)) =
+    lift_bin
+      ((fn x => fn y => Le (lin_add (x, lin_neg y), Cst 0)),
+        (linearize it1, linearize it2))
+  | linform (Eq (it1, it2)) =
+    lift_bin
+      ((fn x => fn y => Eq (lin_add (x, lin_neg y), Cst 0)),
+        (linearize it1, linearize it2))
+  | linform (Divides (d, t)) =
+    (case linearize d of None => None
+      | Some x =>
+          (case x of
+            Cst xa =>
+              (if (xa = 0) then None
+                else (case linearize t of None => None
+                       | Some xa => Some (Divides (x, xa))))
+            | Var xa => None | Neg xa => None | Add (xa, xb) => None
+            | Sub (xa, xb) => None | Mult (xa, xb) => None))
+  | linform T = Some T
+  | linform F = Some F
+  | linform (NOT p) = lift_un NOT (linform p)
+  | linform (And (p, q)) =
+    lift_bin ((fn f => fn g => And (f, g)), (linform p, linform q))
+  | linform (Or (p, q)) =
+    lift_bin ((fn f => fn g => Or (f, g)), (linform p, linform q));
+
+fun nnf (Lt (it1, it2)) = Le (Sub (it1, it2), Cst (~ 1))
+  | nnf (Gt (it1, it2)) = Le (Sub (it2, it1), Cst (~ 1))
+  | nnf (Le (it1, it2)) = Le (it1, it2)
+  | nnf (Ge (it1, it2)) = Le (it2, it1)
+  | nnf (Eq (it1, it2)) = Eq (it2, it1)
+  | nnf (Divides (d, t)) = Divides (d, t)
+  | nnf T = T
+  | nnf F = F
+  | nnf (And (p, q)) = And (nnf p, nnf q)
+  | nnf (Or (p, q)) = Or (nnf p, nnf q)
+  | nnf (Imp (p, q)) = Or (nnf (NOT p), nnf q)
+  | nnf (Equ (p, q)) = Or (And (nnf p, nnf q), And (nnf (NOT p), nnf (NOT q)))
+  | nnf (NOT (Lt (it1, it2))) = Le (it2, it1)
+  | nnf (NOT (Gt (it1, it2))) = Le (it1, it2)
+  | nnf (NOT (Le (it1, it2))) = Le (Sub (it2, it1), Cst (~ 1))
+  | nnf (NOT (Ge (it1, it2))) = Le (Sub (it1, it2), Cst (~ 1))
+  | nnf (NOT (Eq (it1, it2))) = NOT (Eq (it1, it2))
+  | nnf (NOT (Divides (d, t))) = NOT (Divides (d, t))
+  | nnf (NOT T) = F
+  | nnf (NOT F) = T
+  | nnf (NOT (NOT p)) = nnf p
+  | nnf (NOT (And (p, q))) = Or (nnf (NOT p), nnf (NOT q))
+  | nnf (NOT (Or (p, q))) = And (nnf (NOT p), nnf (NOT q))
+  | nnf (NOT (Imp (p, q))) = And (nnf p, nnf (NOT q))
+  | nnf (NOT (Equ (p, q))) =
+    Or (And (nnf p, nnf (NOT q)), And (nnf (NOT p), nnf q));
+
+fun op_45_def2 z w = (z + ~ w);
+
+fun op_45_def0 m n = nat (op_45_def2 (m) (n));
+
+val id_1_def0 : int = (0 + 1);
+
+fun decrvarsI (Cst i) = Cst i
+  | decrvarsI (Var n) = Var (op_45_def0 n id_1_def0)
+  | decrvarsI (Neg a) = Neg (decrvarsI a)
+  | decrvarsI (Add (a, b)) = Add (decrvarsI a, decrvarsI b)
+  | decrvarsI (Sub (a, b)) = Sub (decrvarsI a, decrvarsI b)
+  | decrvarsI (Mult (a, b)) = Mult (decrvarsI a, decrvarsI b);
+
+fun decrvars (Lt (a, b)) = Lt (decrvarsI a, decrvarsI b)
+  | decrvars (Gt (a, b)) = Gt (decrvarsI a, decrvarsI b)
+  | decrvars (Le (a, b)) = Le (decrvarsI a, decrvarsI b)
+  | decrvars (Ge (a, b)) = Ge (decrvarsI a, decrvarsI b)
+  | decrvars (Eq (a, b)) = Eq (decrvarsI a, decrvarsI b)
+  | decrvars (Divides (a, b)) = Divides (decrvarsI a, decrvarsI b)
+  | decrvars T = T
+  | decrvars F = F
+  | decrvars (NOT p) = NOT (decrvars p)
+  | decrvars (And (p, q)) = And (decrvars p, decrvars q)
+  | decrvars (Or (p, q)) = Or (decrvars p, decrvars q)
+  | decrvars (Imp (p, q)) = Imp (decrvars p, decrvars q)
+  | decrvars (Equ (p, q)) = Equ (decrvars p, decrvars q);
+
+fun op_64 [] ys = ys
+  | op_64 (x :: xs) ys = (x :: op_64 xs ys);
+
+fun map f [] = []
+  | map f (x :: xs) = (f x :: map f xs);
+
+fun iupto (i, j) = (if (j < i) then [] else (i :: iupto ((i + 1), j)));
+
+fun all_sums (j, []) = []
+  | all_sums (j, (i :: is)) =
+    op_64 (map (fn x => lin_add (i, Cst x)) (iupto (1, j))) (all_sums (j, is));
+
+fun split x = (fn p => x (fst p) (snd p));
+
+fun negateSnd x = split (fn q => fn r => (q, ~ r)) x;
+
+fun adjust b =
+  (fn (q, r) =>
+    (if (0 <= op_45_def2 r b) then (((2 * q) + 1), op_45_def2 r b)
+      else ((2 * q), r)));
+
+fun negDivAlg (a, b) =
+    (if ((0 <= (a + b)) orelse (b <= 0)) then (~1, (a + b))
+      else adjust b (negDivAlg (a, (2 * b))));
+
+fun posDivAlg (a, b) =
+    (if ((a < b) orelse (b <= 0)) then (0, a)
+      else adjust b (posDivAlg (a, (2 * b))));
+
+fun divAlg x =
+  split (fn a => fn b =>
+          (if (0 <= a)
+            then (if (0 <= 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)))))
+    x;
+
+fun op_mod_def1 a b = snd (divAlg (a, b));
+
+fun op_dvd m n = (op_mod_def1 n m = 0);
+
+fun psimpl (Le (l, r)) =
+    (case lift_bin
+            ((fn x => fn y => lin_add (x, lin_neg y)),
+              (linearize l, linearize r)) of
+      None => Le (l, r)
+      | Some x =>
+          (case x of Cst xa => (if (xa <= 0) then T else F)
+            | Var xa => Le (x, Cst 0) | Neg xa => Le (x, Cst 0)
+            | Add (xa, xb) => Le (x, Cst 0) | Sub (xa, xb) => Le (x, Cst 0)
+            | Mult (xa, xb) => Le (x, Cst 0)))
+  | psimpl (Eq (l, r)) =
+    (case lift_bin
+            ((fn x => fn y => lin_add (x, lin_neg y)),
+              (linearize l, linearize r)) of
+      None => Eq (l, r)
+      | Some x =>
+          (case x of Cst xa => (if (xa = 0) then T else F)
+            | Var xa => Eq (x, Cst 0) | Neg xa => Eq (x, Cst 0)
+            | Add (xa, xb) => Eq (x, Cst 0) | Sub (xa, xb) => Eq (x, Cst 0)
+            | Mult (xa, xb) => Eq (x, Cst 0)))
+  | psimpl (Divides (Cst d, t)) =
+    (case linearize t of None => Divides (Cst d, t)
+      | Some x =>
+          (case x of Cst xa => (if op_dvd d xa then T else F)
+            | Var xa => Divides (Cst d, x) | Neg xa => Divides (Cst d, x)
+            | Add (xa, xb) => Divides (Cst d, x)
+            | Sub (xa, xb) => Divides (Cst d, x)
+            | Mult (xa, xb) => Divides (Cst d, x)))
+  | psimpl (Equ (p, q)) =
+    let val p' = psimpl p; val q' = psimpl q
+    in (case p' of
+         Lt (x, xa) =>
+           (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+             | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+             | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+             | T => p' | F => NOT p' | NOT x => Equ (p', q')
+             | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+             | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+             | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Gt (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Le (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Ge (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Eq (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Divides (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | T => q'
+         | F => (case q' of Lt (x, xa) => NOT q' | Gt (x, xa) => NOT q'
+                  | Le (x, xa) => NOT q' | Ge (x, xa) => NOT q'
+                  | Eq (x, xa) => NOT q' | Divides (x, xa) => NOT q' | T => F
+                  | F => T | NOT x => x | And (x, xa) => NOT q'
+                  | Or (x, xa) => NOT q' | Imp (x, xa) => NOT q'
+                  | Equ (x, xa) => NOT q' | QAll x => NOT q' | QEx x => NOT q')
+         | NOT x =>
+             (case q' of Lt (xa, xb) => Equ (p', q')
+               | Gt (xa, xb) => Equ (p', q') | Le (xa, xb) => Equ (p', q')
+               | Ge (xa, xb) => Equ (p', q') | Eq (xa, xb) => Equ (p', q')
+               | Divides (xa, xb) => Equ (p', q') | T => p' | F => x
+               | NOT xa => Equ (x, xa) | And (xa, xb) => Equ (p', q')
+               | Or (xa, xb) => Equ (p', q') | Imp (xa, xb) => Equ (p', q')
+               | Equ (xa, xb) => Equ (p', q') | QAll xa => Equ (p', q')
+               | QEx xa => Equ (p', q'))
+         | And (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Or (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Imp (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Equ (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | QAll x =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | QEx x =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q')))
+    end
+  | psimpl (NOT p) =
+    let val p' = psimpl p
+    in (case p' of Lt (x, xa) => NOT p' | Gt (x, xa) => NOT p'
+         | Le (x, xa) => NOT p' | Ge (x, xa) => NOT p' | Eq (x, xa) => NOT p'
+         | Divides (x, xa) => NOT p' | T => F | F => T | NOT x => x
+         | And (x, xa) => NOT p' | Or (x, xa) => NOT p' | Imp (x, xa) => NOT p'
+         | Equ (x, xa) => NOT p' | QAll x => NOT p' | QEx x => NOT p')
+    end
+  | psimpl (Lt (u, v)) = Lt (u, v)
+  | psimpl (Gt (w, x)) = Gt (w, x)
+  | psimpl (Ge (aa, ab)) = Ge (aa, ab)
+  | psimpl (Divides (Var bp, af)) = Divides (Var bp, af)
+  | psimpl (Divides (Neg bq, af)) = Divides (Neg bq, af)
+  | psimpl (Divides (Add (br, bs), af)) = Divides (Add (br, bs), af)
+  | psimpl (Divides (Sub (bt, bu), af)) = Divides (Sub (bt, bu), af)
+  | psimpl (Divides (Mult (bv, bw), af)) = Divides (Mult (bv, bw), af)
+  | psimpl T = T
+  | psimpl F = F
+  | psimpl (QAll ap) = QAll ap
+  | psimpl (QEx aq) = QEx aq
+  | psimpl (And (p, q)) =
+    let val p' = psimpl p
+    in (case p' of
+         Lt (x, xa) =>
+           let val q' = psimpl q
+           in (case q' of Lt (x, xa) => And (p', q')
+                | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                | Divides (x, xa) => And (p', q') | T => p' | F => F
+                | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                | QEx x => And (p', q'))
+           end
+         | Gt (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | Le (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | Ge (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | Eq (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | Divides (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | T => psimpl q | F => F
+         | NOT x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | And (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | Or (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | Imp (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | Equ (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | QAll x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | QEx x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end)
+    end
+  | psimpl (Or (p, q)) =
+    let val p' = psimpl p
+    in (case p' of
+         Lt (x, xa) =>
+           let val q' = psimpl q
+           in (case q' of Lt (x, xa) => Or (p', q') | Gt (x, xa) => Or (p', q')
+                | Le (x, xa) => Or (p', q') | Ge (x, xa) => Or (p', q')
+                | Eq (x, xa) => Or (p', q') | Divides (x, xa) => Or (p', q')
+                | T => T | F => p' | NOT x => Or (p', q')
+                | And (x, xa) => Or (p', q') | Or (x, xa) => Or (p', q')
+                | Imp (x, xa) => Or (p', q') | Equ (x, xa) => Or (p', q')
+                | QAll x => Or (p', q') | QEx x => Or (p', q'))
+           end
+         | Gt (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | Le (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | Ge (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | Eq (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | Divides (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | T => T | F => psimpl q
+         | NOT x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | And (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | Or (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | Imp (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | Equ (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | QAll x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | QEx x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end)
+    end
+  | psimpl (Imp (p, q)) =
+    let val p' = psimpl p
+    in (case p' of
+         Lt (x, xa) =>
+           let val q' = psimpl q
+           in (case q' of Lt (x, xa) => Imp (p', q')
+                | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                | QEx x => Imp (p', q'))
+           end
+         | Gt (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | Le (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | Ge (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | Eq (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | Divides (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | T => psimpl q | F => T
+         | NOT x =>
+             let val q' = psimpl q
+             in (case q' of Lt (xa, xb) => Or (x, q')
+                  | Gt (xa, xb) => Or (x, q') | Le (xa, xb) => Or (x, q')
+                  | Ge (xa, xb) => Or (x, q') | Eq (xa, xb) => Or (x, q')
+                  | Divides (xa, xb) => Or (x, q') | T => T | F => x
+                  | NOT xa => Or (x, q') | And (xa, xb) => Or (x, q')
+                  | Or (xa, xb) => Or (x, q') | Imp (xa, xb) => Or (x, q')
+                  | Equ (xa, xb) => Or (x, q') | QAll xa => Or (x, q')
+                  | QEx xa => Or (x, q'))
+             end
+         | And (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | Or (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | Imp (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | Equ (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | QAll x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | QEx x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end)
+    end;
+
+fun subst_it i (Cst b) = Cst b
+  | subst_it i (Var n) = (if (n = 0) then i else Var n)
+  | subst_it i (Neg it) = Neg (subst_it i it)
+  | subst_it i (Add (it1, it2)) = Add (subst_it i it1, subst_it i it2)
+  | subst_it i (Sub (it1, it2)) = Sub (subst_it i it1, subst_it i it2)
+  | subst_it i (Mult (it1, it2)) = Mult (subst_it i it1, subst_it i it2);
+
+fun subst_p i (Le (it1, it2)) = Le (subst_it i it1, subst_it i it2)
+  | subst_p i (Lt (it1, it2)) = Lt (subst_it i it1, subst_it i it2)
+  | subst_p i (Ge (it1, it2)) = Ge (subst_it i it1, subst_it i it2)
+  | subst_p i (Gt (it1, it2)) = Gt (subst_it i it1, subst_it i it2)
+  | subst_p i (Eq (it1, it2)) = Eq (subst_it i it1, subst_it i it2)
+  | subst_p i (Divides (d, t)) = Divides (subst_it i d, subst_it i t)
+  | subst_p i T = T
+  | subst_p i F = F
+  | subst_p i (And (p, q)) = And (subst_p i p, subst_p i q)
+  | subst_p i (Or (p, q)) = Or (subst_p i p, subst_p i q)
+  | subst_p i (Imp (p, q)) = Imp (subst_p i p, subst_p i q)
+  | subst_p i (Equ (p, q)) = Equ (subst_p i p, subst_p i q)
+  | subst_p i (NOT p) = NOT (subst_p i p);
+
+fun explode_disj ([], p) = F
+  | explode_disj ((i :: is), p) =
+    let val pi = psimpl (subst_p i p)
+    in (case pi of
+         Lt (x, xa) =>
+           let val r = explode_disj (is, p)
+           in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                | T => T | F => pi | NOT x => Or (pi, r)
+                | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+           end
+         | Gt (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | Le (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | Ge (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | Eq (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | Divides (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | T => T | F => explode_disj (is, p)
+         | NOT x =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | And (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | Or (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | Imp (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | Equ (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | QAll x =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | QEx x =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end)
+    end;
+
+fun minusinf (And (p, q)) = And (minusinf p, minusinf q)
+  | minusinf (Or (p, q)) = Or (minusinf p, minusinf q)
+  | minusinf (Lt (u, v)) = Lt (u, v)
+  | minusinf (Gt (w, x)) = Gt (w, x)
+  | minusinf (Le (Cst bo, z)) = Le (Cst bo, z)
+  | minusinf (Le (Var bp, z)) = Le (Var bp, z)
+  | minusinf (Le (Neg bq, z)) = Le (Neg bq, z)
+  | minusinf (Le (Add (Cst cg, bs), z)) = Le (Add (Cst cg, bs), z)
+  | minusinf (Le (Add (Var ch, bs), z)) = Le (Add (Var ch, bs), z)
+  | minusinf (Le (Add (Neg ci, bs), z)) = Le (Add (Neg ci, bs), z)
+  | minusinf (Le (Add (Add (cj, ck), bs), z)) = Le (Add (Add (cj, ck), bs), z)
+  | minusinf (Le (Add (Sub (cl, cm), bs), z)) = Le (Add (Sub (cl, cm), bs), z)
+  | minusinf (Le (Add (Mult (Cst cy, Cst dq), bs), z)) =
+    Le (Add (Mult (Cst cy, Cst dq), bs), z)
+  | minusinf (Le (Add (Mult (Cst cy, Var ei), bs), z)) =
+    (if (ei = 0) then (if (cy < 0) then F else T)
+      else Le (Add (Mult (Cst cy, Var (op_45_def0 ei id_1_def0 + 1)), bs), z))
+  | minusinf (Le (Add (Mult (Cst cy, Neg ds), bs), z)) =
+    Le (Add (Mult (Cst cy, Neg ds), bs), z)
+  | minusinf (Le (Add (Mult (Cst cy, Add (dt, du)), bs), z)) =
+    Le (Add (Mult (Cst cy, Add (dt, du)), bs), z)
+  | minusinf (Le (Add (Mult (Cst cy, Sub (dv, dw)), bs), z)) =
+    Le (Add (Mult (Cst cy, Sub (dv, dw)), bs), z)
+  | minusinf (Le (Add (Mult (Cst cy, Mult (dx, dy)), bs), z)) =
+    Le (Add (Mult (Cst cy, Mult (dx, dy)), bs), z)
+  | minusinf (Le (Add (Mult (Var cz, co), bs), z)) =
+    Le (Add (Mult (Var cz, co), bs), z)
+  | minusinf (Le (Add (Mult (Neg da, co), bs), z)) =
+    Le (Add (Mult (Neg da, co), bs), z)
+  | minusinf (Le (Add (Mult (Add (db, dc), co), bs), z)) =
+    Le (Add (Mult (Add (db, dc), co), bs), z)
+  | minusinf (Le (Add (Mult (Sub (dd, de), co), bs), z)) =
+    Le (Add (Mult (Sub (dd, de), co), bs), z)
+  | minusinf (Le (Add (Mult (Mult (df, dg), co), bs), z)) =
+    Le (Add (Mult (Mult (df, dg), co), bs), z)
+  | minusinf (Le (Sub (bt, bu), z)) = Le (Sub (bt, bu), z)
+  | minusinf (Le (Mult (bv, bw), z)) = Le (Mult (bv, bw), z)
+  | minusinf (Ge (aa, ab)) = Ge (aa, ab)
+  | minusinf (Eq (Cst ek, ad)) = Eq (Cst ek, ad)
+  | minusinf (Eq (Var el, ad)) = Eq (Var el, ad)
+  | minusinf (Eq (Neg em, ad)) = Eq (Neg em, ad)
+  | minusinf (Eq (Add (Cst fc, eo), ad)) = Eq (Add (Cst fc, eo), ad)
+  | minusinf (Eq (Add (Var fd, eo), ad)) = Eq (Add (Var fd, eo), ad)
+  | minusinf (Eq (Add (Neg fe, eo), ad)) = Eq (Add (Neg fe, eo), ad)
+  | minusinf (Eq (Add (Add (ff, fg), eo), ad)) = Eq (Add (Add (ff, fg), eo), ad)
+  | minusinf (Eq (Add (Sub (fh, fi), eo), ad)) = Eq (Add (Sub (fh, fi), eo), ad)
+  | minusinf (Eq (Add (Mult (Cst fu, Cst gm), eo), ad)) =
+    Eq (Add (Mult (Cst fu, Cst gm), eo), ad)
+  | minusinf (Eq (Add (Mult (Cst fu, Var he), eo), ad)) =
+    (if (he = 0) then F
+      else Eq (Add (Mult (Cst fu, Var (op_45_def0 he id_1_def0 + 1)), eo), ad))
+  | minusinf (Eq (Add (Mult (Cst fu, Neg go), eo), ad)) =
+    Eq (Add (Mult (Cst fu, Neg go), eo), ad)
+  | minusinf (Eq (Add (Mult (Cst fu, Add (gp, gq)), eo), ad)) =
+    Eq (Add (Mult (Cst fu, Add (gp, gq)), eo), ad)
+  | minusinf (Eq (Add (Mult (Cst fu, Sub (gr, gs)), eo), ad)) =
+    Eq (Add (Mult (Cst fu, Sub (gr, gs)), eo), ad)
+  | minusinf (Eq (Add (Mult (Cst fu, Mult (gt, gu)), eo), ad)) =
+    Eq (Add (Mult (Cst fu, Mult (gt, gu)), eo), ad)
+  | minusinf (Eq (Add (Mult (Var fv, fk), eo), ad)) =
+    Eq (Add (Mult (Var fv, fk), eo), ad)
+  | minusinf (Eq (Add (Mult (Neg fw, fk), eo), ad)) =
+    Eq (Add (Mult (Neg fw, fk), eo), ad)
+  | minusinf (Eq (Add (Mult (Add (fx, fy), fk), eo), ad)) =
+    Eq (Add (Mult (Add (fx, fy), fk), eo), ad)
+  | minusinf (Eq (Add (Mult (Sub (fz, ga), fk), eo), ad)) =
+    Eq (Add (Mult (Sub (fz, ga), fk), eo), ad)
+  | minusinf (Eq (Add (Mult (Mult (gb, gc), fk), eo), ad)) =
+    Eq (Add (Mult (Mult (gb, gc), fk), eo), ad)
+  | minusinf (Eq (Sub (ep, eq), ad)) = Eq (Sub (ep, eq), ad)
+  | minusinf (Eq (Mult (er, es), ad)) = Eq (Mult (er, es), ad)
+  | minusinf (Divides (ae, af)) = Divides (ae, af)
+  | minusinf T = T
+  | minusinf F = F
+  | minusinf (NOT (Lt (hg, hh))) = NOT (Lt (hg, hh))
+  | minusinf (NOT (Gt (hi, hj))) = NOT (Gt (hi, hj))
+  | minusinf (NOT (Le (hk, hl))) = NOT (Le (hk, hl))
+  | minusinf (NOT (Ge (hm, hn))) = NOT (Ge (hm, hn))
+  | minusinf (NOT (Eq (Cst ja, hp))) = NOT (Eq (Cst ja, hp))
+  | minusinf (NOT (Eq (Var jb, hp))) = NOT (Eq (Var jb, hp))
+  | minusinf (NOT (Eq (Neg jc, hp))) = NOT (Eq (Neg jc, hp))
+  | minusinf (NOT (Eq (Add (Cst js, je), hp))) = NOT (Eq (Add (Cst js, je), hp))
+  | minusinf (NOT (Eq (Add (Var jt, je), hp))) = NOT (Eq (Add (Var jt, je), hp))
+  | minusinf (NOT (Eq (Add (Neg ju, je), hp))) = NOT (Eq (Add (Neg ju, je), hp))
+  | minusinf (NOT (Eq (Add (Add (jv, jw), je), hp))) =
+    NOT (Eq (Add (Add (jv, jw), je), hp))
+  | minusinf (NOT (Eq (Add (Sub (jx, jy), je), hp))) =
+    NOT (Eq (Add (Sub (jx, jy), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Cst kk, Cst lc), je), hp))) =
+    NOT (Eq (Add (Mult (Cst kk, Cst lc), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Cst kk, Var lu), je), hp))) =
+    (if (lu = 0) then T
+      else NOT (Eq (Add (Mult (Cst kk, Var (op_45_def0 lu id_1_def0 + 1)), je),
+                     hp)))
+  | minusinf (NOT (Eq (Add (Mult (Cst kk, Neg le), je), hp))) =
+    NOT (Eq (Add (Mult (Cst kk, Neg le), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Cst kk, Add (lf, lg)), je), hp))) =
+    NOT (Eq (Add (Mult (Cst kk, Add (lf, lg)), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Cst kk, Sub (lh, li)), je), hp))) =
+    NOT (Eq (Add (Mult (Cst kk, Sub (lh, li)), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Cst kk, Mult (lj, lk)), je), hp))) =
+    NOT (Eq (Add (Mult (Cst kk, Mult (lj, lk)), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Var kl, ka), je), hp))) =
+    NOT (Eq (Add (Mult (Var kl, ka), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Neg km, ka), je), hp))) =
+    NOT (Eq (Add (Mult (Neg km, ka), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Add (kn, ko), ka), je), hp))) =
+    NOT (Eq (Add (Mult (Add (kn, ko), ka), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Sub (kp, kq), ka), je), hp))) =
+    NOT (Eq (Add (Mult (Sub (kp, kq), ka), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Mult (kr, ks), ka), je), hp))) =
+    NOT (Eq (Add (Mult (Mult (kr, ks), ka), je), hp))
+  | minusinf (NOT (Eq (Sub (jf, jg), hp))) = NOT (Eq (Sub (jf, jg), hp))
+  | minusinf (NOT (Eq (Mult (jh, ji), hp))) = NOT (Eq (Mult (jh, ji), hp))
+  | minusinf (NOT (Divides (hq, hr))) = NOT (Divides (hq, hr))
+  | minusinf (NOT T) = NOT T
+  | minusinf (NOT F) = NOT F
+  | minusinf (NOT (NOT hs)) = NOT (NOT hs)
+  | minusinf (NOT (And (ht, hu))) = NOT (And (ht, hu))
+  | minusinf (NOT (Or (hv, hw))) = NOT (Or (hv, hw))
+  | minusinf (NOT (Imp (hx, hy))) = NOT (Imp (hx, hy))
+  | minusinf (NOT (Equ (hz, ia))) = NOT (Equ (hz, ia))
+  | minusinf (NOT (QAll ib)) = NOT (QAll ib)
+  | minusinf (NOT (QEx ic)) = NOT (QEx ic)
+  | minusinf (Imp (al, am)) = Imp (al, am)
+  | minusinf (Equ (an, ao)) = Equ (an, ao)
+  | minusinf (QAll ap) = QAll ap
+  | minusinf (QEx aq) = QEx aq;
+
+fun abs i = (if (i < 0) then ~ i else i);
+
+fun op_div_def1 a b = fst (divAlg (a, b));
+
+fun op_mod_def0 m n = nat (op_mod_def1 (m) (n));
+
+fun ngcd (m, n) = (if (n = 0) then m else ngcd (n, op_mod_def0 m n));
+
+fun igcd x = split (fn a => fn b => (ngcd (nat (abs a), nat (abs b)))) x;
+
+fun ilcm a b = op_div_def1 (a * b) (igcd (a, b));
+
+fun divlcm (NOT p) = divlcm p
+  | divlcm (And (p, q)) = ilcm (divlcm p) (divlcm q)
+  | divlcm (Or (p, q)) = ilcm (divlcm p) (divlcm q)
+  | divlcm (Lt (u, v)) = 1
+  | divlcm (Gt (w, x)) = 1
+  | divlcm (Le (y, z)) = 1
+  | divlcm (Ge (aa, ab)) = 1
+  | divlcm (Eq (ac, ad)) = 1
+  | divlcm (Divides (Cst bo, Cst cg)) = 1
+  | divlcm (Divides (Cst bo, Var ch)) = 1
+  | divlcm (Divides (Cst bo, Neg ci)) = 1
+  | divlcm (Divides (Cst bo, Add (Cst cy, ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Var cz, ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Neg da, ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Add (db, dc), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Sub (dd, de), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Cst dq, Cst ei), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Cst dq, Var fa), ck))) =
+    (if (fa = 0) then abs bo else 1)
+  | divlcm (Divides (Cst bo, Add (Mult (Cst dq, Neg ek), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Cst dq, Add (el, em)), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Cst dq, Sub (en, eo)), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Cst dq, Mult (ep, eq)), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Var dr, dg), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Neg ds, dg), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Add (dt, du), dg), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Sub (dv, dw), dg), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Mult (dx, dy), dg), ck))) = 1
+  | divlcm (Divides (Cst bo, Sub (cl, cm))) = 1
+  | divlcm (Divides (Cst bo, Mult (cn, co))) = 1
+  | divlcm (Divides (Var bp, af)) = 1
+  | divlcm (Divides (Neg bq, af)) = 1
+  | divlcm (Divides (Add (br, bs), af)) = 1
+  | divlcm (Divides (Sub (bt, bu), af)) = 1
+  | divlcm (Divides (Mult (bv, bw), af)) = 1
+  | divlcm T = 1
+  | divlcm F = 1
+  | divlcm (Imp (al, am)) = 1
+  | divlcm (Equ (an, ao)) = 1
+  | divlcm (QAll ap) = 1
+  | divlcm (QEx aq) = 1;
+
+fun explode_minf (q, B) =
+    let val d = divlcm q; val pm = minusinf q;
+        val dj1 = explode_disj (map (fn x => Cst x) (iupto (1, d)), pm)
+    in (case dj1 of
+         Lt (x, xa) =>
+           let val dj2 = explode_disj (all_sums (d, B), q)
+           in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                | QEx x => Or (dj1, dj2))
+           end
+         | Gt (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | Le (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | Ge (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | Eq (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | Divides (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | T => T | F => explode_disj (all_sums (d, B), q)
+         | NOT x =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | And (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | Or (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | Imp (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | Equ (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | QAll x =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | QEx x =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end)
+    end;
+
+fun mirror (And (p, q)) = And (mirror p, mirror q)
+  | mirror (Or (p, q)) = Or (mirror p, mirror q)
+  | mirror (Lt (u, v)) = Lt (u, v)
+  | mirror (Gt (w, x)) = Gt (w, x)
+  | mirror (Le (Cst bp, aa)) = Le (Cst bp, aa)
+  | mirror (Le (Var bq, aa)) = Le (Var bq, aa)
+  | mirror (Le (Neg br, aa)) = Le (Neg br, aa)
+  | mirror (Le (Add (Cst ch, bt), aa)) = Le (Add (Cst ch, bt), aa)
+  | mirror (Le (Add (Var ci, bt), aa)) = Le (Add (Var ci, bt), aa)
+  | mirror (Le (Add (Neg cj, bt), aa)) = Le (Add (Neg cj, bt), aa)
+  | mirror (Le (Add (Add (ck, cl), bt), aa)) = Le (Add (Add (ck, cl), bt), aa)
+  | mirror (Le (Add (Sub (cm, cn), bt), aa)) = Le (Add (Sub (cm, cn), bt), aa)
+  | mirror (Le (Add (Mult (Cst cz, Cst dr), bt), aa)) =
+    Le (Add (Mult (Cst cz, Cst dr), bt), aa)
+  | mirror (Le (Add (Mult (Cst cz, Var ej), bt), aa)) =
+    (if (ej = 0) then Le (Add (Mult (Cst (~ cz), Var 0), bt), aa)
+      else Le (Add (Mult (Cst cz, Var (op_45_def0 ej id_1_def0 + 1)), bt), aa))
+  | mirror (Le (Add (Mult (Cst cz, Neg dt), bt), aa)) =
+    Le (Add (Mult (Cst cz, Neg dt), bt), aa)
+  | mirror (Le (Add (Mult (Cst cz, Add (du, dv)), bt), aa)) =
+    Le (Add (Mult (Cst cz, Add (du, dv)), bt), aa)
+  | mirror (Le (Add (Mult (Cst cz, Sub (dw, dx)), bt), aa)) =
+    Le (Add (Mult (Cst cz, Sub (dw, dx)), bt), aa)
+  | mirror (Le (Add (Mult (Cst cz, Mult (dy, dz)), bt), aa)) =
+    Le (Add (Mult (Cst cz, Mult (dy, dz)), bt), aa)
+  | mirror (Le (Add (Mult (Var da, cp), bt), aa)) =
+    Le (Add (Mult (Var da, cp), bt), aa)
+  | mirror (Le (Add (Mult (Neg db, cp), bt), aa)) =
+    Le (Add (Mult (Neg db, cp), bt), aa)
+  | mirror (Le (Add (Mult (Add (dc, dd), cp), bt), aa)) =
+    Le (Add (Mult (Add (dc, dd), cp), bt), aa)
+  | mirror (Le (Add (Mult (Sub (de, df), cp), bt), aa)) =
+    Le (Add (Mult (Sub (de, df), cp), bt), aa)
+  | mirror (Le (Add (Mult (Mult (dg, dh), cp), bt), aa)) =
+    Le (Add (Mult (Mult (dg, dh), cp), bt), aa)
+  | mirror (Le (Sub (bu, bv), aa)) = Le (Sub (bu, bv), aa)
+  | mirror (Le (Mult (bw, bx), aa)) = Le (Mult (bw, bx), aa)
+  | mirror (Ge (ab, ac)) = Ge (ab, ac)
+  | mirror (Eq (Cst el, ae)) = Eq (Cst el, ae)
+  | mirror (Eq (Var em, ae)) = Eq (Var em, ae)
+  | mirror (Eq (Neg en, ae)) = Eq (Neg en, ae)
+  | mirror (Eq (Add (Cst fd, ep), ae)) = Eq (Add (Cst fd, ep), ae)
+  | mirror (Eq (Add (Var fe, ep), ae)) = Eq (Add (Var fe, ep), ae)
+  | mirror (Eq (Add (Neg ff, ep), ae)) = Eq (Add (Neg ff, ep), ae)
+  | mirror (Eq (Add (Add (fg, fh), ep), ae)) = Eq (Add (Add (fg, fh), ep), ae)
+  | mirror (Eq (Add (Sub (fi, fj), ep), ae)) = Eq (Add (Sub (fi, fj), ep), ae)
+  | mirror (Eq (Add (Mult (Cst fv, Cst gn), ep), ae)) =
+    Eq (Add (Mult (Cst fv, Cst gn), ep), ae)
+  | mirror (Eq (Add (Mult (Cst fv, Var hf), ep), ae)) =
+    (if (hf = 0) then Eq (Add (Mult (Cst (~ fv), Var 0), ep), ae)
+      else Eq (Add (Mult (Cst fv, Var (op_45_def0 hf id_1_def0 + 1)), ep), ae))
+  | mirror (Eq (Add (Mult (Cst fv, Neg gp), ep), ae)) =
+    Eq (Add (Mult (Cst fv, Neg gp), ep), ae)
+  | mirror (Eq (Add (Mult (Cst fv, Add (gq, gr)), ep), ae)) =
+    Eq (Add (Mult (Cst fv, Add (gq, gr)), ep), ae)
+  | mirror (Eq (Add (Mult (Cst fv, Sub (gs, gt)), ep), ae)) =
+    Eq (Add (Mult (Cst fv, Sub (gs, gt)), ep), ae)
+  | mirror (Eq (Add (Mult (Cst fv, Mult (gu, gv)), ep), ae)) =
+    Eq (Add (Mult (Cst fv, Mult (gu, gv)), ep), ae)
+  | mirror (Eq (Add (Mult (Var fw, fl), ep), ae)) =
+    Eq (Add (Mult (Var fw, fl), ep), ae)
+  | mirror (Eq (Add (Mult (Neg fx, fl), ep), ae)) =
+    Eq (Add (Mult (Neg fx, fl), ep), ae)
+  | mirror (Eq (Add (Mult (Add (fy, fz), fl), ep), ae)) =
+    Eq (Add (Mult (Add (fy, fz), fl), ep), ae)
+  | mirror (Eq (Add (Mult (Sub (ga, gb), fl), ep), ae)) =
+    Eq (Add (Mult (Sub (ga, gb), fl), ep), ae)
+  | mirror (Eq (Add (Mult (Mult (gc, gd), fl), ep), ae)) =
+    Eq (Add (Mult (Mult (gc, gd), fl), ep), ae)
+  | mirror (Eq (Sub (eq, er), ae)) = Eq (Sub (eq, er), ae)
+  | mirror (Eq (Mult (es, et), ae)) = Eq (Mult (es, et), ae)
+  | mirror (Divides (Cst hh, Cst hz)) = Divides (Cst hh, Cst hz)
+  | mirror (Divides (Cst hh, Var ia)) = Divides (Cst hh, Var ia)
+  | mirror (Divides (Cst hh, Neg ib)) = Divides (Cst hh, Neg ib)
+  | mirror (Divides (Cst hh, Add (Cst ir, id))) =
+    Divides (Cst hh, Add (Cst ir, id))
+  | mirror (Divides (Cst hh, Add (Var is, id))) =
+    Divides (Cst hh, Add (Var is, id))
+  | mirror (Divides (Cst hh, Add (Neg it, id))) =
+    Divides (Cst hh, Add (Neg it, id))
+  | mirror (Divides (Cst hh, Add (Add (iu, iv), id))) =
+    Divides (Cst hh, Add (Add (iu, iv), id))
+  | mirror (Divides (Cst hh, Add (Sub (iw, ix), id))) =
+    Divides (Cst hh, Add (Sub (iw, ix), id))
+  | mirror (Divides (Cst hh, Add (Mult (Cst jj, Cst kb), id))) =
+    Divides (Cst hh, Add (Mult (Cst jj, Cst kb), id))
+  | mirror (Divides (Cst hh, Add (Mult (Cst jj, Var kt), id))) =
+    (if (kt = 0) then Divides (Cst hh, Add (Mult (Cst (~ jj), Var 0), id))
+      else Divides
+             (Cst hh,
+               Add (Mult (Cst jj, Var (op_45_def0 kt id_1_def0 + 1)), id)))
+  | mirror (Divides (Cst hh, Add (Mult (Cst jj, Neg kd), id))) =
+    Divides (Cst hh, Add (Mult (Cst jj, Neg kd), id))
+  | mirror (Divides (Cst hh, Add (Mult (Cst jj, Add (ke, kf)), id))) =
+    Divides (Cst hh, Add (Mult (Cst jj, Add (ke, kf)), id))
+  | mirror (Divides (Cst hh, Add (Mult (Cst jj, Sub (kg, kh)), id))) =
+    Divides (Cst hh, Add (Mult (Cst jj, Sub (kg, kh)), id))
+  | mirror (Divides (Cst hh, Add (Mult (Cst jj, Mult (ki, kj)), id))) =
+    Divides (Cst hh, Add (Mult (Cst jj, Mult (ki, kj)), id))
+  | mirror (Divides (Cst hh, Add (Mult (Var jk, iz), id))) =
+    Divides (Cst hh, Add (Mult (Var jk, iz), id))
+  | mirror (Divides (Cst hh, Add (Mult (Neg jl, iz), id))) =
+    Divides (Cst hh, Add (Mult (Neg jl, iz), id))
+  | mirror (Divides (Cst hh, Add (Mult (Add (jm, jn), iz), id))) =
+    Divides (Cst hh, Add (Mult (Add (jm, jn), iz), id))
+  | mirror (Divides (Cst hh, Add (Mult (Sub (jo, jp), iz), id))) =
+    Divides (Cst hh, Add (Mult (Sub (jo, jp), iz), id))
+  | mirror (Divides (Cst hh, Add (Mult (Mult (jq, jr), iz), id))) =
+    Divides (Cst hh, Add (Mult (Mult (jq, jr), iz), id))
+  | mirror (Divides (Cst hh, Sub (ie, if'))) = Divides (Cst hh, Sub (ie, if'))
+  | mirror (Divides (Cst hh, Mult (ig, ih))) = Divides (Cst hh, Mult (ig, ih))
+  | mirror (Divides (Var hi, ag)) = Divides (Var hi, ag)
+  | mirror (Divides (Neg hj, ag)) = Divides (Neg hj, ag)
+  | mirror (Divides (Add (hk, hl), ag)) = Divides (Add (hk, hl), ag)
+  | mirror (Divides (Sub (hm, hn), ag)) = Divides (Sub (hm, hn), ag)
+  | mirror (Divides (Mult (ho, hp), ag)) = Divides (Mult (ho, hp), ag)
+  | mirror T = T
+  | mirror F = F
+  | mirror (NOT (Lt (kv, kw))) = NOT (Lt (kv, kw))
+  | mirror (NOT (Gt (kx, ky))) = NOT (Gt (kx, ky))
+  | mirror (NOT (Le (kz, la))) = NOT (Le (kz, la))
+  | mirror (NOT (Ge (lb, lc))) = NOT (Ge (lb, lc))
+  | mirror (NOT (Eq (Cst mp, le))) = NOT (Eq (Cst mp, le))
+  | mirror (NOT (Eq (Var mq, le))) = NOT (Eq (Var mq, le))
+  | mirror (NOT (Eq (Neg mr, le))) = NOT (Eq (Neg mr, le))
+  | mirror (NOT (Eq (Add (Cst nh, mt), le))) = NOT (Eq (Add (Cst nh, mt), le))
+  | mirror (NOT (Eq (Add (Var ni, mt), le))) = NOT (Eq (Add (Var ni, mt), le))
+  | mirror (NOT (Eq (Add (Neg nj, mt), le))) = NOT (Eq (Add (Neg nj, mt), le))
+  | mirror (NOT (Eq (Add (Add (nk, nl), mt), le))) =
+    NOT (Eq (Add (Add (nk, nl), mt), le))
+  | mirror (NOT (Eq (Add (Sub (nm, nn), mt), le))) =
+    NOT (Eq (Add (Sub (nm, nn), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Cst nz, Cst or), mt), le))) =
+    NOT (Eq (Add (Mult (Cst nz, Cst or), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Cst nz, Var pj), mt), le))) =
+    (if (pj = 0) then NOT (Eq (Add (Mult (Cst (~ nz), Var 0), mt), le))
+      else NOT (Eq (Add (Mult (Cst nz, Var (op_45_def0 pj id_1_def0 + 1)), mt),
+                     le)))
+  | mirror (NOT (Eq (Add (Mult (Cst nz, Neg ot), mt), le))) =
+    NOT (Eq (Add (Mult (Cst nz, Neg ot), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Cst nz, Add (ou, ov)), mt), le))) =
+    NOT (Eq (Add (Mult (Cst nz, Add (ou, ov)), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Cst nz, Sub (ow, ox)), mt), le))) =
+    NOT (Eq (Add (Mult (Cst nz, Sub (ow, ox)), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Cst nz, Mult (oy, oz)), mt), le))) =
+    NOT (Eq (Add (Mult (Cst nz, Mult (oy, oz)), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Var oa, np), mt), le))) =
+    NOT (Eq (Add (Mult (Var oa, np), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Neg ob, np), mt), le))) =
+    NOT (Eq (Add (Mult (Neg ob, np), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Add (oc, od), np), mt), le))) =
+    NOT (Eq (Add (Mult (Add (oc, od), np), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Sub (oe, of'), np), mt), le))) =
+    NOT (Eq (Add (Mult (Sub (oe, of'), np), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Mult (og, oh), np), mt), le))) =
+    NOT (Eq (Add (Mult (Mult (og, oh), np), mt), le))
+  | mirror (NOT (Eq (Sub (mu, mv), le))) = NOT (Eq (Sub (mu, mv), le))
+  | mirror (NOT (Eq (Mult (mw, mx), le))) = NOT (Eq (Mult (mw, mx), le))
+  | mirror (NOT (Divides (Cst pl, Cst qd))) = NOT (Divides (Cst pl, Cst qd))
+  | mirror (NOT (Divides (Cst pl, Var qe))) = NOT (Divides (Cst pl, Var qe))
+  | mirror (NOT (Divides (Cst pl, Neg qf))) = NOT (Divides (Cst pl, Neg qf))
+  | mirror (NOT (Divides (Cst pl, Add (Cst qv, qh)))) =
+    NOT (Divides (Cst pl, Add (Cst qv, qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Var qw, qh)))) =
+    NOT (Divides (Cst pl, Add (Var qw, qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Neg qx, qh)))) =
+    NOT (Divides (Cst pl, Add (Neg qx, qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Add (qy, qz), qh)))) =
+    NOT (Divides (Cst pl, Add (Add (qy, qz), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Sub (ra, rb), qh)))) =
+    NOT (Divides (Cst pl, Add (Sub (ra, rb), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Cst rn, Cst sf), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Cst rn, Cst sf), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Cst rn, Var sx), qh)))) =
+    (if (sx = 0)
+      then NOT (Divides (Cst pl, Add (Mult (Cst (~ rn), Var 0), qh)))
+      else NOT (Divides
+                  (Cst pl,
+                    Add (Mult (Cst rn, Var (op_45_def0 sx id_1_def0 + 1)),
+                          qh))))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Cst rn, Neg sh), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Cst rn, Neg sh), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Cst rn, Add (si, sj)), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Cst rn, Add (si, sj)), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Cst rn, Sub (sk, sl)), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Cst rn, Sub (sk, sl)), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Cst rn, Mult (sm, sn)), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Cst rn, Mult (sm, sn)), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Var ro, rd), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Var ro, rd), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Neg rp, rd), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Neg rp, rd), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Add (rq, rr), rd), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Add (rq, rr), rd), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Sub (rs, rt), rd), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Sub (rs, rt), rd), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Mult (ru, rv), rd), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Mult (ru, rv), rd), qh)))
+  | mirror (NOT (Divides (Cst pl, Sub (qi, qj)))) =
+    NOT (Divides (Cst pl, Sub (qi, qj)))
+  | mirror (NOT (Divides (Cst pl, Mult (qk, ql)))) =
+    NOT (Divides (Cst pl, Mult (qk, ql)))
+  | mirror (NOT (Divides (Var pm, lg))) = NOT (Divides (Var pm, lg))
+  | mirror (NOT (Divides (Neg pn, lg))) = NOT (Divides (Neg pn, lg))
+  | mirror (NOT (Divides (Add (po, pp), lg))) = NOT (Divides (Add (po, pp), lg))
+  | mirror (NOT (Divides (Sub (pq, pr), lg))) = NOT (Divides (Sub (pq, pr), lg))
+  | mirror (NOT (Divides (Mult (ps, pt), lg))) =
+    NOT (Divides (Mult (ps, pt), lg))
+  | mirror (NOT T) = NOT T
+  | mirror (NOT F) = NOT F
+  | mirror (NOT (NOT lh)) = NOT (NOT lh)
+  | mirror (NOT (And (li, lj))) = NOT (And (li, lj))
+  | mirror (NOT (Or (lk, ll))) = NOT (Or (lk, ll))
+  | mirror (NOT (Imp (lm, ln))) = NOT (Imp (lm, ln))
+  | mirror (NOT (Equ (lo, lp))) = NOT (Equ (lo, lp))
+  | mirror (NOT (QAll lq)) = NOT (QAll lq)
+  | mirror (NOT (QEx lr)) = NOT (QEx lr)
+  | mirror (Imp (am, an)) = Imp (am, an)
+  | mirror (Equ (ao, ap)) = Equ (ao, ap)
+  | mirror (QAll aq) = QAll aq
+  | mirror (QEx ar) = QEx ar;
+
+fun op_43_def0 m n = nat ((m) + (n));
+
+fun size_def1 [] = 0
+  | size_def1 (a :: list) = op_43_def0 (size_def1 list) (0 + 1);
+
+fun aset (And (p, q)) = op_64 (aset p) (aset q)
+  | aset (Or (p, q)) = op_64 (aset p) (aset q)
+  | aset (Lt (u, v)) = []
+  | aset (Gt (w, x)) = []
+  | aset (Le (Cst bo, z)) = []
+  | aset (Le (Var bp, z)) = []
+  | aset (Le (Neg bq, z)) = []
+  | aset (Le (Add (Cst cg, bs), z)) = []
+  | aset (Le (Add (Var ch, bs), z)) = []
+  | aset (Le (Add (Neg ci, bs), z)) = []
+  | aset (Le (Add (Add (cj, ck), bs), z)) = []
+  | aset (Le (Add (Sub (cl, cm), bs), z)) = []
+  | aset (Le (Add (Mult (Cst cy, Cst dq), bs), z)) = []
+  | aset (Le (Add (Mult (Cst cy, Var ei), bs), z)) =
+    (if (ei = 0)
+      then (if (cy < 0) then [lin_add (bs, Cst 1)]
+             else [lin_neg bs, lin_add (lin_neg bs, Cst 1)])
+      else [])
+  | aset (Le (Add (Mult (Cst cy, Neg ds), bs), z)) = []
+  | aset (Le (Add (Mult (Cst cy, Add (dt, du)), bs), z)) = []
+  | aset (Le (Add (Mult (Cst cy, Sub (dv, dw)), bs), z)) = []
+  | aset (Le (Add (Mult (Cst cy, Mult (dx, dy)), bs), z)) = []
+  | aset (Le (Add (Mult (Var cz, co), bs), z)) = []
+  | aset (Le (Add (Mult (Neg da, co), bs), z)) = []
+  | aset (Le (Add (Mult (Add (db, dc), co), bs), z)) = []
+  | aset (Le (Add (Mult (Sub (dd, de), co), bs), z)) = []
+  | aset (Le (Add (Mult (Mult (df, dg), co), bs), z)) = []
+  | aset (Le (Sub (bt, bu), z)) = []
+  | aset (Le (Mult (bv, bw), z)) = []
+  | aset (Ge (aa, ab)) = []
+  | aset (Eq (Cst ek, ad)) = []
+  | aset (Eq (Var el, ad)) = []
+  | aset (Eq (Neg em, ad)) = []
+  | aset (Eq (Add (Cst fc, eo), ad)) = []
+  | aset (Eq (Add (Var fd, eo), ad)) = []
+  | aset (Eq (Add (Neg fe, eo), ad)) = []
+  | aset (Eq (Add (Add (ff, fg), eo), ad)) = []
+  | aset (Eq (Add (Sub (fh, fi), eo), ad)) = []
+  | aset (Eq (Add (Mult (Cst fu, Cst gm), eo), ad)) = []
+  | aset (Eq (Add (Mult (Cst fu, Var he), eo), ad)) =
+    (if (he = 0)
+      then (if (fu < 0) then [lin_add (eo, Cst 1)]
+             else [lin_add (lin_neg eo, Cst 1)])
+      else [])
+  | aset (Eq (Add (Mult (Cst fu, Neg go), eo), ad)) = []
+  | aset (Eq (Add (Mult (Cst fu, Add (gp, gq)), eo), ad)) = []
+  | aset (Eq (Add (Mult (Cst fu, Sub (gr, gs)), eo), ad)) = []
+  | aset (Eq (Add (Mult (Cst fu, Mult (gt, gu)), eo), ad)) = []
+  | aset (Eq (Add (Mult (Var fv, fk), eo), ad)) = []
+  | aset (Eq (Add (Mult (Neg fw, fk), eo), ad)) = []
+  | aset (Eq (Add (Mult (Add (fx, fy), fk), eo), ad)) = []
+  | aset (Eq (Add (Mult (Sub (fz, ga), fk), eo), ad)) = []
+  | aset (Eq (Add (Mult (Mult (gb, gc), fk), eo), ad)) = []
+  | aset (Eq (Sub (ep, eq), ad)) = []
+  | aset (Eq (Mult (er, es), ad)) = []
+  | aset (Divides (ae, af)) = []
+  | aset T = []
+  | aset F = []
+  | aset (NOT (Lt (hg, hh))) = []
+  | aset (NOT (Gt (hi, hj))) = []
+  | aset (NOT (Le (hk, hl))) = []
+  | aset (NOT (Ge (hm, hn))) = []
+  | aset (NOT (Eq (Cst ja, hp))) = []
+  | aset (NOT (Eq (Var jb, hp))) = []
+  | aset (NOT (Eq (Neg jc, hp))) = []
+  | aset (NOT (Eq (Add (Cst js, je), hp))) = []
+  | aset (NOT (Eq (Add (Var jt, je), hp))) = []
+  | aset (NOT (Eq (Add (Neg ju, je), hp))) = []
+  | aset (NOT (Eq (Add (Add (jv, jw), je), hp))) = []
+  | aset (NOT (Eq (Add (Sub (jx, jy), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Cst kk, Cst lc), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Cst kk, Var lu), je), hp))) =
+    (if (lu = 0) then (if (kk < 0) then [je] else [lin_neg je]) else [])
+  | aset (NOT (Eq (Add (Mult (Cst kk, Neg le), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Cst kk, Add (lf, lg)), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Cst kk, Sub (lh, li)), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Cst kk, Mult (lj, lk)), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Var kl, ka), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Neg km, ka), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Add (kn, ko), ka), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Sub (kp, kq), ka), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Mult (kr, ks), ka), je), hp))) = []
+  | aset (NOT (Eq (Sub (jf, jg), hp))) = []
+  | aset (NOT (Eq (Mult (jh, ji), hp))) = []
+  | aset (NOT (Divides (hq, hr))) = []
+  | aset (NOT T) = []
+  | aset (NOT F) = []
+  | aset (NOT (NOT hs)) = []
+  | aset (NOT (And (ht, hu))) = []
+  | aset (NOT (Or (hv, hw))) = []
+  | aset (NOT (Imp (hx, hy))) = []
+  | aset (NOT (Equ (hz, ia))) = []
+  | aset (NOT (QAll ib)) = []
+  | aset (NOT (QEx ic)) = []
+  | aset (Imp (al, am)) = []
+  | aset (Equ (an, ao)) = []
+  | aset (QAll ap) = []
+  | aset (QEx aq) = [];
+
+fun op_mem x [] = false
+  | op_mem x (y :: ys) = (if (y = x) then true else op_mem x ys);
+
+fun list_insert x xs = (if op_mem x xs then xs else (x :: xs));
+
+fun list_set [] = []
+  | list_set (x :: xs) = list_insert x (list_set xs);
+
+fun bset (And (p, q)) = op_64 (bset p) (bset q)
+  | bset (Or (p, q)) = op_64 (bset p) (bset q)
+  | bset (Lt (u, v)) = []
+  | bset (Gt (w, x)) = []
+  | bset (Le (Cst bo, z)) = []
+  | bset (Le (Var bp, z)) = []
+  | bset (Le (Neg bq, z)) = []
+  | bset (Le (Add (Cst cg, bs), z)) = []
+  | bset (Le (Add (Var ch, bs), z)) = []
+  | bset (Le (Add (Neg ci, bs), z)) = []
+  | bset (Le (Add (Add (cj, ck), bs), z)) = []
+  | bset (Le (Add (Sub (cl, cm), bs), z)) = []
+  | bset (Le (Add (Mult (Cst cy, Cst dq), bs), z)) = []
+  | bset (Le (Add (Mult (Cst cy, Var ei), bs), z)) =
+    (if (ei = 0)
+      then (if (cy < 0) then [lin_add (bs, Cst ~1), bs]
+             else [lin_add (lin_neg bs, Cst ~1)])
+      else [])
+  | bset (Le (Add (Mult (Cst cy, Neg ds), bs), z)) = []
+  | bset (Le (Add (Mult (Cst cy, Add (dt, du)), bs), z)) = []
+  | bset (Le (Add (Mult (Cst cy, Sub (dv, dw)), bs), z)) = []
+  | bset (Le (Add (Mult (Cst cy, Mult (dx, dy)), bs), z)) = []
+  | bset (Le (Add (Mult (Var cz, co), bs), z)) = []
+  | bset (Le (Add (Mult (Neg da, co), bs), z)) = []
+  | bset (Le (Add (Mult (Add (db, dc), co), bs), z)) = []
+  | bset (Le (Add (Mult (Sub (dd, de), co), bs), z)) = []
+  | bset (Le (Add (Mult (Mult (df, dg), co), bs), z)) = []
+  | bset (Le (Sub (bt, bu), z)) = []
+  | bset (Le (Mult (bv, bw), z)) = []
+  | bset (Ge (aa, ab)) = []
+  | bset (Eq (Cst ek, ad)) = []
+  | bset (Eq (Var el, ad)) = []
+  | bset (Eq (Neg em, ad)) = []
+  | bset (Eq (Add (Cst fc, eo), ad)) = []
+  | bset (Eq (Add (Var fd, eo), ad)) = []
+  | bset (Eq (Add (Neg fe, eo), ad)) = []
+  | bset (Eq (Add (Add (ff, fg), eo), ad)) = []
+  | bset (Eq (Add (Sub (fh, fi), eo), ad)) = []
+  | bset (Eq (Add (Mult (Cst fu, Cst gm), eo), ad)) = []
+  | bset (Eq (Add (Mult (Cst fu, Var he), eo), ad)) =
+    (if (he = 0)
+      then (if (fu < 0) then [lin_add (eo, Cst ~1)]
+             else [lin_add (lin_neg eo, Cst ~1)])
+      else [])
+  | bset (Eq (Add (Mult (Cst fu, Neg go), eo), ad)) = []
+  | bset (Eq (Add (Mult (Cst fu, Add (gp, gq)), eo), ad)) = []
+  | bset (Eq (Add (Mult (Cst fu, Sub (gr, gs)), eo), ad)) = []
+  | bset (Eq (Add (Mult (Cst fu, Mult (gt, gu)), eo), ad)) = []
+  | bset (Eq (Add (Mult (Var fv, fk), eo), ad)) = []
+  | bset (Eq (Add (Mult (Neg fw, fk), eo), ad)) = []
+  | bset (Eq (Add (Mult (Add (fx, fy), fk), eo), ad)) = []
+  | bset (Eq (Add (Mult (Sub (fz, ga), fk), eo), ad)) = []
+  | bset (Eq (Add (Mult (Mult (gb, gc), fk), eo), ad)) = []
+  | bset (Eq (Sub (ep, eq), ad)) = []
+  | bset (Eq (Mult (er, es), ad)) = []
+  | bset (Divides (ae, af)) = []
+  | bset T = []
+  | bset F = []
+  | bset (NOT (Lt (hg, hh))) = []
+  | bset (NOT (Gt (hi, hj))) = []
+  | bset (NOT (Le (hk, hl))) = []
+  | bset (NOT (Ge (hm, hn))) = []
+  | bset (NOT (Eq (Cst ja, hp))) = []
+  | bset (NOT (Eq (Var jb, hp))) = []
+  | bset (NOT (Eq (Neg jc, hp))) = []
+  | bset (NOT (Eq (Add (Cst js, je), hp))) = []
+  | bset (NOT (Eq (Add (Var jt, je), hp))) = []
+  | bset (NOT (Eq (Add (Neg ju, je), hp))) = []
+  | bset (NOT (Eq (Add (Add (jv, jw), je), hp))) = []
+  | bset (NOT (Eq (Add (Sub (jx, jy), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Cst kk, Cst lc), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Cst kk, Var lu), je), hp))) =
+    (if (lu = 0) then (if (kk < 0) then [je] else [lin_neg je]) else [])
+  | bset (NOT (Eq (Add (Mult (Cst kk, Neg le), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Cst kk, Add (lf, lg)), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Cst kk, Sub (lh, li)), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Cst kk, Mult (lj, lk)), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Var kl, ka), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Neg km, ka), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Add (kn, ko), ka), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Sub (kp, kq), ka), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Mult (kr, ks), ka), je), hp))) = []
+  | bset (NOT (Eq (Sub (jf, jg), hp))) = []
+  | bset (NOT (Eq (Mult (jh, ji), hp))) = []
+  | bset (NOT (Divides (hq, hr))) = []
+  | bset (NOT T) = []
+  | bset (NOT F) = []
+  | bset (NOT (NOT hs)) = []
+  | bset (NOT (And (ht, hu))) = []
+  | bset (NOT (Or (hv, hw))) = []
+  | bset (NOT (Imp (hx, hy))) = []
+  | bset (NOT (Equ (hz, ia))) = []
+  | bset (NOT (QAll ib)) = []
+  | bset (NOT (QEx ic)) = []
+  | bset (Imp (al, am)) = []
+  | bset (Equ (an, ao)) = []
+  | bset (QAll ap) = []
+  | bset (QEx aq) = [];
+
+fun adjustcoeff (l, Le (Add (Mult (Cst c, Var 0), r), Cst i)) =
+    (if (c <= 0)
+      then Le (Add (Mult (Cst ~1, Var 0), lin_mul (~ (op_div_def1 l c), r)),
+                Cst 0)
+      else Le (Add (Mult (Cst 1, Var 0), lin_mul (op_div_def1 l c, r)), Cst 0))
+  | adjustcoeff (l, Eq (Add (Mult (Cst c, Var 0), r), Cst i)) =
+    Eq (Add (Mult (Cst 1, Var 0), lin_mul (op_div_def1 l c, r)), Cst 0)
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst c, Var 0), r), Cst i))) =
+    NOT (Eq (Add (Mult (Cst 1, Var 0), lin_mul (op_div_def1 l c, r)), Cst 0))
+  | adjustcoeff (l, And (p, q)) = And (adjustcoeff (l, p), adjustcoeff (l, q))
+  | adjustcoeff (l, Or (p, q)) = Or (adjustcoeff (l, p), adjustcoeff (l, q))
+  | adjustcoeff (l, Lt (w, x)) = Lt (w, x)
+  | adjustcoeff (l, Gt (y, z)) = Gt (y, z)
+  | adjustcoeff (l, Le (Cst bq, ab)) = Le (Cst bq, ab)
+  | adjustcoeff (l, Le (Var br, ab)) = Le (Var br, ab)
+  | adjustcoeff (l, Le (Neg bs, ab)) = Le (Neg bs, ab)
+  | adjustcoeff (l, Le (Add (Cst ci, bu), ab)) = Le (Add (Cst ci, bu), ab)
+  | adjustcoeff (l, Le (Add (Var cj, bu), ab)) = Le (Add (Var cj, bu), ab)
+  | adjustcoeff (l, Le (Add (Neg ck, bu), ab)) = Le (Add (Neg ck, bu), ab)
+  | adjustcoeff (l, Le (Add (Add (cl, cm), bu), ab)) =
+    Le (Add (Add (cl, cm), bu), ab)
+  | adjustcoeff (l, Le (Add (Sub (cn, co), bu), ab)) =
+    Le (Add (Sub (cn, co), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Cst ds), bu), ab)) =
+    Le (Add (Mult (Cst da, Cst ds), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Var 0), bu), Var en)) =
+    Le (Add (Mult (Cst da, Var 0), bu), Var en)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Var 0), bu), Neg eo)) =
+    Le (Add (Mult (Cst da, Var 0), bu), Neg eo)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Var 0), bu), Add (ep, eq))) =
+    Le (Add (Mult (Cst da, Var 0), bu), Add (ep, eq))
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Var 0), bu), Sub (er, es))) =
+    Le (Add (Mult (Cst da, Var 0), bu), Sub (er, es))
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Var 0), bu), Mult (et, eu))) =
+    Le (Add (Mult (Cst da, Var 0), bu), Mult (et, eu))
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Var ek), bu), ab)) =
+    Le (Add (Mult (Cst da, Var ek), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Neg du), bu), ab)) =
+    Le (Add (Mult (Cst da, Neg du), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Add (dv, dw)), bu), ab)) =
+    Le (Add (Mult (Cst da, Add (dv, dw)), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Sub (dx, dy)), bu), ab)) =
+    Le (Add (Mult (Cst da, Sub (dx, dy)), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Mult (dz, ea)), bu), ab)) =
+    Le (Add (Mult (Cst da, Mult (dz, ea)), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Var db, cq), bu), ab)) =
+    Le (Add (Mult (Var db, cq), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Neg dc, cq), bu), ab)) =
+    Le (Add (Mult (Neg dc, cq), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Add (dd, de), cq), bu), ab)) =
+    Le (Add (Mult (Add (dd, de), cq), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Sub (df, dg), cq), bu), ab)) =
+    Le (Add (Mult (Sub (df, dg), cq), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Mult (dh, di), cq), bu), ab)) =
+    Le (Add (Mult (Mult (dh, di), cq), bu), ab)
+  | adjustcoeff (l, Le (Sub (bv, bw), ab)) = Le (Sub (bv, bw), ab)
+  | adjustcoeff (l, Le (Mult (bx, by), ab)) = Le (Mult (bx, by), ab)
+  | adjustcoeff (l, Ge (ac, ad)) = Ge (ac, ad)
+  | adjustcoeff (l, Eq (Cst fe, af)) = Eq (Cst fe, af)
+  | adjustcoeff (l, Eq (Var ff, af)) = Eq (Var ff, af)
+  | adjustcoeff (l, Eq (Neg fg, af)) = Eq (Neg fg, af)
+  | adjustcoeff (l, Eq (Add (Cst fw, fi), af)) = Eq (Add (Cst fw, fi), af)
+  | adjustcoeff (l, Eq (Add (Var fx, fi), af)) = Eq (Add (Var fx, fi), af)
+  | adjustcoeff (l, Eq (Add (Neg fy, fi), af)) = Eq (Add (Neg fy, fi), af)
+  | adjustcoeff (l, Eq (Add (Add (fz, ga), fi), af)) =
+    Eq (Add (Add (fz, ga), fi), af)
+  | adjustcoeff (l, Eq (Add (Sub (gb, gc), fi), af)) =
+    Eq (Add (Sub (gb, gc), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Cst hg), fi), af)) =
+    Eq (Add (Mult (Cst go, Cst hg), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Var 0), fi), Var ib)) =
+    Eq (Add (Mult (Cst go, Var 0), fi), Var ib)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Var 0), fi), Neg ic)) =
+    Eq (Add (Mult (Cst go, Var 0), fi), Neg ic)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Var 0), fi), Add (id, ie))) =
+    Eq (Add (Mult (Cst go, Var 0), fi), Add (id, ie))
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Var 0), fi), Sub (if', ig))) =
+    Eq (Add (Mult (Cst go, Var 0), fi), Sub (if', ig))
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Var 0), fi), Mult (ih, ii))) =
+    Eq (Add (Mult (Cst go, Var 0), fi), Mult (ih, ii))
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Var hy), fi), af)) =
+    Eq (Add (Mult (Cst go, Var hy), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Neg hi), fi), af)) =
+    Eq (Add (Mult (Cst go, Neg hi), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Add (hj, hk)), fi), af)) =
+    Eq (Add (Mult (Cst go, Add (hj, hk)), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Sub (hl, hm)), fi), af)) =
+    Eq (Add (Mult (Cst go, Sub (hl, hm)), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Mult (hn, ho)), fi), af)) =
+    Eq (Add (Mult (Cst go, Mult (hn, ho)), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Var gp, ge), fi), af)) =
+    Eq (Add (Mult (Var gp, ge), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Neg gq, ge), fi), af)) =
+    Eq (Add (Mult (Neg gq, ge), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Add (gr, gs), ge), fi), af)) =
+    Eq (Add (Mult (Add (gr, gs), ge), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Sub (gt, gu), ge), fi), af)) =
+    Eq (Add (Mult (Sub (gt, gu), ge), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Mult (gv, gw), ge), fi), af)) =
+    Eq (Add (Mult (Mult (gv, gw), ge), fi), af)
+  | adjustcoeff (l, Eq (Sub (fj, fk), af)) = Eq (Sub (fj, fk), af)
+  | adjustcoeff (l, Eq (Mult (fl, fm), af)) = Eq (Mult (fl, fm), af)
+  | adjustcoeff (l, Divides (Cst is, Cst jk)) = Divides (Cst is, Cst jk)
+  | adjustcoeff (l, Divides (Cst is, Var jl)) = Divides (Cst is, Var jl)
+  | adjustcoeff (l, Divides (Cst is, Neg jm)) = Divides (Cst is, Neg jm)
+  | adjustcoeff (l, Divides (Cst is, Add (Cst kc, jo))) =
+    Divides (Cst is, Add (Cst kc, jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Var kd, jo))) =
+    Divides (Cst is, Add (Var kd, jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Neg ke, jo))) =
+    Divides (Cst is, Add (Neg ke, jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Add (kf, kg), jo))) =
+    Divides (Cst is, Add (Add (kf, kg), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Sub (kh, ki), jo))) =
+    Divides (Cst is, Add (Sub (kh, ki), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Cst ku, Cst lm), jo))) =
+    Divides (Cst is, Add (Mult (Cst ku, Cst lm), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Cst ku, Var me), jo))) =
+    (if (me = 0)
+      then Divides
+             (Cst (op_div_def1 l ku * is),
+               Add (Mult (Cst 1, Var 0), lin_mul (op_div_def1 l ku, jo)))
+      else Divides
+             (Cst is,
+               Add (Mult (Cst ku, Var (op_45_def0 me id_1_def0 + 1)), jo)))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Cst ku, Neg lo), jo))) =
+    Divides (Cst is, Add (Mult (Cst ku, Neg lo), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Cst ku, Add (lp, lq)), jo))) =
+    Divides (Cst is, Add (Mult (Cst ku, Add (lp, lq)), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Cst ku, Sub (lr, ls)), jo))) =
+    Divides (Cst is, Add (Mult (Cst ku, Sub (lr, ls)), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Cst ku, Mult (lt, lu)), jo))) =
+    Divides (Cst is, Add (Mult (Cst ku, Mult (lt, lu)), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Var kv, kk), jo))) =
+    Divides (Cst is, Add (Mult (Var kv, kk), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Neg kw, kk), jo))) =
+    Divides (Cst is, Add (Mult (Neg kw, kk), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Add (kx, ky), kk), jo))) =
+    Divides (Cst is, Add (Mult (Add (kx, ky), kk), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Sub (kz, la), kk), jo))) =
+    Divides (Cst is, Add (Mult (Sub (kz, la), kk), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Mult (lb, lc), kk), jo))) =
+    Divides (Cst is, Add (Mult (Mult (lb, lc), kk), jo))
+  | adjustcoeff (l, Divides (Cst is, Sub (jp, jq))) =
+    Divides (Cst is, Sub (jp, jq))
+  | adjustcoeff (l, Divides (Cst is, Mult (jr, js))) =
+    Divides (Cst is, Mult (jr, js))
+  | adjustcoeff (l, Divides (Var it, ah)) = Divides (Var it, ah)
+  | adjustcoeff (l, Divides (Neg iu, ah)) = Divides (Neg iu, ah)
+  | adjustcoeff (l, Divides (Add (iv, iw), ah)) = Divides (Add (iv, iw), ah)
+  | adjustcoeff (l, Divides (Sub (ix, iy), ah)) = Divides (Sub (ix, iy), ah)
+  | adjustcoeff (l, Divides (Mult (iz, ja), ah)) = Divides (Mult (iz, ja), ah)
+  | adjustcoeff (l, T) = T
+  | adjustcoeff (l, F) = F
+  | adjustcoeff (l, NOT (Lt (mg, mh))) = NOT (Lt (mg, mh))
+  | adjustcoeff (l, NOT (Gt (mi, mj))) = NOT (Gt (mi, mj))
+  | adjustcoeff (l, NOT (Le (mk, ml))) = NOT (Le (mk, ml))
+  | adjustcoeff (l, NOT (Ge (mm, mn))) = NOT (Ge (mm, mn))
+  | adjustcoeff (l, NOT (Eq (Cst oa, mp))) = NOT (Eq (Cst oa, mp))
+  | adjustcoeff (l, NOT (Eq (Var ob, mp))) = NOT (Eq (Var ob, mp))
+  | adjustcoeff (l, NOT (Eq (Neg oc, mp))) = NOT (Eq (Neg oc, mp))
+  | adjustcoeff (l, NOT (Eq (Add (Cst os, oe), mp))) =
+    NOT (Eq (Add (Cst os, oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Var ot, oe), mp))) =
+    NOT (Eq (Add (Var ot, oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Neg ou, oe), mp))) =
+    NOT (Eq (Add (Neg ou, oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Add (ov, ow), oe), mp))) =
+    NOT (Eq (Add (Add (ov, ow), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Sub (ox, oy), oe), mp))) =
+    NOT (Eq (Add (Sub (ox, oy), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Cst qc), oe), mp))) =
+    NOT (Eq (Add (Mult (Cst pk, Cst qc), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Var qx))) =
+    NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Var qx))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Neg qy))) =
+    NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Neg qy))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Add (qz, ra)))) =
+    NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Add (qz, ra)))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Sub (rb, rc)))) =
+    NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Sub (rb, rc)))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Mult (rd, re)))) =
+    NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Mult (rd, re)))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Var qu), oe), mp))) =
+    NOT (Eq (Add (Mult (Cst pk, Var qu), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Neg qe), oe), mp))) =
+    NOT (Eq (Add (Mult (Cst pk, Neg qe), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Add (qf, qg)), oe), mp))) =
+    NOT (Eq (Add (Mult (Cst pk, Add (qf, qg)), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Sub (qh, qi)), oe), mp))) =
+    NOT (Eq (Add (Mult (Cst pk, Sub (qh, qi)), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Mult (qj, qk)), oe), mp))) =
+    NOT (Eq (Add (Mult (Cst pk, Mult (qj, qk)), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Var pl, pa), oe), mp))) =
+    NOT (Eq (Add (Mult (Var pl, pa), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Neg pm, pa), oe), mp))) =
+    NOT (Eq (Add (Mult (Neg pm, pa), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Add (pn, po), pa), oe), mp))) =
+    NOT (Eq (Add (Mult (Add (pn, po), pa), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Sub (pp, pq), pa), oe), mp))) =
+    NOT (Eq (Add (Mult (Sub (pp, pq), pa), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Mult (pr, ps), pa), oe), mp))) =
+    NOT (Eq (Add (Mult (Mult (pr, ps), pa), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Sub (of', og), mp))) = NOT (Eq (Sub (of', og), mp))
+  | adjustcoeff (l, NOT (Eq (Mult (oh, oi), mp))) = NOT (Eq (Mult (oh, oi), mp))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Cst sg))) =
+    NOT (Divides (Cst ro, Cst sg))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Var sh))) =
+    NOT (Divides (Cst ro, Var sh))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Neg si))) =
+    NOT (Divides (Cst ro, Neg si))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Cst sy, sk)))) =
+    NOT (Divides (Cst ro, Add (Cst sy, sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Var sz, sk)))) =
+    NOT (Divides (Cst ro, Add (Var sz, sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Neg ta, sk)))) =
+    NOT (Divides (Cst ro, Add (Neg ta, sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Add (tb, tc), sk)))) =
+    NOT (Divides (Cst ro, Add (Add (tb, tc), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Sub (td, te), sk)))) =
+    NOT (Divides (Cst ro, Add (Sub (td, te), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Mult (Cst tq, Cst ui), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Cst tq, Cst ui), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Mult (Cst tq, Var va), sk)))) =
+    (if (va = 0)
+      then NOT (Divides
+                  (Cst (op_div_def1 l tq * ro),
+                    Add (Mult (Cst 1, Var 0), lin_mul (op_div_def1 l tq, sk))))
+      else NOT (Divides
+                  (Cst ro,
+                    Add (Mult (Cst tq, Var (op_45_def0 va id_1_def0 + 1)),
+                          sk))))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Mult (Cst tq, Neg uk), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Cst tq, Neg uk), sk)))
+  | adjustcoeff
+      (l, NOT (Divides (Cst ro, Add (Mult (Cst tq, Add (ul, um)), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Cst tq, Add (ul, um)), sk)))
+  | adjustcoeff
+      (l, NOT (Divides (Cst ro, Add (Mult (Cst tq, Sub (un, uo)), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Cst tq, Sub (un, uo)), sk)))
+  | adjustcoeff
+      (l, NOT (Divides (Cst ro, Add (Mult (Cst tq, Mult (up, uq)), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Cst tq, Mult (up, uq)), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Mult (Var tr, tg), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Var tr, tg), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Mult (Neg ts, tg), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Neg ts, tg), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Mult (Add (tt, tu), tg), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Add (tt, tu), tg), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Mult (Sub (tv, tw), tg), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Sub (tv, tw), tg), sk)))
+  | adjustcoeff
+      (l, NOT (Divides (Cst ro, Add (Mult (Mult (tx, ty), tg), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Mult (tx, ty), tg), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Sub (sl, sm)))) =
+    NOT (Divides (Cst ro, Sub (sl, sm)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Mult (sn, so)))) =
+    NOT (Divides (Cst ro, Mult (sn, so)))
+  | adjustcoeff (l, NOT (Divides (Var rp, mr))) = NOT (Divides (Var rp, mr))
+  | adjustcoeff (l, NOT (Divides (Neg rq, mr))) = NOT (Divides (Neg rq, mr))
+  | adjustcoeff (l, NOT (Divides (Add (rr, rs), mr))) =
+    NOT (Divides (Add (rr, rs), mr))
+  | adjustcoeff (l, NOT (Divides (Sub (rt, ru), mr))) =
+    NOT (Divides (Sub (rt, ru), mr))
+  | adjustcoeff (l, NOT (Divides (Mult (rv, rw), mr))) =
+    NOT (Divides (Mult (rv, rw), mr))
+  | adjustcoeff (l, NOT T) = NOT T
+  | adjustcoeff (l, NOT F) = NOT F
+  | adjustcoeff (l, NOT (NOT ms)) = NOT (NOT ms)
+  | adjustcoeff (l, NOT (And (mt, mu))) = NOT (And (mt, mu))
+  | adjustcoeff (l, NOT (Or (mv, mw))) = NOT (Or (mv, mw))
+  | adjustcoeff (l, NOT (Imp (mx, my))) = NOT (Imp (mx, my))
+  | adjustcoeff (l, NOT (Equ (mz, na))) = NOT (Equ (mz, na))
+  | adjustcoeff (l, NOT (QAll nb)) = NOT (QAll nb)
+  | adjustcoeff (l, NOT (QEx nc)) = NOT (QEx nc)
+  | adjustcoeff (l, Imp (an, ao)) = Imp (an, ao)
+  | adjustcoeff (l, Equ (ap, aq)) = Equ (ap, aq)
+  | adjustcoeff (l, QAll ar) = QAll ar
+  | adjustcoeff (l, QEx as') = QEx as';
+
+fun formlcm (Le (Add (Mult (Cst c, Var 0), r), Cst i)) = abs c
+  | formlcm (Eq (Add (Mult (Cst c, Var 0), r), Cst i)) = abs c
+  | formlcm (NOT p) = formlcm p
+  | formlcm (And (p, q)) = ilcm (formlcm p) (formlcm q)
+  | formlcm (Or (p, q)) = ilcm (formlcm p) (formlcm q)
+  | formlcm (Lt (u, v)) = 1
+  | formlcm (Gt (w, x)) = 1
+  | formlcm (Le (Cst bo, z)) = 1
+  | formlcm (Le (Var bp, z)) = 1
+  | formlcm (Le (Neg bq, z)) = 1
+  | formlcm (Le (Add (Cst cg, bs), z)) = 1
+  | formlcm (Le (Add (Var ch, bs), z)) = 1
+  | formlcm (Le (Add (Neg ci, bs), z)) = 1
+  | formlcm (Le (Add (Add (cj, ck), bs), z)) = 1
+  | formlcm (Le (Add (Sub (cl, cm), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Cst dq), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Var 0), bs), Var el)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Var 0), bs), Neg em)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Var 0), bs), Add (en, eo))) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Var 0), bs), Sub (ep, eq))) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Var 0), bs), Mult (er, es))) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Var ei ), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Neg ds), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Add (dt, du)), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Sub (dv, dw)), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Mult (dx, dy)), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Var cz, co), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Neg da, co), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Add (db, dc), co), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Sub (dd, de), co), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Mult (df, dg), co), bs), z)) = 1
+  | formlcm (Le (Sub (bt, bu), z)) = 1
+  | formlcm (Le (Mult (bv, bw), z)) = 1
+  | formlcm (Ge (aa, ab)) = 1
+  | formlcm (Eq (Cst fc, ad)) = 1
+  | formlcm (Eq (Var fd, ad)) = 1
+  | formlcm (Eq (Neg fe, ad)) = 1
+  | formlcm (Eq (Add (Cst fu, fg), ad)) = 1
+  | formlcm (Eq (Add (Var fv, fg), ad)) = 1
+  | formlcm (Eq (Add (Neg fw, fg), ad)) = 1
+  | formlcm (Eq (Add (Add (fx, fy), fg), ad)) = 1
+  | formlcm (Eq (Add (Sub (fz, ga), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Cst he), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Var 0), fg), Var hz)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Var 0), fg), Neg ia)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Var 0), fg), Add (ib, ic))) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Var 0), fg), Sub (id, ie))) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Var 0), fg), Mult (if', ig))) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Var hw), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Neg hg), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Add (hh, hi)), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Sub (hj, hk)), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Mult (hl, hm)), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Var gn, gc), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Neg go, gc), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Add (gp, gq), gc), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Sub (gr, gs), gc), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Mult (gt, gu), gc), fg), ad)) = 1
+  | formlcm (Eq (Sub (fh, fi), ad)) = 1
+  | formlcm (Eq (Mult (fj, fk), ad)) = 1
+  | formlcm (Divides (Cst iq, Cst ji)) = 1
+  | formlcm (Divides (Cst iq, Var jj)) = 1
+  | formlcm (Divides (Cst iq, Neg jk)) = 1
+  | formlcm (Divides (Cst iq, Add (Cst ka, jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Var kb, jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Neg kc, jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Add (kd, ke), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Sub (kf, kg), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Cst ks, Cst lk), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Cst ks, Var mc), jm))) =
+    (if (mc = 0) then abs ks else 1)
+  | formlcm (Divides (Cst iq, Add (Mult (Cst ks, Neg lm), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Cst ks, Add (ln, lo)), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Cst ks, Sub (lp, lq)), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Cst ks, Mult (lr, ls)), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Var kt, ki), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Neg ku, ki), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Add (kv, kw), ki), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Sub (kx, ky), ki), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Mult (kz, la), ki), jm))) = 1
+  | formlcm (Divides (Cst iq, Sub (jn, jo))) = 1
+  | formlcm (Divides (Cst iq, Mult (jp, jq))) = 1
+  | formlcm (Divides (Var ir, af)) = 1
+  | formlcm (Divides (Neg is, af)) = 1
+  | formlcm (Divides (Add (it, iu), af)) = 1
+  | formlcm (Divides (Sub (iv, iw), af)) = 1
+  | formlcm (Divides (Mult (ix, iy), af)) = 1
+  | formlcm T = 1
+  | formlcm F = 1
+  | formlcm (Imp (al, am)) = 1
+  | formlcm (Equ (an, ao)) = 1
+  | formlcm (QAll ap) = 1
+  | formlcm (QEx aq) = 1;
+
+fun unitycoeff p =
+  let val l = formlcm p; val p' = adjustcoeff (l, p)
+  in (if (l = 1) then p'
+       else And (Divides (Cst l, Add (Mult (Cst 1, Var 0), Cst 0)), p'))
+  end;
+
+fun unify p =
+  let val q = unitycoeff p; val B = list_set (bset q); val A = list_set (aset q)
+  in (if op_60_61_def0 (size_def1 B) (size_def1 A) then (q, B)
+       else (mirror q, map lin_neg A))
+  end;
+
+fun cooper p =
+  lift_un (fn q => decrvars (explode_minf (unify q))) (linform (nnf p));
+
+fun pa p = lift_un psimpl (qelim (cooper, p));
+
+val test = pa;
+
+end;
diff -r afaa031ed4da -r 105519771c67 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Wed Sep 14 10:24:39 2005 +0200
+++ b/src/HOL/IsaMakefile	Wed Sep 14 17:25:52 2005 +0200
@@ -78,13 +78,15 @@
   $(SRC)/TFL/casesplit.ML $(SRC)/TFL/dcterm.ML $(SRC)/TFL/post.ML		\
   $(SRC)/TFL/rules.ML $(SRC)/TFL/tfl.ML $(SRC)/TFL/thms.ML			\
   $(SRC)/TFL/thry.ML $(SRC)/TFL/usyntax.ML $(SRC)/TFL/utils.ML			\
-  Binomial.thy Datatype.ML Datatype.thy Datatype_Universe.thy Divides.thy \
+  Binomial.thy Commutative_Ring.thy Datatype.ML Datatype.thy \
+  Datatype_Universe.thy Divides.thy \
   Equiv_Relations.thy Extraction.thy Finite_Set.ML Finite_Set.thy		\
   FixedPoint.thy Fun.thy HOL.ML HOL.thy Hilbert_Choice.thy Inductive.thy		\
   Infinite_Set.thy Integ/IntArith.thy Integ/IntDef.thy Integ/IntDiv.thy		\
   Integ/NatBin.thy Integ/NatSimprocs.thy Integ/Numeral.thy			\
   Integ/Parity.thy Integ/Presburger.thy Integ/cooper_dec.ML			\
-  Integ/cooper_proof.ML Integ/int_arith1.ML Integ/int_factor_simprocs.ML	\
+  Integ/cooper_proof.ML Integ/reflected_presburger.ML \
+  Integ/reflected_cooper.ML Integ/int_arith1.ML Integ/int_factor_simprocs.ML	\
   Integ/nat_simprocs.ML Integ/presburger.ML Integ/qelim.ML LOrder.thy		\
   Lattice_Locales.thy List.ML List.thy Main.ML Main.thy Map.thy		\
   Nat.ML Nat.thy NatArith.thy OrderedGroup.ML OrderedGroup.thy			\
@@ -96,7 +98,7 @@
   Tools/ATP/recon_order_clauses.ML Tools/ATP/recon_parse.ML			\
   Tools/ATP/recon_prelim.ML Tools/ATP/recon_transfer_proof.ML			\
   Tools/ATP/recon_translate_proof.ML Tools/ATP/res_clasimpset.ML		\
-  Tools/ATP/watcher.ML 					\
+  Tools/ATP/watcher.ML 	Tools/comm_ring.ML				\
   Tools/datatype_abs_proofs.ML Tools/datatype_aux.ML				\
   Tools/datatype_codegen.ML Tools/datatype_package.ML				\
   Tools/datatype_prop.ML Tools/datatype_realizer.ML				\
@@ -584,13 +586,15 @@
 HOL-ex: HOL $(LOG)/HOL-ex.gz
 
 $(LOG)/HOL-ex.gz: $(OUT)/HOL ex/Antiquote.thy \
-  ex/BT.thy ex/BinEx.thy ex/Higher_Order_Logic.thy \
+  ex/BT.thy ex/BinEx.thy ex/Commutative_RingEx.thy \
+  ex/Commutative_Ring_Complete.thy ex/Higher_Order_Logic.thy \
   ex/Hilbert_Classical.thy ex/InSort.thy \
   ex/InductiveInvariant.thy  ex/InductiveInvariant_examples.thy\
   ex/Intuitionistic.thy \
   ex/Lagrange.thy ex/Locales.thy ex/MergeSort.thy \
   ex/MT.ML ex/MT.thy ex/MonoidGroup.thy ex/Multiquote.thy \
-  ex/NatSum.thy ex/PER.thy ex/PresburgerEx.thy ex/Primrec.thy ex/Puzzle.thy \
+  ex/NatSum.thy ex/PER.thy ex/PresburgerEx.thy ex/Reflected_Presburger.thy \
+  ex/Primrec.thy ex/Puzzle.thy \
   ex/Qsort.thy ex/Quickcheck_Examples.thy ex/ROOT.ML ex/Recdefs.thy ex/Records.thy \
   ex/Refute_Examples.thy \
   ex/StringEx.thy ex/SVC_Oracle.ML ex/SVC_Oracle.thy \
diff -r afaa031ed4da -r 105519771c67 src/HOL/Presburger.thy
--- a/src/HOL/Presburger.thy	Wed Sep 14 10:24:39 2005 +0200
+++ b/src/HOL/Presburger.thy	Wed Sep 14 17:25:52 2005 +0200
@@ -10,7 +10,8 @@
 
 theory Presburger
 imports NatSimprocs SetInterval
-uses ("cooper_dec.ML") ("cooper_proof.ML") ("qelim.ML") ("presburger.ML")
+uses ("cooper_dec.ML") ("cooper_proof.ML") ("qelim.ML") 
+	("reflected_presburger.ML") ("reflected_cooper.ML") ("presburger.ML")
 begin
 
 text {* Theorem for unitifying the coeffitients of @{text x} in an existential formula*}
@@ -982,8 +983,10 @@
   by (simp cong: conj_cong)
 
 use "cooper_dec.ML"
+use "reflected_presburger.ML" 
+use "reflected_cooper.ML"
 oracle
-  presburger_oracle ("term") = CooperDec.presburger_oracle
+  presburger_oracle ("term") = ReflectedCooper.presburger_oracle
 
 use "cooper_proof.ML"
 use "qelim.ML"
diff -r afaa031ed4da -r 105519771c67 src/HOL/Tools/Presburger/cooper_dec.ML
--- a/src/HOL/Tools/Presburger/cooper_dec.ML	Wed Sep 14 10:24:39 2005 +0200
+++ b/src/HOL/Tools/Presburger/cooper_dec.ML	Wed Sep 14 17:25:52 2005 +0200
@@ -44,7 +44,6 @@
   val evalc : term -> term
   val cooper_w : string list -> term -> (term option * term)
   val integer_qelim : Term.term -> Term.term
-  val presburger_oracle : theory -> term -> term
 end;
 
 structure  CooperDec : COOPER_DEC =
@@ -938,8 +937,4 @@
 
 val integer_qelim = simpl o evalc o (lift_qelim linform (cnnf posineq o evalc) cooper is_arith_rel) ; 
 
-fun presburger_oracle thy t =
-    if (!quick_and_dirty) 
-    then HOLogic.mk_Trueprop (HOLogic.mk_eq(t,integer_qelim t))
-    else error "Presburger oracle: not in quick_and_dirty mode"
 end;
diff -r afaa031ed4da -r 105519771c67 src/HOL/Tools/Presburger/reflected_cooper.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/Presburger/reflected_cooper.ML	Wed Sep 14 17:25:52 2005 +0200
@@ -0,0 +1,124 @@
+(* $Id$ *)
+(* The oracle for Presburger arithmetic based on the verified Code *)
+    (* in HOL/ex/Reflected_Presburger.thy *)
+
+structure ReflectedCooper =
+struct
+
+open Generated;
+(* pseudo reification : term -> intterm *)
+
+fun i_of_term vs t = 
+    case t of
+	Free(xn,xT) => (case assoc(vs,t) of 
+			   NONE   => error "Variable not found in the list!!"
+			 | SOME n => Var n)
+      | Const("0",iT) => Cst 0
+      | Const("1",iT) => Cst 1
+      | Bound i => Var (nat i)
+      | Const("uminus",_)$t' => Neg (i_of_term vs t')
+      | Const ("op +",_)$t1$t2 => Add (i_of_term vs t1,i_of_term vs t2)
+      | Const ("op -",_)$t1$t2 => Sub (i_of_term vs t1,i_of_term vs t2)
+      | Const ("op *",_)$t1$t2 => Mult (i_of_term vs t1,i_of_term vs t2)
+      | Const ("Numeral.number_of",_)$t' => Cst (HOLogic.dest_binum t')
+      | _ => error "i_of_term: unknown term";
+	
+
+(* pseudo reification : term -> QF *)
+fun qf_of_term vs t = 
+    case t of 
+	Const("True",_) => T
+      | Const("False",_) => F
+      | Const("op <",_)$t1$t2 => Lt (i_of_term vs t1,i_of_term vs t2)
+      | Const("op <=",_)$t1$t2 => Le (i_of_term vs t1,i_of_term vs t2)
+      | Const ("Divides.op dvd",_)$t1$t2 => 
+	Divides(i_of_term vs t1,i_of_term vs t2)
+      | Const("op =",eqT)$t1$t2 => 
+	if (domain_type eqT = HOLogic.intT)
+	then let val i1 = i_of_term vs t1
+		 val i2 = i_of_term vs t2
+	     in	Eq (i1,i2)
+	     end 
+	else Equ(qf_of_term vs t1,qf_of_term vs t2)
+      | Const("op &",_)$t1$t2 => And(qf_of_term vs t1,qf_of_term vs t2)
+      | Const("op |",_)$t1$t2 => Or(qf_of_term vs t1,qf_of_term vs t2)
+      | Const("op -->",_)$t1$t2 => Imp(qf_of_term vs t1,qf_of_term vs t2)
+      | Const("Not",_)$t' => NOT(qf_of_term vs t')
+      | Const("Ex",_)$Abs(xn,xT,p) => 
+	QEx(qf_of_term (map (fn(v,n) => (v,n + 1)) vs) p)
+      | Const("All",_)$Abs(xn,xT,p) => 
+	QAll(qf_of_term (map (fn(v,n) => (v,n + 1)) vs) p)
+      | _ => error "qf_of_term : unknown term!";
+
+(*
+fun parse s = term_of (read_cterm (sign_of Main.thy) (s,HOLogic.boolT));
+
+val t = parse "ALL (i::int) (j::int). i < 8* j --> (i - 1 = j + 3 + 2*j) & (j <= -i + k ) | 4 = i | 5 dvd i";
+*)
+fun zip [] [] = []
+  | zip (x::xs) (y::ys) = (x,y)::(zip xs ys);
+
+
+fun start_vs t =
+    let val fs = term_frees t
+    in zip fs (map nat (0 upto (length fs - 1)))
+    end ;
+
+(* transform intterm and QF back to terms *)
+val iT = HOLogic.intT;
+val bT = HOLogic.boolT;
+fun myassoc2 l v =
+    case l of
+	[] => NONE
+      | (x,v')::xs => if v = v' then SOME x
+		      else myassoc2 xs v;
+
+fun term_of_i vs t =
+    case t of 
+	Cst i => CooperDec.mk_numeral i
+      | Var n => valOf (myassoc2 vs n)
+      | Neg t' => Const("uminus",iT --> iT)$(term_of_i vs t')
+      | Add(t1,t2) => Const("op +",[iT,iT] ---> iT)$
+			   (term_of_i vs t1)$(term_of_i vs t2)
+      | Sub(t1,t2) => Const("op -",[iT,iT] ---> iT)$
+			   (term_of_i vs t1)$(term_of_i vs t2)
+      | Mult(t1,t2) => Const("op *",[iT,iT] ---> iT)$
+			   (term_of_i vs t1)$(term_of_i vs t2);
+
+fun term_of_qf vs t = 
+    case t of 
+	T => HOLogic.true_const 
+      | F => HOLogic.false_const
+      | Lt(t1,t2) => Const("op <",[iT,iT] ---> bT)$
+			   (term_of_i vs t1)$(term_of_i vs t2)
+      | Le(t1,t2) => Const("op <=",[iT,iT] ---> bT)$
+			  (term_of_i vs t1)$(term_of_i vs t2)
+      | Gt(t1,t2) => Const("op <",[iT,iT] ---> bT)$
+			   (term_of_i vs t2)$(term_of_i vs t1)
+      | Ge(t1,t2) => Const("op <=",[iT,iT] ---> bT)$
+			  (term_of_i vs t2)$(term_of_i vs t1)
+      | Eq(t1,t2) => Const("op =",[iT,iT] ---> bT)$
+			   (term_of_i vs t1)$(term_of_i vs t2)
+      | Divides(t1,t2) => Const("Divides.op dvd",[iT,iT] ---> bT)$
+			       (term_of_i vs t1)$(term_of_i vs t2)
+      | NOT t' => HOLogic.Not$(term_of_qf vs t')
+      | And(t1,t2) => HOLogic.conj$(term_of_qf vs t1)$(term_of_qf vs t2)
+      | Or(t1,t2) => HOLogic.disj$(term_of_qf vs t1)$(term_of_qf vs t2)
+      | Imp(t1,t2) => HOLogic.imp$(term_of_qf vs t1)$(term_of_qf vs t2)
+      | Equ(t1,t2) => (HOLogic.eq_const bT)$(term_of_qf vs t1)$
+					   (term_of_qf vs t2)
+      | _ => error "If this is raised, Isabelle/HOL or generate_code is inconsistent!";
+
+(* The oracle *)
+ exception COOPER; 
+
+fun presburger_oracle thy t =
+    let val vs = start_vs t
+	val result = lift_un (term_of_qf vs) (pa (qf_of_term vs t))
+    in 
+    case result of 
+	None => raise COOPER
+      | Some t' => HOLogic.mk_Trueprop (HOLogic.mk_eq(t,t'))
+    end ;
+ 
+end;
\ No newline at end of file
diff -r afaa031ed4da -r 105519771c67 src/HOL/Tools/Presburger/reflected_presburger.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/Presburger/reflected_presburger.ML	Wed Sep 14 17:25:52 2005 +0200
@@ -0,0 +1,2172 @@
+(* $Id$ *)
+
+    (* Caution: This file should not be modified. *)
+    (* It is autmatically generated from HOL/ex/Reflected_Presburger.thy *)
+fun nat i = if i < 0 then 0 else i;
+structure Generated =
+struct
+
+datatype intterm = Cst of int | Var of int | Neg of intterm
+  | Add of intterm * intterm | Sub of intterm * intterm
+  | Mult of intterm * intterm;
+
+datatype QF = Lt of intterm * intterm | Gt of intterm * intterm
+  | Le of intterm * intterm | Ge of intterm * intterm | Eq of intterm * intterm
+  | Divides of intterm * intterm | T | F | NOT of QF | And of QF * QF
+  | Or of QF * QF | Imp of QF * QF | Equ of QF * QF | QAll of QF | QEx of QF;
+
+datatype 'a option = None | Some of 'a;
+
+fun lift_un c None = None
+  | lift_un c (Some p) = Some (c p);
+
+fun lift_bin (c, (Some a, Some b)) = Some (c a b)
+  | lift_bin (c, (None, y)) = None
+  | lift_bin (c, (Some y, None)) = None;
+
+fun lift_qe qe None = None
+  | lift_qe qe (Some p) = qe p;
+
+fun qelim (qe, QAll p) = lift_un NOT (lift_qe qe (lift_un NOT (qelim (qe, p))))
+  | qelim (qe, QEx p) = lift_qe qe (qelim (qe, p))
+  | qelim (qe, And (p, q)) =
+    lift_bin ((fn x => fn xa => And (x, xa)), (qelim (qe, p), qelim (qe, q)))
+  | qelim (qe, Or (p, q)) =
+    lift_bin ((fn x => fn xa => Or (x, xa)), (qelim (qe, p), qelim (qe, q)))
+  | qelim (qe, Imp (p, q)) =
+    lift_bin ((fn x => fn xa => Imp (x, xa)), (qelim (qe, p), qelim (qe, q)))
+  | qelim (qe, Equ (p, q)) =
+    lift_bin ((fn x => fn xa => Equ (x, xa)), (qelim (qe, p), qelim (qe, q)))
+  | qelim (qe, NOT p) = lift_un NOT (qelim (qe, p))
+  | qelim (qe, Lt (w, x)) = Some (Lt (w, x))
+  | qelim (qe, Gt (y, z)) = Some (Gt (y, z))
+  | qelim (qe, Le (aa, ab)) = Some (Le (aa, ab))
+  | qelim (qe, Ge (ac, ad)) = Some (Ge (ac, ad))
+  | qelim (qe, Eq (ae, af)) = Some (Eq (ae, af))
+  | qelim (qe, Divides (ag, ah)) = Some (Divides (ag, ah))
+  | qelim (qe, T) = Some T
+  | qelim (qe, F) = Some F;
+
+fun lin_mul (c, Cst i) = Cst (c * i)
+  | lin_mul (c, Add (Mult (Cst c', Var n), r)) =
+    (if (c = 0) then Cst 0
+      else Add (Mult (Cst (c * c'), Var n), lin_mul (c, r)));
+
+fun op_60_def0 m n = ((m) < (n));
+
+fun op_60_61_def0 m n = not (op_60_def0 n m);
+
+fun lin_add (Add (Mult (Cst c1, Var n1), r1), Add (Mult (Cst c2, Var n2), r2)) =
+    (if (n1 = n2)
+      then let val c = Cst (c1 + c2)
+           in (if ((c1 + c2) = 0) then lin_add (r1, r2)
+                else Add (Mult (c, Var n1), lin_add (r1, r2)))
+           end
+      else (if op_60_61_def0 n1 n2
+             then Add (Mult (Cst c1, Var n1),
+                        lin_add (r1, Add (Mult (Cst c2, Var n2), r2)))
+             else Add (Mult (Cst c2, Var n2),
+                        lin_add (Add (Mult (Cst c1, Var n1), r1), r2))))
+  | lin_add (Add (Mult (Cst c1, Var n1), r1), Cst b) =
+    Add (Mult (Cst c1, Var n1), lin_add (r1, Cst b))
+  | lin_add (Cst x, Add (Mult (Cst c2, Var n2), r2)) =
+    Add (Mult (Cst c2, Var n2), lin_add (Cst x, r2))
+  | lin_add (Cst b1, Cst b2) = Cst (b1 + b2);
+
+fun lin_neg i = lin_mul (~1, i);
+
+fun linearize (Cst b) = Some (Cst b)
+  | linearize (Var n) = Some (Add (Mult (Cst 1, Var n), Cst 0))
+  | linearize (Neg i) = lift_un lin_neg (linearize i)
+  | linearize (Add (i, j)) =
+    lift_bin ((fn x => fn y => lin_add (x, y)), (linearize i, linearize j))
+  | linearize (Sub (i, j)) =
+    lift_bin
+      ((fn x => fn y => lin_add (x, lin_neg y)), (linearize i, linearize j))
+  | linearize (Mult (i, j)) =
+    (case linearize i of None => None
+      | Some x =>
+          (case x of
+            Cst xa =>
+              (case linearize j of None => None
+                | Some x => Some (lin_mul (xa, x)))
+            | Var xa =>
+                (case linearize j of None => None
+                  | Some xa =>
+                      (case xa of Cst xa => Some (lin_mul (xa, x))
+                        | Var xa => None | Neg xa => None | Add (xa, xb) => None
+                        | Sub (xa, xb) => None | Mult (xa, xb) => None))
+            | Neg xa =>
+                (case linearize j of None => None
+                  | Some xa =>
+                      (case xa of Cst xa => Some (lin_mul (xa, x))
+                        | Var xa => None | Neg xa => None | Add (xa, xb) => None
+                        | Sub (xa, xb) => None | Mult (xa, xb) => None))
+            | Add (xa, xb) =>
+                (case linearize j of None => None
+                  | Some xa =>
+                      (case xa of Cst xa => Some (lin_mul (xa, x))
+                        | Var xa => None | Neg xa => None | Add (xa, xb) => None
+                        | Sub (xa, xb) => None | Mult (xa, xb) => None))
+            | Sub (xa, xb) =>
+                (case linearize j of None => None
+                  | Some xa =>
+                      (case xa of Cst xa => Some (lin_mul (xa, x))
+                        | Var xa => None | Neg xa => None | Add (xa, xb) => None
+                        | Sub (xa, xb) => None | Mult (xa, xb) => None))
+            | Mult (xa, xb) =>
+                (case linearize j of None => None
+                  | Some xa =>
+                      (case xa of Cst xa => Some (lin_mul (xa, x))
+                        | Var xa => None | Neg xa => None | Add (xa, xb) => None
+                        | Sub (xa, xb) => None | Mult (xa, xb) => None))));
+
+fun linform (Le (it1, it2)) =
+    lift_bin
+      ((fn x => fn y => Le (lin_add (x, lin_neg y), Cst 0)),
+        (linearize it1, linearize it2))
+  | linform (Eq (it1, it2)) =
+    lift_bin
+      ((fn x => fn y => Eq (lin_add (x, lin_neg y), Cst 0)),
+        (linearize it1, linearize it2))
+  | linform (Divides (d, t)) =
+    (case linearize d of None => None
+      | Some x =>
+          (case x of
+            Cst xa =>
+              (if (xa = 0) then None
+                else (case linearize t of None => None
+                       | Some xa => Some (Divides (x, xa))))
+            | Var xa => None | Neg xa => None | Add (xa, xb) => None
+            | Sub (xa, xb) => None | Mult (xa, xb) => None))
+  | linform T = Some T
+  | linform F = Some F
+  | linform (NOT p) = lift_un NOT (linform p)
+  | linform (And (p, q)) =
+    lift_bin ((fn f => fn g => And (f, g)), (linform p, linform q))
+  | linform (Or (p, q)) =
+    lift_bin ((fn f => fn g => Or (f, g)), (linform p, linform q));
+
+fun nnf (Lt (it1, it2)) = Le (Sub (it1, it2), Cst (~ 1))
+  | nnf (Gt (it1, it2)) = Le (Sub (it2, it1), Cst (~ 1))
+  | nnf (Le (it1, it2)) = Le (it1, it2)
+  | nnf (Ge (it1, it2)) = Le (it2, it1)
+  | nnf (Eq (it1, it2)) = Eq (it2, it1)
+  | nnf (Divides (d, t)) = Divides (d, t)
+  | nnf T = T
+  | nnf F = F
+  | nnf (And (p, q)) = And (nnf p, nnf q)
+  | nnf (Or (p, q)) = Or (nnf p, nnf q)
+  | nnf (Imp (p, q)) = Or (nnf (NOT p), nnf q)
+  | nnf (Equ (p, q)) = Or (And (nnf p, nnf q), And (nnf (NOT p), nnf (NOT q)))
+  | nnf (NOT (Lt (it1, it2))) = Le (it2, it1)
+  | nnf (NOT (Gt (it1, it2))) = Le (it1, it2)
+  | nnf (NOT (Le (it1, it2))) = Le (Sub (it2, it1), Cst (~ 1))
+  | nnf (NOT (Ge (it1, it2))) = Le (Sub (it1, it2), Cst (~ 1))
+  | nnf (NOT (Eq (it1, it2))) = NOT (Eq (it1, it2))
+  | nnf (NOT (Divides (d, t))) = NOT (Divides (d, t))
+  | nnf (NOT T) = F
+  | nnf (NOT F) = T
+  | nnf (NOT (NOT p)) = nnf p
+  | nnf (NOT (And (p, q))) = Or (nnf (NOT p), nnf (NOT q))
+  | nnf (NOT (Or (p, q))) = And (nnf (NOT p), nnf (NOT q))
+  | nnf (NOT (Imp (p, q))) = And (nnf p, nnf (NOT q))
+  | nnf (NOT (Equ (p, q))) =
+    Or (And (nnf p, nnf (NOT q)), And (nnf (NOT p), nnf q));
+
+fun op_45_def2 z w = (z + ~ w);
+
+fun op_45_def0 m n = nat (op_45_def2 (m) (n));
+
+val id_1_def0 : int = (0 + 1);
+
+fun decrvarsI (Cst i) = Cst i
+  | decrvarsI (Var n) = Var (op_45_def0 n id_1_def0)
+  | decrvarsI (Neg a) = Neg (decrvarsI a)
+  | decrvarsI (Add (a, b)) = Add (decrvarsI a, decrvarsI b)
+  | decrvarsI (Sub (a, b)) = Sub (decrvarsI a, decrvarsI b)
+  | decrvarsI (Mult (a, b)) = Mult (decrvarsI a, decrvarsI b);
+
+fun decrvars (Lt (a, b)) = Lt (decrvarsI a, decrvarsI b)
+  | decrvars (Gt (a, b)) = Gt (decrvarsI a, decrvarsI b)
+  | decrvars (Le (a, b)) = Le (decrvarsI a, decrvarsI b)
+  | decrvars (Ge (a, b)) = Ge (decrvarsI a, decrvarsI b)
+  | decrvars (Eq (a, b)) = Eq (decrvarsI a, decrvarsI b)
+  | decrvars (Divides (a, b)) = Divides (decrvarsI a, decrvarsI b)
+  | decrvars T = T
+  | decrvars F = F
+  | decrvars (NOT p) = NOT (decrvars p)
+  | decrvars (And (p, q)) = And (decrvars p, decrvars q)
+  | decrvars (Or (p, q)) = Or (decrvars p, decrvars q)
+  | decrvars (Imp (p, q)) = Imp (decrvars p, decrvars q)
+  | decrvars (Equ (p, q)) = Equ (decrvars p, decrvars q);
+
+fun op_64 [] ys = ys
+  | op_64 (x :: xs) ys = (x :: op_64 xs ys);
+
+fun map f [] = []
+  | map f (x :: xs) = (f x :: map f xs);
+
+fun iupto (i, j) = (if (j < i) then [] else (i :: iupto ((i + 1), j)));
+
+fun all_sums (j, []) = []
+  | all_sums (j, (i :: is)) =
+    op_64 (map (fn x => lin_add (i, Cst x)) (iupto (1, j))) (all_sums (j, is));
+
+fun split x = (fn p => x (fst p) (snd p));
+
+fun negateSnd x = split (fn q => fn r => (q, ~ r)) x;
+
+fun adjust b =
+  (fn (q, r) =>
+    (if (0 <= op_45_def2 r b) then (((2 * q) + 1), op_45_def2 r b)
+      else ((2 * q), r)));
+
+fun negDivAlg (a, b) =
+    (if ((0 <= (a + b)) orelse (b <= 0)) then (~1, (a + b))
+      else adjust b (negDivAlg (a, (2 * b))));
+
+fun posDivAlg (a, b) =
+    (if ((a < b) orelse (b <= 0)) then (0, a)
+      else adjust b (posDivAlg (a, (2 * b))));
+
+fun divAlg x =
+  split (fn a => fn b =>
+          (if (0 <= a)
+            then (if (0 <= 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)))))
+    x;
+
+fun op_mod_def1 a b = snd (divAlg (a, b));
+
+fun op_dvd m n = (op_mod_def1 n m = 0);
+
+fun psimpl (Le (l, r)) =
+    (case lift_bin
+            ((fn x => fn y => lin_add (x, lin_neg y)),
+              (linearize l, linearize r)) of
+      None => Le (l, r)
+      | Some x =>
+          (case x of Cst xa => (if (xa <= 0) then T else F)
+            | Var xa => Le (x, Cst 0) | Neg xa => Le (x, Cst 0)
+            | Add (xa, xb) => Le (x, Cst 0) | Sub (xa, xb) => Le (x, Cst 0)
+            | Mult (xa, xb) => Le (x, Cst 0)))
+  | psimpl (Eq (l, r)) =
+    (case lift_bin
+            ((fn x => fn y => lin_add (x, lin_neg y)),
+              (linearize l, linearize r)) of
+      None => Eq (l, r)
+      | Some x =>
+          (case x of Cst xa => (if (xa = 0) then T else F)
+            | Var xa => Eq (x, Cst 0) | Neg xa => Eq (x, Cst 0)
+            | Add (xa, xb) => Eq (x, Cst 0) | Sub (xa, xb) => Eq (x, Cst 0)
+            | Mult (xa, xb) => Eq (x, Cst 0)))
+  | psimpl (Divides (Cst d, t)) =
+    (case linearize t of None => Divides (Cst d, t)
+      | Some x =>
+          (case x of Cst xa => (if op_dvd d xa then T else F)
+            | Var xa => Divides (Cst d, x) | Neg xa => Divides (Cst d, x)
+            | Add (xa, xb) => Divides (Cst d, x)
+            | Sub (xa, xb) => Divides (Cst d, x)
+            | Mult (xa, xb) => Divides (Cst d, x)))
+  | psimpl (Equ (p, q)) =
+    let val p' = psimpl p; val q' = psimpl q
+    in (case p' of
+         Lt (x, xa) =>
+           (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+             | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+             | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+             | T => p' | F => NOT p' | NOT x => Equ (p', q')
+             | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+             | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+             | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Gt (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Le (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Ge (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Eq (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Divides (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | T => q'
+         | F => (case q' of Lt (x, xa) => NOT q' | Gt (x, xa) => NOT q'
+                  | Le (x, xa) => NOT q' | Ge (x, xa) => NOT q'
+                  | Eq (x, xa) => NOT q' | Divides (x, xa) => NOT q' | T => F
+                  | F => T | NOT x => x | And (x, xa) => NOT q'
+                  | Or (x, xa) => NOT q' | Imp (x, xa) => NOT q'
+                  | Equ (x, xa) => NOT q' | QAll x => NOT q' | QEx x => NOT q')
+         | NOT x =>
+             (case q' of Lt (xa, xb) => Equ (p', q')
+               | Gt (xa, xb) => Equ (p', q') | Le (xa, xb) => Equ (p', q')
+               | Ge (xa, xb) => Equ (p', q') | Eq (xa, xb) => Equ (p', q')
+               | Divides (xa, xb) => Equ (p', q') | T => p' | F => x
+               | NOT xa => Equ (x, xa) | And (xa, xb) => Equ (p', q')
+               | Or (xa, xb) => Equ (p', q') | Imp (xa, xb) => Equ (p', q')
+               | Equ (xa, xb) => Equ (p', q') | QAll xa => Equ (p', q')
+               | QEx xa => Equ (p', q'))
+         | And (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Or (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Imp (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | Equ (x, xa) =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | QAll x =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q'))
+         | QEx x =>
+             (case q' of Lt (x, xa) => Equ (p', q') | Gt (x, xa) => Equ (p', q')
+               | Le (x, xa) => Equ (p', q') | Ge (x, xa) => Equ (p', q')
+               | Eq (x, xa) => Equ (p', q') | Divides (x, xa) => Equ (p', q')
+               | T => p' | F => NOT p' | NOT x => Equ (p', q')
+               | And (x, xa) => Equ (p', q') | Or (x, xa) => Equ (p', q')
+               | Imp (x, xa) => Equ (p', q') | Equ (x, xa) => Equ (p', q')
+               | QAll x => Equ (p', q') | QEx x => Equ (p', q')))
+    end
+  | psimpl (NOT p) =
+    let val p' = psimpl p
+    in (case p' of Lt (x, xa) => NOT p' | Gt (x, xa) => NOT p'
+         | Le (x, xa) => NOT p' | Ge (x, xa) => NOT p' | Eq (x, xa) => NOT p'
+         | Divides (x, xa) => NOT p' | T => F | F => T | NOT x => x
+         | And (x, xa) => NOT p' | Or (x, xa) => NOT p' | Imp (x, xa) => NOT p'
+         | Equ (x, xa) => NOT p' | QAll x => NOT p' | QEx x => NOT p')
+    end
+  | psimpl (Lt (u, v)) = Lt (u, v)
+  | psimpl (Gt (w, x)) = Gt (w, x)
+  | psimpl (Ge (aa, ab)) = Ge (aa, ab)
+  | psimpl (Divides (Var bp, af)) = Divides (Var bp, af)
+  | psimpl (Divides (Neg bq, af)) = Divides (Neg bq, af)
+  | psimpl (Divides (Add (br, bs), af)) = Divides (Add (br, bs), af)
+  | psimpl (Divides (Sub (bt, bu), af)) = Divides (Sub (bt, bu), af)
+  | psimpl (Divides (Mult (bv, bw), af)) = Divides (Mult (bv, bw), af)
+  | psimpl T = T
+  | psimpl F = F
+  | psimpl (QAll ap) = QAll ap
+  | psimpl (QEx aq) = QEx aq
+  | psimpl (And (p, q)) =
+    let val p' = psimpl p
+    in (case p' of
+         Lt (x, xa) =>
+           let val q' = psimpl q
+           in (case q' of Lt (x, xa) => And (p', q')
+                | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                | Divides (x, xa) => And (p', q') | T => p' | F => F
+                | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                | QEx x => And (p', q'))
+           end
+         | Gt (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | Le (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | Ge (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | Eq (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | Divides (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | T => psimpl q | F => F
+         | NOT x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | And (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | Or (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | Imp (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | Equ (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | QAll x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end
+         | QEx x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => And (p', q')
+                  | Gt (x, xa) => And (p', q') | Le (x, xa) => And (p', q')
+                  | Ge (x, xa) => And (p', q') | Eq (x, xa) => And (p', q')
+                  | Divides (x, xa) => And (p', q') | T => p' | F => F
+                  | NOT x => And (p', q') | And (x, xa) => And (p', q')
+                  | Or (x, xa) => And (p', q') | Imp (x, xa) => And (p', q')
+                  | Equ (x, xa) => And (p', q') | QAll x => And (p', q')
+                  | QEx x => And (p', q'))
+             end)
+    end
+  | psimpl (Or (p, q)) =
+    let val p' = psimpl p
+    in (case p' of
+         Lt (x, xa) =>
+           let val q' = psimpl q
+           in (case q' of Lt (x, xa) => Or (p', q') | Gt (x, xa) => Or (p', q')
+                | Le (x, xa) => Or (p', q') | Ge (x, xa) => Or (p', q')
+                | Eq (x, xa) => Or (p', q') | Divides (x, xa) => Or (p', q')
+                | T => T | F => p' | NOT x => Or (p', q')
+                | And (x, xa) => Or (p', q') | Or (x, xa) => Or (p', q')
+                | Imp (x, xa) => Or (p', q') | Equ (x, xa) => Or (p', q')
+                | QAll x => Or (p', q') | QEx x => Or (p', q'))
+           end
+         | Gt (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | Le (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | Ge (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | Eq (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | Divides (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | T => T | F => psimpl q
+         | NOT x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | And (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | Or (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | Imp (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | Equ (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | QAll x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end
+         | QEx x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Or (p', q')
+                  | Gt (x, xa) => Or (p', q') | Le (x, xa) => Or (p', q')
+                  | Ge (x, xa) => Or (p', q') | Eq (x, xa) => Or (p', q')
+                  | Divides (x, xa) => Or (p', q') | T => T | F => p'
+                  | NOT x => Or (p', q') | And (x, xa) => Or (p', q')
+                  | Or (x, xa) => Or (p', q') | Imp (x, xa) => Or (p', q')
+                  | Equ (x, xa) => Or (p', q') | QAll x => Or (p', q')
+                  | QEx x => Or (p', q'))
+             end)
+    end
+  | psimpl (Imp (p, q)) =
+    let val p' = psimpl p
+    in (case p' of
+         Lt (x, xa) =>
+           let val q' = psimpl q
+           in (case q' of Lt (x, xa) => Imp (p', q')
+                | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                | QEx x => Imp (p', q'))
+           end
+         | Gt (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | Le (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | Ge (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | Eq (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | Divides (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | T => psimpl q | F => T
+         | NOT x =>
+             let val q' = psimpl q
+             in (case q' of Lt (xa, xb) => Or (x, q')
+                  | Gt (xa, xb) => Or (x, q') | Le (xa, xb) => Or (x, q')
+                  | Ge (xa, xb) => Or (x, q') | Eq (xa, xb) => Or (x, q')
+                  | Divides (xa, xb) => Or (x, q') | T => T | F => x
+                  | NOT xa => Or (x, q') | And (xa, xb) => Or (x, q')
+                  | Or (xa, xb) => Or (x, q') | Imp (xa, xb) => Or (x, q')
+                  | Equ (xa, xb) => Or (x, q') | QAll xa => Or (x, q')
+                  | QEx xa => Or (x, q'))
+             end
+         | And (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | Or (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | Imp (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | Equ (x, xa) =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | QAll x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end
+         | QEx x =>
+             let val q' = psimpl q
+             in (case q' of Lt (x, xa) => Imp (p', q')
+                  | Gt (x, xa) => Imp (p', q') | Le (x, xa) => Imp (p', q')
+                  | Ge (x, xa) => Imp (p', q') | Eq (x, xa) => Imp (p', q')
+                  | Divides (x, xa) => Imp (p', q') | T => T | F => NOT p'
+                  | NOT x => Imp (p', q') | And (x, xa) => Imp (p', q')
+                  | Or (x, xa) => Imp (p', q') | Imp (x, xa) => Imp (p', q')
+                  | Equ (x, xa) => Imp (p', q') | QAll x => Imp (p', q')
+                  | QEx x => Imp (p', q'))
+             end)
+    end;
+
+fun subst_it i (Cst b) = Cst b
+  | subst_it i (Var n) = (if (n = 0) then i else Var n)
+  | subst_it i (Neg it) = Neg (subst_it i it)
+  | subst_it i (Add (it1, it2)) = Add (subst_it i it1, subst_it i it2)
+  | subst_it i (Sub (it1, it2)) = Sub (subst_it i it1, subst_it i it2)
+  | subst_it i (Mult (it1, it2)) = Mult (subst_it i it1, subst_it i it2);
+
+fun subst_p i (Le (it1, it2)) = Le (subst_it i it1, subst_it i it2)
+  | subst_p i (Lt (it1, it2)) = Lt (subst_it i it1, subst_it i it2)
+  | subst_p i (Ge (it1, it2)) = Ge (subst_it i it1, subst_it i it2)
+  | subst_p i (Gt (it1, it2)) = Gt (subst_it i it1, subst_it i it2)
+  | subst_p i (Eq (it1, it2)) = Eq (subst_it i it1, subst_it i it2)
+  | subst_p i (Divides (d, t)) = Divides (subst_it i d, subst_it i t)
+  | subst_p i T = T
+  | subst_p i F = F
+  | subst_p i (And (p, q)) = And (subst_p i p, subst_p i q)
+  | subst_p i (Or (p, q)) = Or (subst_p i p, subst_p i q)
+  | subst_p i (Imp (p, q)) = Imp (subst_p i p, subst_p i q)
+  | subst_p i (Equ (p, q)) = Equ (subst_p i p, subst_p i q)
+  | subst_p i (NOT p) = NOT (subst_p i p);
+
+fun explode_disj ([], p) = F
+  | explode_disj ((i :: is), p) =
+    let val pi = psimpl (subst_p i p)
+    in (case pi of
+         Lt (x, xa) =>
+           let val r = explode_disj (is, p)
+           in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                | T => T | F => pi | NOT x => Or (pi, r)
+                | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+           end
+         | Gt (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | Le (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | Ge (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | Eq (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | Divides (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | T => T | F => explode_disj (is, p)
+         | NOT x =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | And (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | Or (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | Imp (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | Equ (x, xa) =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | QAll x =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end
+         | QEx x =>
+             let val r = explode_disj (is, p)
+             in (case r of Lt (x, xa) => Or (pi, r) | Gt (x, xa) => Or (pi, r)
+                  | Le (x, xa) => Or (pi, r) | Ge (x, xa) => Or (pi, r)
+                  | Eq (x, xa) => Or (pi, r) | Divides (x, xa) => Or (pi, r)
+                  | T => T | F => pi | NOT x => Or (pi, r)
+                  | And (x, xa) => Or (pi, r) | Or (x, xa) => Or (pi, r)
+                  | Imp (x, xa) => Or (pi, r) | Equ (x, xa) => Or (pi, r)
+                  | QAll x => Or (pi, r) | QEx x => Or (pi, r))
+             end)
+    end;
+
+fun minusinf (And (p, q)) = And (minusinf p, minusinf q)
+  | minusinf (Or (p, q)) = Or (minusinf p, minusinf q)
+  | minusinf (Lt (u, v)) = Lt (u, v)
+  | minusinf (Gt (w, x)) = Gt (w, x)
+  | minusinf (Le (Cst bo, z)) = Le (Cst bo, z)
+  | minusinf (Le (Var bp, z)) = Le (Var bp, z)
+  | minusinf (Le (Neg bq, z)) = Le (Neg bq, z)
+  | minusinf (Le (Add (Cst cg, bs), z)) = Le (Add (Cst cg, bs), z)
+  | minusinf (Le (Add (Var ch, bs), z)) = Le (Add (Var ch, bs), z)
+  | minusinf (Le (Add (Neg ci, bs), z)) = Le (Add (Neg ci, bs), z)
+  | minusinf (Le (Add (Add (cj, ck), bs), z)) = Le (Add (Add (cj, ck), bs), z)
+  | minusinf (Le (Add (Sub (cl, cm), bs), z)) = Le (Add (Sub (cl, cm), bs), z)
+  | minusinf (Le (Add (Mult (Cst cy, Cst dq), bs), z)) =
+    Le (Add (Mult (Cst cy, Cst dq), bs), z)
+  | minusinf (Le (Add (Mult (Cst cy, Var ei), bs), z)) =
+    (if (ei = 0) then (if (cy < 0) then F else T)
+      else Le (Add (Mult (Cst cy, Var (op_45_def0 ei id_1_def0 + 1)), bs), z))
+  | minusinf (Le (Add (Mult (Cst cy, Neg ds), bs), z)) =
+    Le (Add (Mult (Cst cy, Neg ds), bs), z)
+  | minusinf (Le (Add (Mult (Cst cy, Add (dt, du)), bs), z)) =
+    Le (Add (Mult (Cst cy, Add (dt, du)), bs), z)
+  | minusinf (Le (Add (Mult (Cst cy, Sub (dv, dw)), bs), z)) =
+    Le (Add (Mult (Cst cy, Sub (dv, dw)), bs), z)
+  | minusinf (Le (Add (Mult (Cst cy, Mult (dx, dy)), bs), z)) =
+    Le (Add (Mult (Cst cy, Mult (dx, dy)), bs), z)
+  | minusinf (Le (Add (Mult (Var cz, co), bs), z)) =
+    Le (Add (Mult (Var cz, co), bs), z)
+  | minusinf (Le (Add (Mult (Neg da, co), bs), z)) =
+    Le (Add (Mult (Neg da, co), bs), z)
+  | minusinf (Le (Add (Mult (Add (db, dc), co), bs), z)) =
+    Le (Add (Mult (Add (db, dc), co), bs), z)
+  | minusinf (Le (Add (Mult (Sub (dd, de), co), bs), z)) =
+    Le (Add (Mult (Sub (dd, de), co), bs), z)
+  | minusinf (Le (Add (Mult (Mult (df, dg), co), bs), z)) =
+    Le (Add (Mult (Mult (df, dg), co), bs), z)
+  | minusinf (Le (Sub (bt, bu), z)) = Le (Sub (bt, bu), z)
+  | minusinf (Le (Mult (bv, bw), z)) = Le (Mult (bv, bw), z)
+  | minusinf (Ge (aa, ab)) = Ge (aa, ab)
+  | minusinf (Eq (Cst ek, ad)) = Eq (Cst ek, ad)
+  | minusinf (Eq (Var el, ad)) = Eq (Var el, ad)
+  | minusinf (Eq (Neg em, ad)) = Eq (Neg em, ad)
+  | minusinf (Eq (Add (Cst fc, eo), ad)) = Eq (Add (Cst fc, eo), ad)
+  | minusinf (Eq (Add (Var fd, eo), ad)) = Eq (Add (Var fd, eo), ad)
+  | minusinf (Eq (Add (Neg fe, eo), ad)) = Eq (Add (Neg fe, eo), ad)
+  | minusinf (Eq (Add (Add (ff, fg), eo), ad)) = Eq (Add (Add (ff, fg), eo), ad)
+  | minusinf (Eq (Add (Sub (fh, fi), eo), ad)) = Eq (Add (Sub (fh, fi), eo), ad)
+  | minusinf (Eq (Add (Mult (Cst fu, Cst gm), eo), ad)) =
+    Eq (Add (Mult (Cst fu, Cst gm), eo), ad)
+  | minusinf (Eq (Add (Mult (Cst fu, Var he), eo), ad)) =
+    (if (he = 0) then F
+      else Eq (Add (Mult (Cst fu, Var (op_45_def0 he id_1_def0 + 1)), eo), ad))
+  | minusinf (Eq (Add (Mult (Cst fu, Neg go), eo), ad)) =
+    Eq (Add (Mult (Cst fu, Neg go), eo), ad)
+  | minusinf (Eq (Add (Mult (Cst fu, Add (gp, gq)), eo), ad)) =
+    Eq (Add (Mult (Cst fu, Add (gp, gq)), eo), ad)
+  | minusinf (Eq (Add (Mult (Cst fu, Sub (gr, gs)), eo), ad)) =
+    Eq (Add (Mult (Cst fu, Sub (gr, gs)), eo), ad)
+  | minusinf (Eq (Add (Mult (Cst fu, Mult (gt, gu)), eo), ad)) =
+    Eq (Add (Mult (Cst fu, Mult (gt, gu)), eo), ad)
+  | minusinf (Eq (Add (Mult (Var fv, fk), eo), ad)) =
+    Eq (Add (Mult (Var fv, fk), eo), ad)
+  | minusinf (Eq (Add (Mult (Neg fw, fk), eo), ad)) =
+    Eq (Add (Mult (Neg fw, fk), eo), ad)
+  | minusinf (Eq (Add (Mult (Add (fx, fy), fk), eo), ad)) =
+    Eq (Add (Mult (Add (fx, fy), fk), eo), ad)
+  | minusinf (Eq (Add (Mult (Sub (fz, ga), fk), eo), ad)) =
+    Eq (Add (Mult (Sub (fz, ga), fk), eo), ad)
+  | minusinf (Eq (Add (Mult (Mult (gb, gc), fk), eo), ad)) =
+    Eq (Add (Mult (Mult (gb, gc), fk), eo), ad)
+  | minusinf (Eq (Sub (ep, eq), ad)) = Eq (Sub (ep, eq), ad)
+  | minusinf (Eq (Mult (er, es), ad)) = Eq (Mult (er, es), ad)
+  | minusinf (Divides (ae, af)) = Divides (ae, af)
+  | minusinf T = T
+  | minusinf F = F
+  | minusinf (NOT (Lt (hg, hh))) = NOT (Lt (hg, hh))
+  | minusinf (NOT (Gt (hi, hj))) = NOT (Gt (hi, hj))
+  | minusinf (NOT (Le (hk, hl))) = NOT (Le (hk, hl))
+  | minusinf (NOT (Ge (hm, hn))) = NOT (Ge (hm, hn))
+  | minusinf (NOT (Eq (Cst ja, hp))) = NOT (Eq (Cst ja, hp))
+  | minusinf (NOT (Eq (Var jb, hp))) = NOT (Eq (Var jb, hp))
+  | minusinf (NOT (Eq (Neg jc, hp))) = NOT (Eq (Neg jc, hp))
+  | minusinf (NOT (Eq (Add (Cst js, je), hp))) = NOT (Eq (Add (Cst js, je), hp))
+  | minusinf (NOT (Eq (Add (Var jt, je), hp))) = NOT (Eq (Add (Var jt, je), hp))
+  | minusinf (NOT (Eq (Add (Neg ju, je), hp))) = NOT (Eq (Add (Neg ju, je), hp))
+  | minusinf (NOT (Eq (Add (Add (jv, jw), je), hp))) =
+    NOT (Eq (Add (Add (jv, jw), je), hp))
+  | minusinf (NOT (Eq (Add (Sub (jx, jy), je), hp))) =
+    NOT (Eq (Add (Sub (jx, jy), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Cst kk, Cst lc), je), hp))) =
+    NOT (Eq (Add (Mult (Cst kk, Cst lc), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Cst kk, Var lu), je), hp))) =
+    (if (lu = 0) then T
+      else NOT (Eq (Add (Mult (Cst kk, Var (op_45_def0 lu id_1_def0 + 1)), je),
+                     hp)))
+  | minusinf (NOT (Eq (Add (Mult (Cst kk, Neg le), je), hp))) =
+    NOT (Eq (Add (Mult (Cst kk, Neg le), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Cst kk, Add (lf, lg)), je), hp))) =
+    NOT (Eq (Add (Mult (Cst kk, Add (lf, lg)), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Cst kk, Sub (lh, li)), je), hp))) =
+    NOT (Eq (Add (Mult (Cst kk, Sub (lh, li)), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Cst kk, Mult (lj, lk)), je), hp))) =
+    NOT (Eq (Add (Mult (Cst kk, Mult (lj, lk)), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Var kl, ka), je), hp))) =
+    NOT (Eq (Add (Mult (Var kl, ka), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Neg km, ka), je), hp))) =
+    NOT (Eq (Add (Mult (Neg km, ka), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Add (kn, ko), ka), je), hp))) =
+    NOT (Eq (Add (Mult (Add (kn, ko), ka), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Sub (kp, kq), ka), je), hp))) =
+    NOT (Eq (Add (Mult (Sub (kp, kq), ka), je), hp))
+  | minusinf (NOT (Eq (Add (Mult (Mult (kr, ks), ka), je), hp))) =
+    NOT (Eq (Add (Mult (Mult (kr, ks), ka), je), hp))
+  | minusinf (NOT (Eq (Sub (jf, jg), hp))) = NOT (Eq (Sub (jf, jg), hp))
+  | minusinf (NOT (Eq (Mult (jh, ji), hp))) = NOT (Eq (Mult (jh, ji), hp))
+  | minusinf (NOT (Divides (hq, hr))) = NOT (Divides (hq, hr))
+  | minusinf (NOT T) = NOT T
+  | minusinf (NOT F) = NOT F
+  | minusinf (NOT (NOT hs)) = NOT (NOT hs)
+  | minusinf (NOT (And (ht, hu))) = NOT (And (ht, hu))
+  | minusinf (NOT (Or (hv, hw))) = NOT (Or (hv, hw))
+  | minusinf (NOT (Imp (hx, hy))) = NOT (Imp (hx, hy))
+  | minusinf (NOT (Equ (hz, ia))) = NOT (Equ (hz, ia))
+  | minusinf (NOT (QAll ib)) = NOT (QAll ib)
+  | minusinf (NOT (QEx ic)) = NOT (QEx ic)
+  | minusinf (Imp (al, am)) = Imp (al, am)
+  | minusinf (Equ (an, ao)) = Equ (an, ao)
+  | minusinf (QAll ap) = QAll ap
+  | minusinf (QEx aq) = QEx aq;
+
+fun abs i = (if (i < 0) then ~ i else i);
+
+fun op_div_def1 a b = fst (divAlg (a, b));
+
+fun op_mod_def0 m n = nat (op_mod_def1 (m) (n));
+
+fun ngcd (m, n) = (if (n = 0) then m else ngcd (n, op_mod_def0 m n));
+
+fun igcd x = split (fn a => fn b => (ngcd (nat (abs a), nat (abs b)))) x;
+
+fun ilcm a b = op_div_def1 (a * b) (igcd (a, b));
+
+fun divlcm (NOT p) = divlcm p
+  | divlcm (And (p, q)) = ilcm (divlcm p) (divlcm q)
+  | divlcm (Or (p, q)) = ilcm (divlcm p) (divlcm q)
+  | divlcm (Lt (u, v)) = 1
+  | divlcm (Gt (w, x)) = 1
+  | divlcm (Le (y, z)) = 1
+  | divlcm (Ge (aa, ab)) = 1
+  | divlcm (Eq (ac, ad)) = 1
+  | divlcm (Divides (Cst bo, Cst cg)) = 1
+  | divlcm (Divides (Cst bo, Var ch)) = 1
+  | divlcm (Divides (Cst bo, Neg ci)) = 1
+  | divlcm (Divides (Cst bo, Add (Cst cy, ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Var cz, ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Neg da, ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Add (db, dc), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Sub (dd, de), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Cst dq, Cst ei), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Cst dq, Var fa), ck))) =
+    (if (fa = 0) then abs bo else 1)
+  | divlcm (Divides (Cst bo, Add (Mult (Cst dq, Neg ek), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Cst dq, Add (el, em)), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Cst dq, Sub (en, eo)), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Cst dq, Mult (ep, eq)), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Var dr, dg), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Neg ds, dg), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Add (dt, du), dg), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Sub (dv, dw), dg), ck))) = 1
+  | divlcm (Divides (Cst bo, Add (Mult (Mult (dx, dy), dg), ck))) = 1
+  | divlcm (Divides (Cst bo, Sub (cl, cm))) = 1
+  | divlcm (Divides (Cst bo, Mult (cn, co))) = 1
+  | divlcm (Divides (Var bp, af)) = 1
+  | divlcm (Divides (Neg bq, af)) = 1
+  | divlcm (Divides (Add (br, bs), af)) = 1
+  | divlcm (Divides (Sub (bt, bu), af)) = 1
+  | divlcm (Divides (Mult (bv, bw), af)) = 1
+  | divlcm T = 1
+  | divlcm F = 1
+  | divlcm (Imp (al, am)) = 1
+  | divlcm (Equ (an, ao)) = 1
+  | divlcm (QAll ap) = 1
+  | divlcm (QEx aq) = 1;
+
+fun explode_minf (q, B) =
+    let val d = divlcm q; val pm = minusinf q;
+        val dj1 = explode_disj (map (fn x => Cst x) (iupto (1, d)), pm)
+    in (case dj1 of
+         Lt (x, xa) =>
+           let val dj2 = explode_disj (all_sums (d, B), q)
+           in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                | QEx x => Or (dj1, dj2))
+           end
+         | Gt (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | Le (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | Ge (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | Eq (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | Divides (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | T => T | F => explode_disj (all_sums (d, B), q)
+         | NOT x =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | And (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | Or (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | Imp (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | Equ (x, xa) =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | QAll x =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end
+         | QEx x =>
+             let val dj2 = explode_disj (all_sums (d, B), q)
+             in (case dj2 of Lt (x, xa) => Or (dj1, dj2)
+                  | Gt (x, xa) => Or (dj1, dj2) | Le (x, xa) => Or (dj1, dj2)
+                  | Ge (x, xa) => Or (dj1, dj2) | Eq (x, xa) => Or (dj1, dj2)
+                  | Divides (x, xa) => Or (dj1, dj2) | T => T | F => dj1
+                  | NOT x => Or (dj1, dj2) | And (x, xa) => Or (dj1, dj2)
+                  | Or (x, xa) => Or (dj1, dj2) | Imp (x, xa) => Or (dj1, dj2)
+                  | Equ (x, xa) => Or (dj1, dj2) | QAll x => Or (dj1, dj2)
+                  | QEx x => Or (dj1, dj2))
+             end)
+    end;
+
+fun mirror (And (p, q)) = And (mirror p, mirror q)
+  | mirror (Or (p, q)) = Or (mirror p, mirror q)
+  | mirror (Lt (u, v)) = Lt (u, v)
+  | mirror (Gt (w, x)) = Gt (w, x)
+  | mirror (Le (Cst bp, aa)) = Le (Cst bp, aa)
+  | mirror (Le (Var bq, aa)) = Le (Var bq, aa)
+  | mirror (Le (Neg br, aa)) = Le (Neg br, aa)
+  | mirror (Le (Add (Cst ch, bt), aa)) = Le (Add (Cst ch, bt), aa)
+  | mirror (Le (Add (Var ci, bt), aa)) = Le (Add (Var ci, bt), aa)
+  | mirror (Le (Add (Neg cj, bt), aa)) = Le (Add (Neg cj, bt), aa)
+  | mirror (Le (Add (Add (ck, cl), bt), aa)) = Le (Add (Add (ck, cl), bt), aa)
+  | mirror (Le (Add (Sub (cm, cn), bt), aa)) = Le (Add (Sub (cm, cn), bt), aa)
+  | mirror (Le (Add (Mult (Cst cz, Cst dr), bt), aa)) =
+    Le (Add (Mult (Cst cz, Cst dr), bt), aa)
+  | mirror (Le (Add (Mult (Cst cz, Var ej), bt), aa)) =
+    (if (ej = 0) then Le (Add (Mult (Cst (~ cz), Var 0), bt), aa)
+      else Le (Add (Mult (Cst cz, Var (op_45_def0 ej id_1_def0 + 1)), bt), aa))
+  | mirror (Le (Add (Mult (Cst cz, Neg dt), bt), aa)) =
+    Le (Add (Mult (Cst cz, Neg dt), bt), aa)
+  | mirror (Le (Add (Mult (Cst cz, Add (du, dv)), bt), aa)) =
+    Le (Add (Mult (Cst cz, Add (du, dv)), bt), aa)
+  | mirror (Le (Add (Mult (Cst cz, Sub (dw, dx)), bt), aa)) =
+    Le (Add (Mult (Cst cz, Sub (dw, dx)), bt), aa)
+  | mirror (Le (Add (Mult (Cst cz, Mult (dy, dz)), bt), aa)) =
+    Le (Add (Mult (Cst cz, Mult (dy, dz)), bt), aa)
+  | mirror (Le (Add (Mult (Var da, cp), bt), aa)) =
+    Le (Add (Mult (Var da, cp), bt), aa)
+  | mirror (Le (Add (Mult (Neg db, cp), bt), aa)) =
+    Le (Add (Mult (Neg db, cp), bt), aa)
+  | mirror (Le (Add (Mult (Add (dc, dd), cp), bt), aa)) =
+    Le (Add (Mult (Add (dc, dd), cp), bt), aa)
+  | mirror (Le (Add (Mult (Sub (de, df), cp), bt), aa)) =
+    Le (Add (Mult (Sub (de, df), cp), bt), aa)
+  | mirror (Le (Add (Mult (Mult (dg, dh), cp), bt), aa)) =
+    Le (Add (Mult (Mult (dg, dh), cp), bt), aa)
+  | mirror (Le (Sub (bu, bv), aa)) = Le (Sub (bu, bv), aa)
+  | mirror (Le (Mult (bw, bx), aa)) = Le (Mult (bw, bx), aa)
+  | mirror (Ge (ab, ac)) = Ge (ab, ac)
+  | mirror (Eq (Cst el, ae)) = Eq (Cst el, ae)
+  | mirror (Eq (Var em, ae)) = Eq (Var em, ae)
+  | mirror (Eq (Neg en, ae)) = Eq (Neg en, ae)
+  | mirror (Eq (Add (Cst fd, ep), ae)) = Eq (Add (Cst fd, ep), ae)
+  | mirror (Eq (Add (Var fe, ep), ae)) = Eq (Add (Var fe, ep), ae)
+  | mirror (Eq (Add (Neg ff, ep), ae)) = Eq (Add (Neg ff, ep), ae)
+  | mirror (Eq (Add (Add (fg, fh), ep), ae)) = Eq (Add (Add (fg, fh), ep), ae)
+  | mirror (Eq (Add (Sub (fi, fj), ep), ae)) = Eq (Add (Sub (fi, fj), ep), ae)
+  | mirror (Eq (Add (Mult (Cst fv, Cst gn), ep), ae)) =
+    Eq (Add (Mult (Cst fv, Cst gn), ep), ae)
+  | mirror (Eq (Add (Mult (Cst fv, Var hf), ep), ae)) =
+    (if (hf = 0) then Eq (Add (Mult (Cst (~ fv), Var 0), ep), ae)
+      else Eq (Add (Mult (Cst fv, Var (op_45_def0 hf id_1_def0 + 1)), ep), ae))
+  | mirror (Eq (Add (Mult (Cst fv, Neg gp), ep), ae)) =
+    Eq (Add (Mult (Cst fv, Neg gp), ep), ae)
+  | mirror (Eq (Add (Mult (Cst fv, Add (gq, gr)), ep), ae)) =
+    Eq (Add (Mult (Cst fv, Add (gq, gr)), ep), ae)
+  | mirror (Eq (Add (Mult (Cst fv, Sub (gs, gt)), ep), ae)) =
+    Eq (Add (Mult (Cst fv, Sub (gs, gt)), ep), ae)
+  | mirror (Eq (Add (Mult (Cst fv, Mult (gu, gv)), ep), ae)) =
+    Eq (Add (Mult (Cst fv, Mult (gu, gv)), ep), ae)
+  | mirror (Eq (Add (Mult (Var fw, fl), ep), ae)) =
+    Eq (Add (Mult (Var fw, fl), ep), ae)
+  | mirror (Eq (Add (Mult (Neg fx, fl), ep), ae)) =
+    Eq (Add (Mult (Neg fx, fl), ep), ae)
+  | mirror (Eq (Add (Mult (Add (fy, fz), fl), ep), ae)) =
+    Eq (Add (Mult (Add (fy, fz), fl), ep), ae)
+  | mirror (Eq (Add (Mult (Sub (ga, gb), fl), ep), ae)) =
+    Eq (Add (Mult (Sub (ga, gb), fl), ep), ae)
+  | mirror (Eq (Add (Mult (Mult (gc, gd), fl), ep), ae)) =
+    Eq (Add (Mult (Mult (gc, gd), fl), ep), ae)
+  | mirror (Eq (Sub (eq, er), ae)) = Eq (Sub (eq, er), ae)
+  | mirror (Eq (Mult (es, et), ae)) = Eq (Mult (es, et), ae)
+  | mirror (Divides (Cst hh, Cst hz)) = Divides (Cst hh, Cst hz)
+  | mirror (Divides (Cst hh, Var ia)) = Divides (Cst hh, Var ia)
+  | mirror (Divides (Cst hh, Neg ib)) = Divides (Cst hh, Neg ib)
+  | mirror (Divides (Cst hh, Add (Cst ir, id))) =
+    Divides (Cst hh, Add (Cst ir, id))
+  | mirror (Divides (Cst hh, Add (Var is, id))) =
+    Divides (Cst hh, Add (Var is, id))
+  | mirror (Divides (Cst hh, Add (Neg it, id))) =
+    Divides (Cst hh, Add (Neg it, id))
+  | mirror (Divides (Cst hh, Add (Add (iu, iv), id))) =
+    Divides (Cst hh, Add (Add (iu, iv), id))
+  | mirror (Divides (Cst hh, Add (Sub (iw, ix), id))) =
+    Divides (Cst hh, Add (Sub (iw, ix), id))
+  | mirror (Divides (Cst hh, Add (Mult (Cst jj, Cst kb), id))) =
+    Divides (Cst hh, Add (Mult (Cst jj, Cst kb), id))
+  | mirror (Divides (Cst hh, Add (Mult (Cst jj, Var kt), id))) =
+    (if (kt = 0) then Divides (Cst hh, Add (Mult (Cst (~ jj), Var 0), id))
+      else Divides
+             (Cst hh,
+               Add (Mult (Cst jj, Var (op_45_def0 kt id_1_def0 + 1)), id)))
+  | mirror (Divides (Cst hh, Add (Mult (Cst jj, Neg kd), id))) =
+    Divides (Cst hh, Add (Mult (Cst jj, Neg kd), id))
+  | mirror (Divides (Cst hh, Add (Mult (Cst jj, Add (ke, kf)), id))) =
+    Divides (Cst hh, Add (Mult (Cst jj, Add (ke, kf)), id))
+  | mirror (Divides (Cst hh, Add (Mult (Cst jj, Sub (kg, kh)), id))) =
+    Divides (Cst hh, Add (Mult (Cst jj, Sub (kg, kh)), id))
+  | mirror (Divides (Cst hh, Add (Mult (Cst jj, Mult (ki, kj)), id))) =
+    Divides (Cst hh, Add (Mult (Cst jj, Mult (ki, kj)), id))
+  | mirror (Divides (Cst hh, Add (Mult (Var jk, iz), id))) =
+    Divides (Cst hh, Add (Mult (Var jk, iz), id))
+  | mirror (Divides (Cst hh, Add (Mult (Neg jl, iz), id))) =
+    Divides (Cst hh, Add (Mult (Neg jl, iz), id))
+  | mirror (Divides (Cst hh, Add (Mult (Add (jm, jn), iz), id))) =
+    Divides (Cst hh, Add (Mult (Add (jm, jn), iz), id))
+  | mirror (Divides (Cst hh, Add (Mult (Sub (jo, jp), iz), id))) =
+    Divides (Cst hh, Add (Mult (Sub (jo, jp), iz), id))
+  | mirror (Divides (Cst hh, Add (Mult (Mult (jq, jr), iz), id))) =
+    Divides (Cst hh, Add (Mult (Mult (jq, jr), iz), id))
+  | mirror (Divides (Cst hh, Sub (ie, if'))) = Divides (Cst hh, Sub (ie, if'))
+  | mirror (Divides (Cst hh, Mult (ig, ih))) = Divides (Cst hh, Mult (ig, ih))
+  | mirror (Divides (Var hi, ag)) = Divides (Var hi, ag)
+  | mirror (Divides (Neg hj, ag)) = Divides (Neg hj, ag)
+  | mirror (Divides (Add (hk, hl), ag)) = Divides (Add (hk, hl), ag)
+  | mirror (Divides (Sub (hm, hn), ag)) = Divides (Sub (hm, hn), ag)
+  | mirror (Divides (Mult (ho, hp), ag)) = Divides (Mult (ho, hp), ag)
+  | mirror T = T
+  | mirror F = F
+  | mirror (NOT (Lt (kv, kw))) = NOT (Lt (kv, kw))
+  | mirror (NOT (Gt (kx, ky))) = NOT (Gt (kx, ky))
+  | mirror (NOT (Le (kz, la))) = NOT (Le (kz, la))
+  | mirror (NOT (Ge (lb, lc))) = NOT (Ge (lb, lc))
+  | mirror (NOT (Eq (Cst mp, le))) = NOT (Eq (Cst mp, le))
+  | mirror (NOT (Eq (Var mq, le))) = NOT (Eq (Var mq, le))
+  | mirror (NOT (Eq (Neg mr, le))) = NOT (Eq (Neg mr, le))
+  | mirror (NOT (Eq (Add (Cst nh, mt), le))) = NOT (Eq (Add (Cst nh, mt), le))
+  | mirror (NOT (Eq (Add (Var ni, mt), le))) = NOT (Eq (Add (Var ni, mt), le))
+  | mirror (NOT (Eq (Add (Neg nj, mt), le))) = NOT (Eq (Add (Neg nj, mt), le))
+  | mirror (NOT (Eq (Add (Add (nk, nl), mt), le))) =
+    NOT (Eq (Add (Add (nk, nl), mt), le))
+  | mirror (NOT (Eq (Add (Sub (nm, nn), mt), le))) =
+    NOT (Eq (Add (Sub (nm, nn), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Cst nz, Cst or), mt), le))) =
+    NOT (Eq (Add (Mult (Cst nz, Cst or), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Cst nz, Var pj), mt), le))) =
+    (if (pj = 0) then NOT (Eq (Add (Mult (Cst (~ nz), Var 0), mt), le))
+      else NOT (Eq (Add (Mult (Cst nz, Var (op_45_def0 pj id_1_def0 + 1)), mt),
+                     le)))
+  | mirror (NOT (Eq (Add (Mult (Cst nz, Neg ot), mt), le))) =
+    NOT (Eq (Add (Mult (Cst nz, Neg ot), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Cst nz, Add (ou, ov)), mt), le))) =
+    NOT (Eq (Add (Mult (Cst nz, Add (ou, ov)), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Cst nz, Sub (ow, ox)), mt), le))) =
+    NOT (Eq (Add (Mult (Cst nz, Sub (ow, ox)), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Cst nz, Mult (oy, oz)), mt), le))) =
+    NOT (Eq (Add (Mult (Cst nz, Mult (oy, oz)), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Var oa, np), mt), le))) =
+    NOT (Eq (Add (Mult (Var oa, np), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Neg ob, np), mt), le))) =
+    NOT (Eq (Add (Mult (Neg ob, np), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Add (oc, od), np), mt), le))) =
+    NOT (Eq (Add (Mult (Add (oc, od), np), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Sub (oe, of'), np), mt), le))) =
+    NOT (Eq (Add (Mult (Sub (oe, of'), np), mt), le))
+  | mirror (NOT (Eq (Add (Mult (Mult (og, oh), np), mt), le))) =
+    NOT (Eq (Add (Mult (Mult (og, oh), np), mt), le))
+  | mirror (NOT (Eq (Sub (mu, mv), le))) = NOT (Eq (Sub (mu, mv), le))
+  | mirror (NOT (Eq (Mult (mw, mx), le))) = NOT (Eq (Mult (mw, mx), le))
+  | mirror (NOT (Divides (Cst pl, Cst qd))) = NOT (Divides (Cst pl, Cst qd))
+  | mirror (NOT (Divides (Cst pl, Var qe))) = NOT (Divides (Cst pl, Var qe))
+  | mirror (NOT (Divides (Cst pl, Neg qf))) = NOT (Divides (Cst pl, Neg qf))
+  | mirror (NOT (Divides (Cst pl, Add (Cst qv, qh)))) =
+    NOT (Divides (Cst pl, Add (Cst qv, qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Var qw, qh)))) =
+    NOT (Divides (Cst pl, Add (Var qw, qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Neg qx, qh)))) =
+    NOT (Divides (Cst pl, Add (Neg qx, qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Add (qy, qz), qh)))) =
+    NOT (Divides (Cst pl, Add (Add (qy, qz), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Sub (ra, rb), qh)))) =
+    NOT (Divides (Cst pl, Add (Sub (ra, rb), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Cst rn, Cst sf), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Cst rn, Cst sf), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Cst rn, Var sx), qh)))) =
+    (if (sx = 0)
+      then NOT (Divides (Cst pl, Add (Mult (Cst (~ rn), Var 0), qh)))
+      else NOT (Divides
+                  (Cst pl,
+                    Add (Mult (Cst rn, Var (op_45_def0 sx id_1_def0 + 1)),
+                          qh))))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Cst rn, Neg sh), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Cst rn, Neg sh), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Cst rn, Add (si, sj)), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Cst rn, Add (si, sj)), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Cst rn, Sub (sk, sl)), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Cst rn, Sub (sk, sl)), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Cst rn, Mult (sm, sn)), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Cst rn, Mult (sm, sn)), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Var ro, rd), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Var ro, rd), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Neg rp, rd), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Neg rp, rd), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Add (rq, rr), rd), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Add (rq, rr), rd), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Sub (rs, rt), rd), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Sub (rs, rt), rd), qh)))
+  | mirror (NOT (Divides (Cst pl, Add (Mult (Mult (ru, rv), rd), qh)))) =
+    NOT (Divides (Cst pl, Add (Mult (Mult (ru, rv), rd), qh)))
+  | mirror (NOT (Divides (Cst pl, Sub (qi, qj)))) =
+    NOT (Divides (Cst pl, Sub (qi, qj)))
+  | mirror (NOT (Divides (Cst pl, Mult (qk, ql)))) =
+    NOT (Divides (Cst pl, Mult (qk, ql)))
+  | mirror (NOT (Divides (Var pm, lg))) = NOT (Divides (Var pm, lg))
+  | mirror (NOT (Divides (Neg pn, lg))) = NOT (Divides (Neg pn, lg))
+  | mirror (NOT (Divides (Add (po, pp), lg))) = NOT (Divides (Add (po, pp), lg))
+  | mirror (NOT (Divides (Sub (pq, pr), lg))) = NOT (Divides (Sub (pq, pr), lg))
+  | mirror (NOT (Divides (Mult (ps, pt), lg))) =
+    NOT (Divides (Mult (ps, pt), lg))
+  | mirror (NOT T) = NOT T
+  | mirror (NOT F) = NOT F
+  | mirror (NOT (NOT lh)) = NOT (NOT lh)
+  | mirror (NOT (And (li, lj))) = NOT (And (li, lj))
+  | mirror (NOT (Or (lk, ll))) = NOT (Or (lk, ll))
+  | mirror (NOT (Imp (lm, ln))) = NOT (Imp (lm, ln))
+  | mirror (NOT (Equ (lo, lp))) = NOT (Equ (lo, lp))
+  | mirror (NOT (QAll lq)) = NOT (QAll lq)
+  | mirror (NOT (QEx lr)) = NOT (QEx lr)
+  | mirror (Imp (am, an)) = Imp (am, an)
+  | mirror (Equ (ao, ap)) = Equ (ao, ap)
+  | mirror (QAll aq) = QAll aq
+  | mirror (QEx ar) = QEx ar;
+
+fun op_43_def0 m n = nat ((m) + (n));
+
+fun size_def1 [] = 0
+  | size_def1 (a :: list) = op_43_def0 (size_def1 list) (0 + 1);
+
+fun aset (And (p, q)) = op_64 (aset p) (aset q)
+  | aset (Or (p, q)) = op_64 (aset p) (aset q)
+  | aset (Lt (u, v)) = []
+  | aset (Gt (w, x)) = []
+  | aset (Le (Cst bo, z)) = []
+  | aset (Le (Var bp, z)) = []
+  | aset (Le (Neg bq, z)) = []
+  | aset (Le (Add (Cst cg, bs), z)) = []
+  | aset (Le (Add (Var ch, bs), z)) = []
+  | aset (Le (Add (Neg ci, bs), z)) = []
+  | aset (Le (Add (Add (cj, ck), bs), z)) = []
+  | aset (Le (Add (Sub (cl, cm), bs), z)) = []
+  | aset (Le (Add (Mult (Cst cy, Cst dq), bs), z)) = []
+  | aset (Le (Add (Mult (Cst cy, Var ei), bs), z)) =
+    (if (ei = 0)
+      then (if (cy < 0) then [lin_add (bs, Cst 1)]
+             else [lin_neg bs, lin_add (lin_neg bs, Cst 1)])
+      else [])
+  | aset (Le (Add (Mult (Cst cy, Neg ds), bs), z)) = []
+  | aset (Le (Add (Mult (Cst cy, Add (dt, du)), bs), z)) = []
+  | aset (Le (Add (Mult (Cst cy, Sub (dv, dw)), bs), z)) = []
+  | aset (Le (Add (Mult (Cst cy, Mult (dx, dy)), bs), z)) = []
+  | aset (Le (Add (Mult (Var cz, co), bs), z)) = []
+  | aset (Le (Add (Mult (Neg da, co), bs), z)) = []
+  | aset (Le (Add (Mult (Add (db, dc), co), bs), z)) = []
+  | aset (Le (Add (Mult (Sub (dd, de), co), bs), z)) = []
+  | aset (Le (Add (Mult (Mult (df, dg), co), bs), z)) = []
+  | aset (Le (Sub (bt, bu), z)) = []
+  | aset (Le (Mult (bv, bw), z)) = []
+  | aset (Ge (aa, ab)) = []
+  | aset (Eq (Cst ek, ad)) = []
+  | aset (Eq (Var el, ad)) = []
+  | aset (Eq (Neg em, ad)) = []
+  | aset (Eq (Add (Cst fc, eo), ad)) = []
+  | aset (Eq (Add (Var fd, eo), ad)) = []
+  | aset (Eq (Add (Neg fe, eo), ad)) = []
+  | aset (Eq (Add (Add (ff, fg), eo), ad)) = []
+  | aset (Eq (Add (Sub (fh, fi), eo), ad)) = []
+  | aset (Eq (Add (Mult (Cst fu, Cst gm), eo), ad)) = []
+  | aset (Eq (Add (Mult (Cst fu, Var he), eo), ad)) =
+    (if (he = 0)
+      then (if (fu < 0) then [lin_add (eo, Cst 1)]
+             else [lin_add (lin_neg eo, Cst 1)])
+      else [])
+  | aset (Eq (Add (Mult (Cst fu, Neg go), eo), ad)) = []
+  | aset (Eq (Add (Mult (Cst fu, Add (gp, gq)), eo), ad)) = []
+  | aset (Eq (Add (Mult (Cst fu, Sub (gr, gs)), eo), ad)) = []
+  | aset (Eq (Add (Mult (Cst fu, Mult (gt, gu)), eo), ad)) = []
+  | aset (Eq (Add (Mult (Var fv, fk), eo), ad)) = []
+  | aset (Eq (Add (Mult (Neg fw, fk), eo), ad)) = []
+  | aset (Eq (Add (Mult (Add (fx, fy), fk), eo), ad)) = []
+  | aset (Eq (Add (Mult (Sub (fz, ga), fk), eo), ad)) = []
+  | aset (Eq (Add (Mult (Mult (gb, gc), fk), eo), ad)) = []
+  | aset (Eq (Sub (ep, eq), ad)) = []
+  | aset (Eq (Mult (er, es), ad)) = []
+  | aset (Divides (ae, af)) = []
+  | aset T = []
+  | aset F = []
+  | aset (NOT (Lt (hg, hh))) = []
+  | aset (NOT (Gt (hi, hj))) = []
+  | aset (NOT (Le (hk, hl))) = []
+  | aset (NOT (Ge (hm, hn))) = []
+  | aset (NOT (Eq (Cst ja, hp))) = []
+  | aset (NOT (Eq (Var jb, hp))) = []
+  | aset (NOT (Eq (Neg jc, hp))) = []
+  | aset (NOT (Eq (Add (Cst js, je), hp))) = []
+  | aset (NOT (Eq (Add (Var jt, je), hp))) = []
+  | aset (NOT (Eq (Add (Neg ju, je), hp))) = []
+  | aset (NOT (Eq (Add (Add (jv, jw), je), hp))) = []
+  | aset (NOT (Eq (Add (Sub (jx, jy), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Cst kk, Cst lc), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Cst kk, Var lu), je), hp))) =
+    (if (lu = 0) then (if (kk < 0) then [je] else [lin_neg je]) else [])
+  | aset (NOT (Eq (Add (Mult (Cst kk, Neg le), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Cst kk, Add (lf, lg)), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Cst kk, Sub (lh, li)), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Cst kk, Mult (lj, lk)), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Var kl, ka), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Neg km, ka), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Add (kn, ko), ka), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Sub (kp, kq), ka), je), hp))) = []
+  | aset (NOT (Eq (Add (Mult (Mult (kr, ks), ka), je), hp))) = []
+  | aset (NOT (Eq (Sub (jf, jg), hp))) = []
+  | aset (NOT (Eq (Mult (jh, ji), hp))) = []
+  | aset (NOT (Divides (hq, hr))) = []
+  | aset (NOT T) = []
+  | aset (NOT F) = []
+  | aset (NOT (NOT hs)) = []
+  | aset (NOT (And (ht, hu))) = []
+  | aset (NOT (Or (hv, hw))) = []
+  | aset (NOT (Imp (hx, hy))) = []
+  | aset (NOT (Equ (hz, ia))) = []
+  | aset (NOT (QAll ib)) = []
+  | aset (NOT (QEx ic)) = []
+  | aset (Imp (al, am)) = []
+  | aset (Equ (an, ao)) = []
+  | aset (QAll ap) = []
+  | aset (QEx aq) = [];
+
+fun op_mem x [] = false
+  | op_mem x (y :: ys) = (if (y = x) then true else op_mem x ys);
+
+fun list_insert x xs = (if op_mem x xs then xs else (x :: xs));
+
+fun list_set [] = []
+  | list_set (x :: xs) = list_insert x (list_set xs);
+
+fun bset (And (p, q)) = op_64 (bset p) (bset q)
+  | bset (Or (p, q)) = op_64 (bset p) (bset q)
+  | bset (Lt (u, v)) = []
+  | bset (Gt (w, x)) = []
+  | bset (Le (Cst bo, z)) = []
+  | bset (Le (Var bp, z)) = []
+  | bset (Le (Neg bq, z)) = []
+  | bset (Le (Add (Cst cg, bs), z)) = []
+  | bset (Le (Add (Var ch, bs), z)) = []
+  | bset (Le (Add (Neg ci, bs), z)) = []
+  | bset (Le (Add (Add (cj, ck), bs), z)) = []
+  | bset (Le (Add (Sub (cl, cm), bs), z)) = []
+  | bset (Le (Add (Mult (Cst cy, Cst dq), bs), z)) = []
+  | bset (Le (Add (Mult (Cst cy, Var ei), bs), z)) =
+    (if (ei = 0)
+      then (if (cy < 0) then [lin_add (bs, Cst ~1), bs]
+             else [lin_add (lin_neg bs, Cst ~1)])
+      else [])
+  | bset (Le (Add (Mult (Cst cy, Neg ds), bs), z)) = []
+  | bset (Le (Add (Mult (Cst cy, Add (dt, du)), bs), z)) = []
+  | bset (Le (Add (Mult (Cst cy, Sub (dv, dw)), bs), z)) = []
+  | bset (Le (Add (Mult (Cst cy, Mult (dx, dy)), bs), z)) = []
+  | bset (Le (Add (Mult (Var cz, co), bs), z)) = []
+  | bset (Le (Add (Mult (Neg da, co), bs), z)) = []
+  | bset (Le (Add (Mult (Add (db, dc), co), bs), z)) = []
+  | bset (Le (Add (Mult (Sub (dd, de), co), bs), z)) = []
+  | bset (Le (Add (Mult (Mult (df, dg), co), bs), z)) = []
+  | bset (Le (Sub (bt, bu), z)) = []
+  | bset (Le (Mult (bv, bw), z)) = []
+  | bset (Ge (aa, ab)) = []
+  | bset (Eq (Cst ek, ad)) = []
+  | bset (Eq (Var el, ad)) = []
+  | bset (Eq (Neg em, ad)) = []
+  | bset (Eq (Add (Cst fc, eo), ad)) = []
+  | bset (Eq (Add (Var fd, eo), ad)) = []
+  | bset (Eq (Add (Neg fe, eo), ad)) = []
+  | bset (Eq (Add (Add (ff, fg), eo), ad)) = []
+  | bset (Eq (Add (Sub (fh, fi), eo), ad)) = []
+  | bset (Eq (Add (Mult (Cst fu, Cst gm), eo), ad)) = []
+  | bset (Eq (Add (Mult (Cst fu, Var he), eo), ad)) =
+    (if (he = 0)
+      then (if (fu < 0) then [lin_add (eo, Cst ~1)]
+             else [lin_add (lin_neg eo, Cst ~1)])
+      else [])
+  | bset (Eq (Add (Mult (Cst fu, Neg go), eo), ad)) = []
+  | bset (Eq (Add (Mult (Cst fu, Add (gp, gq)), eo), ad)) = []
+  | bset (Eq (Add (Mult (Cst fu, Sub (gr, gs)), eo), ad)) = []
+  | bset (Eq (Add (Mult (Cst fu, Mult (gt, gu)), eo), ad)) = []
+  | bset (Eq (Add (Mult (Var fv, fk), eo), ad)) = []
+  | bset (Eq (Add (Mult (Neg fw, fk), eo), ad)) = []
+  | bset (Eq (Add (Mult (Add (fx, fy), fk), eo), ad)) = []
+  | bset (Eq (Add (Mult (Sub (fz, ga), fk), eo), ad)) = []
+  | bset (Eq (Add (Mult (Mult (gb, gc), fk), eo), ad)) = []
+  | bset (Eq (Sub (ep, eq), ad)) = []
+  | bset (Eq (Mult (er, es), ad)) = []
+  | bset (Divides (ae, af)) = []
+  | bset T = []
+  | bset F = []
+  | bset (NOT (Lt (hg, hh))) = []
+  | bset (NOT (Gt (hi, hj))) = []
+  | bset (NOT (Le (hk, hl))) = []
+  | bset (NOT (Ge (hm, hn))) = []
+  | bset (NOT (Eq (Cst ja, hp))) = []
+  | bset (NOT (Eq (Var jb, hp))) = []
+  | bset (NOT (Eq (Neg jc, hp))) = []
+  | bset (NOT (Eq (Add (Cst js, je), hp))) = []
+  | bset (NOT (Eq (Add (Var jt, je), hp))) = []
+  | bset (NOT (Eq (Add (Neg ju, je), hp))) = []
+  | bset (NOT (Eq (Add (Add (jv, jw), je), hp))) = []
+  | bset (NOT (Eq (Add (Sub (jx, jy), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Cst kk, Cst lc), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Cst kk, Var lu), je), hp))) =
+    (if (lu = 0) then (if (kk < 0) then [je] else [lin_neg je]) else [])
+  | bset (NOT (Eq (Add (Mult (Cst kk, Neg le), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Cst kk, Add (lf, lg)), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Cst kk, Sub (lh, li)), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Cst kk, Mult (lj, lk)), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Var kl, ka), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Neg km, ka), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Add (kn, ko), ka), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Sub (kp, kq), ka), je), hp))) = []
+  | bset (NOT (Eq (Add (Mult (Mult (kr, ks), ka), je), hp))) = []
+  | bset (NOT (Eq (Sub (jf, jg), hp))) = []
+  | bset (NOT (Eq (Mult (jh, ji), hp))) = []
+  | bset (NOT (Divides (hq, hr))) = []
+  | bset (NOT T) = []
+  | bset (NOT F) = []
+  | bset (NOT (NOT hs)) = []
+  | bset (NOT (And (ht, hu))) = []
+  | bset (NOT (Or (hv, hw))) = []
+  | bset (NOT (Imp (hx, hy))) = []
+  | bset (NOT (Equ (hz, ia))) = []
+  | bset (NOT (QAll ib)) = []
+  | bset (NOT (QEx ic)) = []
+  | bset (Imp (al, am)) = []
+  | bset (Equ (an, ao)) = []
+  | bset (QAll ap) = []
+  | bset (QEx aq) = [];
+
+fun adjustcoeff (l, Le (Add (Mult (Cst c, Var 0), r), Cst i)) =
+    (if (c <= 0)
+      then Le (Add (Mult (Cst ~1, Var 0), lin_mul (~ (op_div_def1 l c), r)),
+                Cst 0)
+      else Le (Add (Mult (Cst 1, Var 0), lin_mul (op_div_def1 l c, r)), Cst 0))
+  | adjustcoeff (l, Eq (Add (Mult (Cst c, Var 0), r), Cst i)) =
+    Eq (Add (Mult (Cst 1, Var 0), lin_mul (op_div_def1 l c, r)), Cst 0)
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst c, Var 0), r), Cst i))) =
+    NOT (Eq (Add (Mult (Cst 1, Var 0), lin_mul (op_div_def1 l c, r)), Cst 0))
+  | adjustcoeff (l, And (p, q)) = And (adjustcoeff (l, p), adjustcoeff (l, q))
+  | adjustcoeff (l, Or (p, q)) = Or (adjustcoeff (l, p), adjustcoeff (l, q))
+  | adjustcoeff (l, Lt (w, x)) = Lt (w, x)
+  | adjustcoeff (l, Gt (y, z)) = Gt (y, z)
+  | adjustcoeff (l, Le (Cst bq, ab)) = Le (Cst bq, ab)
+  | adjustcoeff (l, Le (Var br, ab)) = Le (Var br, ab)
+  | adjustcoeff (l, Le (Neg bs, ab)) = Le (Neg bs, ab)
+  | adjustcoeff (l, Le (Add (Cst ci, bu), ab)) = Le (Add (Cst ci, bu), ab)
+  | adjustcoeff (l, Le (Add (Var cj, bu), ab)) = Le (Add (Var cj, bu), ab)
+  | adjustcoeff (l, Le (Add (Neg ck, bu), ab)) = Le (Add (Neg ck, bu), ab)
+  | adjustcoeff (l, Le (Add (Add (cl, cm), bu), ab)) =
+    Le (Add (Add (cl, cm), bu), ab)
+  | adjustcoeff (l, Le (Add (Sub (cn, co), bu), ab)) =
+    Le (Add (Sub (cn, co), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Cst ds), bu), ab)) =
+    Le (Add (Mult (Cst da, Cst ds), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Var 0), bu), Var en)) =
+    Le (Add (Mult (Cst da, Var 0), bu), Var en)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Var 0), bu), Neg eo)) =
+    Le (Add (Mult (Cst da, Var 0), bu), Neg eo)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Var 0), bu), Add (ep, eq))) =
+    Le (Add (Mult (Cst da, Var 0), bu), Add (ep, eq))
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Var 0), bu), Sub (er, es))) =
+    Le (Add (Mult (Cst da, Var 0), bu), Sub (er, es))
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Var 0), bu), Mult (et, eu))) =
+    Le (Add (Mult (Cst da, Var 0), bu), Mult (et, eu))
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Var ek), bu), ab)) =
+    Le (Add (Mult (Cst da, Var ek), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Neg du), bu), ab)) =
+    Le (Add (Mult (Cst da, Neg du), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Add (dv, dw)), bu), ab)) =
+    Le (Add (Mult (Cst da, Add (dv, dw)), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Sub (dx, dy)), bu), ab)) =
+    Le (Add (Mult (Cst da, Sub (dx, dy)), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Cst da, Mult (dz, ea)), bu), ab)) =
+    Le (Add (Mult (Cst da, Mult (dz, ea)), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Var db, cq), bu), ab)) =
+    Le (Add (Mult (Var db, cq), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Neg dc, cq), bu), ab)) =
+    Le (Add (Mult (Neg dc, cq), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Add (dd, de), cq), bu), ab)) =
+    Le (Add (Mult (Add (dd, de), cq), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Sub (df, dg), cq), bu), ab)) =
+    Le (Add (Mult (Sub (df, dg), cq), bu), ab)
+  | adjustcoeff (l, Le (Add (Mult (Mult (dh, di), cq), bu), ab)) =
+    Le (Add (Mult (Mult (dh, di), cq), bu), ab)
+  | adjustcoeff (l, Le (Sub (bv, bw), ab)) = Le (Sub (bv, bw), ab)
+  | adjustcoeff (l, Le (Mult (bx, by), ab)) = Le (Mult (bx, by), ab)
+  | adjustcoeff (l, Ge (ac, ad)) = Ge (ac, ad)
+  | adjustcoeff (l, Eq (Cst fe, af)) = Eq (Cst fe, af)
+  | adjustcoeff (l, Eq (Var ff, af)) = Eq (Var ff, af)
+  | adjustcoeff (l, Eq (Neg fg, af)) = Eq (Neg fg, af)
+  | adjustcoeff (l, Eq (Add (Cst fw, fi), af)) = Eq (Add (Cst fw, fi), af)
+  | adjustcoeff (l, Eq (Add (Var fx, fi), af)) = Eq (Add (Var fx, fi), af)
+  | adjustcoeff (l, Eq (Add (Neg fy, fi), af)) = Eq (Add (Neg fy, fi), af)
+  | adjustcoeff (l, Eq (Add (Add (fz, ga), fi), af)) =
+    Eq (Add (Add (fz, ga), fi), af)
+  | adjustcoeff (l, Eq (Add (Sub (gb, gc), fi), af)) =
+    Eq (Add (Sub (gb, gc), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Cst hg), fi), af)) =
+    Eq (Add (Mult (Cst go, Cst hg), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Var 0), fi), Var ib)) =
+    Eq (Add (Mult (Cst go, Var 0), fi), Var ib)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Var 0), fi), Neg ic)) =
+    Eq (Add (Mult (Cst go, Var 0), fi), Neg ic)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Var 0), fi), Add (id, ie))) =
+    Eq (Add (Mult (Cst go, Var 0), fi), Add (id, ie))
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Var 0), fi), Sub (if', ig))) =
+    Eq (Add (Mult (Cst go, Var 0), fi), Sub (if', ig))
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Var 0), fi), Mult (ih, ii))) =
+    Eq (Add (Mult (Cst go, Var 0), fi), Mult (ih, ii))
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Var hy), fi), af)) =
+    Eq (Add (Mult (Cst go, Var hy), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Neg hi), fi), af)) =
+    Eq (Add (Mult (Cst go, Neg hi), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Add (hj, hk)), fi), af)) =
+    Eq (Add (Mult (Cst go, Add (hj, hk)), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Sub (hl, hm)), fi), af)) =
+    Eq (Add (Mult (Cst go, Sub (hl, hm)), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Cst go, Mult (hn, ho)), fi), af)) =
+    Eq (Add (Mult (Cst go, Mult (hn, ho)), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Var gp, ge), fi), af)) =
+    Eq (Add (Mult (Var gp, ge), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Neg gq, ge), fi), af)) =
+    Eq (Add (Mult (Neg gq, ge), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Add (gr, gs), ge), fi), af)) =
+    Eq (Add (Mult (Add (gr, gs), ge), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Sub (gt, gu), ge), fi), af)) =
+    Eq (Add (Mult (Sub (gt, gu), ge), fi), af)
+  | adjustcoeff (l, Eq (Add (Mult (Mult (gv, gw), ge), fi), af)) =
+    Eq (Add (Mult (Mult (gv, gw), ge), fi), af)
+  | adjustcoeff (l, Eq (Sub (fj, fk), af)) = Eq (Sub (fj, fk), af)
+  | adjustcoeff (l, Eq (Mult (fl, fm), af)) = Eq (Mult (fl, fm), af)
+  | adjustcoeff (l, Divides (Cst is, Cst jk)) = Divides (Cst is, Cst jk)
+  | adjustcoeff (l, Divides (Cst is, Var jl)) = Divides (Cst is, Var jl)
+  | adjustcoeff (l, Divides (Cst is, Neg jm)) = Divides (Cst is, Neg jm)
+  | adjustcoeff (l, Divides (Cst is, Add (Cst kc, jo))) =
+    Divides (Cst is, Add (Cst kc, jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Var kd, jo))) =
+    Divides (Cst is, Add (Var kd, jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Neg ke, jo))) =
+    Divides (Cst is, Add (Neg ke, jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Add (kf, kg), jo))) =
+    Divides (Cst is, Add (Add (kf, kg), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Sub (kh, ki), jo))) =
+    Divides (Cst is, Add (Sub (kh, ki), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Cst ku, Cst lm), jo))) =
+    Divides (Cst is, Add (Mult (Cst ku, Cst lm), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Cst ku, Var me), jo))) =
+    (if (me = 0)
+      then Divides
+             (Cst (op_div_def1 l ku * is),
+               Add (Mult (Cst 1, Var 0), lin_mul (op_div_def1 l ku, jo)))
+      else Divides
+             (Cst is,
+               Add (Mult (Cst ku, Var (op_45_def0 me id_1_def0 + 1)), jo)))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Cst ku, Neg lo), jo))) =
+    Divides (Cst is, Add (Mult (Cst ku, Neg lo), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Cst ku, Add (lp, lq)), jo))) =
+    Divides (Cst is, Add (Mult (Cst ku, Add (lp, lq)), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Cst ku, Sub (lr, ls)), jo))) =
+    Divides (Cst is, Add (Mult (Cst ku, Sub (lr, ls)), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Cst ku, Mult (lt, lu)), jo))) =
+    Divides (Cst is, Add (Mult (Cst ku, Mult (lt, lu)), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Var kv, kk), jo))) =
+    Divides (Cst is, Add (Mult (Var kv, kk), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Neg kw, kk), jo))) =
+    Divides (Cst is, Add (Mult (Neg kw, kk), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Add (kx, ky), kk), jo))) =
+    Divides (Cst is, Add (Mult (Add (kx, ky), kk), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Sub (kz, la), kk), jo))) =
+    Divides (Cst is, Add (Mult (Sub (kz, la), kk), jo))
+  | adjustcoeff (l, Divides (Cst is, Add (Mult (Mult (lb, lc), kk), jo))) =
+    Divides (Cst is, Add (Mult (Mult (lb, lc), kk), jo))
+  | adjustcoeff (l, Divides (Cst is, Sub (jp, jq))) =
+    Divides (Cst is, Sub (jp, jq))
+  | adjustcoeff (l, Divides (Cst is, Mult (jr, js))) =
+    Divides (Cst is, Mult (jr, js))
+  | adjustcoeff (l, Divides (Var it, ah)) = Divides (Var it, ah)
+  | adjustcoeff (l, Divides (Neg iu, ah)) = Divides (Neg iu, ah)
+  | adjustcoeff (l, Divides (Add (iv, iw), ah)) = Divides (Add (iv, iw), ah)
+  | adjustcoeff (l, Divides (Sub (ix, iy), ah)) = Divides (Sub (ix, iy), ah)
+  | adjustcoeff (l, Divides (Mult (iz, ja), ah)) = Divides (Mult (iz, ja), ah)
+  | adjustcoeff (l, T) = T
+  | adjustcoeff (l, F) = F
+  | adjustcoeff (l, NOT (Lt (mg, mh))) = NOT (Lt (mg, mh))
+  | adjustcoeff (l, NOT (Gt (mi, mj))) = NOT (Gt (mi, mj))
+  | adjustcoeff (l, NOT (Le (mk, ml))) = NOT (Le (mk, ml))
+  | adjustcoeff (l, NOT (Ge (mm, mn))) = NOT (Ge (mm, mn))
+  | adjustcoeff (l, NOT (Eq (Cst oa, mp))) = NOT (Eq (Cst oa, mp))
+  | adjustcoeff (l, NOT (Eq (Var ob, mp))) = NOT (Eq (Var ob, mp))
+  | adjustcoeff (l, NOT (Eq (Neg oc, mp))) = NOT (Eq (Neg oc, mp))
+  | adjustcoeff (l, NOT (Eq (Add (Cst os, oe), mp))) =
+    NOT (Eq (Add (Cst os, oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Var ot, oe), mp))) =
+    NOT (Eq (Add (Var ot, oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Neg ou, oe), mp))) =
+    NOT (Eq (Add (Neg ou, oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Add (ov, ow), oe), mp))) =
+    NOT (Eq (Add (Add (ov, ow), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Sub (ox, oy), oe), mp))) =
+    NOT (Eq (Add (Sub (ox, oy), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Cst qc), oe), mp))) =
+    NOT (Eq (Add (Mult (Cst pk, Cst qc), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Var qx))) =
+    NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Var qx))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Neg qy))) =
+    NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Neg qy))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Add (qz, ra)))) =
+    NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Add (qz, ra)))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Sub (rb, rc)))) =
+    NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Sub (rb, rc)))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Mult (rd, re)))) =
+    NOT (Eq (Add (Mult (Cst pk, Var 0), oe), Mult (rd, re)))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Var qu), oe), mp))) =
+    NOT (Eq (Add (Mult (Cst pk, Var qu), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Neg qe), oe), mp))) =
+    NOT (Eq (Add (Mult (Cst pk, Neg qe), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Add (qf, qg)), oe), mp))) =
+    NOT (Eq (Add (Mult (Cst pk, Add (qf, qg)), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Sub (qh, qi)), oe), mp))) =
+    NOT (Eq (Add (Mult (Cst pk, Sub (qh, qi)), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Cst pk, Mult (qj, qk)), oe), mp))) =
+    NOT (Eq (Add (Mult (Cst pk, Mult (qj, qk)), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Var pl, pa), oe), mp))) =
+    NOT (Eq (Add (Mult (Var pl, pa), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Neg pm, pa), oe), mp))) =
+    NOT (Eq (Add (Mult (Neg pm, pa), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Add (pn, po), pa), oe), mp))) =
+    NOT (Eq (Add (Mult (Add (pn, po), pa), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Sub (pp, pq), pa), oe), mp))) =
+    NOT (Eq (Add (Mult (Sub (pp, pq), pa), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Add (Mult (Mult (pr, ps), pa), oe), mp))) =
+    NOT (Eq (Add (Mult (Mult (pr, ps), pa), oe), mp))
+  | adjustcoeff (l, NOT (Eq (Sub (of', og), mp))) = NOT (Eq (Sub (of', og), mp))
+  | adjustcoeff (l, NOT (Eq (Mult (oh, oi), mp))) = NOT (Eq (Mult (oh, oi), mp))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Cst sg))) =
+    NOT (Divides (Cst ro, Cst sg))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Var sh))) =
+    NOT (Divides (Cst ro, Var sh))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Neg si))) =
+    NOT (Divides (Cst ro, Neg si))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Cst sy, sk)))) =
+    NOT (Divides (Cst ro, Add (Cst sy, sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Var sz, sk)))) =
+    NOT (Divides (Cst ro, Add (Var sz, sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Neg ta, sk)))) =
+    NOT (Divides (Cst ro, Add (Neg ta, sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Add (tb, tc), sk)))) =
+    NOT (Divides (Cst ro, Add (Add (tb, tc), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Sub (td, te), sk)))) =
+    NOT (Divides (Cst ro, Add (Sub (td, te), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Mult (Cst tq, Cst ui), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Cst tq, Cst ui), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Mult (Cst tq, Var va), sk)))) =
+    (if (va = 0)
+      then NOT (Divides
+                  (Cst (op_div_def1 l tq * ro),
+                    Add (Mult (Cst 1, Var 0), lin_mul (op_div_def1 l tq, sk))))
+      else NOT (Divides
+                  (Cst ro,
+                    Add (Mult (Cst tq, Var (op_45_def0 va id_1_def0 + 1)),
+                          sk))))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Mult (Cst tq, Neg uk), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Cst tq, Neg uk), sk)))
+  | adjustcoeff
+      (l, NOT (Divides (Cst ro, Add (Mult (Cst tq, Add (ul, um)), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Cst tq, Add (ul, um)), sk)))
+  | adjustcoeff
+      (l, NOT (Divides (Cst ro, Add (Mult (Cst tq, Sub (un, uo)), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Cst tq, Sub (un, uo)), sk)))
+  | adjustcoeff
+      (l, NOT (Divides (Cst ro, Add (Mult (Cst tq, Mult (up, uq)), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Cst tq, Mult (up, uq)), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Mult (Var tr, tg), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Var tr, tg), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Mult (Neg ts, tg), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Neg ts, tg), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Mult (Add (tt, tu), tg), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Add (tt, tu), tg), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Add (Mult (Sub (tv, tw), tg), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Sub (tv, tw), tg), sk)))
+  | adjustcoeff
+      (l, NOT (Divides (Cst ro, Add (Mult (Mult (tx, ty), tg), sk)))) =
+    NOT (Divides (Cst ro, Add (Mult (Mult (tx, ty), tg), sk)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Sub (sl, sm)))) =
+    NOT (Divides (Cst ro, Sub (sl, sm)))
+  | adjustcoeff (l, NOT (Divides (Cst ro, Mult (sn, so)))) =
+    NOT (Divides (Cst ro, Mult (sn, so)))
+  | adjustcoeff (l, NOT (Divides (Var rp, mr))) = NOT (Divides (Var rp, mr))
+  | adjustcoeff (l, NOT (Divides (Neg rq, mr))) = NOT (Divides (Neg rq, mr))
+  | adjustcoeff (l, NOT (Divides (Add (rr, rs), mr))) =
+    NOT (Divides (Add (rr, rs), mr))
+  | adjustcoeff (l, NOT (Divides (Sub (rt, ru), mr))) =
+    NOT (Divides (Sub (rt, ru), mr))
+  | adjustcoeff (l, NOT (Divides (Mult (rv, rw), mr))) =
+    NOT (Divides (Mult (rv, rw), mr))
+  | adjustcoeff (l, NOT T) = NOT T
+  | adjustcoeff (l, NOT F) = NOT F
+  | adjustcoeff (l, NOT (NOT ms)) = NOT (NOT ms)
+  | adjustcoeff (l, NOT (And (mt, mu))) = NOT (And (mt, mu))
+  | adjustcoeff (l, NOT (Or (mv, mw))) = NOT (Or (mv, mw))
+  | adjustcoeff (l, NOT (Imp (mx, my))) = NOT (Imp (mx, my))
+  | adjustcoeff (l, NOT (Equ (mz, na))) = NOT (Equ (mz, na))
+  | adjustcoeff (l, NOT (QAll nb)) = NOT (QAll nb)
+  | adjustcoeff (l, NOT (QEx nc)) = NOT (QEx nc)
+  | adjustcoeff (l, Imp (an, ao)) = Imp (an, ao)
+  | adjustcoeff (l, Equ (ap, aq)) = Equ (ap, aq)
+  | adjustcoeff (l, QAll ar) = QAll ar
+  | adjustcoeff (l, QEx as') = QEx as';
+
+fun formlcm (Le (Add (Mult (Cst c, Var 0), r), Cst i)) = abs c
+  | formlcm (Eq (Add (Mult (Cst c, Var 0), r), Cst i)) = abs c
+  | formlcm (NOT p) = formlcm p
+  | formlcm (And (p, q)) = ilcm (formlcm p) (formlcm q)
+  | formlcm (Or (p, q)) = ilcm (formlcm p) (formlcm q)
+  | formlcm (Lt (u, v)) = 1
+  | formlcm (Gt (w, x)) = 1
+  | formlcm (Le (Cst bo, z)) = 1
+  | formlcm (Le (Var bp, z)) = 1
+  | formlcm (Le (Neg bq, z)) = 1
+  | formlcm (Le (Add (Cst cg, bs), z)) = 1
+  | formlcm (Le (Add (Var ch, bs), z)) = 1
+  | formlcm (Le (Add (Neg ci, bs), z)) = 1
+  | formlcm (Le (Add (Add (cj, ck), bs), z)) = 1
+  | formlcm (Le (Add (Sub (cl, cm), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Cst dq), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Var 0), bs), Var el)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Var 0), bs), Neg em)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Var 0), bs), Add (en, eo))) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Var 0), bs), Sub (ep, eq))) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Var 0), bs), Mult (er, es))) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Var ei ), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Neg ds), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Add (dt, du)), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Sub (dv, dw)), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Cst cy, Mult (dx, dy)), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Var cz, co), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Neg da, co), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Add (db, dc), co), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Sub (dd, de), co), bs), z)) = 1
+  | formlcm (Le (Add (Mult (Mult (df, dg), co), bs), z)) = 1
+  | formlcm (Le (Sub (bt, bu), z)) = 1
+  | formlcm (Le (Mult (bv, bw), z)) = 1
+  | formlcm (Ge (aa, ab)) = 1
+  | formlcm (Eq (Cst fc, ad)) = 1
+  | formlcm (Eq (Var fd, ad)) = 1
+  | formlcm (Eq (Neg fe, ad)) = 1
+  | formlcm (Eq (Add (Cst fu, fg), ad)) = 1
+  | formlcm (Eq (Add (Var fv, fg), ad)) = 1
+  | formlcm (Eq (Add (Neg fw, fg), ad)) = 1
+  | formlcm (Eq (Add (Add (fx, fy), fg), ad)) = 1
+  | formlcm (Eq (Add (Sub (fz, ga), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Cst he), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Var 0), fg), Var hz)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Var 0), fg), Neg ia)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Var 0), fg), Add (ib, ic))) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Var 0), fg), Sub (id, ie))) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Var 0), fg), Mult (if', ig))) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Var hw), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Neg hg), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Add (hh, hi)), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Sub (hj, hk)), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Cst gm, Mult (hl, hm)), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Var gn, gc), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Neg go, gc), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Add (gp, gq), gc), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Sub (gr, gs), gc), fg), ad)) = 1
+  | formlcm (Eq (Add (Mult (Mult (gt, gu), gc), fg), ad)) = 1
+  | formlcm (Eq (Sub (fh, fi), ad)) = 1
+  | formlcm (Eq (Mult (fj, fk), ad)) = 1
+  | formlcm (Divides (Cst iq, Cst ji)) = 1
+  | formlcm (Divides (Cst iq, Var jj)) = 1
+  | formlcm (Divides (Cst iq, Neg jk)) = 1
+  | formlcm (Divides (Cst iq, Add (Cst ka, jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Var kb, jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Neg kc, jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Add (kd, ke), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Sub (kf, kg), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Cst ks, Cst lk), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Cst ks, Var mc), jm))) =
+    (if (mc = 0) then abs ks else 1)
+  | formlcm (Divides (Cst iq, Add (Mult (Cst ks, Neg lm), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Cst ks, Add (ln, lo)), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Cst ks, Sub (lp, lq)), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Cst ks, Mult (lr, ls)), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Var kt, ki), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Neg ku, ki), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Add (kv, kw), ki), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Sub (kx, ky), ki), jm))) = 1
+  | formlcm (Divides (Cst iq, Add (Mult (Mult (kz, la), ki), jm))) = 1
+  | formlcm (Divides (Cst iq, Sub (jn, jo))) = 1
+  | formlcm (Divides (Cst iq, Mult (jp, jq))) = 1
+  | formlcm (Divides (Var ir, af)) = 1
+  | formlcm (Divides (Neg is, af)) = 1
+  | formlcm (Divides (Add (it, iu), af)) = 1
+  | formlcm (Divides (Sub (iv, iw), af)) = 1
+  | formlcm (Divides (Mult (ix, iy), af)) = 1
+  | formlcm T = 1
+  | formlcm F = 1
+  | formlcm (Imp (al, am)) = 1
+  | formlcm (Equ (an, ao)) = 1
+  | formlcm (QAll ap) = 1
+  | formlcm (QEx aq) = 1;
+
+fun unitycoeff p =
+  let val l = formlcm p; val p' = adjustcoeff (l, p)
+  in (if (l = 1) then p'
+       else And (Divides (Cst l, Add (Mult (Cst 1, Var 0), Cst 0)), p'))
+  end;
+
+fun unify p =
+  let val q = unitycoeff p; val B = list_set (bset q); val A = list_set (aset q)
+  in (if op_60_61_def0 (size_def1 B) (size_def1 A) then (q, B)
+       else (mirror q, map lin_neg A))
+  end;
+
+fun cooper p =
+  lift_un (fn q => decrvars (explode_minf (unify q))) (linform (nnf p));
+
+fun pa p = lift_un psimpl (qelim (cooper, p));
+
+val test = pa;
+
+end;
diff -r afaa031ed4da -r 105519771c67 src/HOL/Tools/comm_ring.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/comm_ring.ML	Wed Sep 14 17:25:52 2005 +0200
@@ -0,0 +1,134 @@
+(*  ID:         $Id$
+    Author:     Amine Chaieb
+
+	     Tactic for solving equalities over commutative rings *)
+signature CRING =
+sig
+    val cring_tac : int -> tactic
+    val cring_method: int -> Proof.method
+    val setup : (theory -> theory) list
+end
+structure CommRing :CRING = 
+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("0",T);
+fun cring_one T = Const("1",T);
+
+(* reification functions *)
+(* add two polynom expressions *)
+fun polT t = Type ("Commutative_Ring.pol",[t]);
+fun  polexT t = Type("Commutative_Ring.polex",[t]);
+val nT = HOLogic.natT;
+fun listT T = Type ("List.list",[T]);
+
+(* Reification of the constructors *)
+(* Nat*)
+val succ = Const("Suc",nT --> nT);
+val zero = Const("0",nT);
+val one = Const("1",nT);
+
+(* Lists *)
+fun reif_list T [] = Const("List.list.Nil",listT T)
+  | reif_list T (x::xs) = Const("List.list.Cons",[T,listT T] ---> listT T)
+			     $x$(reif_list T xs);
+
+(* pol*)
+fun pol_Pc t = Const("Commutative_Ring.pol.Pc",t --> polT t);
+fun pol_Pinj t = Const("Commutative_Ring.pol.Pinj",[nT,polT t] ---> polT t);
+fun pol_PX t = Const("Commutative_Ring.pol.PX",[polT t, nT, 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, nT] ---> polexT t);
+(* reification of natural numbers *)
+fun reif_nat n = 
+    if n>0 then succ$(reif_nat (n-1))
+    else if n=0 then zero
+    else raise CRing "ring_tac: reif_nat negative n";
+
+(* reification of polynoms : primitive cring expressions *)
+fun reif_pol T vs t =
+    case t of
+       Free(_,_) =>
+	let val i = find_index_eq t vs 
+	in if i = 0 
+	   then (pol_PX T)$((pol_Pc T)$ (cring_one T))
+			  $one$((pol_Pc T)$(cring_zero T)) 
+	   else (pol_Pinj T)$(reif_nat i)$
+			    ((pol_PX T)$((pol_Pc T)$ (cring_one T))
+				       $one$
+				       ((pol_Pc T)$(cring_zero T)))
+	end 
+      | _ => (pol_Pc T)$ t;
+
+
+(* reification of polynom expressions *)
+fun reif_polex T vs t =
+    case t of
+	Const("op +",_)$a$b => (polex_add T) 
+				   $ (reif_polex T vs a)$(reif_polex T vs b)
+      |	Const("op -",_)$a$b => (polex_sub T) 
+				   $ (reif_polex T vs a)$(reif_polex T vs b)
+      | Const("op *",_)$a$b => 	(polex_mul T) 
+				    $ (reif_polex T vs a)$ (reif_polex T vs b)
+      | Const("uminus",_)$a => (polex_neg T) 
+				   $ (reif_polex T vs a)
+      | (Const("Nat.power",_)$a$n) => (polex_pow T) $ (reif_polex T vs a) $ n
+
+      | _ => (polex_pol T) $ (reif_pol T vs t);
+
+(* reification of the equation *)
+val cr_sort = Sign.read_sort (the_context ()) "{comm_ring,recpower}";
+fun reif_eq sg (eq as Const("op =",Type("fun",a::_))$lhs$rhs) = 
+    if Sign.of_sort (the_context()) (a,cr_sort)
+    then 
+	let val fs = term_frees eq
+	    val cvs = cterm_of sg (reif_list a fs)
+	    val clhs = cterm_of sg (reif_polex a fs lhs)
+	    val crhs = cterm_of sg (reif_polex a fs rhs)
+	    val ca = ctyp_of sg a
+	in (ca,cvs,clhs, crhs)
+	end
+    else raise CRing "reif_eq: not an equation over comm_ring + recpower"
+  | reif_eq sg _ = 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_ss = simpset_of (the_context())
+			   addsimps 
+			   (map thm ["mkPX_def", "mkPinj_def","sub_def",
+				     "power_add","even_def","pow_if"])
+			   addsimps [sym OF [thm "power_add"]];
+
+val norm_eq = thm "norm_eq"
+fun comm_ring_tac i =(fn st => 
+    let
+	val g = BasisLibrary.List.nth (prems_of st, i - 1)
+	val sg = sign_of_thm st
+	val (ca,cvs,clhs,crhs) = reif_eq sg (HOLogic.dest_Trueprop g)
+	val norm_eq_th = simplify cring_ss 
+			(instantiate' [SOME ca] [SOME clhs, SOME crhs, SOME cvs] 
+						norm_eq)
+    in ((cut_rules_tac [norm_eq_th] i) 
+	    THEN (simp_tac cring_ss i) 
+	    THEN (simp_tac cring_ss i)) st
+    end);
+
+fun cring_method i = Method.METHOD (fn facts =>
+  Method.insert_tac facts 1 THEN comm_ring_tac i);
+
+val setup =
+  [Method.add_method ("comm_ring",
+     Method.no_args (cring_method 1),
+     "reflective decision procedure for equalities over commutative rings")];
+
+end;
\ No newline at end of file
diff -r afaa031ed4da -r 105519771c67 src/HOL/ex/Commutative_RingEx.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Commutative_RingEx.thy	Wed Sep 14 17:25:52 2005 +0200
@@ -0,0 +1,47 @@
+ (* $Id$ *)
+theory Commutative_RingEx
+imports Main
+begin
+
+  (* Some examples demonstrating the comm_ring method *)
+
+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
\ No newline at end of file
diff -r afaa031ed4da -r 105519771c67 src/HOL/ex/Commutative_Ring_Complete.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Commutative_Ring_Complete.thy	Wed Sep 14 17:25:52 2005 +0200
@@ -0,0 +1,384 @@
+(*  ID:         $Id$
+    Author:     Bernhard Haeupler
+
+  This theory is about the relative completeness of the tactic 
+  As long as the reified atomic polynomials of type 'a pol 
+  are in normal form, the cring method is complete *)	
+
+theory Commutative_Ring_Complete
+imports Main
+begin
+	
+  (* Fromalization of normal form *)
+consts isnorm :: "('a::{comm_ring,recpower}) pol \<Rightarrow> bool"
+recdef isnorm "measure size"
+  "isnorm (Pc c) = True"
+  "isnorm (Pinj i (Pc c)) = False"
+  "isnorm (Pinj i (Pinj j Q)) = False"
+  "isnorm (Pinj 0 P) = False"
+  "isnorm (Pinj i (PX Q1 j Q2)) = isnorm (PX Q1 j Q2)"
+  "isnorm (PX P 0 Q) = False"
+  "isnorm (PX (Pc c) i Q) = (c \<noteq> 0 & isnorm Q)"
+  "isnorm (PX (PX P1 j (Pc c)) i Q) = (c\<noteq>0 \<and> isnorm(PX P1 j (Pc c))\<and>isnorm Q)"
+  "isnorm (PX P i Q) = (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
+
+    (* mkPX conserves normalizedness (_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
+
+    (* add conserves normalizedness *)
+lemma add_cn:"\<lbrakk>isnorm P; (isnorm Q)\<rbrakk> \<Longrightarrow> isnorm (add (P, 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 (add (R, 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( add (R, Pinj (x - 1) P2))" "isnorm(PX Q2 y R)" by simp
+    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 (add (R, 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( add (R, Pinj (x - 1) P2))" "isnorm(PX Q2 y R)" by simp
+    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 (add (P2, Q2))" "isnorm (add (PX P1 (x - y) (Pc 0), 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 (add (P2, Q2))" "isnorm (add (P1, 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 (add (P2, Q2))" "isnorm (add (PX Q1 (y - x) (Pc 0), P1))" by auto
+    with X0 sm1 sm2 have ?case by (simp add: mkPX_cn)}
+  ultimately show ?case by blast
+qed(simp)
+
+    (* mul concerves normalizedness *)
+lemma mul_cn :"\<lbrakk>isnorm P; (isnorm Q)\<rbrakk> \<Longrightarrow> isnorm (mul (P, 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 (mul (R, 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 (mul (R, Pinj (x - 1) P2))" "isnorm (mul (Pinj x P2, 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 (mul (R, 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 (mul (R, Pinj (x - 1) P2))" "isnorm (mul (Pinj x P2, 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 (mul (P1, Q1))" "isnorm (mul (P2, Q2))" "isnorm (mul (P1, mkPinj 1 Q2))" "isnorm (mul (Q1, mkPinj 1 P2))" 
+    by (auto simp add: mkPinj_cn)
+  with prems X0 Y0 have "isnorm (mkPX (mul (P1, Q1)) (x + y) (mul (P2, Q2)))" "isnorm (mkPX (mul (P1, mkPinj (Suc 0) Q2)) x (Pc 0))"  
+    "isnorm (mkPX (mul (Q1, mkPinj (Suc 0) P2)) y (Pc 0))" 
+    by (auto simp add: mkPX_cn)
+  thus ?case by (simp add: add_cn)
+qed(simp)
+
+    (* neg conserves normalizedness *)
+lemma neg_cn: "isnorm P \<Longrightarrow> isnorm (neg P)"
+proof(induct P rule: neg.induct)
+  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)
+
+    (* sub conserves normalizedness *)
+lemma sub_cn:"\<lbrakk>isnorm p; isnorm q\<rbrakk> \<Longrightarrow> isnorm (sub p q)"
+by (simp add: sub_def add_cn neg_cn)
+
+  (* 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 (mul (mul (Pc ((1\<Colon>'a) + (1\<Colon>'a)), P1), mkPinj (Suc 0) P2)) x (Pc (0\<Colon>'a)))"
+              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)
+
+
+    (* pow conserves normalizedness  *)
+lemma pow_cn:"!! P. \<lbrakk>isnorm P\<rbrakk> \<Longrightarrow> isnorm (pow (P, n))"
+proof(induct n rule: nat_less_induct)
+  case (1 k)
+  show ?case 
+  proof(cases "k=0")
+    case False
+    hence 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
\ No newline at end of file
diff -r afaa031ed4da -r 105519771c67 src/HOL/ex/ROOT.ML
--- a/src/HOL/ex/ROOT.ML	Wed Sep 14 10:24:39 2005 +0200
+++ b/src/HOL/ex/ROOT.ML	Wed Sep 14 17:25:52 2005 +0200
@@ -25,6 +25,7 @@
 time_use_thy "CTL";
 time_use_thy "mesontest2";
 time_use_thy "PresburgerEx";
+time_use_thy "Reflected_Presburger";
 time_use_thy "BT";
 time_use_thy "InSort";
 time_use_thy "Qsort";
@@ -33,6 +34,8 @@
 
 
 time_use_thy "Lagrange";
+time_use_thy "Commutative_RingEx";
+time_use_thy "Commutative_Ring_Complete";
 
 time_use_thy "set";
 time_use_thy "MT";
diff -r afaa031ed4da -r 105519771c67 src/HOL/ex/Reflected_Presburger.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/ex/Reflected_Presburger.thy	Wed Sep 14 17:25:52 2005 +0200
@@ -0,0 +1,5726 @@
+(* A formalization of quantifier elimination for Presburger arithmetic*)
+(* based on a generic quantifier elimination algorithm and 
+   cooper's methos to eliminate on \<exists> *)
+
+theory Reflected_Presburger
+imports Main
+uses ("rcooper.ML") ("rpresbtac.ML")
+begin
+
+(* Shadow syntax for integer terms *)
+datatype intterm =
+    Cst int
+  | Var nat
+  | Neg intterm
+  | Add intterm intterm 
+  | Sub intterm intterm
+  | Mult intterm intterm
+
+(* interpretatation of intterms , takes bound variables and free variables *)
+consts I_intterm :: "int list \<Rightarrow> intterm \<Rightarrow> int"
+primrec 
+"I_intterm ats (Cst b) = b"
+"I_intterm ats (Var n) = (ats!n)"
+"I_intterm ats (Neg it) = -(I_intterm ats it)"
+"I_intterm ats (Add it1 it2) = (I_intterm ats it1) + (I_intterm ats it2)" 
+"I_intterm ats (Sub it1 it2) = (I_intterm ats it1) - (I_intterm ats it2)"
+"I_intterm ats (Mult it1 it2) = (I_intterm ats it1) * (I_intterm ats it2)"
+
+(*Shadow syntax for Presburger formulae *)
+datatype QF = 
+   Lt intterm intterm
+  |Gt intterm intterm
+  |Le intterm intterm
+  |Ge intterm intterm
+  |Eq intterm intterm
+  |Divides intterm intterm
+  |T
+  |F
+  |NOT QF
+  |And QF QF
+  |Or QF QF
+  |Imp QF QF
+  |Equ QF QF
+  |QAll QF
+  |QEx QF
+
+(* Interpretation of Presburger  formulae *)
+consts qinterp :: "int list \<Rightarrow> QF \<Rightarrow> bool"
+primrec
+"qinterp ats (Lt it1 it2) = (I_intterm ats it1 < I_intterm ats it2)"
+"qinterp ats (Gt it1 it2) = (I_intterm ats it1 > I_intterm ats it2)"
+"qinterp ats (Le it1 it2) = (I_intterm ats it1 \<le> I_intterm ats it2)"
+"qinterp ats (Ge it1 it2) = (I_intterm ats it1 \<ge> I_intterm ats it2)"
+"qinterp ats (Divides it1 it2) = (I_intterm ats it1 dvd I_intterm ats it2)"
+"qinterp ats (Eq it1 it2) = (I_intterm ats it1 = I_intterm ats it2)"
+"qinterp ats T = True"
+"qinterp ats F = False"
+"qinterp ats (NOT p) = (\<not>(qinterp ats p))"
+"qinterp ats (And p q) = (qinterp ats p \<and> qinterp ats q)"
+"qinterp ats (Or p q) = (qinterp ats p \<or> qinterp ats q)"
+"qinterp ats (Imp p q) = (qinterp ats p \<longrightarrow> qinterp ats q)"
+"qinterp ats (Equ p q) = (qinterp ats p = qinterp ats q)"
+"qinterp ats (QAll p) = (\<forall>x. qinterp (x#ats) p)"
+"qinterp ats (QEx p) = (\<exists>x. qinterp (x#ats) p)"
+
+(* quantifier elimination based on qe, quantifier elimination for an
+   existential formula, with quantifier free body 
+   Since quantifier elimination for one formula is allowed to fail, 
+   the reult is of type QF option *)
+
+consts lift_bin:: "('a \<Rightarrow> 'a \<Rightarrow> 'b) \<times> 'a option \<times> 'a option \<Rightarrow> 'b option"
+recdef lift_bin "measure (\<lambda>(c,a,b). size a)"
+"lift_bin (c,Some a,Some b) = Some (c a b)"
+"lift_bin (c,x, y)  = None"
+
+lemma lift_bin_Some:
+  assumes ls: "lift_bin (c,x,y) = Some t"
+  shows "(\<exists>a. x = Some a) \<and> (\<exists>b. y = Some b)"
+  using ls 
+  by (cases "x", auto) (cases "y", auto)+
+
+consts lift_un:: "('a \<Rightarrow> 'b) \<Rightarrow> 'a option \<Rightarrow> 'b option"
+primrec
+"lift_un c None = None"
+"lift_un c (Some p) = Some (c p)"
+
+consts lift_qe:: "('a \<Rightarrow> 'b option) \<Rightarrow> 'a option \<Rightarrow> 'b option"
+primrec
+"lift_qe qe None = None"
+"lift_qe qe (Some p) = qe p"
+
+(* generic quantifier elimination *)
+consts qelim :: "(QF \<Rightarrow> QF option) \<times> QF \<Rightarrow> QF option"
+recdef qelim "measure (\<lambda>(qe,p). size p)"
+"qelim (qe, (QAll p)) = lift_un NOT (lift_qe qe (lift_un NOT (qelim (qe ,p))))"
+"qelim (qe, (QEx p)) = lift_qe qe (qelim (qe,p))"
+"qelim (qe, (And p q)) = lift_bin (And, (qelim (qe, p)), (qelim (qe, q)))"
+"qelim (qe, (Or p q)) = lift_bin (Or, (qelim (qe, p)), (qelim (qe, q)))"
+"qelim (qe, (Imp p q)) = lift_bin (Imp, (qelim (qe, p)), (qelim (qe, q)))"
+"qelim (qe, (Equ p q)) = lift_bin (Equ, (qelim (qe, p)), (qelim (qe, q)))"
+"qelim (qe,NOT p) = lift_un NOT (qelim (qe,p))"
+"qelim (qe, p) = Some p"
+
+(* quantifier free-ness *)
+consts isqfree :: "QF \<Rightarrow> bool"
+recdef isqfree "measure size"
+"isqfree (QAll p) = False"
+"isqfree (QEx p) = False"
+"isqfree (And p q) = (isqfree p \<and> isqfree q)"
+"isqfree (Or p q) = (isqfree p \<and> isqfree q)"
+"isqfree (Imp p q) = (isqfree p \<and> isqfree q)"
+"isqfree (Equ p q) = (isqfree p \<and> isqfree q)"
+"isqfree (NOT p) = isqfree p"
+"isqfree p = True"
+
+(* qelim lifts quantifierfreeness*)
+lemma qelim_qfree: 
+  assumes qeqf: "(\<And> q q'. \<lbrakk>isqfree q ; qe q = Some q'\<rbrakk> \<Longrightarrow>  isqfree q')"
+  shows qff:"\<And> p'. qelim (qe, p) = Some p' \<Longrightarrow> isqfree p'"
+  using qeqf
+proof (induct p)
+  case (Lt a b)
+  have "qelim (qe, Lt a b) = Some (Lt a b)" by simp
+  moreover have "qelim (qe,Lt a b) = Some p'" . 
+  ultimately have "p' = Lt a b" by simp
+  moreover have "isqfree (Lt a b)" by simp
+  ultimately 
+  show ?case  by simp
+next  
+  case (Gt a b)
+  have "qelim (qe, Gt a b) = Some (Gt a b)" by simp
+  moreover have "qelim (qe,Gt a b) = Some p'" . 
+  ultimately have "p' = Gt a b" by simp
+  moreover have "isqfree (Gt a b)" by simp
+  ultimately 
+  show ?case  by simp
+next  
+  case (Le a b)
+  have "qelim (qe, Le a b) = Some (Le a b)" by simp
+  moreover have "qelim (qe,Le a b) = Some p'" . 
+  ultimately have "p' = Le a b" by simp
+  moreover have "isqfree (Le a b)" by simp
+  ultimately 
+  show ?case  by simp
+next  
+  case (Ge a b)
+  have "qelim (qe, Ge a b) = Some (Ge a b)" by simp
+  moreover have "qelim (qe,Ge a b) = Some p'" . 
+  ultimately have "p' = Ge a b" by simp
+  moreover have "isqfree (Ge a b)" by simp
+  ultimately 
+  show ?case  by simp
+next  
+  case (Eq a b)
+  have "qelim (qe, Eq a b) = Some (Eq a b)" by simp
+  moreover have "qelim (qe,Eq a b) = Some p'" . 
+  ultimately have "p' = Eq a b" by simp
+  moreover have "isqfree (Eq a b)" by simp
+  ultimately 
+  show ?case  by simp
+next  
+  case (Divides a b)
+  have "qelim (qe, Divides a b) = Some (Divides a b)" by simp
+  moreover have "qelim (qe,Divides a b) = Some p'" . 
+  ultimately have "p' = Divides a b" by simp
+  moreover have "isqfree (Divides a b)" by simp
+  ultimately 
+  show ?case  by simp
+next  
+  case T 
+  have "qelim(qe,T) = Some T" by simp
+  moreover have "qelim(qe,T) = Some p'" .
+  ultimately have "p' = T" by simp
+  moreover have "isqfree T" by simp
+  ultimately show ?case by simp
+next  
+  case F
+  have "qelim(qe,F) = Some F" by simp
+  moreover have "qelim(qe,F) = Some p'" .
+  ultimately have "p' = F" by simp
+  moreover have "isqfree F" by simp
+  ultimately show ?case by simp
+next  
+  case (NOT p)
+  from "NOT.prems" have "\<exists> p1. qelim(qe,p) = Some p1"
+    by  (cases "qelim(qe,p)") simp_all
+  then obtain "p1" where p1_def: "qelim(qe,p) = Some p1" by blast
+  from "NOT.prems" have "\<And>q q'. \<lbrakk>isqfree q; qe q = Some q'\<rbrakk> \<Longrightarrow> isqfree q'" 
+    by blast
+  with "NOT.hyps" p1_def have p1qf: "isqfree p1" by blast
+  then have "p' = NOT p1" using "NOT.prems" p1_def
+    by (cases "qelim(qe,NOT p)") simp_all
+  then show ?case using p1qf by simp
+next  
+  case (And p q) 
+  from "And.prems" have p1q1: "(\<exists>p1. qelim(qe,p) = Some p1) \<and> 
+    (\<exists>q1. qelim(qe,q) = Some q1)" using lift_bin_Some[where c="And"] by simp
+  from p1q1 obtain "p1" and "q1" 
+    where p1_def: "qelim(qe,p) = Some p1" 
+    and q1_def: "qelim(qe,q) = Some q1" by blast
+  from prems have qf1:"isqfree p1"
+    using p1_def by blast
+  from prems have qf2:"isqfree q1"
+    using q1_def by blast
+  from "And.prems" have "qelim(qe,And p q) = Some p'" by blast
+  then have "p' = And p1 q1" using p1_def q1_def by simp
+  then 
+  show ?case using qf1 qf2 by simp
+next  
+  case (Or p q)
+  from "Or.prems" have p1q1: "(\<exists>p1. qelim(qe,p) = Some p1) \<and> 
+    (\<exists>q1. qelim(qe,q) = Some q1)" using lift_bin_Some[where c="Or"] by simp
+  from p1q1 obtain "p1" and "q1" 
+    where p1_def: "qelim(qe,p) = Some p1" 
+    and q1_def: "qelim(qe,q) = Some q1" by blast
+  from prems have qf1:"isqfree p1"
+    using p1_def by blast
+  from prems have qf2:"isqfree q1"
+    using q1_def by blast
+  from "Or.prems" have "qelim(qe,Or p q) = Some p'" by blast
+  then have "p' = Or p1 q1" using p1_def q1_def by simp
+  then 
+  show ?case using qf1 qf2 by simp
+next  
+  case (Imp p q)
+  from "Imp.prems" have p1q1: "(\<exists>p1. qelim(qe,p) = Some p1) \<and> 
+    (\<exists>q1. qelim(qe,q) = Some q1)" using lift_bin_Some[where c="Imp"] by simp
+  from p1q1 obtain "p1" and "q1" 
+    where p1_def: "qelim(qe,p) = Some p1" 
+    and q1_def: "qelim(qe,q) = Some q1" by blast
+  from prems have qf1:"isqfree p1"
+    using p1_def by blast
+  from prems have qf2:"isqfree q1"
+    using q1_def by blast
+  from "Imp.prems" have "qelim(qe,Imp p q) = Some p'" by blast
+  then have "p' = Imp p1 q1" using p1_def q1_def by simp
+  then 
+  show ?case using qf1 qf2 by simp
+next  
+  case (Equ p q)
+  from "Equ.prems" have p1q1: "(\<exists>p1. qelim(qe,p) = Some p1) \<and> 
+    (\<exists>q1. qelim(qe,q) = Some q1)" using lift_bin_Some[where c="Equ"] by simp
+  from p1q1 obtain "p1" and "q1" 
+    where p1_def: "qelim(qe,p) = Some p1" 
+    and q1_def: "qelim(qe,q) = Some q1" by blast
+  from prems have qf1:"isqfree p1"
+    using p1_def by blast
+  from prems have qf2:"isqfree q1"
+    using q1_def by blast
+  from "Equ.prems" have "qelim(qe,Equ p q) = Some p'" by blast
+  then have "p' = Equ p1 q1" using p1_def q1_def by simp
+  then 
+  show ?case using qf1 qf2 by simp
+next 
+  case (QEx p)
+  from "QEx.prems" have "\<exists> p1. qelim(qe,p) = Some p1"
+    by  (cases "qelim(qe,p)") simp_all
+  then obtain "p1" where p1_def: "qelim(qe,p) = Some p1" by blast
+  from "QEx.prems" have "\<And>q q'. \<lbrakk>isqfree q; qe q = Some q'\<rbrakk> \<Longrightarrow> isqfree q'" 
+    by blast
+  with "QEx.hyps" p1_def have p1qf: "isqfree p1" by blast
+  from "QEx.prems" have "qe p1 = Some p'" using p1_def by simp
+  with "QEx.prems" show ?case  using p1qf 
+    by simp
+next 
+  case (QAll p) 
+  from "QAll.prems"
+  have "\<exists> p1. lift_qe qe (lift_un NOT (qelim (qe ,p))) = Some p1" 
+    by (cases "lift_qe qe (lift_un NOT (qelim (qe ,p)))") simp_all
+  then obtain "p1" where 
+    p1_def:"lift_qe qe (lift_un NOT (qelim (qe ,p))) = Some p1" by blast
+  then have "\<exists> p2. lift_un NOT (qelim (qe ,p)) = Some p2"
+    by (cases "qelim (qe ,p)") simp_all
+  then obtain "p2" 
+    where p2_def:"lift_un NOT (qelim (qe ,p)) = Some p2" by blast
+  then have "\<exists> p3. qelim(qe,p) = Some p3" by (cases "qelim(qe,p)") simp_all
+  then obtain "p3" where p3_def: "qelim(qe,p) = Some p3" by blast
+  with prems have qf3: "isqfree p3" by blast
+  have p2_def2: "p2 = NOT p3" using p2_def p3_def by simp
+  then have qf2: "isqfree p2" using qf3 by simp
+  have p1_edf2: "qe p2 = Some p1" using p1_def p2_def by simp
+  with "QAll.prems" have qf1: "isqfree p1" using qf2 by blast
+  from "QAll.prems" have "p' = NOT p1" using p1_def by simp
+  with qf1 show ?case by simp
+qed
+
+(* qelim lifts semantical equivalence *)
+lemma qelim_corr: 
+  assumes qecorr: "(\<And> q q' ats. \<lbrakk>isqfree q ; qe q = Some q'\<rbrakk> \<Longrightarrow>  (qinterp ats (QEx q)) = (qinterp ats q'))"
+  and qeqf: "(\<And> q q'. \<lbrakk>isqfree q ; qe q = Some q'\<rbrakk> \<Longrightarrow>  isqfree q')"
+  shows qff:"\<And> p' ats. qelim (qe, p) = Some p' \<Longrightarrow> (qinterp ats p = qinterp ats p')" (is "\<And> p' ats. ?Qe p p' \<Longrightarrow> (?F ats p = ?F ats p')")
+  using qeqf qecorr
+proof (induct p)
+  case (NOT f)  
+  from "NOT.prems" have "\<exists>f'. ?Qe f f' " by (cases "qelim(qe,f)") simp_all
+  then obtain "f'" where df': "?Qe f f'" by blast
+  with prems have feqf': "?F ats f = ?F ats f'" by blast
+  from "NOT.prems" df' have "p' = NOT f'" by simp
+  with feqf' show ?case by simp
+
+next
+  case (And f g) 
+  from "And.prems" have f1g1: "(\<exists>f1. qelim(qe,f) = Some f1) \<and> 
+    (\<exists>g1. qelim(qe,g) = Some g1)" using lift_bin_Some[where c="And"] by simp
+  from f1g1 obtain "f1" and "g1" 
+    where f1_def: "qelim(qe, f) = Some f1" 
+    and g1_def: "qelim(qe,g) = Some g1" by blast
+  from prems f1_def have feqf1: "?F ats f = ?F ats f1" by blast
+  from prems g1_def have geqg1: "?F ats g = ?F ats g1" by blast
+  from "And.prems" f1_def g1_def have "p' = And f1 g1" by simp
+  with feqf1 geqg1 show ?case by simp
+
+next
+  case (Or f g) 
+  from "Or.prems" have f1g1: "(\<exists>f1. qelim(qe,f) = Some f1) \<and> 
+    (\<exists>g1. qelim(qe,g) = Some g1)" using lift_bin_Some[where c="Or"] by simp
+  from f1g1 obtain "f1" and "g1" 
+    where f1_def: "qelim(qe, f) = Some f1" 
+    and g1_def: "qelim(qe,g) = Some g1" by blast
+  from prems f1_def have feqf1: "?F ats f = ?F ats  f1" by blast
+  from prems g1_def have geqg1: "?F ats g = ?F ats g1" by blast
+  from "Or.prems" f1_def g1_def have "p' = Or f1 g1" by simp
+  with feqf1 geqg1 show ?case by simp
+next
+  case (Imp f g)
+  from "Imp.prems" have f1g1: "(\<exists>f1. qelim(qe,f) = Some f1) \<and> 
+    (\<exists>g1. qelim(qe,g) = Some g1)" using lift_bin_Some[where c="Imp"] by simp
+  from f1g1 obtain "f1" and "g1" 
+    where f1_def: "qelim(qe, f) = Some f1" 
+    and g1_def: "qelim(qe,g) = Some g1" by blast
+  from prems f1_def have feqf1: "?F ats f = ?F ats f1" by blast
+  from prems g1_def have geqg1: "?F ats g = ?F ats g1" by blast
+  from "Imp.prems" f1_def g1_def have "p' = Imp f1 g1" by simp
+  with feqf1 geqg1 show ?case by simp
+next
+  case (Equ f g)
+  from "Equ.prems" have f1g1: "(\<exists>f1. qelim(qe,f) = Some f1) \<and> 
+    (\<exists>g1. qelim(qe,g) = Some g1)" using lift_bin_Some[where c="Equ"] by simp
+  from f1g1 obtain "f1" and "g1" 
+    where f1_def: "qelim(qe, f) = Some f1" 
+    and g1_def: "qelim(qe,g) = Some g1" by blast
+  from prems f1_def have feqf1: "?F ats f = ?F ats f1" by blast
+  from prems g1_def have geqg1: "?F ats g = ?F ats g1" by blast
+  from "Equ.prems" f1_def g1_def have "p' = Equ f1 g1" by simp
+  with feqf1 geqg1 show ?case by simp
+next
+  case (QEx f) 
+    from "QEx.prems" have "\<exists> f1. ?Qe f f1"
+    by  (cases "qelim(qe,f)") simp_all
+  then obtain "f1" where f1_def: "qelim(qe,f) = Some f1" by blast
+  from prems have qf1:"isqfree f1" using qelim_qfree by blast
+  from prems have feqf1: "\<forall> ats. qinterp ats f = qinterp ats f1"
+    using f1_def qf1 by blast
+  then  have "?F ats (QEx f) = ?F ats (QEx f1)" 
+    by simp 
+  from prems have "qelim (qe,QEx f) = Some p'" by blast
+  then  have "\<exists> f'. qe f1 = Some f'" using f1_def by simp
+  then obtain "f'" where fdef': "qe f1 = Some f'" by blast
+  with prems have exf1: "?F ats (QEx f1) = ?F ats f'" using qf1 by blast
+  have fp: "?Qe (QEx f) f'" using f1_def fdef' by simp
+  from prems have "?Qe (QEx f) p'" by blast 
+  then have "p' = f'" using fp by simp
+  then show ?case using feqf1 exf1 by simp
+next
+  case (QAll f)
+  from "QAll.prems"
+  have "\<exists> f0. lift_un NOT (lift_qe qe (lift_un NOT (qelim (qe ,f)))) = 
+    Some f0"
+    by (cases "lift_un NOT (lift_qe qe (lift_un NOT (qelim (qe ,f))))") 
+      simp_all
+  then obtain "f0" 
+    where f0_def: "lift_un NOT (lift_qe qe (lift_un NOT (qelim (qe ,f)))) = 
+    Some f0" by blast
+  then have "\<exists> f1. lift_qe qe (lift_un NOT (qelim (qe ,f))) = Some f1" 
+    by (cases "lift_qe qe (lift_un NOT (qelim (qe ,f)))") simp_all
+  then obtain "f1" where 
+    f1_def:"lift_qe qe (lift_un NOT (qelim (qe ,f))) = Some f1" by blast
+  then have "\<exists> f2. lift_un NOT (qelim (qe ,f)) = Some f2"
+    by (cases "qelim (qe ,f)") simp_all
+  then obtain "f2" 
+    where f2_def:"lift_un NOT (qelim (qe ,f)) = Some f2" by blast
+  then have "\<exists> f3. qelim(qe,f) = Some f3" by (cases "qelim(qe,f)") simp_all
+  then obtain "f3" where f3_def: "qelim(qe,f) = Some f3" by blast
+  from prems have qf3:"isqfree f3" using qelim_qfree by blast
+  from prems have feqf3: "\<forall> ats. qinterp ats f = qinterp ats f3"
+    using f3_def qf3 by blast
+  have f23:"f2 = NOT f3" using f2_def f3_def by simp
+  then have feqf2: "\<forall> ats. qinterp ats f = qinterp ats (NOT f2)"
+    using feqf3 by simp
+  have qf2: "isqfree f2" using f23 qf3 by simp
+  have "qe f2 = Some f1" using f1_def f2_def f23 by simp
+  with prems have exf2eqf1: "?F ats (QEx f2) = ?F ats f1" using qf2 by blast
+  have "f0 = NOT f1" using f0_def f1_def by simp
+  then have f0eqf1: "?F ats f0 = ?F ats (NOT f1)" by simp
+  from prems have "qelim (qe, QAll f) = Some p'" by blast
+  then have f0eqp': "p' = f0" using f0_def by simp
+  have "?F ats (QAll f) = (\<forall>x. ?F (x#ats) f)" by simp
+  also have "\<dots> = (\<not> (\<exists> x. ?F (x#ats) (NOT f)))" by simp
+  also have "\<dots> = (\<not> (\<exists> x. ?F (x#ats) (NOT (NOT f2))))" using feqf2
+    by auto
+  also have "\<dots> = (\<not> (\<exists> x. ?F (x#ats) f2))" by simp
+  also have "\<dots> = (\<not> (?F ats f1))" using exf2eqf1 by simp
+  finally show ?case using f0eqp' f0eqf1 by simp
+qed simp_all
+
+(* Cooper's algorithm *)
+
+
+(* Transform an intform into NNF *)
+consts lgth :: "QF \<Rightarrow> nat"
+       nnf :: "QF \<Rightarrow> QF"    
+primrec
+"lgth (Lt it1 it2) = 1"
+"lgth (Gt it1 it2) = 1"
+"lgth (Le it1 it2) = 1"
+"lgth (Ge it1 it2) = 1"
+"lgth (Eq it1 it2) = 1"
+"lgth (Divides it1 it2) = 1"
+"lgth T = 1"
+"lgth F = 1"
+"lgth (NOT p) = 1 + lgth p"
+"lgth (And p q) = 1 + lgth p + lgth q"
+"lgth (Or p q) = 1 + lgth p + lgth q"
+"lgth (Imp p q) = 1 + lgth p + lgth q"
+"lgth (Equ p q) = 1 + lgth p + lgth q" 
+"lgth (QAll p) = 1 + lgth p" 
+"lgth (QEx p) = 1 + lgth p" 
+
+lemma [simp] :"0 < lgth q"
+apply (induct_tac q)
+apply(auto)
+done
+
+(* NNF *)
+recdef nnf "measure (\<lambda>p. lgth p)"
+  "nnf (Lt it1 it2) = Le (Sub it1 it2) (Cst (- 1))"
+  "nnf (Gt it1 it2) = Le (Sub it2 it1) (Cst (- 1))"
+  "nnf (Le it1 it2) = Le it1 it2 "
+  "nnf (Ge it1 it2) = Le it2 it1"
+  "nnf (Eq it1 it2) = Eq it2 it1"
+  "nnf (Divides d t) = Divides d t"
+  "nnf T = T"
+  "nnf F = F"
+  "nnf (And p q) = And (nnf p) (nnf q)"
+  "nnf (Or p q) = Or (nnf p) (nnf q)"
+  "nnf (Imp p q) = Or (nnf (NOT p)) (nnf q)"
+  "nnf (Equ p q) = Or (And (nnf p) (nnf q)) 
+  (And (nnf (NOT p)) (nnf (NOT q)))"
+  "nnf (NOT (Lt it1 it2)) = (Le it2 it1)"
+  "nnf (NOT (Gt it1 it2))  = (Le it1 it2)"
+  "nnf (NOT (Le it1 it2)) = (Le (Sub it2 it1) (Cst (- 1)))"
+  "nnf (NOT (Ge it1 it2)) = (Le (Sub it1 it2) (Cst (- 1)))"
+  "nnf (NOT (Eq it1 it2)) = (NOT (Eq it1 it2))"
+  "nnf (NOT (Divides d t)) = (NOT (Divides d t))"
+  "nnf (NOT T) = F"
+  "nnf (NOT F) = T"
+  "nnf (NOT (NOT p)) = (nnf p)"
+  "nnf (NOT (And p q)) = (Or (nnf (NOT p)) (nnf (NOT q)))"
+  "nnf (NOT (Or p q)) = (And (nnf (NOT p)) (nnf (NOT q)))"
+  "nnf (NOT (Imp p q)) = (And (nnf p) (nnf (NOT q)))"
+  "nnf (NOT (Equ p q)) = (Or (And (nnf p) (nnf (NOT q))) (And (nnf (NOT p)) (nnf q)))"
+
+consts isnnf :: "QF \<Rightarrow> bool"
+recdef isnnf "measure (\<lambda>p. lgth p)"
+  "isnnf (Le it1 it2) = True"
+  "isnnf (Eq it1 it2) = True"
+  "isnnf (Divides d t) = True"
+  "isnnf T = True"
+  "isnnf F = True"
+  "isnnf (And p q) = (isnnf p \<and> isnnf q)"
+  "isnnf (Or p q) = (isnnf p \<and> isnnf q)"
+  "isnnf (NOT (Divides d t)) = True" 
+  "isnnf (NOT (Eq it1 it2)) = True" 
+  "isnnf p = False"
+
+(* nnf preserves semantics *)
+lemma nnf_corr: "isqfree p \<Longrightarrow> qinterp ats p = qinterp ats (nnf p)"
+by (induct p rule: nnf.induct,simp_all) 
+(arith, arith, arith, arith, arith, arith, arith, arith, arith, blast)
+
+
+(* the result of nnf is in NNF *)
+lemma nnf_isnnf : "isqfree p \<Longrightarrow> isnnf (nnf p)"
+by (induct p rule: nnf.induct, auto)
+
+lemma nnf_isqfree: "isnnf p \<Longrightarrow> isqfree p"
+by (induct p rule: isnnf.induct) auto
+
+(* nnf preserves quantifier freeness *)
+lemma nnf_qfree: "isqfree p \<Longrightarrow> isqfree(nnf p)"
+  using nnf_isqfree nnf_isnnf by simp
+
+(* Linearization and normalization of formulae *)
+(* Definition of linearity of an intterm *)
+
+consts islinintterm :: "intterm \<Rightarrow> bool"
+recdef islinintterm "measure size"
+"islinintterm (Cst i) = True"
+"islinintterm (Add (Mult (Cst i) (Var n)) (Cst i')) = (i \<noteq> 0)"
+"islinintterm (Add (Mult (Cst i) (Var n)) (Add (Mult (Cst i') (Var n')) r)) = ( i \<noteq> 0 \<and> i' \<noteq> 0 \<and> n < n' \<and> islinintterm  (Add (Mult (Cst i') (Var n')) r))"
+"islinintterm i = False"
+
+(* subterms of linear terms are linear *)
+lemma islinintterm_subt:
+  assumes lr: "islinintterm (Add (Mult (Cst i) (Var n)) r)"
+  shows "islinintterm r"
+using lr
+by (induct r rule: islinintterm.induct) auto
+
+(* c \<noteq> 0 for linear term c.n + r*)
+lemma islinintterm_cnz:
+  assumes lr: "islinintterm (Add (Mult (Cst i) (Var n)) r)"
+  shows "i \<noteq> 0"
+using lr
+by (induct r rule: islinintterm.induct) auto
+
+lemma islininttermc0r: "islinintterm (Add (Mult (Cst c) (Var n)) r) \<Longrightarrow> (c \<noteq> 0 \<and> islinintterm r)"
+by (induct r rule: islinintterm.induct, simp_all)
+
+(* An alternative linearity definition *)
+consts islintn :: "(nat \<times> intterm) \<Rightarrow> bool"
+recdef islintn "measure (\<lambda> (n,t). (size t))"
+"islintn (n0, Cst i) = True"
+"islintn (n0, Add (Mult (Cst i) (Var n)) r) = (i \<noteq> 0 \<and> n0 \<le> n \<and> islintn (n+1,r))"
+"islintn (n0, t) = False"
+
+constdefs islint :: "intterm \<Rightarrow> bool"
+  "islint t \<equiv> islintn(0,t)"
+
+(* And the equivalence to the first definition *)
+lemma islinintterm_eq_islint: "islinintterm t = islint t"
+  using islint_def
+by (induct t rule: islinintterm.induct) auto
+
+(* monotony *)
+lemma islintn_mon:
+  assumes lin: "islintn (n,t)"
+  and mgen: "m \<le> n"
+  shows "islintn(m,t)"
+  using lin mgen 
+by (induct t rule: islintn.induct) auto
+
+lemma islintn_subt:
+  assumes lint: "islintn(n,Add (Mult (Cst i) (Var m)) r)"
+  shows "islintn (m+1,r)"
+using lint
+by auto
+
+(* List indexin for n > 0 *)
+lemma nth_pos: "0 < n \<longrightarrow> (x#xs) ! n = (y#xs) ! n"
+using Nat.gr0_conv_Suc 
+by clarsimp 
+
+lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
+using Nat.gr0_conv_Suc
+by clarsimp
+
+lemma intterm_novar0: 
+  assumes lin: "islinintterm (Add (Mult (Cst i) (Var n)) r)"
+  shows "I_intterm (x#ats) r = I_intterm (y#ats) r"
+using lin
+by (induct r rule: islinintterm.induct) (simp_all add: nth_pos2)
+(* a simple version of a general theorem: Interpretation doese not depend 
+   on the first variable if it doese not occur in the term *)
+
+lemma linterm_novar0:
+  assumes lin: "islintn (n,t)"
+  and npos: "0 < n"
+  shows "I_intterm (x#ats) t = I_intterm (y#ats) t"
+using lin npos
+by (induct n t rule: islintn.induct) (simp_all add: nth_pos2)
+
+(* Periodicity of dvd *)
+lemma dvd_period:
+  assumes advdd: "(a::int) dvd d"
+  shows "(a dvd (x + t)) = (a dvd ((x+ c*d) + t))"
+using advdd  
+proof-
+  from advdd  have "\<forall>x.\<forall>k. (((a::int) dvd (x + t)) = (a dvd
+ (x+k*d + t)))" by (rule dvd_modd_pinf)
+  then show ?thesis by simp
+qed
+
+(* lin_ad adds two linear terms*)
+consts lin_add :: "intterm \<times> intterm \<Rightarrow> intterm"
+recdef lin_add "measure (\<lambda>(x,y). ((size x) + (size y)))"
+"lin_add (Add (Mult (Cst c1) (Var n1)) (r1),Add (Mult (Cst c2) (Var n2)) (r2)) =
+  (if n1=n2 then 
+  (let c = Cst (c1 + c2) 
+   in (if c1+c2=0 then lin_add(r1,r2) else Add (Mult c (Var n1)) (lin_add (r1,r2))))
+  else if n1 \<le> n2 then (Add (Mult (Cst c1) (Var n1)) (lin_add (r1,Add (Mult (Cst c2) (Var n2)) (r2)))) 
+  else (Add (Mult (Cst c2) (Var n2)) (lin_add (Add (Mult (Cst c1) (Var n1)) r1,r2))))"
+"lin_add (Add (Mult (Cst c1) (Var n1)) (r1),Cst b) = 
+  (Add (Mult (Cst c1) (Var n1)) (lin_add (r1, Cst b)))"  
+"lin_add (Cst x,Add (Mult (Cst c2) (Var n2)) (r2)) = 
+  Add (Mult (Cst c2) (Var n2)) (lin_add (Cst x,r2))" 
+"lin_add (Cst b1, Cst b2) = Cst (b1+b2)"
+
+lemma lin_add_cst_corr: 
+  assumes blin : "islintn(n0,b)"
+  shows "I_intterm ats (lin_add (Cst a,b)) = (I_intterm ats (Add (Cst a) b))"
+using blin
+by (induct n0 b rule: islintn.induct) auto
+
+lemma lin_add_cst_corr2: 
+  assumes blin : "islintn(n0,b)"
+  shows "I_intterm ats (lin_add (b,Cst a)) = (I_intterm ats (Add b (Cst a)))"
+using blin
+by (induct n0 b rule: islintn.induct) auto
+
+lemma lin_add_corrh: "\<And> n01 n02. \<lbrakk> islintn (n01,a) ; islintn (n02,b)\<rbrakk> 
+  \<Longrightarrow> I_intterm ats (lin_add(a,b)) = I_intterm ats (Add a b)"
+proof(induct a b rule: lin_add.induct)
+  case (58 i n r j m s) 
+  have "(n = m \<and> i+j = 0) \<or> (n = m \<and> i+j \<noteq> 0) \<or> n < m \<or> m < n " by arith
+  moreover
+  {assume "n=m\<and>i+j=0" hence ?case using prems by (auto simp add: sym[OF zadd_zmult_distrib]) }
+  moreover
+  {assume "n=m\<and>i+j\<noteq>0" hence ?case using prems by (auto simp add: Let_def zadd_zmult_distrib)}
+  moreover
+  {assume "n < m" hence ?case using prems by auto }
+  moreover
+  {assume "n > m" hence ?case using prems by auto }
+  ultimately show ?case by blast
+qed (auto simp add: lin_add_cst_corr lin_add_cst_corr2 Let_def)
+
+(* lin_add has the semantics of Add*)
+lemma lin_add_corr:
+  assumes lina: "islinintterm a"
+  and linb: "islinintterm b"
+  shows "I_intterm ats (lin_add (a,b)) = (I_intterm ats (Add a b))"
+using lina linb islinintterm_eq_islint islint_def lin_add_corrh
+by blast
+
+lemma lin_add_cst_lint:
+  assumes lin: "islintn (n0,b)"
+  shows "islintn (n0, lin_add (Cst i, b))"
+using lin
+by (induct n0 b rule: islintn.induct) auto
+
+lemma lin_add_cst_lint2:
+  assumes lin: "islintn (n0,b)"
+  shows "islintn (n0, lin_add (b,Cst i))"
+using lin
+by (induct n0 b rule: islintn.induct) auto
+
+(* lin_add preserves linearity..*)
+lemma lin_add_lint: "\<And> n0 n01 n02. \<lbrakk> islintn (n01,a) ; islintn (n02,b); n0 \<le>  min n01 n02 \<rbrakk> 
+  \<Longrightarrow> islintn (n0, lin_add (a,b))"
+proof (induct a b rule: lin_add.induct)
+  case (58 i n r j m s)
+  have "(n =m \<and> i + j = 0) \<or> (n = m \<and> i+j \<noteq> 0) \<or> n <m \<or> m < n" by arith
+  moreover 
+  { assume "n = m"
+      and  "i+j = 0"
+    hence ?case using "58" islintn_mon[where m = "n01" and n = "Suc m"]
+      islintn_mon[where m = "n02" and n = "Suc m"] by auto }
+  moreover 
+  { assume  "n = m"
+      and "i+j \<noteq> 0"
+    hence ?case using "58" islintn_mon[where m = "n01" and n = "Suc m"]
+      islintn_mon[where m = "n02" and n = "Suc m"] by (auto simp add: Let_def) }
+  moreover
+  { assume "n < m" hence ?case using 58 by force }
+moreover
+  { assume "m < n"
+    hence ?case using 58 
+      apply (auto simp add: Let_def) 
+      apply (erule allE[where x = "Suc m" ] )
+      by (erule allE[where x = "Suc m" ] ) simp }
+  ultimately show ?case by blast
+qed(simp_all add: Let_def lin_add_cst_lint lin_add_cst_lint2)
+
+lemma lin_add_lin:
+  assumes lina: "islinintterm a"
+  and linb: "islinintterm b"
+  shows "islinintterm (lin_add (a,b))"
+using islinintterm_eq_islint islint_def lin_add_lint lina linb by auto
+
+(* lin_mul multiplies a linear term by a constant *)
+consts lin_mul :: "int \<times> intterm \<Rightarrow> intterm"
+recdef lin_mul "measure (\<lambda>(c,t). size t)"
+"lin_mul (c,Cst i) = (Cst (c*i))"
+"lin_mul (c,Add (Mult (Cst c') (Var n)) r)  = 
+  (if c = 0 then (Cst 0) else
+  (Add (Mult (Cst (c*c')) (Var n)) (lin_mul (c,r))))"
+
+lemma zmult_zadd_distrib[simp]: "(a::int) * (b+c) = a*b + a*c"
+proof-
+  have "a*(b+c) = (b+c)*a" by simp
+  moreover have "(b+c)*a = b*a + c*a" by (simp add: zadd_zmult_distrib)
+  ultimately show ?thesis by simp
+qed
+
+(* lin_mul has the semantics of Mult *)
+lemma lin_mul_corr: 
+  assumes lint: "islinintterm  t"
+  shows "I_intterm ats (lin_mul (c,t)) = I_intterm ats (Mult (Cst c) t)"
+using lint
+proof (induct c t rule: lin_mul.induct)
+  case (21 c c' n r)
+  have "islinintterm (Add (Mult (Cst c') (Var n)) r)" .
+  then have "islinintterm r" 
+    by (rule islinintterm_subt[of "c'" "n" "r"])
+  then show ?case  using "21.hyps" "21.prems" by simp
+qed(auto)
+
+(* lin_mul preserves linearity *)
+lemma lin_mul_lin:
+  assumes lint: "islinintterm t"
+  shows "islinintterm (lin_mul(c,t))"
+using lint
+by (induct t rule: islinintterm.induct) auto
+
+lemma lin_mul0:
+  assumes lint: "islinintterm t"
+  shows "lin_mul(0,t) = Cst 0"
+  using lint
+  by (induct t rule: islinintterm.induct) auto
+
+lemma lin_mul_lintn:
+  "\<And> m. islintn(m,t) \<Longrightarrow> islintn(m,lin_mul(l,t))"
+  by (induct l t rule: lin_mul.induct) simp_all
+
+(* lin_neg neagtes a linear term *)
+constdefs lin_neg :: "intterm \<Rightarrow> intterm"
+"lin_neg i == lin_mul ((-1::int),i)"
+
+(* lin_neg has the semantics of Neg *)
+lemma lin_neg_corr:
+  assumes lint: "islinintterm  t"
+  shows "I_intterm ats (lin_neg t) = I_intterm ats (Neg t)"
+  using lint lin_mul_corr
+  by (simp add: lin_neg_def lin_mul_corr)
+
+(* lin_neg preserves linearity *)
+lemma lin_neg_lin:
+  assumes lint: "islinintterm  t"
+  shows "islinintterm (lin_neg t)"
+using lint
+by (simp add: lin_mul_lin lin_neg_def)
+
+(* Some properties about lin_add and lin-neg should be moved above *)
+
+lemma lin_neg_idemp: 
+  assumes lini: "islinintterm i"
+  shows "lin_neg (lin_neg i) = i"
+using lini
+by (induct i rule: islinintterm.induct) (auto simp add: lin_neg_def)
+
+lemma lin_neg_lin_add_distrib:
+  assumes lina : "islinintterm a"
+  and linb :"islinintterm b"
+  shows "lin_neg (lin_add(a,b)) = lin_add (lin_neg a, lin_neg b)"
+using lina linb
+proof (induct a b rule: lin_add.induct)
+  case (58 c1 n1 r1 c2 n2 r2)
+  from prems have lincnr1:"islinintterm (Add (Mult (Cst c1) (Var n1)) r1)" by simp
+  have linr1: "islinintterm r1" by (rule islinintterm_subt[OF lincnr1])
+  from prems have lincnr2: "islinintterm (Add (Mult (Cst c2) (Var n2)) r2)" by simp
+  have linr2: "islinintterm r2" by (rule islinintterm_subt[OF lincnr2])
+  have "n1 = n2 \<or> n1 < n2 \<or> n1 > n2" by arith
+  show ?case using prems linr1 linr2 by (simp_all add: lin_neg_def Let_def)
+next
+  case (59 c n r b) 
+  from prems have lincnr: "islinintterm (Add (Mult (Cst c) (Var n)) r)" by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  show ?case using prems linr by (simp add: lin_neg_def Let_def)
+next
+  case (60 b c n r)
+  from prems have lincnr: "islinintterm (Add (Mult (Cst c) (Var n)) r)" by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  show ?case  using prems linr by (simp add: lin_neg_def Let_def)
+qed (simp_all add: lin_neg_def)
+
+(* linearize tries to linearize a term *)
+consts linearize :: "intterm \<Rightarrow> intterm option"
+recdef linearize "measure (\<lambda>t. size t)"
+"linearize (Cst b) = Some (Cst b)"
+"linearize (Var n) = Some (Add (Mult (Cst 1) (Var n)) (Cst 0))"
+"linearize (Neg i) = lift_un lin_neg (linearize i)"
+ "linearize (Add i j) = lift_bin(\<lambda> x. \<lambda> y. lin_add(x,y), linearize i, linearize j)"
+"linearize (Sub i j) = 
+  lift_bin(\<lambda> x. \<lambda> y. lin_add(x,lin_neg y), linearize i, linearize j)"
+"linearize (Mult i j) = 
+  (case linearize i of
+  None \<Rightarrow> None
+  | Some li \<Rightarrow> (case li of 
+     Cst b \<Rightarrow> (case linearize j of
+      None \<Rightarrow> None
+     | (Some lj) \<Rightarrow> Some (lin_mul(b,lj)))
+  | _ \<Rightarrow> (case linearize j of
+      None \<Rightarrow> None
+    | (Some lj) \<Rightarrow> (case lj of 
+        Cst b \<Rightarrow> Some (lin_mul (b,li))
+      | _ \<Rightarrow> None))))"
+
+lemma linearize_linear1:
+  assumes lin: "linearize t \<noteq> None"
+  shows "islinintterm (the (linearize t))"
+using lin
+proof (induct t rule: linearize.induct)
+  case (1 b) show ?case by simp  
+next 
+  case (2 n) show ?case by simp 
+next 
+  case (3 i) show ?case 
+    proof-
+    have "(linearize i = None) \<or> (\<exists>li. linearize i = Some li)" by auto
+    moreover 
+    { assume "linearize i = None" with prems have ?thesis by auto}
+    moreover 
+    { assume lini: "\<exists>li. linearize i = Some li"
+      from lini obtain "li" where  "linearize i = Some li" by blast
+      have linli: "islinintterm li" by (simp!)
+      moreover have "linearize (Neg i) = Some (lin_neg li)" using prems by simp
+      moreover from linli have "islinintterm(lin_neg li)" by (simp add: lin_neg_lin)
+      ultimately have ?thesis by simp
+    }
+    ultimately show ?thesis by blast
+  qed
+next 
+  case (4 i j) show ?case 
+    proof-
+    have "(linearize i = None) \<or> ((\<exists> li. linearize i = Some li) \<and> linearize j = None) \<or> ((\<exists> li. linearize i = Some li) \<and> (\<exists> lj. linearize j = Some lj))" by auto 
+    moreover 
+    {
+      assume nlini: "linearize i = None"
+      from nlini have "linearize (Add i j) = None" 
+	by (simp add: Let_def measure_def inv_image_def) then have ?thesis using prems by auto}
+    moreover 
+    { assume nlinj: "linearize j = None"
+	and lini: "\<exists> li. linearize i = Some li"
+      from nlinj lini have "linearize (Add i j) = None" 
+	by (simp add: Let_def measure_def inv_image_def, auto) with prems  have ?thesis by auto}
+    moreover 
+    { assume lini: "\<exists>li. linearize i = Some li"
+	and linj: "\<exists>lj. linearize j = Some lj"
+      from lini obtain "li" where  "linearize i = Some li" by blast
+      have linli: "islinintterm li" by (simp!)
+      from linj obtain "lj" where  "linearize j = Some lj" by blast
+      have linlj: "islinintterm lj" by (simp!)
+      moreover from lini linj have "linearize (Add i j) = Some (lin_add (li,lj))" 
+	by (simp add: measure_def inv_image_def, auto!)
+      moreover from linli linlj have "islinintterm(lin_add (li,lj))" by (simp add: lin_add_lin)
+      ultimately have ?thesis by simp  }
+    ultimately show ?thesis by blast
+  qed
+next 
+  case (5 i j)show ?case 
+    proof-
+    have "(linearize i = None) \<or> ((\<exists> li. linearize i = Some li) \<and> linearize j = None) \<or> ((\<exists> li. linearize i = Some li) \<and> (\<exists> lj. linearize j = Some lj))" by auto 
+    moreover 
+    {
+      assume nlini: "linearize i = None"
+      from nlini have "linearize (Sub i j) = None" by (simp add: Let_def measure_def inv_image_def) then have ?thesis by (auto!)
+    }
+    moreover 
+    {
+      assume lini: "\<exists> li. linearize i = Some li"
+	and nlinj: "linearize j = None"
+      from nlinj lini have "linearize (Sub i j) = None" 
+	by (simp add: Let_def measure_def inv_image_def, auto) then have ?thesis by (auto!)
+    }
+    moreover 
+    {
+      assume lini: "\<exists>li. linearize i = Some li"
+	and linj: "\<exists>lj. linearize j = Some lj"
+      from lini obtain "li" where  "linearize i = Some li" by blast
+      have linli: "islinintterm li" by (simp!)
+      from linj obtain "lj" where  "linearize j = Some lj" by blast
+      have linlj: "islinintterm lj" by (simp!)
+      moreover from lini linj have "linearize (Sub i j) = Some (lin_add (li,lin_neg lj))" 
+	by (simp add: measure_def inv_image_def, auto!)
+      moreover from linli linlj have "islinintterm(lin_add (li,lin_neg lj))" by (simp add: lin_add_lin lin_neg_lin)
+      ultimately have ?thesis by simp
+    }
+    ultimately show ?thesis by blast
+  qed
+next
+  case (6 i j)show ?case 
+    proof-
+      have cses: "(linearize i = None) \<or> 
+	((\<exists> li. linearize i = Some li) \<and> linearize j = None) \<or> 
+	((\<exists> li. linearize i = Some li) \<and> (\<exists> bj. linearize j = Some (Cst bj)))
+	\<or> ((\<exists> bi. linearize i = Some (Cst bi)) \<and> (\<exists> lj. linearize j = Some lj))
+	\<or> ((\<exists> li. linearize i = Some li \<and> \<not> (\<exists> bi. li = Cst bi)) \<and> (\<exists> lj. linearize j = Some lj \<and> \<not> (\<exists> bj. lj = Cst bj)))" by auto 
+    moreover 
+    {
+      assume nlini: "linearize i = None"
+      from nlini have "linearize (Mult i j) = None" 
+	by (simp add: Let_def measure_def inv_image_def)  
+      with prems have ?thesis by auto }
+    moreover 
+    {  assume lini: "\<exists> li. linearize i = Some li"
+	and nlinj: "linearize j = None"
+      from lini obtain "li" where "linearize i = Some li" by blast 
+      moreover from nlinj lini have "linearize (Mult i j) = None"
+	using prems
+	by (cases li ) (auto simp add: Let_def measure_def inv_image_def)
+      with prems have ?thesis by auto}
+    moreover 
+    { assume lini: "\<exists>li. linearize i = Some li"
+	and linj: "\<exists>bj. linearize j = Some (Cst bj)"
+      from lini obtain "li" where  li_def: "linearize i = Some li" by blast
+      from prems have linli: "islinintterm li" by simp
+      moreover 
+      from linj  obtain "bj" where  bj_def: "linearize j = Some (Cst bj)" by blast
+      have linlj: "islinintterm (Cst bj)" by simp 
+      moreover from lini linj prems 
+      have "linearize (Mult i j) = Some (lin_mul (bj,li))"
+	by (cases li) (auto simp add: measure_def inv_image_def)
+      moreover from linli linlj have "islinintterm(lin_mul (bj,li))" by (simp add: lin_mul_lin)
+      ultimately have ?thesis by simp  }
+    moreover 
+    { assume lini: "\<exists>bi. linearize i = Some (Cst bi)"
+	and linj: "\<exists>lj. linearize j = Some lj"
+      from lini obtain "bi" where  "linearize i = Some (Cst bi)" by blast
+      from prems have linli: "islinintterm (Cst bi)" by simp
+      moreover 
+      from linj  obtain "lj" where  "linearize j = Some lj" by blast
+      from prems have linlj: "islinintterm lj" by simp
+      moreover from lini linj prems have "linearize (Mult i j) = Some (lin_mul (bi,lj))" 
+	by (simp add: measure_def inv_image_def) 
+      moreover from linli linlj have "islinintterm(lin_mul (bi,lj))" by (simp add: lin_mul_lin)
+      ultimately have ?thesis by simp }
+    moreover 
+    { assume linc: "\<exists> li. linearize i = Some li \<and> \<not> (\<exists> bi. li = Cst bi)"
+	and ljnc: "\<exists> lj. linearize j = Some lj \<and> \<not> (\<exists> bj. lj = Cst bj)"
+      from linc obtain "li" where "linearize i = Some li \<and> \<not> (\<exists> bi. li = Cst bi)" by blast
+      moreover 
+      from ljnc obtain "lj" where "linearize j = Some lj \<and> \<not> (\<exists> bj. lj = Cst bj)" by blast
+      ultimately have "linearize (Mult i j) = None"
+	by (cases li, auto simp add: measure_def inv_image_def) (cases lj, auto)+
+      with prems have ?thesis by simp }
+    ultimately show ?thesis by blast
+  qed
+qed  
+
+(* the result of linearize, when successful, is a linear term*)
+lemma linearize_linear: "\<And> t'. linearize t = Some t' \<Longrightarrow> islinintterm t'"
+proof-
+  fix t'
+  assume lint: "linearize t = Some t'"
+  from lint have lt: "linearize t \<noteq> None" by auto
+  then have "islinintterm (the (linearize t))" by (rule_tac  linearize_linear1[OF lt])
+  with lint show "islinintterm t'" by simp
+qed
+
+lemma linearize_corr1: 
+  assumes lin: "linearize t \<noteq> None"
+  shows "I_intterm ats t = I_intterm ats (the (linearize t))"
+using lin
+proof (induct t rule: linearize.induct)
+  case (3 i) show ?case 
+    proof-
+    have "(linearize i = None) \<or> (\<exists>li. linearize i = Some li)" by auto
+    moreover 
+    {
+      assume "linearize i = None"
+      have ?thesis using prems by simp
+    }
+    moreover 
+    {
+      assume lini: "\<exists>li. linearize i = Some li"
+      from lini have lini2: "linearize i \<noteq> None" by simp
+      from lini obtain "li" where  "linearize i = Some li" by blast
+      from lini2 lini have "islinintterm (the (linearize i))"
+	by (simp add: linearize_linear1[OF lini2])
+      then have linli: "islinintterm li" using prems by simp
+      have ieqli: "I_intterm ats i = I_intterm ats li" using prems by simp
+      moreover have "linearize (Neg i) = Some (lin_neg li)" using prems by simp
+      moreover from ieqli linli have "I_intterm ats (Neg i) = I_intterm ats (lin_neg li)" by (simp add: lin_neg_corr[OF linli])
+      ultimately have ?thesis using prems by (simp add: lin_neg_corr)
+    }
+    ultimately show ?thesis by blast
+  qed
+next 
+  case (4 i j) show ?case 
+    proof-
+    have "(linearize i = None) \<or> ((\<exists> li. linearize i = Some li) \<and> linearize j = None) \<or> ((\<exists> li. linearize i = Some li) \<and> (\<exists> lj. linearize j = Some lj))" by auto 
+    moreover 
+    {
+      assume nlini: "linearize i = None"
+      from nlini have "linearize (Add i j) = None" by (simp add: Let_def measure_def inv_image_def) then have ?thesis using prems by auto
+    }
+    moreover 
+    {
+      assume nlinj: "linearize j = None"
+	and lini: "\<exists> li. linearize i = Some li"
+      from nlinj lini have "linearize (Add i j) = None" 
+	by (simp add: Let_def measure_def inv_image_def, auto) 
+      then have ?thesis using prems by auto
+    }
+    moreover 
+    {
+      assume lini: "\<exists>li. linearize i = Some li"
+	and linj: "\<exists>lj. linearize j = Some lj"
+      from lini have lini2: "linearize i \<noteq> None" by simp
+      from linj have linj2: "linearize j \<noteq> None" by simp
+      from lini obtain "li" where  "linearize i = Some li" by blast
+      from lini2 have "islinintterm (the (linearize i))" by (simp add: linearize_linear1)
+      then have linli: "islinintterm li" using prems by simp
+      from linj obtain "lj" where  "linearize j = Some lj" by blast
+      from linj2 have "islinintterm (the (linearize j))" by (simp add: linearize_linear1)
+      then have linlj: "islinintterm lj" using prems by simp
+      moreover from lini linj have "linearize (Add i j) = Some (lin_add (li,lj))"
+	using prems by (simp add: measure_def inv_image_def)
+      moreover from linli linlj have "I_intterm ats (lin_add (li,lj)) = I_intterm ats (Add li lj)" by (simp add: lin_add_corr)
+      ultimately have ?thesis using prems by simp
+    }
+    ultimately show ?thesis by blast
+  qed
+next 
+  case (5 i j)show ?case 
+    proof-
+    have "(linearize i = None) \<or> ((\<exists> li. linearize i = Some li) \<and> linearize j = None) \<or> ((\<exists> li. linearize i = Some li) \<and> (\<exists> lj. linearize j = Some lj))" by auto 
+    moreover 
+    {
+      assume nlini: "linearize i = None"
+      from nlini have "linearize (Sub i j) = None" by (simp add: Let_def measure_def inv_image_def) then have ?thesis using prems by auto
+    }
+    moreover 
+    {
+      assume lini: "\<exists> li. linearize i = Some li"
+	and nlinj: "linearize j = None"
+      from nlinj lini have "linearize (Sub i j) = None" 
+	by (simp add: Let_def measure_def inv_image_def, auto) with prems have ?thesis by auto
+    }
+    moreover 
+    {
+      assume lini: "\<exists>li. linearize i = Some li"
+	and linj: "\<exists>lj. linearize j = Some lj"
+      from lini have lini2: "linearize i \<noteq> None" by simp
+      from linj have linj2: "linearize j \<noteq> None" by simp
+      from lini obtain "li" where  "linearize i = Some li" by blast
+      from lini2 have "islinintterm (the (linearize i))" by (simp add: linearize_linear1)
+      with prems have linli: "islinintterm li" by simp
+      from linj obtain "lj" where  "linearize j = Some lj" by blast
+      from linj2 have "islinintterm (the (linearize j))" by (simp add: linearize_linear1)
+      with prems have linlj: "islinintterm lj" by simp
+      moreover from prems have "linearize (Sub i j) = Some (lin_add (li,lin_neg lj))" 
+	by (simp add: measure_def inv_image_def)
+      moreover from linlj have linnlj:"islinintterm (lin_neg lj)" by (simp add: lin_neg_lin)
+      moreover from linli linnlj have "I_intterm ats (lin_add (li,lin_neg lj)) = I_intterm ats (Add li (lin_neg lj))" by (simp only: lin_add_corr[OF linli linnlj])
+      moreover from linli linlj linnlj have "I_intterm ats (Add li (lin_neg lj)) = I_intterm ats (Sub li lj)" 
+	by (simp add: lin_neg_corr)
+      ultimately have ?thesis using prems by simp    
+    }
+    ultimately show ?thesis by blast
+  qed
+next
+  case (6 i j)show ?case 
+    proof-
+      have cses: "(linearize i = None) \<or> 
+	((\<exists> li. linearize i = Some li) \<and> linearize j = None) \<or> 
+	((\<exists> li. linearize i = Some li) \<and> (\<exists> bj. linearize j = Some (Cst bj)))
+	\<or> ((\<exists> bi. linearize i = Some (Cst bi)) \<and> (\<exists> lj. linearize j = Some lj))
+	\<or> ((\<exists> li. linearize i = Some li \<and> \<not> (\<exists> bi. li = Cst bi)) \<and> (\<exists> lj. linearize j = Some lj \<and> \<not> (\<exists> bj. lj = Cst bj)))" by auto 
+    moreover 
+    {
+      assume nlini: "linearize i = None"
+      from nlini have "linearize (Mult i j) = None" by (simp add: Let_def measure_def inv_image_def) with prems  have ?thesis by auto
+    }
+    moreover 
+    {
+      assume lini: "\<exists> li. linearize i = Some li"
+	and nlinj: "linearize j = None"
+
+      from lini obtain "li" where "linearize i = Some li" by blast 
+      moreover from prems have "linearize (Mult i j) = None" 
+	by (cases li) (simp_all add: Let_def measure_def inv_image_def)
+      with prems have ?thesis by auto
+    }
+    moreover 
+    {
+      assume lini: "\<exists>li. linearize i = Some li"
+	and linj: "\<exists>bj. linearize j = Some (Cst bj)"
+      from lini have lini2: "linearize i \<noteq> None" by simp
+      from linj have linj2: "linearize j \<noteq> None" by auto
+      from lini obtain "li" where  "linearize i = Some li" by blast
+      from lini2 have "islinintterm (the (linearize i))" by (simp add: linearize_linear1)
+      with prems  have linli: "islinintterm li" by simp
+      moreover 
+      from linj  obtain "bj" where  "linearize j = Some (Cst bj)" by blast
+      have linlj: "islinintterm (Cst bj)" by simp
+      moreover from prems have "linearize (Mult i j) = Some (lin_mul (bj,li))"
+ 	by (cases li) (auto simp add: measure_def inv_image_def) 
+      then have lm1: "I_intterm ats (the(linearize (Mult i j))) = I_intterm ats (lin_mul (bj,li))" by simp
+      moreover from linli linlj have "I_intterm ats (lin_mul(bj,li)) = I_intterm ats (Mult li (Cst bj))" by (simp add: lin_mul_corr)
+      with prems 
+      have "I_intterm ats (lin_mul(bj,li)) = I_intterm ats (Mult li (the (linearize j)))" 
+	by auto
+      moreover have "I_intterm ats (Mult li (the (linearize j))) = I_intterm ats (Mult i (the (linearize j)))" using prems  by simp
+      moreover have "I_intterm ats i = I_intterm ats (the (linearize i))"  
+	using lini2 lini "6.hyps" by simp
+	moreover have "I_intterm ats j = I_intterm ats (the (linearize j))"
+	  using prems by (cases li) (auto simp add: measure_def inv_image_def)
+      ultimately have ?thesis by auto }
+    moreover 
+    { assume lini: "\<exists>bi. linearize i = Some (Cst bi)"
+	and linj: "\<exists>lj. linearize j = Some lj"
+      from lini have lini2 : "linearize i \<noteq> None" by auto
+      from linj have linj2 : "linearize j \<noteq> None" by auto      
+      from lini obtain "bi" where  "linearize i = Some (Cst bi)" by blast
+      have linli: "islinintterm (Cst bi)" using prems by simp
+      moreover 
+      from linj  obtain "lj" where  "linearize j = Some lj" by blast
+      from linj2 have "islinintterm (the (linearize j))" by (simp add: linearize_linear1) 
+      then have linlj: "islinintterm lj" by (simp!)
+      moreover from linli lini linj have "linearize (Mult i j) = Some (lin_mul (bi,lj))" 	apply (simp add: measure_def inv_image_def) 
+	apply auto by (case_tac "li::intterm",auto!)
+      then have lm1: "I_intterm ats (the(linearize (Mult i j))) = I_intterm ats (lin_mul (bi,lj))" by simp
+      moreover from linli linlj have "I_intterm ats (lin_mul(bi,lj)) = I_intterm ats (Mult (Cst bi) lj)" by (simp add: lin_mul_corr)
+      then have "I_intterm ats (lin_mul(bi,lj)) = I_intterm ats (Mult (the (linearize i)) lj)" by (auto!)
+      moreover have "I_intterm ats (Mult (the (linearize i)) lj) = I_intterm ats (Mult (the (linearize i)) j)" using lini lini2  by (simp!)
+      moreover have "I_intterm ats i = I_intterm ats (the (linearize i))"  
+	using lini2 lini "6.hyps" by simp
+	moreover have "I_intterm ats j = I_intterm ats (the (linearize j))"
+	  using linj linj2 lini lini2 linli linlj "6.hyps" by (auto!)
+
+      ultimately have ?thesis by auto }
+    moreover 
+    { assume linc: "\<exists> li. linearize i = Some li \<and> \<not> (\<exists> bi. li = Cst bi)"
+	and ljnc: "\<exists> lj. linearize j = Some lj \<and> \<not> (\<exists> bj. lj = Cst bj)"
+      from linc obtain "li" where "\<exists> li. linearize i = Some li \<and> \<not> (\<exists> bi. li = Cst bi)" by blast
+      moreover 
+      from ljnc obtain "lj" where "\<exists> lj. linearize j = Some lj \<and> \<not> (\<exists> bj. lj = Cst bj)" by blast
+      ultimately have "linearize (Mult i j) = None"
+	apply (simp add: measure_def inv_image_def)
+	apply (case_tac "linearize i", auto)
+	apply (case_tac a)
+	apply (auto!)
+	by (case_tac "lj",auto)+
+      then have ?thesis by (simp!) }
+    ultimately show ?thesis by blast
+  qed
+qed  simp_all
+
+
+(* linearize, when successfull, preserves semantics *)
+lemma linearize_corr: "\<And> t'. linearize t = Some t' \<Longrightarrow> I_intterm ats t = I_intterm ats t' "
+proof-
+  fix t'
+  assume lint: "linearize t = Some t'"
+  show  "I_intterm ats t = I_intterm ats t'"
+  proof-
+    from lint have lt: "linearize t \<noteq> None" by simp 
+    then have "I_intterm ats t = I_intterm ats (the (linearize t))" 
+      by (rule_tac linearize_corr1[OF lt])
+    with lint show ?thesis by simp
+  qed
+qed
+
+(* tries to linearize a formula *)
+consts linform :: "QF \<Rightarrow> QF option"
+primrec
+"linform (Le it1 it2) =  
+  lift_bin(\<lambda>x. \<lambda>y. Le (lin_add(x,lin_neg y)) (Cst 0),linearize it1, linearize it2)"
+"linform (Eq it1 it2) =  
+  lift_bin(\<lambda>x. \<lambda>y. Eq (lin_add(x,lin_neg y)) (Cst 0),linearize it1, linearize it2)"
+"linform (Divides d t) =  
+  (case linearize d of
+    None \<Rightarrow> None
+   | Some ld \<Rightarrow> (case ld of
+          Cst b \<Rightarrow> 
+               (if (b=0) then None
+               else 
+                (case linearize t of 
+                 None \<Rightarrow> None
+               | Some lt \<Rightarrow> Some (Divides ld lt)))
+         | _ \<Rightarrow> None))"
+"linform  T = Some T"
+"linform  F = Some F"
+"linform (NOT p) = lift_un NOT (linform p)"
+"linform (And p q)= lift_bin(\<lambda>f. \<lambda>g. And f g, linform p, linform q)"
+"linform (Or p q) = lift_bin(\<lambda>f. \<lambda>g. Or f g, linform p, linform q)"
+
+(* linearity of formulae *)
+consts islinform :: "QF \<Rightarrow> bool"
+recdef islinform "measure size"
+"islinform (Le it (Cst i)) = (i=0 \<and> islinintterm it )"
+"islinform (Eq it (Cst i)) = (i=0 \<and> islinintterm it)"
+"islinform (Divides (Cst d) t) = (d \<noteq> 0 \<and> islinintterm t)"
+"islinform  T = True"
+"islinform  F = True"
+"islinform (NOT (Divides (Cst d) t)) = (d \<noteq> 0 \<and> islinintterm t)"
+"islinform (NOT (Eq it (Cst i))) = (i=0 \<and> islinintterm it)"
+"islinform (And p q)= ((islinform p) \<and> (islinform q))"
+"islinform (Or p q) = ((islinform p) \<and> (islinform q))"
+"islinform p = False"
+
+(* linform preserves nnf, if successful *)
+lemma linform_nnf: 
+  assumes nnfp: "isnnf p" 
+  shows "\<And> p'. \<lbrakk>linform p = Some p'\<rbrakk> \<Longrightarrow> isnnf p'"
+using nnfp
+proof (induct p rule: isnnf.induct, simp_all)
+  case (goal1 a b p')
+  show ?case 
+    using prems 
+    by (cases "linearize a", auto) (cases "linearize b", auto)
+next 
+  case (goal2 a b p')
+  show ?case 
+    using prems 
+    by (cases "linearize a", auto) (cases "linearize b", auto)
+next 
+  case (goal3 d t p')
+  show ?case 
+    using prems
+    apply (cases "linearize d", auto)
+    apply (case_tac "a",auto)
+    apply (case_tac "int=0",auto)
+    by (cases "linearize t",auto)
+next
+  case (goal4 f g p') show ?case 
+    using prems
+    by (cases "linform f", auto) (cases "linform g", auto)
+next
+  case (goal5 f g p') show ?case 
+    using prems
+    by (cases "linform f", auto) (cases "linform g", auto)
+next
+  case (goal6 d t p') show ?case 
+    using prems
+    apply (cases "linearize d", auto)
+    apply (case_tac "a", auto)
+    apply (case_tac "int = 0",auto)
+    by (cases "linearize t", auto)
+next 
+  case (goal7 a b p')
+  show ?case 
+    using prems 
+    by (cases "linearize a", auto) (cases "linearize b", auto)
+
+
+qed
+
+
+lemma linform_corr: "\<And> lp. \<lbrakk> isnnf p ; linform p = Some lp \<rbrakk> \<Longrightarrow> 
+                     (qinterp ats p = qinterp ats lp)"
+proof (induct p rule: linform.induct)
+  case (Le x y)
+  show ?case
+    using "Le.prems"
+  proof-
+    have "(\<exists> lx ly. linearize x = Some lx \<and> linearize y = Some ly) \<or> 
+      (linearize x = None) \<or> (linearize y = None)"by auto
+    moreover 
+    {
+      assume linxy: "\<exists> lx ly. linearize x = Some lx \<and> linearize y = Some ly"
+      from linxy obtain "lx" "ly" 
+	where lxly:"linearize x = Some lx \<and> linearize y = Some ly" by blast
+      then 
+      have lxeqx: "I_intterm ats x = I_intterm ats lx" 
+	by (simp add: linearize_corr)
+      from lxly have lxlin: "islinintterm lx" 
+	by (auto simp add: linearize_linear)
+      from lxly have lyeqy: "I_intterm ats y = I_intterm ats ly"
+	by (simp add: linearize_corr)
+      from lxly have lylin: "islinintterm ly" 
+	by (auto simp add: linearize_linear)
+      from "prems"
+      have lpeqle: "lp =  (Le (lin_add(lx,lin_neg ly)) (Cst 0))"
+	by auto
+      moreover
+      have lin1: "islinintterm (Cst 1)" by simp
+      then
+      have ?thesis  
+	using lxlin lylin lin1 lin_add_lin lin_neg_lin "prems" lxly lpeqle
+	by (simp add: lin_add_corr lin_neg_corr lxeqx lyeqy)
+      
+    }
+    
+    moreover
+    {
+      assume "linearize x = None"
+      have ?thesis using "prems" by simp
+    }
+    
+    moreover
+    {
+      assume "linearize y = None"
+      then have ?thesis using "prems"
+	by (case_tac "linearize x", auto)
+    }
+    ultimately show ?thesis by blast
+  qed
+  
+next 
+  case (Eq x y)
+  show ?case
+    using "Eq.prems"
+  proof-
+    have "(\<exists> lx ly. linearize x = Some lx \<and> linearize y = Some ly) \<or> 
+      (linearize x = None) \<or> (linearize y = None)"by auto
+    moreover 
+    {
+      assume linxy: "\<exists> lx ly. linearize x = Some lx \<and> linearize y = Some ly"
+      from linxy obtain "lx" "ly" 
+	where lxly:"linearize x = Some lx \<and> linearize y = Some ly" by blast
+      then 
+      have lxeqx: "I_intterm ats x = I_intterm ats lx" 
+	by (simp add: linearize_corr)
+      from lxly have lxlin: "islinintterm lx" 
+	by (auto simp add: linearize_linear)
+      from lxly have lyeqy: "I_intterm ats y = I_intterm ats ly"
+	by (simp add: linearize_corr)
+      from lxly have lylin: "islinintterm ly" 
+	by (auto simp add: linearize_linear)
+      from "prems"
+      have lpeqle: "lp =  (Eq (lin_add(lx,lin_neg ly)) (Cst 0))"
+	by auto
+      moreover
+      have lin1: "islinintterm (Cst 1)" by simp
+      then
+      have ?thesis  
+	using lxlin lylin lin1 lin_add_lin lin_neg_lin "prems" lxly lpeqle
+	by (simp add: lin_add_corr lin_neg_corr lxeqx lyeqy)
+      
+    }
+    
+    moreover
+    {
+      assume "linearize x = None"
+      have ?thesis using "prems" by simp
+    }
+    
+    moreover
+    {
+      assume "linearize y = None"
+      then have ?thesis using "prems"
+	by (case_tac "linearize x", auto)
+    }
+    ultimately show ?thesis by blast
+  qed
+  
+next 
+  case (Divides d t)
+  show ?case
+    using "Divides.prems"
+    apply (case_tac "linearize d",auto)
+    apply (case_tac a, auto)
+    apply (case_tac "int = 0", auto)
+    apply (case_tac "linearize t", auto)
+    apply (simp add: linearize_corr)
+    apply (case_tac a, auto)
+    apply (case_tac "int = 0", auto)
+    by (case_tac "linearize t", auto simp add: linearize_corr)
+next
+  case (NOT f) show ?case
+    using "prems"
+  proof-
+    have "(\<exists> lf. linform f = Some lf) \<or> (linform f = None)" by auto
+    moreover 
+    {
+      assume linf: "\<exists> lf. linform f = Some lf"
+      from prems have "isnnf (NOT f)" by simp
+      then have fnnf: "isnnf f" by (cases f) auto
+      from linf obtain "lf" where lf: "linform f = Some lf" by blast
+      then have "lp = NOT lf" using "prems" by auto
+      with "NOT.prems" "NOT.hyps" lf fnnf
+      have ?case by simp
+    }
+    moreover 
+    {
+      assume "linform f = None"
+      then 
+      have "linform (NOT f) = None" by simp
+      then 
+      have ?thesis  using "NOT.prems" by simp
+    }
+    ultimately show ?thesis by blast
+  qed
+next
+  case (Or f g) 
+  show ?case using "Or.hyps"
+  proof -
+    have "((\<exists> lf. linform f = Some lf ) \<and> (\<exists> lg. linform g = Some lg)) \<or> 
+      (linform f = None) \<or> (linform g = None)" by auto
+    moreover
+    {
+      assume linf: "\<exists> lf. linform f = Some lf"
+	and ling: "\<exists> lg. linform g = Some lg"
+      from linf obtain "lf" where lf: "linform f = Some lf" by blast
+      from ling obtain "lg" where lg: "linform g = Some lg" by blast
+      from lf lg have "linform (Or f g) = Some (Or lf lg)" by simp
+      then have "lp = Or lf lg" using lf lg "prems"  by simp
+      with lf lg "prems" have ?thesis by simp
+    }
+    moreover
+    {
+      assume "linform f = None"
+      then have ?thesis using "Or.prems"  by auto
+    }
+    moreover
+    {
+      assume "linform g = None"
+      then have ?thesis using "Or.prems"  by (case_tac "linform f", auto)
+      
+    }
+    ultimately show ?thesis by blast
+  qed
+next
+  case (And f g) 
+  show ?case using "And.hyps"
+  proof -
+    have "((\<exists> lf. linform f = Some lf ) \<and> (\<exists> lg. linform g = Some lg)) \<or> 
+      (linform f = None) \<or> (linform g = None)" by auto
+    moreover
+    {
+      assume linf: "\<exists> lf. linform f = Some lf"
+	and ling: "\<exists> lg. linform g = Some lg"
+      from linf obtain "lf" where lf: "linform f = Some lf" by blast
+      from ling obtain "lg" where lg: "linform g = Some lg" by blast
+      from lf lg have "linform (And f g) = Some (And lf lg)" by simp
+      then have "lp = And lf lg" using lf lg "prems"  by simp
+      with lf lg "prems" have ?thesis by simp
+    }
+    moreover
+    {
+      assume "linform f = None"
+      then have ?thesis using "And.prems"  by auto
+    }
+    moreover
+    {
+      assume "linform g = None"
+      then have ?thesis using "And.prems"  by (case_tac "linform f", auto)
+      
+    }
+    ultimately show ?thesis by blast
+  qed
+
+qed simp_all
+
+
+(* the result of linform is a linear formula *)
+lemma linform_lin: "\<And> lp. \<lbrakk> isnnf p ; linform p = Some lp\<rbrakk> \<Longrightarrow> islinform lp"
+proof (induct p rule: linform.induct)
+   case (Le x y)
+  have "((\<exists> lx. linearize x = Some lx) \<and> (\<exists> ly. linearize y = Some ly)) \<or> 
+    (linearize x = None) \<or> (linearize y = None) " by clarsimp
+  moreover 
+  {
+    assume linx: "\<exists> lx. linearize x = Some lx"
+      and liny: "\<exists> ly. linearize y = Some ly"
+    from linx obtain "lx" where lx: "linearize x = Some lx" by blast
+    from liny obtain "ly" where ly: "linearize y = Some ly" by blast
+    from lx have lxlin: "islinintterm lx" by (simp add: linearize_linear)
+    from ly have lylin: "islinintterm ly" by (simp add: linearize_linear)    
+    have lin1:"islinintterm (Cst 1)" by simp
+    have lin0: "islinintterm (Cst 0)" by simp
+    from "prems"  have "lp = Le (lin_add(lx,lin_neg ly)) (Cst 0)"
+      by auto
+    with lin0 lin1 lxlin lylin "prems" 
+    have ?case by (simp add: lin_add_lin lin_neg_lin)
+    
+  }
+
+  moreover 
+  {
+    assume "linearize x = None"
+    then have ?case using "prems" by simp
+  }
+  moreover 
+  {
+    assume "linearize y = None"
+    then have ?case using "prems" by (case_tac "linearize x",simp_all)
+  }
+  ultimately show ?case by blast
+next
+   case (Eq x y)
+  have "((\<exists> lx. linearize x = Some lx) \<and> (\<exists> ly. linearize y = Some ly)) \<or> 
+    (linearize x = None) \<or> (linearize y = None) " by clarsimp
+  moreover 
+  {
+    assume linx: "\<exists> lx. linearize x = Some lx"
+      and liny: "\<exists> ly. linearize y = Some ly"
+    from linx obtain "lx" where lx: "linearize x = Some lx" by blast
+    from liny obtain "ly" where ly: "linearize y = Some ly" by blast
+    from lx have lxlin: "islinintterm lx" by (simp add: linearize_linear)
+    from ly have lylin: "islinintterm ly" by (simp add: linearize_linear)    
+    have lin1:"islinintterm (Cst 1)" by simp
+    have lin0: "islinintterm (Cst 0)" by simp
+    from "prems"  have "lp = Eq (lin_add(lx,lin_neg ly)) (Cst 0)"
+      by auto
+    with lin0 lin1 lxlin lylin "prems" 
+    have ?case by (simp add: lin_add_lin lin_neg_lin)
+    
+  }
+
+  moreover 
+  {
+    assume "linearize x = None"
+    then have ?case using "prems" by simp
+  }
+  moreover 
+  {
+    assume "linearize y = None"
+    then have ?case using "prems" by (case_tac "linearize x",simp_all)
+  }
+  ultimately show ?case by blast
+next
+   case (Divides d t)
+   show ?case 
+     using prems
+     apply (case_tac "linearize d", auto)
+     apply (case_tac a, auto)
+     apply (case_tac "int = 0", auto)
+
+     by (case_tac "linearize t",auto simp add: linearize_linear)
+next
+  case (Or f g)
+ show ?case using "Or.hyps"
+  proof -
+    have "((\<exists> lf. linform f = Some lf ) \<and> (\<exists> lg. linform g = Some lg)) \<or> 
+      (linform f = None) \<or> (linform g = None)" by auto
+    moreover
+    {
+      assume linf: "\<exists> lf. linform f = Some lf"
+	and ling: "\<exists> lg. linform g = Some lg"
+      from linf obtain "lf" where lf: "linform f = Some lf" by blast
+      from ling obtain "lg" where lg: "linform g = Some lg" by blast
+      from lf lg have "linform (Or f g) = Some (Or lf lg)" by simp
+      then have "lp = Or lf lg" using lf lg "prems"  by simp
+      with lf lg "prems" have ?thesis by simp
+    }
+    moreover
+    {
+      assume "linform f = None"
+      then have ?thesis using "Or.prems"  by auto
+    }
+    moreover
+    {
+      assume "linform g = None"
+      then have ?thesis using "Or.prems"  by (case_tac "linform f", auto)
+      
+    }
+    ultimately show ?thesis by blast
+  qed
+next
+  case (And f g) 
+  show ?case using "And.hyps"
+  proof -
+    have "((\<exists> lf. linform f = Some lf ) \<and> (\<exists> lg. linform g = Some lg)) \<or> 
+      (linform f = None) \<or> (linform g = None)" by auto
+    moreover
+    {
+      assume linf: "\<exists> lf. linform f = Some lf"
+	and ling: "\<exists> lg. linform g = Some lg"
+      from linf obtain "lf" where lf: "linform f = Some lf" by blast
+      from ling obtain "lg" where lg: "linform g = Some lg" by blast
+      from lf lg have "linform (And f g) = Some (And lf lg)" by simp
+      then have "lp = And lf lg" using lf lg "prems"  by simp
+      with lf lg "prems" have ?thesis by simp
+    }
+    moreover
+    {
+      assume "linform f = None"
+      then have ?thesis using "And.prems"  by auto
+    }
+    moreover
+    {
+      assume "linform g = None"
+      then have ?thesis using "And.prems"  by (case_tac "linform f", auto)
+      
+    }
+    ultimately show ?thesis by blast
+  qed
+next
+  case (NOT f) show ?case
+    using "prems"
+  proof-
+    have "(\<exists> lf. linform f = Some lf) \<or> (linform f = None)" by auto
+    moreover 
+    {
+      assume linf: "\<exists> lf. linform f = Some lf"
+      from prems have "isnnf (NOT f)" by simp
+      then have fnnf: "isnnf f" by (cases f) auto
+      from linf obtain "lf" where lf: "linform f = Some lf" by blast
+      then have "lp = NOT lf" using "prems" by auto
+      with "NOT.prems" "NOT.hyps" lf fnnf
+      have ?thesis 
+	using fnnf
+	apply (cases f, auto) 
+	prefer 2
+	apply (case_tac "linearize intterm1",auto)
+	apply (case_tac a, auto)
+	apply (case_tac "int = 0", auto)
+	apply (case_tac "linearize intterm2") 
+	apply (auto simp add: linearize_linear)
+	apply (case_tac "linearize intterm1",auto)
+	by (case_tac "linearize intterm2") 
+      (auto simp add: linearize_linear lin_add_lin lin_neg_lin)
+    }
+    moreover 
+    {
+      assume "linform f = None"
+      then 
+      have "linform (NOT f) = None" by simp
+      then 
+      have ?thesis  using "NOT.prems" by simp
+    }
+    ultimately show ?thesis by blast
+  qed
+qed (simp_all)
+
+
+(* linform, if successfull, preserves quantifier freeness *)
+lemma linform_isnnf: "islinform p \<Longrightarrow> isnnf p"
+by (induct p rule: islinform.induct) auto
+
+lemma linform_isqfree: "islinform p \<Longrightarrow> isqfree p"
+using linform_isnnf nnf_isqfree by simp
+
+lemma linform_qfree: "\<And> p'. \<lbrakk> isnnf p ; linform p = Some p'\<rbrakk> \<Longrightarrow> isqfree p'"
+using linform_isqfree linform_lin 
+by simp
+
+(* Definitions and lemmas about gcd and lcm *)
+constdefs lcm :: "nat \<times> nat \<Rightarrow> nat"
+  "lcm \<equiv> (\<lambda>(m,n). m*n div gcd(m,n))"
+
+constdefs ilcm :: "int \<Rightarrow> int \<Rightarrow> int"
+  "ilcm \<equiv> \<lambda>i.\<lambda>j. int (lcm(nat(abs i),nat(abs j)))"
+
+(* ilcm_dvd12 are needed later *)
+lemma lcm_dvd1: 
+  assumes mpos: " m >0"
+  and npos: "n>0"
+  shows "m dvd (lcm(m,n))"
+proof-
+  have "gcd(m,n) dvd n" by simp
+  then obtain "k" where "n = gcd(m,n) * k" using dvd_def by auto
+  then have "m*n div gcd(m,n) = m*(gcd(m,n)*k) div gcd(m,n)" by (simp add: mult_ac)
+  also have "\<dots> = m*k" using mpos npos gcd_zero by simp
+  finally show ?thesis by (simp add: lcm_def)
+qed
+
+lemma lcm_dvd2: 
+  assumes mpos: " m >0"
+  and npos: "n>0"
+  shows "n dvd (lcm(m,n))"
+proof-
+  have "gcd(m,n) dvd m" by simp
+  then obtain "k" where "m = gcd(m,n) * k" using dvd_def by auto
+  then have "m*n div gcd(m,n) = (gcd(m,n)*k)*n div gcd(m,n)" by (simp add: mult_ac)
+  also have "\<dots> = n*k" using mpos npos gcd_zero by simp
+  finally show ?thesis by (simp add: lcm_def)
+qed
+
+lemma ilcm_dvd1: 
+assumes anz: "a \<noteq> 0" 
+  and bnz: "b \<noteq> 0"
+  shows "a dvd (ilcm a b)"
+proof-
+  let ?na = "nat (abs a)"
+  let ?nb = "nat (abs b)"
+  have nap: "?na >0" using anz by simp
+  have nbp: "?nb >0" using bnz by simp
+  from nap nbp have "?na dvd lcm(?na,?nb)" using lcm_dvd1 by simp
+  thus ?thesis by (simp add: ilcm_def dvd_int_iff)
+qed
+
+
+lemma ilcm_dvd2: 
+assumes anz: "a \<noteq> 0" 
+  and bnz: "b \<noteq> 0"
+  shows "b dvd (ilcm a b)"
+proof-
+  let ?na = "nat (abs a)"
+  let ?nb = "nat (abs b)"
+  have nap: "?na >0" using anz by simp
+  have nbp: "?nb >0" using bnz by simp
+  from nap nbp have "?nb dvd lcm(?na,?nb)" using lcm_dvd2 by simp
+  thus ?thesis by (simp add: ilcm_def dvd_int_iff)
+qed
+
+lemma zdvd_self_abs1: "(d::int) dvd (abs d)"
+by (case_tac "d <0", simp_all)
+
+lemma zdvd_self_abs2: "(abs (d::int)) dvd d"
+by (case_tac "d<0", simp_all)
+
+(* lcm a b is positive for positive a and b *)
+
+lemma lcm_pos: 
+  assumes mpos: "m > 0"
+  and npos: "n>0"
+  shows "lcm (m,n) > 0"
+
+proof(rule ccontr, simp add: lcm_def gcd_zero)
+assume h:"m*n div gcd(m,n) = 0"
+from mpos npos have "gcd (m,n) \<noteq> 0" using gcd_zero by simp
+hence gcdp: "gcd(m,n) > 0" by simp
+with h
+have "m*n < gcd(m,n)"
+  by (cases "m * n < gcd (m, n)") (auto simp add: div_if[OF gcdp, where m="m*n"])
+moreover 
+have "gcd(m,n) dvd m" by simp
+ with mpos dvd_imp_le have t1:"gcd(m,n) \<le> m" by simp
+ with npos have t1:"gcd(m,n)*n \<le> m*n" by simp
+ have "gcd(m,n) \<le> gcd(m,n)*n" using npos by simp
+ with t1 have "gcd(m,n) \<le> m*n" by arith
+ultimately show "False" by simp
+qed
+
+lemma ilcm_pos: 
+  assumes apos: " 0 < a"
+  and bpos: "0 < b" 
+  shows "0 < ilcm  a b"
+proof-
+  let ?na = "nat (abs a)"
+  let ?nb = "nat (abs b)"
+  have nap: "?na >0" using apos by simp
+  have nbp: "?nb >0" using bpos by simp
+  have "0 < lcm (?na,?nb)" by (rule lcm_pos[OF nap nbp])
+  thus ?thesis by (simp add: ilcm_def)
+qed
+
+(* fomlcm computes the lcm of all c, where c is the coeffitient of Var 0 *)
+consts formlcm :: "QF \<Rightarrow> int"
+recdef formlcm "measure size"
+"formlcm (Le (Add (Mult (Cst c) (Var 0)) r) (Cst i)) = abs c "
+"formlcm (Eq (Add (Mult (Cst c) (Var 0)) r) (Cst i)) = abs c "
+"formlcm (Divides (Cst d) (Add (Mult (Cst c) (Var 0)) r)) = abs c"
+"formlcm (NOT p) = formlcm p"
+"formlcm (And p q)= ilcm (formlcm p) (formlcm q)"
+"formlcm (Or p q) = ilcm (formlcm p) (formlcm q)"
+"formlcm p = 1"
+
+(* the property that formlcm should fullfill *)
+consts divideallc:: "int \<times> QF \<Rightarrow> bool"
+recdef divideallc "measure (\<lambda>(i,p). size p)"
+"divideallc (l,Le (Add (Mult (Cst c) (Var 0)) r) (Cst i)) = (c dvd l)"
+"divideallc (l,Eq (Add (Mult (Cst c) (Var 0)) r) (Cst i)) = (c dvd l)"
+"divideallc(l,Divides (Cst d) (Add (Mult (Cst c) (Var 0)) r)) = (c dvd l)"
+"divideallc (l,NOT p) = divideallc(l,p)"
+"divideallc (l,And p q) = (divideallc (l,p) \<and> divideallc (l,q))"
+"divideallc (l,Or p q) = (divideallc (l,p) \<and> divideallc (l,q))"
+"divideallc p = True"
+
+(* formlcm retuns a positive integer *)
+lemma formlcm_pos: 
+  assumes linp: "islinform p"
+  shows "0 < formlcm p"
+using linp
+proof (induct p rule: formlcm.induct, simp_all add: ilcm_pos)
+  case (goal1 c r i)
+  have "i=0 \<or> i \<noteq> 0" by simp
+  moreover
+  {
+    assume "i \<noteq> 0" then have ?case using prems by simp
+  }
+  moreover 
+  {
+    assume iz: "i = 0"
+    then have "islinintterm (Add (Mult (Cst c) (Var 0)) r)" using prems by simp
+    then have "c\<noteq>0" 
+      using prems
+      by (simp add: islininttermc0r[where c="c" and n="0" and r="r"])
+    then have ?case by simp
+  }
+  ultimately 
+  show ?case by blast
+next 
+  case (goal2 c r i)
+  have "i=0 \<or> i \<noteq> 0" by simp
+  moreover
+  {
+    assume "i \<noteq> 0" then have ?case using prems by simp
+  }
+  moreover 
+  {
+    assume iz: "i = 0"
+    then have "islinintterm (Add (Mult (Cst c) (Var 0)) r)" using prems by simp
+    then have "c\<noteq>0" 
+      using prems
+      by (simp add: islininttermc0r[where c="c" and n="0" and r="r"])
+    then have ?case by simp
+  }
+  ultimately 
+  show ?case by blast
+
+next 
+  case (goal3 d c r)
+  show ?case using prems by (simp add: islininttermc0r[where c="c" and n="0" and r="r"])
+next
+  case (goal4 f)
+  show ?case using prems 
+    by (cases f,auto) (case_tac "intterm2", auto,case_tac "intterm1", auto)
+qed
+
+lemma divideallc_mono: "\<And> c. \<lbrakk> divideallc(c,p) ; c dvd d\<rbrakk> \<Longrightarrow> divideallc (d,p)"
+proof (induct d p rule: divideallc.induct, simp_all)
+  case (goal1 l a b) show ?case by ( rule zdvd_trans [where m="a" and n="b" and k="l"])
+next
+  case (goal2 l a b) show ?case by ( rule zdvd_trans [where m="a" and n="b" and k="l"])
+next
+ case (goal3 l a b) show ?case by ( rule zdvd_trans [where m="a" and n="b" and k="l"])
+next
+  case (goal4 l f g k)
+  have  "divideallc (l,g)" using prems by clarsimp
+  moreover have "divideallc (l,f)" using prems by clarsimp
+  ultimately
+  show ?case  by simp
+next 
+  case (goal5 l f g k)
+  have  "divideallc (l,g)" using prems by clarsimp
+  moreover have "divideallc (l,f)" using prems by clarsimp
+  ultimately
+  show ?case  by simp
+  
+qed
+
+(* fomlcm retuns a number all coeffitients of Var 0 divide *)
+
+lemma formlcm_divideallc: 
+  assumes linp: "islinform p"
+  shows "divideallc(formlcm p, p)"
+using linp
+proof (induct p rule: formlcm.induct, simp_all add: zdvd_self_abs1)
+  case (goal1 f)
+  show ?case using prems
+    by (cases f,auto) (case_tac "intterm2", auto, case_tac "intterm1",auto)
+next 
+  case (goal2 f g)
+  have "formlcm f >0" using formlcm_pos prems by simp 
+    hence "formlcm f \<noteq> 0" by simp
+  moreover have "formlcm g > 0" using formlcm_pos prems by simp
+  hence "formlcm g \<noteq> 0" by simp
+  ultimately
+  show ?case using prems formlcm_pos
+     by (simp add: ilcm_dvd1 ilcm_dvd2 
+       divideallc_mono[where c="formlcm f" and d="ilcm (formlcm f) (formlcm g)"]  
+       divideallc_mono[where c="formlcm g" and d="ilcm (formlcm f) (formlcm g)"])
+next 
+  case (goal3 f g)
+  have "formlcm f >0" using formlcm_pos prems by simp 
+    hence "formlcm f \<noteq> 0" by simp
+  moreover have "formlcm g > 0" using formlcm_pos prems by simp
+  hence "formlcm g \<noteq> 0" by simp
+  ultimately
+  show ?case using prems 
+    by (simp add: ilcm_dvd1 ilcm_dvd2 
+      divideallc_mono[where c="formlcm f" and d="ilcm (formlcm f) (formlcm g)"]  
+      divideallc_mono[where c="formlcm g" and d="ilcm (formlcm f) (formlcm g)"])
+qed
+
+(* adjustcoeff transforms the formula given an l , look at correctness thm*)
+consts adjustcoeff :: "int \<times> QF \<Rightarrow> QF"
+recdef adjustcoeff "measure (\<lambda>(l,p). size p)"
+"adjustcoeff (l,(Le (Add (Mult (Cst c) (Var 0)) r) (Cst i))) = 
+  (if c\<le>0 then 
+  Le (Add (Mult (Cst -1) (Var 0)) (lin_mul (- (l div c), r))) (Cst (0::int))
+  else
+  Le (Add (Mult (Cst 1) (Var 0)) (lin_mul (l div c, r))) (Cst (0::int)))"
+"adjustcoeff (l,(Eq (Add (Mult (Cst c) (Var 0)) r) (Cst i))) = 
+  (Eq (Add (Mult (Cst 1) (Var 0)) (lin_mul (l div c, r))) (Cst (0::int)))"
+"adjustcoeff (l,Divides (Cst d) (Add (Mult (Cst c) (Var 0)) r)) = 
+  Divides (Cst ((l div c) * d))
+  (Add (Mult (Cst 1) (Var 0)) (lin_mul (l div c, r)))"
+"adjustcoeff (l,NOT (Divides (Cst d) (Add (Mult (Cst c) (Var 0)) r))) = NOT (Divides (Cst ((l div c) * d))
+  (Add (Mult (Cst 1) (Var 0)) (lin_mul (l div c, r))))"
+"adjustcoeff (l,(NOT(Eq (Add (Mult (Cst c) (Var 0)) r) (Cst i)))) = 
+  (NOT(Eq (Add (Mult (Cst 1) (Var 0)) (lin_mul (l div c, r))) (Cst (0::int))))"
+"adjustcoeff (l,And p q) = And (adjustcoeff (l,p)) (adjustcoeff(l,q))"
+"adjustcoeff (l,Or p q) = Or (adjustcoeff (l,p)) (adjustcoeff(l,q))"
+"adjustcoeff (l,p) = p"
+
+
+(* unitycoeff expects a quantifier free formula an transforms it to an equivalent formula where the bound variable occurs only with coeffitient 1  or -1 *)
+constdefs unitycoeff :: "QF \<Rightarrow> QF"
+  "unitycoeff p == 
+  (let l = formlcm p;
+       p' = adjustcoeff (l,p)
+   in (if l=1 then p' else 
+      (And (Divides (Cst l) (Add (Mult (Cst 1) (Var 0)) (Cst 0))) p')))"
+
+(* what is a unified formula *)
+consts isunified :: "QF \<Rightarrow> bool"
+recdef isunified "measure size"
+"isunified (Le (Add (Mult (Cst i) (Var 0)) r) (Cst z)) = 
+  ((abs i) = 1  \<and> (islinform(Le (Add (Mult (Cst i) (Var 0)) r) (Cst z))))"
+"isunified (Eq (Add (Mult (Cst i) (Var 0)) r) (Cst z)) = 
+  ((abs i) = 1  \<and> (islinform(Le (Add (Mult (Cst i) (Var 0)) r) (Cst z))))"
+"isunified (NOT(Eq (Add (Mult (Cst i) (Var 0)) r) (Cst z))) = 
+  ((abs i) = 1  \<and> (islinform(Le (Add (Mult (Cst i) (Var 0)) r) (Cst z))))"
+"isunified (Divides (Cst d) (Add (Mult (Cst i) (Var 0)) r)) = 
+  ((abs i) = 1 \<and> (islinform(Divides (Cst d) (Add (Mult (Cst i) (Var 0)) r))))"
+"isunified (NOT(Divides (Cst d) (Add (Mult (Cst i) (Var 0)) r))) = 
+  ((abs i) = 1 \<and> (islinform(NOT(Divides (Cst d) (Add (Mult (Cst i) (Var 0)) r)))))"
+"isunified (And p q) = (isunified p \<and> isunified q)"
+"isunified (Or p q) = (isunified p \<and> isunified q)"
+"isunified p = islinform p"
+
+lemma unified_islinform: "isunified p \<Longrightarrow> islinform p"
+by (induct p rule: isunified.induct) auto
+
+lemma adjustcoeff_lenpos: 
+  "0 < n \<Longrightarrow> adjustcoeff (l, Le (Add (Mult (Cst i) (Var n)) r) (Cst c)) =
+    Le (Add (Mult (Cst i) (Var n)) r) (Cst c)"
+by (cases n, auto)
+
+lemma adjustcoeff_eqnpos: 
+  "0 < n \<Longrightarrow> adjustcoeff (l, Eq (Add (Mult (Cst i) (Var n)) r) (Cst c)) =
+    Eq (Add (Mult (Cst i) (Var n)) r) (Cst c)"
+by (cases n, auto)
+
+
+(* Properties of adjustcoeff and unitycoeff *)
+
+(* Some simple lemmas used afterwards *)
+lemma zmult_zle_mono: "(i::int) \<le> j \<Longrightarrow> 0 \<le> k \<Longrightarrow> k * i \<le> k * j"
+  apply (erule order_le_less [THEN iffD1, THEN disjE, of "0::int"])
+  apply (erule order_le_less [THEN iffD1, THEN disjE])
+  apply (rule order_less_imp_le)
+  apply (rule zmult_zless_mono2)
+  apply simp_all
+  done
+
+lemma zmult_zle_mono_eq:
+  assumes kpos: "0 < k"
+  shows "((i::int) \<le> j) = (k*i \<le> k*j)" (is "?P = ?Q")
+proof
+  assume P: ?P
+  from kpos have kge0: "0 \<le> k" by simp
+  show ?Q
+    by (rule zmult_zle_mono[OF P kge0])
+next 
+  assume ?Q
+  then have "k*i - k*j \<le> 0" by simp
+  then have le1: "k*(i-j) \<le> k*0"
+    by (simp add: zdiff_zmult_distrib2)
+  have "i -j \<le> 0" 
+    by (rule mult_left_le_imp_le[OF le1 kpos])
+  then 
+  show ?P by simp
+qed
+  
+
+lemma adjustcoeff_le_corr:
+  assumes lpos: "0 < l"
+  and ipos: "0 < (i::int)"
+  and dvd: "i dvd l"
+  shows "(i*x + r \<le> 0) = (l*x + ((l div i)*r) \<le> 0)"
+proof-
+  from lpos ipos have ilel: "i\<le>l" by (simp add: zdvd_imp_le [OF dvd lpos])
+  from ipos have inz: "i \<noteq> 0" by simp
+  have "i div i\<le> l div i"
+    by (simp add: zdiv_mono1[OF ilel ipos])
+  then have ldivipos:"0 < l div i" 
+    by (simp add: zdiv_self[OF inz])
+  
+  from dvd have "\<exists>i'. i*i' = l" by (auto simp add: dvd_def)
+  then obtain "i'" where ii'eql: "i*i' = l" by blast
+  have "(i * x + r \<le> 0) = (l div i * (i * x + r) \<le> l div i * 0)"
+    by (rule zmult_zle_mono_eq[OF ldivipos, where i="i*x + r" and j="0"])
+  also
+  have "(l div i * (i * x + r) \<le> l div i * 0) = ((l div i * i) * x + ((l div i)*r) \<le> 0)"
+    by (simp add: mult_ac)
+  also have "((l div i * i) * x + ((l div i)*r) \<le> 0) = (l*x + ((l div i)*r) \<le> 0)"
+    using sym[OF ii'eql] inz
+    by (simp add: zmult_ac)
+  finally  
+  show ?thesis
+    by simp
+qed
+
+lemma adjustcoeff_le_corr2:
+  assumes lpos: "0 < l"
+  and ineg: "(i::int) < 0"
+  and dvd: "i dvd l"
+  shows "(i*x + r \<le> 0) = ((-l)*x + ((-(l div i))*r) \<le> 0)"
+proof-
+  from dvd have midvdl: "-i dvd l" by simp
+  from ineg have mipos: "0 < -i" by simp
+  from lpos ineg have milel: "-i\<le>l" by (simp add: zdvd_imp_le [OF midvdl lpos])
+  from ineg have inz: "i \<noteq> 0" by simp
+  have "l div i\<le> -i div i"
+    by (simp add: zdiv_mono1_neg[OF milel ineg])
+  then have "l div i \<le> -1" 
+    apply (simp add: zdiv_zminus1_eq_if[OF inz, where a="i"])
+    by (simp add: zdiv_self[OF inz])
+  then have ldivineg: "l div i < 0" by simp
+  then have mldivipos: "0 < - (l div i)" by simp
+  
+  from dvd have "\<exists>i'. i*i' = l" by (auto simp add: dvd_def)
+  then obtain "i'" where ii'eql: "i*i' = l" by blast
+  have "(i * x + r \<le> 0) = (- (l div i) * (i * x + r) \<le> - (l div i) * 0)"
+    by (rule zmult_zle_mono_eq[OF mldivipos, where i="i*x + r" and j="0"])
+  also
+  have "(- (l div i) * (i * x + r) \<le> - (l div i) * 0) = (-((l div i) * i) * x \<le> (l div i)*r)"
+    by (simp add: mult_ac)
+  also have " (-((l div i) * i) * x \<le> (l div i)*r) = (- (l*x) \<le> (l div i)*r)"
+    using sym[OF ii'eql] inz
+    by (simp add: zmult_ac)
+  finally  
+  show ?thesis
+    by simp
+qed
+
+(* FIXME : Move this theorem above, it simplifies the 2 theorems above : adjustcoeff_le_corr1,2 *)
+lemma dvd_div_pos: 
+  assumes bpos: "0 < (b::int)"
+  and anz: "a\<noteq>0"
+  and dvd: "a dvd b"
+  shows "(b div a)*a = b"
+proof-
+  from anz have "0 < a \<or> a < 0" by arith
+  moreover
+  {
+    assume apos: "0 < a" 
+    from bpos apos have aleb: "a\<le>b" by (simp add: zdvd_imp_le [OF dvd bpos])
+    have "a div a\<le> b div a"
+      by (simp add: zdiv_mono1[OF aleb apos])
+    then have bdivapos:"0 < b div a" 
+      by (simp add: zdiv_self[OF anz])
+    
+    from dvd have "\<exists>a'. a*a' = b" by (auto simp add: dvd_def)
+    then obtain "a'" where aa'eqb: "a*a' = b" by blast
+    then have ?thesis  using anz sym[OF aa'eqb] by simp
+    
+  }
+  moreover
+  {
+    assume aneg: "a < 0"
+    from dvd have midvdb: "-a dvd b" by simp
+    from aneg have mapos: "0 < -a" by simp
+    from bpos aneg have maleb: "-a\<le>b" by (simp add: zdvd_imp_le [OF midvdb bpos])
+    from aneg have anz: "a \<noteq> 0" by simp
+    have "b div a\<le> -a div a"
+      by (simp add: zdiv_mono1_neg[OF maleb aneg])
+    then have "b div a \<le> -1" 
+      apply (simp add: zdiv_zminus1_eq_if[OF anz, where a="a"])
+      by (simp add: zdiv_self[OF anz])
+    then have bdivaneg: "b div a < 0" by simp
+    then have mbdivapos: "0 < - (b div a)" by simp
+    
+    from dvd have "\<exists>a'. a*a' = b" by (auto simp add: dvd_def)
+    then obtain "a'" where aa'eqb: "a*a' = b" by blast
+    then have ?thesis using anz sym[OF aa'eqb] by (simp)
+  }
+  ultimately show ?thesis by blast
+qed
+
+lemma adjustcoeff_eq_corr: 
+  assumes lpos: "(0::int) < l"
+  and inz: "i \<noteq> 0"
+  and dvd: "i dvd l"
+  shows "(i*x + r = 0) = (l*x + ((l div i)*r) = 0)"
+proof-
+  thm dvd_div_pos[OF lpos inz dvd]
+  have ldvdii: "(l div i)*i = l" by (rule dvd_div_pos[OF lpos inz dvd])
+  have ldivinz: "l div i \<noteq> 0" using inz ldvdii lpos by auto
+  have "(i*x + r = 0) = ((l div i)*(i*x + r) = (l div i)*0)"
+    using ldivinz by arith
+  also have "\<dots> = (((l div i)*i)*x + (l div i)*r = 0)"
+    by (simp add: zmult_ac)
+  finally show ?thesis using ldvdii by simp
+qed
+
+
+
+(* Correctness theorem for adjustcoeff *)
+lemma adjustcoeff_corr:
+  assumes linp: "islinform p"
+  and alldvd: "divideallc (l,p)"
+  and lpos: "0 < l"
+  shows "qinterp (a#ats) p = qinterp ((a*l)#ats) (adjustcoeff(l, p))"
+using linp alldvd
+proof (induct p rule: islinform.induct,simp_all)
+  case (goal1 t c)
+  from prems have cz: "c=0" by simp
+    then have ?case
+      using prems
+    proof(induct t rule: islinintterm.induct)
+      case (2 i n i') show ?case using prems
+	proof-
+	  from prems have "i\<noteq>0" by simp
+	  then 
+	  have "(n=0 \<and> i < 0) \<or> (n=0 \<and> i > 0) \<or> n\<noteq>0" by arith
+	  moreover 
+	  {
+	    assume "n\<noteq>0" then have ?thesis 
+	      by (simp add: nth_pos2 adjustcoeff_lenpos)
+	  }
+	  moreover
+	  {
+	    assume nz: "n=0"
+	      and ipos: "0 < i"
+	    from prems nz have idvdl: "i dvd l" by simp
+	    have "(i*a + i' \<le> 0) = (l*a+ ((l div i)*i') \<le> 0)" 
+	      by (rule adjustcoeff_le_corr[OF lpos ipos idvdl])
+	    then 
+	    have ?thesis using prems by (simp add: mult_ac)
+	  }
+	  moreover
+	  {
+	    assume nz: "n=0"
+	      and ineg: "i < 0"
+	    from prems nz have idvdl: "i dvd l" by simp
+	    have "(i*a+i' \<le> 0) = (-l*a + (-(l div i) * i') \<le> 0)"
+	      by (rule adjustcoeff_le_corr2[OF lpos ineg idvdl])
+	    then 
+	    have ?thesis using prems
+	      by (simp add: zmult_ac)
+	  }
+	  ultimately show ?thesis by blast
+	qed
+      next
+	case (3 i n i' n' r) show ?case  using prems
+	proof-
+	  from prems 
+	  have lininrp: "islinintterm (Add (Mult (Cst i') (Var n')) r)" 
+	    by simp
+	  then
+	  have "islint (Add (Mult (Cst i') (Var n')) (r))" 
+	    by (simp add: islinintterm_eq_islint)
+	  then have linr: "islintn(Suc n',r)"
+	    by (simp add: islinintterm_subt[OF lininrp] islinintterm_eq_islint islint_def)
+	  from lininrp have linr2: "islinintterm r"
+	    by (simp add: islinintterm_subt[OF lininrp])
+	  from prems have "n < n'" by simp
+	  then have nppos: "0 < n'" by simp
+	  from prems have "i\<noteq>0" by simp
+	  then 
+	  have "(n=0 \<and> i < 0) \<or> (n=0 \<and> i > 0) \<or> n\<noteq>0" by arith
+	  moreover 
+	  {
+	    assume nnz: "n\<noteq>0"
+	    from linr have ?thesis using nppos nnz intterm_novar0[OF lininrp] prems
+	      apply (simp add: adjustcoeff_lenpos linterm_novar0[OF linr, where x="a" and y="a*l"])
+	      by (simp add: nth_pos2)
+	      
+	  }
+	  moreover
+	  {
+	    assume nz: "n=0"
+	      and ipos: "0 < i"
+	    from prems nz have idvdl: "i dvd l" by simp
+	    have "(i * a + (i' * (a # ats) ! n' + I_intterm (a # ats) r) \<le> 0) =
+	      (l * a + l div i * (i' * (a # ats) ! n' + I_intterm (a # ats) r) \<le> 0)"
+	      by (rule adjustcoeff_le_corr[OF lpos ipos idvdl])
+	    then 
+	    have ?thesis using prems linr linr2
+	      by (simp add: mult_ac nth_pos2 lin_mul_corr 
+		linterm_novar0[OF linr, where x="a" and y="a*l"])
+	  }
+	  moreover
+	  {
+	    assume nz: "n=0"
+	      and ineg: "i < 0"
+	    from prems nz have idvdl: "i dvd l" by simp
+	    have "(i * a + (i' * (a # ats) ! n' + I_intterm (a # ats) r) \<le> 0) =
+	      (- l * a + - (l div i) * (i' * (a # ats) ! n' + I_intterm (a # ats) r) \<le> 0)"
+	      by (rule adjustcoeff_le_corr2[OF lpos ineg idvdl, where  x="a" and r="(i'* (a#ats) ! n' + I_intterm (a#ats) r )"])
+	    then 
+	    have ?thesis using prems linr linr2
+	      by (simp add: zmult_ac nth_pos2 lin_mul_corr 
+		linterm_novar0[OF linr, where x="a" and y="a*l"] )
+	  }
+	  ultimately show ?thesis by blast
+	qed	  
+    qed simp_all
+    then show ?case by simp 
+  
+next
+  case (goal2 t c)
+  from prems have cz: "c=0" by simp
+    then have ?case
+      using prems
+    proof(induct t rule: islinintterm.induct)
+      case (2 i n i') show ?case using prems
+	proof-
+	  from prems have inz: "i\<noteq>0" by simp
+	  then 
+	  have "n=0 \<or> n\<noteq>0" by arith
+	  moreover 
+	  {
+	    assume "n\<noteq>0" then have ?thesis 
+	      by (simp add: nth_pos2 adjustcoeff_eqnpos)
+	  }
+	  moreover
+	  {
+	    assume nz: "n=0"
+	    from prems nz have idvdl: "i dvd l" by simp
+	    have "(i*a + i' = 0) = (l*a+ ((l div i)*i') = 0)" 
+	      by (rule adjustcoeff_eq_corr[OF lpos inz idvdl])
+	    then 
+	    have ?thesis using prems by (simp add: mult_ac)
+	  }
+	  ultimately show ?thesis by blast
+	qed
+      next
+	case (3 i n i' n' r) show ?case  using prems
+	proof-
+	  from prems 
+	  have lininrp: "islinintterm (Add (Mult (Cst i') (Var n')) r)" 
+	    by simp
+	  then
+	  have "islint (Add (Mult (Cst i') (Var n')) (r))" 
+	    by (simp add: islinintterm_eq_islint)
+	  then have linr: "islintn(Suc n',r)"
+	    by (simp add: islinintterm_subt[OF lininrp] islinintterm_eq_islint islint_def)
+	  from lininrp have linr2: "islinintterm r"
+	    by (simp add: islinintterm_subt[OF lininrp])
+	  from prems have "n < n'" by simp
+	  then have nppos: "0 < n'" by simp
+	  from prems have "i\<noteq>0" by simp
+	  then 
+	  have "n=0 \<or> n\<noteq>0" by arith
+	  moreover 
+	  {
+	    assume nnz: "n\<noteq>0"
+	    from linr have ?thesis using nppos nnz intterm_novar0[OF lininrp] prems
+	      apply (simp add: adjustcoeff_eqnpos linterm_novar0[OF linr, where x="a" and y="a*l"])
+	      by (simp add: nth_pos2)
+	      
+	  }
+	  moreover
+	  {
+	    assume nz: "n=0"
+	    from prems have inz: "i \<noteq> 0" by auto
+	    from prems nz have idvdl: "i dvd l" by simp
+	    thm adjustcoeff_eq_corr[OF lpos inz idvdl]
+	    have "(i * a + (i' * (a # ats) ! n' + I_intterm (a # ats) r) = 0) =
+	      (l * a + l div i * (i' * (a # ats) ! n' + I_intterm (a # ats) r) = 0)"
+	      by (rule adjustcoeff_eq_corr[OF lpos inz idvdl])
+	    then 
+	    have ?thesis using prems linr linr2
+	      by (simp add: mult_ac nth_pos2 lin_mul_corr 
+		linterm_novar0[OF linr, where x="a" and y="a*l"])
+	  }
+	  ultimately show ?thesis by blast
+	qed	  
+    qed simp_all
+    then show ?case by simp 
+  
+next
+  case (goal3 d t) show ?case
+    using prems
+    proof (induct t rule: islinintterm.induct)
+      case (2 i n i') 
+      have "n=0 \<or> (\<exists>m. (n = Suc m))" by arith
+      moreover
+      {
+	assume "\<exists>m. n = Suc m"
+	then have ?case using prems  by auto
+      }
+      moreover 
+      {
+	assume nz: "n=0"
+	from prems have inz: "i\<noteq>0" by simp
+	from prems have idvdl: "i dvd l" by simp
+	have ldiviieql: "l div i * i = l" by (rule dvd_div_pos[OF lpos inz idvdl])
+	with lpos have ldivinz: "0 \<noteq> l div i" by auto
+	  
+	then have ?case using prems
+	  apply simp
+	  apply (simp add: 
+	    ac_dvd_eq[OF ldivinz, where m="d" and c="i" and n="a" and t="i'"] 
+	    ldiviieql)
+	  by (simp add: zmult_commute)
+      }
+      ultimately show ?case by blast
+
+    next 
+      case (3 i n i' n' r)
+      from prems 
+      have lininrp: "islinintterm (Add (Mult (Cst i') (Var n')) r)" 
+	by simp
+      then
+      have "islint (Add (Mult (Cst i') (Var n')) (r))" 
+	by (simp add: islinintterm_eq_islint)
+      then have linr: "islintn(Suc n',r)"
+	by (simp add: islinintterm_subt[OF lininrp] islinintterm_eq_islint islint_def)
+      from lininrp have linr2: "islinintterm r"
+	by (simp add: islinintterm_subt[OF lininrp])
+      from prems have "n < n'" by simp
+      then have nppos: "0 < n'" by simp
+      from prems have inz: "i\<noteq>0" by simp
+      
+      have "n=0 \<or> (\<exists>m. (n = Suc m))" by arith
+      moreover
+      {
+	assume "\<exists>m. n = Suc m"
+	then have npos: "0 < n" by arith
+	have ?case using nppos intterm_novar0[OF lininrp] prems
+	  apply (auto simp add: linterm_novar0[OF linr, where x="a" and y="a*l"])
+	  by (simp_all add: nth_pos2)
+      }
+      moreover 
+      {
+	assume nz: "n=0"
+	from prems have idvdl: "i dvd l" by simp
+	have ldiviieql: "l div i * i = l" by (rule dvd_div_pos[OF lpos inz idvdl])
+	with lpos have ldivinz: "0 \<noteq> l div i" by auto
+	  
+	then have ?case using prems linr2 linr
+	  apply (simp add: nth_pos2 lin_mul_corr linterm_novar0)
+	  
+	  apply (simp add: ac_dvd_eq[OF ldivinz, where m="d" and c="i" and n="a" and t="(i' * ats ! (n' - Suc 0) + I_intterm (a # ats) r)"] ldiviieql)
+	  by (simp add: zmult_ac linterm_novar0[OF linr, where x="a" and y="a*l"])
+      }
+      ultimately show ?case by blast
+      
+    qed simp_all
+next
+  case (goal4 d t) show ?case
+    using prems
+    proof (induct t rule: islinintterm.induct)
+      case (2 i n i') 
+      have "n=0 \<or> (\<exists>m. (n = Suc m))" by arith
+      moreover
+      {
+	assume "\<exists>m. n = Suc m"
+	then have ?case using prems  by auto
+      }
+      moreover 
+      {
+	assume nz: "n=0"
+	from prems have inz: "i\<noteq>0" by simp
+	from prems have idvdl: "i dvd l" by simp
+	have ldiviieql: "l div i * i = l" by (rule dvd_div_pos[OF lpos inz idvdl])
+	with lpos have ldivinz: "0 \<noteq> l div i" by auto
+	  
+	then have ?case using prems
+	  apply simp
+	  apply (simp add: 
+	    ac_dvd_eq[OF ldivinz, where m="d" and c="i" and n="a" and t="i'"] 
+	    ldiviieql)
+	  by (simp add: zmult_commute)
+      }
+      ultimately show ?case by blast
+
+    next 
+      case (3 i n i' n' r)
+      from prems 
+      have lininrp: "islinintterm (Add (Mult (Cst i') (Var n')) r)" 
+	by simp
+      then
+      have "islint (Add (Mult (Cst i') (Var n')) (r))" 
+	by (simp add: islinintterm_eq_islint)
+      then have linr: "islintn(Suc n',r)"
+	by (simp add: islinintterm_subt[OF lininrp] islinintterm_eq_islint islint_def)
+      from lininrp have linr2: "islinintterm r"
+	by (simp add: islinintterm_subt[OF lininrp])
+      from prems have "n < n'" by simp
+      then have nppos: "0 < n'" by simp
+      from prems have inz: "i\<noteq>0" by simp
+      
+      have "n=0 \<or> (\<exists>m. (n = Suc m))" by arith
+      moreover
+      {
+	assume "\<exists>m. n = Suc m"
+	then have npos: "0 < n" by arith
+	have ?case using nppos intterm_novar0[OF lininrp] prems
+	  apply (auto simp add: linterm_novar0[OF linr, where x="a" and y="a*l"])
+	  by (simp_all add: nth_pos2)
+      }
+      moreover 
+      {
+	assume nz: "n=0"
+	from prems have idvdl: "i dvd l" by simp
+	have ldiviieql: "l div i * i = l" by (rule dvd_div_pos[OF lpos inz idvdl])
+	with lpos have ldivinz: "0 \<noteq> l div i" by auto
+	  
+	then have ?case using prems linr2 linr
+	  apply (simp add: nth_pos2 lin_mul_corr linterm_novar0)
+	  
+	  apply (simp add: ac_dvd_eq[OF ldivinz, where m="d" and c="i" and n="a" and t="(i' * ats ! (n' - Suc 0) + I_intterm (a # ats) r)"] ldiviieql)
+	  by (simp add: zmult_ac linterm_novar0[OF linr, where x="a" and y="a*l"])
+      }
+      ultimately show ?case by blast
+      
+    qed simp_all
+next
+    case (goal5 t c)
+  from prems have cz: "c=0" by simp
+    then have ?case
+      using prems
+    proof(induct t rule: islinintterm.induct)
+      case (2 i n i') show ?case using prems
+	proof-
+	  from prems have inz: "i\<noteq>0" by simp
+	  then 
+	  have "n=0 \<or> n\<noteq>0" by arith
+	  moreover 
+	  {
+	    assume "n\<noteq>0" then have ?thesis
+	      using prems
+	      by (cases n, simp_all)
+	  }
+	  moreover
+	  {
+	    assume nz: "n=0"
+	    from prems nz have idvdl: "i dvd l" by simp
+	    have "(i*a + i' = 0) = (l*a+ ((l div i)*i') = 0)" 
+	      by (rule adjustcoeff_eq_corr[OF lpos inz idvdl])
+	    then 
+	    have ?thesis using prems by (simp add: mult_ac)
+	  }
+	  ultimately show ?thesis by blast
+	qed
+      next
+	case (3 i n i' n' r) show ?case  using prems
+	proof-
+	  from prems 
+	  have lininrp: "islinintterm (Add (Mult (Cst i') (Var n')) r)" 
+	    by simp
+	  then
+	  have "islint (Add (Mult (Cst i') (Var n')) (r))" 
+	    by (simp add: islinintterm_eq_islint)
+	  then have linr: "islintn(Suc n',r)"
+	    by (simp add: islinintterm_subt[OF lininrp] islinintterm_eq_islint islint_def)
+	  from lininrp have linr2: "islinintterm r"
+	    by (simp add: islinintterm_subt[OF lininrp])
+	  from prems have "n < n'" by simp
+	  then have nppos: "0 < n'" by simp
+	  from prems have "i\<noteq>0" by simp
+	  then 
+	  have "n=0 \<or> n\<noteq>0" by arith
+	  moreover 
+	  {
+	    assume nnz: "n\<noteq>0"
+	    then have ?thesis using prems linr nppos nnz intterm_novar0[OF lininrp]
+	      by (cases n, simp_all)
+	    (simp add: nth_pos2 linterm_novar0[OF linr, where x="a" and y="a*l"])
+	  }
+	  moreover
+	  {
+	    assume nz: "n=0"
+	    from prems have inz: "i \<noteq> 0" by auto
+	    from prems nz have idvdl: "i dvd l" by simp
+	    have "(i * a + (i' * (a # ats) ! n' + I_intterm (a # ats) r) = 0) =
+	      (l * a + l div i * (i' * (a # ats) ! n' + I_intterm (a # ats) r) = 0)"
+	      by (rule adjustcoeff_eq_corr[OF lpos inz idvdl])
+	    then 
+	    have ?thesis using prems linr linr2
+	      by (simp add: mult_ac nth_pos2 lin_mul_corr 
+		linterm_novar0[OF linr, where x="a" and y="a*l"])
+	  }
+	  ultimately show ?thesis by blast
+	qed	  
+    qed simp_all
+    then show ?case by simp 
+  
+qed
+
+(* unitycoeff preserves semantics *)
+lemma unitycoeff_corr:
+  assumes linp: "islinform p"
+  shows "qinterp ats (QEx p) = qinterp ats (QEx (unitycoeff p))"
+proof-
+  
+  have lpos: "0 < formlcm p" by (rule formlcm_pos[OF linp])
+  have dvd : "divideallc (formlcm p, p)" by (rule formlcm_divideallc[OF linp])
+  show ?thesis  using prems lpos dvd 
+  proof (simp add: unitycoeff_def Let_def,case_tac "formlcm p = 1",
+      simp_all add: adjustcoeff_corr)
+    show "(\<exists>x. qinterp (x * formlcm p # ats) (adjustcoeff (formlcm p, p))) =
+      (\<exists>x. formlcm p dvd x \<and>
+      qinterp (x # ats) (adjustcoeff (formlcm p, p)))"
+      (is "(\<exists>x. ?P(x* (formlcm p))) = (\<exists>x. formlcm p dvd x \<and> ?P x)")
+    proof-
+      have "(\<exists>x. ?P(x* (formlcm p))) = (\<exists>x. ?P((formlcm p)*x))"
+	by (simp add: mult_commute)
+      also have "(\<exists>x. ?P((formlcm p)*x)) = (\<exists>x. (formlcm p dvd x) \<and> ?P x)"
+	by (simp add: unity_coeff_ex[where P="?P"])
+      finally show ?thesis by simp
+    qed
+  qed
+qed
+
+(* the resul of adjustcoeff is unified for all l with divideallc (l,p) *)
+lemma adjustcoeff_unified: 
+  assumes linp: "islinform p"
+  and dvdc: "divideallc(l,p)"
+  and lpos: "l > 0"
+  shows "isunified (adjustcoeff(l, p))"
+  using linp dvdc lpos
+  proof(induct l p rule: adjustcoeff.induct,simp_all add: lin_mul_lintn islinintterm_eq_islint islint_def)
+    case (goal1 l d c r)
+    from prems have "c >0 \<or> c < 0" by auto
+    moreover {
+      assume cpos: "c > 0 "
+      from prems have lp: "l > 0" by simp
+      from prems have cdvdl: "c dvd l" by simp
+      have clel: "c \<le> l" by (rule zdvd_imp_le[OF cdvdl lp])
+      have "c div c \<le>  l div c" by (rule zdiv_mono1[OF clel cpos])
+      then have ?case using cpos by (simp add: zdiv_self)      
+    }
+    moreover {
+      assume cneg: "c < 0"
+      
+     have mcpos: "-c > 0" by simp
+      then have mcnz: "-c \<noteq> 0" by simp
+      from prems have mcdvdl: "-c dvd l" 
+	by simp 
+      then have l1:"l mod -c = 0" by (simp add: zdvd_iff_zmod_eq_0)
+      from prems have lp: "l >0" by simp
+      have mclel: "-c \<le> l" by (rule zdvd_imp_le[OF mcdvdl lp])
+      have "l div c = (-l div -c)"  by simp
+      also have "\<dots> = - (l div -c)" using l1
+	by (simp only: zdiv_zminus1_eq_if[OF mcnz, where a="l"]) simp
+      finally have diveq: "l div c = - (l div -c)" by simp
+      
+      have "-c div -c \<le> l div -c" by (rule zdiv_mono1[OF mclel mcpos])
+      then have "0 < l div -c" using cneg
+	by (simp add: zdiv_self)
+      then have ?case using diveq by simp
+    }
+    ultimately  show ?case by blast
+  next
+    case (goal2 l p)    from prems have "c >0 \<or> c < 0" by auto
+    moreover {
+      assume cpos: "c > 0 "
+      from prems have lp: "l > 0" by simp
+      from prems have cdvdl: "c dvd l" by simp
+      have clel: "c \<le> l" by (rule zdvd_imp_le[OF cdvdl lp])
+      have "c div c \<le>  l div c" by (rule zdiv_mono1[OF clel cpos])
+      then have ?case using cpos by (simp add: zdiv_self)      
+    }
+    moreover {
+      assume cneg: "c < 0"
+      
+     have mcpos: "-c > 0" by simp
+      then have mcnz: "-c \<noteq> 0" by simp
+      from prems have mcdvdl: "-c dvd l" 
+	by simp 
+      then have l1:"l mod -c = 0" by (simp add: zdvd_iff_zmod_eq_0)
+      from prems have lp: "l >0" by simp
+      have mclel: "-c \<le> l" by (rule zdvd_imp_le[OF mcdvdl lp])
+      have "l div c = (-l div -c)"  by simp
+      also have "\<dots> = - (l div -c)" using l1
+	by (simp only: zdiv_zminus1_eq_if[OF mcnz, where a="l"]) simp
+      finally have diveq: "l div c = - (l div -c)" by simp
+      
+      have "-c div -c \<le> l div -c" by (rule zdiv_mono1[OF mclel mcpos])
+      then have "0 < l div -c" using cneg
+	by (simp add: zdiv_self)
+      then have ?case using diveq by simp
+    }
+    ultimately  show ?case by blast
+  qed
+
+lemma adjustcoeff_lcm_unified:
+  assumes linp: "islinform p"
+  shows "isunified (adjustcoeff(formlcm p, p))"
+using linp adjustcoeff_unified formlcm_pos formlcm_divideallc
+by simp
+
+(* the result of unitycoeff is unified *)
+lemma unitycoeff_unified:
+  assumes linp: "islinform p"
+  shows "isunified (unitycoeff p)"
+using linp formlcm_pos[OF linp]
+proof (auto simp add: unitycoeff_def Let_def adjustcoeff_lcm_unified)
+  assume f1: "formlcm p = 1"
+  have "isunified (adjustcoeff(formlcm p, p))" 
+    by (rule adjustcoeff_lcm_unified[OF linp])
+  with f1 
+  show "isunified (adjustcoeff(1,p))" by simp
+qed
+
+lemma unified_isnnf: 
+  assumes unifp: "isunified p"
+  shows "isnnf p"
+  using unified_islinform[OF unifp] linform_isnnf
+  by simp
+
+lemma unified_isqfree: "isunified p\<Longrightarrow> isqfree p"
+using unified_islinform linform_isqfree
+by auto
+
+(* Plus/Minus infinity , B and A set definitions *)
+
+consts minusinf :: "QF \<Rightarrow> QF"
+       plusinf  :: "QF \<Rightarrow> QF"
+       aset     :: "QF \<Rightarrow> intterm list"
+       bset     :: "QF \<Rightarrow> intterm list"
+
+recdef minusinf "measure size"
+"minusinf (Le (Add (Mult (Cst c) (Var 0)) r) z) =
+  (if c < 0 then F else T)"
+"minusinf (Eq (Add (Mult (Cst c) (Var 0)) r) z) = F"
+"minusinf (NOT(Eq (Add (Mult (Cst c) (Var 0)) r) z)) = T"
+"minusinf (And p q) = And (minusinf p) (minusinf q)"
+"minusinf (Or p q) = Or (minusinf p) (minusinf q)"
+"minusinf p = p"
+
+recdef plusinf "measure size"
+"plusinf (Le (Add (Mult (Cst c) (Var 0)) r) z) =
+  (if c < 0 then T else F)"
+"plusinf (Eq (Add (Mult (Cst c) (Var 0)) r) z) = F"
+"plusinf (NOT (Eq (Add (Mult (Cst c) (Var 0)) r) z)) = T"
+"plusinf (And p q) = And (plusinf p) (plusinf q)"
+"plusinf (Or p q) = Or (plusinf p) (plusinf q)"
+"plusinf p = p"
+
+recdef bset "measure size"
+"bset (Le (Add (Mult (Cst c) (Var 0)) r) z) = 
+ (if c < 0 then [lin_add(r,(Cst -1)), r]
+         else [lin_add(lin_neg r,(Cst -1))])"
+"bset (Eq (Add (Mult (Cst c) (Var 0)) r) z) =  
+  (if c < 0 then [lin_add(r,(Cst -1))]
+         else [lin_add(lin_neg r,(Cst -1))])"
+"bset (NOT(Eq (Add (Mult (Cst c) (Var 0)) r) z)) =  
+  (if c < 0 then [r]
+         else [lin_neg r])"
+"bset (And p q) = (bset p) @ (bset q)"
+"bset (Or p q) = (bset p) @ (bset q)"
+"bset p = []"
+
+recdef aset "measure size"
+"aset (Le (Add (Mult (Cst c) (Var 0)) r) z) = 
+  (if c < 0 then [lin_add (r, Cst 1)]
+         else [lin_add (lin_neg r, Cst 1), lin_neg r])"
+"aset (Eq (Add (Mult (Cst c) (Var 0)) r) z) = 
+  (if c < 0 then [lin_add(r,(Cst 1))]
+       else [lin_add(lin_neg r,(Cst 1))])"
+"aset (NOT(Eq (Add (Mult (Cst c) (Var 0)) r) z)) = 
+  (if c < 0 then [r] 
+      else [lin_neg r])"
+"aset (And p q) = (aset p) @ (aset q)"
+"aset (Or p q) = (aset p) @ (aset q)"
+"aset p = []"
+
+(* divlcm computes \<delta> = lcm d , where d | x +t occurs in p *)
+consts divlcm :: "QF \<Rightarrow> int"
+recdef divlcm "measure size"
+"divlcm (Divides (Cst d) (Add (Mult (Cst c) (Var 0)) r)) = (abs d)"
+"divlcm (NOT p) = divlcm p"
+"divlcm (And p q)= ilcm (divlcm p) (divlcm q)"
+"divlcm (Or p q) = ilcm (divlcm p) (divlcm q)"
+"divlcm p = 1"
+
+(* the preoperty of \<delta> *)
+consts alldivide :: "int \<times> QF \<Rightarrow> bool"
+recdef alldivide "measure (%(d,p). size p)"
+"alldivide (d,(Divides (Cst d') (Add (Mult (Cst c) (Var 0)) r))) = 
+  (d' dvd d)"
+"alldivide (d,(NOT p)) = alldivide (d,p)"
+"alldivide (d,(And p q)) = (alldivide (d,p) \<and> alldivide (d,q))"
+"alldivide (d,(Or p q)) = ((alldivide (d,p)) \<and> (alldivide (d,q)))"
+"alldivide (d,p) = True"
+
+(* alldivide is monotone *)
+lemma alldivide_mono: "\<And> d'. \<lbrakk> alldivide (d,p) ; d dvd d'\<rbrakk> \<Longrightarrow> alldivide (d',p)"
+proof(induct d p rule: alldivide.induct, simp_all add: ilcm_dvd1 ilcm_dvd2)
+  fix "d1" "d2" "d3"
+  assume th1:"d2 dvd (d1::int)"
+    and th2: "d1 dvd d3"
+  show "d2 dvd d3" by (rule zdvd_trans[OF th1 th2])
+qed
+
+(* Some simple lemmas *)
+lemma zdvd_eq_zdvd_abs: " (d::int) dvd d' = (d dvd (abs d')) "
+proof-
+  have "d' < 0 \<or> d' \<ge> 0" by arith
+  moreover
+  {
+    assume dn': "d' < 0"
+    then have "abs d' = - d'" by simp
+    then 
+    have ?thesis by (simp)
+  }
+  moreover 
+  {
+    assume dp': "d' \<ge> 0"
+    then have "abs d' = d'" by simp
+    then have ?thesis  by simp
+  }
+    ultimately show ?thesis by blast
+qed
+
+lemma zdvd_refl_abs: "(d::int) dvd (abs d)"
+proof-
+  have "d dvd d" by simp
+  then show ?thesis by (simp add: iffD1 [OF zdvd_eq_zdvd_abs [where d = "d" and d'="d"]])
+qed
+
+(* \<delta> > 0*)
+lemma divlcm_pos: 
+  assumes 
+  linp: "islinform p"
+  shows "0 < divlcm p"
+using linp
+proof (induct p rule: divlcm.induct,simp_all add: ilcm_pos)
+  case (goal1 f) show ?case 
+    using prems 
+    by (cases f, auto) (case_tac "intterm1", auto)
+qed
+
+lemma nz_le: "(x::int) > 0 \<Longrightarrow> x \<noteq> 0" by auto
+(* divlcm is correct *)
+lemma divlcm_corr:
+  assumes 
+  linp: "islinform p"
+  shows "alldivide (divlcm p,p)"
+thm nz_le[OF divlcm_pos[OF linp]]
+  using linp divlcm_pos
+proof (induct p rule: divlcm.induct,simp_all add: zdvd_refl_abs,clarsimp simp add: Nat.gr0_conv_Suc)
+  case (goal1 f)
+  have "islinform f" using prems  
+    by (cases f, auto) (case_tac "intterm2", auto,case_tac "intterm1", auto)
+  then have "alldivide (divlcm f, f)"  using prems by simp
+  moreover have "divlcm (NOT f) = divlcm f" by simp
+  moreover have "alldivide (x,f) = alldivide (x,NOT f)" by simp
+  ultimately show ?case by simp
+next
+  case (goal2 f g)
+  have dvd1: "(divlcm f) dvd (ilcm (divlcm f) (divlcm g))" 
+    using prems by(simp add: ilcm_dvd1 nz_le)
+  have dvd2: "(divlcm g) dvd (ilcm (divlcm f) (divlcm g))" 
+    using prems by (simp add: ilcm_dvd2 nz_le)
+  from dvd1 prems 
+  have "alldivide (ilcm (divlcm f) (divlcm g), f)" 
+    by (simp add: alldivide_mono[where d= "divlcm f" and p="f" and d' ="ilcm (divlcm f) (divlcm g)"])
+  moreover   from dvd2 prems 
+   have "alldivide (ilcm (divlcm f) (divlcm g), g)" 
+    by (simp add: alldivide_mono[where d= "divlcm g" and p="g" and d' ="ilcm (divlcm f) (divlcm g)"])
+  ultimately show ?case by simp
+next
+  case (goal3 f g)
+  have dvd1: "(divlcm f) dvd (ilcm (divlcm f) (divlcm g))" 
+    using prems by (simp add: nz_le ilcm_dvd1)
+  have dvd2: "(divlcm g) dvd (ilcm (divlcm f) (divlcm g))" 
+    using prems by (simp add: nz_le ilcm_dvd2)
+  from dvd1 prems 
+  have "alldivide (ilcm (divlcm f) (divlcm g), f)" 
+    by (simp add: alldivide_mono[where d= "divlcm f" and p="f" and d' ="ilcm (divlcm f) (divlcm g)"])
+  moreover   from dvd2 prems 
+   have "alldivide (ilcm (divlcm f) (divlcm g), g)" 
+    by (simp add: alldivide_mono[where d= "divlcm g" and p="g" and d' ="ilcm (divlcm f) (divlcm g)"])
+  ultimately show ?case by simp
+qed
+
+
+(* Properties of  minusinf and plusinf*)
+
+(* minusinf p and p are the same for minusinfity \<dots> *)
+lemma minusinf_eq: 
+  assumes unifp: "isunified p" 
+  shows "\<exists> z. \<forall> x. x < z \<longrightarrow> (qinterp (x#ats) p = qinterp (x#ats) (minusinf p))"
+using unifp unified_islinform[OF unifp]
+proof (induct p rule: minusinf.induct)
+  case (1 c r z)
+  have "c <0 \<or> 0 \<le> c" by arith
+  moreover 
+  {
+    assume cneg: " c < 0"
+    from prems have z0: "z= Cst 0" 
+      by (cases z,auto)
+    with prems have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" 
+      by simp
+
+    from prems z0 have ?case 
+      proof-
+	show ?thesis
+	  using prems z0
+      apply auto
+      apply (rule exI[where x="I_intterm (a # ats) r"])
+      apply (rule allI)
+      proof-
+	fix x
+	show "x < I_intterm (a # ats) r \<longrightarrow> \<not> - x + I_intterm (x # ats) r \<le> 0"
+	  by (simp add: intterm_novar0[OF lincnr, where x="a" and y="x"])
+      qed
+    qed
+  }
+  moreover
+  {
+    assume cpos: "0 \<le> c"
+    from prems have z0: "z= Cst 0" 
+      by (cases z) auto
+    with prems have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" 
+      by simp
+    
+    from prems z0 have ?case
+      proof-
+	show ?thesis
+	  using prems z0
+      apply auto
+      apply (rule exI[where x="-(I_intterm (a # ats) r)"])
+      apply (rule allI)
+      proof-
+	fix x
+	show "x < - I_intterm (a # ats) r \<longrightarrow> x + I_intterm (x # ats) r \<le> 0"
+	  by (simp add: intterm_novar0[OF lincnr, where x="a" and y="x"])
+      qed
+    qed
+  }
+    
+    ultimately show ?case by blast
+next
+  case (2 c r z)
+  from prems have z0: "z= Cst 0" 
+    by (cases z,auto)
+  with prems have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" 
+    by simp
+  have "c <0 \<or> 0 \<le> c" by arith
+  moreover 
+  {
+    assume cneg: " c < 0"
+    from prems z0 have ?case 
+      proof-
+	show ?thesis
+	  using prems z0
+      apply auto
+      apply (rule exI[where x="I_intterm (a # ats) r"])
+      apply (rule allI)
+      proof-
+	fix x
+	show "x < I_intterm (a # ats) r \<longrightarrow> \<not> - x + I_intterm (x # ats) r = 0"
+	  by (simp add: intterm_novar0[OF lincnr, where x="a" and y="x"])
+      qed
+    qed
+  }
+  moreover
+  {
+    assume cpos: "0 \<le> c"
+    from prems z0 have ?case
+      proof-
+	show ?thesis
+	  using prems z0
+      apply auto
+      apply (rule exI[where x="-(I_intterm (a # ats) r)"])
+      apply (rule allI)
+      proof-
+	fix x
+	show "x < - I_intterm (a # ats) r \<longrightarrow> x + I_intterm (x # ats) r \<noteq> 0"
+	  by (simp add: intterm_novar0[OF lincnr, where x="a" and y="x"])
+      qed
+    qed
+  }
+    
+    ultimately show ?case by blast
+next
+  case (3 c r z)
+  from prems have z0: "z= Cst 0" 
+    by (cases z,auto)
+  with prems have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" 
+    by simp
+  have "c <0 \<or> 0 \<le> c" by arith
+  moreover 
+  {
+    assume cneg: " c < 0"
+    from prems z0 have ?case 
+      proof-
+	show ?thesis
+	  using prems z0
+      apply auto
+      apply (rule exI[where x="I_intterm (a # ats) r"])
+      apply (rule allI)
+      proof-
+	fix x
+	show "x < I_intterm (a # ats) r \<longrightarrow> \<not> - x + I_intterm (x # ats) r = 0"
+	  by (simp add: intterm_novar0[OF lincnr, where x="a" and y="x"])
+      qed
+    qed
+  }
+  moreover
+  {
+    assume cpos: "0 \<le> c"
+    from prems z0 have ?case
+      proof-
+	show ?thesis
+	  using prems z0
+      apply auto
+      apply (rule exI[where x="-(I_intterm (a # ats) r)"])
+      apply (rule allI)
+      proof-
+	fix x
+	show "x < - I_intterm (a # ats) r \<longrightarrow> x + I_intterm (x # ats) r \<noteq> 0"
+	  by (simp add: intterm_novar0[OF lincnr, where x="a" and y="x"])
+      qed
+    qed
+  }
+    
+    ultimately show ?case by blast
+next
+  
+  case (4 f g) 
+  from prems obtain "zf" where 
+    zf:"\<forall>x<zf. qinterp (x # ats) f = qinterp (x # ats) (minusinf f)" by auto
+  from prems obtain "zg" where 
+    zg:"\<forall>x<zg. qinterp (x # ats) g = qinterp (x # ats) (minusinf g)" by auto
+  from zf zg show ?case 
+    apply auto
+    apply (rule exI[where x="min zf zg"])
+    by simp
+  
+next case (5 f g)  
+  from prems obtain "zf" where 
+    zf:"\<forall>x<zf. qinterp (x # ats) f = qinterp (x # ats) (minusinf f)" by auto
+  from prems obtain "zg" where 
+    zg:"\<forall>x<zg. qinterp (x # ats) g = qinterp (x # ats) (minusinf g)" by auto
+  from zf zg show ?case 
+    apply auto
+    apply (rule exI[where x="min zf zg"])
+    by simp
+  
+qed simp_all
+
+(* miusinf p behaves periodically*)
+lemma minusinf_repeats: 
+  assumes alldvd: "alldivide (d,p)"
+  and unity: "isunified p"
+  shows "qinterp (x#ats) (minusinf p) = qinterp ((x + c*d)#ats) (minusinf p)"
+  using alldvd unity unified_islinform[OF unity]
+proof(induct p rule: islinform.induct, simp_all)
+  case (goal1 t a)
+  show ?case
+    using prems
+    apply (cases t, simp_all add: nth_pos2)
+    apply (case_tac "intterm1", simp_all)
+    apply (case_tac "intterm1a",simp_all)
+    by (case_tac "intterm2a",simp_all)
+  (case_tac "nat",simp_all add: nth_pos2 intterm_novar0[where x="x" and y="x+c*d"])
+next 
+  case (goal2 t a)
+  show ?case
+    using prems
+    apply (cases t, simp_all add: nth_pos2)
+    apply (case_tac "intterm1", simp_all)
+    apply (case_tac "intterm1a",simp_all)
+    by (case_tac "intterm2a",simp_all)
+  (case_tac "nat",simp_all add: nth_pos2 intterm_novar0[where x="x" and y="x+c*d"])
+next 
+  case (goal3 a t)
+  show ?case using prems
+
+  proof(induct t rule: islinintterm.induct, simp_all add: nth_pos2)
+    case (goal1 i n i')
+    show ?case
+      using prems
+    proof(cases n, simp_all, case_tac "i=1", simp,
+	simp add: dvd_period[where a="a" and d="d" and x="x" and c="c"])
+      case goal1
+      from prems have "(abs i = 1) \<and> i \<noteq> 1" by auto 
+      then  have im1: "i=-1" by arith
+      then have "(a dvd i*x + i') = (a dvd x + (-i'))" 
+	by (simp add: uminus_dvd_conv'[where d="a" and t="-x +i'"])
+      moreover 
+      from im1 have "(a dvd i*x + (i*(c * d)) + i') = (a dvd (x + c*d - i'))"
+	apply simp
+	apply (simp add: uminus_dvd_conv'[where d="a" and t="-x - c * d + i'"])
+	by (simp add: zadd_ac)
+      ultimately 
+      have eq1:"((a dvd i*x + i') = (a dvd i*x + (i*(c * d)) + i')) = 
+	((a dvd x + (-i'))  = (a dvd (x + c*d - i')))" by simp
+      moreover 
+      have dvd2: "(a dvd x + (-i')) = (a dvd x + c * d + (-i'))"
+	by (rule dvd_period[where a="a" and d="d" and x="x" and c="c"], assumption)
+      ultimately show ?case by simp
+    qed
+  next
+    case (goal2 i n i' n' r)
+    have "n = 0 \<or> 0 < n" by arith
+    moreover 
+    {
+      assume npos: "0 < n"
+      from prems have "n < n'" by simp then have "0 < n'" by simp
+      moreover from prems
+      have linr: "islinintterm (Add (Mult (Cst i') (Var n')) r)" by simp
+      ultimately have ?case 
+	using prems npos
+	by (simp add: nth_pos2 intterm_novar0[OF linr,where x="x" and y="x + c*d"])
+    }
+    moreover 
+    {
+      assume n0: "n=0"
+      from prems have lin2: "islinintterm (Add (Mult (Cst i') (Var n')) r)" by simp
+      from prems have "n < n'" by simp then have npos': "0 < n'" by simp
+      with prems have ?case
+      proof(simp add: intterm_novar0[OF lin2, where x="x" and y="x+c*d"] 
+	  nth_pos2 dvd_period,case_tac "i=1",
+	  simp add: dvd_period[where a="a" and d="d" and x="x" and c="c"], simp)
+	case goal1
+	from prems have "abs i = 1 \<and> i\<noteq>1" by auto
+	then have mi: "i = -1" by arith
+	have "(a dvd -x + (i' * ats ! (n' - Suc 0) + I_intterm ((x + c * d) # ats) r)) = 
+	  (a dvd x + (-i' * ats ! (n' - Suc 0) - I_intterm ((x + c * d) # ats) r))" 
+	  by (simp add: 
+	    uminus_dvd_conv'[where d="a" and 
+	    t="-x + (i' * ats ! (n' - Suc 0) + I_intterm ((x + c * d) # ats) r)"])
+	also 
+	have "(a dvd x + (-i' * ats ! (n' - Suc 0) - I_intterm ((x + c * d) # ats) r)) = 
+	  (a dvd x +c*d + (-i' * ats ! (n' - Suc 0) - I_intterm ((x + c * d) # ats) r))"
+	  by (rule dvd_period[where a="a" and d="d" and x="x" and c="c"], assumption)
+	also 
+	have "(a dvd x +c*d + 
+	  (-i' * ats ! (n' - Suc 0) - I_intterm ((x + c * d) # ats) r)) = 
+	  (a dvd -(x +c*d + 
+	  (-i' * ats ! (n' - Suc 0) - I_intterm ((x + c * d) # ats) r)))"
+	  by (rule uminus_dvd_conv'[where d="a" and 
+	    t="x +c*d + (-i' * ats ! (n' - Suc 0) - I_intterm ((x + c * d) # ats) r)"])
+	also
+	have "(a dvd -(x +c*d + 
+	  (-i' * ats ! (n' - Suc 0) - I_intterm ((x + c * d) # ats) r)))
+	  = (a dvd
+          - x - c * d + (i' * ats ! (n' - Suc 0) + I_intterm ((x + c * d) # ats) r))" 
+	  by (auto,simp_all add: zadd_ac)
+	finally show ?case using mi by auto
+      qed
+    }
+    ultimately show ?case by blast
+  qed
+next 
+  case (goal4 a t)
+  show ?case using prems 
+  proof(induct t rule: islinintterm.induct, simp_all,case_tac "n=0",
+      simp_all add: nth_pos2)
+    case (goal1 i n i')
+    show ?case
+      using prems
+    proof(case_tac "i=1", simp,
+	simp add: dvd_period[where a="a" and d="d" and x="x" and c="c"])
+      case goal1
+      from prems have "abs i = 1 \<and> i\<noteq>1" by auto 
+      then have im1: "i=-1" by arith
+      then have "(a dvd i*x + i') = (a dvd x + (-i'))" 
+	by (simp add: uminus_dvd_conv'[where d="a" and t="-x +i'"])
+      moreover 
+      from im1 have "(a dvd i*x + (i*(c * d)) + i') = (a dvd (x + c*d - i'))"
+	apply simp
+	apply (simp add: uminus_dvd_conv'[where d="a" and t="-x - c * d + i'"])
+	by (simp add: zadd_ac)
+      ultimately 
+      have eq1:"((a dvd i*x + i') = (a dvd i*x + (i*(c * d)) + i')) = 
+	((a dvd x + (-i'))  = (a dvd (x + c*d - i')))" by simp
+      moreover 
+      have dvd2: "(a dvd x + (-i')) = (a dvd x + c * d + (-i'))"
+	by (rule dvd_period[where a="a" and d="d" and x="x" and c="c"], assumption)
+      ultimately show ?thesis by simp
+    qed
+  next
+    case (goal2 i n i' n' r)
+    have "n = 0 \<or> 0 < n" by arith
+    moreover 
+    {
+      assume npos: "0 < n"
+      from prems have "n < n'" by simp then have "0 < n'" by simp
+      moreover from prems
+      have linr: "islinintterm (Add (Mult (Cst i') (Var n')) r)" by simp
+      ultimately have ?case 
+	using prems npos
+	by (simp add: nth_pos2 intterm_novar0[OF linr,where x="x" and y="x + c*d"])
+    }
+    moreover 
+    {
+      assume n0: "n=0"
+      from prems have lin2: "islinintterm (Add (Mult (Cst i') (Var n')) r)" by simp
+      from prems have "n < n'" by simp then have npos': "0 < n'" by simp
+      with prems have ?case
+      proof(simp add: intterm_novar0[OF lin2, where x="x" and y="x+c*d"] 
+	  nth_pos2 dvd_period,case_tac "i=1",
+	  simp add: dvd_period[where a="a" and d="d" and x="x" and c="c"], simp)
+	case goal1
+	from prems have "abs i = 1 \<and> i\<noteq>1" by auto
+	then have mi: "i = -1" by arith
+	have "(a dvd -x + (i' * ats ! (n' - Suc 0) + I_intterm ((x + c * d) # ats) r)) = 
+	  (a dvd x + (-i' * ats ! (n' - Suc 0) - I_intterm ((x + c * d) # ats) r))" 
+	  by (simp add: 
+	    uminus_dvd_conv'[where d="a" and 
+	    t="-x + (i' * ats ! (n' - Suc 0) + I_intterm ((x + c * d) # ats) r)"])
+	also 
+	have "(a dvd x + (-i' * ats ! (n' - Suc 0) - I_intterm ((x + c * d) # ats) r)) = 
+	  (a dvd x +c*d + (-i' * ats ! (n' - Suc 0) - I_intterm ((x + c * d) # ats) r))"
+	  by (rule dvd_period[where a="a" and d="d" and x="x" and c="c"], assumption)
+	also 
+	have "(a dvd x +c*d + 
+	  (-i' * ats ! (n' - Suc 0) - I_intterm ((x + c * d) # ats) r)) = 
+	  (a dvd -(x +c*d + 
+	  (-i' * ats ! (n' - Suc 0) - I_intterm ((x + c * d) # ats) r)))"
+	  by (rule uminus_dvd_conv'[where d="a" and 
+	    t="x +c*d + (-i' * ats ! (n' - Suc 0) - I_intterm ((x + c * d) # ats) r)"])
+	also
+	have "(a dvd -(x +c*d + 
+	  (-i' * ats ! (n' - Suc 0) - I_intterm ((x + c * d) # ats) r)))
+	  = (a dvd
+          - x - c * d + (i' * ats ! (n' - Suc 0) + I_intterm ((x + c * d) # ats) r))" 
+	  by (auto,simp_all add: zadd_ac)
+	finally show ?case using mi by auto
+      qed
+    }
+    ultimately show ?case by blast
+  qed
+next 
+  case (goal5 t a)
+  show ?case
+    using prems
+    apply (cases t, simp_all add: nth_pos2)
+    apply (case_tac "intterm1", simp_all)
+    apply (case_tac "intterm1a",simp_all)
+    by (case_tac "intterm2a",simp_all)
+  (case_tac "nat",simp_all add: nth_pos2 intterm_novar0[where x="x" and y="x+c*d"])
+qed
+
+lemma minusinf_repeats2:
+  assumes alldvd: "alldivide (d,p)"
+  and unity: "isunified p"
+  shows "\<forall> x k. (qinterp (x#ats) (minusinf p) = qinterp ((x - k*d)#ats) (minusinf p))" 
+  (is "\<forall> x k. ?P x = ?P (x - k*d)")
+proof(rule allI, rule allI)
+  fix x k
+  show "?P x = ?P (x - k*d)"
+  proof-
+    have "?P x = ?P (x + (-k)*d)" by (rule minusinf_repeats[OF alldvd unity])
+    then have "?P x = ?P (x - (k*d))" by simp
+    then show ?thesis by blast 
+  qed
+qed
+
+
+(* existence for minusinf p is existence for p *)
+lemma minusinf_lemma:
+  assumes unifp: "isunified p"
+  and exminf: "\<exists> j \<in> {1 ..d}. qinterp (j#ats) (minusinf p)" (is "\<exists> j \<in> {1 .. d}. ?P1 j")
+  shows "\<exists> x. qinterp (x#ats) p" (is "\<exists> x. ?P x")
+proof-
+  from exminf obtain "j" where P1j: "?P1 j" by blast
+  have ePeqP1: "\<exists>z. \<forall> x. x < z \<longrightarrow> (?P x = ?P1 x)"
+    by (rule minusinf_eq[OF unifp])
+  then obtain "z" where P1eqP : "\<forall> x. x < z \<longrightarrow> (?P x = ?P1 x)" by blast
+  let ?d = "divlcm p"
+  have alldvd: "alldivide (?d,p)" using unified_islinform[OF unifp] divlcm_corr
+    by auto
+  have dpos: "0 < ?d" using unified_islinform[OF unifp] divlcm_pos
+    by simp
+  have P1eqP1 : "\<forall> x k. ?P1 x = ?P1 (x - k*(?d))"
+    by (rule minusinf_repeats2[OF alldvd unifp])
+  let ?w = "j - (abs (j-z) +1)* ?d"
+  show "\<exists> x. ?P x"
+  proof
+    have w: "?w < z" 
+      by (rule decr_lemma[OF dpos])
+    
+    have "?P1 j = ?P1 ?w" using P1eqP1 by blast
+    also have "\<dots> = ?P ?w"  using w P1eqP by blast
+    finally show "?P ?w" using P1j by blast
+  qed
+qed
+
+(* limited search for the withness for minusinf p, due to peridicity *)
+lemma minusinf_disj:
+  assumes unifp: "isunified p"
+  shows "(\<exists> x. qinterp (x#ats) (minusinf p)) = 
+  (\<exists> j \<in> { 1.. divlcm p}. qinterp (j#ats) (minusinf p))" 
+  (is "(\<exists> x. ?P x) = (\<exists> j \<in> { 1.. ?d}. ?P j)")
+proof
+  have linp: "islinform p" by (rule unified_islinform[OF unifp])
+  have dpos: "0 < ?d" by (rule divlcm_pos[OF linp])
+  have alldvd: "alldivide(?d,p)" by (rule divlcm_corr[OF linp])
+  {
+    assume "\<exists> j\<in> {1 .. ?d}. ?P j"
+    then show "\<exists> x. ?P x" using dpos  by auto
+  next
+    assume "\<exists> x. ?P x"
+    then obtain "x" where P: "?P x" by blast
+    have modd: "\<forall>x k. ?P x = ?P (x - k*?d)"
+      by (rule minusinf_repeats2[OF alldvd unifp])
+    
+    have "x mod ?d = x - (x div ?d)*?d"
+      by(simp add:zmod_zdiv_equality mult_ac eq_diff_eq)
+    hence Pmod: "?P x = ?P (x mod ?d)" using modd by simp
+    show "\<exists> j\<in> {1 .. ?d}. ?P j"
+    proof (cases)
+      assume "x mod ?d = 0"
+      hence "?P 0" using P Pmod by simp
+      moreover have "?P 0 = ?P (0 - (-1)*?d)" using modd by blast
+      ultimately have "?P ?d" by simp
+      moreover have "?d \<in> {1 .. ?d}" using dpos 
+	by (simp add:atLeastAtMost_iff)
+      ultimately show "\<exists> j\<in> {1 .. ?d}. ?P j" ..
+    next 
+      assume not0: "x mod ?d \<noteq> 0"
+      have "?P(x mod ?d)" using dpos P Pmod by(simp add:pos_mod_sign pos_mod_bound)
+      moreover have "x mod ?d : {1 .. ?d}"
+      proof -
+	have "0 \<le> x mod ?d" by(rule pos_mod_sign[OF dpos])
+	moreover have "x mod ?d < ?d"  by(rule pos_mod_bound[OF dpos])
+	ultimately show ?thesis using not0 by(simp add:atLeastAtMost_iff)
+      qed
+      ultimately show "\<exists> j\<in> {1 .. ?d}. ?P j" ..
+    qed
+  }
+qed
+
+lemma minusinf_qfree:
+  assumes linp : "islinform p"
+  shows "isqfree (minusinf p)"
+  using linp
+ by (induct p rule: minusinf.induct) auto 
+
+(* Properties of bset and a set *)
+
+(* The elements of a bset are linear *) 
+lemma bset_lin:
+  assumes unifp: "isunified p"
+  shows "\<forall> b \<in> set (bset p). islinintterm b"
+using unifp unified_islinform[OF unifp]
+proof (induct p rule: bset.induct, auto)
+  case (goal1 c r z)
+  from prems have "z = Cst 0" by (cases z, simp_all)
+  then have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" using prems by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  have "islinintterm (Cst -1)" by simp
+  then show ?case using linr lin_add_lin by simp
+next 
+  case (goal2 c r z)
+  from prems have "z = Cst 0" by (cases z, simp_all)
+  then have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" using prems by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  show ?case by (rule linr)
+next
+  case (goal3 c r z)
+  from prems have "z = Cst 0" by (cases z, simp_all) 
+  then have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" using prems by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  have "islinintterm (Cst -1)" by simp
+  then show ?case using linr lin_add_lin lin_neg_lin by simp
+next
+  case (goal4 c r z)
+  from prems have "z = Cst 0" by (cases z, simp_all) 
+  then have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" using prems by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  have "islinintterm (Cst -1)" by simp
+  then show ?case using linr lin_add_lin lin_neg_lin by simp
+next
+  case (goal5 c r z)
+  from prems have "z = Cst 0" by (cases z, simp_all) 
+  then have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" using prems by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  have "islinintterm (Cst -1)" by simp
+  then show ?case using linr lin_add_lin lin_neg_lin by simp
+next
+  case (goal6 c r z)
+  from prems have "z = Cst 0" by (cases z, simp_all) 
+  then have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" using prems by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  have "islinintterm (Cst -1)" by simp
+  then show ?case using linr lin_add_lin lin_neg_lin by simp
+next
+  case (goal7 c r z)
+  from prems have "z = Cst 0" by (cases z, simp_all) 
+  then have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" using prems by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  have "islinintterm (Cst -1)" by simp
+  then show ?case using linr lin_add_lin lin_neg_lin by simp
+qed
+
+(* The third lemma in Norrisch's Paper *)
+lemma bset_disj_repeat:
+  assumes unifp: "isunified p"
+  and alldvd: "alldivide (d,p)"
+  and dpos: "0 < d"
+  and nob: "(qinterp (x#ats) q) \<and> \<not>(\<exists>j\<in> {1 .. d}. \<exists> b \<in> set (bset p). (qinterp (((I_intterm (a#ats) b) + j)#ats) q)) \<and>(qinterp (x#ats) p)" 
+  (is "?Q x \<and> \<not>(\<exists> j\<in> {1.. d}. \<exists> b\<in> ?B. ?Q (?I a b + j)) \<and> ?P x") 
+    shows "?P (x -d)"  
+  using unifp nob alldvd unified_islinform[OF unifp]
+proof (induct p rule: islinform.induct,auto)
+  case (goal1 t)
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have "n=0 \<or> n>0" by arith
+    moreover {assume "n>0" then have ?case 
+	using prems
+	by (simp add: nth_pos2 
+	  intterm_novar0[OF lininr, where x="x" and y="x-d"]) }
+    moreover 
+    {assume nz: "n = 0"
+      from prems have "abs i = 1" by auto 
+      then have "i = -1 \<or> i =1" by arith
+      moreover
+      {
+	assume i1: "i=1"
+	have ?case  using dpos prems  
+	  by (auto simp add: intterm_novar0[OF lininr, where x="x" and y="x - d"])
+      }
+      moreover 
+      {
+	assume im1: "i = -1"
+	have ?case 
+	  using prems 
+	proof(auto simp add: intterm_novar0[OF lininr, where x="x - d" and y="x"], cases)
+	  assume "- x + d +  ?I x r \<le> 0"
+	  then show "- x + d + ?I x r \<le> 0" .
+	next 
+	  assume np: "\<not> - x + d +  ?I x r \<le> 0"
+	  then have ltd:"x - ?I x r \<le> d - 1" by simp 
+	  from prems have "-x + ?I x r \<le> 0" by simp
+	  then have ge0: "x - ?I x r \<ge> 0" 
+	    by simp
+	  from ltd ge0 have "x - ?I x r = 0 \<or> (1 \<le> x - ?I x r \<and> x - ?I x r \<le> d - 1) " by arith
+	  moreover
+	  {
+	    assume "x - ?I x r = 0"
+	    then have xeqr: "x = ?I x r" by simp
+	    from prems have "?Q x" by simp
+	    with xeqr have qr:"?Q (?I x r)" by simp
+	    from prems have lininr: "islinintterm (Add (Mult (Cst i) (Var 0)) r)" by simp
+	    have "islinintterm r" by (rule islinintterm_subt[OF lininr])
+	    from prems 
+	    have "\<forall>j\<in>{1..d}. \<not> ?Q (?I a r + -1 + j)"
+	      using linr by (auto simp add: lin_add_corr)
+	    moreover from dpos have "1 \<in> {1..d}" by simp
+	    ultimately have " \<not> ?Q (?I a r + -1 + 1)" by blast
+	    with dpos linr have "\<not> ?Q (?I x r)"
+	      by (simp add: intterm_novar0[OF lininr, where x="x" and y="a"] lin_add_corr)
+	    with qr have "- x + d + ?I x r \<le> 0" by simp
+	  }
+	  moreover
+	  {
+	    assume gt0: "1 \<le> x - ?I x r \<and> x - ?I x r \<le> d - 1"
+	    then have "\<exists> j\<in> {1 .. d - 1}. x - ?I x r =  j" by simp
+	    then have "\<exists> j\<in> {1 .. d}. x - ?I x r =  j" by auto
+	    then obtain  "j" where con: "1\<le>j \<and> j \<le> d  \<and> x - ?I x r = j" by auto
+	    then have xeqr: "x = ?I x r + j" by auto
+	    with prems have "?Q (?I x r + j)" by simp
+	    with con have qrpj: "\<exists> j\<in> {1 .. d}. ?Q (?I x r + j)" by auto
+	    from prems have "\<forall>j\<in>{1..d}. \<not> ?Q (?I a r + j)" by auto
+	    then have "\<not> (\<exists> j\<in>{1..d}. ?Q (?I x r + j))" 
+	      by (simp add: intterm_novar0[OF lininr, where x="x" and y="a"])
+	    with qrpj prems have "- x + d + ?I x r \<le> 0" by simp 
+	    
+	  }
+	  ultimately show "- x + d + ?I x r \<le> 0" by blast
+	qed
+      }
+      ultimately have ?case by blast
+    }
+    ultimately have ?case by blast
+  }
+  ultimately show ?case by blast
+next  
+  case (goal3 a t)
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have "n=0 \<or> n>0" by arith
+    moreover {assume "n>0" then have ?case using prems 
+	by (simp add: nth_pos2 
+	  intterm_novar0[OF lininr, where x="x" and y="x-d"]) }
+    moreover {
+      assume nz: "n=0"
+      from prems have "abs i = 1" by auto
+      then have ipm: "i=1 \<or> i = -1" by arith
+      from nz prems have advdixr: "a dvd (i * x) + I_intterm (x # ats) r" 
+	by simp
+      from prems have "a dvd d" by simp
+      then have advdid: "a dvd i*d" using ipm by auto  
+      have ?case
+      using prems ipm 
+      by (auto simp add: intterm_novar0[OF lininr, where x="x-d" and y="x"] dvd_period[OF advdid, where x="i*x" and c="-1"])
+  }
+  ultimately have ?case by blast
+  } ultimately show ?case by blast
+next
+
+  case (goal4 a t)
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+
+    have "n=0 \<or> n>0" by arith
+    moreover {assume "n>0" then have ?case using prems 
+	by (simp add: nth_pos2 
+	  intterm_novar0[OF lininr, where x="x" and y="x-d"]) }
+    moreover {
+      assume nz: "n=0"
+      from prems have "abs i = 1" by auto
+      then have ipm: "i =1 \<or> i = -1" by arith
+      from nz prems have advdixr: "\<not> (a dvd (i * x) + I_intterm (x # ats) r)" 
+	by simp
+      from prems have "a dvd d" by simp
+      then have advdid: "a dvd i*d" using ipm by auto
+      have ?case
+      using prems ipm 
+      by (auto simp add: intterm_novar0[OF lininr, where x="x-d" and y="x"] dvd_period[OF advdid, where x="i*x" and c="-1"])
+  }
+  ultimately have ?case by blast
+  } ultimately show ?case by blast
+next 
+  case (goal2 t)
+  from prems
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have "n=0 \<or> n>0" by arith
+    moreover {assume "n>0" then have ?case 
+	using prems
+	by (simp add: nth_pos2 
+	  intterm_novar0[OF lininr, where x="x" and y="x-d"]) }
+    moreover 
+    {assume nz: "n = 0"
+      from prems have "abs i = 1" by auto 
+      then have "i = -1 \<or> i =1" by arith
+      moreover
+      {
+	assume i1: "i=1"
+	with prems have px: "x + ?I x r = 0" by simp
+	then have "x = (- ?I x r - 1) + 1" by simp
+	hence q1: "?Q ((- ?I x r - 1) + 1)" by simp
+	from prems have "\<not> (?Q ((?I a (lin_add(lin_neg r, Cst -1))) + 1))"
+	  by auto
+	hence "\<not> (?Q ((- ?I a r - 1) + 1))" 
+	  using lin_add_corr lin_neg_corr linr lin_neg_lin
+	  by simp
+	hence "\<not> (?Q ((- ?I x r - 1) + 1))" 
+	  using intterm_novar0[OF lininr, where x="x" and y="a"]
+	  by simp
+	with q1 have  ?case by simp
+      }
+      moreover 
+      {
+	assume im1: "i = -1"
+	with prems have px: "-x + ?I x r = 0" by simp
+	then have "x = ?I x r" by simp
+	hence q1: "?Q (?I x r)" by simp
+	from prems have "\<not> (?Q ((?I a (lin_add(r, Cst -1))) + 1))"
+	  by auto
+	hence "\<not> (?Q (?I a r))" 
+	  using lin_add_corr lin_neg_corr linr lin_neg_lin
+	  by simp
+	hence "\<not> (?Q (?I x r ))" 
+	  using intterm_novar0[OF lininr, where x="x" and y="a"]
+	  by simp
+	with q1 have  ?case by simp
+      }
+      ultimately have ?case by blast
+    }
+    ultimately have ?case by blast
+  }
+  ultimately show ?case by blast
+next
+  case (goal5 t)
+  from prems
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have "n=0 \<or> n>0" by arith
+    moreover {assume "n>0" then have ?case 
+	using prems
+	by (simp add: nth_pos2 
+	  intterm_novar0[OF lininr, where x="x" and y="x-d"]) }
+    moreover 
+    {assume nz: "n = 0"
+      from prems have "abs i = 1" by auto 
+      then have "i = -1 \<or> i =1" by arith
+      moreover
+      {
+	assume i1: "i=1"
+	with prems have px: "x -d + ?I (x-d) r = 0" by simp
+	hence "x = (- ?I x r) + d" 
+	  using intterm_novar0[OF lininr, where x="x" and y="x-d"]
+	  by simp
+	hence q1: "?Q (- ?I x r + d)" by simp
+	from prems have "\<not> (?Q ((?I a (lin_neg r)) + d))"
+	  by auto
+	hence "\<not> (?Q (- ?I a r + d))" 
+	  using lin_neg_corr linr by simp
+	hence "\<not> (?Q ((- ?I x r + d)))" 
+	  using intterm_novar0[OF lininr, where x="x" and y="a"]
+	  by simp
+	with q1 have  ?case by simp
+      }
+      moreover 
+      {
+	assume im1: "i = -1"
+	with prems have px: "- (x -d) + ?I (x - d) r = 0" by simp
+	then have "x = ?I x r + d "
+ 	  using intterm_novar0[OF lininr, where x="x" and y="x-d"]
+	  by simp
+	hence q1: "?Q (?I x r + d)" by simp
+	from prems have "\<not> (?Q ((?I a r) + d))"
+	  by auto
+	hence "\<not> (?Q (?I x r + d))" 
+	  using intterm_novar0[OF lininr, where x="x" and y="a"]
+	  by simp
+	with q1 have  ?case by simp
+      }
+      ultimately have ?case by blast
+    }
+    ultimately have ?case by blast
+  }
+  ultimately show ?case by blast
+  
+qed
+  
+lemma bset_disj_repeat2:
+  assumes unifp: "isunified p"
+
+  shows "\<forall> x. \<not>(\<exists>j\<in> {1 .. (divlcm p)}. \<exists> b \<in> set (bset p). 
+  (qinterp (((I_intterm (a#ats) b) + j)#ats) p))  
+  \<longrightarrow> (qinterp (x#ats) p) \<longrightarrow> (qinterp ((x - (divlcm p))#ats) p)" 
+  (is "\<forall> x. \<not>(\<exists> j\<in> {1 .. ?d}. \<exists> b\<in> ?B. ?P (?I a b + j)) \<longrightarrow> ?P x \<longrightarrow> ?P (x - ?d)")
+proof
+  fix x
+  have linp: "islinform p" by (rule unified_islinform[OF unifp])
+  have dpos: "?d > 0" by (rule divlcm_pos[OF linp])
+  have alldvd: "alldivide(?d,p)" by (rule divlcm_corr[OF linp])
+    show "\<not>(\<exists> j\<in> {1 .. ?d}. \<exists> b\<in> ?B. ?P (?I a b + j)) \<longrightarrow> ?P x \<longrightarrow> ?P (x - ?d)"
+    using prems bset_disj_repeat[OF unifp alldvd dpos]
+    by blast
+qed
+
+(* Cooper's theorem in the minusinfinity version *)
+lemma cooper_mi_eq: 
+  assumes unifp : "isunified p"
+  shows "(\<exists> x. qinterp (x#ats) p) = 
+  ((\<exists> j \<in> {1 .. (divlcm p)}. qinterp (j#ats) (minusinf p)) \<or> 
+  (\<exists> j \<in> {1 .. (divlcm p)}. \<exists> b \<in> set (bset p). 
+  qinterp (((I_intterm (a#ats) b) + j)#ats) p))"
+  (is "(\<exists> x. ?P x) = ((\<exists> j\<in> {1 .. ?d}. ?MP j) \<or> (\<exists> j \<in> ?D. \<exists> b\<in> ?B. ?P (?I a b + j)))")
+proof-
+  have linp :"islinform p" by (rule unified_islinform[OF unifp])
+  have dpos: "?d > 0" by (rule divlcm_pos[OF linp])
+  have alldvd: "alldivide(?d,p)" by (rule divlcm_corr[OF linp])
+  have eMPimpeP: "(\<exists>j \<in> ?D. ?MP j) \<longrightarrow> (\<exists>x. ?P x)"
+    by (simp add: minusinf_lemma[OF unifp, where d="?d" and ats="ats"])
+  have ePimpeP: "(\<exists> j \<in> ?D. \<exists> b\<in> ?B. ?P (?I a b + j)) \<longrightarrow> (\<exists> x. ?P x)"
+    by blast
+  have bst_rep: "\<forall> x. \<not> (\<exists> j \<in> ?D. \<exists> b \<in> ?B. ?P (?I a b + j)) \<longrightarrow> ?P x \<longrightarrow> ?P (x - ?d)"
+    by (rule bset_disj_repeat2[OF unifp])
+  have MPrep: "\<forall> x k. ?MP x = ?MP (x- k*?d)"
+    by (rule minusinf_repeats2[OF alldvd unifp])
+  have MPeqP: "\<exists> z. \<forall>  x < z. ?P x = ?MP x"
+    by (rule minusinf_eq[OF unifp])
+  let ?B'= "{?I a b| b. b\<in> ?B}"
+  from bst_rep have bst_rep2: "\<forall>x. \<not> (\<exists>j\<in>?D. \<exists>b\<in> ?B'. ?P (b+j)) \<longrightarrow> ?P x \<longrightarrow> ?P (x - ?d)"
+    by auto
+  show ?thesis 
+  using cpmi_eq[OF dpos MPeqP bst_rep2 MPrep]
+  by auto
+qed
+
+(* A formalized analogy between aset, bset, plusinfinity and minusinfinity *)
+
+consts mirror:: "QF \<Rightarrow> QF"
+recdef mirror "measure size"
+"mirror (Le (Add (Mult (Cst c) (Var 0)) r) z) =
+  (Le (Add (Mult (Cst (- c)) (Var 0)) r) z)"
+"mirror (Eq (Add (Mult (Cst c) (Var 0)) r) z) =
+  (Eq (Add (Mult (Cst (- c)) (Var 0)) r) z)"
+"mirror (Divides (Cst d) (Add (Mult (Cst c) (Var 0)) r)) = 
+  (Divides (Cst d) (Add (Mult (Cst (- c)) (Var 0)) r))"
+"mirror (NOT(Divides (Cst d) (Add (Mult (Cst c) (Var 0)) r))) = 
+  (NOT(Divides (Cst d) (Add (Mult (Cst (- c)) (Var 0)) r)))"
+"mirror (NOT(Eq (Add (Mult (Cst c) (Var 0)) r) z)) =
+  (NOT(Eq (Add (Mult (Cst (- c)) (Var 0)) r) z))"
+"mirror (And p q) = And (mirror p) (mirror q)"
+"mirror (Or p q) = Or (mirror p) (mirror q)"
+"mirror p = p"
+(* mirror preserves unifiedness *)
+
+lemma[simp]: "(abs (i::int) = 1) = (i =1 \<or> i = -1)"  by arith
+lemma mirror_unified:
+  assumes unif: "isunified p"
+  shows "isunified (mirror p)"
+  using unif
+proof (induct p rule: mirror.induct, simp_all)
+  case (goal1 c r z)
+  from prems have zz: "z = Cst 0" by (cases z, simp_all) 
+  then show ?case using prems 
+    by (auto simp add: islinintterm_eq_islint islint_def)
+next 
+  case (goal2 c r z)
+  from prems have zz: "z = Cst 0" by (cases z, simp_all) 
+  then show ?case using prems 
+    by (auto simp add: islinintterm_eq_islint islint_def)
+next
+  case (goal3 d c r) show ?case using prems by (auto simp add: islinintterm_eq_islint islint_def) 
+next 
+  case (goal4 d c r) show ?case using prems  by (auto simp add: islinintterm_eq_islint islint_def)
+next 
+ case (goal5 c r z)
+  from prems have zz: "z = Cst 0" by (cases z, simp_all) 
+  then show ?case using prems 
+    by (auto simp add: islinintterm_eq_islint islint_def)
+qed
+
+(* relationship between plusinf and minusinf *)
+lemma plusinf_eq_minusinf_mirror:
+  assumes unifp: "isunified p"
+  shows "(qinterp (x#ats) (plusinf p)) = (qinterp ((- x)#ats) (minusinf (mirror p)))"
+using unifp unified_islinform[OF unifp]
+proof (induct p rule: islinform.induct, simp_all)
+  case (goal1 t z)
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have ?case using prems 
+      by (cases n, auto simp add: nth_pos2 
+	  intterm_novar0[OF lininr, where x="x" and y="-x"] )}
+  ultimately show ?case by blast
+    
+next
+  case (goal2 t z)
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have ?case using prems 
+      by (cases n, auto simp add: nth_pos2 
+	  intterm_novar0[OF lininr, where x="x" and y="-x"] )}
+  ultimately show ?case by blast
+next
+  case (goal3 d t)
+  
+ from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+
+    have ?case using prems 
+      by (cases n, simp_all add: nth_pos2 
+	  intterm_novar0[OF lininr, where x="x" and y="-x"] )}
+  ultimately show ?case by blast
+next
+
+  case (goal4 d t)
+  
+ from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+
+    have ?case using prems 
+      by (cases n, simp_all add: nth_pos2 
+	  intterm_novar0[OF lininr, where x="x" and y="-x"] )}
+  ultimately show ?case by blast
+next
+  case (goal5 t z)
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have ?case using prems 
+      by (cases n, auto simp add: nth_pos2 
+	  intterm_novar0[OF lininr, where x="x" and y="-x"] )}
+  ultimately show ?case by blast
+qed
+
+(* relationship between aset abd bset *)
+lemma aset_eq_bset_mirror: 
+  assumes unifp: "isunified p"
+  shows "set (aset p) = set (map lin_neg (bset (mirror p)))"
+using unifp
+proof(induct p rule: mirror.induct)
+  case (1 c r z) 
+  from prems have zz: "z = Cst 0"
+    by (cases z, auto)
+  from prems zz have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  have neg1eqm1: "Cst 1 = lin_neg (Cst -1)" by (simp add: lin_neg_def)
+  have negm1eq1: "Cst -1 = lin_neg (Cst 1)" by (simp add: lin_neg_def)
+  show ?case  using prems linr zz apply (auto simp add: lin_neg_lin_add_distrib lin_neg_idemp neg1eqm1)
+    by (simp add: negm1eq1 lin_neg_idemp sym[OF lin_neg_lin_add_distrib] lin_add_lin)
+next
+  case (2 c r z)   from prems have zz: "z = Cst 0"
+    by (cases z, auto)
+  from prems zz have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  have neg1eqm1: "Cst 1 = lin_neg (Cst -1)" by (simp add: lin_neg_def)
+  have negm1eq1: "Cst -1 = lin_neg (Cst 1)" by (simp add: lin_neg_def)
+  show ?case  using prems linr zz
+    by (auto simp add: lin_neg_lin_add_distrib lin_neg_idemp neg1eqm1)
+  (simp add: negm1eq1 lin_neg_idemp sym[OF lin_neg_lin_add_distrib] lin_add_lin lin_neg_lin)
+
+next
+  case (5 c r z)  from prems have zz: "z = Cst 0"
+    by (cases z, auto)
+  from prems zz have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  have neg1eqm1: "Cst 1 = lin_neg (Cst -1)" by (simp add: lin_neg_def)
+  have negm1eq1: "Cst -1 = lin_neg (Cst 1)" by (simp add: lin_neg_def)
+  show ?case  using prems linr zz
+    by(auto simp add: lin_neg_lin_add_distrib lin_neg_idemp neg1eqm1)
+  
+qed simp_all
+
+(* relationship between aset abd bset 2*)
+lemma aset_eq_bset_mirror2: 
+  assumes unifp: "isunified p"
+  shows "aset p = map lin_neg (bset (mirror p))"
+using unifp
+proof(induct p rule: mirror.induct)
+  case (1 c r z) 
+  from prems have zz: "z = Cst 0"
+    by (cases z, auto)
+  from prems zz have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  have neg1eqm1: "Cst 1 = lin_neg (Cst -1)" by (simp add: lin_neg_def)
+  have negm1eq1: "Cst -1 = lin_neg (Cst 1)" by (simp add: lin_neg_def)
+  show ?case  using prems linr zz
+    apply (simp add: lin_neg_lin_add_distrib lin_neg_idemp neg1eqm1)
+    apply (simp add: negm1eq1 lin_neg_idemp sym[OF lin_neg_lin_add_distrib] lin_add_lin)
+    by arith
+next
+  case (2 c r z)   from prems have zz: "z = Cst 0"
+    by (cases z, auto)
+  from prems zz have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  have neg1eqm1: "Cst 1 = lin_neg (Cst -1)" by (simp add: lin_neg_def)
+  have negm1eq1: "Cst -1 = lin_neg (Cst 1)" by (simp add: lin_neg_def)
+  show ?case  using prems linr zz
+    by(auto simp add: lin_neg_lin_add_distrib lin_neg_idemp neg1eqm1)
+    (simp add: negm1eq1 lin_neg_idemp sym[OF lin_neg_lin_add_distrib] lin_add_lin lin_neg_lin)
+
+next
+  case (5 c r z)  from prems have zz: "z = Cst 0"
+    by (cases z, auto)
+  from prems zz have lincnr: "islinintterm (Add (Mult (Cst c) (Var 0)) r)" by simp
+  have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  have neg1eqm1: "Cst 1 = lin_neg (Cst -1)" by (simp add: lin_neg_def)
+  have negm1eq1: "Cst -1 = lin_neg (Cst 1)" by (simp add: lin_neg_def)
+  show ?case  using prems linr zz
+    by(auto simp add: lin_neg_lin_add_distrib lin_neg_idemp neg1eqm1)
+  
+qed simp_all
+
+(* mirror preserves divlcm *)
+lemma divlcm_mirror_eq:
+  assumes unifp: "isunified p"
+  shows "divlcm p = divlcm (mirror p)"
+  using unifp
+by (induct p rule: mirror.induct) auto
+
+(* mirror almost preserves semantics *)  
+lemma mirror_interp: 
+  assumes unifp: "isunified p"
+  shows "(qinterp (x#ats) p) = (qinterp ((- x)#ats) (mirror p))" (is "?P x = ?MP (-x)")
+using unifp unified_islinform[OF unifp]
+proof (induct p rule: islinform.induct)
+  case (1 t z)
+  from prems have zz: "z = 0" by simp
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have ?case using prems zz
+      by (cases n) (simp_all add: nth_pos2 
+	intterm_novar0[OF lininr, where x="x" and y="-x"])
+  }
+  ultimately show ?case by blast
+next
+  case (2 t z)
+  from prems have zz: "z = 0" by simp
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have ?case using prems zz
+      by (cases n) (simp_all add: nth_pos2 
+	intterm_novar0[OF lininr, where x="x" and y="-x"])
+  }
+  ultimately show ?case by blast
+next
+  case (3 d t) 
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have ?case
+      using prems linr 
+      by (cases n) (simp_all add: nth_pos2
+	intterm_novar0[OF lininr, where x="x" and y="-x"])
+  }
+  ultimately show ?case by blast
+next
+
+  case (6 d t) 
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have ?case
+      using prems linr 
+      by (cases n) (simp_all add: nth_pos2
+	intterm_novar0[OF lininr, where x="x" and y="-x"])
+  }
+  ultimately show ?case by blast
+next 
+  case (7 t z)
+  from prems have zz: "z = 0" by simp
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have ?case using prems zz
+      by (cases n) (simp_all add: nth_pos2 
+	intterm_novar0[OF lininr, where x="x" and y="-x"])
+  }
+  ultimately show ?case by blast 
+qed simp_all
+
+
+lemma mirror_interp2: 
+  assumes unifp: "islinform p"
+  shows "(qinterp (x#ats) p) = (qinterp ((- x)#ats) (mirror p))" (is "?P x = ?MP (-x)")
+using unifp 
+proof (induct p rule: islinform.induct)
+  case (1 t z)
+  from prems have zz: "z = 0" by simp
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have ?case using prems zz
+      by (cases n) (simp_all add: nth_pos2 
+	intterm_novar0[OF lininr, where x="x" and y="-x"])
+  }
+  ultimately show ?case by blast
+next
+  case (2 t z)
+  from prems have zz: "z = 0" by simp
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have ?case using prems zz
+      by (cases n) (simp_all add: nth_pos2 
+	intterm_novar0[OF lininr, where x="x" and y="-x"])
+  }
+  ultimately show ?case by blast
+next
+  case (3 d t) 
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have ?case
+      using prems linr 
+      by (cases n) (simp_all add: nth_pos2
+	intterm_novar0[OF lininr, where x="x" and y="-x"])
+  }
+  ultimately show ?case by blast
+next
+
+  case (6 d t) 
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have ?case
+      using prems linr 
+      by (cases n) (simp_all add: nth_pos2
+	intterm_novar0[OF lininr, where x="x" and y="-x"])
+  }
+  ultimately show ?case by blast
+next 
+  case (7 t z)
+  from prems have zz: "z = 0" by simp
+  from prems 
+  have lint: "islinintterm t" by simp
+  then have "(\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r) \<or> (\<exists> i. t = Cst i)"
+    by (induct t rule: islinintterm.induct) auto
+  moreover{ assume "\<exists> i. t = Cst i" then have ?case using prems by auto }
+  moreover
+  { assume "\<exists> i n r. t = Add (Mult (Cst i) (Var n) ) r"
+    then obtain "i" "n" "r" where 
+      inr_def: "t = Add (Mult (Cst i) (Var n) ) r" 
+      by blast
+    with lint have lininr: "islinintterm (Add (Mult (Cst i) (Var n) ) r)" 
+      by simp
+    have linr: "islinintterm r" 
+      by (rule islinintterm_subt[OF lininr])
+    have ?case using prems zz
+      by (cases n) (simp_all add: nth_pos2 
+	intterm_novar0[OF lininr, where x="x" and y="-x"])
+  }
+  ultimately show ?case by blast 
+qed simp_all
+
+(* mirror preserves existence *)
+lemma mirror_ex: 
+  assumes unifp: "isunified p"
+  shows "(\<exists> x. (qinterp (x#ats) p)) = (\<exists> y. (qinterp (y#ats) (mirror p)))" 
+  (is "(\<exists> x. ?P x) = (\<exists> y. ?MP y)")
+proof
+  assume "\<exists> x. ?P x"
+  then obtain "x" where px:"?P x" by blast
+  have "?MP (-x)" 
+    using px
+    by(simp add: mirror_interp[OF unifp, where x="x"])
+  then show "\<exists> y. ?MP y" by blast
+next 
+  assume "\<exists> y. ?MP y"
+  then obtain "y" where mpy: "?MP y" by blast
+  have "?P (-y)"
+    using mpy
+    by (simp add: mirror_interp[OF unifp, where x="-y"])
+  then show "\<exists> x. ?P x" by blast
+qed
+
+lemma mirror_ex2: 
+  assumes unifp: "isunified p"
+  shows "qinterp ats (QEx p) = qinterp ats (QEx (mirror p))"
+using mirror_ex[OF unifp] by simp
+
+  
+(* Cooper's theorem in its plusinfinity version *)
+lemma cooper_pi_eq:
+  assumes unifp : "isunified p"
+  shows "(\<exists> x. qinterp (x#ats) p) = 
+  ((\<exists> j \<in> {1 .. (divlcm p)}. qinterp (-j#ats) (plusinf p)) \<or> 
+  (\<exists> j \<in> {1 .. (divlcm p)}. \<exists> b \<in> set (aset p). 
+  qinterp (((I_intterm (a#ats) b) - j)#ats) p))"
+  (is "(\<exists> x. ?P x) = ((\<exists> j\<in> {1 .. ?d}. ?PP (-j)) \<or> (\<exists> j \<in> ?D. \<exists> b\<in> ?A. ?P (?I a b - j)))")
+proof-
+  have unifmp: "isunified (mirror p)" by (rule mirror_unified[OF unifp])
+  have th1: 
+    "(\<exists> j\<in> {1 .. ?d}. ?PP (-j)) = (\<exists> j\<in> {1..?d}.  qinterp (j # ats) (minusinf (mirror p)))"
+    by (simp add: plusinf_eq_minusinf_mirror[OF unifp])
+  have dth: "?d = divlcm (mirror p)"
+    by (rule divlcm_mirror_eq[OF unifp])
+  have "(\<exists> j \<in> ?D. \<exists> b\<in> ?A. ?P (?I a b - j)) = 
+    (\<exists> j\<in> ?D. \<exists> b \<in> set (map lin_neg (bset (mirror p))). ?P (?I a b - j))"
+    by (simp only: aset_eq_bset_mirror[OF unifp])
+  also have "\<dots> = (\<exists> j\<in> ?D. \<exists> b \<in> set (bset (mirror p)). ?P (?I a (lin_neg b) - j))"
+    by simp
+  also have "\<dots> = (\<exists> j\<in> ?D. \<exists> b \<in> set (bset (mirror p)). ?P (-(?I a b + j)))"
+  proof
+    assume "\<exists>j\<in>{1..divlcm p}.
+      \<exists>b\<in>set (bset (mirror p)). qinterp ((I_intterm (a # ats) (lin_neg b) - j) # ats) p"
+    then
+    obtain "j" and "b" where 
+      pbmj: "j\<in> ?D \<and> b\<in> set (bset (mirror p)) \<and> ?P (?I a (lin_neg b) - j)" by blast
+    then have linb: "islinintterm b" 
+      by (auto simp add:bset_lin[OF unifmp])
+    from linb pbmj have "?P (-(?I a b + j))" by (simp add: lin_neg_corr)
+    then show "\<exists> j\<in> ?D. \<exists> b \<in> set (bset (mirror p)). ?P (-(?I a b + j))"
+      using pbmj
+      by auto
+  next 
+    assume "\<exists> j\<in> ?D. \<exists> b \<in> set (bset (mirror p)). ?P (-(?I a b + j))"
+    then obtain "j" and "b" where 
+      pbmj: "j\<in> ?D \<and> b\<in> set (bset (mirror p)) \<and> ?P (-(?I a b + j))"
+      by blast
+    then have linb: "islinintterm b" 
+      by (auto simp add:bset_lin[OF unifmp])
+    from linb pbmj have "?P (?I a (lin_neg b) - j)"  
+      by (simp add: lin_neg_corr)
+    then show "\<exists> j\<in> ?D. \<exists> b \<in> set (bset (mirror p)). ?P (?I a (lin_neg b) - j)"
+      using pbmj by auto
+  qed
+  finally 
+  have bth: "(\<exists> j\<in> ?D. \<exists> b\<in> ?A. ?P (?I a b - j)) =
+    (\<exists>j\<in> ?D. \<exists> b\<in>set (bset (mirror p)). 
+    qinterp ((I_intterm (a # ats) b + j) # ats) (mirror p))"
+    by (simp add: mirror_interp[OF unifp] zadd_ac)
+  from bth dth th1
+  have "(\<exists> x. ?P x) = (\<exists> x. qinterp (x#ats) (mirror p))"
+    by (simp add: mirror_ex[OF unifp])
+  also have "\<dots> = ((\<exists>j\<in>{1..divlcm (mirror p)}. qinterp (j # ats) (minusinf (mirror p))) \<or>
+    (\<exists>j\<in>{1..divlcm (mirror p)}.
+    \<exists>b\<in>set (bset (mirror p)). qinterp ((I_intterm (a # ats) b + j) # ats) (mirror p)))"
+    (is "(\<exists> x. ?MP x) = ((\<exists> j\<in> ?DM. ?MPM j) \<or> (\<exists> j \<in> ?DM. \<exists> b\<in> ?BM. ?MP (?I a b + j)))")
+    by (rule cooper_mi_eq[OF unifmp])
+  also 
+  have "\<dots> = ((\<exists> j\<in> ?D. ?PP (-j)) \<or> (\<exists> j \<in> ?D. \<exists> b\<in> ?BM. ?MP (?I a b + j)))"
+    using bth th1 dth by simp
+  finally  show ?thesis using sym[OF bth] by simp
+qed
+   
+
+(* substitution of a term into a Qfree formula, substitution of Bound 0 by i*)
+
+consts subst_it:: "intterm \<Rightarrow> intterm \<Rightarrow> intterm"
+primrec
+"subst_it i (Cst b) = Cst b"
+"subst_it i (Var n) = (if n = 0 then i else Var n)"
+"subst_it i (Neg it) = Neg (subst_it i it)"
+"subst_it i (Add it1 it2) = Add (subst_it i it1) (subst_it i it2)" 
+"subst_it i (Sub it1 it2) = Sub (subst_it i it1) (subst_it i it2)"
+"subst_it i (Mult it1 it2) = Mult (subst_it i it1) (subst_it i it2)"
+
+
+(* subst_it preserves semantics *)
+lemma subst_it_corr: 
+"I_intterm (a#ats) (subst_it i t) = I_intterm ((I_intterm (a#ats) i)#ats) t"
+by (induct t rule: subst_it.induct, simp_all add: nth_pos2)
+
+consts subst_p:: "intterm \<Rightarrow> QF \<Rightarrow> QF"
+primrec
+"subst_p i (Le it1 it2) = Le (subst_it i it1) (subst_it i it2)"
+"subst_p i (Lt it1 it2) = Lt (subst_it i it1) (subst_it i it2)"
+"subst_p i (Ge it1 it2) = Ge (subst_it i it1) (subst_it i it2)"
+"subst_p i (Gt it1 it2) = Gt (subst_it i it1) (subst_it i it2)"
+"subst_p i (Eq it1 it2) = Eq (subst_it i it1) (subst_it i it2)"
+"subst_p i (Divides d t) = Divides (subst_it i d) (subst_it i t)"
+"subst_p i T = T"
+"subst_p i F = F"
+"subst_p i (And p q) = And (subst_p i p) (subst_p i q)"
+"subst_p i (Or p q) = Or (subst_p i p) (subst_p i q)"
+"subst_p i (Imp p q) = Imp (subst_p i p) (subst_p i q)"
+"subst_p i (Equ p q) = Equ (subst_p i p) (subst_p i q)"
+"subst_p i (NOT p) = (NOT (subst_p i p))"
+
+(* subs_p preserves correctness *)
+lemma subst_p_corr: 
+  assumes qf: "isqfree p" 
+  shows "qinterp (a # ats) (subst_p i p) = qinterp ((I_intterm (a#ats) i)#ats) p "
+  using qf
+by (induct p rule: subst_p.induct) (simp_all add: subst_it_corr)
+
+(* novar0 p is true if the fomula doese not depend on the quantified variable*)
+consts novar0I:: "intterm \<Rightarrow> bool"
+primrec
+"novar0I (Cst i) = True"
+"novar0I (Var n) = (n > 0)"
+"novar0I (Neg a) = (novar0I a)"
+"novar0I (Add a b) = (novar0I a \<and> novar0I b)"
+"novar0I (Sub a b) = (novar0I a \<and> novar0I b)"
+"novar0I (Mult a b) = (novar0I a \<and> novar0I b)"
+
+consts novar0:: "QF \<Rightarrow> bool"
+recdef novar0 "measure size"
+"novar0 (Lt a b) = (novar0I a \<and> novar0I b)"
+"novar0 (Gt a b) = (novar0I a \<and> novar0I b)"
+"novar0 (Le a b) = (novar0I a \<and> novar0I b)"
+"novar0 (Ge a b) = (novar0I a \<and> novar0I b)"
+"novar0 (Eq a b) = (novar0I a \<and> novar0I b)"
+"novar0 (Divides a b) = (novar0I a \<and> novar0I b)"
+"novar0 T = True" 
+"novar0 F = True"
+"novar0 (NOT p) = novar0 p" 
+"novar0 (And p q) = (novar0 p \<and> novar0 q)"
+"novar0 (Or p q)  = (novar0 p \<and> novar0 q)"
+"novar0 (Imp p q) = (novar0 p \<and> novar0 q)"
+"novar0 (Equ p q) = (novar0 p \<and> novar0 q)"
+"novar0 p = False"
+
+(* Interpretation of terms, that doese not depend on Var 0 *)
+lemma I_intterm_novar0:
+  assumes nov0: "novar0I x"
+  shows "I_intterm (a#ats) x = I_intterm (b#ats) x"
+using nov0
+by (induct x) (auto simp add: nth_pos2)
+
+(* substition is meaningless for term independent of Var 0*)
+lemma subst_p_novar0_corr:
+assumes qfp: "isqfree p"
+  and nov0: "novar0I i"
+  shows "qinterp (a#ats) (subst_p i p) = qinterp (I_intterm (b#ats) i#ats) p"
+proof-
+  have "qinterp (a#ats) (subst_p i p) = qinterp (I_intterm (a#ats) i#ats) p"
+    by (rule subst_p_corr[OF qfp])
+  moreover have "I_intterm (a#ats) i#ats = I_intterm (b#ats) i#ats"
+    by (simp add: I_intterm_novar0[OF nov0, where a="a" and b="b"])
+  ultimately show ?thesis by simp
+qed
+
+(* linearity and independence on Var 0*)
+lemma lin_novar0: 
+  assumes linx: "islinintterm x"
+  and nov0: "novar0I x"
+  shows "\<exists> n > 0. islintn(n,x)"
+using linx nov0
+by (induct x rule: islinintterm.induct) auto
+
+lemma lintnpos_novar0:
+ assumes  npos: "n > 0"
+  and linx: "islintn(n,x)"
+  shows "novar0I x"
+using npos linx
+by (induct n x rule: islintn.induct) auto
+
+(* lin_add preserves independence on Var 0*)
+lemma lin_add_novar0:
+  assumes nov0a: "novar0I a"
+  and nov0b : "novar0I b"
+  and lina : "islinintterm a"
+  and linb: "islinintterm b"
+  shows "novar0I (lin_add (a,b))"
+proof-
+  have "\<exists> na > 0. islintn(na, a)" by (rule lin_novar0[OF lina nov0a]) 
+  then obtain "na" where na: "na > 0 \<and> islintn(na,a)" by blast
+  have "\<exists> nb > 0. islintn(nb, b)" by (rule lin_novar0[OF linb nov0b]) 
+  then obtain "nb" where nb: "nb > 0 \<and> islintn(nb,b)" by blast
+  from na have napos: "na > 0" by simp
+  from na have linna: "islintn(na,a)" by simp
+  from nb have nbpos: "nb > 0" by simp
+  from nb have linnb: "islintn(nb,b)" by simp
+  have "min na nb \<le> min na nb" by simp
+  then have "islintn (min na nb, lin_add(a,b))" by (simp add: lin_add_lint[OF linna linnb])
+  moreover have "min na nb > 0" using napos nbpos by (simp add: min_def)
+  ultimately show ?thesis by (simp only: lintnpos_novar0)
+qed
+
+(* lin__mul preserves independence on Var 0*)
+lemma lin_mul_novar0:
+  assumes linx: "islinintterm x"
+  and nov0: "novar0I x"
+  shows "novar0I (lin_mul(i,x))"
+  using linx nov0
+proof (induct i x rule: lin_mul.induct, auto)
+  case (goal1 c c' n r)
+  from prems have lincnr: "islinintterm (Add (Mult (Cst c') (Var n)) r)" by simp
+  have "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+  then show ?case using prems by simp
+qed
+    
+(* lin_neg preserves indepenednce on Var 0*)
+lemma lin_neg_novar0:
+  assumes linx: "islinintterm x"
+  and nov0: "novar0I x"
+  shows "novar0I (lin_neg x)"
+by (auto simp add: lin_mul_novar0 linx nov0 lin_neg_def)
+
+(* subterms of linear terms are independent on Var 0*)
+lemma intterm_subt_novar0:
+  assumes lincnr: "islinintterm (Add (Mult (Cst c) (Var n)) r)"
+  shows "novar0I r"
+proof-
+  have cnz: "c \<noteq> 0" by (rule islinintterm_cnz[OF lincnr])
+  have "islintn(0,Add (Mult (Cst c) (Var n)) r)" using lincnr
+    by (simp only: islinintterm_eq_islint islint_def)
+  then have "islintn (n+1,r)" by auto
+  moreover have "n+1 >0 " by arith
+  ultimately show ?thesis 
+    using lintnpos_novar0
+    by auto
+qed
+
+(* decrease the De-Bruijn indices*)
+consts decrvarsI:: "intterm \<Rightarrow> intterm"
+primrec
+"decrvarsI (Cst i) = (Cst i)"
+"decrvarsI (Var n) = (Var (n - 1))"
+"decrvarsI (Neg a) = (Neg (decrvarsI a))"
+"decrvarsI (Add a b) = (Add (decrvarsI a) (decrvarsI b))"
+"decrvarsI (Sub a b) = (Sub (decrvarsI a) (decrvarsI b))"
+"decrvarsI (Mult a b) = (Mult (decrvarsI a) (decrvarsI b))"
+
+(* One can decrease the indics for terms and formulae independent on Var 0*)
+lemma intterm_decrvarsI:
+  assumes nov0: "novar0I t"
+  shows "I_intterm (a#ats) t = I_intterm ats (decrvarsI t)"
+using nov0
+by (induct t) (auto simp add: nth_pos2)
+
+consts decrvars:: "QF \<Rightarrow> QF"
+primrec
+"decrvars (Lt a b) = (Lt (decrvarsI a) (decrvarsI b))"
+"decrvars (Gt a b) = (Gt (decrvarsI a) (decrvarsI b))"
+"decrvars (Le a b) = (Le (decrvarsI a) (decrvarsI b))"
+"decrvars (Ge a b) = (Ge (decrvarsI a) (decrvarsI b))"
+"decrvars (Eq a b) = (Eq (decrvarsI a) (decrvarsI b))"
+"decrvars (Divides a b) = (Divides (decrvarsI a) (decrvarsI b))"
+"decrvars T = T" 
+"decrvars F = F"
+"decrvars (NOT p) = (NOT (decrvars p))" 
+"decrvars (And p q) = (And (decrvars p) (decrvars q))"
+"decrvars (Or p q)  = (Or (decrvars p) (decrvars q))"
+"decrvars (Imp p q) = (Imp (decrvars p) (decrvars q))"
+"decrvars (Equ p q) = (Equ (decrvars p) (decrvars q))"
+
+(* decrvars preserves quantifier freeness*)
+lemma decrvars_qfree: "isqfree p \<Longrightarrow> isqfree (decrvars p)"
+by (induct p rule: isqfree.induct, auto)
+
+lemma novar0_qfree: "novar0 p \<Longrightarrow> isqfree p"
+by (induct p) auto
+
+lemma qinterp_novar0:
+  assumes nov0: "novar0 p"
+  shows "qinterp (a#ats) p = qinterp ats (decrvars p)"
+using nov0
+by(induct p) (simp_all add: intterm_decrvarsI)
+
+(* All elements of bset p doese not depend on Var 0*)
+lemma bset_novar0:
+  assumes unifp: "isunified p"
+  shows "\<forall> b\<in> set (bset p). novar0I b "
+  using unifp
+proof(induct p rule: bset.induct)
+  case (1 c r z) 
+  from prems have zz: "z = Cst 0" by (cases "z", auto) 
+    from prems zz have lincnr: "islinintterm(Add (Mult (Cst c) (Var 0)) r)" by simp
+    have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+    have novar0r: "novar0I r" by (rule intterm_subt_novar0[OF lincnr])
+    from prems zz have "c = 1 \<or> c = -1" by auto
+    moreover 
+    {
+      assume c1: "c=1"
+      have lin1: "islinintterm (Cst 1)" by simp
+      have novar01: "novar0I (Cst 1)" by simp
+      then have ?case 
+	using prems zz novar0r lin1 novar01
+	by (auto simp add: lin_add_novar0 lin_neg_novar0 linr lin_neg_lin)
+    }
+    moreover 
+    {
+      assume c1: "c= -1"
+      have lin1: "islinintterm (Cst -1)" by simp
+      have novar01: "novar0I (Cst -1)" by simp
+      then have ?case 
+	using prems zz novar0r lin1 novar01
+	by (auto simp add: lin_add_novar0 lin_neg_novar0 linr lin_neg_lin)
+    }
+    ultimately show ?case by blast
+next 
+  case (2 c r z) 
+  from prems have zz: "z = Cst 0" by (cases "z", auto) 
+    from prems zz have lincnr: "islinintterm(Add (Mult (Cst c) (Var 0)) r)" by simp
+    have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+    have novar0r: "novar0I r" by (rule intterm_subt_novar0[OF lincnr])
+    from prems zz have "c = 1 \<or> c = -1" by auto
+    moreover 
+    {
+      assume c1: "c=1"
+      have lin1: "islinintterm (Cst 1)" by simp
+      have novar01: "novar0I (Cst 1)" by simp
+      then have ?case 
+	using prems zz novar0r lin1 novar01
+	by (auto simp add: lin_add_novar0 lin_neg_novar0 linr lin_neg_lin)
+    }
+    moreover 
+    {
+      assume c1: "c= -1"
+      have lin1: "islinintterm (Cst -1)" by simp
+      have novar01: "novar0I (Cst -1)" by simp
+      then have ?case 
+	using prems zz novar0r lin1 novar01
+	by (auto simp add: lin_add_novar0 lin_neg_novar0 linr lin_neg_lin)
+    }
+    ultimately show ?case by blast
+next 
+  case (3 c r z) 
+  from prems have zz: "z = Cst 0" by (cases "z", auto) 
+    from prems zz have lincnr: "islinintterm(Add (Mult (Cst c) (Var 0)) r)" by simp
+    have linr: "islinintterm r" by (rule islinintterm_subt[OF lincnr])
+    have novar0r: "novar0I r" by (rule intterm_subt_novar0[OF lincnr])
+    from prems zz have "c = 1 \<or> c = -1" by auto
+    moreover 
+    {
+      assume c1: "c=1"
+      have lin1: "islinintterm (Cst 1)" by simp
+      have novar01: "novar0I (Cst 1)" by simp
+      then have ?case 
+	using prems zz novar0r lin1 novar01
+	by (auto simp add: lin_add_novar0 lin_neg_novar0 linr lin_neg_lin)
+    }
+    moreover 
+    {
+      assume c1: "c= -1"
+      have lin1: "islinintterm (Cst -1)" by simp
+      have novar01: "novar0I (Cst -1)" by simp
+      then have ?case 
+	using prems zz novar0r lin1 novar01
+	by (auto simp add: lin_add_novar0 lin_neg_novar0 linr lin_neg_lin)
+    }
+    ultimately show ?case by blast
+qed auto
+
+(* substitution preserves independence on Var 0*)
+lemma subst_it_novar0:
+  assumes nov0x: "novar0I x"
+  shows "novar0I (subst_it x t)"
+  using nov0x
+  by (induct t) auto
+
+lemma subst_p_novar0:
+  assumes nov0x:"novar0I x"
+  and qfp: "isqfree p"
+  shows "novar0 (subst_p x p)"
+  using nov0x qfp
+  by (induct p rule: novar0.induct) (simp_all add: subst_it_novar0)
+
+(* linearize preserves independence on Var 0 *)
+lemma linearize_novar0: 
+  assumes nov0t: "novar0I t "
+  shows "\<And> t'. linearize t = Some t' \<Longrightarrow> novar0I t'"
+using nov0t
+proof(induct t rule: novar0I.induct)
+  case (Neg a)
+  let ?la = "linearize a"
+  from prems have "\<exists> a'. ?la = Some a'" by (cases ?la, auto)
+  then obtain "a'" where "?la = Some a'" by blast
+  with prems have nv0a':"novar0I a'" by simp
+  have "islinintterm a'" using prems by (simp add: linearize_linear)
+  with nv0a' have "novar0I (lin_neg a')" 
+    by (simp add: lin_neg_novar0)
+  then 
+  show ?case using prems by simp 
+next 
+  case (Add a b) 
+  let ?la = "linearize a"
+  let ?lb = "linearize b"
+  from prems have linab: "linearize (Add a b) = Some t'" by simp
+  then have "\<exists> a'. ?la = Some a'" by (cases ?la) auto
+  then obtain "a'" where "?la = Some a'" by blast
+  with prems have nv0a':"novar0I a'" by simp
+  have lina': "islinintterm a'" using prems by (simp add: linearize_linear)
+  from linab have "\<exists> b'. ?lb = Some b'"
+    by (cases ?la, auto simp add: measure_def inv_image_def) (cases ?lb, auto)
+  then obtain "b'" where "?lb = Some b'" by blast
+  with prems have nv0b':"novar0I b'" by simp
+  have linb': "islinintterm b'" using prems by (simp add: linearize_linear)
+  then show ?case using prems lina' linb' nv0a' nv0b'
+    by (auto simp add: measure_def inv_image_def lin_add_novar0)
+next 
+  case (Sub a b)
+    let ?la = "linearize a"
+  let ?lb = "linearize b"
+  from prems have linab: "linearize (Sub a b) = Some t'" by simp
+  then have "\<exists> a'. ?la = Some a'" by (cases ?la) auto
+  then obtain "a'" where "?la = Some a'" by blast
+  with prems have nv0a':"novar0I a'" by simp
+  have lina': "islinintterm a'" using prems by (simp add: linearize_linear)
+  from linab have "\<exists> b'. ?lb = Some b'"
+    by (cases ?la, auto simp add: measure_def inv_image_def) (cases ?lb, auto)
+  then obtain "b'" where "?lb = Some b'" by blast
+  with prems have nv0b':"novar0I b'" by simp
+  have linb': "islinintterm b'" using prems by (simp add: linearize_linear)
+  then show ?case using prems lina' linb' nv0a' nv0b'
+    by (auto simp add: 
+      measure_def inv_image_def lin_add_novar0 lin_neg_novar0 lin_neg_lin)
+next 
+  case (Mult a b)     
+  let ?la = "linearize a"
+  let ?lb = "linearize b"
+  from prems have linab: "linearize (Mult a b) = Some t'" by simp
+  then have "\<exists> a'. ?la = Some a'"
+    by (cases ?la, auto simp add: measure_def inv_image_def)
+  then obtain "a'" where "?la = Some a'" by blast
+  with prems have nv0a':"novar0I a'" by simp
+  have lina': "islinintterm a'" using prems by (simp add: linearize_linear)
+  from prems linab have "\<exists> b'. ?lb = Some b'"
+    apply (cases ?la, auto simp add: measure_def inv_image_def) 
+    by (cases "a'",auto simp add: measure_def inv_image_def) (cases ?lb, auto)+
+  then obtain "b'" where "?lb = Some b'" by blast
+  with prems have nv0b':"novar0I b'" by simp
+  have linb': "islinintterm b'" using prems by (simp add: linearize_linear)
+  then show ?case using prems lina' linb' nv0a' nv0b' 
+    by (cases "a'",auto simp add: measure_def inv_image_def lin_mul_novar0)
+  (cases "b'",auto simp add: measure_def inv_image_def lin_mul_novar0)
+qed auto
+
+
+(* simplification of formulae *)
+consts psimpl :: "QF \<Rightarrow> QF"
+recdef psimpl "measure size"
+"psimpl (Le l r) = 
+  (case (linearize (Sub l r)) of
+   None \<Rightarrow> Le l r
+ | Some x \<Rightarrow> (case x of 
+       Cst i \<Rightarrow> (if i \<le> 0 then T else F)
+     | _ \<Rightarrow> (Le x (Cst 0))))"
+"psimpl (Eq l r) = 
+  (case (linearize (Sub l r)) of
+   None \<Rightarrow> Eq l r
+ | Some x \<Rightarrow> (case x of 
+       Cst i \<Rightarrow> (if i = 0 then T else F)
+     | _ \<Rightarrow> (Eq x (Cst 0))))"
+
+"psimpl (Divides (Cst d) t) = 
+  (case (linearize t) of
+  None \<Rightarrow> (Divides (Cst d) t)
+  | Some c \<Rightarrow> (case c of
+     Cst i \<Rightarrow> (if d dvd i then T else F)
+   | _ \<Rightarrow>  (Divides (Cst d) c)))"
+
+"psimpl (And p q) = 
+  (let p'= psimpl p
+  in (case p' of 
+       F \<Rightarrow> F
+      |T \<Rightarrow> psimpl q
+      | _ \<Rightarrow> let q' = psimpl q
+             in (case q' of
+                     F \<Rightarrow> F
+                   | T \<Rightarrow> p'
+                   | _ \<Rightarrow> (And p' q'))))"
+
+"psimpl (Or p q) = 
+  (let p'= psimpl p
+  in (case p' of 
+        T \<Rightarrow> T
+      | F \<Rightarrow> psimpl q
+      | _ \<Rightarrow> let q' = psimpl q
+             in (case q' of
+                     T \<Rightarrow> T
+                   | F \<Rightarrow> p'
+                   | _ \<Rightarrow> (Or p' q'))))"
+
+"psimpl (Imp p q) = 
+  (let p'= psimpl p
+  in (case p' of 
+       F \<Rightarrow> T
+      |T \<Rightarrow> psimpl q
+      | NOT p1 \<Rightarrow> let q' = psimpl q
+             in (case q' of
+                     F \<Rightarrow> p1
+                   | T \<Rightarrow> T
+                   | _ \<Rightarrow> (Or p1 q'))
+      | _ \<Rightarrow> let q' = psimpl q
+             in (case q' of
+                     F \<Rightarrow> NOT p'
+                   | T \<Rightarrow> T
+                   | _ \<Rightarrow> (Imp p' q'))))"
+
+"psimpl (Equ p q) = 
+  (let p'= psimpl p ; q' = psimpl q
+  in (case p' of 
+        T \<Rightarrow> q'
+      | F \<Rightarrow> (case q' of
+                  T \<Rightarrow> F
+                | F \<Rightarrow> T
+                | NOT q1 \<Rightarrow> q1
+                | _ \<Rightarrow> NOT q')
+      | NOT p1 \<Rightarrow>  (case q' of
+                  T \<Rightarrow> p'
+                | F \<Rightarrow> p1
+                | NOT q1 \<Rightarrow> (Equ p1 q1)
+                | _ \<Rightarrow> (Equ p' q'))
+      | _ \<Rightarrow> (case q' of
+                  T \<Rightarrow> p'
+                | F \<Rightarrow> NOT p'
+                | _ \<Rightarrow> (Equ p' q'))))"
+
+"psimpl (NOT p) = 
+  (let p' = psimpl p
+  in ( case p' of 
+       F \<Rightarrow> T
+     | T \<Rightarrow> F
+     | NOT p1 \<Rightarrow> p1 
+     | _ \<Rightarrow> (NOT p')))"
+"psimpl p = p"
+
+(* psimpl preserves semantics *)
+lemma psimpl_corr: "qinterp ats p = qinterp ats (psimpl p)"
+proof(induct p rule: psimpl.induct)
+  case (1 l r)
+  have "(\<exists> lx. linearize (Sub l r) = Some lx) \<or> (linearize (Sub l r) = None)" by auto
+  moreover
+  {
+    assume lin: "\<exists> lx. linearize (Sub l r) = Some lx"
+    from lin obtain "lx" where lx: "linearize (Sub l r) = Some lx" by blast
+    from lx have "I_intterm ats (Sub l r) = I_intterm ats lx"
+      by (rule linearize_corr[where t="Sub l r" and t'= "lx"])
+    then have feq: "qinterp ats (Le l r) = qinterp ats (Le lx (Cst 0))" by (simp , arith)
+    from lx have lxlin: "islinintterm lx" by (rule linearize_linear)
+    from lxlin feq have ?case 
+      proof-
+	have "(\<exists> i. lx = Cst i) \<or> (\<not> (\<exists> i. lx = Cst i))" by blast
+	moreover
+	{
+	  assume lxcst: "\<exists> i. lx = Cst i"
+	  from lxcst obtain "i" where lxi: "lx = Cst i" by blast
+	  with feq have "qinterp ats (Le l r) = (i \<le> 0)" by simp
+	  then have ?case using prems by (simp add: measure_def inv_image_def)
+	}
+	moreover 
+	{
+	  assume "(\<not> (\<exists> i. lx = Cst i))"
+	  then have "(case lx of 
+	    Cst i \<Rightarrow> (if i \<le> 0 then T else F)
+	    | _ \<Rightarrow> (Le lx (Cst 0))) = (Le lx (Cst 0))" 
+	    by (case_tac "lx::intterm", auto)
+	  with prems lxlin feq have ?case by (auto simp add: measure_def inv_image_def)
+	}
+	ultimately show ?thesis  by blast
+      qed
+  }
+  moreover
+  {
+    assume "linearize (Sub l r) = None"
+    then have ?case using prems by simp
+  }
+  ultimately show ?case by blast
+  
+next 
+  case (2 l r)
+  have "(\<exists> lx. linearize (Sub l r) = Some lx) \<or> (linearize (Sub l r) = None)" by auto
+  moreover
+  {
+    assume lin: "\<exists> lx. linearize (Sub l r) = Some lx"
+    from lin obtain "lx" where lx: "linearize (Sub l r) = Some lx" by blast
+    from lx have "I_intterm ats (Sub l r) = I_intterm ats lx"
+      by (rule linearize_corr[where t="Sub l r" and t'= "lx"])
+    then have feq: "qinterp ats (Eq l r) = qinterp ats (Eq lx (Cst 0))" by (simp , arith)
+    from lx have lxlin: "islinintterm lx" by (rule linearize_linear)
+    from lxlin feq have ?case 
+      proof-
+	have "(\<exists> i. lx = Cst i) \<or> (\<not> (\<exists> i. lx = Cst i))" by blast
+	moreover
+	{
+	  assume lxcst: "\<exists> i. lx = Cst i"
+	  from lxcst obtain "i" where lxi: "lx = Cst i" by blast
+	  with feq have "qinterp ats (Eq l r) = (i = 0)" by simp
+	  then have ?case using prems by (simp add: measure_def inv_image_def)
+	}
+	moreover 
+	{
+	  assume "(\<not> (\<exists> i. lx = Cst i))"
+	  then have "(case lx of 
+	    Cst i \<Rightarrow> (if i = 0 then T else F)
+	    | _ \<Rightarrow> (Eq lx (Cst 0))) = (Eq lx (Cst 0))" 
+	    by (case_tac "lx::intterm", auto)
+	  with prems lxlin feq have ?case by (auto simp add: measure_def inv_image_def)
+	}
+	ultimately show ?thesis  by blast
+      qed
+  }
+  moreover
+  {
+    assume "linearize (Sub l r) = None"
+    then have ?case using prems by simp
+  }
+  ultimately show ?case by blast
+  
+next 
+    
+  case (3 d t)  
+  have "(\<exists> lt. linearize t = Some lt) \<or> (linearize t = None)" by auto
+  moreover
+  {
+    assume lin: "\<exists> lt. linearize t  = Some lt"
+    from lin obtain "lt" where lt: "linearize t = Some lt" by blast
+    from lt have "I_intterm ats t = I_intterm ats lt"
+      by (rule linearize_corr[where t="t" and t'= "lt"])
+    then have feq: "qinterp ats (Divides (Cst d) t) = qinterp ats (Divides (Cst d) lt)" by (simp)
+    from lt have ltlin: "islinintterm lt" by (rule linearize_linear)
+    from ltlin feq have ?case using prems  apply simp by (case_tac "lt::intterm", simp_all)
+  }
+  moreover
+  {
+    assume "linearize t = None"
+    then have ?case using prems by simp
+  }
+  ultimately show ?case by blast
+  
+next 
+  case (4 f g)
+
+    let ?sf = "psimpl f"
+  let ?sg = "psimpl g"
+  show ?case using prems 
+    by (cases ?sf, simp_all add: Let_def measure_def inv_image_def) 
+  (cases ?sg, simp_all)+
+next
+  case (5 f g)
+      let ?sf = "psimpl f"
+  let ?sg = "psimpl g"
+  show ?case using prems
+    apply (cases ?sf, simp_all add: Let_def measure_def inv_image_def) 
+    apply (cases ?sg, simp_all)
+    apply (cases ?sg, simp_all)
+    apply (cases ?sg, simp_all)
+    apply (cases ?sg, simp_all)
+    apply (cases ?sg, simp_all)
+    apply (cases ?sg, simp_all)
+    apply (cases ?sg, simp_all)
+    apply blast
+    apply (cases ?sg, simp_all)
+    apply (cases ?sg, simp_all)
+     apply (cases ?sg, simp_all)
+   apply blast
+    apply (cases ?sg, simp_all)
+    by (cases ?sg, simp_all) (cases ?sg, simp_all)
+next
+  case (6 f g)
+  let ?sf = "psimpl f"
+  let ?sg = "psimpl g"
+  show ?case using prems 
+    apply(simp add: Let_def measure_def inv_image_def)
+    apply(cases ?sf,simp_all)
+    apply (simp_all add: Let_def measure_def inv_image_def)
+    apply(cases ?sg, simp_all)
+    apply(cases ?sg, simp_all)
+    apply(cases ?sg, simp_all)
+    apply(cases ?sg, simp_all)
+    apply(cases ?sg, simp_all)
+    apply(cases ?sg, simp_all)
+    apply(cases ?sg, simp_all)
+    apply blast
+    apply blast
+    apply blast
+    apply blast
+    apply blast
+    apply blast
+    apply blast
+    apply blast
+    apply blast
+    apply blast
+    apply blast
+    apply blast
+    apply blast
+    apply(cases ?sg, simp_all)
+    apply(cases ?sg, simp_all)
+    apply(cases ?sg, simp_all)
+    apply(cases ?sg, simp_all)
+    apply(cases ?sg, simp_all)
+    apply(cases ?sg, simp_all)
+    done
+next
+  case (7 f g)
+  let ?sf = "psimpl f"
+  let ?sg = "psimpl g"
+  show ?case 
+    using prems
+    by (cases ?sf, simp_all add: Let_def) (cases ?sg, simp_all)+
+next
+  case (8 f) show ?case 
+    using prems
+    apply (simp add: Let_def)
+    by (case_tac "psimpl f", simp_all)
+qed simp_all
+
+(* psimpl preserves independence on Var 0*)
+lemma psimpl_novar0:
+  assumes nov0p: "novar0 p"
+  shows "novar0 (psimpl p)"
+  using nov0p
+proof (induct p rule: psimpl.induct)
+  case (1 l r)
+  let ?ls = "linearize (Sub l r)"
+  have "?ls = None \<or> (\<exists> x. ?ls = Some x)" by auto
+  moreover
+  {
+    assume "?ls = None" then have ?case 
+      using prems by (simp add: measure_def inv_image_def)
+  }
+  moreover {
+    assume "\<exists> x. ?ls = Some x"
+    then obtain "x" where ls_d: "?ls = Some x" by blast
+    from prems have "novar0I l" by simp
+    moreover from prems have "novar0I r" by simp
+    ultimately have nv0s: "novar0I (Sub l r)" by simp
+    from prems have "novar0I x" 
+      by (simp add: linearize_novar0[OF nv0s, where t'="x"])
+    then have ?case
+      using prems
+      by (cases "x") (auto simp add: measure_def inv_image_def)
+  }
+  ultimately show ?case by blast
+next
+  case (2 l r)
+  let ?ls = "linearize (Sub l r)"
+  have "?ls = None \<or> (\<exists> x. ?ls = Some x)" by auto
+  moreover
+  {
+    assume "?ls = None" then have ?case 
+      using prems by (simp add: measure_def inv_image_def)
+  }
+  moreover {
+    assume "\<exists> x. ?ls = Some x"
+    then obtain "x" where ls_d: "?ls = Some x" by blast
+    from prems have "novar0I l" by simp
+    moreover from prems have "novar0I r" by simp
+    ultimately have nv0s: "novar0I (Sub l r)" by simp
+    from prems have "novar0I x" 
+      by (simp add: linearize_novar0[OF nv0s, where t'="x"])
+    then have ?case
+      using prems
+      by (cases "x") (auto simp add: measure_def inv_image_def)
+  }
+  ultimately show ?case by blast
+next
+  case (3 d t)
+  let ?lt = "linearize t"
+  have "?lt = None \<or> (\<exists> x. ?lt = Some x)"  by auto
+  moreover 
+  { assume "?lt = None" then have ?case using prems by simp }
+  moreover {
+    assume "\<exists>x. ?lt = Some x"
+    then obtain "x" where x_d: "?lt = Some x" by blast
+    from prems have nv0t: "novar0I t" by simp
+    with x_d have "novar0I x" 
+      by (simp add: linearize_novar0[OF nv0t])
+    with prems have ?case 
+      by (cases "x") simp_all
+  }
+  ultimately show ?case by blast
+next
+  case (4 f g)
+  let ?sf = "psimpl f"
+  let ?sg = "psimpl g"
+  show ?case 
+    using prems 
+    by (cases ?sf, simp_all add: Let_def measure_def inv_image_def)
+  (cases ?sg,simp_all)+
+next
+  case (5 f g)
+  let ?sf = "psimpl f"
+  let ?sg = "psimpl g"
+  show ?case 
+    using prems 
+    by (cases ?sf, simp_all add: Let_def measure_def inv_image_def)
+  (cases ?sg,simp_all)+
+next
+  case (6 f g)
+  let ?sf = "psimpl f"
+  let ?sg = "psimpl g"
+  show ?case 
+    using prems 
+    by (cases ?sf, simp_all add: Let_def measure_def inv_image_def)
+  (cases ?sg,simp_all)+
+next
+  case (7 f g)
+  let ?sf = "psimpl f"
+  let ?sg = "psimpl g"
+  show ?case 
+    using prems 
+    by (cases ?sf, simp_all add: Let_def measure_def inv_image_def)
+  (cases ?sg,simp_all)+
+
+next
+  case (8 f)
+  let ?sf = "psimpl f"
+  from prems have nv0sf:"novar0 ?sf" by simp
+  show ?case using prems nv0sf 
+    by (cases ?sf, auto simp add: Let_def measure_def inv_image_def)
+qed simp_all
+
+(* implements a disj of p applied to all elements of the list*)
+consts explode_disj :: "(intterm list \<times> QF) \<Rightarrow> QF"
+recdef explode_disj "measure (\<lambda>(is,p). length is)"
+"explode_disj ([],p) = F"
+"explode_disj (i#is,p) = 
+  (let pi = psimpl (subst_p i p)
+   in ( case pi of
+        T \<Rightarrow> T 
+       | F \<Rightarrow> explode_disj (is,p)
+       | _ \<Rightarrow> (let r = explode_disj (is,p)
+               in (case r of
+                      T \<Rightarrow> T
+                    | F \<Rightarrow> pi
+                    | _ \<Rightarrow> Or pi r))))"
+
+(* correctness theorem for one iteration of explode_disj *)
+lemma explode_disj_disj: 
+  assumes qfp: "isqfree p"
+  shows "(qinterp (x#xs) (explode_disj(i#is,p))) = 
+  (qinterp (x#xs) (subst_p i p) \<or> (qinterp (x#xs) (explode_disj(is,p))))"
+  using qfp
+proof-
+  let ?pi = "psimpl (subst_p i p)"
+  have pi: "qinterp (x#xs) ?pi = qinterp (x#xs) (subst_p i p)"
+    by (simp add: psimpl_corr[where p="(subst_p i p)"])
+  let ?dp = "explode_disj(is,p)"
+  show ?thesis using pi
+  proof (cases)
+    assume "?pi= T \<or> ?pi = F"
+    then show ?thesis using pi by (case_tac "?pi::QF", auto)
+    
+  next
+    assume notTF: "\<not> (?pi = T \<or> ?pi = F)" 
+    let ?dp = "explode_disj(is,p)"
+    have dp_cases: "explode_disj(i#is,p) = 
+      (case (explode_disj(is,p)) of
+      T \<Rightarrow> T
+      | F \<Rightarrow> psimpl (subst_p i p)
+      | _ \<Rightarrow> Or (psimpl (subst_p i p)) (explode_disj(is,p)))" using notTF
+      by (cases "?pi")
+    (simp_all add: Let_def cong del: QF.weak_case_cong)
+    show ?thesis using pi dp_cases notTF
+    proof(cases)
+      assume "?dp = T \<or> ?dp = F"
+      then show ?thesis 
+	using pi dp_cases
+	by (cases "?dp") auto
+    next
+      assume "\<not> (?dp = T \<or> ?dp = F)"
+      then show ?thesis using pi dp_cases notTF
+	by (cases ?dp) auto 
+    qed
+  qed
+qed
+
+(* correctness theorem for explode_disj *)
+lemma explode_disj_corr: 
+  assumes qfp: "isqfree p"
+  shows "(\<exists> x \<in> set xs. qinterp (a#ats) (subst_p x p)) = 
+  (qinterp (a#ats) (explode_disj(xs,p)))" (is "(\<exists> x \<in> set xs. ?P x) = (?DP a xs )")
+  using qfp
+  proof (induct xs)
+    case Nil show ?case by simp
+  next 
+    case (Cons y ys)
+    have "(\<exists> x \<in> set (y#ys). ?P x) = (?P y \<or> (\<exists> x\<in> set ys. ?P x))"
+      by auto
+    also have "\<dots> = (?P y \<or> ?DP a ys)" using "Cons.hyps" qfp by auto 
+    also have "\<dots> = ?DP a (y#ys)" using explode_disj_disj[OF qfp] by auto
+    finally show ?case by simp
+qed
+
+(* explode_disj preserves independence on Var 0*)
+lemma explode_disj_novar0:
+  assumes nov0xs: "\<forall>x \<in> set xs. novar0I x"
+  and qfp: "isqfree p"
+  shows "novar0 (explode_disj (xs,p))"
+  using nov0xs qfp
+proof (induct xs, auto simp add: Let_def)
+  case (goal1 a as)
+  let ?q = "subst_p a p"
+  let ?qs = "psimpl ?q"
+  have "?qs = T \<or> ?qs = F \<or> (?qs \<noteq> T \<or> ?qs \<noteq> F)" by simp
+  moreover
+  { assume "?qs = T"  then have ?case  by simp }
+  moreover
+  { assume "?qs = F"  then have ?case by simp }
+  moreover
+  {
+    assume qsnTF: "?qs \<noteq> T \<and> ?qs \<noteq> F"
+    let ?r = "explode_disj (as,p)"
+    have nov0qs: "novar0 ?qs"
+      using prems
+      by (auto simp add: psimpl_novar0 subst_p_novar0)
+    have "?r = T \<or> ?r = F \<or> (?r \<noteq> T \<or> ?r \<noteq> F)" by simp
+    moreover
+    { assume "?r = T" then have ?case by (cases ?qs) auto  }
+    moreover
+    { assume "?r = F"  then have ?case  using nov0qs by (cases ?qs, auto)  }
+    moreover
+    { assume "?r \<noteq> T \<and> ?r \<noteq> F"  then have ?case using nov0qs prems qsnTF
+	by (cases ?qs, auto simp add: Let_def) (cases ?r,auto)+
+    }
+    ultimately have ?case by blast
+  }
+  ultimately show ?case by blast
+qed  
+  
+(* Some simple lemmas used for technical reasons *)
+lemma eval_Or_cases: 
+  "qinterp (a#ats) (case f of 
+       T \<Rightarrow> T
+       | F \<Rightarrow> g
+       | _ \<Rightarrow> (case g of 
+                     T \<Rightarrow> T
+                   | F \<Rightarrow> f
+                   | _ \<Rightarrow> Or f g)) = (qinterp (a#ats) f \<or> qinterp (a#ats) g)"
+proof-
+  let ?result = "
+    (case f of 
+    T \<Rightarrow> T
+    | F \<Rightarrow> g
+    | _ \<Rightarrow> (case g of 
+    T \<Rightarrow> T
+    | F \<Rightarrow> f
+    | _ \<Rightarrow> Or f g))"
+  have "f = T \<or> f = F \<or> (f \<noteq> T \<and> f\<noteq> F)" by auto
+  moreover 
+  {
+    assume fT: "f = T"
+    then have ?thesis by auto
+  }
+  moreover 
+  {
+    assume "f=F"
+    then have ?thesis by auto
+  }
+  moreover 
+  {
+    assume fnT: "f\<noteq>T"
+      and fnF: "f\<noteq>F"
+    have "g = T \<or> g = F \<or> (g \<noteq> T \<and> g\<noteq> F)" by auto
+    moreover 
+    {
+      assume "g=T"
+      then have ?thesis using fnT fnF by (cases f, auto)
+    }
+    moreover 
+    {
+      assume "g=F"
+      then have ?thesis using fnT fnF by (cases f, auto)
+    }
+    moreover 
+    {
+      assume gnT: "g\<noteq>T"
+	and gnF: "g\<noteq>F"
+      then have "?result = (case g of 
+        T \<Rightarrow> T
+        | F \<Rightarrow> f
+        | _ \<Rightarrow> Or f g)"
+	using fnT fnF
+	by (cases f, auto)
+      also have "\<dots> = Or f g"
+	using gnT gnF
+	by (cases g, auto)
+      finally have "?result = Or f g" by simp
+      then
+      have  ?thesis by simp
+    }
+    ultimately have ?thesis by blast
+	   
+  }
+  
+  ultimately show ?thesis by blast
+qed
+
+lemma or_case_novar0:
+  assumes fnTF: "f \<noteq> T \<and> f \<noteq> F"
+  and gnTF: "g \<noteq> T \<and> g \<noteq> F"
+  and f0: "novar0 f"
+  and g0: "novar0 g"
+  shows "novar0 
+     (case f of T \<Rightarrow> T | F \<Rightarrow> g
+     | _ \<Rightarrow> (case g of T \<Rightarrow> T | F \<Rightarrow> f | _ \<Rightarrow> Or f g))"
+using fnTF gnTF f0 g0
+by (cases f, auto) (cases g, auto)+
+
+
+(* An implementation of sets trough lists *)
+constdefs list_insert :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list"
+  "list_insert x xs \<equiv> (if x mem xs then xs else x#xs)"
+
+lemma list_insert_set: "set (list_insert x xs) = set (x#xs)"
+by(induct xs) (auto simp add: list_insert_def)
+
+consts list_union :: "('a list \<times> 'a list) \<Rightarrow> 'a list"
+
+recdef list_union "measure (\<lambda>(xs,ys). length xs)"
+"list_union ([], ys) = ys"
+"list_union (xs, []) = xs"
+"list_union (x#xs,ys) = list_insert x (list_union (xs,ys))"
+
+lemma list_union_set: "set (list_union(xs,ys)) = set (xs@ys)"
+  by(induct xs ys rule: list_union.induct, auto simp add:list_insert_set)
+
+
+consts list_set ::"'a list \<Rightarrow> 'a list"
+primrec 
+  "list_set [] = []"
+  "list_set (x#xs) = list_insert x (list_set xs)"
+
+lemma list_set_set: "set xs = set (list_set xs)"
+by (induct xs) (auto simp add: list_insert_set)
+
+consts iupto :: "int \<times> int \<Rightarrow> int list"
+recdef iupto "measure (\<lambda> (i,j). nat (j - i +1))"
+"iupto(i,j) = (if j<i then [] else (i#(iupto(i+1,j))))"
+
+(* correctness theorem for iupto *)
+lemma iupto_set: "set (iupto(i,j)) = {i .. j}"
+proof(induct rule: iupto.induct)
+  case (1 a b)
+  show ?case
+    using prems by (simp add: simp_from_to)
+qed
+
+consts all_sums :: "int \<times> intterm list \<Rightarrow> intterm list"
+recdef all_sums "measure (\<lambda>(i,is). length is)"
+"all_sums (j,[]) = []"
+"all_sums (j,i#is) = (map (\<lambda>x. lin_add (i,(Cst x))) (iupto(1,j))@(all_sums (j,is)))"
+(* all_sums preserves independence on Var 0*)
+lemma all_sums_novar0:
+  assumes nov0xs: "\<forall> x\<in> set xs. novar0I x"
+  and linxs: "\<forall> x\<in> set xs. islinintterm x "
+  shows "\<forall> x\<in> set (all_sums (d,xs)). novar0I x"
+  using nov0xs linxs
+proof(induct d xs rule: all_sums.induct)
+  case 1 show ?case by simp
+next 
+  case (2 j a as)
+  have lina: "islinintterm a" using "2.prems" by auto
+  have nov0a: "novar0I a" using "2.prems" by auto
+  let ?ys = "map (\<lambda>x. lin_add (a,(Cst x))) (iupto(1,j))"
+  have nov0ys: "\<forall> y\<in> set ?ys. novar0I y"
+  proof-
+    have linx: "\<forall> x \<in> set (iupto(1,j)). islinintterm (Cst x)" by simp
+    have nov0x: "\<forall> x \<in> set (iupto(1,j)). novar0I (Cst x)" by simp
+    with nov0a lina linx have "\<forall> x\<in> set (iupto(1,j)). novar0I (lin_add (a,Cst x))" 
+      by (simp add: lin_add_novar0)
+    then show ?thesis by auto
+  qed
+  from "2.prems"
+  have linas: "\<forall>u\<in>set as. islinintterm u" by auto
+  from "2.prems" have nov0as: "\<forall>u\<in>set as. novar0I u" by auto
+  from "2.hyps" linas nov0as have nov0alls: "\<forall>u\<in>set (all_sums (j, as)). novar0I u" by simp
+  from nov0alls nov0ys have 
+    cs: "(\<forall> u\<in> set (?ys@ (all_sums (j,as))). novar0I u)"
+    by (simp only: sym[OF list_all_iff]) auto
+  
+  have "all_sums(j,a#as) = ?ys@(all_sums(j,as))"
+    by simp
+  then 
+  have "?case = (\<forall> x\<in> set (?ys@ (all_sums (j,as))). novar0I x)"
+    by auto
+  with cs show ?case by blast
+qed
+
+(* correctness theorem for all_sums*)
+lemma all_sums_ex: 
+  "(\<exists> j\<in> {1..d}. \<exists> b\<in> (set xs). P (lin_add(b,Cst j))) = 
+  (\<exists> x\<in> set (all_sums (d,xs)). P x)"
+proof(induct d xs rule: all_sums.induct)
+  case (1 a) show ?case by simp
+next 
+  case (2 a y ys)
+  have "(\<exists> x\<in> set (map (\<lambda>x. lin_add (y,(Cst x))) (iupto(1,a))) . P x) = 
+    (\<exists> j\<in> set (iupto(1,a)). P (lin_add(y,Cst j)))" 
+    by auto
+  also have "\<dots> = (\<exists> j\<in> {1..a}. P (lin_add(y,Cst j)))" 
+    by (simp only : iupto_set)
+  finally
+  have dsj1:"(\<exists>j\<in>{1..a}. P (lin_add (y, Cst j))) = (\<exists>x\<in>set (map (\<lambda>x. lin_add (y, Cst x)) (iupto (1, a))). P x)" by simp
+  
+  from prems have "(\<exists> j\<in> {1..a}. \<exists> b\<in> (set (y#ys)). P (lin_add(b,Cst j))) = 
+    ((\<exists> j\<in> {1..a}. P (lin_add(y,Cst j))) \<or> (\<exists> j\<in> {1..a}. \<exists> b \<in> set ys. P (lin_add(b,Cst j))))" by auto
+  also
+  have " \<dots> = ((\<exists> j\<in> {1..a}. P (lin_add(y,Cst j))) \<or> (\<exists> x\<in> set (all_sums(a, ys)). P x))" using prems by simp
+  also have "\<dots> = ((\<exists>x\<in>set (map (\<lambda>x. lin_add (y, Cst x)) (iupto (1, a))). P x) \<or> (\<exists>x\<in>set (all_sums (a, ys)). P x))" using dsj1 by simp
+  also have "\<dots> = (\<exists> x\<in> (set (map (\<lambda>x. lin_add (y, Cst x)) (iupto (1, a)))) \<union> (set (all_sums(a, ys))). P x)" by blast
+  finally show ?case by simp
+qed
+
+
+
+(* explode_minf (p,B)  assumes that p is unified and B = bset p, it computes the rhs of cooper_mi_eq*)
+
+consts explode_minf :: "(QF \<times> intterm list) \<Rightarrow> QF"
+recdef explode_minf "measure size"
+"explode_minf (q,B) = 
+  (let d = divlcm q;
+       pm = minusinf q;
+        dj1 = explode_disj ((map Cst (iupto (1, d))),pm)
+   in (case dj1 of 
+         T \<Rightarrow> T
+       | F \<Rightarrow> explode_disj (all_sums (d,B),q)
+        | _ \<Rightarrow> (let dj2 = explode_disj (all_sums (d,B),q)
+              in ( case dj2 of 
+                     T \<Rightarrow> T
+                   | F \<Rightarrow> dj1
+                   | _ \<Rightarrow> Or dj1 dj2))))"
+
+(* The result of the rhs of cooper's theorem doese not depend on Var 0*)
+lemma explode_minf_novar0:
+  assumes unifp : "isunified p"
+  and bst: "set (bset p) = set B"
+  shows "novar0 (explode_minf (p,B))"
+proof-
+  let ?d = "divlcm p"
+  let ?pm = "minusinf p"
+  let ?dj1 = "explode_disj (map Cst (iupto(1,?d)),?pm)"
+  
+  have qfpm: "isqfree ?pm"  using unified_islinform[OF unifp] minusinf_qfree by simp
+  have dpos: "?d >0" using unified_islinform[OF unifp] divlcm_pos by simp 
+  have "\<forall> x\<in> set (map Cst (iupto(1,?d))). novar0I x" by auto
+  then have dj1_nov0: "novar0 ?dj1" using qfpm explode_disj_novar0 by simp
+  
+  let ?dj2 = "explode_disj (all_sums (?d,B),p)"
+  have 
+    bstlin: "\<forall>b\<in>set B. islinintterm b"
+    using bset_lin[OF unifp] bst
+    by simp
+  
+  have bstnov0: "\<forall>b\<in>set B. novar0I b"
+    using bst bset_novar0[OF unifp] by simp
+  have allsnov0: "\<forall>x\<in>set (all_sums(?d,B)). novar0I x "
+    by (simp add:all_sums_novar0[OF bstnov0 bstlin] )
+  then have dj2_nov0: "novar0 ?dj2" 
+    using explode_disj_novar0 unified_isqfree[OF unifp] bst by simp
+  have "?dj1 = T \<or> ?dj1 = F \<or> (?dj1 \<noteq> T \<and> ?dj1 \<noteq> F)" by auto
+  moreover
+  { assume "?dj1 = T" then have ?thesis by simp }
+  moreover
+  { assume "?dj1 = F" then have ?thesis using bst dj2_nov0 by (simp add: Let_def)}
+  moreover
+  {
+    assume dj1nFT:"?dj1 \<noteq> T \<and> ?dj1 \<noteq> F"
+    
+    have "?dj2 = T \<or> ?dj2 = F \<or> (?dj2 \<noteq> T \<and> ?dj2 \<noteq> F)" by auto
+    moreover
+    { assume "?dj2 = T" then have ?thesis by (cases ?dj1) simp_all }
+    moreover
+    { assume "?dj2 = F" then have ?thesis using dj1_nov0 bst
+	by (cases ?dj1) (simp_all add: Let_def)}
+    moreover
+    {
+      assume dj2_nTF:"?dj2 \<noteq> T \<and> ?dj2 \<noteq> F"
+      let ?res = "\<lambda>f. \<lambda>g. (case f of T \<Rightarrow> T | F \<Rightarrow> g
+	| _ \<Rightarrow> (case g of T \<Rightarrow> T| F \<Rightarrow> f| _ \<Rightarrow> Or f g))"
+      have expth: "explode_minf (p,B) = ?res ?dj1 ?dj2"
+	by (simp add: Let_def del: iupto.simps split del: split_if
+	  cong del: QF.weak_case_cong)
+      then have ?thesis
+	using prems or_case_novar0 [OF dj1nFT dj2_nTF dj1_nov0 dj2_nov0]
+	by (simp add: Let_def del: iupto.simps cong del: QF.weak_case_cong)
+    }
+    ultimately have ?thesis by blast
+  }
+  ultimately show ?thesis by blast
+qed
+  
+(* explode_minf computes the rhs of cooper's thm*)
+lemma explode_minf_corr:
+  assumes unifp : "isunified p"
+  and bst: "set (bset p) = set B"
+  shows "(\<exists> x . qinterp (x#ats) p) = (qinterp (a#ats) (explode_minf (p,B)))"
+  (is "(\<exists> x. ?P x) = (?EXP a p)")
+proof-
+  let ?d = "divlcm p"
+  let ?pm = "minusinf p"
+  let ?dj1 = "explode_disj (map Cst (iupto(1,?d)),?pm)"
+  have qfpm: "isqfree ?pm"  using unified_islinform[OF unifp] minusinf_qfree by simp 
+  have nnfp: "isnnf p" by (rule unified_isnnf[OF unifp])
+
+  have "(\<exists>j\<in>{1..?d}. qinterp (j # ats) (minusinf p))
+    = (\<exists>j\<in> set (iupto(1,?d)). qinterp (j#ats) (minusinf p))"
+    (is "(\<exists> j\<in> {1..?d}. ?QM j) = \<dots>")
+    by (simp add: sym[OF iupto_set] )
+  also
+  have "\<dots> =(\<exists>j\<in> set (iupto(1,?d)). qinterp ((I_intterm (a#ats) (Cst j))#ats) (minusinf p))"
+    by simp
+  also have 
+    "\<dots> = (\<exists>j\<in> set (map Cst (iupto(1,?d))). qinterp ((I_intterm (a#ats) j)#ats) (minusinf p))" by simp
+  also have 
+    "\<dots> = 
+    (\<exists>j\<in> set (map Cst (iupto(1,?d))). qinterp (a#ats) (subst_p j (minusinf p)))"
+    by (simp add: subst_p_corr[OF qfpm])
+  finally have dj1_thm: 
+    "(\<exists> j\<in> {1..?d}. ?QM j) = (qinterp (a#ats) ?dj1)"
+    by (simp only: explode_disj_corr[OF qfpm])
+  let ?dj2 = "explode_disj (all_sums (?d,B),p)"
+  have 
+    bstlin: "\<forall>b\<in>set B. islinintterm b" 
+    using bst by (simp add: bset_lin[OF unifp])
+  have bstnov0: "\<forall>b\<in>set B. novar0I b" 
+    using bst by (simp add: bset_novar0[OF unifp])
+  have allsnov0: "\<forall>x\<in>set (all_sums(?d,B)). novar0I x "
+    by (simp add:all_sums_novar0[OF bstnov0 bstlin] )
+  have "(\<exists> j\<in> {1..?d}. \<exists> b\<in> set B. ?P (I_intterm (a#ats) b + j)) = 
+   (\<exists> j\<in> {1..?d}. \<exists> b\<in> set B. ?P (I_intterm (a#ats) (lin_add(b,Cst j))))"
+    using bst by (auto simp add: lin_add_corr bset_lin[OF unifp])
+  also have "\<dots> = (\<exists> x \<in> set (all_sums (?d, B)). ?P (I_intterm (a#ats) x))"
+    by (simp add: all_sums_ex[where P="\<lambda> t. ?P (I_intterm (a#ats) t)"])
+  finally 
+  have "(\<exists> j\<in> {1..?d}. \<exists> b\<in> set B. ?P (I_intterm (a#ats) b + j)) = 
+    (\<exists> x \<in> set (all_sums (?d, B)). qinterp (a#ats) (subst_p x p))"
+    using allsnov0 prems linform_isqfree unified_islinform[OF unifp]
+    by (simp add: all_sums_ex subst_p_corr)
+  also have "\<dots> = (qinterp (a#ats) ?dj2)"
+    using linform_isqfree unified_islinform[OF unifp]
+    by (simp add: explode_disj_corr)
+  finally have dj2th: 
+    "(\<exists> j\<in> {1..?d}. \<exists> b\<in> set B. ?P (I_intterm (a#ats) b + j)) =  
+    (qinterp (a#ats) ?dj2)" by simp
+  let ?result = "\<lambda>f. \<lambda>g. 
+    (case f of 
+    T \<Rightarrow> T
+    | F \<Rightarrow> g
+    | _ \<Rightarrow> (case g of 
+    T \<Rightarrow> T
+    | F \<Rightarrow> f
+    | _ \<Rightarrow> Or f g))"
+  have "?EXP a p =  qinterp (a#ats) (?result ?dj1 ?dj2)"
+    by (simp only: explode_minf.simps Let_def)
+  also
+  have "\<dots> = (qinterp (a#ats) ?dj1 \<or> qinterp (a#ats) ?dj2)" 
+    by (rule eval_Or_cases[where f="?dj1" and g="?dj2" and a="a" and ats="ats"])
+  also 
+  have "\<dots> = ((\<exists> j\<in> {1..?d}. ?QM j) \<or> 
+    (\<exists> j\<in> {1..?d}. \<exists> b\<in> set B. ?P (I_intterm (a#ats) b + j)))"
+    by (simp add: dj1_thm dj2th)
+  also
+  have "\<dots> = (\<exists> x. ?P x)"
+    using bst sym[OF cooper_mi_eq[OF unifp]] by simp
+  finally show ?thesis by simp
+qed
+
+
+lemma explode_minf_corr2:
+  assumes unifp : "isunified p"
+  and bst: "set (bset p) = set B"
+  shows "(qinterp ats (QEx p)) = (qinterp ats (decrvars(explode_minf (p,B))))"
+  (is "?P = (?Qe p)")
+proof-
+  have "?P = (\<exists>x. qinterp (x#ats) p)" by simp
+  also have "\<dots>  = (qinterp (a # ats) (explode_minf (p,B)))"
+    using unifp bst explode_minf_corr by simp
+  finally have ex: "?P = (qinterp (a # ats) (explode_minf (p,B)))" .
+  have nv0: "novar0 (explode_minf (p,B))"
+    by (rule explode_minf_novar0[OF unifp])
+  show ?thesis
+    using qinterp_novar0[OF nv0] ex by simp
+qed
+
+(* An implementation of cooper's method for both plus/minus/infinity *)
+
+(* unify the formula *)
+constdefs unify:: "QF \<Rightarrow> (QF \<times> intterm list)"
+  "unify p \<equiv> 
+  (let q = unitycoeff p;
+       B = list_set(bset q);
+       A = list_set (aset q)
+  in
+  if (length B \<le> length A)
+             then (q,B)
+             else (mirror q, map lin_neg A))"
+  
+(* unify behaves like unitycoeff *)
+lemma unify_ex:
+  assumes linp: "islinform p"
+  shows "qinterp ats (QEx p) = qinterp ats (QEx (fst (unify p)))"
+proof-
+  have "length (list_set(bset (unitycoeff p))) \<le> length (list_set (aset (unitycoeff p))) \<or> length (list_set(bset (unitycoeff p))) > length (list_set (aset (unitycoeff p)))" by arith
+  moreover
+  {
+    assume "length (list_set(bset (unitycoeff p))) \<le> length (list_set (aset (unitycoeff p)))"
+    then have "fst (unify p) = unitycoeff p" using unify_def by (simp add: Let_def)
+    then have ?thesis using unitycoeff_corr[OF linp]
+      by simp
+  }
+  moreover 
+  {
+    assume "length (list_set(bset (unitycoeff p))) > length (list_set (aset (unitycoeff p)))"
+    then have unif: "fst(unify p) = mirror (unitycoeff p)"
+      using unify_def by (simp add: Let_def)
+    let ?q ="unitycoeff p"
+    have unifq: "isunified ?q" by(rule unitycoeff_unified[OF linp])
+    have linq: "islinform ?q" by (rule unified_islinform[OF unifq])
+    have "qinterp ats (QEx ?q) = qinterp ats (QEx (mirror ?q))" 
+      by (rule mirror_ex2[OF unifq])
+    moreover have "qinterp ats (QEx p) = qinterp ats (QEx ?q)"
+      using unitycoeff_corr linp by simp
+    ultimately have ?thesis using prems unif by simp
+  }
+  ultimately show ?thesis by blast
+qed
+
+(* unify's result is a unified formula *)
+lemma unify_unified: 
+  assumes linp: "islinform p"
+  shows "isunified (fst (unify p))"
+  using linp unitycoeff_unified mirror_unified unify_def unified_islinform
+  by (auto simp add: Let_def)
+
+
+(* unify preserves quantifier-freeness*)
+lemma unify_qfree:
+  assumes linp: "islinform p"
+  shows "isqfree (fst(unify p))"
+  using linp unify_unified unified_isqfree by simp
+
+lemma unify_bst: 
+  assumes linp: " islinform p" 
+  and unif: "unify p = (q,B)"
+  shows "set B = set (bset q)" 
+proof-
+  let ?q = "unitycoeff p"
+  let ?a = "aset ?q"
+  let ?b = "bset ?q"
+  let ?la = "list_set ?a"
+  let ?lb = "list_set ?b"
+  have " length ?lb \<le> length ?la \<or> length ?lb > length ?la" by arith
+  moreover 
+  {
+    assume "length ?lb \<le> length ?la"
+    then
+    have "unify p = (?q,?lb)"using unify_def prems by (simp add: Let_def)
+    then 
+    have ?thesis using prems by (simp add: sym[OF list_set_set])
+  }
+  moreover
+  {    assume "length ?lb > length ?la"
+    have r: "unify p = (mirror ?q,map lin_neg ?la)"using unify_def prems by (simp add: Let_def)
+    have lin: "\<forall> x\<in> set (bset (mirror ?q)). islinintterm x"
+      using bset_lin mirror_unified unitycoeff_unified[OF linp] by auto
+    with r prems aset_eq_bset_mirror lin_neg_idemp unitycoeff_unified linp
+    have "set B = set (map lin_neg (map lin_neg (bset (mirror (unitycoeff p)))))"
+       by (simp add: sym[OF list_set_set])
+     also have "\<dots> = set (map (\<lambda>x. lin_neg (lin_neg x)) (bset (mirror (unitycoeff p))))"
+       by auto
+     also have "\<dots> = set (bset (mirror (unitycoeff p)))"
+       using lin lin_neg_idemp  by (auto simp add: map_idI)
+     finally
+     have ?thesis using r prems aset_eq_bset_mirror lin_neg_idemp unitycoeff_unified linp
+       by (simp add: sym[OF list_set_set])}
+  ultimately show ?thesis by blast
+qed
+
+lemma explode_minf_unify_novar0: 
+  assumes linp: "islinform p"
+  shows "novar0 (explode_minf (unify p))"
+proof-
+  have "\<exists> q B. unify p = (q,B)" by simp
+  then obtain "q" "B" where qB_def: "unify p = (q,B)" by blast
+  have unifq: "isunified q" using unify_unified[OF linp] qB_def by simp
+  have bst: "set B = set (bset q)" using unify_bst linp qB_def by simp
+  from unifq bst explode_minf_novar0 show ?thesis
+    using qB_def by simp
+qed
+
+lemma explode_minf_unify_corr2:
+  assumes linp: "islinform p"
+  shows "qinterp ats (QEx p) = qinterp ats (decrvars(explode_minf(unify p)))"
+proof-
+  have "\<exists> q B. unify p = (q,B)" by simp
+  then obtain "q" "B" where qB_def: "unify p = (q,B)" by blast
+  have unifq: "isunified q" using unify_unified[OF linp] qB_def by simp
+  have bst: "set (bset q) = set B" using unify_bst linp qB_def by simp
+  from explode_minf_corr2[OF unifq bst] unify_ex[OF linp] show ?thesis
+    using qB_def by simp
+qed
+(* An implementation of cooper's method *)
+constdefs cooper:: "QF \<Rightarrow> QF option"
+"cooper p \<equiv> lift_un (\<lambda>q. decrvars(explode_minf (unify q))) (linform (nnf p))"
+
+(* cooper eliminates quantifiers *)
+lemma cooper_qfree: "(\<And> q q'. \<lbrakk>isqfree q ; cooper q = Some q'\<rbrakk> \<Longrightarrow>  isqfree q')"
+proof-
+  fix "q" "q'"
+  assume qfq: "isqfree q"
+    and qeq: "cooper q = Some q'"
+  from qeq have "\<exists>p. linform (nnf q) = Some p"
+    by (cases "linform (nnf q)") (simp_all add: cooper_def)
+  then obtain "p" where p_def: "linform (nnf q) = Some p" by blast
+  have linp: "islinform p" using p_def linform_lin nnf_isnnf qfq 
+    by auto
+  have nnfq: "isnnf (nnf q)" using nnf_isnnf qfq by simp
+  then have nnfp: "isnnf p" using linform_nnf[OF nnfq] p_def by auto
+  have qfp: "isqfree p" using linp linform_isqfree by simp 
+  have "cooper q = Some (decrvars(explode_minf (unify p)))" using p_def 
+    by (simp add: cooper_def del: explode_minf.simps)
+  then have "q' = decrvars (explode_minf (unify p))" using qeq by simp
+  with linp qfp nnfp  unify_unified unify_qfree unified_islinform 
+  show "isqfree q'"
+    using novar0_qfree explode_minf_unify_novar0 decrvars_qfree
+    by simp
+qed
+
+(* cooper preserves semantics *)
+lemma cooper_corr: "(\<And> q q' ats. \<lbrakk>isqfree q ; cooper q = Some q'\<rbrakk> \<Longrightarrow>  (qinterp ats (QEx q)) = (qinterp ats q'))"  (is "\<And> q q' ats. \<lbrakk> _ ; _ \<rbrakk> \<Longrightarrow> (?P ats (QEx q) = ?P ats q')")
+proof-
+  fix "q" "q'" "ats"
+  assume qfq: "isqfree q"
+    and qeq: "cooper q = Some q'"
+  from qeq have "\<exists>p. linform (nnf q) = Some p"
+    by (cases "linform (nnf q)") (simp_all add: cooper_def)
+  then obtain "p" where p_def: "linform (nnf q) = Some p" by blast
+  have linp: "islinform p" using p_def linform_lin nnf_isnnf qfq by auto
+  have qfp: "isqfree p" using linp linform_isqfree by simp 
+  have nnfq: "isnnf (nnf q)" using nnf_isnnf qfq by simp
+  then have nnfp: "isnnf p" using linform_nnf[OF nnfq] p_def by auto
+  have "\<forall> ats. ?P ats q = ?P ats (nnf q)" using nnf_corr qfq by auto
+  then have qeqp: "\<forall> ats. ?P ats q = ?P ats p"
+    using linform_corr p_def nnf_isnnf qfq
+    by auto
+
+  have "cooper q = Some (decrvars (explode_minf (unify p)))" using p_def 
+    by (simp add: cooper_def del: explode_minf.simps)
+  then have decr: "q' = decrvars(explode_minf (unify p))" using qeq by simp
+  have eqq:"?P ats (QEx q) = ?P ats (QEx p)" using qeqp by auto
+  with decr explode_minf_unify_corr2 unified_islinform unify_unified linp 
+  show "?P ats (QEx q) = ?P ats q'" by simp
+qed  
+
+(* A decision procedure for Presburger Arithmetics *)
+constdefs pa:: "QF \<Rightarrow> QF option"
+"pa p \<equiv> lift_un psimpl (qelim(cooper, p))"
+
+lemma psimpl_qfree: "isqfree p \<Longrightarrow> isqfree (psimpl p)"
+apply(induct p rule: isqfree.induct)
+apply(auto simp add: Let_def measure_def inv_image_def)
+apply (simp_all cong del: QF.weak_case_cong add: Let_def)
+apply (case_tac "psimpl p", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl p", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl p", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl p", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+apply (case_tac "psimpl q", auto)
+
+apply (case_tac "psimpl p", auto)
+apply (case_tac "lift_bin (\<lambda>x y. lin_add (x, lin_neg y), linearize y,
+                   linearize z)", auto)
+apply (case_tac "a",auto)
+apply (case_tac "lift_bin (\<lambda>x y. lin_add (x, lin_neg y), linearize ac,
+                   linearize ad)", auto)
+apply (case_tac "a",auto)
+apply (case_tac "ae", auto)
+apply (case_tac "linearize af", auto)
+by (case_tac "a", auto)
+
+(* pa eliminates quantifiers *)
+theorem pa_qfree: "\<And> p'. pa p = Some p' \<Longrightarrow> isqfree p'"
+proof(simp only: pa_def)
+fix "p'"
+assume qep: "lift_un psimpl (qelim (cooper, p)) = Some p'"
+then have "\<exists> q. qelim (cooper, p) = Some q"
+  by (cases "qelim(cooper, p)") auto
+then obtain "q" where q_def: "qelim (cooper, p) = Some q" by blast
+have "\<And>q q'. \<lbrakk>isqfree q; cooper q = Some q'\<rbrakk> \<Longrightarrow> isqfree q'" using cooper_qfree by blast
+with q_def
+have "isqfree q" using qelim_qfree by blast
+then have "isqfree (psimpl q)" using psimpl_qfree
+  by auto
+then show "isqfree p'"
+  using prems 
+  by simp
+
+qed
+
+(* pa preserves semantics *)
+theorem pa_corr: 
+  "\<And> p'. pa p = Some p' \<Longrightarrow> (qinterp ats p = qinterp ats p')"
+proof(simp only: pa_def)
+  fix "p'"
+  assume qep: "lift_un psimpl (qelim(cooper, p)) = Some p'"
+ then have "\<exists> q. qelim (cooper, p) = Some q"
+  by (cases "qelim(cooper, p)") auto
+then obtain "q" where q_def: "qelim (cooper, p) = Some q" by blast 
+  have cp1:"\<And>q q' ats. 
+    \<lbrakk>isqfree q; cooper q = Some q'\<rbrakk> \<Longrightarrow> qinterp ats (QEx q) = qinterp ats q'"
+    using cooper_corr by blast
+  moreover have cp2: "\<And>q q'. \<lbrakk>isqfree q; cooper q = Some q'\<rbrakk> \<Longrightarrow> isqfree q'"
+    using cooper_qfree by blast
+  ultimately have "qinterp ats p = qinterp ats q" using qelim_corr qep psimpl_corr q_def
+    by blast
+  then have "qinterp ats p = qinterp ats (psimpl q)" using psimpl_corr q_def
+    by auto
+  then show "qinterp ats p = qinterp ats p'" using prems 
+    by simp
+qed
+
+lemma [code]: "linearize (Mult i j) = 
+  (case linearize i of
+  None \<Rightarrow> None
+  | Some li \<Rightarrow> (case li of 
+     Cst b \<Rightarrow> (case linearize j of
+      None \<Rightarrow> None
+     | (Some lj) \<Rightarrow> Some (lin_mul(b,lj)))
+  | _ \<Rightarrow> (case linearize j of
+      None \<Rightarrow> None
+    | (Some lj) \<Rightarrow> (case lj of 
+        Cst b \<Rightarrow> Some (lin_mul (b,li))
+      | _ \<Rightarrow> None))))"
+by (simp add: measure_def inv_image_def)
+
+lemma [code]: "psimpl (And p q) = 
+  (let p'= psimpl p
+  in (case p' of 
+       F \<Rightarrow> F
+      |T \<Rightarrow> psimpl q
+      | _ \<Rightarrow> let q' = psimpl q
+             in (case q' of
+                     F \<Rightarrow> F
+                   | T \<Rightarrow> p'
+                   | _ \<Rightarrow> (And p' q'))))"
+
+by (simp add: measure_def inv_image_def)
+
+lemma [code]: "psimpl (Or p q) = 
+  (let p'= psimpl p
+  in (case p' of 
+        T \<Rightarrow> T
+      | F \<Rightarrow> psimpl q
+      | _ \<Rightarrow> let q' = psimpl q
+             in (case q' of
+                     T \<Rightarrow> T
+                   | F \<Rightarrow> p'
+                   | _ \<Rightarrow> (Or p' q'))))"
+
+by (simp add: measure_def inv_image_def)
+
+lemma [code]: "psimpl (Imp p q) = 
+  (let p'= psimpl p
+  in (case p' of 
+       F \<Rightarrow> T
+      |T \<Rightarrow> psimpl q
+      | NOT p1 \<Rightarrow> let q' = psimpl q
+             in (case q' of
+                     F \<Rightarrow> p1
+                   | T \<Rightarrow> T
+                   | _ \<Rightarrow> (Or p1 q'))
+      | _ \<Rightarrow> let q' = psimpl q
+             in (case q' of
+                     F \<Rightarrow> NOT p'
+                   | T \<Rightarrow> T
+                   | _ \<Rightarrow> (Imp p' q'))))"
+by (simp add: measure_def inv_image_def)
+
+declare zdvd_iff_zmod_eq_0 [code]
+
+(*
+generate_code ("presburger.ML") test = "pa"
+*)
+use "rcooper.ML"
+oracle rpresburger_oracle ("term") = RCooper.rpresburger_oracle
+use "rpresbtac.ML"
+setup RPresburger.setup
+
+end