replaced code generator framework for reflected cooper

--- a/src/HOL/Tools/Qelim/cooper.ML	Tue Jul 10 09:23:14 2007 +0200
+++ b/src/HOL/Tools/Qelim/cooper.ML	Tue Jul 10 09:23:15 2007 +0200
@@ -11,11 +11,13 @@
 
 structure Cooper: COOPER =
 struct
+
 open Conv;
+open Normalizer;
 structure Integertab = TableFun(type key = Integer.int val ord = Integer.ord);
+
 exception COOPER of string * exn;
 val simp_thms_conv = Simplifier.rewrite (HOL_basic_ss addsimps simp_thms);
-
 val FWD = Drule.implies_elim_list;
 
 val true_tm = @{cterm "True"};
@@ -23,7 +25,7 @@
 val zdvd1_eq = @{thm "zdvd1_eq"};
 val presburger_ss = @{simpset} addsimps [zdvd1_eq];
 val lin_ss = presburger_ss addsimps (@{thm "dvd_eq_mod_eq_0"}::zdvd1_eq::@{thms zadd_ac});
-(* Some types and constants *)
+
 val iT = HOLogic.intT
 val bT = HOLogic.boolT;
 val dest_numeral = HOLogic.dest_number #> snd;
@@ -59,7 +61,7 @@
 val eval_ss = presburger_ss addsimps [simp_from_to] delsimps [insert_iff,bex_triv];
 val eval_conv = Simplifier.rewrite eval_ss;
 
-(* recongnising cterm without moving to terms *)
+(* recognising cterm without moving to terms *)
 
 datatype fm = And of cterm*cterm| Or of cterm*cterm| Eq of cterm | NEq of cterm 
             | Lt of cterm | Le of cterm | Gt of cterm | Ge of cterm
@@ -78,9 +80,9 @@
 | Const ("Orderings.ord_class.less_eq",_)$y$z => 
    if term_of x aconv y then Le (Thm.dest_arg ct) 
    else if term_of x aconv z then Ge (Thm.dest_arg1 ct) else Nox
-| Const ("Divides.dvd",_)$_$(Const(@{const_name "HOL.plus"},_)$y$_) =>
+| Const (@{const_name Divides.dvd},_)$_$(Const(@{const_name "HOL.plus"},_)$y$_) =>
    if term_of x aconv y then Dvd (Thm.dest_binop ct ||> Thm.dest_arg) else Nox 
-| Const("Not",_) $ (Const ("Divides.dvd",_)$_$(Const(@{const_name "HOL.plus"},_)$y$_)) =>
+| Const("Not",_) $ (Const (@{const_name Divides.dvd},_)$_$(Const(@{const_name "HOL.plus"},_)$y$_)) =>
    if term_of x aconv y then 
    NDvd (Thm.dest_binop (Thm.dest_arg ct) ||> Thm.dest_arg) else Nox 
 | _ => Nox)
@@ -117,10 +119,10 @@
 val cadd =  @{cterm "op + :: int => _"}
 val cmulC =  @{cterm "op * :: int => _"}
 val cminus =  @{cterm "op - :: int => _"}
-val cone =  @{cterm "1:: int"}
+val cone =  @{cterm "1 :: int"}
 val cneg = @{cterm "uminus :: int => _"}
 val [addC, mulC, subC, negC] = map term_of [cadd, cmulC, cminus, cneg]
-val [zero, one] = [@{term "0::int"}, @{term "1::int"}];
+val [zero, one] = [@{term "0 :: int"}, @{term "1 :: int"}];
 
 val is_numeral = can dest_numeral; 
 
@@ -225,8 +227,8 @@
   | lin (vs as x::_) (Const("Not",_)$(Const("Orderings.ord_class.less_eq",T)$s$t)) = 
     lin vs (Const("Orderings.ord_class.less",T)$t$s)
   | lin vs (Const ("Not",T)$t) = Const ("Not",T)$ (lin vs t)
-  | lin (vs as x::_) (Const("Divides.dvd",_)$d$t) = 
-    HOLogic.mk_binrel "Divides.dvd" (numeral1 abs d, lint vs t)
+  | lin (vs as x::_) (Const(@{const_name Divides.dvd},_)$d$t) = 
+    HOLogic.mk_binrel @{const_name Divides.dvd} (numeral1 abs d, lint vs t)
   | lin (vs as x::_) ((b as Const("op =",_))$s$t) = 
      (case lint vs (subC$t$s) of 
       (t as a$(m$c$y)$r) => 
@@ -255,7 +257,7 @@
  
 fun linearize_conv ctxt vs ct =  
  case (term_of ct) of 
-  Const("Divides.dvd",_)$d$t => 
+  Const(@{const_name Divides.dvd},_)$d$t => 
   let 
     val th = binop_conv (lint_conv ctxt vs) ct
     val (d',t') = Thm.dest_binop (Thm.rhs_of th)
@@ -280,7 +282,7 @@
       | _ => dth
      end
   end
-| Const("Not",_)$(Const("Divides.dvd",_)$_$_) => arg_conv (linearize_conv ctxt vs) ct
+| Const("Not",_)$(Const(@{const_name Divides.dvd},_)$_$_) => arg_conv (linearize_conv ctxt vs) ct
 | t => if is_intrel t 
       then (provelin ctxt ((HOLogic.eq_const bT)$t$(lin vs t) |> HOLogic.mk_Trueprop))
        RS eq_reflection
@@ -303,7 +305,7 @@
     if x aconv y 
        andalso s mem ["Orderings.ord_class.less", "Orderings.ord_class.less_eq"] 
     then (ins (dest_numeral c) acc, dacc) else (acc,dacc)
-  | Const("Divides.dvd",_)$_$(Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c$y)$_) => 
+  | Const(@{const_name Divides.dvd},_)$_$(Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c$y)$_) => 
     if x aconv y then (acc,ins (dest_numeral c) dacc) else (acc,dacc)
   | Const("op &",_)$_$_ => h (h (acc,dacc) (Thm.dest_arg1 t)) (Thm.dest_arg t)
   | Const("op |",_)$_$_ => h (h (acc,dacc) (Thm.dest_arg1 t)) (Thm.dest_arg t)
@@ -320,8 +322,8 @@
     val th = 
      Simplifier.rewrite lin_ss 
       (Thm.capply @{cterm Trueprop} (Thm.capply @{cterm "Not"} 
-         (Thm.capply (Thm.capply @{cterm "op = :: int => _"} (Numeral.mk_cnumber @{ctyp "int"} x)) 
-         @{cterm "0::int"})))
+           (Thm.capply (Thm.capply @{cterm "op = :: int => _"} (Numeral.mk_cnumber @{ctyp "int"} x)) 
+           @{cterm "0::int"})))
    in equal_elim (Thm.symmetric th) TrueI end;
   val notz = let val tab = fold Integertab.update 
                                (ds ~~ (map (fn x => nzprop (Integer.div l x)) ds)) Integertab.empty 
@@ -341,7 +343,7 @@
   | Const(s,_)$_$(Const("HOL.times_class.times",_)$c$y) => 
     if x=y andalso s mem ["Orderings.ord_class.less", "Orderings.ord_class.less_eq"] 
     then cv (Integer.div l (dest_numeral c)) t else Thm.reflexive t
-  | Const("Divides.dvd",_)$d$(r as (Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c$y)$_)) => 
+  | Const(@{const_name Divides.dvd},_)$d$(r as (Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c$y)$_)) => 
     if x=y then 
       let 
        val k = Integer.div l (dest_numeral c)
@@ -357,7 +359,7 @@
     else Thm.reflexive t
   | _ => Thm.reflexive t
   val uth = unit_conv p
-  val clt = Numeral.mk_cnumber @{ctyp "int"} l
+  val clt =  Numeral.mk_cnumber @{ctyp "int"} l
   val ltx = Thm.capply (Thm.capply cmulC clt) cx
   val th = Drule.arg_cong_rule e (Thm.abstract_rule (fst (dest_Free x )) cx uth)
   val th' = inst' [Thm.cabs ltx (Thm.rhs_of uth), clt] unity_coeff_ex
@@ -508,7 +510,7 @@
 fun integer_nnf_conv ctxt env =
  nnf_conv then_conv literals_conv [HOLogic.conj, HOLogic.disj] [] env (linearize_conv ctxt);
 
-(* val my_term = ref (@{cterm "NOTHING"}); *)
+(* val my_term = ref (@{cterm "NotaHING"}); *)
 local
  val pcv = Simplifier.rewrite 
      (HOL_basic_ss addsimps (simp_thms @ (List.take(ex_simps,4)) 
@@ -533,7 +535,8 @@
 structure Coopereif =
 struct
 
-open GeneratedCooper;
+open GeneratedCooper.Reflected_Presburger;
+
 fun cooper s = raise Cooper.COOPER ("Cooper Oracle Failed", ERROR s);
 fun i_of_term vs t = 
     case t of
@@ -560,22 +563,22 @@
       | Const("False",_) => F
       | Const(@{const_name "Orderings.less"},_)$t1$t2 => Lt (Sub (i_of_term vs t1,i_of_term vs t2))
       | Const(@{const_name "Orderings.less_eq"},_)$t1$t2 => Le (Sub(i_of_term vs t1,i_of_term vs t2))
-      | Const(@{const_name "Divides.dvd"},_)$t1$t2 => 
+      | Const(@{const_name Divides.dvd},_)$t1$t2 => 
 	(Dvd(HOLogic.dest_number t1 |> snd |> Integer.machine_int, i_of_term vs t2) handle _ => cooper "Reification: unsupported dvd")
       | @{term "op = :: int => _"}$t1$t2 => Eq (Sub (i_of_term vs t1,i_of_term vs t2))
-      | @{term "op = :: bool => _ "}$t1$t2 => Iff(qf_of_term ps vs t1,qf_of_term ps vs t2)
+      | @{term "op = :: bool => _ "}$t1$t2 => Iffa(qf_of_term ps vs t1,qf_of_term ps vs t2)
       | Const("op &",_)$t1$t2 => And(qf_of_term ps vs t1,qf_of_term ps vs t2)
       | Const("op |",_)$t1$t2 => Or(qf_of_term ps vs t1,qf_of_term ps vs t2)
-      | Const("op -->",_)$t1$t2 => Imp(qf_of_term ps vs t1,qf_of_term ps vs t2)
-      | Const("Not",_)$t' => NOT(qf_of_term ps vs t')
+      | Const("op -->",_)$t1$t2 => Impa(qf_of_term ps vs t1,qf_of_term ps vs t2)
+      | Const("Not",_)$t' => Nota(qf_of_term ps vs t')
       | Const("Ex",_)$Abs(xn,xT,p) => 
          let val (xn',p') = variant_abs (xn,xT,p)
-             val vs' = (Free (xn',xT), nat 0) :: (map (fn(v,n) => (v,1+ n)) vs)
+             val vs' = (Free (xn',xT), 0) :: (map (fn(v,n) => (v,1+ n)) vs)
          in E (qf_of_term ps vs' p')
          end
       | Const("All",_)$Abs(xn,xT,p) => 
          let val (xn',p') = variant_abs (xn,xT,p)
-             val vs' = (Free (xn',xT), nat 0) :: (map (fn(v,n) => (v,1+ n)) vs)
+             val vs' = (Free (xn',xT), 0) :: (map (fn(v,n) => (v,1+ n)) vs)
          in A (qf_of_term ps vs' p')
          end
       | _ =>(case AList.lookup (op aconv) ps t of 
@@ -612,7 +615,7 @@
 fun myassoc2 l v =
     case l of
 	[] => NONE
-      | (x,v': int)::xs => if v = v' then SOME x
+      | (x,v')::xs => if v = v' then SOME x
 		      else myassoc2 xs v;
 
 fun term_of_i vs t =
@@ -625,7 +628,7 @@
 			   (term_of_i vs t1)$(term_of_i vs t2)
       | Mul(i,t2) => Const(@{const_name "HOL.times"},[iT,iT] ---> iT)$
 			   (HOLogic.mk_number HOLogic.intT (Integer.int i))$(term_of_i vs t2)
-      | CX(i,t')=> term_of_i vs (Add(Mul (i,Bound (nat 0)),t'));
+      | Cx(i,t')=> term_of_i vs (Add(Mul (i, Bound 0),t'));
 
 fun term_of_qf ps vs t = 
  case t of 
@@ -636,17 +639,17 @@
  | Gt t' => @{term "op < :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
  | Ge t' => @{term "op <= :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
  | Eq t' => @{term "op = :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
- | NEq t' => term_of_qf ps vs (NOT(Eq t'))
+ | NEq t' => term_of_qf ps vs (Nota(Eq t'))
  | Dvd(i,t') => @{term "op dvd :: int => _ "}$ 
                  (HOLogic.mk_number HOLogic.intT (Integer.int i))$(term_of_i vs t')
- | NDvd(i,t')=> term_of_qf ps vs (NOT(Dvd(i,t')))
- | NOT t' => HOLogic.Not$(term_of_qf ps vs t')
+ | NDvd(i,t')=> term_of_qf ps vs (Nota(Dvd(i,t')))
+ | Nota t' => HOLogic.Not$(term_of_qf ps vs t')
  | And(t1,t2) => HOLogic.conj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
  | Or(t1,t2) => HOLogic.disj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
- | Imp(t1,t2) => HOLogic.imp$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
- | Iff(t1,t2) => (HOLogic.eq_const bT)$(term_of_qf ps vs t1)$ (term_of_qf ps vs t2)
+ | Impa(t1,t2) => HOLogic.imp$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
+ | Iffa(t1,t2) => (HOLogic.eq_const bT)$(term_of_qf ps vs t1)$ (term_of_qf ps vs t2)
  | Closed n => valOf (myassoc2 ps n)
- | NClosed n => term_of_qf ps vs (NOT (Closed n))
+ | NClosed n => term_of_qf ps vs (Nota (Closed n))
  | _ => cooper "If this is raised, Isabelle/HOL or generate_code is inconsistent!";
 
 (* The oracle *)

--- a/src/HOL/Tools/Qelim/generated_cooper.ML	Tue Jul 10 09:23:14 2007 +0200
+++ b/src/HOL/Tools/Qelim/generated_cooper.ML	Tue Jul 10 09:23:15 2007 +0200
@@ -1,1693 +1,2244 @@
-structure GeneratedCooper =
+(*  Title:      HOL/Tools/Presburger/generated_cooper.ML
+    ID:         $Id$
+
+This file is generated from HOL/ex/Reflected_Presburger.thy.  DO NOT EDIT.
+*)
+
+structure GeneratedCooper = 
 struct
-nonfix oo;
-fun nat i = if i < 0 then 0 else i;
 
-val one_def0 : int = (0 + 1);
+structure Product_Type = 
+struct
+
+fun fst (y, b) = y;
 
-datatype num = C of int | Bound of int | CX of int * num | Neg of num
-  | Add of num * num | Sub of num * num | Mul of int * num;
+fun snd (a, y) = y;
+
+end; (*struct Product_Type*)
+
+structure Integer = 
+struct
 
-fun snd (a, b) = b;
+datatype bit = B0 | B1;
+
+fun suc n = (IntInf.+ (n, (1 : IntInf.int)));
 
-fun negateSnd x = (fn (q, r) => (q, ~ r)) x;
+val zero_nat : IntInf.int = (0 : IntInf.int);
 
-fun minus_def2 z w = (z + ~ w);
+fun nat k = (if IntInf.< (k, (0 : IntInf.int)) then zero_nat else k);
 
 fun adjust b =
-  (fn (q, r) =>
-    (if (0 <= minus_def2 r b) then (((2 * q) + 1), minus_def2 r b)
-      else ((2 * q), r)));
+  (fn a as (q, r) =>
+    (if IntInf.<= ((0 : IntInf.int), IntInf.- (r, b))
+      then (IntInf.+ (IntInf.* ((2 : IntInf.int), q), (1 : IntInf.int)),
+             IntInf.- (r, b))
+      else (IntInf.* ((2 : IntInf.int), q), r)));
 
 fun negDivAlg a b =
-    (if ((0 <= (a + b)) orelse (b <= 0)) then (~1, (a + b))
-      else adjust b (negDivAlg a (2 * b)));
+  (if IntInf.<= ((0 : IntInf.int), IntInf.+ (a, b)) orelse
+        IntInf.<= (b, (0 : IntInf.int))
+    then ((~1 : IntInf.int), IntInf.+ (a, b))
+    else adjust b (negDivAlg a (IntInf.* ((2 : IntInf.int), b))));
+
+val negateSnd : IntInf.int * IntInf.int -> IntInf.int * IntInf.int =
+  (fn a as (q, r) => (q, IntInf.~ r));
 
 fun posDivAlg a b =
-    (if ((a < b) orelse (b <= 0)) then (0, a)
-      else adjust b (posDivAlg a (2 * b)));
+  (if IntInf.< (a, b) orelse IntInf.<= (b, (0 : IntInf.int))
+    then ((0 : IntInf.int), a)
+    else adjust b (posDivAlg a (IntInf.* ((2 : IntInf.int), b))));
+
+val divAlg : IntInf.int * IntInf.int -> IntInf.int * IntInf.int =
+  (fn a as (aa, b) =>
+    (if IntInf.<= ((0 : IntInf.int), aa)
+      then (if IntInf.<= ((0 : IntInf.int), b) then posDivAlg aa b
+             else (if ((aa : IntInf.int) = (0 : IntInf.int))
+                    then ((0 : IntInf.int), (0 : IntInf.int))
+                    else negateSnd (negDivAlg (IntInf.~ aa) (IntInf.~ b))))
+      else (if IntInf.< ((0 : IntInf.int), b) then negDivAlg aa b
+             else negateSnd (posDivAlg (IntInf.~ aa) (IntInf.~ b)))));
+
+fun abs_int i = (if IntInf.< (i, (0 : IntInf.int)) then IntInf.~ i else i);
+
+fun div_int a b = Product_Type.fst (divAlg (a, b));
+
+fun mod_int a b = Product_Type.snd (divAlg (a, b));
+
+fun dvd_int m n = (((mod_int n m) : IntInf.int) = (0 : IntInf.int));
+
+fun eq_bit B0 B0 = true
+  | eq_bit B1 B1 = true
+  | eq_bit B0 B1 = false
+  | eq_bit B1 B0 = false;
+
+fun int_aux i n =
+  (if ((n : IntInf.int) = (0 : IntInf.int)) then i
+    else int_aux (IntInf.+ (i, (1 : IntInf.int)))
+           (IntInf.- (n, (1 : IntInf.int))));
+
+end; (*struct Integer*)
+
+structure Nat = 
+struct
+
+fun div_nat m k = (Product_Type.fst (Integer.divAlg (m, k)));
+
+fun mod_nat m k = (Product_Type.snd (Integer.divAlg (m, k)));
+
+end; (*struct Nat*)
+
+structure GCD = 
+struct
+
+fun gcd (m, n) =
+  (if ((n : IntInf.int) = Integer.zero_nat) then m
+    else gcd (n, Nat.mod_nat m n));
+
+val lcm : IntInf.int * IntInf.int -> IntInf.int =
+  (fn a as (m, n) => Nat.div_nat (IntInf.* (m, n)) (gcd (m, n)));
+
+val ilcm : IntInf.int -> IntInf.int -> IntInf.int =
+  (fn i => fn j =>
+    Integer.int_aux (0 : IntInf.int)
+      (lcm (Integer.nat (Integer.abs_int i), Integer.nat (Integer.abs_int j))));
+
+end; (*struct GCD*)
+
+structure HOL = 
+struct
+
+type 'a eq = {eq : 'a -> 'a -> bool};
+fun eq (A_:'a eq) = #eq A_;
+
+end; (*struct HOL*)
+
+structure List = 
+struct
+
+fun map f (x :: xs) = f x :: map f xs
+  | map f [] = [];
+
+fun foldr f (x :: xs) a = f x (foldr f xs a)
+  | foldr f [] y = y;
+
+fun append (x :: xs) ys = x :: append xs ys
+  | append [] y = y;
+
+fun memberl A_ x (y :: ys) = HOL.eq A_ x y orelse memberl A_ x ys
+  | memberl A_ x [] = false;
+
+fun remdups A_ (x :: xs) =
+  (if memberl A_ x xs then remdups A_ xs else x :: remdups A_ xs)
+  | remdups A_ [] = [];
+
+fun allpairs f (x :: xs) ys = append (map (f x) ys) (allpairs f xs ys)
+  | allpairs f [] ys = [];
 
-fun divAlg x =
-  (fn (a, 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 size_list (a :: lista) =
+  (IntInf.+ ((size_list lista), (Integer.suc Integer.zero_nat)))
+  | size_list [] = Integer.zero_nat;
+
+end; (*struct List*)
+
+structure Reflected_Presburger = 
+struct
+
+datatype num = C of IntInf.int | Bound of IntInf.int | Cx of IntInf.int * num |
+  Neg of num | Add of num * num | Sub of num * num | Mul of IntInf.int * num;
+
+datatype fm = T | F | Lt of num | Le of num | Gt of num | Ge of num | Eq of num
+  | NEq of num | Dvd of IntInf.int * num | NDvd of IntInf.int * num | Nota of fm
+  | And of fm * fm | Or of fm * fm | Impa of fm * fm | Iffa of fm * fm | E of fm
+  | A of fm | Closed of IntInf.int | NClosed of IntInf.int;
+
+fun disjuncts (NClosed aq) = [NClosed aq]
+  | disjuncts (Closed ap) = [Closed ap]
+  | disjuncts (A ao) = [A ao]
+  | disjuncts (E an) = [E an]
+  | disjuncts (Iffa (al, am)) = [Iffa (al, am)]
+  | disjuncts (Impa (aj, ak)) = [Impa (aj, ak)]
+  | disjuncts (And (af, ag)) = [And (af, ag)]
+  | disjuncts (Nota ae) = [Nota ae]
+  | disjuncts (NDvd (ac, ad)) = [NDvd (ac, ad)]
+  | disjuncts (Dvd (aa, ab)) = [Dvd (aa, ab)]
+  | disjuncts (NEq z) = [NEq z]
+  | disjuncts (Eq y) = [Eq y]
+  | disjuncts (Ge x) = [Ge x]
+  | disjuncts (Gt w) = [Gt w]
+  | disjuncts (Le v) = [Le v]
+  | disjuncts (Lt u) = [Lt u]
+  | disjuncts T = [T]
+  | disjuncts F = []
+  | disjuncts (Or (p, q)) = List.append (disjuncts p) (disjuncts q);
 
-fun mod_def1 a b = snd (divAlg (a, b));
-
-fun dvd m n = (mod_def1 n m = 0);
-
-fun abs i = (if (i < 0) then ~ i else i);
-
-fun less_def3 m n = ((m) < (n));
-
-fun less_eq_def3 m n = Bool.not (less_def3 n m);
+fun eq_num (C int) (C int') = ((int : IntInf.int) = int')
+  | eq_num (Bound nat) (Bound nat') = ((nat : IntInf.int) = nat')
+  | eq_num (Cx (int, num)) (Cx (int', num')) =
+    ((int : IntInf.int) = int') andalso eq_num num num'
+  | eq_num (Neg num) (Neg num') = eq_num num num'
+  | eq_num (Add (num1, num2)) (Add (num1', num2')) =
+    eq_num num1 num1' andalso eq_num num2 num2'
+  | eq_num (Sub (num1, num2)) (Sub (num1', num2')) =
+    eq_num num1 num1' andalso eq_num num2 num2'
+  | eq_num (Mul (int, num)) (Mul (int', num')) =
+    ((int : IntInf.int) = int') andalso eq_num num num'
+  | eq_num (C a) (Bound b) = false
+  | eq_num (C a) (Cx (b, c)) = false
+  | eq_num (C a) (Neg b) = false
+  | eq_num (C a) (Add (b, c)) = false
+  | eq_num (C a) (Sub (b, c)) = false
+  | eq_num (C a) (Mul (b, c)) = false
+  | eq_num (Bound a) (Cx (b, c)) = false
+  | eq_num (Bound a) (Neg b) = false
+  | eq_num (Bound a) (Add (b, c)) = false
+  | eq_num (Bound a) (Sub (b, c)) = false
+  | eq_num (Bound a) (Mul (b, c)) = false
+  | eq_num (Cx (a, b)) (Neg c) = false
+  | eq_num (Cx (a, b)) (Add (c, d)) = false
+  | eq_num (Cx (a, b)) (Sub (c, d)) = false
+  | eq_num (Cx (a, b)) (Mul (c, d)) = false
+  | eq_num (Neg a) (Add (b, c)) = false
+  | eq_num (Neg a) (Sub (b, c)) = false
+  | eq_num (Neg a) (Mul (b, c)) = false
+  | eq_num (Add (a, b)) (Sub (c, d)) = false
+  | eq_num (Add (a, b)) (Mul (c, d)) = false
+  | eq_num (Sub (a, b)) (Mul (c, d)) = false
+  | eq_num (Bound b) (C a) = false
+  | eq_num (Cx (b, c)) (C a) = false
+  | eq_num (Neg b) (C a) = false
+  | eq_num (Add (b, c)) (C a) = false
+  | eq_num (Sub (b, c)) (C a) = false
+  | eq_num (Mul (b, c)) (C a) = false
+  | eq_num (Cx (b, c)) (Bound a) = false
+  | eq_num (Neg b) (Bound a) = false
+  | eq_num (Add (b, c)) (Bound a) = false
+  | eq_num (Sub (b, c)) (Bound a) = false
+  | eq_num (Mul (b, c)) (Bound a) = false
+  | eq_num (Neg c) (Cx (a, b)) = false
+  | eq_num (Add (c, d)) (Cx (a, b)) = false
+  | eq_num (Sub (c, d)) (Cx (a, b)) = false
+  | eq_num (Mul (c, d)) (Cx (a, b)) = false
+  | eq_num (Add (b, c)) (Neg a) = false
+  | eq_num (Sub (b, c)) (Neg a) = false
+  | eq_num (Mul (b, c)) (Neg a) = false
+  | eq_num (Sub (c, d)) (Add (a, b)) = false
+  | eq_num (Mul (c, d)) (Add (a, b)) = false
+  | eq_num (Mul (c, d)) (Sub (a, b)) = false;
 
-fun numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (c2, Bound n2), r2)) =
-    (if (n1 = n2)
-      then let val c = (c1 + c2)
-           in (if (c = 0) then numadd (r1, r2)
-                else Add (Mul (c, Bound n1), numadd (r1, r2)))
-           end
-      else (if less_eq_def3 n1 n2
-             then Add (Mul (c1, Bound n1),
-                        numadd (r1, Add (Mul (c2, Bound n2), r2)))
-             else Add (Mul (c2, Bound n2),
-                        numadd (Add (Mul (c1, Bound n1), r1), r2))))
-  | numadd (Add (Mul (c1, Bound n1), r1), C afq) =
-    Add (Mul (c1, Bound n1), numadd (r1, C afq))
-  | numadd (Add (Mul (c1, Bound n1), r1), Bound afr) =
-    Add (Mul (c1, Bound n1), numadd (r1, Bound afr))
-  | numadd (Add (Mul (c1, Bound n1), r1), CX (afs, aft)) =
-    Add (Mul (c1, Bound n1), numadd (r1, CX (afs, aft)))
-  | numadd (Add (Mul (c1, Bound n1), r1), Neg afu) =
-    Add (Mul (c1, Bound n1), numadd (r1, Neg afu))
-  | numadd (Add (Mul (c1, Bound n1), r1), Add (C agx, afw)) =
-    Add (Mul (c1, Bound n1), numadd (r1, Add (C agx, afw)))
-  | numadd (Add (Mul (c1, Bound n1), r1), Add (Bound agy, afw)) =
-    Add (Mul (c1, Bound n1), numadd (r1, Add (Bound agy, afw)))
-  | numadd (Add (Mul (c1, Bound n1), r1), Add (CX (agz, aha), afw)) =
-    Add (Mul (c1, Bound n1), numadd (r1, Add (CX (agz, aha), afw)))
-  | numadd (Add (Mul (c1, Bound n1), r1), Add (Neg ahb, afw)) =
-    Add (Mul (c1, Bound n1), numadd (r1, Add (Neg ahb, afw)))
-  | numadd (Add (Mul (c1, Bound n1), r1), Add (Add (ahc, ahd), afw)) =
-    Add (Mul (c1, Bound n1), numadd (r1, Add (Add (ahc, ahd), afw)))
-  | numadd (Add (Mul (c1, Bound n1), r1), Add (Sub (ahe, ahf), afw)) =
-    Add (Mul (c1, Bound n1), numadd (r1, Add (Sub (ahe, ahf), afw)))
-  | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, C aie), afw)) =
-    Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, C aie), afw)))
-  | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, CX (aig, aih)), afw)) =
-    Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, CX (aig, aih)), afw)))
-  | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Neg aii), afw)) =
-    Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Neg aii), afw)))
-  | numadd
-      (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Add (aij, aik)), afw)) =
-    Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Add (aij, aik)), afw)))
-  | numadd
-      (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Sub (ail, aim)), afw)) =
-    Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Sub (ail, aim)), afw)))
+fun eq_fm T T = true
+  | eq_fm F F = true
+  | eq_fm (Lt num) (Lt num') = eq_num num num'
+  | eq_fm (Le num) (Le num') = eq_num num num'
+  | eq_fm (Gt num) (Gt num') = eq_num num num'
+  | eq_fm (Ge num) (Ge num') = eq_num num num'
+  | eq_fm (Eq num) (Eq num') = eq_num num num'
+  | eq_fm (NEq num) (NEq num') = eq_num num num'
+  | eq_fm (Dvd (int, num)) (Dvd (int', num')) =
+    ((int : IntInf.int) = int') andalso eq_num num num'
+  | eq_fm (NDvd (int, num)) (NDvd (int', num')) =
+    ((int : IntInf.int) = int') andalso eq_num num num'
+  | eq_fm (Nota fm) (Nota fm') = eq_fm fm fm'
+  | eq_fm (And (fm1, fm2)) (And (fm1', fm2')) =
+    eq_fm fm1 fm1' andalso eq_fm fm2 fm2'
+  | eq_fm (Or (fm1, fm2)) (Or (fm1', fm2')) =
+    eq_fm fm1 fm1' andalso eq_fm fm2 fm2'
+  | eq_fm (Impa (fm1, fm2)) (Impa (fm1', fm2')) =
+    eq_fm fm1 fm1' andalso eq_fm fm2 fm2'
+  | eq_fm (Iffa (fm1, fm2)) (Iffa (fm1', fm2')) =
+    eq_fm fm1 fm1' andalso eq_fm fm2 fm2'
+  | eq_fm (E fm) (E fm') = eq_fm fm fm'
+  | eq_fm (A fm) (A fm') = eq_fm fm fm'
+  | eq_fm (Closed nat) (Closed nat') = ((nat : IntInf.int) = nat')
+  | eq_fm (NClosed nat) (NClosed nat') = ((nat : IntInf.int) = nat')
+  | eq_fm T F = false
+  | eq_fm T (Lt a) = false
+  | eq_fm T (Le a) = false
+  | eq_fm T (Gt a) = false
+  | eq_fm T (Ge a) = false
+  | eq_fm T (Eq a) = false
+  | eq_fm T (NEq a) = false
+  | eq_fm T (Dvd (a, b)) = false
+  | eq_fm T (NDvd (a, b)) = false
+  | eq_fm T (Nota a) = false
+  | eq_fm T (And (a, b)) = false
+  | eq_fm T (Or (a, b)) = false
+  | eq_fm T (Impa (a, b)) = false
+  | eq_fm T (Iffa (a, b)) = false
+  | eq_fm T (E a) = false
+  | eq_fm T (A a) = false
+  | eq_fm T (Closed a) = false
+  | eq_fm T (NClosed a) = false
+  | eq_fm F (Lt a) = false
+  | eq_fm F (Le a) = false
+  | eq_fm F (Gt a) = false
+  | eq_fm F (Ge a) = false
+  | eq_fm F (Eq a) = false
+  | eq_fm F (NEq a) = false
+  | eq_fm F (Dvd (a, b)) = false
+  | eq_fm F (NDvd (a, b)) = false
+  | eq_fm F (Nota a) = false
+  | eq_fm F (And (a, b)) = false
+  | eq_fm F (Or (a, b)) = false
+  | eq_fm F (Impa (a, b)) = false
+  | eq_fm F (Iffa (a, b)) = false
+  | eq_fm F (E a) = false
+  | eq_fm F (A a) = false
+  | eq_fm F (Closed a) = false
+  | eq_fm F (NClosed a) = false
+  | eq_fm (Lt a) (Le b) = false
+  | eq_fm (Lt a) (Gt b) = false
+  | eq_fm (Lt a) (Ge b) = false
+  | eq_fm (Lt a) (Eq b) = false
+  | eq_fm (Lt a) (NEq b) = false
+  | eq_fm (Lt a) (Dvd (b, c)) = false
+  | eq_fm (Lt a) (NDvd (b, c)) = false
+  | eq_fm (Lt a) (Nota b) = false
+  | eq_fm (Lt a) (And (b, c)) = false
+  | eq_fm (Lt a) (Or (b, c)) = false
+  | eq_fm (Lt a) (Impa (b, c)) = false
+  | eq_fm (Lt a) (Iffa (b, c)) = false
+  | eq_fm (Lt a) (E b) = false
+  | eq_fm (Lt a) (A b) = false
+  | eq_fm (Lt a) (Closed b) = false
+  | eq_fm (Lt a) (NClosed b) = false
+  | eq_fm (Le a) (Gt b) = false
+  | eq_fm (Le a) (Ge b) = false
+  | eq_fm (Le a) (Eq b) = false
+  | eq_fm (Le a) (NEq b) = false
+  | eq_fm (Le a) (Dvd (b, c)) = false
+  | eq_fm (Le a) (NDvd (b, c)) = false
+  | eq_fm (Le a) (Nota b) = false
+  | eq_fm (Le a) (And (b, c)) = false
+  | eq_fm (Le a) (Or (b, c)) = false
+  | eq_fm (Le a) (Impa (b, c)) = false
+  | eq_fm (Le a) (Iffa (b, c)) = false
+  | eq_fm (Le a) (E b) = false
+  | eq_fm (Le a) (A b) = false
+  | eq_fm (Le a) (Closed b) = false
+  | eq_fm (Le a) (NClosed b) = false
+  | eq_fm (Gt a) (Ge b) = false
+  | eq_fm (Gt a) (Eq b) = false
+  | eq_fm (Gt a) (NEq b) = false
+  | eq_fm (Gt a) (Dvd (b, c)) = false
+  | eq_fm (Gt a) (NDvd (b, c)) = false
+  | eq_fm (Gt a) (Nota b) = false
+  | eq_fm (Gt a) (And (b, c)) = false
+  | eq_fm (Gt a) (Or (b, c)) = false
+  | eq_fm (Gt a) (Impa (b, c)) = false
+  | eq_fm (Gt a) (Iffa (b, c)) = false
+  | eq_fm (Gt a) (E b) = false
+  | eq_fm (Gt a) (A b) = false
+  | eq_fm (Gt a) (Closed b) = false
+  | eq_fm (Gt a) (NClosed b) = false
+  | eq_fm (Ge a) (Eq b) = false
+  | eq_fm (Ge a) (NEq b) = false
+  | eq_fm (Ge a) (Dvd (b, c)) = false
+  | eq_fm (Ge a) (NDvd (b, c)) = false
+  | eq_fm (Ge a) (Nota b) = false
+  | eq_fm (Ge a) (And (b, c)) = false
+  | eq_fm (Ge a) (Or (b, c)) = false
+  | eq_fm (Ge a) (Impa (b, c)) = false
+  | eq_fm (Ge a) (Iffa (b, c)) = false
+  | eq_fm (Ge a) (E b) = false
+  | eq_fm (Ge a) (A b) = false
+  | eq_fm (Ge a) (Closed b) = false
+  | eq_fm (Ge a) (NClosed b) = false
+  | eq_fm (Eq a) (NEq b) = false
+  | eq_fm (Eq a) (Dvd (b, c)) = false
+  | eq_fm (Eq a) (NDvd (b, c)) = false
+  | eq_fm (Eq a) (Nota b) = false
+  | eq_fm (Eq a) (And (b, c)) = false
+  | eq_fm (Eq a) (Or (b, c)) = false
+  | eq_fm (Eq a) (Impa (b, c)) = false
+  | eq_fm (Eq a) (Iffa (b, c)) = false
+  | eq_fm (Eq a) (E b) = false
+  | eq_fm (Eq a) (A b) = false
+  | eq_fm (Eq a) (Closed b) = false
+  | eq_fm (Eq a) (NClosed b) = false
+  | eq_fm (NEq a) (Dvd (b, c)) = false
+  | eq_fm (NEq a) (NDvd (b, c)) = false
+  | eq_fm (NEq a) (Nota b) = false
+  | eq_fm (NEq a) (And (b, c)) = false
+  | eq_fm (NEq a) (Or (b, c)) = false
+  | eq_fm (NEq a) (Impa (b, c)) = false
+  | eq_fm (NEq a) (Iffa (b, c)) = false
+  | eq_fm (NEq a) (E b) = false
+  | eq_fm (NEq a) (A b) = false
+  | eq_fm (NEq a) (Closed b) = false
+  | eq_fm (NEq a) (NClosed b) = false
+  | eq_fm (Dvd (a, b)) (NDvd (c, d)) = false
+  | eq_fm (Dvd (a, b)) (Nota c) = false
+  | eq_fm (Dvd (a, b)) (And (c, d)) = false
+  | eq_fm (Dvd (a, b)) (Or (c, d)) = false
+  | eq_fm (Dvd (a, b)) (Impa (c, d)) = false
+  | eq_fm (Dvd (a, b)) (Iffa (c, d)) = false
+  | eq_fm (Dvd (a, b)) (E c) = false
+  | eq_fm (Dvd (a, b)) (A c) = false
+  | eq_fm (Dvd (a, b)) (Closed c) = false
+  | eq_fm (Dvd (a, b)) (NClosed c) = false
+  | eq_fm (NDvd (a, b)) (Nota c) = false
+  | eq_fm (NDvd (a, b)) (And (c, d)) = false
+  | eq_fm (NDvd (a, b)) (Or (c, d)) = false
+  | eq_fm (NDvd (a, b)) (Impa (c, d)) = false
+  | eq_fm (NDvd (a, b)) (Iffa (c, d)) = false
+  | eq_fm (NDvd (a, b)) (E c) = false
+  | eq_fm (NDvd (a, b)) (A c) = false
+  | eq_fm (NDvd (a, b)) (Closed c) = false
+  | eq_fm (NDvd (a, b)) (NClosed c) = false
+  | eq_fm (Nota a) (And (b, c)) = false
+  | eq_fm (Nota a) (Or (b, c)) = false
+  | eq_fm (Nota a) (Impa (b, c)) = false
+  | eq_fm (Nota a) (Iffa (b, c)) = false
+  | eq_fm (Nota a) (E b) = false
+  | eq_fm (Nota a) (A b) = false
+  | eq_fm (Nota a) (Closed b) = false
+  | eq_fm (Nota a) (NClosed b) = false
+  | eq_fm (And (a, b)) (Or (c, d)) = false
+  | eq_fm (And (a, b)) (Impa (c, d)) = false
+  | eq_fm (And (a, b)) (Iffa (c, d)) = false
+  | eq_fm (And (a, b)) (E c) = false
+  | eq_fm (And (a, b)) (A c) = false
+  | eq_fm (And (a, b)) (Closed c) = false
+  | eq_fm (And (a, b)) (NClosed c) = false
+  | eq_fm (Or (a, b)) (Impa (c, d)) = false
+  | eq_fm (Or (a, b)) (Iffa (c, d)) = false
+  | eq_fm (Or (a, b)) (E c) = false
+  | eq_fm (Or (a, b)) (A c) = false
+  | eq_fm (Or (a, b)) (Closed c) = false
+  | eq_fm (Or (a, b)) (NClosed c) = false
+  | eq_fm (Impa (a, b)) (Iffa (c, d)) = false
+  | eq_fm (Impa (a, b)) (E c) = false
+  | eq_fm (Impa (a, b)) (A c) = false
+  | eq_fm (Impa (a, b)) (Closed c) = false
+  | eq_fm (Impa (a, b)) (NClosed c) = false
+  | eq_fm (Iffa (a, b)) (E c) = false
+  | eq_fm (Iffa (a, b)) (A c) = false
+  | eq_fm (Iffa (a, b)) (Closed c) = false
+  | eq_fm (Iffa (a, b)) (NClosed c) = false
+  | eq_fm (E a) (A b) = false
+  | eq_fm (E a) (Closed b) = false
+  | eq_fm (E a) (NClosed b) = false
+  | eq_fm (A a) (Closed b) = false
+  | eq_fm (A a) (NClosed b) = false
+  | eq_fm (Closed a) (NClosed b) = false
+  | eq_fm F T = false
+  | eq_fm (Lt a) T = false
+  | eq_fm (Le a) T = false
+  | eq_fm (Gt a) T = false
+  | eq_fm (Ge a) T = false
+  | eq_fm (Eq a) T = false
+  | eq_fm (NEq a) T = false
+  | eq_fm (Dvd (a, b)) T = false
+  | eq_fm (NDvd (a, b)) T = false
+  | eq_fm (Nota a) T = false
+  | eq_fm (And (a, b)) T = false
+  | eq_fm (Or (a, b)) T = false
+  | eq_fm (Impa (a, b)) T = false
+  | eq_fm (Iffa (a, b)) T = false
+  | eq_fm (E a) T = false
+  | eq_fm (A a) T = false
+  | eq_fm (Closed a) T = false
+  | eq_fm (NClosed a) T = false
+  | eq_fm (Lt a) F = false
+  | eq_fm (Le a) F = false
+  | eq_fm (Gt a) F = false
+  | eq_fm (Ge a) F = false
+  | eq_fm (Eq a) F = false
+  | eq_fm (NEq a) F = false
+  | eq_fm (Dvd (a, b)) F = false
+  | eq_fm (NDvd (a, b)) F = false
+  | eq_fm (Nota a) F = false
+  | eq_fm (And (a, b)) F = false
+  | eq_fm (Or (a, b)) F = false
+  | eq_fm (Impa (a, b)) F = false
+  | eq_fm (Iffa (a, b)) F = false
+  | eq_fm (E a) F = false
+  | eq_fm (A a) F = false
+  | eq_fm (Closed a) F = false
+  | eq_fm (NClosed a) F = false
+  | eq_fm (Le b) (Lt a) = false
+  | eq_fm (Gt b) (Lt a) = false
+  | eq_fm (Ge b) (Lt a) = false
+  | eq_fm (Eq b) (Lt a) = false
+  | eq_fm (NEq b) (Lt a) = false
+  | eq_fm (Dvd (b, c)) (Lt a) = false
+  | eq_fm (NDvd (b, c)) (Lt a) = false
+  | eq_fm (Nota b) (Lt a) = false
+  | eq_fm (And (b, c)) (Lt a) = false
+  | eq_fm (Or (b, c)) (Lt a) = false
+  | eq_fm (Impa (b, c)) (Lt a) = false
+  | eq_fm (Iffa (b, c)) (Lt a) = false
+  | eq_fm (E b) (Lt a) = false
+  | eq_fm (A b) (Lt a) = false
+  | eq_fm (Closed b) (Lt a) = false
+  | eq_fm (NClosed b) (Lt a) = false
+  | eq_fm (Gt b) (Le a) = false
+  | eq_fm (Ge b) (Le a) = false
+  | eq_fm (Eq b) (Le a) = false
+  | eq_fm (NEq b) (Le a) = false
+  | eq_fm (Dvd (b, c)) (Le a) = false
+  | eq_fm (NDvd (b, c)) (Le a) = false
+  | eq_fm (Nota b) (Le a) = false
+  | eq_fm (And (b, c)) (Le a) = false
+  | eq_fm (Or (b, c)) (Le a) = false
+  | eq_fm (Impa (b, c)) (Le a) = false
+  | eq_fm (Iffa (b, c)) (Le a) = false
+  | eq_fm (E b) (Le a) = false
+  | eq_fm (A b) (Le a) = false
+  | eq_fm (Closed b) (Le a) = false
+  | eq_fm (NClosed b) (Le a) = false
+  | eq_fm (Ge b) (Gt a) = false
+  | eq_fm (Eq b) (Gt a) = false
+  | eq_fm (NEq b) (Gt a) = false
+  | eq_fm (Dvd (b, c)) (Gt a) = false
+  | eq_fm (NDvd (b, c)) (Gt a) = false
+  | eq_fm (Nota b) (Gt a) = false
+  | eq_fm (And (b, c)) (Gt a) = false
+  | eq_fm (Or (b, c)) (Gt a) = false
+  | eq_fm (Impa (b, c)) (Gt a) = false
+  | eq_fm (Iffa (b, c)) (Gt a) = false
+  | eq_fm (E b) (Gt a) = false
+  | eq_fm (A b) (Gt a) = false
+  | eq_fm (Closed b) (Gt a) = false
+  | eq_fm (NClosed b) (Gt a) = false
+  | eq_fm (Eq b) (Ge a) = false
+  | eq_fm (NEq b) (Ge a) = false
+  | eq_fm (Dvd (b, c)) (Ge a) = false
+  | eq_fm (NDvd (b, c)) (Ge a) = false
+  | eq_fm (Nota b) (Ge a) = false
+  | eq_fm (And (b, c)) (Ge a) = false
+  | eq_fm (Or (b, c)) (Ge a) = false
+  | eq_fm (Impa (b, c)) (Ge a) = false
+  | eq_fm (Iffa (b, c)) (Ge a) = false
+  | eq_fm (E b) (Ge a) = false
+  | eq_fm (A b) (Ge a) = false
+  | eq_fm (Closed b) (Ge a) = false
+  | eq_fm (NClosed b) (Ge a) = false
+  | eq_fm (NEq b) (Eq a) = false
+  | eq_fm (Dvd (b, c)) (Eq a) = false
+  | eq_fm (NDvd (b, c)) (Eq a) = false
+  | eq_fm (Nota b) (Eq a) = false
+  | eq_fm (And (b, c)) (Eq a) = false
+  | eq_fm (Or (b, c)) (Eq a) = false
+  | eq_fm (Impa (b, c)) (Eq a) = false
+  | eq_fm (Iffa (b, c)) (Eq a) = false
+  | eq_fm (E b) (Eq a) = false
+  | eq_fm (A b) (Eq a) = false
+  | eq_fm (Closed b) (Eq a) = false
+  | eq_fm (NClosed b) (Eq a) = false
+  | eq_fm (Dvd (b, c)) (NEq a) = false
+  | eq_fm (NDvd (b, c)) (NEq a) = false
+  | eq_fm (Nota b) (NEq a) = false
+  | eq_fm (And (b, c)) (NEq a) = false
+  | eq_fm (Or (b, c)) (NEq a) = false
+  | eq_fm (Impa (b, c)) (NEq a) = false
+  | eq_fm (Iffa (b, c)) (NEq a) = false
+  | eq_fm (E b) (NEq a) = false
+  | eq_fm (A b) (NEq a) = false
+  | eq_fm (Closed b) (NEq a) = false
+  | eq_fm (NClosed b) (NEq a) = false
+  | eq_fm (NDvd (c, d)) (Dvd (a, b)) = false
+  | eq_fm (Nota c) (Dvd (a, b)) = false
+  | eq_fm (And (c, d)) (Dvd (a, b)) = false
+  | eq_fm (Or (c, d)) (Dvd (a, b)) = false
+  | eq_fm (Impa (c, d)) (Dvd (a, b)) = false
+  | eq_fm (Iffa (c, d)) (Dvd (a, b)) = false
+  | eq_fm (E c) (Dvd (a, b)) = false
+  | eq_fm (A c) (Dvd (a, b)) = false
+  | eq_fm (Closed c) (Dvd (a, b)) = false
+  | eq_fm (NClosed c) (Dvd (a, b)) = false
+  | eq_fm (Nota c) (NDvd (a, b)) = false
+  | eq_fm (And (c, d)) (NDvd (a, b)) = false
+  | eq_fm (Or (c, d)) (NDvd (a, b)) = false
+  | eq_fm (Impa (c, d)) (NDvd (a, b)) = false
+  | eq_fm (Iffa (c, d)) (NDvd (a, b)) = false
+  | eq_fm (E c) (NDvd (a, b)) = false
+  | eq_fm (A c) (NDvd (a, b)) = false
+  | eq_fm (Closed c) (NDvd (a, b)) = false
+  | eq_fm (NClosed c) (NDvd (a, b)) = false
+  | eq_fm (And (b, c)) (Nota a) = false
+  | eq_fm (Or (b, c)) (Nota a) = false
+  | eq_fm (Impa (b, c)) (Nota a) = false
+  | eq_fm (Iffa (b, c)) (Nota a) = false
+  | eq_fm (E b) (Nota a) = false
+  | eq_fm (A b) (Nota a) = false
+  | eq_fm (Closed b) (Nota a) = false
+  | eq_fm (NClosed b) (Nota a) = false
+  | eq_fm (Or (c, d)) (And (a, b)) = false
+  | eq_fm (Impa (c, d)) (And (a, b)) = false
+  | eq_fm (Iffa (c, d)) (And (a, b)) = false
+  | eq_fm (E c) (And (a, b)) = false
+  | eq_fm (A c) (And (a, b)) = false
+  | eq_fm (Closed c) (And (a, b)) = false
+  | eq_fm (NClosed c) (And (a, b)) = false
+  | eq_fm (Impa (c, d)) (Or (a, b)) = false
+  | eq_fm (Iffa (c, d)) (Or (a, b)) = false
+  | eq_fm (E c) (Or (a, b)) = false
+  | eq_fm (A c) (Or (a, b)) = false
+  | eq_fm (Closed c) (Or (a, b)) = false
+  | eq_fm (NClosed c) (Or (a, b)) = false
+  | eq_fm (Iffa (c, d)) (Impa (a, b)) = false
+  | eq_fm (E c) (Impa (a, b)) = false
+  | eq_fm (A c) (Impa (a, b)) = false
+  | eq_fm (Closed c) (Impa (a, b)) = false
+  | eq_fm (NClosed c) (Impa (a, b)) = false
+  | eq_fm (E c) (Iffa (a, b)) = false
+  | eq_fm (A c) (Iffa (a, b)) = false
+  | eq_fm (Closed c) (Iffa (a, b)) = false
+  | eq_fm (NClosed c) (Iffa (a, b)) = false
+  | eq_fm (A b) (E a) = false
+  | eq_fm (Closed b) (E a) = false
+  | eq_fm (NClosed b) (E a) = false
+  | eq_fm (Closed b) (A a) = false
+  | eq_fm (NClosed b) (A a) = false
+  | eq_fm (NClosed b) (Closed a) = false;
+
+fun djf f p q =
+  (if eq_fm q T then T
+    else (if eq_fm q F then f p
+           else (case f p of T => T | F => q | Lt num => Or (f p, q)
+                   | Le num => Or (f p, q) | Gt num => Or (f p, q)
+                   | Ge num => Or (f p, q) | Eq num => Or (f p, q)
+                   | NEq num => Or (f p, q) | Dvd (int, num) => Or (f p, q)
+                   | NDvd (int, num) => Or (f p, q) | Nota fm => Or (f p, q)
+                   | And (fm1, fm2) => Or (f p, q)
+                   | Or (fm1, fm2) => Or (f p, q)
+                   | Impa (fm1, fm2) => Or (f p, q)
+                   | Iffa (fm1, fm2) => Or (f p, q) | E fm => Or (f p, q)
+                   | A fm => Or (f p, q) | Closed nat => Or (f p, q)
+                   | NClosed nat => Or (f p, q))));
+
+fun evaldjf f ps = List.foldr (djf f) ps F;
+
+fun dj f p = evaldjf f (disjuncts p);
+
+fun zsplit0 (Mul (i, a)) =
+  let
+    val (i', a') = zsplit0 a;
+  in
+    (IntInf.* (i, i'), Mul (i, a'))
+  end
+  | zsplit0 (Sub (a, b)) =
+    let
+      val (ia, a') = zsplit0 a;
+      val (ib, b') = zsplit0 b;
+    in
+      (IntInf.- (ia, ib), Sub (a', b'))
+    end
+  | zsplit0 (Add (a, b)) =
+    let
+      val (ia, a') = zsplit0 a;
+      val (ib, b') = zsplit0 b;
+    in
+      (IntInf.+ (ia, ib), Add (a', b'))
+    end
+  | zsplit0 (Neg a) =
+    let
+      val (i', a') = zsplit0 a;
+    in
+      (IntInf.~ i', Neg a')
+    end
+  | zsplit0 (Cx (i, a)) =
+    let
+      val (i', aa) = zsplit0 a;
+    in
+      (IntInf.+ (i, i'), aa)
+    end
+  | zsplit0 (Bound n) =
+    (if ((n : IntInf.int) = Integer.zero_nat)
+      then ((1 : IntInf.int), C (0 : IntInf.int))
+      else ((0 : IntInf.int), Bound n))
+  | zsplit0 (C c) = ((0 : IntInf.int), C c);
+
+fun zlfm (NClosed ar) = NClosed ar
+  | zlfm (Closed aq) = Closed aq
+  | zlfm (A ap) = A ap
+  | zlfm (E ao) = E ao
+  | zlfm (Nota (A cj)) = Nota (A cj)
+  | zlfm (Nota (E ci)) = Nota (E ci)
+  | zlfm F = F
+  | zlfm T = T
+  | zlfm (Nota (NClosed p)) = Closed p
+  | zlfm (Nota (Closed p)) = NClosed p
+  | zlfm (Nota (NDvd (i, a))) = zlfm (Dvd (i, a))
+  | zlfm (Nota (Dvd (i, a))) = zlfm (NDvd (i, a))
+  | zlfm (Nota (NEq a)) = zlfm (Eq a)
+  | zlfm (Nota (Eq a)) = zlfm (NEq a)
+  | zlfm (Nota (Ge a)) = zlfm (Lt a)
+  | zlfm (Nota (Gt a)) = zlfm (Le a)
+  | zlfm (Nota (Le a)) = zlfm (Gt a)
+  | zlfm (Nota (Lt a)) = zlfm (Ge a)
+  | zlfm (Nota F) = T
+  | zlfm (Nota T) = F
+  | zlfm (Nota (Nota p)) = zlfm p
+  | zlfm (Nota (Iffa (p, q))) =
+    Or (And (zlfm p, zlfm (Nota q)), And (zlfm (Nota p), zlfm q))
+  | zlfm (Nota (Impa (p, q))) = And (zlfm p, zlfm (Nota q))
+  | zlfm (Nota (Or (p, q))) = And (zlfm (Nota p), zlfm (Nota q))
+  | zlfm (Nota (And (p, q))) = Or (zlfm (Nota p), zlfm (Nota q))
+  | zlfm (NDvd (i, a)) =
+    (if ((i : IntInf.int) = (0 : IntInf.int)) then zlfm (NEq a)
+      else let
+             val (c, r) = zsplit0 a;
+           in
+             (if ((c : IntInf.int) = (0 : IntInf.int))
+               then NDvd (Integer.abs_int i, r)
+               else (if IntInf.< ((0 : IntInf.int), c)
+                      then NDvd (Integer.abs_int i, Cx (c, r))
+                      else NDvd (Integer.abs_int i, Cx (IntInf.~ c, Neg r))))
+           end)
+  | zlfm (Dvd (i, a)) =
+    (if ((i : IntInf.int) = (0 : IntInf.int)) then zlfm (Eq a)
+      else let
+             val (c, r) = zsplit0 a;
+           in
+             (if ((c : IntInf.int) = (0 : IntInf.int))
+               then Dvd (Integer.abs_int i, r)
+               else (if IntInf.< ((0 : IntInf.int), c)
+                      then Dvd (Integer.abs_int i, Cx (c, r))
+                      else Dvd (Integer.abs_int i, Cx (IntInf.~ c, Neg r))))
+           end)
+  | zlfm (NEq a) =
+    let
+      val (c, r) = zsplit0 a;
+    in
+      (if ((c : IntInf.int) = (0 : IntInf.int)) then NEq r
+        else (if IntInf.< ((0 : IntInf.int), c) then NEq (Cx (c, r))
+               else NEq (Cx (IntInf.~ c, Neg r))))
+    end
+  | zlfm (Eq a) =
+    let
+      val (c, r) = zsplit0 a;
+    in
+      (if ((c : IntInf.int) = (0 : IntInf.int)) then Eq r
+        else (if IntInf.< ((0 : IntInf.int), c) then Eq (Cx (c, r))
+               else Eq (Cx (IntInf.~ c, Neg r))))
+    end
+  | zlfm (Ge a) =
+    let
+      val (c, r) = zsplit0 a;
+    in
+      (if ((c : IntInf.int) = (0 : IntInf.int)) then Ge r
+        else (if IntInf.< ((0 : IntInf.int), c) then Ge (Cx (c, r))
+               else Le (Cx (IntInf.~ c, Neg r))))
+    end
+  | zlfm (Gt a) =
+    let
+      val (c, r) = zsplit0 a;
+    in
+      (if ((c : IntInf.int) = (0 : IntInf.int)) then Gt r
+        else (if IntInf.< ((0 : IntInf.int), c) then Gt (Cx (c, r))
+               else Lt (Cx (IntInf.~ c, Neg r))))
+    end
+  | zlfm (Le a) =
+    let
+      val (c, r) = zsplit0 a;
+    in
+      (if ((c : IntInf.int) = (0 : IntInf.int)) then Le r
+        else (if IntInf.< ((0 : IntInf.int), c) then Le (Cx (c, r))
+               else Ge (Cx (IntInf.~ c, Neg r))))
+    end
+  | zlfm (Lt a) =
+    let
+      val (c, r) = zsplit0 a;
+    in
+      (if ((c : IntInf.int) = (0 : IntInf.int)) then Lt r
+        else (if IntInf.< ((0 : IntInf.int), c) then Lt (Cx (c, r))
+               else Gt (Cx (IntInf.~ c, Neg r))))
+    end
+  | zlfm (Iffa (p, q)) =
+    Or (And (zlfm p, zlfm q), And (zlfm (Nota p), zlfm (Nota q)))
+  | zlfm (Impa (p, q)) = Or (zlfm (Nota p), zlfm q)
+  | zlfm (Or (p, q)) = Or (zlfm p, zlfm q)
+  | zlfm (And (p, q)) = And (zlfm p, zlfm q);
+
+fun zeta (NClosed aq) = (1 : IntInf.int)
+  | zeta (Closed ap) = (1 : IntInf.int)
+  | zeta (A ao) = (1 : IntInf.int)
+  | zeta (E an) = (1 : IntInf.int)
+  | zeta (Iffa (al, am)) = (1 : IntInf.int)
+  | zeta (Impa (aj, ak)) = (1 : IntInf.int)
+  | zeta (Nota ae) = (1 : IntInf.int)
+  | zeta (NDvd (ac, Mul (hv, hw))) = (1 : IntInf.int)
+  | zeta (NDvd (ac, Sub (ht, hu))) = (1 : IntInf.int)
+  | zeta (NDvd (ac, Add (hr, hs))) = (1 : IntInf.int)
+  | zeta (NDvd (ac, Neg hq)) = (1 : IntInf.int)
+  | zeta (NDvd (ac, Bound hn)) = (1 : IntInf.int)
+  | zeta (NDvd (ac, C hm)) = (1 : IntInf.int)
+  | zeta (Dvd (aa, Mul (gz, ha))) = (1 : IntInf.int)
+  | zeta (Dvd (aa, Sub (gx, gy))) = (1 : IntInf.int)
+  | zeta (Dvd (aa, Add (gv, gw))) = (1 : IntInf.int)
+  | zeta (Dvd (aa, Neg gu)) = (1 : IntInf.int)
+  | zeta (Dvd (aa, Bound gr)) = (1 : IntInf.int)
+  | zeta (Dvd (aa, C gq)) = (1 : IntInf.int)
+  | zeta (NEq (Mul (gd, ge))) = (1 : IntInf.int)
+  | zeta (NEq (Sub (gb, gc))) = (1 : IntInf.int)
+  | zeta (NEq (Add (fz, ga))) = (1 : IntInf.int)
+  | zeta (NEq (Neg fy)) = (1 : IntInf.int)
+  | zeta (NEq (Bound fv)) = (1 : IntInf.int)
+  | zeta (NEq (C fu)) = (1 : IntInf.int)
+  | zeta (Eq (Mul (fh, fi))) = (1 : IntInf.int)
+  | zeta (Eq (Sub (ff, fg))) = (1 : IntInf.int)
+  | zeta (Eq (Add (fd, fe))) = (1 : IntInf.int)
+  | zeta (Eq (Neg fc)) = (1 : IntInf.int)
+  | zeta (Eq (Bound ez)) = (1 : IntInf.int)
+  | zeta (Eq (C ey)) = (1 : IntInf.int)
+  | zeta (Ge (Mul (el, em))) = (1 : IntInf.int)
+  | zeta (Ge (Sub (ej, ek))) = (1 : IntInf.int)
+  | zeta (Ge (Add (eh, ei))) = (1 : IntInf.int)
+  | zeta (Ge (Neg eg)) = (1 : IntInf.int)
+  | zeta (Ge (Bound ed)) = (1 : IntInf.int)
+  | zeta (Ge (C ec)) = (1 : IntInf.int)
+  | zeta (Gt (Mul (dp, dq))) = (1 : IntInf.int)
+  | zeta (Gt (Sub (dn, doa))) = (1 : IntInf.int)
+  | zeta (Gt (Add (dl, dm))) = (1 : IntInf.int)
+  | zeta (Gt (Neg dk)) = (1 : IntInf.int)
+  | zeta (Gt (Bound dh)) = (1 : IntInf.int)
+  | zeta (Gt (C dg)) = (1 : IntInf.int)
+  | zeta (Le (Mul (ct, cu))) = (1 : IntInf.int)
+  | zeta (Le (Sub (cr, cs))) = (1 : IntInf.int)
+  | zeta (Le (Add (cp, cq))) = (1 : IntInf.int)
+  | zeta (Le (Neg co)) = (1 : IntInf.int)
+  | zeta (Le (Bound cl)) = (1 : IntInf.int)
+  | zeta (Le (C ck)) = (1 : IntInf.int)
+  | zeta (Lt (Mul (bx, by))) = (1 : IntInf.int)
+  | zeta (Lt (Sub (bv, bw))) = (1 : IntInf.int)
+  | zeta (Lt (Add (bt, bu))) = (1 : IntInf.int)
+  | zeta (Lt (Neg bs)) = (1 : IntInf.int)
+  | zeta (Lt (Bound bp)) = (1 : IntInf.int)
+  | zeta (Lt (C bo)) = (1 : IntInf.int)
+  | zeta F = (1 : IntInf.int)
+  | zeta T = (1 : IntInf.int)
+  | zeta (NDvd (i, Cx (y, e))) = y
+  | zeta (Dvd (i, Cx (y, e))) = y
+  | zeta (Ge (Cx (y, e))) = y
+  | zeta (Gt (Cx (y, e))) = y
+  | zeta (Le (Cx (y, e))) = y
+  | zeta (Lt (Cx (y, e))) = y
+  | zeta (NEq (Cx (y, e))) = y
+  | zeta (Eq (Cx (y, e))) = y
+  | zeta (Or (p, q)) = GCD.ilcm (zeta p) (zeta q)
+  | zeta (And (p, q)) = GCD.ilcm (zeta p) (zeta q);
+
+fun a_beta (NClosed aq) = (fn k => NClosed aq)
+  | a_beta (Closed ap) = (fn k => Closed ap)
+  | a_beta (A ao) = (fn k => A ao)
+  | a_beta (E an) = (fn k => E an)
+  | a_beta (Iffa (al, am)) = (fn k => Iffa (al, am))
+  | a_beta (Impa (aj, ak)) = (fn k => Impa (aj, ak))
+  | a_beta (Nota ae) = (fn k => Nota ae)
+  | a_beta (NDvd (ac, Mul (hv, hw))) = (fn k => NDvd (ac, Mul (hv, hw)))
+  | a_beta (NDvd (ac, Sub (ht, hu))) = (fn k => NDvd (ac, Sub (ht, hu)))
+  | a_beta (NDvd (ac, Add (hr, hs))) = (fn k => NDvd (ac, Add (hr, hs)))
+  | a_beta (NDvd (ac, Neg hq)) = (fn k => NDvd (ac, Neg hq))
+  | a_beta (NDvd (ac, Bound hn)) = (fn k => NDvd (ac, Bound hn))
+  | a_beta (NDvd (ac, C hm)) = (fn k => NDvd (ac, C hm))
+  | a_beta (Dvd (aa, Mul (gz, ha))) = (fn k => Dvd (aa, Mul (gz, ha)))
+  | a_beta (Dvd (aa, Sub (gx, gy))) = (fn k => Dvd (aa, Sub (gx, gy)))
+  | a_beta (Dvd (aa, Add (gv, gw))) = (fn k => Dvd (aa, Add (gv, gw)))
+  | a_beta (Dvd (aa, Neg gu)) = (fn k => Dvd (aa, Neg gu))
+  | a_beta (Dvd (aa, Bound gr)) = (fn k => Dvd (aa, Bound gr))
+  | a_beta (Dvd (aa, C gq)) = (fn k => Dvd (aa, C gq))
+  | a_beta (NEq (Mul (gd, ge))) = (fn k => NEq (Mul (gd, ge)))
+  | a_beta (NEq (Sub (gb, gc))) = (fn k => NEq (Sub (gb, gc)))
+  | a_beta (NEq (Add (fz, ga))) = (fn k => NEq (Add (fz, ga)))
+  | a_beta (NEq (Neg fy)) = (fn k => NEq (Neg fy))
+  | a_beta (NEq (Bound fv)) = (fn k => NEq (Bound fv))
+  | a_beta (NEq (C fu)) = (fn k => NEq (C fu))
+  | a_beta (Eq (Mul (fh, fi))) = (fn k => Eq (Mul (fh, fi)))
+  | a_beta (Eq (Sub (ff, fg))) = (fn k => Eq (Sub (ff, fg)))
+  | a_beta (Eq (Add (fd, fe))) = (fn k => Eq (Add (fd, fe)))
+  | a_beta (Eq (Neg fc)) = (fn k => Eq (Neg fc))
+  | a_beta (Eq (Bound ez)) = (fn k => Eq (Bound ez))
+  | a_beta (Eq (C ey)) = (fn k => Eq (C ey))
+  | a_beta (Ge (Mul (el, em))) = (fn k => Ge (Mul (el, em)))
+  | a_beta (Ge (Sub (ej, ek))) = (fn k => Ge (Sub (ej, ek)))
+  | a_beta (Ge (Add (eh, ei))) = (fn k => Ge (Add (eh, ei)))
+  | a_beta (Ge (Neg eg)) = (fn k => Ge (Neg eg))
+  | a_beta (Ge (Bound ed)) = (fn k => Ge (Bound ed))
+  | a_beta (Ge (C ec)) = (fn k => Ge (C ec))
+  | a_beta (Gt (Mul (dp, dq))) = (fn k => Gt (Mul (dp, dq)))
+  | a_beta (Gt (Sub (dn, doa))) = (fn k => Gt (Sub (dn, doa)))
+  | a_beta (Gt (Add (dl, dm))) = (fn k => Gt (Add (dl, dm)))
+  | a_beta (Gt (Neg dk)) = (fn k => Gt (Neg dk))
+  | a_beta (Gt (Bound dh)) = (fn k => Gt (Bound dh))
+  | a_beta (Gt (C dg)) = (fn k => Gt (C dg))
+  | a_beta (Le (Mul (ct, cu))) = (fn k => Le (Mul (ct, cu)))
+  | a_beta (Le (Sub (cr, cs))) = (fn k => Le (Sub (cr, cs)))
+  | a_beta (Le (Add (cp, cq))) = (fn k => Le (Add (cp, cq)))
+  | a_beta (Le (Neg co)) = (fn k => Le (Neg co))
+  | a_beta (Le (Bound cl)) = (fn k => Le (Bound cl))
+  | a_beta (Le (C ck)) = (fn k => Le (C ck))
+  | a_beta (Lt (Mul (bx, by))) = (fn k => Lt (Mul (bx, by)))
+  | a_beta (Lt (Sub (bv, bw))) = (fn k => Lt (Sub (bv, bw)))
+  | a_beta (Lt (Add (bt, bu))) = (fn k => Lt (Add (bt, bu)))
+  | a_beta (Lt (Neg bs)) = (fn k => Lt (Neg bs))
+  | a_beta (Lt (Bound bp)) = (fn k => Lt (Bound bp))
+  | a_beta (Lt (C bo)) = (fn k => Lt (C bo))
+  | a_beta F = (fn k => F)
+  | a_beta T = (fn k => T)
+  | a_beta (NDvd (i, Cx (c, e))) =
+    (fn k =>
+      NDvd (IntInf.* (Integer.div_int k c, i),
+             Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e))))
+  | a_beta (Dvd (i, Cx (c, e))) =
+    (fn k =>
+      Dvd (IntInf.* (Integer.div_int k c, i),
+            Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e))))
+  | a_beta (Ge (Cx (c, e))) =
+    (fn k => Ge (Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e))))
+  | a_beta (Gt (Cx (c, e))) =
+    (fn k => Gt (Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e))))
+  | a_beta (Le (Cx (c, e))) =
+    (fn k => Le (Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e))))
+  | a_beta (Lt (Cx (c, e))) =
+    (fn k => Lt (Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e))))
+  | a_beta (NEq (Cx (c, e))) =
+    (fn k => NEq (Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e))))
+  | a_beta (Eq (Cx (c, e))) =
+    (fn k => Eq (Cx ((1 : IntInf.int), Mul (Integer.div_int k c, e))))
+  | a_beta (Or (p, q)) = (fn k => Or (a_beta p k, a_beta q k))
+  | a_beta (And (p, q)) = (fn k => And (a_beta p k, a_beta q k));
+
+fun delta (NClosed aq) = (1 : IntInf.int)
+  | delta (Closed ap) = (1 : IntInf.int)
+  | delta (A ao) = (1 : IntInf.int)
+  | delta (E an) = (1 : IntInf.int)
+  | delta (Iffa (al, am)) = (1 : IntInf.int)
+  | delta (Impa (aj, ak)) = (1 : IntInf.int)
+  | delta (Nota ae) = (1 : IntInf.int)
+  | delta (NDvd (ac, Mul (ct, cu))) = (1 : IntInf.int)
+  | delta (NDvd (ac, Sub (cr, cs))) = (1 : IntInf.int)
+  | delta (NDvd (ac, Add (cp, cq))) = (1 : IntInf.int)
+  | delta (NDvd (ac, Neg co)) = (1 : IntInf.int)
+  | delta (NDvd (ac, Bound cl)) = (1 : IntInf.int)
+  | delta (NDvd (ac, C ck)) = (1 : IntInf.int)
+  | delta (Dvd (aa, Mul (bx, by))) = (1 : IntInf.int)
+  | delta (Dvd (aa, Sub (bv, bw))) = (1 : IntInf.int)
+  | delta (Dvd (aa, Add (bt, bu))) = (1 : IntInf.int)
+  | delta (Dvd (aa, Neg bs)) = (1 : IntInf.int)
+  | delta (Dvd (aa, Bound bp)) = (1 : IntInf.int)
+  | delta (Dvd (aa, C bo)) = (1 : IntInf.int)
+  | delta (NEq z) = (1 : IntInf.int)
+  | delta (Eq y) = (1 : IntInf.int)
+  | delta (Ge x) = (1 : IntInf.int)
+  | delta (Gt w) = (1 : IntInf.int)
+  | delta (Le v) = (1 : IntInf.int)
+  | delta (Lt u) = (1 : IntInf.int)
+  | delta F = (1 : IntInf.int)
+  | delta T = (1 : IntInf.int)
+  | delta (NDvd (y, Cx (c, e))) = y
+  | delta (Dvd (y, Cx (c, e))) = y
+  | delta (Or (p, q)) = GCD.ilcm (delta p) (delta q)
+  | delta (And (p, q)) = GCD.ilcm (delta p) (delta q);
+
+fun beta (NClosed aq) = []
+  | beta (Closed ap) = []
+  | beta (A ao) = []
+  | beta (E an) = []
+  | beta (Iffa (al, am)) = []
+  | beta (Impa (aj, ak)) = []
+  | beta (Nota ae) = []
+  | beta (NDvd (ac, ad)) = []
+  | beta (Dvd (aa, ab)) = []
+  | beta (NEq (Mul (gd, ge))) = []
+  | beta (NEq (Sub (gb, gc))) = []
+  | beta (NEq (Add (fz, ga))) = []
+  | beta (NEq (Neg fy)) = []
+  | beta (NEq (Bound fv)) = []
+  | beta (NEq (C fu)) = []
+  | beta (Eq (Mul (fh, fi))) = []
+  | beta (Eq (Sub (ff, fg))) = []
+  | beta (Eq (Add (fd, fe))) = []
+  | beta (Eq (Neg fc)) = []
+  | beta (Eq (Bound ez)) = []
+  | beta (Eq (C ey)) = []
+  | beta (Ge (Mul (el, em))) = []
+  | beta (Ge (Sub (ej, ek))) = []
+  | beta (Ge (Add (eh, ei))) = []
+  | beta (Ge (Neg eg)) = []
+  | beta (Ge (Bound ed)) = []
+  | beta (Ge (C ec)) = []
+  | beta (Gt (Mul (dp, dq))) = []
+  | beta (Gt (Sub (dn, doa))) = []
+  | beta (Gt (Add (dl, dm))) = []
+  | beta (Gt (Neg dk)) = []
+  | beta (Gt (Bound dh)) = []
+  | beta (Gt (C dg)) = []
+  | beta (Le (Mul (ct, cu))) = []
+  | beta (Le (Sub (cr, cs))) = []
+  | beta (Le (Add (cp, cq))) = []
+  | beta (Le (Neg co)) = []
+  | beta (Le (Bound cl)) = []
+  | beta (Le (C ck)) = []
+  | beta (Lt (Mul (bx, by))) = []
+  | beta (Lt (Sub (bv, bw))) = []
+  | beta (Lt (Add (bt, bu))) = []
+  | beta (Lt (Neg bs)) = []
+  | beta (Lt (Bound bp)) = []
+  | beta (Lt (C bo)) = []
+  | beta F = []
+  | beta T = []
+  | beta (Ge (Cx (c, e))) = [Sub (C (~1 : IntInf.int), e)]
+  | beta (Gt (Cx (c, e))) = [Neg e]
+  | beta (Le (Cx (c, e))) = []
+  | beta (Lt (Cx (c, e))) = []
+  | beta (NEq (Cx (c, e))) = [Neg e]
+  | beta (Eq (Cx (c, e))) = [Sub (C (~1 : IntInf.int), e)]
+  | beta (Or (p, q)) = List.append (beta p) (beta q)
+  | beta (And (p, q)) = List.append (beta p) (beta q);
+
+fun alpha (NClosed aq) = []
+  | alpha (Closed ap) = []
+  | alpha (A ao) = []
+  | alpha (E an) = []
+  | alpha (Iffa (al, am)) = []
+  | alpha (Impa (aj, ak)) = []
+  | alpha (Nota ae) = []
+  | alpha (NDvd (ac, ad)) = []
+  | alpha (Dvd (aa, ab)) = []
+  | alpha (NEq (Mul (gd, ge))) = []
+  | alpha (NEq (Sub (gb, gc))) = []
+  | alpha (NEq (Add (fz, ga))) = []
+  | alpha (NEq (Neg fy)) = []
+  | alpha (NEq (Bound fv)) = []
+  | alpha (NEq (C fu)) = []
+  | alpha (Eq (Mul (fh, fi))) = []
+  | alpha (Eq (Sub (ff, fg))) = []
+  | alpha (Eq (Add (fd, fe))) = []
+  | alpha (Eq (Neg fc)) = []
+  | alpha (Eq (Bound ez)) = []
+  | alpha (Eq (C ey)) = []
+  | alpha (Ge (Mul (el, em))) = []
+  | alpha (Ge (Sub (ej, ek))) = []
+  | alpha (Ge (Add (eh, ei))) = []
+  | alpha (Ge (Neg eg)) = []
+  | alpha (Ge (Bound ed)) = []
+  | alpha (Ge (C ec)) = []
+  | alpha (Gt (Mul (dp, dq))) = []
+  | alpha (Gt (Sub (dn, doa))) = []
+  | alpha (Gt (Add (dl, dm))) = []
+  | alpha (Gt (Neg dk)) = []
+  | alpha (Gt (Bound dh)) = []
+  | alpha (Gt (C dg)) = []
+  | alpha (Le (Mul (ct, cu))) = []
+  | alpha (Le (Sub (cr, cs))) = []
+  | alpha (Le (Add (cp, cq))) = []
+  | alpha (Le (Neg co)) = []
+  | alpha (Le (Bound cl)) = []
+  | alpha (Le (C ck)) = []
+  | alpha (Lt (Mul (bx, by))) = []
+  | alpha (Lt (Sub (bv, bw))) = []
+  | alpha (Lt (Add (bt, bu))) = []
+  | alpha (Lt (Neg bs)) = []
+  | alpha (Lt (Bound bp)) = []
+  | alpha (Lt (C bo)) = []
+  | alpha F = []
+  | alpha T = []
+  | alpha (Ge (Cx (c, e))) = []
+  | alpha (Gt (Cx (c, e))) = []
+  | alpha (Le (Cx (c, e))) = [Add (C (~1 : IntInf.int), e)]
+  | alpha (Lt (Cx (c, e))) = [e]
+  | alpha (NEq (Cx (c, e))) = [e]
+  | alpha (Eq (Cx (c, e))) = [Add (C (~1 : IntInf.int), e)]
+  | alpha (Or (p, q)) = List.append (alpha p) (alpha q)
+  | alpha (And (p, q)) = List.append (alpha p) (alpha q);
+
+fun numadd (Mul (ar, asa), Mul (aza, azb)) = Add (Mul (ar, asa), Mul (aza, azb))
+  | numadd (Mul (ar, asa), Sub (ayy, ayz)) = Add (Mul (ar, asa), Sub (ayy, ayz))
+  | numadd (Mul (ar, asa), Add (Mul (azw, Mul (bas, bat)), ayx)) =
+    Add (Mul (ar, asa), Add (Mul (azw, Mul (bas, bat)), ayx))
+  | numadd (Mul (ar, asa), Add (Mul (azw, Sub (baq, bar)), ayx)) =
+    Add (Mul (ar, asa), Add (Mul (azw, Sub (baq, bar)), ayx))
+  | numadd (Mul (ar, asa), Add (Mul (azw, Add (bao, bap)), ayx)) =
+    Add (Mul (ar, asa), Add (Mul (azw, Add (bao, bap)), ayx))
+  | numadd (Mul (ar, asa), Add (Mul (azw, Neg ban), ayx)) =
+    Add (Mul (ar, asa), Add (Mul (azw, Neg ban), ayx))
+  | numadd (Mul (ar, asa), Add (Mul (azw, Cx (bal, bam)), ayx)) =
+    Add (Mul (ar, asa), Add (Mul (azw, Cx (bal, bam)), ayx))
+  | numadd (Mul (ar, asa), Add (Mul (azw, C baj), ayx)) =
+    Add (Mul (ar, asa), Add (Mul (azw, C baj), ayx))
+  | numadd (Mul (ar, asa), Add (Sub (azu, azv), ayx)) =
+    Add (Mul (ar, asa), Add (Sub (azu, azv), ayx))
+  | numadd (Mul (ar, asa), Add (Add (azs, azt), ayx)) =
+    Add (Mul (ar, asa), Add (Add (azs, azt), ayx))
+  | numadd (Mul (ar, asa), Add (Neg azr, ayx)) =
+    Add (Mul (ar, asa), Add (Neg azr, ayx))
+  | numadd (Mul (ar, asa), Add (Cx (azp, azq), ayx)) =
+    Add (Mul (ar, asa), Add (Cx (azp, azq), ayx))
+  | numadd (Mul (ar, asa), Add (Bound azo, ayx)) =
+    Add (Mul (ar, asa), Add (Bound azo, ayx))
+  | numadd (Mul (ar, asa), Add (C azn, ayx)) =
+    Add (Mul (ar, asa), Add (C azn, ayx))
+  | numadd (Mul (ar, asa), Neg ayv) = Add (Mul (ar, asa), Neg ayv)
+  | numadd (Mul (ar, asa), Cx (ayt, ayu)) = Add (Mul (ar, asa), Cx (ayt, ayu))
+  | numadd (Mul (ar, asa), Bound ays) = Add (Mul (ar, asa), Bound ays)
+  | numadd (Mul (ar, asa), C ayr) = Add (Mul (ar, asa), C ayr)
+  | numadd (Sub (ap, aq), Mul (awm, awn)) = Add (Sub (ap, aq), Mul (awm, awn))
+  | numadd (Sub (ap, aq), Sub (awk, awl)) = Add (Sub (ap, aq), Sub (awk, awl))
+  | numadd (Sub (ap, aq), Add (Mul (axi, Mul (aye, ayf)), awj)) =
+    Add (Sub (ap, aq), Add (Mul (axi, Mul (aye, ayf)), awj))
+  | numadd (Sub (ap, aq), Add (Mul (axi, Sub (ayc, ayd)), awj)) =
+    Add (Sub (ap, aq), Add (Mul (axi, Sub (ayc, ayd)), awj))
+  | numadd (Sub (ap, aq), Add (Mul (axi, Add (aya, ayb)), awj)) =
+    Add (Sub (ap, aq), Add (Mul (axi, Add (aya, ayb)), awj))
+  | numadd (Sub (ap, aq), Add (Mul (axi, Neg axz), awj)) =
+    Add (Sub (ap, aq), Add (Mul (axi, Neg axz), awj))
+  | numadd (Sub (ap, aq), Add (Mul (axi, Cx (axx, axy)), awj)) =
+    Add (Sub (ap, aq), Add (Mul (axi, Cx (axx, axy)), awj))
+  | numadd (Sub (ap, aq), Add (Mul (axi, C axv), awj)) =
+    Add (Sub (ap, aq), Add (Mul (axi, C axv), awj))
+  | numadd (Sub (ap, aq), Add (Sub (axg, axh), awj)) =
+    Add (Sub (ap, aq), Add (Sub (axg, axh), awj))
+  | numadd (Sub (ap, aq), Add (Add (axe, axf), awj)) =
+    Add (Sub (ap, aq), Add (Add (axe, axf), awj))
+  | numadd (Sub (ap, aq), Add (Neg axd, awj)) =
+    Add (Sub (ap, aq), Add (Neg axd, awj))
+  | numadd (Sub (ap, aq), Add (Cx (axb, axc), awj)) =
+    Add (Sub (ap, aq), Add (Cx (axb, axc), awj))
+  | numadd (Sub (ap, aq), Add (Bound axa, awj)) =
+    Add (Sub (ap, aq), Add (Bound axa, awj))
+  | numadd (Sub (ap, aq), Add (C awz, awj)) =
+    Add (Sub (ap, aq), Add (C awz, awj))
+  | numadd (Sub (ap, aq), Neg awh) = Add (Sub (ap, aq), Neg awh)
+  | numadd (Sub (ap, aq), Cx (awf, awg)) = Add (Sub (ap, aq), Cx (awf, awg))
+  | numadd (Sub (ap, aq), Bound awe) = Add (Sub (ap, aq), Bound awe)
+  | numadd (Sub (ap, aq), C awd) = Add (Sub (ap, aq), C awd)
+  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Mul (aty, atz)) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao), Mul (aty, atz))
+  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Sub (atw, atx)) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao), Sub (atw, atx))
   | numadd
-      (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Mul (ain, aio)), afw)) =
-    Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Mul (ain, aio)), afw)))
-  | numadd (Add (Mul (c1, Bound n1), r1), Sub (afx, afy)) =
-    Add (Mul (c1, Bound n1), numadd (r1, Sub (afx, afy)))
-  | numadd (Add (Mul (c1, Bound n1), r1), Mul (afz, aga)) =
-    Add (Mul (c1, Bound n1), numadd (r1, Mul (afz, aga)))
-  | numadd (C w, Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (C w, r2))
-  | numadd (Bound x, Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Bound x, r2))
-  | numadd (CX (y, z), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (CX (y, z), r2))
-  | numadd (Neg ab, Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Neg ab, r2))
-  | numadd (Add (C li, ad), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Add (C li, ad), r2))
-  | numadd (Add (Bound lj, ad), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Add (Bound lj, ad), r2))
-  | numadd (Add (CX (lk, ll), ad), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Add (CX (lk, ll), ad), r2))
-  | numadd (Add (Neg lm, ad), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Add (Neg lm, ad), r2))
-  | numadd (Add (Add (ln, lo), ad), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Add (Add (ln, lo), ad), r2))
-  | numadd (Add (Sub (lp, lq), ad), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Add (Sub (lp, lq), ad), r2))
-  | numadd (Add (Mul (lr, C abv), ad), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, C abv), ad), r2))
-  | numadd (Add (Mul (lr, CX (abx, aby)), ad), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, CX (abx, aby)), ad), r2))
-  | numadd (Add (Mul (lr, Neg abz), ad), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Neg abz), ad), r2))
-  | numadd (Add (Mul (lr, Add (aca, acb)), ad), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Add (aca, acb)), ad), r2))
-  | numadd (Add (Mul (lr, Sub (acc, acd)), ad), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Sub (acc, acd)), ad), r2))
-  | numadd (Add (Mul (lr, Mul (ace, acf)), ad), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Mul (ace, acf)), ad), r2))
-  | numadd (Sub (ae, af), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Sub (ae, af), r2))
-  | numadd (Mul (ag, ah), Add (Mul (c2, Bound n2), r2)) =
-    Add (Mul (c2, Bound n2), numadd (Mul (ag, ah), r2))
-  | numadd (C b1, C b2) = C (b1 + b2)
-  | numadd (C ai, Bound bf) = Add (C ai, Bound bf)
-  | numadd (C ai, CX (bg, bh)) = Add (C ai, CX (bg, bh))
-  | numadd (C ai, Neg bi) = Add (C ai, Neg bi)
-  | numadd (C ai, Add (C ca, bk)) = Add (C ai, Add (C ca, bk))
-  | numadd (C ai, Add (Bound cb, bk)) = Add (C ai, Add (Bound cb, bk))
-  | numadd (C ai, Add (CX (cc, cd), bk)) = Add (C ai, Add (CX (cc, cd), bk))
-  | numadd (C ai, Add (Neg ce, bk)) = Add (C ai, Add (Neg ce, bk))
-  | numadd (C ai, Add (Add (cf, cg), bk)) = Add (C ai, Add (Add (cf, cg), bk))
-  | numadd (C ai, Add (Sub (ch, ci), bk)) = Add (C ai, Add (Sub (ch, ci), bk))
-  | numadd (C ai, Add (Mul (cj, C cw), bk)) =
-    Add (C ai, Add (Mul (cj, C cw), bk))
-  | numadd (C ai, Add (Mul (cj, CX (cy, cz)), bk)) =
-    Add (C ai, Add (Mul (cj, CX (cy, cz)), bk))
-  | numadd (C ai, Add (Mul (cj, Neg da), bk)) =
-    Add (C ai, Add (Mul (cj, Neg da), bk))
-  | numadd (C ai, Add (Mul (cj, Add (db, dc)), bk)) =
-    Add (C ai, Add (Mul (cj, Add (db, dc)), bk))
-  | numadd (C ai, Add (Mul (cj, Sub (dd, de)), bk)) =
-    Add (C ai, Add (Mul (cj, Sub (dd, de)), bk))
-  | numadd (C ai, Add (Mul (cj, Mul (df, dg)), bk)) =
-    Add (C ai, Add (Mul (cj, Mul (df, dg)), bk))
-  | numadd (C ai, Sub (bl, bm)) = Add (C ai, Sub (bl, bm))
-  | numadd (C ai, Mul (bn, bo)) = Add (C ai, Mul (bn, bo))
-  | numadd (Bound aj, C ds) = Add (Bound aj, C ds)
-  | numadd (Bound aj, Bound dt) = Add (Bound aj, Bound dt)
-  | numadd (Bound aj, CX (du, dv)) = Add (Bound aj, CX (du, dv))
-  | numadd (Bound aj, Neg dw) = Add (Bound aj, Neg dw)
-  | numadd (Bound aj, Add (C eo, dy)) = Add (Bound aj, Add (C eo, dy))
-  | numadd (Bound aj, Add (Bound ep, dy)) = Add (Bound aj, Add (Bound ep, dy))
-  | numadd (Bound aj, Add (CX (eq, er), dy)) =
-    Add (Bound aj, Add (CX (eq, er), dy))
-  | numadd (Bound aj, Add (Neg es, dy)) = Add (Bound aj, Add (Neg es, dy))
-  | numadd (Bound aj, Add (Add (et, eu), dy)) =
-    Add (Bound aj, Add (Add (et, eu), dy))
-  | numadd (Bound aj, Add (Sub (ev, ew), dy)) =
-    Add (Bound aj, Add (Sub (ev, ew), dy))
-  | numadd (Bound aj, Add (Mul (ex, C fk), dy)) =
-    Add (Bound aj, Add (Mul (ex, C fk), dy))
-  | numadd (Bound aj, Add (Mul (ex, CX (fm, fn')), dy)) =
-    Add (Bound aj, Add (Mul (ex, CX (fm, fn')), dy))
-  | numadd (Bound aj, Add (Mul (ex, Neg fo), dy)) =
-    Add (Bound aj, Add (Mul (ex, Neg fo), dy))
-  | numadd (Bound aj, Add (Mul (ex, Add (fp, fq)), dy)) =
-    Add (Bound aj, Add (Mul (ex, Add (fp, fq)), dy))
-  | numadd (Bound aj, Add (Mul (ex, Sub (fr, fs)), dy)) =
-    Add (Bound aj, Add (Mul (ex, Sub (fr, fs)), dy))
-  | numadd (Bound aj, Add (Mul (ex, Mul (ft, fu)), dy)) =
-    Add (Bound aj, Add (Mul (ex, Mul (ft, fu)), dy))
-  | numadd (Bound aj, Sub (dz, ea)) = Add (Bound aj, Sub (dz, ea))
-  | numadd (Bound aj, Mul (eb, ec)) = Add (Bound aj, Mul (eb, ec))
-  | numadd (CX (ak, al), C gg) = Add (CX (ak, al), C gg)
-  | numadd (CX (ak, al), Bound gh) = Add (CX (ak, al), Bound gh)
-  | numadd (CX (ak, al), CX (gi, gj)) = Add (CX (ak, al), CX (gi, gj))
-  | numadd (CX (ak, al), Neg gk) = Add (CX (ak, al), Neg gk)
-  | numadd (CX (ak, al), Add (C hc, gm)) = Add (CX (ak, al), Add (C hc, gm))
-  | numadd (CX (ak, al), Add (Bound hd, gm)) =
-    Add (CX (ak, al), Add (Bound hd, gm))
-  | numadd (CX (ak, al), Add (CX (he, hf), gm)) =
-    Add (CX (ak, al), Add (CX (he, hf), gm))
-  | numadd (CX (ak, al), Add (Neg hg, gm)) = Add (CX (ak, al), Add (Neg hg, gm))
-  | numadd (CX (ak, al), Add (Add (hh, hi), gm)) =
-    Add (CX (ak, al), Add (Add (hh, hi), gm))
-  | numadd (CX (ak, al), Add (Sub (hj, hk), gm)) =
-    Add (CX (ak, al), Add (Sub (hj, hk), gm))
-  | numadd (CX (ak, al), Add (Mul (hl, C hy), gm)) =
-    Add (CX (ak, al), Add (Mul (hl, C hy), gm))
-  | numadd (CX (ak, al), Add (Mul (hl, CX (ia, ib)), gm)) =
-    Add (CX (ak, al), Add (Mul (hl, CX (ia, ib)), gm))
-  | numadd (CX (ak, al), Add (Mul (hl, Neg ic), gm)) =
-    Add (CX (ak, al), Add (Mul (hl, Neg ic), gm))
-  | numadd (CX (ak, al), Add (Mul (hl, Add (id, ie)), gm)) =
-    Add (CX (ak, al), Add (Mul (hl, Add (id, ie)), gm))
-  | numadd (CX (ak, al), Add (Mul (hl, Sub (if', ig)), gm)) =
-    Add (CX (ak, al), Add (Mul (hl, Sub (if', ig)), gm))
-  | numadd (CX (ak, al), Add (Mul (hl, Mul (ih, ii)), gm)) =
-    Add (CX (ak, al), Add (Mul (hl, Mul (ih, ii)), gm))
-  | numadd (CX (ak, al), Sub (gn, go)) = Add (CX (ak, al), Sub (gn, go))
-  | numadd (CX (ak, al), Mul (gp, gq)) = Add (CX (ak, al), Mul (gp, gq))
-  | numadd (Neg am, C iu) = Add (Neg am, C iu)
-  | numadd (Neg am, Bound iv) = Add (Neg am, Bound iv)
-  | numadd (Neg am, CX (iw, ix)) = Add (Neg am, CX (iw, ix))
-  | numadd (Neg am, Neg iy) = Add (Neg am, Neg iy)
-  | numadd (Neg am, Add (C jq, ja)) = Add (Neg am, Add (C jq, ja))
-  | numadd (Neg am, Add (Bound jr, ja)) = Add (Neg am, Add (Bound jr, ja))
-  | numadd (Neg am, Add (CX (js, jt), ja)) = Add (Neg am, Add (CX (js, jt), ja))
-  | numadd (Neg am, Add (Neg ju, ja)) = Add (Neg am, Add (Neg ju, ja))
-  | numadd (Neg am, Add (Add (jv, jw), ja)) =
-    Add (Neg am, Add (Add (jv, jw), ja))
-  | numadd (Neg am, Add (Sub (jx, jy), ja)) =
-    Add (Neg am, Add (Sub (jx, jy), ja))
-  | numadd (Neg am, Add (Mul (jz, C km), ja)) =
-    Add (Neg am, Add (Mul (jz, C km), ja))
-  | numadd (Neg am, Add (Mul (jz, CX (ko, kp)), ja)) =
-    Add (Neg am, Add (Mul (jz, CX (ko, kp)), ja))
-  | numadd (Neg am, Add (Mul (jz, Neg kq), ja)) =
-    Add (Neg am, Add (Mul (jz, Neg kq), ja))
-  | numadd (Neg am, Add (Mul (jz, Add (kr, ks)), ja)) =
-    Add (Neg am, Add (Mul (jz, Add (kr, ks)), ja))
-  | numadd (Neg am, Add (Mul (jz, Sub (kt, ku)), ja)) =
-    Add (Neg am, Add (Mul (jz, Sub (kt, ku)), ja))
-  | numadd (Neg am, Add (Mul (jz, Mul (kv, kw)), ja)) =
-    Add (Neg am, Add (Mul (jz, Mul (kv, kw)), ja))
-  | numadd (Neg am, Sub (jb, jc)) = Add (Neg am, Sub (jb, jc))
-  | numadd (Neg am, Mul (jd, je)) = Add (Neg am, Mul (jd, je))
-  | numadd (Add (C lt, ao), C mp) = Add (Add (C lt, ao), C mp)
-  | numadd (Add (C lt, ao), Bound mq) = Add (Add (C lt, ao), Bound mq)
-  | numadd (Add (C lt, ao), CX (mr, ms)) = Add (Add (C lt, ao), CX (mr, ms))
-  | numadd (Add (C lt, ao), Neg mt) = Add (Add (C lt, ao), Neg mt)
-  | numadd (Add (C lt, ao), Add (C nl, mv)) =
-    Add (Add (C lt, ao), Add (C nl, mv))
-  | numadd (Add (C lt, ao), Add (Bound nm, mv)) =
-    Add (Add (C lt, ao), Add (Bound nm, mv))
-  | numadd (Add (C lt, ao), Add (CX (nn, no), mv)) =
-    Add (Add (C lt, ao), Add (CX (nn, no), mv))
-  | numadd (Add (C lt, ao), Add (Neg np, mv)) =
-    Add (Add (C lt, ao), Add (Neg np, mv))
-  | numadd (Add (C lt, ao), Add (Add (nq, nr), mv)) =
-    Add (Add (C lt, ao), Add (Add (nq, nr), mv))
-  | numadd (Add (C lt, ao), Add (Sub (ns, nt), mv)) =
-    Add (Add (C lt, ao), Add (Sub (ns, nt), mv))
-  | numadd (Add (C lt, ao), Add (Mul (nu, C oh), mv)) =
-    Add (Add (C lt, ao), Add (Mul (nu, C oh), mv))
-  | numadd (Add (C lt, ao), Add (Mul (nu, CX (oj, ok)), mv)) =
-    Add (Add (C lt, ao), Add (Mul (nu, CX (oj, ok)), mv))
-  | numadd (Add (C lt, ao), Add (Mul (nu, Neg ol), mv)) =
-    Add (Add (C lt, ao), Add (Mul (nu, Neg ol), mv))
-  | numadd (Add (C lt, ao), Add (Mul (nu, Add (om, on)), mv)) =
-    Add (Add (C lt, ao), Add (Mul (nu, Add (om, on)), mv))
-  | numadd (Add (C lt, ao), Add (Mul (nu, Sub (oo, op')), mv)) =
-    Add (Add (C lt, ao), Add (Mul (nu, Sub (oo, op')), mv))
-  | numadd (Add (C lt, ao), Add (Mul (nu, Mul (oq, or)), mv)) =
-    Add (Add (C lt, ao), Add (Mul (nu, Mul (oq, or)), mv))
-  | numadd (Add (C lt, ao), Sub (mw, mx)) = Add (Add (C lt, ao), Sub (mw, mx))
-  | numadd (Add (C lt, ao), Mul (my, mz)) = Add (Add (C lt, ao), Mul (my, mz))
-  | numadd (Add (Bound lu, ao), C pd) = Add (Add (Bound lu, ao), C pd)
-  | numadd (Add (Bound lu, ao), Bound pe) = Add (Add (Bound lu, ao), Bound pe)
-  | numadd (Add (Bound lu, ao), CX (pf, pg)) =
-    Add (Add (Bound lu, ao), CX (pf, pg))
-  | numadd (Add (Bound lu, ao), Neg ph) = Add (Add (Bound lu, ao), Neg ph)
-  | numadd (Add (Bound lu, ao), Add (C pz, pj)) =
-    Add (Add (Bound lu, ao), Add (C pz, pj))
-  | numadd (Add (Bound lu, ao), Add (Bound qa, pj)) =
-    Add (Add (Bound lu, ao), Add (Bound qa, pj))
-  | numadd (Add (Bound lu, ao), Add (CX (qb, qc), pj)) =
-    Add (Add (Bound lu, ao), Add (CX (qb, qc), pj))
-  | numadd (Add (Bound lu, ao), Add (Neg qd, pj)) =
-    Add (Add (Bound lu, ao), Add (Neg qd, pj))
-  | numadd (Add (Bound lu, ao), Add (Add (qe, qf), pj)) =
-    Add (Add (Bound lu, ao), Add (Add (qe, qf), pj))
-  | numadd (Add (Bound lu, ao), Add (Sub (qg, qh), pj)) =
-    Add (Add (Bound lu, ao), Add (Sub (qg, qh), pj))
-  | numadd (Add (Bound lu, ao), Add (Mul (qi, C qv), pj)) =
-    Add (Add (Bound lu, ao), Add (Mul (qi, C qv), pj))
-  | numadd (Add (Bound lu, ao), Add (Mul (qi, CX (qx, qy)), pj)) =
-    Add (Add (Bound lu, ao), Add (Mul (qi, CX (qx, qy)), pj))
-  | numadd (Add (Bound lu, ao), Add (Mul (qi, Neg qz), pj)) =
-    Add (Add (Bound lu, ao), Add (Mul (qi, Neg qz), pj))
-  | numadd (Add (Bound lu, ao), Add (Mul (qi, Add (ra, rb)), pj)) =
-    Add (Add (Bound lu, ao), Add (Mul (qi, Add (ra, rb)), pj))
-  | numadd (Add (Bound lu, ao), Add (Mul (qi, Sub (rc, rd)), pj)) =
-    Add (Add (Bound lu, ao), Add (Mul (qi, Sub (rc, rd)), pj))
-  | numadd (Add (Bound lu, ao), Add (Mul (qi, Mul (re, rf)), pj)) =
-    Add (Add (Bound lu, ao), Add (Mul (qi, Mul (re, rf)), pj))
-  | numadd (Add (Bound lu, ao), Sub (pk, pl)) =
-    Add (Add (Bound lu, ao), Sub (pk, pl))
-  | numadd (Add (Bound lu, ao), Mul (pm, pn)) =
-    Add (Add (Bound lu, ao), Mul (pm, pn))
-  | numadd (Add (CX (lv, lw), ao), C rr) = Add (Add (CX (lv, lw), ao), C rr)
-  | numadd (Add (CX (lv, lw), ao), Bound rs) =
-    Add (Add (CX (lv, lw), ao), Bound rs)
-  | numadd (Add (CX (lv, lw), ao), CX (rt, ru)) =
-    Add (Add (CX (lv, lw), ao), CX (rt, ru))
-  | numadd (Add (CX (lv, lw), ao), Neg rv) = Add (Add (CX (lv, lw), ao), Neg rv)
-  | numadd (Add (CX (lv, lw), ao), Add (C sn, rx)) =
-    Add (Add (CX (lv, lw), ao), Add (C sn, rx))
-  | numadd (Add (CX (lv, lw), ao), Add (Bound so, rx)) =
-    Add (Add (CX (lv, lw), ao), Add (Bound so, rx))
-  | numadd (Add (CX (lv, lw), ao), Add (CX (sp, sq), rx)) =
-    Add (Add (CX (lv, lw), ao), Add (CX (sp, sq), rx))
-  | numadd (Add (CX (lv, lw), ao), Add (Neg sr, rx)) =
-    Add (Add (CX (lv, lw), ao), Add (Neg sr, rx))
-  | numadd (Add (CX (lv, lw), ao), Add (Add (ss, st), rx)) =
-    Add (Add (CX (lv, lw), ao), Add (Add (ss, st), rx))
-  | numadd (Add (CX (lv, lw), ao), Add (Sub (su, sv), rx)) =
-    Add (Add (CX (lv, lw), ao), Add (Sub (su, sv), rx))
-  | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, C tj), rx)) =
-    Add (Add (CX (lv, lw), ao), Add (Mul (sw, C tj), rx))
-  | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, CX (tl, tm)), rx)) =
-    Add (Add (CX (lv, lw), ao), Add (Mul (sw, CX (tl, tm)), rx))
-  | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, Neg tn), rx)) =
-    Add (Add (CX (lv, lw), ao), Add (Mul (sw, Neg tn), rx))
-  | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, Add (to, tp)), rx)) =
-    Add (Add (CX (lv, lw), ao), Add (Mul (sw, Add (to, tp)), rx))
-  | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, Sub (tq, tr)), rx)) =
-    Add (Add (CX (lv, lw), ao), Add (Mul (sw, Sub (tq, tr)), rx))
-  | numadd (Add (CX (lv, lw), ao), Add (Mul (sw, Mul (ts, tt)), rx)) =
-    Add (Add (CX (lv, lw), ao), Add (Mul (sw, Mul (ts, tt)), rx))
-  | numadd (Add (CX (lv, lw), ao), Sub (ry, rz)) =
-    Add (Add (CX (lv, lw), ao), Sub (ry, rz))
-  | numadd (Add (CX (lv, lw), ao), Mul (sa, sb)) =
-    Add (Add (CX (lv, lw), ao), Mul (sa, sb))
-  | numadd (Add (Neg lx, ao), C uf) = Add (Add (Neg lx, ao), C uf)
-  | numadd (Add (Neg lx, ao), Bound ug) = Add (Add (Neg lx, ao), Bound ug)
-  | numadd (Add (Neg lx, ao), CX (uh, ui)) = Add (Add (Neg lx, ao), CX (uh, ui))
-  | numadd (Add (Neg lx, ao), Neg uj) = Add (Add (Neg lx, ao), Neg uj)
-  | numadd (Add (Neg lx, ao), Add (C vb, ul)) =
-    Add (Add (Neg lx, ao), Add (C vb, ul))
-  | numadd (Add (Neg lx, ao), Add (Bound vc, ul)) =
-    Add (Add (Neg lx, ao), Add (Bound vc, ul))
-  | numadd (Add (Neg lx, ao), Add (CX (vd, ve), ul)) =
-    Add (Add (Neg lx, ao), Add (CX (vd, ve), ul))
-  | numadd (Add (Neg lx, ao), Add (Neg vf, ul)) =
-    Add (Add (Neg lx, ao), Add (Neg vf, ul))
-  | numadd (Add (Neg lx, ao), Add (Add (vg, vh), ul)) =
-    Add (Add (Neg lx, ao), Add (Add (vg, vh), ul))
-  | numadd (Add (Neg lx, ao), Add (Sub (vi, vj), ul)) =
-    Add (Add (Neg lx, ao), Add (Sub (vi, vj), ul))
-  | numadd (Add (Neg lx, ao), Add (Mul (vk, C vx), ul)) =
-    Add (Add (Neg lx, ao), Add (Mul (vk, C vx), ul))
-  | numadd (Add (Neg lx, ao), Add (Mul (vk, CX (vz, wa)), ul)) =
-    Add (Add (Neg lx, ao), Add (Mul (vk, CX (vz, wa)), ul))
-  | numadd (Add (Neg lx, ao), Add (Mul (vk, Neg wb), ul)) =
-    Add (Add (Neg lx, ao), Add (Mul (vk, Neg wb), ul))
-  | numadd (Add (Neg lx, ao), Add (Mul (vk, Add (wc, wd)), ul)) =
-    Add (Add (Neg lx, ao), Add (Mul (vk, Add (wc, wd)), ul))
-  | numadd (Add (Neg lx, ao), Add (Mul (vk, Sub (we, wf)), ul)) =
-    Add (Add (Neg lx, ao), Add (Mul (vk, Sub (we, wf)), ul))
-  | numadd (Add (Neg lx, ao), Add (Mul (vk, Mul (wg, wh)), ul)) =
-    Add (Add (Neg lx, ao), Add (Mul (vk, Mul (wg, wh)), ul))
-  | numadd (Add (Neg lx, ao), Sub (um, un)) =
-    Add (Add (Neg lx, ao), Sub (um, un))
-  | numadd (Add (Neg lx, ao), Mul (uo, up)) =
-    Add (Add (Neg lx, ao), Mul (uo, up))
-  | numadd (Add (Add (ly, lz), ao), C wt) = Add (Add (Add (ly, lz), ao), C wt)
-  | numadd (Add (Add (ly, lz), ao), Bound wu) =
-    Add (Add (Add (ly, lz), ao), Bound wu)
-  | numadd (Add (Add (ly, lz), ao), CX (wv, ww)) =
-    Add (Add (Add (ly, lz), ao), CX (wv, ww))
-  | numadd (Add (Add (ly, lz), ao), Neg wx) =
-    Add (Add (Add (ly, lz), ao), Neg wx)
-  | numadd (Add (Add (ly, lz), ao), Add (C xp, wz)) =
-    Add (Add (Add (ly, lz), ao), Add (C xp, wz))
-  | numadd (Add (Add (ly, lz), ao), Add (Bound xq, wz)) =
-    Add (Add (Add (ly, lz), ao), Add (Bound xq, wz))
-  | numadd (Add (Add (ly, lz), ao), Add (CX (xr, xs), wz)) =
-    Add (Add (Add (ly, lz), ao), Add (CX (xr, xs), wz))
-  | numadd (Add (Add (ly, lz), ao), Add (Neg xt, wz)) =
-    Add (Add (Add (ly, lz), ao), Add (Neg xt, wz))
-  | numadd (Add (Add (ly, lz), ao), Add (Add (xu, xv), wz)) =
-    Add (Add (Add (ly, lz), ao), Add (Add (xu, xv), wz))
-  | numadd (Add (Add (ly, lz), ao), Add (Sub (xw, xx), wz)) =
-    Add (Add (Add (ly, lz), ao), Add (Sub (xw, xx), wz))
-  | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, C yl), wz)) =
-    Add (Add (Add (ly, lz), ao), Add (Mul (xy, C yl), wz))
-  | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, CX (yn, yo)), wz)) =
-    Add (Add (Add (ly, lz), ao), Add (Mul (xy, CX (yn, yo)), wz))
-  | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Neg yp), wz)) =
-    Add (Add (Add (ly, lz), ao), Add (Mul (xy, Neg yp), wz))
-  | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Add (yq, yr)), wz)) =
-    Add (Add (Add (ly, lz), ao), Add (Mul (xy, Add (yq, yr)), wz))
-  | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Sub (ys, yt)), wz)) =
-    Add (Add (Add (ly, lz), ao), Add (Mul (xy, Sub (ys, yt)), wz))
-  | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Mul (yu, yv)), wz)) =
-    Add (Add (Add (ly, lz), ao), Add (Mul (xy, Mul (yu, yv)), wz))
-  | numadd (Add (Add (ly, lz), ao), Sub (xa, xb)) =
-    Add (Add (Add (ly, lz), ao), Sub (xa, xb))
-  | numadd (Add (Add (ly, lz), ao), Mul (xc, xd)) =
-    Add (Add (Add (ly, lz), ao), Mul (xc, xd))
-  | numadd (Add (Sub (ma, mb), ao), C zh) = Add (Add (Sub (ma, mb), ao), C zh)
-  | numadd (Add (Sub (ma, mb), ao), Bound zi) =
-    Add (Add (Sub (ma, mb), ao), Bound zi)
-  | numadd (Add (Sub (ma, mb), ao), CX (zj, zk)) =
-    Add (Add (Sub (ma, mb), ao), CX (zj, zk))
-  | numadd (Add (Sub (ma, mb), ao), Neg zl) =
-    Add (Add (Sub (ma, mb), ao), Neg zl)
-  | numadd (Add (Sub (ma, mb), ao), Add (C aad, zn)) =
-    Add (Add (Sub (ma, mb), ao), Add (C aad, zn))
-  | numadd (Add (Sub (ma, mb), ao), Add (Bound aae, zn)) =
-    Add (Add (Sub (ma, mb), ao), Add (Bound aae, zn))
-  | numadd (Add (Sub (ma, mb), ao), Add (CX (aaf, aag), zn)) =
-    Add (Add (Sub (ma, mb), ao), Add (CX (aaf, aag), zn))
-  | numadd (Add (Sub (ma, mb), ao), Add (Neg aah, zn)) =
-    Add (Add (Sub (ma, mb), ao), Add (Neg aah, zn))
-  | numadd (Add (Sub (ma, mb), ao), Add (Add (aai, aaj), zn)) =
-    Add (Add (Sub (ma, mb), ao), Add (Add (aai, aaj), zn))
-  | numadd (Add (Sub (ma, mb), ao), Add (Sub (aak, aal), zn)) =
-    Add (Add (Sub (ma, mb), ao), Add (Sub (aak, aal), zn))
-  | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, C aaz), zn)) =
-    Add (Add (Sub (ma, mb), ao), Add (Mul (aam, C aaz), zn))
-  | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, CX (abb, abc)), zn)) =
-    Add (Add (Sub (ma, mb), ao), Add (Mul (aam, CX (abb, abc)), zn))
-  | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Neg abd), zn)) =
-    Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Neg abd), zn))
-  | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Add (abe, abf)), zn)) =
-    Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Add (abe, abf)), zn))
-  | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Sub (abg, abh)), zn)) =
-    Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Sub (abg, abh)), zn))
-  | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Mul (abi, abj)), zn)) =
-    Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Mul (abi, abj)), zn))
-  | numadd (Add (Sub (ma, mb), ao), Sub (zo, zp)) =
-    Add (Add (Sub (ma, mb), ao), Sub (zo, zp))
-  | numadd (Add (Sub (ma, mb), ao), Mul (zq, zr)) =
-    Add (Add (Sub (ma, mb), ao), Mul (zq, zr))
-  | numadd (Add (Mul (mc, C acg), ao), C adc) =
-    Add (Add (Mul (mc, C acg), ao), C adc)
-  | numadd (Add (Mul (mc, C acg), ao), Bound add) =
-    Add (Add (Mul (mc, C acg), ao), Bound add)
-  | numadd (Add (Mul (mc, C acg), ao), CX (ade, adf)) =
-    Add (Add (Mul (mc, C acg), ao), CX (ade, adf))
-  | numadd (Add (Mul (mc, C acg), ao), Neg adg) =
-    Add (Add (Mul (mc, C acg), ao), Neg adg)
-  | numadd (Add (Mul (mc, C acg), ao), Add (C ady, adi)) =
-    Add (Add (Mul (mc, C acg), ao), Add (C ady, adi))
-  | numadd (Add (Mul (mc, C acg), ao), Add (Bound adz, adi)) =
-    Add (Add (Mul (mc, C acg), ao), Add (Bound adz, adi))
-  | numadd (Add (Mul (mc, C acg), ao), Add (CX (aea, aeb), adi)) =
-    Add (Add (Mul (mc, C acg), ao), Add (CX (aea, aeb), adi))
-  | numadd (Add (Mul (mc, C acg), ao), Add (Neg aec, adi)) =
-    Add (Add (Mul (mc, C acg), ao), Add (Neg aec, adi))
-  | numadd (Add (Mul (mc, C acg), ao), Add (Add (aed, aee), adi)) =
-    Add (Add (Mul (mc, C acg), ao), Add (Add (aed, aee), adi))
-  | numadd (Add (Mul (mc, C acg), ao), Add (Sub (aef, aeg), adi)) =
-    Add (Add (Mul (mc, C acg), ao), Add (Sub (aef, aeg), adi))
-  | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, C aeu), adi)) =
-    Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, C aeu), adi))
-  | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, CX (aew, aex)), adi)) =
-    Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, CX (aew, aex)), adi))
-  | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Neg aey), adi)) =
-    Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Neg aey), adi))
-  | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Add (aez, afa)), adi)) =
-    Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Add (aez, afa)), adi))
-  | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Sub (afb, afc)), adi)) =
-    Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Sub (afb, afc)), adi))
-  | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Mul (afd, afe)), adi)) =
-    Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Mul (afd, afe)), adi))
-  | numadd (Add (Mul (mc, C acg), ao), Sub (adj, adk)) =
-    Add (Add (Mul (mc, C acg), ao), Sub (adj, adk))
-  | numadd (Add (Mul (mc, C acg), ao), Mul (adl, adm)) =
-    Add (Add (Mul (mc, C acg), ao), Mul (adl, adm))
-  | numadd (Add (Mul (mc, CX (aci, acj)), ao), C ajl) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), C ajl)
-  | numadd (Add (Mul (mc, CX (aci, acj)), ao), Bound ajm) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), Bound ajm)
-  | numadd (Add (Mul (mc, CX (aci, acj)), ao), CX (ajn, ajo)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), CX (ajn, ajo))
-  | numadd (Add (Mul (mc, CX (aci, acj)), ao), Neg ajp) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), Neg ajp)
-  | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (C akh, ajr)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), Add (C akh, ajr))
-  | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Bound aki, ajr)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Bound aki, ajr))
-  | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (CX (akj, akk), ajr)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), Add (CX (akj, akk), ajr))
-  | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Neg akl, ajr)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Neg akl, ajr))
-  | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Add (akm, akn), ajr)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Add (akm, akn), ajr))
-  | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Sub (ako, akp), ajr)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Sub (ako, akp), ajr))
-  | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, C ald), ajr)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, C ald), ajr))
+    (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Mul (avq, avr)), atv)) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao),
+          Add (Mul (auu, Mul (avq, avr)), atv))
   | numadd
-      (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, CX (alf, alg)), ajr)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, CX (alf, alg)), ajr))
-  | numadd (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, Neg alh), ajr)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), Add (Mul (akq, Neg alh), ajr))
-  | numadd
-      (Add (Mul (mc, CX (aci, acj)), ao),
-        Add (Mul (akq, Add (ali, alj)), ajr)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao),
-          Add (Mul (akq, Add (ali, alj)), ajr))
-  | numadd
-      (Add (Mul (mc, CX (aci, acj)), ao),
-        Add (Mul (akq, Sub (alk, all)), ajr)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao),
-          Add (Mul (akq, Sub (alk, all)), ajr))
-  | numadd
-      (Add (Mul (mc, CX (aci, acj)), ao),
-        Add (Mul (akq, Mul (alm, aln)), ajr)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao),
-          Add (Mul (akq, Mul (alm, aln)), ajr))
-  | numadd (Add (Mul (mc, CX (aci, acj)), ao), Sub (ajs, ajt)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), Sub (ajs, ajt))
-  | numadd (Add (Mul (mc, CX (aci, acj)), ao), Mul (aju, ajv)) =
-    Add (Add (Mul (mc, CX (aci, acj)), ao), Mul (aju, ajv))
-  | numadd (Add (Mul (mc, Neg ack), ao), C alz) =
-    Add (Add (Mul (mc, Neg ack), ao), C alz)
-  | numadd (Add (Mul (mc, Neg ack), ao), Bound ama) =
-    Add (Add (Mul (mc, Neg ack), ao), Bound ama)
-  | numadd (Add (Mul (mc, Neg ack), ao), CX (amb, amc)) =
-    Add (Add (Mul (mc, Neg ack), ao), CX (amb, amc))
-  | numadd (Add (Mul (mc, Neg ack), ao), Neg amd) =
-    Add (Add (Mul (mc, Neg ack), ao), Neg amd)
-  | numadd (Add (Mul (mc, Neg ack), ao), Add (C amv, amf)) =
-    Add (Add (Mul (mc, Neg ack), ao), Add (C amv, amf))
-  | numadd (Add (Mul (mc, Neg ack), ao), Add (Bound amw, amf)) =
-    Add (Add (Mul (mc, Neg ack), ao), Add (Bound amw, amf))
-  | numadd (Add (Mul (mc, Neg ack), ao), Add (CX (amx, amy), amf)) =
-    Add (Add (Mul (mc, Neg ack), ao), Add (CX (amx, amy), amf))
-  | numadd (Add (Mul (mc, Neg ack), ao), Add (Neg amz, amf)) =
-    Add (Add (Mul (mc, Neg ack), ao), Add (Neg amz, amf))
-  | numadd (Add (Mul (mc, Neg ack), ao), Add (Add (ana, anb), amf)) =
-    Add (Add (Mul (mc, Neg ack), ao), Add (Add (ana, anb), amf))
-  | numadd (Add (Mul (mc, Neg ack), ao), Add (Sub (anc, and'), amf)) =
-    Add (Add (Mul (mc, Neg ack), ao), Add (Sub (anc, and'), amf))
-  | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, C anr), amf)) =
-    Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, C anr), amf))
-  | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, CX (ant, anu)), amf)) =
-    Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, CX (ant, anu)), amf))
-  | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Neg anv), amf)) =
-    Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Neg anv), amf))
-  | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Add (anw, anx)), amf)) =
-    Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Add (anw, anx)), amf))
-  | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Sub (any, anz)), amf)) =
-    Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Sub (any, anz)), amf))
-  | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Mul (aoa, aob)), amf)) =
-    Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Mul (aoa, aob)), amf))
-  | numadd (Add (Mul (mc, Neg ack), ao), Sub (amg, amh)) =
-    Add (Add (Mul (mc, Neg ack), ao), Sub (amg, amh))
-  | numadd (Add (Mul (mc, Neg ack), ao), Mul (ami, amj)) =
-    Add (Add (Mul (mc, Neg ack), ao), Mul (ami, amj))
-  | numadd (Add (Mul (mc, Add (acl, acm)), ao), C aon) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao), C aon)
-  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Bound aoo) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao), Bound aoo)
-  | numadd (Add (Mul (mc, Add (acl, acm)), ao), CX (aop, aoq)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao), CX (aop, aoq))
-  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Neg aor) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao), Neg aor)
-  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (C apj, aot)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (C apj, aot))
-  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Bound apk, aot)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Bound apk, aot))
-  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (CX (apl, apm), aot)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (CX (apl, apm), aot))
-  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Neg apn, aot)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Neg apn, aot))
-  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Add (apo, app), aot)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Add (apo, app), aot))
-  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Sub (apq, apr), aot)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Sub (apq, apr), aot))
-  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, C aqf), aot)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, C aqf), aot))
-  | numadd
-      (Add (Mul (mc, Add (acl, acm)), ao),
-        Add (Mul (aps, CX (aqh, aqi)), aot)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao),
-          Add (Mul (aps, CX (aqh, aqi)), aot))
-  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Neg aqj), aot)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Neg aqj), aot))
-  | numadd
-      (Add (Mul (mc, Add (acl, acm)), ao),
-        Add (Mul (aps, Add (aqk, aql)), aot)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao),
-          Add (Mul (aps, Add (aqk, aql)), aot))
-  | numadd
-      (Add (Mul (mc, Add (acl, acm)), ao),
-        Add (Mul (aps, Sub (aqm, aqn)), aot)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao),
-          Add (Mul (aps, Sub (aqm, aqn)), aot))
-  | numadd
-      (Add (Mul (mc, Add (acl, acm)), ao),
-        Add (Mul (aps, Mul (aqo, aqp)), aot)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao),
-          Add (Mul (aps, Mul (aqo, aqp)), aot))
-  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Sub (aou, aov)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao), Sub (aou, aov))
-  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Mul (aow, aox)) =
-    Add (Add (Mul (mc, Add (acl, acm)), ao), Mul (aow, aox))
-  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), C arb) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao), C arb)
-  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Bound arc) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao), Bound arc)
-  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), CX (ard, are)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao), CX (ard, are))
-  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Neg arf) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao), Neg arf)
-  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (C arx, arh)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (C arx, arh))
-  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Bound ary, arh)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Bound ary, arh))
-  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (CX (arz, asa), arh)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (CX (arz, asa), arh))
-  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Neg asb, arh)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Neg asb, arh))
-  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Add (asc, asd), arh)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Add (asc, asd), arh))
-  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Sub (ase, asf), arh)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Sub (ase, asf), arh))
-  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, C ast), arh)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, C ast), arh))
-  | numadd
-      (Add (Mul (mc, Sub (acn, aco)), ao),
-        Add (Mul (asg, CX (asv, asw)), arh)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao),
-          Add (Mul (asg, CX (asv, asw)), arh))
-  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Neg asx), arh)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Neg asx), arh))
-  | numadd
-      (Add (Mul (mc, Sub (acn, aco)), ao),
-        Add (Mul (asg, Add (asy, asz)), arh)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao),
-          Add (Mul (asg, Add (asy, asz)), arh))
-  | numadd
-      (Add (Mul (mc, Sub (acn, aco)), ao),
-        Add (Mul (asg, Sub (ata, atb)), arh)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao),
-          Add (Mul (asg, Sub (ata, atb)), arh))
-  | numadd
-      (Add (Mul (mc, Sub (acn, aco)), ao),
-        Add (Mul (asg, Mul (atc, atd)), arh)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao),
-          Add (Mul (asg, Mul (atc, atd)), arh))
-  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Sub (ari, arj)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao), Sub (ari, arj))
-  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Mul (ark, arl)) =
-    Add (Add (Mul (mc, Sub (acn, aco)), ao), Mul (ark, arl))
-  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), C atp) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao), C atp)
-  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Bound atq) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao), Bound atq)
-  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), CX (atr, ats)) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao), CX (atr, ats))
-  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Neg att) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao), Neg att)
-  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (C aul, atv)) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (C aul, atv))
-  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Bound aum, atv)) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Bound aum, atv))
-  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (CX (aun, auo), atv)) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (CX (aun, auo), atv))
-  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Neg aup, atv)) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Neg aup, atv))
-  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Add (auq, aur), atv)) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Add (auq, aur), atv))
-  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Sub (aus, aut), atv)) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Sub (aus, aut), atv))
-  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, C avh), atv)) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, C avh), atv))
-  | numadd
-      (Add (Mul (mc, Mul (acp, acq)), ao),
-        Add (Mul (auu, CX (avj, avk)), atv)) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao),
-          Add (Mul (auu, CX (avj, avk)), atv))
-  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Neg avl), atv)) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Neg avl), atv))
-  | numadd
-      (Add (Mul (mc, Mul (acp, acq)), ao),
-        Add (Mul (auu, Add (avm, avn)), atv)) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao),
-          Add (Mul (auu, Add (avm, avn)), atv))
-  | numadd
-      (Add (Mul (mc, Mul (acp, acq)), ao),
-        Add (Mul (auu, Sub (avo, avp)), atv)) =
+    (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Sub (avo, avp)), atv)) =
     Add (Add (Mul (mc, Mul (acp, acq)), ao),
           Add (Mul (auu, Sub (avo, avp)), atv))
   | numadd
-      (Add (Mul (mc, Mul (acp, acq)), ao),
-        Add (Mul (auu, Mul (avq, avr)), atv)) =
+    (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Add (avm, avn)), atv)) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao),
+          Add (Mul (auu, Add (avm, avn)), atv))
+  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Neg avl), atv)) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Neg avl), atv))
+  | numadd
+    (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, Cx (avj, avk)), atv)) =
     Add (Add (Mul (mc, Mul (acp, acq)), ao),
-          Add (Mul (auu, Mul (avq, avr)), atv))
-  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Sub (atw, atx)) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao), Sub (atw, atx))
-  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Mul (aty, atz)) =
-    Add (Add (Mul (mc, Mul (acp, acq)), ao), Mul (aty, atz))
-  | numadd (Sub (ap, aq), C awd) = Add (Sub (ap, aq), C awd)
-  | numadd (Sub (ap, aq), Bound awe) = Add (Sub (ap, aq), Bound awe)
-  | numadd (Sub (ap, aq), CX (awf, awg)) = Add (Sub (ap, aq), CX (awf, awg))
-  | numadd (Sub (ap, aq), Neg awh) = Add (Sub (ap, aq), Neg awh)
-  | numadd (Sub (ap, aq), Add (C awz, awj)) =
-    Add (Sub (ap, aq), Add (C awz, awj))
-  | numadd (Sub (ap, aq), Add (Bound axa, awj)) =
-    Add (Sub (ap, aq), Add (Bound axa, awj))
-  | numadd (Sub (ap, aq), Add (CX (axb, axc), awj)) =
-    Add (Sub (ap, aq), Add (CX (axb, axc), awj))
-  | numadd (Sub (ap, aq), Add (Neg axd, awj)) =
-    Add (Sub (ap, aq), Add (Neg axd, awj))
-  | numadd (Sub (ap, aq), Add (Add (axe, axf), awj)) =
-    Add (Sub (ap, aq), Add (Add (axe, axf), awj))
-  | numadd (Sub (ap, aq), Add (Sub (axg, axh), awj)) =
-    Add (Sub (ap, aq), Add (Sub (axg, axh), awj))
-  | numadd (Sub (ap, aq), Add (Mul (axi, C axv), awj)) =
-    Add (Sub (ap, aq), Add (Mul (axi, C axv), awj))
-  | numadd (Sub (ap, aq), Add (Mul (axi, CX (axx, axy)), awj)) =
-    Add (Sub (ap, aq), Add (Mul (axi, CX (axx, axy)), awj))
-  | numadd (Sub (ap, aq), Add (Mul (axi, Neg axz), awj)) =
-    Add (Sub (ap, aq), Add (Mul (axi, Neg axz), awj))
-  | numadd (Sub (ap, aq), Add (Mul (axi, Add (aya, ayb)), awj)) =
-    Add (Sub (ap, aq), Add (Mul (axi, Add (aya, ayb)), awj))
-  | numadd (Sub (ap, aq), Add (Mul (axi, Sub (ayc, ayd)), awj)) =
-    Add (Sub (ap, aq), Add (Mul (axi, Sub (ayc, ayd)), awj))
-  | numadd (Sub (ap, aq), Add (Mul (axi, Mul (aye, ayf)), awj)) =
-    Add (Sub (ap, aq), Add (Mul (axi, Mul (aye, ayf)), awj))
-  | numadd (Sub (ap, aq), Sub (awk, awl)) = Add (Sub (ap, aq), Sub (awk, awl))
-  | numadd (Sub (ap, aq), Mul (awm, awn)) = Add (Sub (ap, aq), Mul (awm, awn))
-  | numadd (Mul (ar, as'), C ayr) = Add (Mul (ar, as'), C ayr)
-  | numadd (Mul (ar, as'), Bound ays) = Add (Mul (ar, as'), Bound ays)
-  | numadd (Mul (ar, as'), CX (ayt, ayu)) = Add (Mul (ar, as'), CX (ayt, ayu))
-  | numadd (Mul (ar, as'), Neg ayv) = Add (Mul (ar, as'), Neg ayv)
-  | numadd (Mul (ar, as'), Add (C azn, ayx)) =
-    Add (Mul (ar, as'), Add (C azn, ayx))
-  | numadd (Mul (ar, as'), Add (Bound azo, ayx)) =
-    Add (Mul (ar, as'), Add (Bound azo, ayx))
-  | numadd (Mul (ar, as'), Add (CX (azp, azq), ayx)) =
-    Add (Mul (ar, as'), Add (CX (azp, azq), ayx))
-  | numadd (Mul (ar, as'), Add (Neg azr, ayx)) =
-    Add (Mul (ar, as'), Add (Neg azr, ayx))
-  | numadd (Mul (ar, as'), Add (Add (azs, azt), ayx)) =
-    Add (Mul (ar, as'), Add (Add (azs, azt), ayx))
-  | numadd (Mul (ar, as'), Add (Sub (azu, azv), ayx)) =
-    Add (Mul (ar, as'), Add (Sub (azu, azv), ayx))
-  | numadd (Mul (ar, as'), Add (Mul (azw, C baj), ayx)) =
-    Add (Mul (ar, as'), Add (Mul (azw, C baj), ayx))
-  | numadd (Mul (ar, as'), Add (Mul (azw, CX (bal, bam)), ayx)) =
-    Add (Mul (ar, as'), Add (Mul (azw, CX (bal, bam)), ayx))
-  | numadd (Mul (ar, as'), Add (Mul (azw, Neg ban), ayx)) =
-    Add (Mul (ar, as'), Add (Mul (azw, Neg ban), ayx))
-  | numadd (Mul (ar, as'), Add (Mul (azw, Add (bao, bap)), ayx)) =
-    Add (Mul (ar, as'), Add (Mul (azw, Add (bao, bap)), ayx))
-  | numadd (Mul (ar, as'), Add (Mul (azw, Sub (baq, bar)), ayx)) =
-    Add (Mul (ar, as'), Add (Mul (azw, Sub (baq, bar)), ayx))
-  | numadd (Mul (ar, as'), Add (Mul (azw, Mul (bas, bat)), ayx)) =
-    Add (Mul (ar, as'), Add (Mul (azw, Mul (bas, bat)), ayx))
-  | numadd (Mul (ar, as'), Sub (ayy, ayz)) = Add (Mul (ar, as'), Sub (ayy, ayz))
-  | numadd (Mul (ar, as'), Mul (aza, azb)) =
-    Add (Mul (ar, as'), Mul (aza, azb));
+          Add (Mul (auu, Cx (avj, avk)), atv))
+  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, C avh), atv)) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Mul (auu, C avh), atv))
+  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Sub (aus, aut), atv)) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Sub (aus, aut), atv))
+  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Add (auq, aur), atv)) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Add (auq, aur), atv))
+  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Neg aup, atv)) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Neg aup, atv))
+  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Cx (aun, auo), atv)) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Cx (aun, auo), atv))
+  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (Bound aum, atv)) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (Bound aum, atv))
+  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Add (C aul, atv)) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao), Add (C aul, atv))
+  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Neg att) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao), Neg att)
+  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Cx (atr, ats)) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao), Cx (atr, ats))
+  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), Bound atq) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao), Bound atq)
+  | numadd (Add (Mul (mc, Mul (acp, acq)), ao), C atp) =
+    Add (Add (Mul (mc, Mul (acp, acq)), ao), C atp)
+  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Mul (ark, arl)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao), Mul (ark, arl))
+  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Sub (ari, arj)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao), Sub (ari, arj))
+  | numadd
+    (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Mul (atc, atd)), arh)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao),
+          Add (Mul (asg, Mul (atc, atd)), arh))
+  | numadd
+    (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Sub (ata, atb)), arh)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao),
+          Add (Mul (asg, Sub (ata, atb)), arh))
+  | numadd
+    (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Add (asy, asz)), arh)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao),
+          Add (Mul (asg, Add (asy, asz)), arh))
+  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Neg asx), arh)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Neg asx), arh))
+  | numadd
+    (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, Cx (asv, asw)), arh)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao),
+          Add (Mul (asg, Cx (asv, asw)), arh))
+  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, C ast), arh)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Mul (asg, C ast), arh))
+  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Sub (ase, asf), arh)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Sub (ase, asf), arh))
+  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Add (asc, asd), arh)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Add (asc, asd), arh))
+  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Neg asb, arh)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Neg asb, arh))
+  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Cx (arz, asa), arh)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Cx (arz, asa), arh))
+  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (Bound ary, arh)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (Bound ary, arh))
+  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Add (C arx, arh)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao), Add (C arx, arh))
+  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Neg arf) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao), Neg arf)
+  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Cx (ard, are)) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao), Cx (ard, are))
+  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), Bound arc) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao), Bound arc)
+  | numadd (Add (Mul (mc, Sub (acn, aco)), ao), C arb) =
+    Add (Add (Mul (mc, Sub (acn, aco)), ao), C arb)
+  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Mul (aow, aox)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao), Mul (aow, aox))
+  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Sub (aou, aov)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao), Sub (aou, aov))
+  | numadd
+    (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Mul (aqo, aqp)), aot)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao),
+          Add (Mul (aps, Mul (aqo, aqp)), aot))
+  | numadd
+    (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Sub (aqm, aqn)), aot)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao),
+          Add (Mul (aps, Sub (aqm, aqn)), aot))
+  | numadd
+    (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Add (aqk, aql)), aot)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao),
+          Add (Mul (aps, Add (aqk, aql)), aot))
+  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Neg aqj), aot)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Neg aqj), aot))
+  | numadd
+    (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, Cx (aqh, aqi)), aot)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao),
+          Add (Mul (aps, Cx (aqh, aqi)), aot))
+  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, C aqf), aot)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Mul (aps, C aqf), aot))
+  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Sub (apq, apr), aot)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Sub (apq, apr), aot))
+  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Add (apo, app), aot)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Add (apo, app), aot))
+  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Neg apn, aot)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Neg apn, aot))
+  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Cx (apl, apm), aot)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Cx (apl, apm), aot))
+  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (Bound apk, aot)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (Bound apk, aot))
+  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Add (C apj, aot)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao), Add (C apj, aot))
+  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Neg aor) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao), Neg aor)
+  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Cx (aop, aoq)) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao), Cx (aop, aoq))
+  | numadd (Add (Mul (mc, Add (acl, acm)), ao), Bound aoo) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao), Bound aoo)
+  | numadd (Add (Mul (mc, Add (acl, acm)), ao), C aon) =
+    Add (Add (Mul (mc, Add (acl, acm)), ao), C aon)
+  | numadd (Add (Mul (mc, Neg ack), ao), Mul (ami, amj)) =
+    Add (Add (Mul (mc, Neg ack), ao), Mul (ami, amj))
+  | numadd (Add (Mul (mc, Neg ack), ao), Sub (amg, amh)) =
+    Add (Add (Mul (mc, Neg ack), ao), Sub (amg, amh))
+  | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Mul (aoa, aob)), amf)) =
+    Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Mul (aoa, aob)), amf))
+  | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Sub (any, anz)), amf)) =
+    Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Sub (any, anz)), amf))
+  | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Add (anw, anx)), amf)) =
+    Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Add (anw, anx)), amf))
+  | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Neg anv), amf)) =
+    Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Neg anv), amf))
+  | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Cx (ant, anu)), amf)) =
+    Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, Cx (ant, anu)), amf))
+  | numadd (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, C anr), amf)) =
+    Add (Add (Mul (mc, Neg ack), ao), Add (Mul (ane, C anr), amf))
+  | numadd (Add (Mul (mc, Neg ack), ao), Add (Sub (anc, anda), amf)) =
+    Add (Add (Mul (mc, Neg ack), ao), Add (Sub (anc, anda), amf))
+  | numadd (Add (Mul (mc, Neg ack), ao), Add (Add (ana, anb), amf)) =
+    Add (Add (Mul (mc, Neg ack), ao), Add (Add (ana, anb), amf))
+  | numadd (Add (Mul (mc, Neg ack), ao), Add (Neg amz, amf)) =
+    Add (Add (Mul (mc, Neg ack), ao), Add (Neg amz, amf))
+  | numadd (Add (Mul (mc, Neg ack), ao), Add (Cx (amx, amy), amf)) =
+    Add (Add (Mul (mc, Neg ack), ao), Add (Cx (amx, amy), amf))
+  | numadd (Add (Mul (mc, Neg ack), ao), Add (Bound amw, amf)) =
+    Add (Add (Mul (mc, Neg ack), ao), Add (Bound amw, amf))
+  | numadd (Add (Mul (mc, Neg ack), ao), Add (C amv, amf)) =
+    Add (Add (Mul (mc, Neg ack), ao), Add (C amv, amf))
+  | numadd (Add (Mul (mc, Neg ack), ao), Neg amd) =
+    Add (Add (Mul (mc, Neg ack), ao), Neg amd)
+  | numadd (Add (Mul (mc, Neg ack), ao), Cx (amb, amc)) =
+    Add (Add (Mul (mc, Neg ack), ao), Cx (amb, amc))
+  | numadd (Add (Mul (mc, Neg ack), ao), Bound ama) =
+    Add (Add (Mul (mc, Neg ack), ao), Bound ama)
+  | numadd (Add (Mul (mc, Neg ack), ao), C alz) =
+    Add (Add (Mul (mc, Neg ack), ao), C alz)
+  | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Mul (aju, ajv)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), Mul (aju, ajv))
+  | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Sub (ajs, ajt)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), Sub (ajs, ajt))
+  | numadd
+    (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, Mul (alm, aln)), ajr)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao),
+          Add (Mul (akq, Mul (alm, aln)), ajr))
+  | numadd
+    (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, Sub (alk, all)), ajr)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao),
+          Add (Mul (akq, Sub (alk, all)), ajr))
+  | numadd
+    (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, Add (ali, alj)), ajr)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao),
+          Add (Mul (akq, Add (ali, alj)), ajr))
+  | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, Neg alh), ajr)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, Neg alh), ajr))
+  | numadd
+    (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, Cx (alf, alg)), ajr)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, Cx (alf, alg)), ajr))
+  | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, C ald), ajr)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Mul (akq, C ald), ajr))
+  | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (Sub (ako, akp), ajr)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Sub (ako, akp), ajr))
+  | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (Add (akm, akn), ajr)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Add (akm, akn), ajr))
+  | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (Neg akl, ajr)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Neg akl, ajr))
+  | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (Cx (akj, akk), ajr)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Cx (akj, akk), ajr))
+  | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (Bound aki, ajr)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (Bound aki, ajr))
+  | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Add (C akh, ajr)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), Add (C akh, ajr))
+  | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Neg ajp) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), Neg ajp)
+  | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Cx (ajn, ajo)) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), Cx (ajn, ajo))
+  | numadd (Add (Mul (mc, Cx (aci, acj)), ao), Bound ajm) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), Bound ajm)
+  | numadd (Add (Mul (mc, Cx (aci, acj)), ao), C ajl) =
+    Add (Add (Mul (mc, Cx (aci, acj)), ao), C ajl)
+  | numadd (Add (Mul (mc, C acg), ao), Mul (adl, adm)) =
+    Add (Add (Mul (mc, C acg), ao), Mul (adl, adm))
+  | numadd (Add (Mul (mc, C acg), ao), Sub (adj, adk)) =
+    Add (Add (Mul (mc, C acg), ao), Sub (adj, adk))
+  | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Mul (afd, afe)), adi)) =
+    Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Mul (afd, afe)), adi))
+  | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Sub (afb, afc)), adi)) =
+    Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Sub (afb, afc)), adi))
+  | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Add (aez, afa)), adi)) =
+    Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Add (aez, afa)), adi))
+  | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Neg aey), adi)) =
+    Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Neg aey), adi))
+  | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Cx (aew, aex)), adi)) =
+    Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, Cx (aew, aex)), adi))
+  | numadd (Add (Mul (mc, C acg), ao), Add (Mul (aeh, C aeu), adi)) =
+    Add (Add (Mul (mc, C acg), ao), Add (Mul (aeh, C aeu), adi))
+  | numadd (Add (Mul (mc, C acg), ao), Add (Sub (aef, aeg), adi)) =
+    Add (Add (Mul (mc, C acg), ao), Add (Sub (aef, aeg), adi))
+  | numadd (Add (Mul (mc, C acg), ao), Add (Add (aed, aee), adi)) =
+    Add (Add (Mul (mc, C acg), ao), Add (Add (aed, aee), adi))
+  | numadd (Add (Mul (mc, C acg), ao), Add (Neg aec, adi)) =
+    Add (Add (Mul (mc, C acg), ao), Add (Neg aec, adi))
+  | numadd (Add (Mul (mc, C acg), ao), Add (Cx (aea, aeb), adi)) =
+    Add (Add (Mul (mc, C acg), ao), Add (Cx (aea, aeb), adi))
+  | numadd (Add (Mul (mc, C acg), ao), Add (Bound adz, adi)) =
+    Add (Add (Mul (mc, C acg), ao), Add (Bound adz, adi))
+  | numadd (Add (Mul (mc, C acg), ao), Add (C ady, adi)) =
+    Add (Add (Mul (mc, C acg), ao), Add (C ady, adi))
+  | numadd (Add (Mul (mc, C acg), ao), Neg adg) =
+    Add (Add (Mul (mc, C acg), ao), Neg adg)
+  | numadd (Add (Mul (mc, C acg), ao), Cx (ade, adf)) =
+    Add (Add (Mul (mc, C acg), ao), Cx (ade, adf))
+  | numadd (Add (Mul (mc, C acg), ao), Bound add) =
+    Add (Add (Mul (mc, C acg), ao), Bound add)
+  | numadd (Add (Mul (mc, C acg), ao), C adc) =
+    Add (Add (Mul (mc, C acg), ao), C adc)
+  | numadd (Add (Sub (ma, mb), ao), Mul (zq, zr)) =
+    Add (Add (Sub (ma, mb), ao), Mul (zq, zr))
+  | numadd (Add (Sub (ma, mb), ao), Sub (zo, zp)) =
+    Add (Add (Sub (ma, mb), ao), Sub (zo, zp))
+  | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Mul (abi, abj)), zn)) =
+    Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Mul (abi, abj)), zn))
+  | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Sub (abg, abh)), zn)) =
+    Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Sub (abg, abh)), zn))
+  | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Add (abe, abf)), zn)) =
+    Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Add (abe, abf)), zn))
+  | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Neg abd), zn)) =
+    Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Neg abd), zn))
+  | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, Cx (abb, abc)), zn)) =
+    Add (Add (Sub (ma, mb), ao), Add (Mul (aam, Cx (abb, abc)), zn))
+  | numadd (Add (Sub (ma, mb), ao), Add (Mul (aam, C aaz), zn)) =
+    Add (Add (Sub (ma, mb), ao), Add (Mul (aam, C aaz), zn))
+  | numadd (Add (Sub (ma, mb), ao), Add (Sub (aak, aal), zn)) =
+    Add (Add (Sub (ma, mb), ao), Add (Sub (aak, aal), zn))
+  | numadd (Add (Sub (ma, mb), ao), Add (Add (aai, aaj), zn)) =
+    Add (Add (Sub (ma, mb), ao), Add (Add (aai, aaj), zn))
+  | numadd (Add (Sub (ma, mb), ao), Add (Neg aah, zn)) =
+    Add (Add (Sub (ma, mb), ao), Add (Neg aah, zn))
+  | numadd (Add (Sub (ma, mb), ao), Add (Cx (aaf, aag), zn)) =
+    Add (Add (Sub (ma, mb), ao), Add (Cx (aaf, aag), zn))
+  | numadd (Add (Sub (ma, mb), ao), Add (Bound aae, zn)) =
+    Add (Add (Sub (ma, mb), ao), Add (Bound aae, zn))
+  | numadd (Add (Sub (ma, mb), ao), Add (C aad, zn)) =
+    Add (Add (Sub (ma, mb), ao), Add (C aad, zn))
+  | numadd (Add (Sub (ma, mb), ao), Neg zl) =
+    Add (Add (Sub (ma, mb), ao), Neg zl)
+  | numadd (Add (Sub (ma, mb), ao), Cx (zj, zk)) =
+    Add (Add (Sub (ma, mb), ao), Cx (zj, zk))
+  | numadd (Add (Sub (ma, mb), ao), Bound zi) =
+    Add (Add (Sub (ma, mb), ao), Bound zi)
+  | numadd (Add (Sub (ma, mb), ao), C zh) = Add (Add (Sub (ma, mb), ao), C zh)
+  | numadd (Add (Add (ly, lz), ao), Mul (xc, xd)) =
+    Add (Add (Add (ly, lz), ao), Mul (xc, xd))
+  | numadd (Add (Add (ly, lz), ao), Sub (xa, xb)) =
+    Add (Add (Add (ly, lz), ao), Sub (xa, xb))
+  | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Mul (yu, yv)), wz)) =
+    Add (Add (Add (ly, lz), ao), Add (Mul (xy, Mul (yu, yv)), wz))
+  | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Sub (ys, yt)), wz)) =
+    Add (Add (Add (ly, lz), ao), Add (Mul (xy, Sub (ys, yt)), wz))
+  | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Add (yq, yr)), wz)) =
+    Add (Add (Add (ly, lz), ao), Add (Mul (xy, Add (yq, yr)), wz))
+  | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Neg yp), wz)) =
+    Add (Add (Add (ly, lz), ao), Add (Mul (xy, Neg yp), wz))
+  | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, Cx (yn, yo)), wz)) =
+    Add (Add (Add (ly, lz), ao), Add (Mul (xy, Cx (yn, yo)), wz))
+  | numadd (Add (Add (ly, lz), ao), Add (Mul (xy, C yl), wz)) =
+    Add (Add (Add (ly, lz), ao), Add (Mul (xy, C yl), wz))
+  | numadd (Add (Add (ly, lz), ao), Add (Sub (xw, xx), wz)) =
+    Add (Add (Add (ly, lz), ao), Add (Sub (xw, xx), wz))
+  | numadd (Add (Add (ly, lz), ao), Add (Add (xu, xv), wz)) =
+    Add (Add (Add (ly, lz), ao), Add (Add (xu, xv), wz))
+  | numadd (Add (Add (ly, lz), ao), Add (Neg xt, wz)) =
+    Add (Add (Add (ly, lz), ao), Add (Neg xt, wz))
+  | numadd (Add (Add (ly, lz), ao), Add (Cx (xr, xs), wz)) =
+    Add (Add (Add (ly, lz), ao), Add (Cx (xr, xs), wz))
+  | numadd (Add (Add (ly, lz), ao), Add (Bound xq, wz)) =
+    Add (Add (Add (ly, lz), ao), Add (Bound xq, wz))
+  | numadd (Add (Add (ly, lz), ao), Add (C xp, wz)) =
+    Add (Add (Add (ly, lz), ao), Add (C xp, wz))
+  | numadd (Add (Add (ly, lz), ao), Neg wx) =
+    Add (Add (Add (ly, lz), ao), Neg wx)
+  | numadd (Add (Add (ly, lz), ao), Cx (wv, ww)) =
+    Add (Add (Add (ly, lz), ao), Cx (wv, ww))
+  | numadd (Add (Add (ly, lz), ao), Bound wu) =
+    Add (Add (Add (ly, lz), ao), Bound wu)
+  | numadd (Add (Add (ly, lz), ao), C wt) = Add (Add (Add (ly, lz), ao), C wt)
+  | numadd (Add (Neg lx, ao), Mul (uo, up)) =
+    Add (Add (Neg lx, ao), Mul (uo, up))
+  | numadd (Add (Neg lx, ao), Sub (um, un)) =
+    Add (Add (Neg lx, ao), Sub (um, un))
+  | numadd (Add (Neg lx, ao), Add (Mul (vk, Mul (wg, wh)), ul)) =
+    Add (Add (Neg lx, ao), Add (Mul (vk, Mul (wg, wh)), ul))
+  | numadd (Add (Neg lx, ao), Add (Mul (vk, Sub (we, wf)), ul)) =
+    Add (Add (Neg lx, ao), Add (Mul (vk, Sub (we, wf)), ul))
+  | numadd (Add (Neg lx, ao), Add (Mul (vk, Add (wc, wd)), ul)) =
+    Add (Add (Neg lx, ao), Add (Mul (vk, Add (wc, wd)), ul))
+  | numadd (Add (Neg lx, ao), Add (Mul (vk, Neg wb), ul)) =
+    Add (Add (Neg lx, ao), Add (Mul (vk, Neg wb), ul))
+  | numadd (Add (Neg lx, ao), Add (Mul (vk, Cx (vz, wa)), ul)) =
+    Add (Add (Neg lx, ao), Add (Mul (vk, Cx (vz, wa)), ul))
+  | numadd (Add (Neg lx, ao), Add (Mul (vk, C vx), ul)) =
+    Add (Add (Neg lx, ao), Add (Mul (vk, C vx), ul))
+  | numadd (Add (Neg lx, ao), Add (Sub (vi, vj), ul)) =
+    Add (Add (Neg lx, ao), Add (Sub (vi, vj), ul))
+  | numadd (Add (Neg lx, ao), Add (Add (vg, vh), ul)) =
+    Add (Add (Neg lx, ao), Add (Add (vg, vh), ul))
+  | numadd (Add (Neg lx, ao), Add (Neg vf, ul)) =
+    Add (Add (Neg lx, ao), Add (Neg vf, ul))
+  | numadd (Add (Neg lx, ao), Add (Cx (vd, ve), ul)) =
+    Add (Add (Neg lx, ao), Add (Cx (vd, ve), ul))
+  | numadd (Add (Neg lx, ao), Add (Bound vc, ul)) =
+    Add (Add (Neg lx, ao), Add (Bound vc, ul))
+  | numadd (Add (Neg lx, ao), Add (C vb, ul)) =
+    Add (Add (Neg lx, ao), Add (C vb, ul))
+  | numadd (Add (Neg lx, ao), Neg uj) = Add (Add (Neg lx, ao), Neg uj)
+  | numadd (Add (Neg lx, ao), Cx (uh, ui)) = Add (Add (Neg lx, ao), Cx (uh, ui))
+  | numadd (Add (Neg lx, ao), Bound ug) = Add (Add (Neg lx, ao), Bound ug)
+  | numadd (Add (Neg lx, ao), C uf) = Add (Add (Neg lx, ao), C uf)
+  | numadd (Add (Cx (lv, lw), ao), Mul (sa, sb)) =
+    Add (Add (Cx (lv, lw), ao), Mul (sa, sb))
+  | numadd (Add (Cx (lv, lw), ao), Sub (ry, rz)) =
+    Add (Add (Cx (lv, lw), ao), Sub (ry, rz))
+  | numadd (Add (Cx (lv, lw), ao), Add (Mul (sw, Mul (ts, tt)), rx)) =
+    Add (Add (Cx (lv, lw), ao), Add (Mul (sw, Mul (ts, tt)), rx))
+  | numadd (Add (Cx (lv, lw), ao), Add (Mul (sw, Sub (tq, tr)), rx)) =
+    Add (Add (Cx (lv, lw), ao), Add (Mul (sw, Sub (tq, tr)), rx))
+  | numadd (Add (Cx (lv, lw), ao), Add (Mul (sw, Add (to, tp)), rx)) =
+    Add (Add (Cx (lv, lw), ao), Add (Mul (sw, Add (to, tp)), rx))
+  | numadd (Add (Cx (lv, lw), ao), Add (Mul (sw, Neg tn), rx)) =
+    Add (Add (Cx (lv, lw), ao), Add (Mul (sw, Neg tn), rx))
+  | numadd (Add (Cx (lv, lw), ao), Add (Mul (sw, Cx (tl, tm)), rx)) =
+    Add (Add (Cx (lv, lw), ao), Add (Mul (sw, Cx (tl, tm)), rx))
+  | numadd (Add (Cx (lv, lw), ao), Add (Mul (sw, C tj), rx)) =
+    Add (Add (Cx (lv, lw), ao), Add (Mul (sw, C tj), rx))
+  | numadd (Add (Cx (lv, lw), ao), Add (Sub (su, sv), rx)) =
+    Add (Add (Cx (lv, lw), ao), Add (Sub (su, sv), rx))
+  | numadd (Add (Cx (lv, lw), ao), Add (Add (ss, st), rx)) =
+    Add (Add (Cx (lv, lw), ao), Add (Add (ss, st), rx))
+  | numadd (Add (Cx (lv, lw), ao), Add (Neg sr, rx)) =
+    Add (Add (Cx (lv, lw), ao), Add (Neg sr, rx))
+  | numadd (Add (Cx (lv, lw), ao), Add (Cx (sp, sq), rx)) =
+    Add (Add (Cx (lv, lw), ao), Add (Cx (sp, sq), rx))
+  | numadd (Add (Cx (lv, lw), ao), Add (Bound so, rx)) =
+    Add (Add (Cx (lv, lw), ao), Add (Bound so, rx))
+  | numadd (Add (Cx (lv, lw), ao), Add (C sn, rx)) =
+    Add (Add (Cx (lv, lw), ao), Add (C sn, rx))
+  | numadd (Add (Cx (lv, lw), ao), Neg rv) = Add (Add (Cx (lv, lw), ao), Neg rv)
+  | numadd (Add (Cx (lv, lw), ao), Cx (rt, ru)) =
+    Add (Add (Cx (lv, lw), ao), Cx (rt, ru))
+  | numadd (Add (Cx (lv, lw), ao), Bound rs) =
+    Add (Add (Cx (lv, lw), ao), Bound rs)
+  | numadd (Add (Cx (lv, lw), ao), C rr) = Add (Add (Cx (lv, lw), ao), C rr)
+  | numadd (Add (Bound lu, ao), Mul (pm, pn)) =
+    Add (Add (Bound lu, ao), Mul (pm, pn))
+  | numadd (Add (Bound lu, ao), Sub (pk, pl)) =
+    Add (Add (Bound lu, ao), Sub (pk, pl))
+  | numadd (Add (Bound lu, ao), Add (Mul (qi, Mul (re, rf)), pj)) =
+    Add (Add (Bound lu, ao), Add (Mul (qi, Mul (re, rf)), pj))
+  | numadd (Add (Bound lu, ao), Add (Mul (qi, Sub (rc, rd)), pj)) =
+    Add (Add (Bound lu, ao), Add (Mul (qi, Sub (rc, rd)), pj))
+  | numadd (Add (Bound lu, ao), Add (Mul (qi, Add (ra, rb)), pj)) =
+    Add (Add (Bound lu, ao), Add (Mul (qi, Add (ra, rb)), pj))
+  | numadd (Add (Bound lu, ao), Add (Mul (qi, Neg qz), pj)) =
+    Add (Add (Bound lu, ao), Add (Mul (qi, Neg qz), pj))
+  | numadd (Add (Bound lu, ao), Add (Mul (qi, Cx (qx, qy)), pj)) =
+    Add (Add (Bound lu, ao), Add (Mul (qi, Cx (qx, qy)), pj))
+  | numadd (Add (Bound lu, ao), Add (Mul (qi, C qv), pj)) =
+    Add (Add (Bound lu, ao), Add (Mul (qi, C qv), pj))
+  | numadd (Add (Bound lu, ao), Add (Sub (qg, qh), pj)) =
+    Add (Add (Bound lu, ao), Add (Sub (qg, qh), pj))
+  | numadd (Add (Bound lu, ao), Add (Add (qe, qf), pj)) =
+    Add (Add (Bound lu, ao), Add (Add (qe, qf), pj))
+  | numadd (Add (Bound lu, ao), Add (Neg qd, pj)) =
+    Add (Add (Bound lu, ao), Add (Neg qd, pj))
+  | numadd (Add (Bound lu, ao), Add (Cx (qb, qc), pj)) =
+    Add (Add (Bound lu, ao), Add (Cx (qb, qc), pj))
+  | numadd (Add (Bound lu, ao), Add (Bound qa, pj)) =
+    Add (Add (Bound lu, ao), Add (Bound qa, pj))
+  | numadd (Add (Bound lu, ao), Add (C pz, pj)) =
+    Add (Add (Bound lu, ao), Add (C pz, pj))
+  | numadd (Add (Bound lu, ao), Neg ph) = Add (Add (Bound lu, ao), Neg ph)
+  | numadd (Add (Bound lu, ao), Cx (pf, pg)) =
+    Add (Add (Bound lu, ao), Cx (pf, pg))
+  | numadd (Add (Bound lu, ao), Bound pe) = Add (Add (Bound lu, ao), Bound pe)
+  | numadd (Add (Bound lu, ao), C pd) = Add (Add (Bound lu, ao), C pd)
+  | numadd (Add (C lt, ao), Mul (my, mz)) = Add (Add (C lt, ao), Mul (my, mz))
+  | numadd (Add (C lt, ao), Sub (mw, mx)) = Add (Add (C lt, ao), Sub (mw, mx))
+  | numadd (Add (C lt, ao), Add (Mul (nu, Mul (oq, or)), mv)) =
+    Add (Add (C lt, ao), Add (Mul (nu, Mul (oq, or)), mv))
+  | numadd (Add (C lt, ao), Add (Mul (nu, Sub (ooa, opa)), mv)) =
+    Add (Add (C lt, ao), Add (Mul (nu, Sub (ooa, opa)), mv))
+  | numadd (Add (C lt, ao), Add (Mul (nu, Add (om, on)), mv)) =
+    Add (Add (C lt, ao), Add (Mul (nu, Add (om, on)), mv))
+  | numadd (Add (C lt, ao), Add (Mul (nu, Neg ol), mv)) =
+    Add (Add (C lt, ao), Add (Mul (nu, Neg ol), mv))
+  | numadd (Add (C lt, ao), Add (Mul (nu, Cx (oj, ok)), mv)) =
+    Add (Add (C lt, ao), Add (Mul (nu, Cx (oj, ok)), mv))
+  | numadd (Add (C lt, ao), Add (Mul (nu, C oh), mv)) =
+    Add (Add (C lt, ao), Add (Mul (nu, C oh), mv))
+  | numadd (Add (C lt, ao), Add (Sub (ns, nt), mv)) =
+    Add (Add (C lt, ao), Add (Sub (ns, nt), mv))
+  | numadd (Add (C lt, ao), Add (Add (nq, nr), mv)) =
+    Add (Add (C lt, ao), Add (Add (nq, nr), mv))
+  | numadd (Add (C lt, ao), Add (Neg np, mv)) =
+    Add (Add (C lt, ao), Add (Neg np, mv))
+  | numadd (Add (C lt, ao), Add (Cx (nn, no), mv)) =
+    Add (Add (C lt, ao), Add (Cx (nn, no), mv))
+  | numadd (Add (C lt, ao), Add (Bound nm, mv)) =
+    Add (Add (C lt, ao), Add (Bound nm, mv))
+  | numadd (Add (C lt, ao), Add (C nl, mv)) =
+    Add (Add (C lt, ao), Add (C nl, mv))
+  | numadd (Add (C lt, ao), Neg mt) = Add (Add (C lt, ao), Neg mt)
+  | numadd (Add (C lt, ao), Cx (mr, ms)) = Add (Add (C lt, ao), Cx (mr, ms))
+  | numadd (Add (C lt, ao), Bound mq) = Add (Add (C lt, ao), Bound mq)
+  | numadd (Add (C lt, ao), C mp) = Add (Add (C lt, ao), C mp)
+  | numadd (Neg am, Mul (jd, je)) = Add (Neg am, Mul (jd, je))
+  | numadd (Neg am, Sub (jb, jc)) = Add (Neg am, Sub (jb, jc))
+  | numadd (Neg am, Add (Mul (jz, Mul (kv, kw)), ja)) =
+    Add (Neg am, Add (Mul (jz, Mul (kv, kw)), ja))
+  | numadd (Neg am, Add (Mul (jz, Sub (kt, ku)), ja)) =
+    Add (Neg am, Add (Mul (jz, Sub (kt, ku)), ja))
+  | numadd (Neg am, Add (Mul (jz, Add (kr, ks)), ja)) =
+    Add (Neg am, Add (Mul (jz, Add (kr, ks)), ja))
+  | numadd (Neg am, Add (Mul (jz, Neg kq), ja)) =
+    Add (Neg am, Add (Mul (jz, Neg kq), ja))
+  | numadd (Neg am, Add (Mul (jz, Cx (ko, kp)), ja)) =
+    Add (Neg am, Add (Mul (jz, Cx (ko, kp)), ja))
+  | numadd (Neg am, Add (Mul (jz, C km), ja)) =
+    Add (Neg am, Add (Mul (jz, C km), ja))
+  | numadd (Neg am, Add (Sub (jx, jy), ja)) =
+    Add (Neg am, Add (Sub (jx, jy), ja))
+  | numadd (Neg am, Add (Add (jv, jw), ja)) =
+    Add (Neg am, Add (Add (jv, jw), ja))
+  | numadd (Neg am, Add (Neg ju, ja)) = Add (Neg am, Add (Neg ju, ja))
+  | numadd (Neg am, Add (Cx (js, jt), ja)) = Add (Neg am, Add (Cx (js, jt), ja))
+  | numadd (Neg am, Add (Bound jr, ja)) = Add (Neg am, Add (Bound jr, ja))
+  | numadd (Neg am, Add (C jq, ja)) = Add (Neg am, Add (C jq, ja))
+  | numadd (Neg am, Neg iy) = Add (Neg am, Neg iy)
+  | numadd (Neg am, Cx (iw, ix)) = Add (Neg am, Cx (iw, ix))
+  | numadd (Neg am, Bound iv) = Add (Neg am, Bound iv)
+  | numadd (Neg am, C iu) = Add (Neg am, C iu)
+  | numadd (Cx (ak, al), Mul (gp, gq)) = Add (Cx (ak, al), Mul (gp, gq))
+  | numadd (Cx (ak, al), Sub (gn, go)) = Add (Cx (ak, al), Sub (gn, go))
+  | numadd (Cx (ak, al), Add (Mul (hl, Mul (ih, ii)), gm)) =
+    Add (Cx (ak, al), Add (Mul (hl, Mul (ih, ii)), gm))
+  | numadd (Cx (ak, al), Add (Mul (hl, Sub (ifa, ig)), gm)) =
+    Add (Cx (ak, al), Add (Mul (hl, Sub (ifa, ig)), gm))
+  | numadd (Cx (ak, al), Add (Mul (hl, Add (id, ie)), gm)) =
+    Add (Cx (ak, al), Add (Mul (hl, Add (id, ie)), gm))
+  | numadd (Cx (ak, al), Add (Mul (hl, Neg ic), gm)) =
+    Add (Cx (ak, al), Add (Mul (hl, Neg ic), gm))
+  | numadd (Cx (ak, al), Add (Mul (hl, Cx (ia, ib)), gm)) =
+    Add (Cx (ak, al), Add (Mul (hl, Cx (ia, ib)), gm))
+  | numadd (Cx (ak, al), Add (Mul (hl, C hy), gm)) =
+    Add (Cx (ak, al), Add (Mul (hl, C hy), gm))
+  | numadd (Cx (ak, al), Add (Sub (hj, hk), gm)) =
+    Add (Cx (ak, al), Add (Sub (hj, hk), gm))
+  | numadd (Cx (ak, al), Add (Add (hh, hi), gm)) =
+    Add (Cx (ak, al), Add (Add (hh, hi), gm))
+  | numadd (Cx (ak, al), Add (Neg hg, gm)) = Add (Cx (ak, al), Add (Neg hg, gm))
+  | numadd (Cx (ak, al), Add (Cx (he, hf), gm)) =
+    Add (Cx (ak, al), Add (Cx (he, hf), gm))
+  | numadd (Cx (ak, al), Add (Bound hd, gm)) =
+    Add (Cx (ak, al), Add (Bound hd, gm))
+  | numadd (Cx (ak, al), Add (C hc, gm)) = Add (Cx (ak, al), Add (C hc, gm))
+  | numadd (Cx (ak, al), Neg gk) = Add (Cx (ak, al), Neg gk)
+  | numadd (Cx (ak, al), Cx (gi, gj)) = Add (Cx (ak, al), Cx (gi, gj))
+  | numadd (Cx (ak, al), Bound gh) = Add (Cx (ak, al), Bound gh)
+  | numadd (Cx (ak, al), C gg) = Add (Cx (ak, al), C gg)
+  | numadd (Bound aj, Mul (eb, ec)) = Add (Bound aj, Mul (eb, ec))
+  | numadd (Bound aj, Sub (dz, ea)) = Add (Bound aj, Sub (dz, ea))
+  | numadd (Bound aj, Add (Mul (ex, Mul (ft, fu)), dy)) =
+    Add (Bound aj, Add (Mul (ex, Mul (ft, fu)), dy))
+  | numadd (Bound aj, Add (Mul (ex, Sub (fr, fs)), dy)) =
+    Add (Bound aj, Add (Mul (ex, Sub (fr, fs)), dy))
+  | numadd (Bound aj, Add (Mul (ex, Add (fp, fq)), dy)) =
+    Add (Bound aj, Add (Mul (ex, Add (fp, fq)), dy))
+  | numadd (Bound aj, Add (Mul (ex, Neg fo), dy)) =
+    Add (Bound aj, Add (Mul (ex, Neg fo), dy))
+  | numadd (Bound aj, Add (Mul (ex, Cx (fm, fna)), dy)) =
+    Add (Bound aj, Add (Mul (ex, Cx (fm, fna)), dy))
+  | numadd (Bound aj, Add (Mul (ex, C fk), dy)) =
+    Add (Bound aj, Add (Mul (ex, C fk), dy))
+  | numadd (Bound aj, Add (Sub (ev, ew), dy)) =
+    Add (Bound aj, Add (Sub (ev, ew), dy))
+  | numadd (Bound aj, Add (Add (et, eu), dy)) =
+    Add (Bound aj, Add (Add (et, eu), dy))
+  | numadd (Bound aj, Add (Neg es, dy)) = Add (Bound aj, Add (Neg es, dy))
+  | numadd (Bound aj, Add (Cx (eq, er), dy)) =
+    Add (Bound aj, Add (Cx (eq, er), dy))
+  | numadd (Bound aj, Add (Bound ep, dy)) = Add (Bound aj, Add (Bound ep, dy))
+  | numadd (Bound aj, Add (C eo, dy)) = Add (Bound aj, Add (C eo, dy))
+  | numadd (Bound aj, Neg dw) = Add (Bound aj, Neg dw)
+  | numadd (Bound aj, Cx (du, dv)) = Add (Bound aj, Cx (du, dv))
+  | numadd (Bound aj, Bound dt) = Add (Bound aj, Bound dt)
+  | numadd (Bound aj, C ds) = Add (Bound aj, C ds)
+  | numadd (C ai, Mul (bn, bo)) = Add (C ai, Mul (bn, bo))
+  | numadd (C ai, Sub (bl, bm)) = Add (C ai, Sub (bl, bm))
+  | numadd (C ai, Add (Mul (cj, Mul (df, dg)), bk)) =
+    Add (C ai, Add (Mul (cj, Mul (df, dg)), bk))
+  | numadd (C ai, Add (Mul (cj, Sub (dd, de)), bk)) =
+    Add (C ai, Add (Mul (cj, Sub (dd, de)), bk))
+  | numadd (C ai, Add (Mul (cj, Add (db, dc)), bk)) =
+    Add (C ai, Add (Mul (cj, Add (db, dc)), bk))
+  | numadd (C ai, Add (Mul (cj, Neg da), bk)) =
+    Add (C ai, Add (Mul (cj, Neg da), bk))
+  | numadd (C ai, Add (Mul (cj, Cx (cy, cz)), bk)) =
+    Add (C ai, Add (Mul (cj, Cx (cy, cz)), bk))
+  | numadd (C ai, Add (Mul (cj, C cw), bk)) =
+    Add (C ai, Add (Mul (cj, C cw), bk))
+  | numadd (C ai, Add (Sub (ch, ci), bk)) = Add (C ai, Add (Sub (ch, ci), bk))
+  | numadd (C ai, Add (Add (cf, cg), bk)) = Add (C ai, Add (Add (cf, cg), bk))
+  | numadd (C ai, Add (Neg ce, bk)) = Add (C ai, Add (Neg ce, bk))
+  | numadd (C ai, Add (Cx (cc, cd), bk)) = Add (C ai, Add (Cx (cc, cd), bk))
+  | numadd (C ai, Add (Bound cb, bk)) = Add (C ai, Add (Bound cb, bk))
+  | numadd (C ai, Add (C ca, bk)) = Add (C ai, Add (C ca, bk))
+  | numadd (C ai, Neg bi) = Add (C ai, Neg bi)
+  | numadd (C ai, Cx (bg, bh)) = Add (C ai, Cx (bg, bh))
+  | numadd (C ai, Bound bf) = Add (C ai, Bound bf)
+  | numadd (C b1, C b2) = C (IntInf.+ (b1, b2))
+  | numadd (Mul (ag, ah), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Mul (ag, ah), r2))
+  | numadd (Sub (ae, af), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Sub (ae, af), r2))
+  | numadd (Add (Mul (lr, Mul (ace, acf)), ad), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Mul (ace, acf)), ad), r2))
+  | numadd (Add (Mul (lr, Sub (acc, acd)), ad), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Sub (acc, acd)), ad), r2))
+  | numadd (Add (Mul (lr, Add (aca, acb)), ad), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Add (aca, acb)), ad), r2))
+  | numadd (Add (Mul (lr, Neg abz), ad), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Neg abz), ad), r2))
+  | numadd (Add (Mul (lr, Cx (abx, aby)), ad), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, Cx (abx, aby)), ad), r2))
+  | numadd (Add (Mul (lr, C abv), ad), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Add (Mul (lr, C abv), ad), r2))
+  | numadd (Add (Sub (lp, lq), ad), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Add (Sub (lp, lq), ad), r2))
+  | numadd (Add (Add (ln, lo), ad), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Add (Add (ln, lo), ad), r2))
+  | numadd (Add (Neg lm, ad), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Add (Neg lm, ad), r2))
+  | numadd (Add (Cx (lk, ll), ad), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Add (Cx (lk, ll), ad), r2))
+  | numadd (Add (Bound lj, ad), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Add (Bound lj, ad), r2))
+  | numadd (Add (C li, ad), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Add (C li, ad), r2))
+  | numadd (Neg ab, Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Neg ab, r2))
+  | numadd (Cx (y, z), Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Cx (y, z), r2))
+  | numadd (Bound x, Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (Bound x, r2))
+  | numadd (C w, Add (Mul (c2, Bound n2), r2)) =
+    Add (Mul (c2, Bound n2), numadd (C w, r2))
+  | numadd (Add (Mul (c1, Bound n1), r1), Mul (afz, aga)) =
+    Add (Mul (c1, Bound n1), numadd (r1, Mul (afz, aga)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Sub (afx, afy)) =
+    Add (Mul (c1, Bound n1), numadd (r1, Sub (afx, afy)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Mul (ain, aio)), afw))
+    = Add (Mul (c1, Bound n1),
+            numadd (r1, Add (Mul (ahg, Mul (ain, aio)), afw)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Sub (ail, aim)), afw))
+    = Add (Mul (c1, Bound n1),
+            numadd (r1, Add (Mul (ahg, Sub (ail, aim)), afw)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Add (aij, aik)), afw))
+    = Add (Mul (c1, Bound n1),
+            numadd (r1, Add (Mul (ahg, Add (aij, aik)), afw)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Neg aii), afw)) =
+    Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Neg aii), afw)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, Cx (aig, aih)), afw)) =
+    Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, Cx (aig, aih)), afw)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (ahg, C aie), afw)) =
+    Add (Mul (c1, Bound n1), numadd (r1, Add (Mul (ahg, C aie), afw)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Add (Sub (ahe, ahf), afw)) =
+    Add (Mul (c1, Bound n1), numadd (r1, Add (Sub (ahe, ahf), afw)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Add (Add (ahc, ahd), afw)) =
+    Add (Mul (c1, Bound n1), numadd (r1, Add (Add (ahc, ahd), afw)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Add (Neg ahb, afw)) =
+    Add (Mul (c1, Bound n1), numadd (r1, Add (Neg ahb, afw)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Add (Cx (agz, aha), afw)) =
+    Add (Mul (c1, Bound n1), numadd (r1, Add (Cx (agz, aha), afw)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Add (Bound agy, afw)) =
+    Add (Mul (c1, Bound n1), numadd (r1, Add (Bound agy, afw)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Add (C agx, afw)) =
+    Add (Mul (c1, Bound n1), numadd (r1, Add (C agx, afw)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Neg afu) =
+    Add (Mul (c1, Bound n1), numadd (r1, Neg afu))
+  | numadd (Add (Mul (c1, Bound n1), r1), Cx (afs, aft)) =
+    Add (Mul (c1, Bound n1), numadd (r1, Cx (afs, aft)))
+  | numadd (Add (Mul (c1, Bound n1), r1), Bound afr) =
+    Add (Mul (c1, Bound n1), numadd (r1, Bound afr))
+  | numadd (Add (Mul (c1, Bound n1), r1), C afq) =
+    Add (Mul (c1, Bound n1), numadd (r1, C afq))
+  | numadd (Add (Mul (c1, Bound n1), r1), Add (Mul (c2, Bound n2), r2)) =
+    (if ((n1 : IntInf.int) = n2)
+      then let
+             val c = IntInf.+ (c1, c2);
+           in
+             (if ((c : IntInf.int) = (0 : IntInf.int)) then numadd (r1, r2)
+               else Add (Mul (c, Bound n1), numadd (r1, r2)))
+           end
+      else (if IntInf.<= (n1, n2)
+             then Add (Mul (c1, Bound n1),
+                        numadd (r1, Add (Mul (c2, Bound n2), r2)))
+             else Add (Mul (c2, Bound n2),
+                        numadd (Add (Mul (c1, Bound n1), r1), r2))));
 
-fun nummul (C j) = (fn i => C (i * j))
-  | nummul (Add (a, b)) = (fn i => numadd (nummul a i, nummul b i))
-  | nummul (Mul (c, t)) = (fn i => nummul t (i * c))
-  | nummul (Bound v) = (fn i => Mul (i, Bound v))
-  | nummul (CX (w, x)) = (fn i => Mul (i, CX (w, x)))
-  | nummul (Neg y) = (fn i => Mul (i, Neg y))
-  | nummul (Sub (ac, ad)) = (fn i => Mul (i, Sub (ac, ad)));
+fun nummul i (Sub (v, va)) = Mul (i, Sub (v, va))
+  | nummul i (Neg v) = Mul (i, Neg v)
+  | nummul i (Cx (v, va)) = Mul (i, Cx (v, va))
+  | nummul i (Bound v) = Mul (i, Bound v)
+  | nummul i (Mul (c, t)) = nummul (IntInf.* (i, c)) t
+  | nummul i (Add (a, b)) = numadd (nummul i a, nummul i b)
+  | nummul i (C j) = C (IntInf.* (i, j));
+
+fun numneg t = nummul (IntInf.~ (1 : IntInf.int)) t;
+
+fun numsub s t =
+  (if eq_num s t then C (0 : IntInf.int) else numadd (s, numneg t));
 
-fun numneg t = nummul t (~ 1);
+fun simpnum (Cx (v, va)) = Cx (v, va)
+  | simpnum (Mul (i, t)) =
+    (if ((i : IntInf.int) = (0 : IntInf.int)) then C (0 : IntInf.int)
+      else nummul i (simpnum t))
+  | simpnum (Sub (t, s)) = numsub (simpnum t) (simpnum s)
+  | simpnum (Add (t, s)) = numadd (simpnum t, simpnum s)
+  | simpnum (Neg t) = numneg (simpnum t)
+  | simpnum (Bound n) =
+    Add (Mul ((1 : IntInf.int), Bound n), C (0 : IntInf.int))
+  | simpnum (C j) = C j;
+
+val eq_numa = {eq = eq_num} : num HOL.eq;
 
-fun numsub s t = (if (s = t) then C 0 else numadd (s, numneg t));
+fun mirror (NClosed aq) = NClosed aq
+  | mirror (Closed ap) = Closed ap
+  | mirror (A ao) = A ao
+  | mirror (E an) = E an
+  | mirror (Iffa (al, am)) = Iffa (al, am)
+  | mirror (Impa (aj, ak)) = Impa (aj, ak)
+  | mirror (Nota ae) = Nota ae
+  | mirror (NDvd (ac, Mul (hv, hw))) = NDvd (ac, Mul (hv, hw))
+  | mirror (NDvd (ac, Sub (ht, hu))) = NDvd (ac, Sub (ht, hu))
+  | mirror (NDvd (ac, Add (hr, hs))) = NDvd (ac, Add (hr, hs))
+  | mirror (NDvd (ac, Neg hq)) = NDvd (ac, Neg hq)
+  | mirror (NDvd (ac, Bound hn)) = NDvd (ac, Bound hn)
+  | mirror (NDvd (ac, C hm)) = NDvd (ac, C hm)
+  | mirror (Dvd (aa, Mul (gz, ha))) = Dvd (aa, Mul (gz, ha))
+  | mirror (Dvd (aa, Sub (gx, gy))) = Dvd (aa, Sub (gx, gy))
+  | mirror (Dvd (aa, Add (gv, gw))) = Dvd (aa, Add (gv, gw))
+  | mirror (Dvd (aa, Neg gu)) = Dvd (aa, Neg gu)
+  | mirror (Dvd (aa, Bound gr)) = Dvd (aa, Bound gr)
+  | mirror (Dvd (aa, C gq)) = Dvd (aa, C gq)
+  | mirror (NEq (Mul (gd, ge))) = NEq (Mul (gd, ge))
+  | mirror (NEq (Sub (gb, gc))) = NEq (Sub (gb, gc))
+  | mirror (NEq (Add (fz, ga))) = NEq (Add (fz, ga))
+  | mirror (NEq (Neg fy)) = NEq (Neg fy)
+  | mirror (NEq (Bound fv)) = NEq (Bound fv)
+  | mirror (NEq (C fu)) = NEq (C fu)
+  | mirror (Eq (Mul (fh, fi))) = Eq (Mul (fh, fi))
+  | mirror (Eq (Sub (ff, fg))) = Eq (Sub (ff, fg))
+  | mirror (Eq (Add (fd, fe))) = Eq (Add (fd, fe))
+  | mirror (Eq (Neg fc)) = Eq (Neg fc)
+  | mirror (Eq (Bound ez)) = Eq (Bound ez)
+  | mirror (Eq (C ey)) = Eq (C ey)
+  | mirror (Ge (Mul (el, em))) = Ge (Mul (el, em))
+  | mirror (Ge (Sub (ej, ek))) = Ge (Sub (ej, ek))
+  | mirror (Ge (Add (eh, ei))) = Ge (Add (eh, ei))
+  | mirror (Ge (Neg eg)) = Ge (Neg eg)
+  | mirror (Ge (Bound ed)) = Ge (Bound ed)
+  | mirror (Ge (C ec)) = Ge (C ec)
+  | mirror (Gt (Mul (dp, dq))) = Gt (Mul (dp, dq))
+  | mirror (Gt (Sub (dn, doa))) = Gt (Sub (dn, doa))
+  | mirror (Gt (Add (dl, dm))) = Gt (Add (dl, dm))
+  | mirror (Gt (Neg dk)) = Gt (Neg dk)
+  | mirror (Gt (Bound dh)) = Gt (Bound dh)
+  | mirror (Gt (C dg)) = Gt (C dg)
+  | mirror (Le (Mul (ct, cu))) = Le (Mul (ct, cu))
+  | mirror (Le (Sub (cr, cs))) = Le (Sub (cr, cs))
+  | mirror (Le (Add (cp, cq))) = Le (Add (cp, cq))
+  | mirror (Le (Neg co)) = Le (Neg co)
+  | mirror (Le (Bound cl)) = Le (Bound cl)
+  | mirror (Le (C ck)) = Le (C ck)
+  | mirror (Lt (Mul (bx, by))) = Lt (Mul (bx, by))
+  | mirror (Lt (Sub (bv, bw))) = Lt (Sub (bv, bw))
+  | mirror (Lt (Add (bt, bu))) = Lt (Add (bt, bu))
+  | mirror (Lt (Neg bs)) = Lt (Neg bs)
+  | mirror (Lt (Bound bp)) = Lt (Bound bp)
+  | mirror (Lt (C bo)) = Lt (C bo)
+  | mirror F = F
+  | mirror T = T
+  | mirror (NDvd (i, Cx (c, e))) = NDvd (i, Cx (c, Neg e))
+  | mirror (Dvd (i, Cx (c, e))) = Dvd (i, Cx (c, Neg e))
+  | mirror (Ge (Cx (c, e))) = Le (Cx (c, Neg e))
+  | mirror (Gt (Cx (c, e))) = Lt (Cx (c, Neg e))
+  | mirror (Le (Cx (c, e))) = Ge (Cx (c, Neg e))
+  | mirror (Lt (Cx (c, e))) = Gt (Cx (c, Neg e))
+  | mirror (NEq (Cx (c, e))) = NEq (Cx (c, Neg e))
+  | mirror (Eq (Cx (c, e))) = Eq (Cx (c, Neg e))
+  | mirror (Or (p, q)) = Or (mirror p, mirror q)
+  | mirror (And (p, q)) = And (mirror p, mirror q);
 
-fun simpnum (C j) = C j
-  | simpnum (Bound n) = Add (Mul (1, Bound n), C 0)
-  | simpnum (Neg t) = numneg (simpnum t)
-  | simpnum (Add (t, s)) = numadd (simpnum t, simpnum s)
-  | simpnum (Sub (t, s)) = numsub (simpnum t) (simpnum s)
-  | simpnum (Mul (i, t)) = (if (i = 0) then C 0 else nummul (simpnum t) i)
-  | simpnum (CX (w, x)) = CX (w, x);
+fun unita p =
+  let
+    val p' = zlfm p;
+    val l = zeta p';
+    val q =
+      And (Dvd (l, Cx ((1 : IntInf.int), C (0 : IntInf.int))), a_beta p' l);
+    val d = delta q;
+    val b = List.remdups eq_numa (List.map simpnum (beta q));
+    val a = List.remdups eq_numa (List.map simpnum (alpha q));
+  in
+    (if IntInf.<= ((List.size_list b), (List.size_list a)) then (q, (b, d))
+      else (mirror q, (a, d)))
+  end;
+
+fun iupt i j =
+  (if IntInf.< (j, i) then []
+    else i :: iupt (IntInf.+ (i, (1 : IntInf.int))) j);
 
-datatype fm = T | F | Lt of num | Le of num | Gt of num | Ge of num | Eq of num
-  | NEq of num | Dvd of int * num | NDvd of int * num | NOT of fm
-  | And of fm * fm | Or of fm * fm | Imp of fm * fm | Iff of fm * fm | E of fm
-  | A of fm | Closed of int | NClosed of int;
+fun minusinf (NClosed aq) = NClosed aq
+  | minusinf (Closed ap) = Closed ap
+  | minusinf (A ao) = A ao
+  | minusinf (E an) = E an
+  | minusinf (Iffa (al, am)) = Iffa (al, am)
+  | minusinf (Impa (aj, ak)) = Impa (aj, ak)
+  | minusinf (Nota ae) = Nota ae
+  | minusinf (NDvd (ac, ad)) = NDvd (ac, ad)
+  | minusinf (Dvd (aa, ab)) = Dvd (aa, ab)
+  | minusinf (NEq (Mul (gd, ge))) = NEq (Mul (gd, ge))
+  | minusinf (NEq (Sub (gb, gc))) = NEq (Sub (gb, gc))
+  | minusinf (NEq (Add (fz, ga))) = NEq (Add (fz, ga))
+  | minusinf (NEq (Neg fy)) = NEq (Neg fy)
+  | minusinf (NEq (Bound fv)) = NEq (Bound fv)
+  | minusinf (NEq (C fu)) = NEq (C fu)
+  | minusinf (Eq (Mul (fh, fi))) = Eq (Mul (fh, fi))
+  | minusinf (Eq (Sub (ff, fg))) = Eq (Sub (ff, fg))
+  | minusinf (Eq (Add (fd, fe))) = Eq (Add (fd, fe))
+  | minusinf (Eq (Neg fc)) = Eq (Neg fc)
+  | minusinf (Eq (Bound ez)) = Eq (Bound ez)
+  | minusinf (Eq (C ey)) = Eq (C ey)
+  | minusinf (Ge (Mul (el, em))) = Ge (Mul (el, em))
+  | minusinf (Ge (Sub (ej, ek))) = Ge (Sub (ej, ek))
+  | minusinf (Ge (Add (eh, ei))) = Ge (Add (eh, ei))
+  | minusinf (Ge (Neg eg)) = Ge (Neg eg)
+  | minusinf (Ge (Bound ed)) = Ge (Bound ed)
+  | minusinf (Ge (C ec)) = Ge (C ec)
+  | minusinf (Gt (Mul (dp, dq))) = Gt (Mul (dp, dq))
+  | minusinf (Gt (Sub (dn, doa))) = Gt (Sub (dn, doa))
+  | minusinf (Gt (Add (dl, dm))) = Gt (Add (dl, dm))
+  | minusinf (Gt (Neg dk)) = Gt (Neg dk)
+  | minusinf (Gt (Bound dh)) = Gt (Bound dh)
+  | minusinf (Gt (C dg)) = Gt (C dg)
+  | minusinf (Le (Mul (ct, cu))) = Le (Mul (ct, cu))
+  | minusinf (Le (Sub (cr, cs))) = Le (Sub (cr, cs))
+  | minusinf (Le (Add (cp, cq))) = Le (Add (cp, cq))
+  | minusinf (Le (Neg co)) = Le (Neg co)
+  | minusinf (Le (Bound cl)) = Le (Bound cl)
+  | minusinf (Le (C ck)) = Le (C ck)
+  | minusinf (Lt (Mul (bx, by))) = Lt (Mul (bx, by))
+  | minusinf (Lt (Sub (bv, bw))) = Lt (Sub (bv, bw))
+  | minusinf (Lt (Add (bt, bu))) = Lt (Add (bt, bu))
+  | minusinf (Lt (Neg bs)) = Lt (Neg bs)
+  | minusinf (Lt (Bound bp)) = Lt (Bound bp)
+  | minusinf (Lt (C bo)) = Lt (C bo)
+  | minusinf F = F
+  | minusinf T = T
+  | minusinf (Ge (Cx (c, e))) = F
+  | minusinf (Gt (Cx (c, e))) = F
+  | minusinf (Le (Cx (c, e))) = T
+  | minusinf (Lt (Cx (c, e))) = T
+  | minusinf (NEq (Cx (c, e))) = T
+  | minusinf (Eq (Cx (c, e))) = F
+  | minusinf (Or (p, q)) = Or (minusinf p, minusinf q)
+  | minusinf (And (p, q)) = And (minusinf p, minusinf q);
 
-fun not (NOT p) = p
-  | not T = F
-  | not F = T
-  | not (Lt u) = NOT (Lt u)
-  | not (Le v) = NOT (Le v)
-  | not (Gt w) = NOT (Gt w)
-  | not (Ge x) = NOT (Ge x)
-  | not (Eq y) = NOT (Eq y)
-  | not (NEq z) = NOT (NEq z)
-  | not (Dvd (aa, ab)) = NOT (Dvd (aa, ab))
-  | not (NDvd (ac, ad)) = NOT (NDvd (ac, ad))
-  | not (And (af, ag)) = NOT (And (af, ag))
-  | not (Or (ah, ai)) = NOT (Or (ah, ai))
-  | not (Imp (aj, ak)) = NOT (Imp (aj, ak))
-  | not (Iff (al, am)) = NOT (Iff (al, am))
-  | not (E an) = NOT (E an)
-  | not (A ao) = NOT (A ao)
-  | not (Closed ap) = NOT (Closed ap)
-  | not (NClosed aq) = NOT (NClosed aq);
+fun numsubst0 t (Mul (i, a)) = Mul (i, numsubst0 t a)
+  | numsubst0 t (Sub (a, b)) = Sub (numsubst0 t a, numsubst0 t b)
+  | numsubst0 t (Add (a, b)) = Add (numsubst0 t a, numsubst0 t b)
+  | numsubst0 t (Neg a) = Neg (numsubst0 t a)
+  | numsubst0 t (Cx (i, a)) = Add (Mul (i, t), numsubst0 t a)
+  | numsubst0 t (Bound n) =
+    (if ((n : IntInf.int) = Integer.zero_nat) then t else Bound n)
+  | numsubst0 t (C c) = C c;
+
+fun subst0 t (NClosed p) = NClosed p
+  | subst0 t (Closed p) = Closed p
+  | subst0 t (Iffa (p, q)) = Iffa (subst0 t p, subst0 t q)
+  | subst0 t (Impa (p, q)) = Impa (subst0 t p, subst0 t q)
+  | subst0 t (Or (p, q)) = Or (subst0 t p, subst0 t q)
+  | subst0 t (And (p, q)) = And (subst0 t p, subst0 t q)
+  | subst0 t (Nota p) = Nota (subst0 t p)
+  | subst0 t (NDvd (i, a)) = NDvd (i, numsubst0 t a)
+  | subst0 t (Dvd (i, a)) = Dvd (i, numsubst0 t a)
+  | subst0 t (NEq a) = NEq (numsubst0 t a)
+  | subst0 t (Eq a) = Eq (numsubst0 t a)
+  | subst0 t (Ge a) = Ge (numsubst0 t a)
+  | subst0 t (Gt a) = Gt (numsubst0 t a)
+  | subst0 t (Le a) = Le (numsubst0 t a)
+  | subst0 t (Lt a) = Lt (numsubst0 t a)
+  | subst0 t F = F
+  | subst0 t T = T;
+
+fun conj p q =
+  (if eq_fm p F orelse eq_fm q F then F
+    else (if eq_fm p T then q else (if eq_fm q T then p else And (p, q))));
+
+fun disj p q =
+  (if eq_fm p T orelse eq_fm q T then T
+    else (if eq_fm p F then q else (if eq_fm q F then p else Or (p, q))));
+
+fun nota (NClosed v) = Nota (NClosed v)
+  | nota (Closed v) = Nota (Closed v)
+  | nota (A v) = Nota (A v)
+  | nota (E v) = Nota (E v)
+  | nota (Iffa (v, va)) = Nota (Iffa (v, va))
+  | nota (Impa (v, va)) = Nota (Impa (v, va))
+  | nota (Or (v, va)) = Nota (Or (v, va))
+  | nota (And (v, va)) = Nota (And (v, va))
+  | nota (NDvd (v, va)) = Nota (NDvd (v, va))
+  | nota (Dvd (v, va)) = Nota (Dvd (v, va))
+  | nota (NEq v) = Nota (NEq v)
+  | nota (Eq v) = Nota (Eq v)
+  | nota (Ge v) = Nota (Ge v)
+  | nota (Gt v) = Nota (Gt v)
+  | nota (Le v) = Nota (Le v)
+  | nota (Lt v) = Nota (Lt v)
+  | nota F = T
+  | nota T = F
+  | nota (Nota y) = y;
+
+fun imp p q =
+  (if eq_fm p F orelse eq_fm q T then T
+    else (if eq_fm p T then q
+           else (if eq_fm q F then nota p else Impa (p, q))));
 
 fun iff p q =
-  (if (p = q) then T
-    else (if ((p = not q) orelse (not p = q)) then F
-           else (if (p = F) then not q
-                  else (if (q = F) then not p
-                         else (if (p = T) then q
-                                else (if (q = T) then p else Iff (p, q)))))));
-
-fun imp p q =
-  (if ((p = F) orelse (q = T)) then T
-    else (if (p = T) then q else (if (q = F) then not p else Imp (p, q))));
-
-fun disj p q =
-  (if ((p = T) orelse (q = T)) then T
-    else (if (p = F) then q else (if (q = F) then p else Or (p, q))));
-
-fun conj p q =
-  (if ((p = F) orelse (q = F)) then F
-    else (if (p = T) then q else (if (q = T) then p else And (p, q))));
+  (if eq_fm p q then T
+    else (if eq_fm p (nota q) orelse eq_fm (nota p) q then F
+           else (if eq_fm p F then nota q
+                  else (if eq_fm q F then nota p
+                         else (if eq_fm p T then q
+                                else (if eq_fm q T then p
+                                       else Iffa (p, q)))))));
 
-fun simpfm (And (p, q)) = conj (simpfm p) (simpfm q)
-  | simpfm (Or (p, q)) = disj (simpfm p) (simpfm q)
-  | simpfm (Imp (p, q)) = imp (simpfm p) (simpfm q)
-  | simpfm (Iff (p, q)) = iff (simpfm p) (simpfm q)
-  | simpfm (NOT p) = not (simpfm p)
-  | simpfm (Lt a) =
-    let val a' = simpnum a
-    in (case a' of C x => (if (x < 0) then T else F) | Bound x => Lt a'
-         | CX (x, xa) => Lt a' | Neg x => Lt a' | Add (x, xa) => Lt a'
-         | Sub (x, xa) => Lt a' | Mul (x, xa) => Lt a')
-    end
-  | simpfm (Le a) =
-    let val a' = simpnum a
-    in (case a' of C x => (if (x <= 0) then T else F) | Bound x => Le a'
-         | CX (x, xa) => Le a' | Neg x => Le a' | Add (x, xa) => Le a'
-         | Sub (x, xa) => Le a' | Mul (x, xa) => Le a')
-    end
-  | simpfm (Gt a) =
-    let val a' = simpnum a
-    in (case a' of C x => (if (0 < x) then T else F) | Bound x => Gt a'
-         | CX (x, xa) => Gt a' | Neg x => Gt a' | Add (x, xa) => Gt a'
-         | Sub (x, xa) => Gt a' | Mul (x, xa) => Gt a')
-    end
-  | simpfm (Ge a) =
-    let val a' = simpnum a
-    in (case a' of C x => (if (0 <= x) then T else F) | Bound x => Ge a'
-         | CX (x, xa) => Ge a' | Neg x => Ge a' | Add (x, xa) => Ge a'
-         | Sub (x, xa) => Ge a' | Mul (x, xa) => Ge a')
+fun simpfm (NClosed v) = NClosed v
+  | simpfm (Closed v) = Closed v
+  | simpfm (A v) = A v
+  | simpfm (E v) = E v
+  | simpfm F = F
+  | simpfm T = T
+  | simpfm (NDvd (i, a)) =
+    (if ((i : IntInf.int) = (0 : IntInf.int)) then simpfm (NEq a)
+      else (if (((Integer.abs_int i) : IntInf.int) = (1 : IntInf.int)) then F
+             else let
+                    val a' = simpnum a;
+                  in
+                    (case a'
+                       of C v => (if not (Integer.dvd_int i v) then T else F)
+                       | Bound nat => NDvd (i, a')
+                       | Cx (int, num) => NDvd (i, a') | Neg num => NDvd (i, a')
+                       | Add (num1, num2) => NDvd (i, a')
+                       | Sub (num1, num2) => NDvd (i, a')
+                       | Mul (int, num) => NDvd (i, a'))
+                  end))
+  | simpfm (Dvd (i, a)) =
+    (if ((i : IntInf.int) = (0 : IntInf.int)) then simpfm (Eq a)
+      else (if (((Integer.abs_int i) : IntInf.int) = (1 : IntInf.int)) then T
+             else let
+                    val a' = simpnum a;
+                  in
+                    (case a' of C v => (if Integer.dvd_int i v then T else F)
+                       | Bound nat => Dvd (i, a') | Cx (int, num) => Dvd (i, a')
+                       | Neg num => Dvd (i, a')
+                       | Add (num1, num2) => Dvd (i, a')
+                       | Sub (num1, num2) => Dvd (i, a')
+                       | Mul (int, num) => Dvd (i, a'))
+                  end))
+  | simpfm (NEq a) =
+    let
+      val a' = simpnum a;
+    in
+      (case a'
+         of C v => (if not ((v : IntInf.int) = (0 : IntInf.int)) then T else F)
+         | Bound nat => NEq a' | Cx (int, num) => NEq a' | Neg num => NEq a'
+         | Add (num1, num2) => NEq a' | Sub (num1, num2) => NEq a'
+         | Mul (int, num) => NEq a')
     end
   | simpfm (Eq a) =
-    let val a' = simpnum a
-    in (case a' of C x => (if (x = 0) then T else F) | Bound x => Eq a'
-         | CX (x, xa) => Eq a' | Neg x => Eq a' | Add (x, xa) => Eq a'
-         | Sub (x, xa) => Eq a' | Mul (x, xa) => Eq a')
+    let
+      val a' = simpnum a;
+    in
+      (case a'
+         of C v => (if ((v : IntInf.int) = (0 : IntInf.int)) then T else F)
+         | Bound nat => Eq a' | Cx (int, num) => Eq a' | Neg num => Eq a'
+         | Add (num1, num2) => Eq a' | Sub (num1, num2) => Eq a'
+         | Mul (int, num) => Eq a')
     end
-  | simpfm (NEq a) =
-    let val a' = simpnum a
-    in (case a' of C x => (if Bool.not (x = 0) then T else F)
-         | Bound x => NEq a' | CX (x, xa) => NEq a' | Neg x => NEq a'
-         | Add (x, xa) => NEq a' | Sub (x, xa) => NEq a'
-         | Mul (x, xa) => NEq a')
+  | simpfm (Ge a) =
+    let
+      val a' = simpnum a;
+    in
+      (case a' of C v => (if IntInf.<= ((0 : IntInf.int), v) then T else F)
+         | Bound nat => Ge a' | Cx (int, num) => Ge a' | Neg num => Ge a'
+         | Add (num1, num2) => Ge a' | Sub (num1, num2) => Ge a'
+         | Mul (int, num) => Ge a')
     end
-  | simpfm (Dvd (i, a)) =
-    (if (i = 0) then simpfm (Eq a)
-      else (if (abs i = 1) then T
-             else let val a' = simpnum a
-                  in (case a' of C x => (if dvd i x then T else F)
-                       | Bound x => Dvd (i, a') | CX (x, xa) => Dvd (i, a')
-                       | Neg x => Dvd (i, a') | Add (x, xa) => Dvd (i, a')
-                       | Sub (x, xa) => Dvd (i, a')
-                       | Mul (x, xa) => Dvd (i, a'))
-                  end))
-  | simpfm (NDvd (i, a)) =
-    (if (i = 0) then simpfm (NEq a)
-      else (if (abs i = 1) then F
-             else let val a' = simpnum a
-                  in (case a' of C x => (if Bool.not (dvd i x) then T else F)
-                       | Bound x => NDvd (i, a') | CX (x, xa) => NDvd (i, a')
-                       | Neg x => NDvd (i, a') | Add (x, xa) => NDvd (i, a')
-                       | Sub (x, xa) => NDvd (i, a')
-                       | Mul (x, xa) => NDvd (i, a'))
-                  end))
-  | simpfm T = T
-  | simpfm F = F
-  | simpfm (E ao) = E ao
-  | simpfm (A ap) = A ap
-  | simpfm (Closed aq) = Closed aq
-  | simpfm (NClosed ar) = NClosed ar;
-
-fun foldr f [] a = a
-  | foldr f (x :: xs) a = f x (foldr f xs a);
-
-fun djf f p q =
-  (if (q = T) then T
-    else (if (q = F) then f p
-           else let val fp = f p
-                in (case fp of T => T | F => q | Lt x => Or (f p, q)
-                     | Le x => Or (f p, q) | Gt x => Or (f p, q)
-                     | Ge x => Or (f p, q) | Eq x => Or (f p, q)
-                     | NEq x => Or (f p, q) | Dvd (x, xa) => Or (f p, q)
-                     | NDvd (x, xa) => Or (f p, q) | NOT x => Or (f p, q)
-                     | And (x, xa) => Or (f p, q) | Or (x, xa) => Or (f p, q)
-                     | Imp (x, xa) => Or (f p, q) | Iff (x, xa) => Or (f p, q)
-                     | E x => Or (f p, q) | A x => Or (f p, q)
-                     | Closed x => Or (f p, q) | NClosed x => Or (f p, q))
-                end));
-
-fun evaldjf f ps = foldr (djf f) ps F;
-
-fun append [] ys = ys
-  | append (x :: xs) ys = (x :: append xs ys);
-
-fun disjuncts (Or (p, q)) = append (disjuncts p) (disjuncts q)
-  | disjuncts F = []
-  | disjuncts T = [T]
-  | disjuncts (Lt u) = [Lt u]
-  | disjuncts (Le v) = [Le v]
-  | disjuncts (Gt w) = [Gt w]
-  | disjuncts (Ge x) = [Ge x]
-  | disjuncts (Eq y) = [Eq y]
-  | disjuncts (NEq z) = [NEq z]
-  | disjuncts (Dvd (aa, ab)) = [Dvd (aa, ab)]
-  | disjuncts (NDvd (ac, ad)) = [NDvd (ac, ad)]
-  | disjuncts (NOT ae) = [NOT ae]
-  | disjuncts (And (af, ag)) = [And (af, ag)]
-  | disjuncts (Imp (aj, ak)) = [Imp (aj, ak)]
-  | disjuncts (Iff (al, am)) = [Iff (al, am)]
-  | disjuncts (E an) = [E an]
-  | disjuncts (A ao) = [A ao]
-  | disjuncts (Closed ap) = [Closed ap]
-  | disjuncts (NClosed aq) = [NClosed aq];
-
-fun DJ f p = evaldjf f (disjuncts p);
-
-fun qelim (E p) = (fn qe => DJ qe (qelim p qe))
-  | qelim (A p) = (fn qe => not (qe (qelim (NOT p) qe)))
-  | qelim (NOT p) = (fn qe => not (qelim p qe))
-  | qelim (And (p, q)) = (fn qe => conj (qelim p qe) (qelim q qe))
-  | qelim (Or (p, q)) = (fn qe => disj (qelim p qe) (qelim q qe))
-  | qelim (Imp (p, q)) = (fn qe => imp (qelim p qe) (qelim q qe))
-  | qelim (Iff (p, q)) = (fn qe => iff (qelim p qe) (qelim q qe))
-  | qelim T = (fn y => simpfm T)
-  | qelim F = (fn y => simpfm F)
-  | qelim (Lt u) = (fn y => simpfm (Lt u))
-  | qelim (Le v) = (fn y => simpfm (Le v))
-  | qelim (Gt w) = (fn y => simpfm (Gt w))
-  | qelim (Ge x) = (fn y => simpfm (Ge x))
-  | qelim (Eq y) = (fn ya => simpfm (Eq y))
-  | qelim (NEq z) = (fn y => simpfm (NEq z))
-  | qelim (Dvd (aa, ab)) = (fn y => simpfm (Dvd (aa, ab)))
-  | qelim (NDvd (ac, ad)) = (fn y => simpfm (NDvd (ac, ad)))
-  | qelim (Closed ap) = (fn y => simpfm (Closed ap))
-  | qelim (NClosed aq) = (fn y => simpfm (NClosed aq));
-
-fun minus_def1 m n = nat (minus_def2 (m) (n));
-
-fun decrnum (Bound n) = Bound (minus_def1 n one_def0)
-  | decrnum (Neg a) = Neg (decrnum a)
-  | decrnum (Add (a, b)) = Add (decrnum a, decrnum b)
-  | decrnum (Sub (a, b)) = Sub (decrnum a, decrnum b)
-  | decrnum (Mul (c, a)) = Mul (c, decrnum a)
-  | decrnum (C u) = C u
-  | decrnum (CX (w, x)) = CX (w, x);
-
-fun decr (Lt a) = Lt (decrnum a)
-  | decr (Le a) = Le (decrnum a)
-  | decr (Gt a) = Gt (decrnum a)
-  | decr (Ge a) = Ge (decrnum a)
-  | decr (Eq a) = Eq (decrnum a)
-  | decr (NEq a) = NEq (decrnum a)
-  | decr (Dvd (i, a)) = Dvd (i, decrnum a)
-  | decr (NDvd (i, a)) = NDvd (i, decrnum a)
-  | decr (NOT p) = NOT (decr p)
-  | decr (And (p, q)) = And (decr p, decr q)
-  | decr (Or (p, q)) = Or (decr p, decr q)
-  | decr (Imp (p, q)) = Imp (decr p, decr q)
-  | decr (Iff (p, q)) = Iff (decr p, decr q)
-  | decr T = T
-  | decr F = F
-  | decr (E ao) = E ao
-  | decr (A ap) = A ap
-  | decr (Closed aq) = Closed aq
-  | decr (NClosed ar) = NClosed ar;
-
-fun map f [] = []
-  | map f (x :: xs) = (f x :: map f xs);
-
-fun allpairs f [] ys = []
-  | allpairs f (x :: xs) ys = append (map (f x) ys) (allpairs f xs ys);
-
-fun numsubst0 t (C c) = C c
-  | numsubst0 t (Bound n) = (if (n = 0) then t else Bound n)
-  | numsubst0 t (CX (i, a)) = Add (Mul (i, t), numsubst0 t a)
-  | numsubst0 t (Neg a) = Neg (numsubst0 t a)
-  | numsubst0 t (Add (a, b)) = Add (numsubst0 t a, numsubst0 t b)
-  | numsubst0 t (Sub (a, b)) = Sub (numsubst0 t a, numsubst0 t b)
-  | numsubst0 t (Mul (i, a)) = Mul (i, numsubst0 t a);
-
-fun subst0 t T = T
-  | subst0 t F = F
-  | subst0 t (Lt a) = Lt (numsubst0 t a)
-  | subst0 t (Le a) = Le (numsubst0 t a)
-  | subst0 t (Gt a) = Gt (numsubst0 t a)
-  | subst0 t (Ge a) = Ge (numsubst0 t a)
-  | subst0 t (Eq a) = Eq (numsubst0 t a)
-  | subst0 t (NEq a) = NEq (numsubst0 t a)
-  | subst0 t (Dvd (i, a)) = Dvd (i, numsubst0 t a)
-  | subst0 t (NDvd (i, a)) = NDvd (i, numsubst0 t a)
-  | subst0 t (NOT p) = NOT (subst0 t p)
-  | subst0 t (And (p, q)) = And (subst0 t p, subst0 t q)
-  | subst0 t (Or (p, q)) = Or (subst0 t p, subst0 t q)
-  | subst0 t (Imp (p, q)) = Imp (subst0 t p, subst0 t q)
-  | subst0 t (Iff (p, q)) = Iff (subst0 t p, subst0 t q)
-  | subst0 t (Closed P) = Closed P
-  | subst0 t (NClosed P) = NClosed P;
-
-fun minusinf (And (p, q)) = And (minusinf p, minusinf q)
-  | minusinf (Or (p, q)) = Or (minusinf p, minusinf q)
-  | minusinf (Eq (CX (c, e))) = F
-  | minusinf (NEq (CX (c, e))) = T
-  | minusinf (Lt (CX (c, e))) = T
-  | minusinf (Le (CX (c, e))) = T
-  | minusinf (Gt (CX (c, e))) = F
-  | minusinf (Ge (CX (c, e))) = F
-  | minusinf T = T
-  | minusinf F = F
-  | minusinf (Lt (C bo)) = Lt (C bo)
-  | minusinf (Lt (Bound bp)) = Lt (Bound bp)
-  | minusinf (Lt (Neg bs)) = Lt (Neg bs)
-  | minusinf (Lt (Add (bt, bu))) = Lt (Add (bt, bu))
-  | minusinf (Lt (Sub (bv, bw))) = Lt (Sub (bv, bw))
-  | minusinf (Lt (Mul (bx, by))) = Lt (Mul (bx, by))
-  | minusinf (Le (C ck)) = Le (C ck)
-  | minusinf (Le (Bound cl)) = Le (Bound cl)
-  | minusinf (Le (Neg co)) = Le (Neg co)
-  | minusinf (Le (Add (cp, cq))) = Le (Add (cp, cq))
-  | minusinf (Le (Sub (cr, cs))) = Le (Sub (cr, cs))
-  | minusinf (Le (Mul (ct, cu))) = Le (Mul (ct, cu))
-  | minusinf (Gt (C dg)) = Gt (C dg)
-  | minusinf (Gt (Bound dh)) = Gt (Bound dh)
-  | minusinf (Gt (Neg dk)) = Gt (Neg dk)
-  | minusinf (Gt (Add (dl, dm))) = Gt (Add (dl, dm))
-  | minusinf (Gt (Sub (dn, do'))) = Gt (Sub (dn, do'))
-  | minusinf (Gt (Mul (dp, dq))) = Gt (Mul (dp, dq))
-  | minusinf (Ge (C ec)) = Ge (C ec)
-  | minusinf (Ge (Bound ed)) = Ge (Bound ed)
-  | minusinf (Ge (Neg eg)) = Ge (Neg eg)
-  | minusinf (Ge (Add (eh, ei))) = Ge (Add (eh, ei))
-  | minusinf (Ge (Sub (ej, ek))) = Ge (Sub (ej, ek))
-  | minusinf (Ge (Mul (el, em))) = Ge (Mul (el, em))
-  | minusinf (Eq (C ey)) = Eq (C ey)
-  | minusinf (Eq (Bound ez)) = Eq (Bound ez)
-  | minusinf (Eq (Neg fc)) = Eq (Neg fc)
-  | minusinf (Eq (Add (fd, fe))) = Eq (Add (fd, fe))
-  | minusinf (Eq (Sub (ff, fg))) = Eq (Sub (ff, fg))
-  | minusinf (Eq (Mul (fh, fi))) = Eq (Mul (fh, fi))
-  | minusinf (NEq (C fu)) = NEq (C fu)
-  | minusinf (NEq (Bound fv)) = NEq (Bound fv)
-  | minusinf (NEq (Neg fy)) = NEq (Neg fy)
-  | minusinf (NEq (Add (fz, ga))) = NEq (Add (fz, ga))
-  | minusinf (NEq (Sub (gb, gc))) = NEq (Sub (gb, gc))
-  | minusinf (NEq (Mul (gd, ge))) = NEq (Mul (gd, ge))
-  | minusinf (Dvd (aa, ab)) = Dvd (aa, ab)
-  | minusinf (NDvd (ac, ad)) = NDvd (ac, ad)
-  | minusinf (NOT ae) = NOT ae
-  | minusinf (Imp (aj, ak)) = Imp (aj, ak)
-  | minusinf (Iff (al, am)) = Iff (al, am)
-  | minusinf (E an) = E an
-  | minusinf (A ao) = A ao
-  | minusinf (Closed ap) = Closed ap
-  | minusinf (NClosed aq) = NClosed aq;
-
-fun iupt (i, j) = (if (j < i) then [] else (i :: iupt ((i + 1), j)));
-
-fun mirror (And (p, q)) = And (mirror p, mirror q)
-  | mirror (Or (p, q)) = Or (mirror p, mirror q)
-  | mirror (Eq (CX (c, e))) = Eq (CX (c, Neg e))
-  | mirror (NEq (CX (c, e))) = NEq (CX (c, Neg e))
-  | mirror (Lt (CX (c, e))) = Gt (CX (c, Neg e))
-  | mirror (Le (CX (c, e))) = Ge (CX (c, Neg e))
-  | mirror (Gt (CX (c, e))) = Lt (CX (c, Neg e))
-  | mirror (Ge (CX (c, e))) = Le (CX (c, Neg e))
-  | mirror (Dvd (i, CX (c, e))) = Dvd (i, CX (c, Neg e))
-  | mirror (NDvd (i, CX (c, e))) = NDvd (i, CX (c, Neg e))
-  | mirror T = T
-  | mirror F = F
-  | mirror (Lt (C bo)) = Lt (C bo)
-  | mirror (Lt (Bound bp)) = Lt (Bound bp)
-  | mirror (Lt (Neg bs)) = Lt (Neg bs)
-  | mirror (Lt (Add (bt, bu))) = Lt (Add (bt, bu))
-  | mirror (Lt (Sub (bv, bw))) = Lt (Sub (bv, bw))
-  | mirror (Lt (Mul (bx, by))) = Lt (Mul (bx, by))
-  | mirror (Le (C ck)) = Le (C ck)
-  | mirror (Le (Bound cl)) = Le (Bound cl)
-  | mirror (Le (Neg co)) = Le (Neg co)
-  | mirror (Le (Add (cp, cq))) = Le (Add (cp, cq))
-  | mirror (Le (Sub (cr, cs))) = Le (Sub (cr, cs))
-  | mirror (Le (Mul (ct, cu))) = Le (Mul (ct, cu))
-  | mirror (Gt (C dg)) = Gt (C dg)
-  | mirror (Gt (Bound dh)) = Gt (Bound dh)
-  | mirror (Gt (Neg dk)) = Gt (Neg dk)
-  | mirror (Gt (Add (dl, dm))) = Gt (Add (dl, dm))
-  | mirror (Gt (Sub (dn, do'))) = Gt (Sub (dn, do'))
-  | mirror (Gt (Mul (dp, dq))) = Gt (Mul (dp, dq))
-  | mirror (Ge (C ec)) = Ge (C ec)
-  | mirror (Ge (Bound ed)) = Ge (Bound ed)
-  | mirror (Ge (Neg eg)) = Ge (Neg eg)
-  | mirror (Ge (Add (eh, ei))) = Ge (Add (eh, ei))
-  | mirror (Ge (Sub (ej, ek))) = Ge (Sub (ej, ek))
-  | mirror (Ge (Mul (el, em))) = Ge (Mul (el, em))
-  | mirror (Eq (C ey)) = Eq (C ey)
-  | mirror (Eq (Bound ez)) = Eq (Bound ez)
-  | mirror (Eq (Neg fc)) = Eq (Neg fc)
-  | mirror (Eq (Add (fd, fe))) = Eq (Add (fd, fe))
-  | mirror (Eq (Sub (ff, fg))) = Eq (Sub (ff, fg))
-  | mirror (Eq (Mul (fh, fi))) = Eq (Mul (fh, fi))
-  | mirror (NEq (C fu)) = NEq (C fu)
-  | mirror (NEq (Bound fv)) = NEq (Bound fv)
-  | mirror (NEq (Neg fy)) = NEq (Neg fy)
-  | mirror (NEq (Add (fz, ga))) = NEq (Add (fz, ga))
-  | mirror (NEq (Sub (gb, gc))) = NEq (Sub (gb, gc))
-  | mirror (NEq (Mul (gd, ge))) = NEq (Mul (gd, ge))
-  | mirror (Dvd (aa, C gq)) = Dvd (aa, C gq)
-  | mirror (Dvd (aa, Bound gr)) = Dvd (aa, Bound gr)
-  | mirror (Dvd (aa, Neg gu)) = Dvd (aa, Neg gu)
-  | mirror (Dvd (aa, Add (gv, gw))) = Dvd (aa, Add (gv, gw))
-  | mirror (Dvd (aa, Sub (gx, gy))) = Dvd (aa, Sub (gx, gy))
-  | mirror (Dvd (aa, Mul (gz, ha))) = Dvd (aa, Mul (gz, ha))
-  | mirror (NDvd (ac, C hm)) = NDvd (ac, C hm)
-  | mirror (NDvd (ac, Bound hn)) = NDvd (ac, Bound hn)
-  | mirror (NDvd (ac, Neg hq)) = NDvd (ac, Neg hq)
-  | mirror (NDvd (ac, Add (hr, hs))) = NDvd (ac, Add (hr, hs))
-  | mirror (NDvd (ac, Sub (ht, hu))) = NDvd (ac, Sub (ht, hu))
-  | mirror (NDvd (ac, Mul (hv, hw))) = NDvd (ac, Mul (hv, hw))
-  | mirror (NOT ae) = NOT ae
-  | mirror (Imp (aj, ak)) = Imp (aj, ak)
-  | mirror (Iff (al, am)) = Iff (al, am)
-  | mirror (E an) = E an
-  | mirror (A ao) = A ao
-  | mirror (Closed ap) = Closed ap
-  | mirror (NClosed aq) = NClosed aq;
-
-fun plus_def0 m n = nat ((m) + (n));
-
-fun size_def9 [] = 0
-  | size_def9 (a :: list) = plus_def0 (size_def9 list) (0 + 1);
-
-fun alpha (And (p, q)) = append (alpha p) (alpha q)
-  | alpha (Or (p, q)) = append (alpha p) (alpha q)
-  | alpha (Eq (CX (c, e))) = [Add (C ~1, e)]
-  | alpha (NEq (CX (c, e))) = [e]
-  | alpha (Lt (CX (c, e))) = [e]
-  | alpha (Le (CX (c, e))) = [Add (C ~1, e)]
-  | alpha (Gt (CX (c, e))) = []
-  | alpha (Ge (CX (c, e))) = []
-  | alpha T = []
-  | alpha F = []
-  | alpha (Lt (C bo)) = []
-  | alpha (Lt (Bound bp)) = []
-  | alpha (Lt (Neg bs)) = []
-  | alpha (Lt (Add (bt, bu))) = []
-  | alpha (Lt (Sub (bv, bw))) = []
-  | alpha (Lt (Mul (bx, by))) = []
-  | alpha (Le (C ck)) = []
-  | alpha (Le (Bound cl)) = []
-  | alpha (Le (Neg co)) = []
-  | alpha (Le (Add (cp, cq))) = []
-  | alpha (Le (Sub (cr, cs))) = []
-  | alpha (Le (Mul (ct, cu))) = []
-  | alpha (Gt (C dg)) = []
-  | alpha (Gt (Bound dh)) = []
-  | alpha (Gt (Neg dk)) = []
-  | alpha (Gt (Add (dl, dm))) = []
-  | alpha (Gt (Sub (dn, do'))) = []
-  | alpha (Gt (Mul (dp, dq))) = []
-  | alpha (Ge (C ec)) = []
-  | alpha (Ge (Bound ed)) = []
-  | alpha (Ge (Neg eg)) = []
-  | alpha (Ge (Add (eh, ei))) = []
-  | alpha (Ge (Sub (ej, ek))) = []
-  | alpha (Ge (Mul (el, em))) = []
-  | alpha (Eq (C ey)) = []
-  | alpha (Eq (Bound ez)) = []
-  | alpha (Eq (Neg fc)) = []
-  | alpha (Eq (Add (fd, fe))) = []
-  | alpha (Eq (Sub (ff, fg))) = []
-  | alpha (Eq (Mul (fh, fi))) = []
-  | alpha (NEq (C fu)) = []
-  | alpha (NEq (Bound fv)) = []
-  | alpha (NEq (Neg fy)) = []
-  | alpha (NEq (Add (fz, ga))) = []
-  | alpha (NEq (Sub (gb, gc))) = []
-  | alpha (NEq (Mul (gd, ge))) = []
-  | alpha (Dvd (aa, ab)) = []
-  | alpha (NDvd (ac, ad)) = []
-  | alpha (NOT ae) = []
-  | alpha (Imp (aj, ak)) = []
-  | alpha (Iff (al, am)) = []
-  | alpha (E an) = []
-  | alpha (A ao) = []
-  | alpha (Closed ap) = []
-  | alpha (NClosed aq) = [];
-
-fun memberl x [] = false
-  | memberl x (y :: ys) = ((x = y) orelse memberl x ys);
+  | simpfm (Gt a) =
+    let
+      val a' = simpnum a;
+    in
+      (case a' of C v => (if IntInf.< ((0 : IntInf.int), v) then T else F)
+         | Bound nat => Gt a' | Cx (int, num) => Gt a' | Neg num => Gt a'
+         | Add (num1, num2) => Gt a' | Sub (num1, num2) => Gt a'
+         | Mul (int, num) => Gt a')
+    end
+  | simpfm (Le a) =
+    let
+      val a' = simpnum a;
+    in
+      (case a' of C v => (if IntInf.<= (v, (0 : IntInf.int)) then T else F)
+         | Bound nat => Le a' | Cx (int, num) => Le a' | Neg num => Le a'
+         | Add (num1, num2) => Le a' | Sub (num1, num2) => Le a'
+         | Mul (int, num) => Le a')
+    end
+  | simpfm (Lt a) =
+    let
+      val a' = simpnum a;
+    in
+      (case a' of C v => (if IntInf.< (v, (0 : IntInf.int)) then T else F)
+         | Bound nat => Lt a' | Cx (int, num) => Lt a' | Neg num => Lt a'
+         | Add (num1, num2) => Lt a' | Sub (num1, num2) => Lt a'
+         | Mul (int, num) => Lt a')
+    end
+  | simpfm (Nota p) = nota (simpfm p)
+  | simpfm (Iffa (p, q)) = iff (simpfm p) (simpfm q)
+  | simpfm (Impa (p, q)) = imp (simpfm p) (simpfm q)
+  | simpfm (Or (p, q)) = disj (simpfm p) (simpfm q)
+  | simpfm (And (p, q)) = conj (simpfm p) (simpfm q);
 
-fun remdups [] = []
-  | remdups (x :: xs) =
-    (if memberl x xs then remdups xs else (x :: remdups xs));
-
-fun beta (And (p, q)) = append (beta p) (beta q)
-  | beta (Or (p, q)) = append (beta p) (beta q)
-  | beta (Eq (CX (c, e))) = [Sub (C ~1, e)]
-  | beta (NEq (CX (c, e))) = [Neg e]
-  | beta (Lt (CX (c, e))) = []
-  | beta (Le (CX (c, e))) = []
-  | beta (Gt (CX (c, e))) = [Neg e]
-  | beta (Ge (CX (c, e))) = [Sub (C ~1, e)]
-  | beta T = []
-  | beta F = []
-  | beta (Lt (C bo)) = []
-  | beta (Lt (Bound bp)) = []
-  | beta (Lt (Neg bs)) = []
-  | beta (Lt (Add (bt, bu))) = []
-  | beta (Lt (Sub (bv, bw))) = []
-  | beta (Lt (Mul (bx, by))) = []
-  | beta (Le (C ck)) = []
-  | beta (Le (Bound cl)) = []
-  | beta (Le (Neg co)) = []
-  | beta (Le (Add (cp, cq))) = []
-  | beta (Le (Sub (cr, cs))) = []
-  | beta (Le (Mul (ct, cu))) = []
-  | beta (Gt (C dg)) = []
-  | beta (Gt (Bound dh)) = []
-  | beta (Gt (Neg dk)) = []
-  | beta (Gt (Add (dl, dm))) = []
-  | beta (Gt (Sub (dn, do'))) = []
-  | beta (Gt (Mul (dp, dq))) = []
-  | beta (Ge (C ec)) = []
-  | beta (Ge (Bound ed)) = []
-  | beta (Ge (Neg eg)) = []
-  | beta (Ge (Add (eh, ei))) = []
-  | beta (Ge (Sub (ej, ek))) = []
-  | beta (Ge (Mul (el, em))) = []
-  | beta (Eq (C ey)) = []
-  | beta (Eq (Bound ez)) = []
-  | beta (Eq (Neg fc)) = []
-  | beta (Eq (Add (fd, fe))) = []
-  | beta (Eq (Sub (ff, fg))) = []
-  | beta (Eq (Mul (fh, fi))) = []
-  | beta (NEq (C fu)) = []
-  | beta (NEq (Bound fv)) = []
-  | beta (NEq (Neg fy)) = []
-  | beta (NEq (Add (fz, ga))) = []
-  | beta (NEq (Sub (gb, gc))) = []
-  | beta (NEq (Mul (gd, ge))) = []
-  | beta (Dvd (aa, ab)) = []
-  | beta (NDvd (ac, ad)) = []
-  | beta (NOT ae) = []
-  | beta (Imp (aj, ak)) = []
-  | beta (Iff (al, am)) = []
-  | beta (E an) = []
-  | beta (A ao) = []
-  | beta (Closed ap) = []
-  | beta (NClosed aq) = [];
-
-fun fst (a, b) = a;
-
-fun div_def1 a b = fst (divAlg (a, b));
-
-fun div_def0 m n = nat (div_def1 (m) (n));
-
-fun mod_def0 m n = nat (mod_def1 (m) (n));
-
-fun gcd (m, n) = (if (n = 0) then m else gcd (n, mod_def0 m n));
-
-fun times_def0 m n = nat ((m) * (n));
-
-fun lcm x = (fn (m, n) => div_def0 (times_def0 m n) (gcd (m, n))) x;
-
-fun ilcm x = (fn j => (lcm (nat (abs x), nat (abs j))));
-
-fun delta (And (p, q)) = ilcm (delta p) (delta q)
-  | delta (Or (p, q)) = ilcm (delta p) (delta q)
-  | delta (Dvd (i, CX (c, e))) = i
-  | delta (NDvd (i, CX (c, e))) = i
-  | delta T = 1
-  | delta F = 1
-  | delta (Lt u) = 1
-  | delta (Le v) = 1
-  | delta (Gt w) = 1
-  | delta (Ge x) = 1
-  | delta (Eq y) = 1
-  | delta (NEq z) = 1
-  | delta (Dvd (aa, C bo)) = 1
-  | delta (Dvd (aa, Bound bp)) = 1
-  | delta (Dvd (aa, Neg bs)) = 1
-  | delta (Dvd (aa, Add (bt, bu))) = 1
-  | delta (Dvd (aa, Sub (bv, bw))) = 1
-  | delta (Dvd (aa, Mul (bx, by))) = 1
-  | delta (NDvd (ac, C ck)) = 1
-  | delta (NDvd (ac, Bound cl)) = 1
-  | delta (NDvd (ac, Neg co)) = 1
-  | delta (NDvd (ac, Add (cp, cq))) = 1
-  | delta (NDvd (ac, Sub (cr, cs))) = 1
-  | delta (NDvd (ac, Mul (ct, cu))) = 1
-  | delta (NOT ae) = 1
-  | delta (Imp (aj, ak)) = 1
-  | delta (Iff (al, am)) = 1
-  | delta (E an) = 1
-  | delta (A ao) = 1
-  | delta (Closed ap) = 1
-  | delta (NClosed aq) = 1;
-
-fun a_beta (And (p, q)) = (fn k => And (a_beta p k, a_beta q k))
-  | a_beta (Or (p, q)) = (fn k => Or (a_beta p k, a_beta q k))
-  | a_beta (Eq (CX (c, e))) = (fn k => Eq (CX (1, Mul (div_def1 k c, e))))
-  | a_beta (NEq (CX (c, e))) = (fn k => NEq (CX (1, Mul (div_def1 k c, e))))
-  | a_beta (Lt (CX (c, e))) = (fn k => Lt (CX (1, Mul (div_def1 k c, e))))
-  | a_beta (Le (CX (c, e))) = (fn k => Le (CX (1, Mul (div_def1 k c, e))))
-  | a_beta (Gt (CX (c, e))) = (fn k => Gt (CX (1, Mul (div_def1 k c, e))))
-  | a_beta (Ge (CX (c, e))) = (fn k => Ge (CX (1, Mul (div_def1 k c, e))))
-  | a_beta (Dvd (i, CX (c, e))) =
-    (fn k => Dvd ((div_def1 k c * i), CX (1, Mul (div_def1 k c, e))))
-  | a_beta (NDvd (i, CX (c, e))) =
-    (fn k => NDvd ((div_def1 k c * i), CX (1, Mul (div_def1 k c, e))))
-  | a_beta T = (fn k => T)
-  | a_beta F = (fn k => F)
-  | a_beta (Lt (C bo)) = (fn k => Lt (C bo))
-  | a_beta (Lt (Bound bp)) = (fn k => Lt (Bound bp))
-  | a_beta (Lt (Neg bs)) = (fn k => Lt (Neg bs))
-  | a_beta (Lt (Add (bt, bu))) = (fn k => Lt (Add (bt, bu)))
-  | a_beta (Lt (Sub (bv, bw))) = (fn k => Lt (Sub (bv, bw)))
-  | a_beta (Lt (Mul (bx, by))) = (fn k => Lt (Mul (bx, by)))
-  | a_beta (Le (C ck)) = (fn k => Le (C ck))
-  | a_beta (Le (Bound cl)) = (fn k => Le (Bound cl))
-  | a_beta (Le (Neg co)) = (fn k => Le (Neg co))
-  | a_beta (Le (Add (cp, cq))) = (fn k => Le (Add (cp, cq)))
-  | a_beta (Le (Sub (cr, cs))) = (fn k => Le (Sub (cr, cs)))
-  | a_beta (Le (Mul (ct, cu))) = (fn k => Le (Mul (ct, cu)))
-  | a_beta (Gt (C dg)) = (fn k => Gt (C dg))
-  | a_beta (Gt (Bound dh)) = (fn k => Gt (Bound dh))
-  | a_beta (Gt (Neg dk)) = (fn k => Gt (Neg dk))
-  | a_beta (Gt (Add (dl, dm))) = (fn k => Gt (Add (dl, dm)))
-  | a_beta (Gt (Sub (dn, do'))) = (fn k => Gt (Sub (dn, do')))
-  | a_beta (Gt (Mul (dp, dq))) = (fn k => Gt (Mul (dp, dq)))
-  | a_beta (Ge (C ec)) = (fn k => Ge (C ec))
-  | a_beta (Ge (Bound ed)) = (fn k => Ge (Bound ed))
-  | a_beta (Ge (Neg eg)) = (fn k => Ge (Neg eg))
-  | a_beta (Ge (Add (eh, ei))) = (fn k => Ge (Add (eh, ei)))
-  | a_beta (Ge (Sub (ej, ek))) = (fn k => Ge (Sub (ej, ek)))
-  | a_beta (Ge (Mul (el, em))) = (fn k => Ge (Mul (el, em)))
-  | a_beta (Eq (C ey)) = (fn k => Eq (C ey))
-  | a_beta (Eq (Bound ez)) = (fn k => Eq (Bound ez))
-  | a_beta (Eq (Neg fc)) = (fn k => Eq (Neg fc))
-  | a_beta (Eq (Add (fd, fe))) = (fn k => Eq (Add (fd, fe)))
-  | a_beta (Eq (Sub (ff, fg))) = (fn k => Eq (Sub (ff, fg)))
-  | a_beta (Eq (Mul (fh, fi))) = (fn k => Eq (Mul (fh, fi)))
-  | a_beta (NEq (C fu)) = (fn k => NEq (C fu))
-  | a_beta (NEq (Bound fv)) = (fn k => NEq (Bound fv))
-  | a_beta (NEq (Neg fy)) = (fn k => NEq (Neg fy))
-  | a_beta (NEq (Add (fz, ga))) = (fn k => NEq (Add (fz, ga)))
-  | a_beta (NEq (Sub (gb, gc))) = (fn k => NEq (Sub (gb, gc)))
-  | a_beta (NEq (Mul (gd, ge))) = (fn k => NEq (Mul (gd, ge)))
-  | a_beta (Dvd (aa, C gq)) = (fn k => Dvd (aa, C gq))
-  | a_beta (Dvd (aa, Bound gr)) = (fn k => Dvd (aa, Bound gr))
-  | a_beta (Dvd (aa, Neg gu)) = (fn k => Dvd (aa, Neg gu))
-  | a_beta (Dvd (aa, Add (gv, gw))) = (fn k => Dvd (aa, Add (gv, gw)))
-  | a_beta (Dvd (aa, Sub (gx, gy))) = (fn k => Dvd (aa, Sub (gx, gy)))
-  | a_beta (Dvd (aa, Mul (gz, ha))) = (fn k => Dvd (aa, Mul (gz, ha)))
-  | a_beta (NDvd (ac, C hm)) = (fn k => NDvd (ac, C hm))
-  | a_beta (NDvd (ac, Bound hn)) = (fn k => NDvd (ac, Bound hn))
-  | a_beta (NDvd (ac, Neg hq)) = (fn k => NDvd (ac, Neg hq))
-  | a_beta (NDvd (ac, Add (hr, hs))) = (fn k => NDvd (ac, Add (hr, hs)))
-  | a_beta (NDvd (ac, Sub (ht, hu))) = (fn k => NDvd (ac, Sub (ht, hu)))
-  | a_beta (NDvd (ac, Mul (hv, hw))) = (fn k => NDvd (ac, Mul (hv, hw)))
-  | a_beta (NOT ae) = (fn k => NOT ae)
-  | a_beta (Imp (aj, ak)) = (fn k => Imp (aj, ak))
-  | a_beta (Iff (al, am)) = (fn k => Iff (al, am))
-  | a_beta (E an) = (fn k => E an)
-  | a_beta (A ao) = (fn k => A ao)
-  | a_beta (Closed ap) = (fn k => Closed ap)
-  | a_beta (NClosed aq) = (fn k => NClosed aq);
+fun decrnum (Cx (w, x)) = Cx (w, x)
+  | decrnum (C u) = C u
+  | decrnum (Mul (c, a)) = Mul (c, decrnum a)
+  | decrnum (Sub (a, b)) = Sub (decrnum a, decrnum b)
+  | decrnum (Add (a, b)) = Add (decrnum a, decrnum b)
+  | decrnum (Neg a) = Neg (decrnum a)
+  | decrnum (Bound n) = Bound (Integer.nat (IntInf.- (n, (1 : IntInf.int))));
 
-fun zeta (And (p, q)) = ilcm (zeta p) (zeta q)
-  | zeta (Or (p, q)) = ilcm (zeta p) (zeta q)
-  | zeta (Eq (CX (c, e))) = c
-  | zeta (NEq (CX (c, e))) = c
-  | zeta (Lt (CX (c, e))) = c
-  | zeta (Le (CX (c, e))) = c
-  | zeta (Gt (CX (c, e))) = c
-  | zeta (Ge (CX (c, e))) = c
-  | zeta (Dvd (i, CX (c, e))) = c
-  | zeta (NDvd (i, CX (c, e))) = c
-  | zeta T = 1
-  | zeta F = 1
-  | zeta (Lt (C bo)) = 1
-  | zeta (Lt (Bound bp)) = 1
-  | zeta (Lt (Neg bs)) = 1
-  | zeta (Lt (Add (bt, bu))) = 1
-  | zeta (Lt (Sub (bv, bw))) = 1
-  | zeta (Lt (Mul (bx, by))) = 1
-  | zeta (Le (C ck)) = 1
-  | zeta (Le (Bound cl)) = 1
-  | zeta (Le (Neg co)) = 1
-  | zeta (Le (Add (cp, cq))) = 1
-  | zeta (Le (Sub (cr, cs))) = 1
-  | zeta (Le (Mul (ct, cu))) = 1
-  | zeta (Gt (C dg)) = 1
-  | zeta (Gt (Bound dh)) = 1
-  | zeta (Gt (Neg dk)) = 1
-  | zeta (Gt (Add (dl, dm))) = 1
-  | zeta (Gt (Sub (dn, do'))) = 1
-  | zeta (Gt (Mul (dp, dq))) = 1
-  | zeta (Ge (C ec)) = 1
-  | zeta (Ge (Bound ed)) = 1
-  | zeta (Ge (Neg eg)) = 1
-  | zeta (Ge (Add (eh, ei))) = 1
-  | zeta (Ge (Sub (ej, ek))) = 1
-  | zeta (Ge (Mul (el, em))) = 1
-  | zeta (Eq (C ey)) = 1
-  | zeta (Eq (Bound ez)) = 1
-  | zeta (Eq (Neg fc)) = 1
-  | zeta (Eq (Add (fd, fe))) = 1
-  | zeta (Eq (Sub (ff, fg))) = 1
-  | zeta (Eq (Mul (fh, fi))) = 1
-  | zeta (NEq (C fu)) = 1
-  | zeta (NEq (Bound fv)) = 1
-  | zeta (NEq (Neg fy)) = 1
-  | zeta (NEq (Add (fz, ga))) = 1
-  | zeta (NEq (Sub (gb, gc))) = 1
-  | zeta (NEq (Mul (gd, ge))) = 1
-  | zeta (Dvd (aa, C gq)) = 1
-  | zeta (Dvd (aa, Bound gr)) = 1
-  | zeta (Dvd (aa, Neg gu)) = 1
-  | zeta (Dvd (aa, Add (gv, gw))) = 1
-  | zeta (Dvd (aa, Sub (gx, gy))) = 1
-  | zeta (Dvd (aa, Mul (gz, ha))) = 1
-  | zeta (NDvd (ac, C hm)) = 1
-  | zeta (NDvd (ac, Bound hn)) = 1
-  | zeta (NDvd (ac, Neg hq)) = 1
-  | zeta (NDvd (ac, Add (hr, hs))) = 1
-  | zeta (NDvd (ac, Sub (ht, hu))) = 1
-  | zeta (NDvd (ac, Mul (hv, hw))) = 1
-  | zeta (NOT ae) = 1
-  | zeta (Imp (aj, ak)) = 1
-  | zeta (Iff (al, am)) = 1
-  | zeta (E an) = 1
-  | zeta (A ao) = 1
-  | zeta (Closed ap) = 1
-  | zeta (NClosed aq) = 1;
-
-fun split x = (fn p => x (fst p) (snd p));
-
-fun zsplit0 (C c) = (0, C c)
-  | zsplit0 (Bound n) = (if (n = 0) then (1, C 0) else (0, Bound n))
-  | zsplit0 (CX (i, a)) = split (fn i' => (fn x => ((i + i'), x))) (zsplit0 a)
-  | zsplit0 (Neg a) = (fn (i', a') => (~ i', Neg a')) (zsplit0 a)
-  | zsplit0 (Add (a, b)) =
-    (fn (ia, a') => (fn (ib, b') => ((ia + ib), Add (a', b'))) (zsplit0 b))
-      (zsplit0 a)
-  | zsplit0 (Sub (a, b)) =
-    (fn (ia, a') =>
-      (fn (ib, b') => (minus_def2 ia ib, Sub (a', b'))) (zsplit0 b))
-      (zsplit0 a)
-  | zsplit0 (Mul (i, a)) = (fn (i', a') => ((i * i'), Mul (i, a'))) (zsplit0 a);
+fun decr (NClosed ar) = NClosed ar
+  | decr (Closed aq) = Closed aq
+  | decr (A ap) = A ap
+  | decr (E ao) = E ao
+  | decr F = F
+  | decr T = T
+  | decr (Iffa (p, q)) = Iffa (decr p, decr q)
+  | decr (Impa (p, q)) = Impa (decr p, decr q)
+  | decr (Or (p, q)) = Or (decr p, decr q)
+  | decr (And (p, q)) = And (decr p, decr q)
+  | decr (Nota p) = Nota (decr p)
+  | decr (NDvd (i, a)) = NDvd (i, decrnum a)
+  | decr (Dvd (i, a)) = Dvd (i, decrnum a)
+  | decr (NEq a) = NEq (decrnum a)
+  | decr (Eq a) = Eq (decrnum a)
+  | decr (Ge a) = Ge (decrnum a)
+  | decr (Gt a) = Gt (decrnum a)
+  | decr (Le a) = Le (decrnum a)
+  | decr (Lt a) = Lt (decrnum a);
 
-fun zlfm (And (p, q)) = And (zlfm p, zlfm q)
-  | zlfm (Or (p, q)) = Or (zlfm p, zlfm q)
-  | zlfm (Imp (p, q)) = Or (zlfm (NOT p), zlfm q)
-  | zlfm (Iff (p, q)) =
-    Or (And (zlfm p, zlfm q), And (zlfm (NOT p), zlfm (NOT q)))
-  | zlfm (Lt a) =
-    let val x = zsplit0 a
-    in (fn (c, r) =>
-         (if (c = 0) then Lt r
-           else (if (0 < c) then Lt (CX (c, r)) else Gt (CX (~ c, Neg r)))))
-         x
-    end
-  | zlfm (Le a) =
-    let val x = zsplit0 a
-    in (fn (c, r) =>
-         (if (c = 0) then Le r
-           else (if (0 < c) then Le (CX (c, r)) else Ge (CX (~ c, Neg r)))))
-         x
-    end
-  | zlfm (Gt a) =
-    let val x = zsplit0 a
-    in (fn (c, r) =>
-         (if (c = 0) then Gt r
-           else (if (0 < c) then Gt (CX (c, r)) else Lt (CX (~ c, Neg r)))))
-         x
-    end
-  | zlfm (Ge a) =
-    let val x = zsplit0 a
-    in (fn (c, r) =>
-         (if (c = 0) then Ge r
-           else (if (0 < c) then Ge (CX (c, r)) else Le (CX (~ c, Neg r)))))
-         x
-    end
-  | zlfm (Eq a) =
-    let val x = zsplit0 a
-    in (fn (c, r) =>
-         (if (c = 0) then Eq r
-           else (if (0 < c) then Eq (CX (c, r)) else Eq (CX (~ c, Neg r)))))
-         x
-    end
-  | zlfm (NEq a) =
-    let val x = zsplit0 a
-    in (fn (c, r) =>
-         (if (c = 0) then NEq r
-           else (if (0 < c) then NEq (CX (c, r)) else NEq (CX (~ c, Neg r)))))
-         x
-    end
-  | zlfm (Dvd (i, a)) =
-    (if (i = 0) then zlfm (Eq a)
-      else let val x = zsplit0 a
-           in (fn (c, r) =>
-                (if (c = 0) then Dvd (abs i, r)
-                  else (if (0 < c) then Dvd (abs i, CX (c, r))
-                         else Dvd (abs i, CX (~ c, Neg r)))))
-                x
+fun cooper p =
+  let
+    val (q, a) = unita p;
+    val (b, d) = a;
+    val js = iupt (1 : IntInf.int) d;
+    val mq = simpfm (minusinf q);
+    val md = evaldjf (fn j => simpfm (subst0 (C j) mq)) js;
+  in
+    (if eq_fm md T then T
+      else let
+             val qd =
+               evaldjf (fn aa as (ba, j) => simpfm (subst0 (Add (ba, C j)) q))
+                 (List.allpairs (fn aa => fn ba => (aa, ba)) b js);
+           in
+             decr (disj md qd)
            end)
-  | zlfm (NDvd (i, a)) =
-    (if (i = 0) then zlfm (NEq a)
-      else let val x = zsplit0 a
-           in (fn (c, r) =>
-                (if (c = 0) then NDvd (abs i, r)
-                  else (if (0 < c) then NDvd (abs i, CX (c, r))
-                         else NDvd (abs i, CX (~ c, Neg r)))))
-                x
-           end)
-  | zlfm (NOT (And (p, q))) = Or (zlfm (NOT p), zlfm (NOT q))
-  | zlfm (NOT (Or (p, q))) = And (zlfm (NOT p), zlfm (NOT q))
-  | zlfm (NOT (Imp (p, q))) = And (zlfm p, zlfm (NOT q))
-  | zlfm (NOT (Iff (p, q))) =
-    Or (And (zlfm p, zlfm (NOT q)), And (zlfm (NOT p), zlfm q))
-  | zlfm (NOT (NOT p)) = zlfm p
-  | zlfm (NOT T) = F
-  | zlfm (NOT F) = T
-  | zlfm (NOT (Lt a)) = zlfm (Ge a)
-  | zlfm (NOT (Le a)) = zlfm (Gt a)
-  | zlfm (NOT (Gt a)) = zlfm (Le a)
-  | zlfm (NOT (Ge a)) = zlfm (Lt a)
-  | zlfm (NOT (Eq a)) = zlfm (NEq a)
-  | zlfm (NOT (NEq a)) = zlfm (Eq a)
-  | zlfm (NOT (Dvd (i, a))) = zlfm (NDvd (i, a))
-  | zlfm (NOT (NDvd (i, a))) = zlfm (Dvd (i, a))
-  | zlfm (NOT (Closed P)) = NClosed P
-  | zlfm (NOT (NClosed P)) = Closed P
-  | zlfm T = T
-  | zlfm F = F
-  | zlfm (NOT (E ci)) = NOT (E ci)
-  | zlfm (NOT (A cj)) = NOT (A cj)
-  | zlfm (E ao) = E ao
-  | zlfm (A ap) = A ap
-  | zlfm (Closed aq) = Closed aq
-  | zlfm (NClosed ar) = NClosed ar;
-
-fun unit p =
-  let val p' = zlfm p; val l = zeta p';
-      val q = And (Dvd (l, CX (1, C 0)), a_beta p' l); val d = delta q;
-      val B = remdups (map simpnum (beta q));
-      val a = remdups (map simpnum (alpha q))
-  in (if less_eq_def3 (size_def9 B) (size_def9 a) then (q, (B, d))
-       else (mirror q, (a, d)))
   end;
 
-fun cooper p =
-  let val (q, (B, d)) = unit p; val js = iupt (1, d);
-      val mq = simpfm (minusinf q);
-      val md = evaldjf (fn j => simpfm (subst0 (C j) mq)) js
-  in (if (md = T) then T
-       else let val qd =
-                  evaldjf (fn (b, j) => simpfm (subst0 (Add (b, C j)) q))
-                    (allpairs (fn x => fn xa => (x, xa)) B js)
-            in decr (disj md qd) end)
-  end;
-
-fun prep (E T) = T
-  | prep (E F) = F
-  | prep (E (Or (p, q))) = Or (prep (E p), prep (E q))
-  | prep (E (Imp (p, q))) = Or (prep (E (NOT p)), prep (E q))
-  | prep (E (Iff (p, q))) =
-    Or (prep (E (And (p, q))), prep (E (And (NOT p, NOT q))))
-  | prep (E (NOT (And (p, q)))) = Or (prep (E (NOT p)), prep (E (NOT q)))
-  | prep (E (NOT (Imp (p, q)))) = prep (E (And (p, NOT q)))
-  | prep (E (NOT (Iff (p, q)))) =
-    Or (prep (E (And (p, NOT q))), prep (E (And (NOT p, q))))
-  | prep (E (Lt ef)) = E (prep (Lt ef))
-  | prep (E (Le eg)) = E (prep (Le eg))
-  | prep (E (Gt eh)) = E (prep (Gt eh))
-  | prep (E (Ge ei)) = E (prep (Ge ei))
-  | prep (E (Eq ej)) = E (prep (Eq ej))
-  | prep (E (NEq ek)) = E (prep (NEq ek))
-  | prep (E (Dvd (el, em))) = E (prep (Dvd (el, em)))
-  | prep (E (NDvd (en, eo))) = E (prep (NDvd (en, eo)))
-  | prep (E (NOT T)) = E (prep (NOT T))
-  | prep (E (NOT F)) = E (prep (NOT F))
-  | prep (E (NOT (Lt gw))) = E (prep (NOT (Lt gw)))
-  | prep (E (NOT (Le gx))) = E (prep (NOT (Le gx)))
-  | prep (E (NOT (Gt gy))) = E (prep (NOT (Gt gy)))
-  | prep (E (NOT (Ge gz))) = E (prep (NOT (Ge gz)))
-  | prep (E (NOT (Eq ha))) = E (prep (NOT (Eq ha)))
-  | prep (E (NOT (NEq hb))) = E (prep (NOT (NEq hb)))
-  | prep (E (NOT (Dvd (hc, hd)))) = E (prep (NOT (Dvd (hc, hd))))
-  | prep (E (NOT (NDvd (he, hf)))) = E (prep (NOT (NDvd (he, hf))))
-  | prep (E (NOT (NOT hg))) = E (prep (NOT (NOT hg)))
-  | prep (E (NOT (Or (hj, hk)))) = E (prep (NOT (Or (hj, hk))))
-  | prep (E (NOT (E hp))) = E (prep (NOT (E hp)))
-  | prep (E (NOT (A hq))) = E (prep (NOT (A hq)))
-  | prep (E (NOT (Closed hr))) = E (prep (NOT (Closed hr)))
-  | prep (E (NOT (NClosed hs))) = E (prep (NOT (NClosed hs)))
-  | prep (E (And (eq, er))) = E (prep (And (eq, er)))
-  | prep (E (E ey)) = E (prep (E ey))
-  | prep (E (A ez)) = E (prep (A ez))
-  | prep (E (Closed fa)) = E (prep (Closed fa))
-  | prep (E (NClosed fb)) = E (prep (NClosed fb))
+fun prep (NClosed aq) = NClosed aq
+  | prep (Closed ap) = Closed ap
+  | prep (NDvd (ac, ad)) = NDvd (ac, ad)
+  | prep (Dvd (aa, ab)) = Dvd (aa, ab)
+  | prep (NEq z) = NEq z
+  | prep (Eq y) = Eq y
+  | prep (Ge x) = Ge x
+  | prep (Gt w) = Gt w
+  | prep (Le v) = Le v
+  | prep (Lt u) = Lt u
+  | prep F = F
+  | prep T = T
+  | prep (Iffa (p, q)) = Or (prep (And (p, q)), prep (And (Nota p, Nota q)))
+  | prep (Impa (p, q)) = prep (Or (Nota p, q))
+  | prep (And (p, q)) = And (prep p, prep q)
+  | prep (Or (p, q)) = Or (prep p, prep q)
+  | prep (Nota (NClosed ck)) = Nota (prep (NClosed ck))
+  | prep (Nota (Closed cj)) = Nota (prep (Closed cj))
+  | prep (Nota (E ch)) = Nota (prep (E ch))
+  | prep (Nota (NDvd (bw, bx))) = Nota (prep (NDvd (bw, bx)))
+  | prep (Nota (Dvd (bu, bv))) = Nota (prep (Dvd (bu, bv)))
+  | prep (Nota (NEq bt)) = Nota (prep (NEq bt))
+  | prep (Nota (Eq bs)) = Nota (prep (Eq bs))
+  | prep (Nota (Ge br)) = Nota (prep (Ge br))
+  | prep (Nota (Gt bq)) = Nota (prep (Gt bq))
+  | prep (Nota (Le bp)) = Nota (prep (Le bp))
+  | prep (Nota (Lt bo)) = Nota (prep (Lt bo))
+  | prep (Nota F) = Nota (prep F)
+  | prep (Nota T) = Nota (prep T)
+  | prep (Nota (Iffa (p, q))) =
+    Or (prep (And (p, Nota q)), prep (And (Nota p, q)))
+  | prep (Nota (Impa (p, q))) = And (prep p, prep (Nota q))
+  | prep (Nota (Or (p, q))) = And (prep (Nota p), prep (Nota q))
+  | prep (Nota (A p)) = prep (E (Nota p))
+  | prep (Nota (And (p, q))) = Or (prep (Nota p), prep (Nota q))
+  | prep (Nota (Nota p)) = prep p
+  | prep (A (NClosed kj)) = prep (Nota (E (Nota (NClosed kj))))
+  | prep (A (Closed ki)) = prep (Nota (E (Nota (Closed ki))))
+  | prep (A (A kh)) = prep (Nota (E (Nota (A kh))))
+  | prep (A (E kg)) = prep (Nota (E (Nota (E kg))))
+  | prep (A (Iffa (ke, kf))) = prep (Nota (E (Nota (Iffa (ke, kf)))))
+  | prep (A (Impa (kc, kd))) = prep (Nota (E (Nota (Impa (kc, kd)))))
+  | prep (A (Or (ka, kb))) = prep (Nota (E (Nota (Or (ka, kb)))))
+  | prep (A (Nota jx)) = prep (Nota (E (Nota (Nota jx))))
+  | prep (A (NDvd (jv, jw))) = prep (Nota (E (Nota (NDvd (jv, jw)))))
+  | prep (A (Dvd (jt, ju))) = prep (Nota (E (Nota (Dvd (jt, ju)))))
+  | prep (A (NEq js)) = prep (Nota (E (Nota (NEq js))))
+  | prep (A (Eq jr)) = prep (Nota (E (Nota (Eq jr))))
+  | prep (A (Ge jq)) = prep (Nota (E (Nota (Ge jq))))
+  | prep (A (Gt jp)) = prep (Nota (E (Nota (Gt jp))))
+  | prep (A (Le jo)) = prep (Nota (E (Nota (Le jo))))
+  | prep (A (Lt jn)) = prep (Nota (E (Nota (Lt jn))))
+  | prep (A F) = prep (Nota (E (Nota F)))
+  | prep (A T) = prep (Nota (E (Nota T)))
   | prep (A (And (p, q))) = And (prep (A p), prep (A q))
-  | prep (A T) = prep (NOT (E (NOT T)))
-  | prep (A F) = prep (NOT (E (NOT F)))
-  | prep (A (Lt jn)) = prep (NOT (E (NOT (Lt jn))))
-  | prep (A (Le jo)) = prep (NOT (E (NOT (Le jo))))
-  | prep (A (Gt jp)) = prep (NOT (E (NOT (Gt jp))))
-  | prep (A (Ge jq)) = prep (NOT (E (NOT (Ge jq))))
-  | prep (A (Eq jr)) = prep (NOT (E (NOT (Eq jr))))
-  | prep (A (NEq js)) = prep (NOT (E (NOT (NEq js))))
-  | prep (A (Dvd (jt, ju))) = prep (NOT (E (NOT (Dvd (jt, ju)))))
-  | prep (A (NDvd (jv, jw))) = prep (NOT (E (NOT (NDvd (jv, jw)))))
-  | prep (A (NOT jx)) = prep (NOT (E (NOT (NOT jx))))
-  | prep (A (Or (ka, kb))) = prep (NOT (E (NOT (Or (ka, kb)))))
-  | prep (A (Imp (kc, kd))) = prep (NOT (E (NOT (Imp (kc, kd)))))
-  | prep (A (Iff (ke, kf))) = prep (NOT (E (NOT (Iff (ke, kf)))))
-  | prep (A (E kg)) = prep (NOT (E (NOT (E kg))))
-  | prep (A (A kh)) = prep (NOT (E (NOT (A kh))))
-  | prep (A (Closed ki)) = prep (NOT (E (NOT (Closed ki))))
-  | prep (A (NClosed kj)) = prep (NOT (E (NOT (NClosed kj))))
-  | prep (NOT (NOT p)) = prep p
-  | prep (NOT (And (p, q))) = Or (prep (NOT p), prep (NOT q))
-  | prep (NOT (A p)) = prep (E (NOT p))
-  | prep (NOT (Or (p, q))) = And (prep (NOT p), prep (NOT q))
-  | prep (NOT (Imp (p, q))) = And (prep p, prep (NOT q))
-  | prep (NOT (Iff (p, q))) = Or (prep (And (p, NOT q)), prep (And (NOT p, q)))
-  | prep (NOT T) = NOT (prep T)
-  | prep (NOT F) = NOT (prep F)
-  | prep (NOT (Lt bo)) = NOT (prep (Lt bo))
-  | prep (NOT (Le bp)) = NOT (prep (Le bp))
-  | prep (NOT (Gt bq)) = NOT (prep (Gt bq))
-  | prep (NOT (Ge br)) = NOT (prep (Ge br))
-  | prep (NOT (Eq bs)) = NOT (prep (Eq bs))
-  | prep (NOT (NEq bt)) = NOT (prep (NEq bt))
-  | prep (NOT (Dvd (bu, bv))) = NOT (prep (Dvd (bu, bv)))
-  | prep (NOT (NDvd (bw, bx))) = NOT (prep (NDvd (bw, bx)))
-  | prep (NOT (E ch)) = NOT (prep (E ch))
-  | prep (NOT (Closed cj)) = NOT (prep (Closed cj))
-  | prep (NOT (NClosed ck)) = NOT (prep (NClosed ck))
-  | prep (Or (p, q)) = Or (prep p, prep q)
-  | prep (And (p, q)) = And (prep p, prep q)
-  | prep (Imp (p, q)) = prep (Or (NOT p, q))
-  | prep (Iff (p, q)) = Or (prep (And (p, q)), prep (And (NOT p, NOT q)))
-  | prep T = T
-  | prep F = F
-  | prep (Lt u) = Lt u
-  | prep (Le v) = Le v
-  | prep (Gt w) = Gt w
-  | prep (Ge x) = Ge x
-  | prep (Eq y) = Eq y
-  | prep (NEq z) = NEq z
-  | prep (Dvd (aa, ab)) = Dvd (aa, ab)
-  | prep (NDvd (ac, ad)) = NDvd (ac, ad)
-  | prep (Closed ap) = Closed ap
-  | prep (NClosed aq) = NClosed aq;
+  | prep (E (NClosed fb)) = E (prep (NClosed fb))
+  | prep (E (Closed fa)) = E (prep (Closed fa))
+  | prep (E (A ez)) = E (prep (A ez))
+  | prep (E (E ey)) = E (prep (E ey))
+  | prep (E (And (eq, er))) = E (prep (And (eq, er)))
+  | prep (E (Nota (NClosed hs))) = E (prep (Nota (NClosed hs)))
+  | prep (E (Nota (Closed hr))) = E (prep (Nota (Closed hr)))
+  | prep (E (Nota (A hq))) = E (prep (Nota (A hq)))
+  | prep (E (Nota (E hp))) = E (prep (Nota (E hp)))
+  | prep (E (Nota (Or (hj, hk)))) = E (prep (Nota (Or (hj, hk))))
+  | prep (E (Nota (Nota hg))) = E (prep (Nota (Nota hg)))
+  | prep (E (Nota (NDvd (he, hf)))) = E (prep (Nota (NDvd (he, hf))))
+  | prep (E (Nota (Dvd (hc, hd)))) = E (prep (Nota (Dvd (hc, hd))))
+  | prep (E (Nota (NEq hb))) = E (prep (Nota (NEq hb)))
+  | prep (E (Nota (Eq ha))) = E (prep (Nota (Eq ha)))
+  | prep (E (Nota (Ge gz))) = E (prep (Nota (Ge gz)))
+  | prep (E (Nota (Gt gy))) = E (prep (Nota (Gt gy)))
+  | prep (E (Nota (Le gx))) = E (prep (Nota (Le gx)))
+  | prep (E (Nota (Lt gw))) = E (prep (Nota (Lt gw)))
+  | prep (E (Nota F)) = E (prep (Nota F))
+  | prep (E (Nota T)) = E (prep (Nota T))
+  | prep (E (NDvd (en, eo))) = E (prep (NDvd (en, eo)))
+  | prep (E (Dvd (el, em))) = E (prep (Dvd (el, em)))
+  | prep (E (NEq ek)) = E (prep (NEq ek))
+  | prep (E (Eq ej)) = E (prep (Eq ej))
+  | prep (E (Ge ei)) = E (prep (Ge ei))
+  | prep (E (Gt eh)) = E (prep (Gt eh))
+  | prep (E (Le eg)) = E (prep (Le eg))
+  | prep (E (Lt ef)) = E (prep (Lt ef))
+  | prep (E (Nota (Iffa (p, q)))) =
+    Or (prep (E (And (p, Nota q))), prep (E (And (Nota p, q))))
+  | prep (E (Nota (Impa (p, q)))) = prep (E (And (p, Nota q)))
+  | prep (E (Nota (And (p, q)))) = Or (prep (E (Nota p)), prep (E (Nota q)))
+  | prep (E (Iffa (p, q))) =
+    Or (prep (E (And (p, q))), prep (E (And (Nota p, Nota q))))
+  | prep (E (Impa (p, q))) = Or (prep (E (Nota p)), prep (E q))
+  | prep (E (Or (p, q))) = Or (prep (E p), prep (E q))
+  | prep (E F) = F
+  | prep (E T) = T;
 
-fun pa x = qelim (prep x) cooper;
-
-val pa = (fn x => pa x);
+fun qelim (NClosed aq) = (fn y => simpfm (NClosed aq))
+  | qelim (Closed ap) = (fn y => simpfm (Closed ap))
+  | qelim (NDvd (ac, ad)) = (fn y => simpfm (NDvd (ac, ad)))
+  | qelim (Dvd (aa, ab)) = (fn y => simpfm (Dvd (aa, ab)))
+  | qelim (NEq z) = (fn y => simpfm (NEq z))
+  | qelim (Eq y) = (fn ya => simpfm (Eq y))
+  | qelim (Ge x) = (fn y => simpfm (Ge x))
+  | qelim (Gt w) = (fn y => simpfm (Gt w))
+  | qelim (Le v) = (fn y => simpfm (Le v))
+  | qelim (Lt u) = (fn y => simpfm (Lt u))
+  | qelim F = (fn y => simpfm F)
+  | qelim T = (fn y => simpfm T)
+  | qelim (Iffa (p, q)) = (fn qe => iff (qelim p qe) (qelim q qe))
+  | qelim (Impa (p, q)) = (fn qe => imp (qelim p qe) (qelim q qe))
+  | qelim (Or (p, q)) = (fn qe => disj (qelim p qe) (qelim q qe))
+  | qelim (And (p, q)) = (fn qe => conj (qelim p qe) (qelim q qe))
+  | qelim (Nota p) = (fn qe => nota (qelim p qe))
+  | qelim (A p) = (fn qe => nota (qe (qelim (Nota p) qe)))
+  | qelim (E p) = (fn qe => dj qe (qelim p qe));
 
-val test =
-  (fn x =>
-    pa (E (A (Imp (Ge (Sub (Bound 0, Bound one_def0)),
-                    E (E (Eq (Sub (Add (Mul (3, Bound one_def0),
- Mul (5, Bound 0)),
-                                    Bound (nat 2))))))))));
+val pa : fm -> fm = (fn p => qelim (prep p) cooper);
 
-end;
+end; (*struct Reflected_Presburger*)
+
+end; (*struct ROOT*)

--- a/src/HOL/ex/Reflected_Presburger.thy	Tue Jul 10 09:23:14 2007 +0200
+++ b/src/HOL/ex/Reflected_Presburger.thy	Tue Jul 10 09:23:15 2007 +0200
@@ -3,62 +3,18 @@
 uses ("coopereif.ML") ("coopertac.ML")
 begin
 
-lemma allpairs_set: "set (allpairs Pair xs ys) = {(x,y). x\<in> set xs \<and> y \<in> set ys}"
-by (induct xs) auto
-
-
-  (* generate a list from i to j*)
-consts iupt :: "int \<times> int \<Rightarrow> int list"
-recdef iupt "measure (\<lambda> (i,j). nat (j-i +1))" 
-  "iupt (i,j) = (if j <i then [] else (i# iupt(i+1, j)))"
-
-lemma iupt_set: "set (iupt(i,j)) = {i .. j}"
-proof(induct rule: iupt.induct)
-  case (1 a b)
-  show ?case
-    using prems by (simp add: simp_from_to)
-qed
-
-lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
-using Nat.gr0_conv_Suc
-by clarsimp
-
+function
+  iupt :: "int \<Rightarrow> int \<Rightarrow> int list"
+where
+  "iupt i j = (if j < i then [] else i # iupt (i+1) j)"
+by pat_completeness auto
+termination by (relation "measure (\<lambda> (i, j). nat (j-i+1))") auto
 
-lemma myl: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a \<le> b) = (0 \<le> b - a)" 
-proof(clarify)
-  fix x y ::"'a"
-  have "(x \<le> y) = (x - y \<le> 0)" by (simp only: le_iff_diff_le_0[where a="x" and b="y"])
-  also have "\<dots> = (- (y - x) \<le> 0)" by simp
-  also have "\<dots> = (0 \<le> y - x)" by (simp only: neg_le_0_iff_le[where a="y-x"])
-  finally show "(x \<le> y) = (0 \<le> y - x)" .
-qed
-
-lemma myless: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a < b) = (0 < b - a)" 
-proof(clarify)
-  fix x y ::"'a"
-  have "(x < y) = (x - y < 0)" by (simp only: less_iff_diff_less_0[where a="x" and b="y"])
-  also have "\<dots> = (- (y - x) < 0)" by simp
-  also have "\<dots> = (0 < y - x)" by (simp only: neg_less_0_iff_less[where a="y-x"])
-  finally show "(x < y) = (0 < y - x)" .
-qed
-
-lemma myeq: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a = b) = (0 = b - a)"
-  by auto
+lemma iupt_set: "set (iupt i j) = {i..j}"
+  by (induct rule: iupt.induct) (simp add: simp_from_to)
 
 (* 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-
-  {fix x k
-    from inf_period(3)[OF advdd, rule_format, where x=x and k="-k"]  
-    have " ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))" by simp}
-  hence "\<forall>x.\<forall>k. ((a::int) dvd (x + t)) = (a dvd (x+k*d + t))"  by simp
-  then show ?thesis by simp
-qed
-
   (*********************************************************************************)
   (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
   (*********************************************************************************)
@@ -198,7 +154,7 @@
   assumes nb: "numbound0 a"
   shows "Inum (b#bs) a = Inum (b'#bs) a"
 using nb
-by (induct a rule: numbound0.induct) (auto simp add: nth_pos2)
+by (induct a rule: numbound0.induct) (auto simp add: gr0_conv_Suc)
 
 primrec
   "bound0 T = True"
@@ -224,7 +180,7 @@
   assumes bp: "bound0 p"
   shows "Ifm bbs (b#bs) p = Ifm bbs (b'#bs) p"
 using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
-by (induct p rule: bound0.induct) (auto simp add: nth_pos2)
+by (induct p rule: bound0.induct) (auto simp add: gr0_conv_Suc)
 
 primrec
   "numsubst0 t (C c) = (C c)"
@@ -237,12 +193,12 @@
 
 lemma numsubst0_I:
   shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b#bs) a)#bs) t"
-  by (induct t) (simp_all add: nth_pos2)
+  by (induct t) (auto simp add: gr0_conv_Suc)
 
 lemma numsubst0_I':
   assumes nb: "numbound0 a"
   shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b'#bs) a)#bs) t"
-  by (induct t) (simp_all add: nth_pos2 numbound0_I[OF nb, where b="b" and b'="b'"])
+  by (induct t) (auto simp add: gr0_conv_Suc numbound0_I[OF nb, where b="b" and b'="b'"])
 
 
 primrec
@@ -267,7 +223,7 @@
 lemma subst0_I: assumes qfp: "qfree p"
   shows "Ifm bbs (b#bs) (subst0 a p) = Ifm bbs ((Inum (b#bs) a)#bs) p"
   using qfp numsubst0_I[where b="b" and bs="bs" and a="a"]
-  by (induct p) (simp_all add: nth_pos2 )
+  by (induct p) (simp_all add: gr0_conv_Suc)
 
 
 consts 
@@ -300,12 +256,12 @@
 
 lemma decrnum: assumes nb: "numbound0 t"
   shows "Inum (x#bs) t = Inum bs (decrnum t)"
-  using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2)
+  using nb by (induct t rule: decrnum.induct, auto simp add: gr0_conv_Suc)
 
 lemma decr: assumes nb: "bound0 p"
   shows "Ifm bbs (x#bs) p = Ifm bbs bs (decr p)"
   using nb 
-  by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum)
+  by (induct p rule: decr.induct, simp_all add: gr0_conv_Suc decrnum)
 
 lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
 by (induct p, simp_all)
@@ -444,10 +400,8 @@
 constdefs lex_bnd :: "num \<Rightarrow> num \<Rightarrow> bool"
   "lex_bnd t s \<equiv> lex_ns (bnds t, bnds s)"
 
-consts simpnum:: "num \<Rightarrow> num"
+consts
   numadd:: "num \<times> num \<Rightarrow> num"
-  nummul:: "num \<Rightarrow> int \<Rightarrow> num"
-  numfloor:: "num \<Rightarrow> num"
 recdef numadd "measure (\<lambda> (t,s). size t + size s)"
   "numadd (Add (Mul c1 (Bound n1)) r1,Add (Mul c2 (Bound n2)) r2) =
   (if n1=n2 then 
@@ -460,6 +414,25 @@
   "numadd (C b1, C b2) = C (b1+b2)"
   "numadd (a,b) = Add a b"
 
+(*function (sequential)
+  numadd :: "num \<Rightarrow> num \<Rightarrow> num"
+where
+  "numadd (Add (Mul c1 (Bound n1)) r1) (Add (Mul c2 (Bound n2)) r2) =
+      (if n1 = n2 then (let c = c1 + c2
+      in (if c = 0 then numadd r1 r2 else
+        Add (Mul c (Bound n1)) (numadd r1 r2)))
+      else if n1 \<le> n2 then
+        Add (Mul c1 (Bound n1)) (numadd r1 (Add (Mul c2 (Bound n2)) r2))
+      else
+        Add (Mul c2 (Bound n2)) (numadd (Add (Mul c1 (Bound n1)) r1) r2))"
+  | "numadd (Add (Mul c1 (Bound n1)) r1) t =
+      Add (Mul c1 (Bound n1)) (numadd r1 t)"  
+  | "numadd t (Add (Mul c2 (Bound n2)) r2) =
+      Add (Mul c2 (Bound n2)) (numadd t r2)" 
+  | "numadd (C b1) (C b2) = C (b1 + b2)"
+  | "numadd a b = Add a b"
+apply pat_completeness apply auto*)
+  
 lemma numadd: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
 apply (induct t s rule: numadd.induct, simp_all add: Let_def)
 apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
@@ -471,23 +444,25 @@
 lemma numadd_nb: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
 by (induct t s rule: numadd.induct, auto simp add: Let_def)
 
-recdef nummul "measure size"
-  "nummul (C j) = (\<lambda> i. C (i*j))"
-  "nummul (Add a b) = (\<lambda> i. numadd (nummul a i, nummul b i))"
-  "nummul (Mul c t) = (\<lambda> i. nummul t (i*c))"
-  "nummul t = (\<lambda> i. Mul i t)"
+fun
+  nummul :: "int \<Rightarrow> num \<Rightarrow> num"
+where
+  "nummul i (C j) = C (i * j)"
+  | "nummul i (Add a b) = numadd (nummul i a, nummul i b)"
+  | "nummul i (Mul c t) = nummul (i * c) t"
+  | "nummul i t = Mul i t"
 
-lemma nummul: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
+lemma nummul: "\<And> i. Inum bs (nummul i t) = Inum bs (Mul i t)"
 by (induct t rule: nummul.induct, auto simp add: ring_simps numadd)
 
-lemma nummul_nb: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
+lemma nummul_nb: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul i t)"
 by (induct t rule: nummul.induct, auto simp add: numadd_nb)
 
 constdefs numneg :: "num \<Rightarrow> num"
-  "numneg t \<equiv> nummul t (- 1)"
+  "numneg t \<equiv> nummul (- 1) t"
 
 constdefs numsub :: "num \<Rightarrow> num \<Rightarrow> num"
-  "numsub s t \<equiv> (if s = t then C 0 else numadd (s,numneg t))"
+  "numsub s t \<equiv> (if s = t then C 0 else numadd (s, numneg t))"
 
 lemma numneg: "Inum bs (numneg t) = Inum bs (Neg t)"
 using numneg_def nummul by simp
@@ -501,14 +476,16 @@
 lemma numsub_nb: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)"
 using numsub_def numadd_nb numneg_nb by simp
 
-recdef simpnum "measure size"
+fun
+  simpnum :: "num \<Rightarrow> num"
+where
   "simpnum (C j) = C j"
-  "simpnum (Bound n) = Add (Mul 1 (Bound n)) (C 0)"
-  "simpnum (Neg t) = numneg (simpnum t)"
-  "simpnum (Add t s) = numadd (simpnum t,simpnum s)"
-  "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
-  "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)"
-  "simpnum t = t"
+  | "simpnum (Bound n) = Add (Mul 1 (Bound n)) (C 0)"
+  | "simpnum (Neg t) = numneg (simpnum t)"
+  | "simpnum (Add t s) = numadd (simpnum t, simpnum s)"
+  | "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
+  | "simpnum (Mul i t) = (if i = 0 then C 0 else nummul i (simpnum t))"
+  | "simpnum t = t"
 
 lemma simpnum_ci: "Inum bs (simpnum t) = Inum bs t"
 by (induct t rule: simpnum.induct, auto simp add: numneg numadd numsub nummul)
@@ -517,12 +494,13 @@
   "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
 by (induct t rule: simpnum.induct, auto simp add: numadd_nb numsub_nb nummul_nb numneg_nb)
 
-consts not:: "fm \<Rightarrow> fm"
-recdef not "measure size"
+fun
+  not :: "fm \<Rightarrow> fm"
+where
   "not (NOT p) = p"
-  "not T = F"
-  "not F = T"
-  "not p = NOT p"
+  | "not T = F"
+  | "not F = T"
+  | "not p = NOT p"
 lemma not: "Ifm bbs bs (not p) = Ifm bbs bs (NOT p)"
 by (cases p) auto
 lemma not_qf: "qfree p \<Longrightarrow> qfree (not p)"
@@ -571,27 +549,31 @@
 lemma iff_nb: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
 using iff_def by (unfold iff_def,cases "p=q", auto simp add: not_bn)
 
-consts simpfm :: "fm \<Rightarrow> fm"
-recdef simpfm "measure fmsize"
+function (sequential)
+  simpfm :: "fm \<Rightarrow> fm"
+where
   "simpfm (And p q) = conj (simpfm p) (simpfm q)"
-  "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
-  "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
-  "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
-  "simpfm (NOT p) = not (simpfm p)"
-  "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F 
-  | _ \<Rightarrow> Lt a')"
-  "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0)  then T else F | _ \<Rightarrow> Le a')"
-  "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0)  then T else F | _ \<Rightarrow> Gt a')"
-  "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0)  then T else F | _ \<Rightarrow> Ge a')"
-  "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0)  then T else F | _ \<Rightarrow> Eq a')"
-  "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0)  then T else F | _ \<Rightarrow> NEq a')"
-  "simpfm (Dvd i a) = (if i=0 then simpfm (Eq a)
+  | "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
+  | "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
+  | "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
+  | "simpfm (NOT p) = not (simpfm p)"
+  | "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F 
+      | _ \<Rightarrow> Lt a')"
+  | "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0)  then T else F | _ \<Rightarrow> Le a')"
+  | "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0)  then T else F | _ \<Rightarrow> Gt a')"
+  | "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0)  then T else F | _ \<Rightarrow> Ge a')"
+  | "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0)  then T else F | _ \<Rightarrow> Eq a')"
+  | "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0)  then T else F | _ \<Rightarrow> NEq a')"
+  | "simpfm (Dvd i a) = (if i=0 then simpfm (Eq a)
              else if (abs i = 1) then T
              else let a' = simpnum a in case a' of C v \<Rightarrow> if (i dvd v)  then T else F | _ \<Rightarrow> Dvd i a')"
-  "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a) 
+  | "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a) 
              else if (abs i = 1) then F
              else let a' = simpnum a in case a' of C v \<Rightarrow> if (\<not>(i dvd v)) then T else F | _ \<Rightarrow> NDvd i a')"
-  "simpfm p = p"
+  | "simpfm p = p"
+by pat_completeness auto
+termination by (relation "measure fmsize") auto
+
 lemma simpfm: "Ifm bbs bs (simpfm p) = Ifm bbs bs p"
 proof(induct p rule: simpfm.induct)
   case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
@@ -727,6 +709,20 @@
   "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
   "qelim p = (\<lambda> y. simpfm p)"
 
+(*function (sequential)
+  qelim :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
+where
+  "qelim qe (E p) = DJ qe (qelim qe p)"
+  | "qelim qe (A p) = not (qe ((qelim qe (NOT p))))"
+  | "qelim qe (NOT p) = not (qelim qe p)"
+  | "qelim qe (And p q) = conj (qelim qe p) (qelim qe q)" 
+  | "qelim qe (Or  p q) = disj (qelim qe p) (qelim qe q)" 
+  | "qelim qe (Imp p q) = imp (qelim qe p) (qelim qe q)"
+  | "qelim qe (Iff p q) = iff (qelim qe p) (qelim qe q)"
+  | "qelim qe p = simpfm p"
+by pat_completeness auto
+termination by (relation "measure (fmsize o snd)") auto*)
+
 lemma qelim_ci:
   assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bbs bs (qe p) = Ifm bbs bs (E p))"
   shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bbs bs (qelim p qe) = Ifm bbs bs p)"
@@ -735,21 +731,21 @@
 (auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf 
   simpfm simpfm_qf simp del: simpfm.simps)
   (* Linearity for fm where Bound 0 ranges over \<int> *)
-consts
+
+fun
   zsplit0 :: "num \<Rightarrow> int \<times> num" (* splits the bounded from the unbounded part*)
-recdef zsplit0 "measure size"
+where
   "zsplit0 (C c) = (0,C c)"
-  "zsplit0 (Bound n) = (if n=0 then (1, C 0) else (0,Bound n))"
-  "zsplit0 (CX i a) = (let (i',a') =  zsplit0 a in (i+i', a'))"
-  "zsplit0 (Neg a) = (let (i',a') =  zsplit0 a in (-i', Neg a'))"
-  "zsplit0 (Add a b) = (let (ia,a') =  zsplit0 a ; 
+  | "zsplit0 (Bound n) = (if n=0 then (1, C 0) else (0,Bound n))"
+  | "zsplit0 (CX i a) = (let (i',a') =  zsplit0 a in (i+i', a'))"
+  | "zsplit0 (Neg a) = (let (i',a') =  zsplit0 a in (-i', Neg a'))"
+  | "zsplit0 (Add a b) = (let (ia,a') =  zsplit0 a ; 
                             (ib,b') =  zsplit0 b 
                             in (ia+ib, Add a' b'))"
-  "zsplit0 (Sub a b) = (let (ia,a') =  zsplit0 a ; 
+  | "zsplit0 (Sub a b) = (let (ia,a') =  zsplit0 a ; 
                             (ib,b') =  zsplit0 b 
                             in (ia-ib, Sub a' b'))"
-  "zsplit0 (Mul i a) = (let (i',a') =  zsplit0 a in (i*i', Mul i a'))"
-(hints simp add: Let_def)
+  | "zsplit0 (Mul i a) = (let (i',a') =  zsplit0 a in (i*i', Mul i a'))"
 
 lemma zsplit0_I:
   shows "\<And> n a. zsplit0 t = (n,a) \<Longrightarrow> (Inum ((x::int) #bs) (CX n a) = Inum (x #bs) t) \<and> numbound0 a"
@@ -788,7 +784,7 @@
   ultimately have th: "a=Add ?as ?at \<and> n=?ns + ?nt" using prems 
     by (simp add: Let_def split_def)
   from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
-  from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (x # bs) (CX xa xb) = Inum (x # bs) t \<and> numbound0 xb)" by simp
+  from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (x # bs) (CX xa xb) = Inum (x # bs) t \<and> numbound0 xb)" by auto
   with bluddy abjt have th3: "(?I x (CX ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
   from abjs prems  have th2: "(?I x (CX ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
   from th3[simplified] th2[simplified] th[simplified] show ?case 
@@ -804,7 +800,7 @@
   ultimately have th: "a=Sub ?as ?at \<and> n=?ns - ?nt" using prems 
     by (simp add: Let_def split_def)
   from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
-  from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (x # bs) (CX xa xb) = Inum (x # bs) t \<and> numbound0 xb)" by simp
+  from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (x # bs) (CX xa xb) = Inum (x # bs) t \<and> numbound0 xb)" by auto
   with bluddy abjt have th3: "(?I x (CX ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
   from abjs prems  have th2: "(?I x (CX ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
   from th3[simplified] th2[simplified] th[simplified] show ?case 
@@ -823,7 +819,6 @@
 
 consts
   iszlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *)
-  zlfm :: "fm \<Rightarrow> fm"       (* Linearity transformation for fm *)
 recdef iszlfm "measure size"
   "iszlfm (And p q) = (iszlfm p \<and> iszlfm q)" 
   "iszlfm (Or p q) = (iszlfm p \<and> iszlfm q)" 
@@ -842,7 +837,8 @@
 lemma zlin_qfree: "iszlfm p \<Longrightarrow> qfree p"
   by (induct p rule: iszlfm.induct) auto
 
-
+consts
+  zlfm :: "fm \<Rightarrow> fm"       (* Linearity transformation for fm *)
 recdef zlfm "measure fmsize"
   "zlfm (And p q) = And (zlfm p) (zlfm q)"
   "zlfm (Or p q) = Or (zlfm p) (zlfm q)"
@@ -1309,7 +1305,7 @@
 	by blast
       thus "i dvd c*x - c*(k*d) + Inum (x # bs) e" by (simp add: dvd_def)
     qed
-qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="x - k*d" and b'="x"])
+qed (auto simp add: gr0_conv_Suc numbound0_I[where bs="bs" and b="x - k*d" and b'="x"])
 
     (* Is'nt this beautiful?*)
 lemma minusinf_ex:
@@ -1374,7 +1370,7 @@
     using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"]
     by simp
   finally show ?case by simp
-qed (auto simp add: numbound0_I[where bs="bs" and b="x" and b'="- x"] nth_pos2)
+qed (auto simp add: numbound0_I[where bs="bs" and b="x" and b'="- x"] gr0_conv_Suc)
 
 lemma mirror_l: "iszlfm p \<and> d\<beta> p 1 
   \<Longrightarrow> iszlfm (mirror p) \<and> d\<beta> (mirror p) 1"
@@ -1571,7 +1567,7 @@
     using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e - j * k"] ldcp by simp
   also have "\<dots> = (\<exists> (k::int). c * x + Inum (x # bs) e = j * k)" by simp
   finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be  mult_strict_mono[OF ldcp jp ldcp ] by (simp add: dvd_def)
-qed (simp_all add: nth_pos2 numbound0_I[where bs="bs" and b="(l * x)" and b'="x"])
+qed (auto simp add: gr0_conv_Suc numbound0_I[where bs="bs" and b="(l * x)" and b'="x"])
 
 lemma a\<beta>_ex: assumes linp: "iszlfm p" and d: "d\<beta> p l" and lp: "l>0"
   shows "(\<exists> x. l dvd x \<and> Ifm bbs (x #bs) (a\<beta> p l)) = (\<exists> (x::int). Ifm bbs (x#bs) p)"
@@ -1665,7 +1661,7 @@
     from prems have id: "j dvd d" by simp
     from c1 have "?p x = (j dvd (x+ ?e))" by simp
     also have "\<dots> = (j dvd x - d + ?e)" 
-      using dvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp
+      using zdvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp
     finally show ?case 
       using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p by simp
 next
@@ -1674,9 +1670,9 @@
     from prems have id: "j dvd d" by simp
     from c1 have "?p x = (\<not> j dvd (x+ ?e))" by simp
     also have "\<dots> = (\<not> j dvd x - d + ?e)" 
-      using dvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp
+      using zdvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp
     finally show ?case using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p by simp
-qed (auto simp add: numbound0_I[where bs="bs" and b="(x - d)" and b'="x"] nth_pos2)
+qed (auto simp add: numbound0_I[where bs="bs" and b="(x - d)" and b'="x"] gr0_conv_Suc)
 
 lemma \<beta>':   
   assumes lp: "iszlfm p"
@@ -1820,7 +1816,7 @@
 
 constdefs cooper :: "fm \<Rightarrow> fm"
   "cooper p \<equiv> 
-  (let (q,B,d) = unit p; js = iupt (1,d);
+  (let (q,B,d) = unit p; js = iupt 1 d;
        mq = simpfm (minusinf q);
        md = evaldjf (\<lambda> j. simpfm (subst0 (C j) mq)) js
    in if md = T then T else
@@ -1836,7 +1832,7 @@
   let ?q = "fst (unit p)"
   let ?B = "fst (snd(unit p))"
   let ?d = "snd (snd (unit p))"
-  let ?js = "iupt (1,?d)"
+  let ?js = "iupt 1 ?d"
   let ?mq = "minusinf ?q"
   let ?smq = "simpfm ?mq"
   let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js"
@@ -1857,7 +1853,7 @@
     by (auto simp add: simpfm_bound0)
   from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp 
   from Bn jsnb have "\<forall> (b,j) \<in> set (allpairs Pair ?B ?js). numbound0 (Add b (C j))"
-    by (simp add: allpairs_set)
+    by simp
   hence "\<forall> (b,j) \<in> set (allpairs Pair ?B ?js). bound0 (subst0 (Add b (C j)) ?q)"
     using subst0_bound0[OF qfq] by blast
   hence "\<forall> (b,j) \<in> set (allpairs Pair ?B ?js). bound0 (simpfm (subst0 (Add b (C j)) ?q))"
@@ -1877,9 +1873,9 @@
   also have "\<dots> = (?I i (evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js) \<or> (\<exists> j\<in> set ?js. \<exists> b\<in> set ?B. ?I i (subst0 (Add b (C j)) ?q)))" 
    by (simp only: evaldjf_ex subst0_I[OF qfq])
  also have "\<dots>= (?I i ?md \<or> (\<exists> (b,j) \<in> set (allpairs Pair ?B ?js). (\<lambda> (b,j). ?I i (simpfm (subst0 (Add b (C j)) ?q))) (b,j)))"
-   by (simp only: simpfm allpairs_set) blast
+   by (simp only: simpfm set_allpairs) blast
  also have "\<dots> = (?I i ?md \<or> (?I i (evaldjf (\<lambda> (b,j). simpfm (subst0 (Add b (C j)) ?q)) (allpairs Pair ?B ?js))))"
-   by (simp only: evaldjf_ex[where bs="i#bs" and f="\<lambda> (b,j). simpfm (subst0 (Add b (C j)) ?q)" and ps="allpairs Pair ?B ?js"]) (auto simp add: split_def) 
+   by (simp only: evaldjf_ex[where bs="i#bs" and f="\<lambda> (b,j). simpfm (subst0 (Add b (C j)) ?q)" and ps="allpairs Pair ?B ?js"]) (auto simp add: split_def)
  finally have mdqd: "?lhs = (?I i ?md \<or> ?I i ?qd)" by simp  
   also have "\<dots> = (?I i (disj ?md ?qd))" by (simp add: disj)
   also have "\<dots> = (Ifm bbs bs (decr (disj ?md ?qd)))" by (simp only: decr [OF mdqdb]) 
@@ -1910,10 +1906,10 @@
     (E (E (Eq (Sub (Add (Mul 3 (Bound 1)) (Mul 5 (Bound 0)))
       (Bound 2))))))))"
 
-code_gen pa cooper_test in SML
+code_gen pa cooper_test in SML to GeneratedCooper
+(*code_reserved SML oo code_gen pa in SML to GeneratedCooper file "~~/src/HOL/Tools/Qelim/raw_generated_cooper.ML"*)
 
-ML {* structure GeneratedCooper = struct open ROOT end *}
-ML {* GeneratedCooper.Reflected_Presburger.cooper_test () *}
+ML {* GeneratedCooper.cooper_test () *}
 use "coopereif.ML"
 oracle linzqe_oracle ("term") = Coopereif.cooper_oracle
 use "coopertac.ML"