# HG changeset patch
# User haftmann
# Date 1273492698 -7200
# Node ID 3981db1621313b094687449d366c80ad29f5a64d
# Parent  cb074cec7a3012f3299207b79422dcb77377fb4b
less complex organization of cooper source code

diff -r cb074cec7a30 -r 3981db162131 src/HOL/Decision_Procs/Cooper.thy
--- a/src/HOL/Decision_Procs/Cooper.thy	Mon May 10 12:25:49 2010 +0200
+++ b/src/HOL/Decision_Procs/Cooper.thy	Mon May 10 13:58:18 2010 +0200
@@ -1910,7 +1910,7 @@
 ML {* @{code cooper_test} () *}
 
 (*
-code_reflect Generated_Cooper
+code_reflect Cooper_Procedure
   functions pa
   file "~~/src/HOL/Tools/Qelim/generated_cooper.ML"
 *)
diff -r cb074cec7a30 -r 3981db162131 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Mon May 10 12:25:49 2010 +0200
+++ b/src/HOL/IsaMakefile	Mon May 10 13:58:18 2010 +0200
@@ -302,7 +302,6 @@
   Tools/Predicate_Compile/predicate_compile_specialisation.ML \
   Tools/Predicate_Compile/predicate_compile_pred.ML \
   Tools/quickcheck_generators.ML \
-  Tools/Qelim/cooper_data.ML \
   Tools/Qelim/cooper.ML \
   Tools/Qelim/generated_cooper.ML \
   Tools/Qelim/presburger.ML \
diff -r cb074cec7a30 -r 3981db162131 src/HOL/Presburger.thy
--- a/src/HOL/Presburger.thy	Mon May 10 12:25:49 2010 +0200
+++ b/src/HOL/Presburger.thy	Mon May 10 13:58:18 2010 +0200
@@ -8,14 +8,11 @@
 imports Groebner_Basis SetInterval
 uses
   "Tools/Qelim/qelim.ML"
-  "Tools/Qelim/cooper_data.ML"
-  "Tools/Qelim/generated_cooper.ML"
+  "Tools/Qelim/cooper_procedure.ML"
   ("Tools/Qelim/cooper.ML")
   ("Tools/Qelim/presburger.ML")
 begin
 
-setup CooperData.setup
-
 subsection{* The @{text "-\<infinity>"} and @{text "+\<infinity>"} Properties *}
 
 
@@ -405,28 +402,9 @@
   "(((number_of v)::int) = (number_of w)) = iszero ((number_of (v + (uminus w)))::int)"
   by (rule eq_number_of_eq)
 
-declare dvd_eq_mod_eq_0[symmetric, presburger]
-declare mod_1[presburger] 
-declare mod_0[presburger]
-declare mod_by_1[presburger]
-declare zmod_zero[presburger]
-declare zmod_self[presburger]
-declare mod_self[presburger]
-declare mod_by_0[presburger]
-declare mod_div_trivial[presburger]
-declare div_mod_equality2[presburger]
-declare div_mod_equality[presburger]
-declare mod_div_equality2[presburger]
-declare mod_div_equality[presburger]
-declare mod_mult_self1[presburger]
-declare mod_mult_self2[presburger]
-declare zdiv_zmod_equality2[presburger]
-declare zdiv_zmod_equality[presburger]
-declare mod2_Suc_Suc[presburger]
-lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a"
-by simp_all
+use "Tools/Qelim/cooper.ML"
 
-use "Tools/Qelim/cooper.ML"
+setup CooperData.setup
 oracle linzqe_oracle = Coopereif.cooper_oracle
 
 use "Tools/Qelim/presburger.ML"
@@ -451,6 +429,27 @@
 end
 *} "Cooper's algorithm for Presburger arithmetic"
 
+declare dvd_eq_mod_eq_0[symmetric, presburger]
+declare mod_1[presburger] 
+declare mod_0[presburger]
+declare mod_by_1[presburger]
+declare zmod_zero[presburger]
+declare zmod_self[presburger]
+declare mod_self[presburger]
+declare mod_by_0[presburger]
+declare mod_div_trivial[presburger]
+declare div_mod_equality2[presburger]
+declare div_mod_equality[presburger]
+declare mod_div_equality2[presburger]
+declare mod_div_equality[presburger]
+declare mod_mult_self1[presburger]
+declare mod_mult_self2[presburger]
+declare zdiv_zmod_equality2[presburger]
+declare zdiv_zmod_equality[presburger]
+declare mod2_Suc_Suc[presburger]
+lemma [presburger]: "(a::int) div 0 = 0" and [presburger]: "a mod 0 = a"
+by simp_all
+
 lemma [presburger, algebra]: "m mod 2 = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
 lemma [presburger, algebra]: "m mod 2 = Suc 0 \<longleftrightarrow> \<not> 2 dvd m " by presburger
 lemma [presburger, algebra]: "m mod (Suc (Suc 0)) = (1::nat) \<longleftrightarrow> \<not> 2 dvd m " by presburger
diff -r cb074cec7a30 -r 3981db162131 src/HOL/Tools/Qelim/cooper.ML
--- a/src/HOL/Tools/Qelim/cooper.ML	Mon May 10 12:25:49 2010 +0200
+++ b/src/HOL/Tools/Qelim/cooper.ML	Mon May 10 13:58:18 2010 +0200
@@ -2,12 +2,101 @@
     Author:     Amine Chaieb, TU Muenchen
 *)
 
+signature COOPER_DATA =
+sig
+  type entry
+  val get: Proof.context -> entry
+  val del: term list -> attribute
+  val add: term list -> attribute 
+  val setup: theory -> theory
+end;
+
 signature COOPER =
 sig
-  val cooper_conv : Proof.context -> conv
+  val cooper_conv: Proof.context -> conv
   exception COOPER of string * exn
 end;
 
+signature COOPER_REIFY =
+sig
+  val cooper_oracle: cterm -> cterm
+end;
+
+structure CooperData : COOPER_DATA =
+struct
+
+type entry = simpset * (term list);
+
+val allowed_consts = 
+  [@{term "op + :: int => _"}, @{term "op + :: nat => _"},
+   @{term "op - :: int => _"}, @{term "op - :: nat => _"},
+   @{term "op * :: int => _"}, @{term "op * :: nat => _"},
+   @{term "op div :: int => _"}, @{term "op div :: nat => _"},
+   @{term "op mod :: int => _"}, @{term "op mod :: nat => _"},
+   @{term "Int.Bit0"}, @{term "Int.Bit1"},
+   @{term "op &"}, @{term "op |"}, @{term "op -->"}, 
+   @{term "op = :: int => _"}, @{term "op = :: nat => _"}, @{term "op = :: bool => _"},
+   @{term "op < :: int => _"}, @{term "op < :: nat => _"},
+   @{term "op <= :: int => _"}, @{term "op <= :: nat => _"},
+   @{term "op dvd :: int => _"}, @{term "op dvd :: nat => _"},
+   @{term "abs :: int => _"},
+   @{term "max :: int => _"}, @{term "max :: nat => _"},
+   @{term "min :: int => _"}, @{term "min :: nat => _"},
+   @{term "uminus :: int => _"}, (*@ {term "uminus :: nat => _"},*)
+   @{term "Not"}, @{term "Suc"},
+   @{term "Ex :: (int => _) => _"}, @{term "Ex :: (nat => _) => _"},
+   @{term "All :: (int => _) => _"}, @{term "All :: (nat => _) => _"},
+   @{term "nat"}, @{term "int"},
+   @{term "Int.Bit0"}, @{term "Int.Bit1"},
+   @{term "Int.Pls"}, @{term "Int.Min"},
+   @{term "Int.number_of :: int => int"}, @{term "Int.number_of :: int => nat"},
+   @{term "0::int"}, @{term "1::int"}, @{term "0::nat"}, @{term "1::nat"},
+   @{term "True"}, @{term "False"}];
+
+structure Data = Generic_Data
+(
+  type T = simpset * term list;
+  val empty = (HOL_ss, allowed_consts);
+  val extend  = I;
+  fun merge ((ss1, ts1), (ss2, ts2)) =
+    (merge_ss (ss1, ss2), Library.merge (op aconv) (ts1, ts2));
+);
+
+val get = Data.get o Context.Proof;
+
+fun add ts = Thm.declaration_attribute (fn th => fn context => 
+  context |> Data.map (fn (ss,ts') => 
+     (ss addsimps [th], merge (op aconv) (ts',ts) ))) 
+
+fun del ts = Thm.declaration_attribute (fn th => fn context => 
+  context |> Data.map (fn (ss,ts') => 
+     (ss delsimps [th], subtract (op aconv) ts' ts ))) 
+
+
+(* theory setup *)
+
+local
+
+fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ();
+
+val constsN = "consts";
+val any_keyword = keyword constsN
+val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
+val terms = thms >> map (term_of o Drule.dest_term);
+
+fun optional scan = Scan.optional scan [];
+
+in
+
+val setup =
+  Attrib.setup @{binding presburger}
+    ((Scan.lift (Args.$$$ "del") |-- optional (keyword constsN |-- terms)) >> del ||
+      optional (keyword constsN |-- terms) >> add) "Cooper data";
+
+end;
+
+end;
+
 structure Cooper: COOPER =
 struct
 
@@ -519,58 +608,54 @@
 end;
 end;
 
-
-
-structure Coopereif =
+structure Coopereif : COOPER_REIFY =
 struct
 
-fun member eq = Library.member eq;
-
 fun cooper s = raise Cooper.COOPER ("Cooper oracle failed", ERROR s);
 fun i_of_term vs t = case t
  of Free (xn, xT) => (case AList.lookup (op aconv) vs t
      of NONE   => cooper "Variable not found in the list!"
-      | SOME n => Generated_Cooper.Bound n)
-  | @{term "0::int"} => Generated_Cooper.C 0
-  | @{term "1::int"} => Generated_Cooper.C 1
-  | Term.Bound i => Generated_Cooper.Bound i
-  | Const(@{const_name Groups.uminus},_)$t' => Generated_Cooper.Neg (i_of_term vs t')
-  | Const(@{const_name Groups.plus},_)$t1$t2 => Generated_Cooper.Add (i_of_term vs t1,i_of_term vs t2)
-  | Const(@{const_name Groups.minus},_)$t1$t2 => Generated_Cooper.Sub (i_of_term vs t1,i_of_term vs t2)
+      | SOME n => Cooper_Procedure.Bound n)
+  | @{term "0::int"} => Cooper_Procedure.C 0
+  | @{term "1::int"} => Cooper_Procedure.C 1
+  | Term.Bound i => Cooper_Procedure.Bound i
+  | Const(@{const_name Groups.uminus},_)$t' => Cooper_Procedure.Neg (i_of_term vs t')
+  | Const(@{const_name Groups.plus},_)$t1$t2 => Cooper_Procedure.Add (i_of_term vs t1,i_of_term vs t2)
+  | Const(@{const_name Groups.minus},_)$t1$t2 => Cooper_Procedure.Sub (i_of_term vs t1,i_of_term vs t2)
   | Const(@{const_name Groups.times},_)$t1$t2 =>
-     (Generated_Cooper.Mul (HOLogic.dest_number t1 |> snd, i_of_term vs t2)
+     (Cooper_Procedure.Mul (HOLogic.dest_number t1 |> snd, i_of_term vs t2)
     handle TERM _ =>
-       (Generated_Cooper.Mul (HOLogic.dest_number t2 |> snd, i_of_term vs t1)
+       (Cooper_Procedure.Mul (HOLogic.dest_number t2 |> snd, i_of_term vs t1)
         handle TERM _ => cooper "Reification: Unsupported kind of multiplication"))
-  | _ => (Generated_Cooper.C (HOLogic.dest_number t |> snd)
+  | _ => (Cooper_Procedure.C (HOLogic.dest_number t |> snd)
            handle TERM _ => cooper "Reification: unknown term");
 
 fun qf_of_term ps vs t =  case t
- of Const("True",_) => Generated_Cooper.T
-  | Const("False",_) => Generated_Cooper.F
-  | Const(@{const_name Orderings.less},_)$t1$t2 => Generated_Cooper.Lt (Generated_Cooper.Sub (i_of_term vs t1,i_of_term vs t2))
-  | Const(@{const_name Orderings.less_eq},_)$t1$t2 => Generated_Cooper.Le (Generated_Cooper.Sub(i_of_term vs t1,i_of_term vs t2))
+ of Const("True",_) => Cooper_Procedure.T
+  | Const("False",_) => Cooper_Procedure.F
+  | Const(@{const_name Orderings.less},_)$t1$t2 => Cooper_Procedure.Lt (Cooper_Procedure.Sub (i_of_term vs t1,i_of_term vs t2))
+  | Const(@{const_name Orderings.less_eq},_)$t1$t2 => Cooper_Procedure.Le (Cooper_Procedure.Sub(i_of_term vs t1,i_of_term vs t2))
   | Const(@{const_name Rings.dvd},_)$t1$t2 =>
-      (Generated_Cooper.Dvd(HOLogic.dest_number t1 |> snd, i_of_term vs t2) handle _ => cooper "Reification: unsupported dvd")  (* FIXME avoid handle _ *)
-  | @{term "op = :: int => _"}$t1$t2 => Generated_Cooper.Eq (Generated_Cooper.Sub (i_of_term vs t1,i_of_term vs t2))
-  | @{term "op = :: bool => _ "}$t1$t2 => Generated_Cooper.Iff(qf_of_term ps vs t1,qf_of_term ps vs t2)
-  | Const("op &",_)$t1$t2 => Generated_Cooper.And(qf_of_term ps vs t1,qf_of_term ps vs t2)
-  | Const("op |",_)$t1$t2 => Generated_Cooper.Or(qf_of_term ps vs t1,qf_of_term ps vs t2)
-  | Const("op -->",_)$t1$t2 => Generated_Cooper.Imp(qf_of_term ps vs t1,qf_of_term ps vs t2)
-  | Const (@{const_name Not},_)$t' => Generated_Cooper.Not(qf_of_term ps vs t')
+      (Cooper_Procedure.Dvd(HOLogic.dest_number t1 |> snd, i_of_term vs t2) handle _ => cooper "Reification: unsupported dvd")  (* FIXME avoid handle _ *)
+  | @{term "op = :: int => _"}$t1$t2 => Cooper_Procedure.Eq (Cooper_Procedure.Sub (i_of_term vs t1,i_of_term vs t2))
+  | @{term "op = :: bool => _ "}$t1$t2 => Cooper_Procedure.Iff(qf_of_term ps vs t1,qf_of_term ps vs t2)
+  | Const("op &",_)$t1$t2 => Cooper_Procedure.And(qf_of_term ps vs t1,qf_of_term ps vs t2)
+  | Const("op |",_)$t1$t2 => Cooper_Procedure.Or(qf_of_term ps vs t1,qf_of_term ps vs t2)
+  | Const("op -->",_)$t1$t2 => Cooper_Procedure.Imp(qf_of_term ps vs t1,qf_of_term ps vs t2)
+  | Const (@{const_name Not},_)$t' => Cooper_Procedure.Not(qf_of_term ps vs t')
   | Const("Ex",_)$Abs(xn,xT,p) =>
      let val (xn',p') = variant_abs (xn,xT,p)
          val vs' = (Free (xn',xT), 0) :: (map (fn(v,n) => (v,1+ n)) vs)
-     in Generated_Cooper.E (qf_of_term ps vs' p')
+     in Cooper_Procedure.E (qf_of_term ps vs' p')
      end
   | Const("All",_)$Abs(xn,xT,p) =>
      let val (xn',p') = variant_abs (xn,xT,p)
          val vs' = (Free (xn',xT), 0) :: (map (fn(v,n) => (v,1+ n)) vs)
-     in Generated_Cooper.A (qf_of_term ps vs' p')
+     in Cooper_Procedure.A (qf_of_term ps vs' p')
      end
   | _ =>(case AList.lookup (op aconv) ps t of
            NONE => cooper "Reification: unknown term!"
-         | SOME n => Generated_Cooper.Closed n);
+         | SOME n => Cooper_Procedure.Closed n);
 
 local
  val ops = [@{term "op &"}, @{term "op |"}, @{term "op -->"}, @{term "op = :: bool => _"},
@@ -596,35 +681,35 @@
           else myassoc2 xs v;
 
 fun term_of_i vs t = case t
- of Generated_Cooper.C i => HOLogic.mk_number HOLogic.intT i
-  | Generated_Cooper.Bound n => the (myassoc2 vs n)
-  | Generated_Cooper.Neg t' => @{term "uminus :: int => _"} $ term_of_i vs t'
-  | Generated_Cooper.Add (t1, t2) => @{term "op + :: int => _"} $ term_of_i vs t1 $ term_of_i vs t2
-  | Generated_Cooper.Sub (t1, t2) => @{term "op - :: int => _"} $ term_of_i vs t1 $ term_of_i vs t2
-  | Generated_Cooper.Mul (i, t2) => @{term "op * :: int => _"} $
+ of Cooper_Procedure.C i => HOLogic.mk_number HOLogic.intT i
+  | Cooper_Procedure.Bound n => the (myassoc2 vs n)
+  | Cooper_Procedure.Neg t' => @{term "uminus :: int => _"} $ term_of_i vs t'
+  | Cooper_Procedure.Add (t1, t2) => @{term "op + :: int => _"} $ term_of_i vs t1 $ term_of_i vs t2
+  | Cooper_Procedure.Sub (t1, t2) => @{term "op - :: int => _"} $ term_of_i vs t1 $ term_of_i vs t2
+  | Cooper_Procedure.Mul (i, t2) => @{term "op * :: int => _"} $
       HOLogic.mk_number HOLogic.intT i $ term_of_i vs t2
-  | Generated_Cooper.Cn (n, i, t') => term_of_i vs (Generated_Cooper.Add (Generated_Cooper.Mul (i, Generated_Cooper.Bound n), t'));
+  | Cooper_Procedure.Cn (n, i, t') => term_of_i vs (Cooper_Procedure.Add (Cooper_Procedure.Mul (i, Cooper_Procedure.Bound n), t'));
 
 fun term_of_qf ps vs t =
  case t of
-   Generated_Cooper.T => HOLogic.true_const
- | Generated_Cooper.F => HOLogic.false_const
- | Generated_Cooper.Lt t' => @{term "op < :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
- | Generated_Cooper.Le t' => @{term "op <= :: int => _ "}$ term_of_i vs t' $ @{term "0::int"}
- | Generated_Cooper.Gt t' => @{term "op < :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
- | Generated_Cooper.Ge t' => @{term "op <= :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
- | Generated_Cooper.Eq t' => @{term "op = :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
- | Generated_Cooper.NEq t' => term_of_qf ps vs (Generated_Cooper.Not (Generated_Cooper.Eq t'))
- | Generated_Cooper.Dvd(i,t') => @{term "op dvd :: int => _ "} $
+   Cooper_Procedure.T => HOLogic.true_const
+ | Cooper_Procedure.F => HOLogic.false_const
+ | Cooper_Procedure.Lt t' => @{term "op < :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
+ | Cooper_Procedure.Le t' => @{term "op <= :: int => _ "}$ term_of_i vs t' $ @{term "0::int"}
+ | Cooper_Procedure.Gt t' => @{term "op < :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
+ | Cooper_Procedure.Ge t' => @{term "op <= :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
+ | Cooper_Procedure.Eq t' => @{term "op = :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
+ | Cooper_Procedure.NEq t' => term_of_qf ps vs (Cooper_Procedure.Not (Cooper_Procedure.Eq t'))
+ | Cooper_Procedure.Dvd(i,t') => @{term "op dvd :: int => _ "} $
     HOLogic.mk_number HOLogic.intT i $ term_of_i vs t'
- | Generated_Cooper.NDvd(i,t')=> term_of_qf ps vs (Generated_Cooper.Not(Generated_Cooper.Dvd(i,t')))
- | Generated_Cooper.Not t' => HOLogic.Not$(term_of_qf ps vs t')
- | Generated_Cooper.And(t1,t2) => HOLogic.conj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
- | Generated_Cooper.Or(t1,t2) => HOLogic.disj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
- | Generated_Cooper.Imp(t1,t2) => HOLogic.imp$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
- | Generated_Cooper.Iff(t1,t2) => @{term "op = :: bool => _"} $ term_of_qf ps vs t1 $ term_of_qf ps vs t2
- | Generated_Cooper.Closed n => the (myassoc2 ps n)
- | Generated_Cooper.NClosed n => term_of_qf ps vs (Generated_Cooper.Not (Generated_Cooper.Closed n))
+ | Cooper_Procedure.NDvd(i,t')=> term_of_qf ps vs (Cooper_Procedure.Not(Cooper_Procedure.Dvd(i,t')))
+ | Cooper_Procedure.Not t' => HOLogic.Not$(term_of_qf ps vs t')
+ | Cooper_Procedure.And(t1,t2) => HOLogic.conj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
+ | Cooper_Procedure.Or(t1,t2) => HOLogic.disj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
+ | Cooper_Procedure.Imp(t1,t2) => HOLogic.imp$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
+ | Cooper_Procedure.Iff(t1,t2) => @{term "op = :: bool => _"} $ term_of_qf ps vs t1 $ term_of_qf ps vs t2
+ | Cooper_Procedure.Closed n => the (myassoc2 ps n)
+ | Cooper_Procedure.NClosed n => term_of_qf ps vs (Cooper_Procedure.Not (Cooper_Procedure.Closed n))
  | _ => cooper "If this is raised, Isabelle/HOL or code generator is inconsistent!";
 
 fun cooper_oracle ct =
@@ -634,7 +719,7 @@
     val (vs, ps) = pairself (map_index swap) (OldTerm.term_frees t, term_bools [] t);
   in
     Thm.cterm_of thy (Logic.mk_equals (HOLogic.mk_Trueprop t,
-      HOLogic.mk_Trueprop (term_of_qf ps vs (Generated_Cooper.pa (qf_of_term ps vs t)))))
+      HOLogic.mk_Trueprop (term_of_qf ps vs (Cooper_Procedure.pa (qf_of_term ps vs t)))))
   end;
 
 end;
diff -r cb074cec7a30 -r 3981db162131 src/HOL/Tools/Qelim/cooper_data.ML
--- a/src/HOL/Tools/Qelim/cooper_data.ML	Mon May 10 12:25:49 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,87 +0,0 @@
-(*  Title:      HOL/Tools/Qelim/cooper_data.ML
-    Author:     Amine Chaieb, TU Muenchen
-*)
-
-signature COOPER_DATA =
-sig
-  type entry
-  val get: Proof.context -> entry
-  val del: term list -> attribute
-  val add: term list -> attribute 
-  val setup: theory -> theory
-end;
-
-structure CooperData : COOPER_DATA =
-struct
-
-type entry = simpset * (term list);
-
-val allowed_consts = 
-  [@{term "op + :: int => _"}, @{term "op + :: nat => _"},
-   @{term "op - :: int => _"}, @{term "op - :: nat => _"},
-   @{term "op * :: int => _"}, @{term "op * :: nat => _"},
-   @{term "op div :: int => _"}, @{term "op div :: nat => _"},
-   @{term "op mod :: int => _"}, @{term "op mod :: nat => _"},
-   @{term "Int.Bit0"}, @{term "Int.Bit1"},
-   @{term "op &"}, @{term "op |"}, @{term "op -->"}, 
-   @{term "op = :: int => _"}, @{term "op = :: nat => _"}, @{term "op = :: bool => _"},
-   @{term "op < :: int => _"}, @{term "op < :: nat => _"},
-   @{term "op <= :: int => _"}, @{term "op <= :: nat => _"},
-   @{term "op dvd :: int => _"}, @{term "op dvd :: nat => _"},
-   @{term "abs :: int => _"},
-   @{term "max :: int => _"}, @{term "max :: nat => _"},
-   @{term "min :: int => _"}, @{term "min :: nat => _"},
-   @{term "uminus :: int => _"}, (*@ {term "uminus :: nat => _"},*)
-   @{term "Not"}, @{term "Suc"},
-   @{term "Ex :: (int => _) => _"}, @{term "Ex :: (nat => _) => _"},
-   @{term "All :: (int => _) => _"}, @{term "All :: (nat => _) => _"},
-   @{term "nat"}, @{term "int"},
-   @{term "Int.Bit0"}, @{term "Int.Bit1"},
-   @{term "Int.Pls"}, @{term "Int.Min"},
-   @{term "Int.number_of :: int => int"}, @{term "Int.number_of :: int => nat"},
-   @{term "0::int"}, @{term "1::int"}, @{term "0::nat"}, @{term "1::nat"},
-   @{term "True"}, @{term "False"}];
-
-structure Data = Generic_Data
-(
-  type T = simpset * term list;
-  val empty = (HOL_ss, allowed_consts);
-  val extend  = I;
-  fun merge ((ss1, ts1), (ss2, ts2)) =
-    (merge_ss (ss1, ss2), Library.merge (op aconv) (ts1, ts2));
-);
-
-val get = Data.get o Context.Proof;
-
-fun add ts = Thm.declaration_attribute (fn th => fn context => 
-  context |> Data.map (fn (ss,ts') => 
-     (ss addsimps [th], merge (op aconv) (ts',ts) ))) 
-
-fun del ts = Thm.declaration_attribute (fn th => fn context => 
-  context |> Data.map (fn (ss,ts') => 
-     (ss delsimps [th], subtract (op aconv) ts' ts ))) 
-
-
-(* theory setup *)
-
-local
-
-fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ();
-
-val constsN = "consts";
-val any_keyword = keyword constsN
-val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat;
-val terms = thms >> map (term_of o Drule.dest_term);
-
-fun optional scan = Scan.optional scan [];
-
-in
-
-val setup =
-  Attrib.setup @{binding presburger}
-    ((Scan.lift (Args.$$$ "del") |-- optional (keyword constsN |-- terms)) >> del ||
-      optional (keyword constsN |-- terms) >> add) "Cooper data";
-
-end;
-
-end;
diff -r cb074cec7a30 -r 3981db162131 src/HOL/Tools/Qelim/cooper_procedure.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/Qelim/cooper_procedure.ML	Mon May 10 13:58:18 2010 +0200
@@ -0,0 +1,2274 @@
+(* Generated from Cooper.thy; DO NOT EDIT! *)
+
+structure Cooper_Procedure : sig
+  type 'a eq
+  val eq : 'a eq -> 'a -> 'a -> bool
+  val eqa : 'a eq -> 'a -> 'a -> bool
+  val leta : 'a -> ('a -> 'b) -> 'b
+  val suc : IntInf.int -> IntInf.int
+  datatype num = C of IntInf.int | Bound of IntInf.int |
+    Cn of IntInf.int * 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
+    | 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 IntInf.int |
+    NClosed of IntInf.int
+  val map : ('a -> 'b) -> 'a list -> 'b list
+  val append : 'a list -> 'a list -> 'a list
+  val disjuncts : fm -> fm list
+  val fm_case :
+    'a -> 'a -> (num -> 'a) ->
+                  (num -> 'a) ->
+                    (num -> 'a) ->
+                      (num -> 'a) ->
+                        (num -> 'a) ->
+                          (num -> 'a) ->
+                            (IntInf.int -> num -> 'a) ->
+                              (IntInf.int -> num -> 'a) ->
+                                (fm -> 'a) ->
+                                  (fm -> fm -> 'a) ->
+                                    (fm -> fm -> 'a) ->
+                                      (fm -> fm -> 'a) ->
+(fm -> fm -> 'a) ->
+  (fm -> 'a) ->
+    (fm -> 'a) -> (IntInf.int -> 'a) -> (IntInf.int -> 'a) -> fm -> 'a
+  val eq_num : num -> num -> bool
+  val eq_fm : fm -> fm -> bool
+  val djf : ('a -> fm) -> 'a -> fm -> fm
+  val foldr : ('a -> 'b -> 'b) -> 'a list -> 'b -> 'b
+  val evaldjf : ('a -> fm) -> 'a list -> fm
+  val dj : (fm -> fm) -> fm -> fm
+  val disj : fm -> fm -> fm
+  val minus_nat : IntInf.int -> IntInf.int -> IntInf.int
+  val decrnum : num -> num
+  val decr : fm -> fm
+  val concat_map : ('a -> 'b list) -> 'a list -> 'b list
+  val numsubst0 : num -> num -> num
+  val subst0 : num -> fm -> fm
+  val minusinf : fm -> fm
+  val eq_int : IntInf.int eq
+  val zero_int : IntInf.int
+  type 'a zero
+  val zero : 'a zero -> 'a
+  val zero_inta : IntInf.int zero
+  type 'a times
+  val times : 'a times -> 'a -> 'a -> 'a
+  type 'a no_zero_divisors
+  val times_no_zero_divisors : 'a no_zero_divisors -> 'a times
+  val zero_no_zero_divisors : 'a no_zero_divisors -> 'a zero
+  val times_int : IntInf.int times
+  val no_zero_divisors_int : IntInf.int no_zero_divisors
+  type 'a one
+  val one : 'a one -> 'a
+  type 'a zero_neq_one
+  val one_zero_neq_one : 'a zero_neq_one -> 'a one
+  val zero_zero_neq_one : 'a zero_neq_one -> 'a zero
+  type 'a semigroup_mult
+  val times_semigroup_mult : 'a semigroup_mult -> 'a times
+  type 'a plus
+  val plus : 'a plus -> 'a -> 'a -> 'a
+  type 'a semigroup_add
+  val plus_semigroup_add : 'a semigroup_add -> 'a plus
+  type 'a ab_semigroup_add
+  val semigroup_add_ab_semigroup_add : 'a ab_semigroup_add -> 'a semigroup_add
+  type 'a semiring
+  val ab_semigroup_add_semiring : 'a semiring -> 'a ab_semigroup_add
+  val semigroup_mult_semiring : 'a semiring -> 'a semigroup_mult
+  type 'a mult_zero
+  val times_mult_zero : 'a mult_zero -> 'a times
+  val zero_mult_zero : 'a mult_zero -> 'a zero
+  type 'a monoid_add
+  val semigroup_add_monoid_add : 'a monoid_add -> 'a semigroup_add
+  val zero_monoid_add : 'a monoid_add -> 'a zero
+  type 'a comm_monoid_add
+  val ab_semigroup_add_comm_monoid_add :
+    'a comm_monoid_add -> 'a ab_semigroup_add
+  val monoid_add_comm_monoid_add : 'a comm_monoid_add -> 'a monoid_add
+  type 'a semiring_0
+  val comm_monoid_add_semiring_0 : 'a semiring_0 -> 'a comm_monoid_add
+  val mult_zero_semiring_0 : 'a semiring_0 -> 'a mult_zero
+  val semiring_semiring_0 : 'a semiring_0 -> 'a semiring
+  type 'a power
+  val one_power : 'a power -> 'a one
+  val times_power : 'a power -> 'a times
+  type 'a monoid_mult
+  val semigroup_mult_monoid_mult : 'a monoid_mult -> 'a semigroup_mult
+  val power_monoid_mult : 'a monoid_mult -> 'a power
+  type 'a semiring_1
+  val monoid_mult_semiring_1 : 'a semiring_1 -> 'a monoid_mult
+  val semiring_0_semiring_1 : 'a semiring_1 -> 'a semiring_0
+  val zero_neq_one_semiring_1 : 'a semiring_1 -> 'a zero_neq_one
+  type 'a cancel_semigroup_add
+  val semigroup_add_cancel_semigroup_add :
+    'a cancel_semigroup_add -> 'a semigroup_add
+  type 'a cancel_ab_semigroup_add
+  val ab_semigroup_add_cancel_ab_semigroup_add :
+    'a cancel_ab_semigroup_add -> 'a ab_semigroup_add
+  val cancel_semigroup_add_cancel_ab_semigroup_add :
+    'a cancel_ab_semigroup_add -> 'a cancel_semigroup_add
+  type 'a cancel_comm_monoid_add
+  val cancel_ab_semigroup_add_cancel_comm_monoid_add :
+    'a cancel_comm_monoid_add -> 'a cancel_ab_semigroup_add
+  val comm_monoid_add_cancel_comm_monoid_add :
+    'a cancel_comm_monoid_add -> 'a comm_monoid_add
+  type 'a semiring_0_cancel
+  val cancel_comm_monoid_add_semiring_0_cancel :
+    'a semiring_0_cancel -> 'a cancel_comm_monoid_add
+  val semiring_0_semiring_0_cancel : 'a semiring_0_cancel -> 'a semiring_0
+  type 'a semiring_1_cancel
+  val semiring_0_cancel_semiring_1_cancel :
+    'a semiring_1_cancel -> 'a semiring_0_cancel
+  val semiring_1_semiring_1_cancel : 'a semiring_1_cancel -> 'a semiring_1
+  type 'a dvd
+  val times_dvd : 'a dvd -> 'a times
+  type 'a ab_semigroup_mult
+  val semigroup_mult_ab_semigroup_mult :
+    'a ab_semigroup_mult -> 'a semigroup_mult
+  type 'a comm_semiring
+  val ab_semigroup_mult_comm_semiring : 'a comm_semiring -> 'a ab_semigroup_mult
+  val semiring_comm_semiring : 'a comm_semiring -> 'a semiring
+  type 'a comm_semiring_0
+  val comm_semiring_comm_semiring_0 : 'a comm_semiring_0 -> 'a comm_semiring
+  val semiring_0_comm_semiring_0 : 'a comm_semiring_0 -> 'a semiring_0
+  type 'a comm_monoid_mult
+  val ab_semigroup_mult_comm_monoid_mult :
+    'a comm_monoid_mult -> 'a ab_semigroup_mult
+  val monoid_mult_comm_monoid_mult : 'a comm_monoid_mult -> 'a monoid_mult
+  type 'a comm_semiring_1
+  val comm_monoid_mult_comm_semiring_1 :
+    'a comm_semiring_1 -> 'a comm_monoid_mult
+  val comm_semiring_0_comm_semiring_1 : 'a comm_semiring_1 -> 'a comm_semiring_0
+  val dvd_comm_semiring_1 : 'a comm_semiring_1 -> 'a dvd
+  val semiring_1_comm_semiring_1 : 'a comm_semiring_1 -> 'a semiring_1
+  type 'a comm_semiring_0_cancel
+  val comm_semiring_0_comm_semiring_0_cancel :
+    'a comm_semiring_0_cancel -> 'a comm_semiring_0
+  val semiring_0_cancel_comm_semiring_0_cancel :
+    'a comm_semiring_0_cancel -> 'a semiring_0_cancel
+  type 'a comm_semiring_1_cancel
+  val comm_semiring_0_cancel_comm_semiring_1_cancel :
+    'a comm_semiring_1_cancel -> 'a comm_semiring_0_cancel
+  val comm_semiring_1_comm_semiring_1_cancel :
+    'a comm_semiring_1_cancel -> 'a comm_semiring_1
+  val semiring_1_cancel_comm_semiring_1_cancel :
+    'a comm_semiring_1_cancel -> 'a semiring_1_cancel
+  type 'a diva
+  val dvd_div : 'a diva -> 'a dvd
+  val diva : 'a diva -> 'a -> 'a -> 'a
+  val moda : 'a diva -> 'a -> 'a -> 'a
+  type 'a semiring_div
+  val div_semiring_div : 'a semiring_div -> 'a diva
+  val comm_semiring_1_cancel_semiring_div :
+    'a semiring_div -> 'a comm_semiring_1_cancel
+  val no_zero_divisors_semiring_div : 'a semiring_div -> 'a no_zero_divisors
+  val one_int : IntInf.int
+  val one_inta : IntInf.int one
+  val zero_neq_one_int : IntInf.int zero_neq_one
+  val semigroup_mult_int : IntInf.int semigroup_mult
+  val plus_int : IntInf.int plus
+  val semigroup_add_int : IntInf.int semigroup_add
+  val ab_semigroup_add_int : IntInf.int ab_semigroup_add
+  val semiring_int : IntInf.int semiring
+  val mult_zero_int : IntInf.int mult_zero
+  val monoid_add_int : IntInf.int monoid_add
+  val comm_monoid_add_int : IntInf.int comm_monoid_add
+  val semiring_0_int : IntInf.int semiring_0
+  val power_int : IntInf.int power
+  val monoid_mult_int : IntInf.int monoid_mult
+  val semiring_1_int : IntInf.int semiring_1
+  val cancel_semigroup_add_int : IntInf.int cancel_semigroup_add
+  val cancel_ab_semigroup_add_int : IntInf.int cancel_ab_semigroup_add
+  val cancel_comm_monoid_add_int : IntInf.int cancel_comm_monoid_add
+  val semiring_0_cancel_int : IntInf.int semiring_0_cancel
+  val semiring_1_cancel_int : IntInf.int semiring_1_cancel
+  val dvd_int : IntInf.int dvd
+  val ab_semigroup_mult_int : IntInf.int ab_semigroup_mult
+  val comm_semiring_int : IntInf.int comm_semiring
+  val comm_semiring_0_int : IntInf.int comm_semiring_0
+  val comm_monoid_mult_int : IntInf.int comm_monoid_mult
+  val comm_semiring_1_int : IntInf.int comm_semiring_1
+  val comm_semiring_0_cancel_int : IntInf.int comm_semiring_0_cancel
+  val comm_semiring_1_cancel_int : IntInf.int comm_semiring_1_cancel
+  val abs_int : IntInf.int -> IntInf.int
+  val split : ('a -> 'b -> 'c) -> 'a * 'b -> 'c
+  val sgn_int : IntInf.int -> IntInf.int
+  val apsnd : ('a -> 'b) -> 'c * 'a -> 'c * 'b
+  val divmod_int : IntInf.int -> IntInf.int -> IntInf.int * IntInf.int
+  val snd : 'a * 'b -> 'b
+  val mod_int : IntInf.int -> IntInf.int -> IntInf.int
+  val fst : 'a * 'b -> 'a
+  val div_int : IntInf.int -> IntInf.int -> IntInf.int
+  val div_inta : IntInf.int diva
+  val semiring_div_int : IntInf.int semiring_div
+  val dvd : 'a semiring_div * 'a eq -> 'a -> 'a -> bool
+  val num_case :
+    (IntInf.int -> 'a) ->
+      (IntInf.int -> 'a) ->
+        (IntInf.int -> IntInf.int -> num -> 'a) ->
+          (num -> 'a) ->
+            (num -> num -> 'a) ->
+              (num -> num -> 'a) -> (IntInf.int -> num -> 'a) -> num -> 'a
+  val nummul : IntInf.int -> num -> num
+  val numneg : num -> num
+  val numadd : num * num -> num
+  val numsub : num -> num -> num
+  val simpnum : num -> num
+  val nota : fm -> fm
+  val iffa : fm -> fm -> fm
+  val impa : fm -> fm -> fm
+  val conj : fm -> fm -> fm
+  val simpfm : fm -> fm
+  val iupt : IntInf.int -> IntInf.int -> IntInf.int list
+  val mirror : fm -> fm
+  val size_list : 'a list -> IntInf.int
+  val alpha : fm -> num list
+  val beta : fm -> num list
+  val eq_numa : num eq
+  val member : 'a eq -> 'a -> 'a list -> bool
+  val remdups : 'a eq -> 'a list -> 'a list
+  val gcd_int : IntInf.int -> IntInf.int -> IntInf.int
+  val lcm_int : IntInf.int -> IntInf.int -> IntInf.int
+  val delta : fm -> IntInf.int
+  val a_beta : fm -> IntInf.int -> fm
+  val zeta : fm -> IntInf.int
+  val zsplit0 : num -> IntInf.int * num
+  val zlfm : fm -> fm
+  val unita : fm -> fm * (num list * IntInf.int)
+  val cooper : fm -> fm
+  val prep : fm -> fm
+  val qelim : fm -> (fm -> fm) -> fm
+  val pa : fm -> fm
+end = struct
+
+type 'a eq = {eq : 'a -> 'a -> bool};
+val eq = #eq : 'a eq -> 'a -> 'a -> bool;
+
+fun eqa A_ a b = eq A_ a b;
+
+fun leta s f = f s;
+
+fun suc n = IntInf.+ (n, (1 : IntInf.int));
+
+datatype num = C of IntInf.int | Bound of IntInf.int |
+  Cn of IntInf.int * 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 | 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 IntInf.int | NClosed of IntInf.int;
+
+fun map f [] = []
+  | map f (x :: xs) = f x :: map f xs;
+
+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 fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+  (NClosed nat) = f19 nat
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (Closed nat) = f18 nat
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (A fm) = f17 fm
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (E fm) = f16 fm
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (Iff (fm1, fm2)) = f15 fm1 fm2
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (Imp (fm1, fm2)) = f14 fm1 fm2
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (Or (fm1, fm2)) = f13 fm1 fm2
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (And (fm1, fm2)) = f12 fm1 fm2
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (Not fm) = f11 fm
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (NDvd (inta, num)) = f10 inta num
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (Dvd (inta, num)) = f9 inta num
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (NEq num) = f8 num
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (Eq num) = f7 num
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (Ge num) = f6 num
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (Gt num) = f5 num
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (Le num) = f4 num
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
+    (Lt num) = f3 num
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 F
+    = f2
+  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 T
+    = f1;
+
+fun eq_num (C intaa) (C inta) = ((intaa : IntInf.int) = inta)
+  | eq_num (Bound nata) (Bound nat) = ((nata : IntInf.int) = nat)
+  | eq_num (Cn (nata, intaa, numa)) (Cn (nat, inta, num)) =
+    ((nata : IntInf.int) = nat) andalso
+      (((intaa : IntInf.int) = inta) andalso eq_num numa num)
+  | eq_num (Neg numa) (Neg num) = eq_num numa num
+  | eq_num (Add (num1a, num2a)) (Add (num1, num2)) =
+    eq_num num1a num1 andalso eq_num num2a num2
+  | eq_num (Sub (num1a, num2a)) (Sub (num1, num2)) =
+    eq_num num1a num1 andalso eq_num num2a num2
+  | eq_num (Mul (intaa, numa)) (Mul (inta, num)) =
+    ((intaa : IntInf.int) = inta) andalso eq_num numa num
+  | eq_num (C inta) (Bound nat) = false
+  | eq_num (Bound nat) (C inta) = false
+  | eq_num (C intaa) (Cn (nat, inta, num)) = false
+  | eq_num (Cn (nat, intaa, num)) (C inta) = false
+  | eq_num (C inta) (Neg num) = false
+  | eq_num (Neg num) (C inta) = false
+  | eq_num (C inta) (Add (num1, num2)) = false
+  | eq_num (Add (num1, num2)) (C inta) = false
+  | eq_num (C inta) (Sub (num1, num2)) = false
+  | eq_num (Sub (num1, num2)) (C inta) = false
+  | eq_num (C intaa) (Mul (inta, num)) = false
+  | eq_num (Mul (intaa, num)) (C inta) = false
+  | eq_num (Bound nata) (Cn (nat, inta, num)) = false
+  | eq_num (Cn (nata, inta, num)) (Bound nat) = false
+  | eq_num (Bound nat) (Neg num) = false
+  | eq_num (Neg num) (Bound nat) = false
+  | eq_num (Bound nat) (Add (num1, num2)) = false
+  | eq_num (Add (num1, num2)) (Bound nat) = false
+  | eq_num (Bound nat) (Sub (num1, num2)) = false
+  | eq_num (Sub (num1, num2)) (Bound nat) = false
+  | eq_num (Bound nat) (Mul (inta, num)) = false
+  | eq_num (Mul (inta, num)) (Bound nat) = false
+  | eq_num (Cn (nat, inta, numa)) (Neg num) = false
+  | eq_num (Neg numa) (Cn (nat, inta, num)) = false
+  | eq_num (Cn (nat, inta, num)) (Add (num1, num2)) = false
+  | eq_num (Add (num1, num2)) (Cn (nat, inta, num)) = false
+  | eq_num (Cn (nat, inta, num)) (Sub (num1, num2)) = false
+  | eq_num (Sub (num1, num2)) (Cn (nat, inta, num)) = false
+  | eq_num (Cn (nat, intaa, numa)) (Mul (inta, num)) = false
+  | eq_num (Mul (intaa, numa)) (Cn (nat, inta, num)) = false
+  | eq_num (Neg num) (Add (num1, num2)) = false
+  | eq_num (Add (num1, num2)) (Neg num) = false
+  | eq_num (Neg num) (Sub (num1, num2)) = false
+  | eq_num (Sub (num1, num2)) (Neg num) = false
+  | eq_num (Neg numa) (Mul (inta, num)) = false
+  | eq_num (Mul (inta, numa)) (Neg num) = false
+  | eq_num (Add (num1a, num2a)) (Sub (num1, num2)) = false
+  | eq_num (Sub (num1a, num2a)) (Add (num1, num2)) = false
+  | eq_num (Add (num1, num2)) (Mul (inta, num)) = false
+  | eq_num (Mul (inta, num)) (Add (num1, num2)) = false
+  | eq_num (Sub (num1, num2)) (Mul (inta, num)) = false
+  | eq_num (Mul (inta, num)) (Sub (num1, num2)) = false;
+
+fun eq_fm T T = true
+  | eq_fm F F = true
+  | eq_fm (Lt numa) (Lt num) = eq_num numa num
+  | eq_fm (Le numa) (Le num) = eq_num numa num
+  | eq_fm (Gt numa) (Gt num) = eq_num numa num
+  | eq_fm (Ge numa) (Ge num) = eq_num numa num
+  | eq_fm (Eq numa) (Eq num) = eq_num numa num
+  | eq_fm (NEq numa) (NEq num) = eq_num numa num
+  | eq_fm (Dvd (intaa, numa)) (Dvd (inta, num)) =
+    ((intaa : IntInf.int) = inta) andalso eq_num numa num
+  | eq_fm (NDvd (intaa, numa)) (NDvd (inta, num)) =
+    ((intaa : IntInf.int) = inta) andalso eq_num numa num
+  | eq_fm (Not fma) (Not fm) = eq_fm fma fm
+  | eq_fm (And (fm1a, fm2a)) (And (fm1, fm2)) =
+    eq_fm fm1a fm1 andalso eq_fm fm2a fm2
+  | eq_fm (Or (fm1a, fm2a)) (Or (fm1, fm2)) =
+    eq_fm fm1a fm1 andalso eq_fm fm2a fm2
+  | eq_fm (Imp (fm1a, fm2a)) (Imp (fm1, fm2)) =
+    eq_fm fm1a fm1 andalso eq_fm fm2a fm2
+  | eq_fm (Iff (fm1a, fm2a)) (Iff (fm1, fm2)) =
+    eq_fm fm1a fm1 andalso eq_fm fm2a fm2
+  | eq_fm (E fma) (E fm) = eq_fm fma fm
+  | eq_fm (A fma) (A fm) = eq_fm fma fm
+  | eq_fm (Closed nata) (Closed nat) = ((nata : IntInf.int) = nat)
+  | eq_fm (NClosed nata) (NClosed nat) = ((nata : IntInf.int) = nat)
+  | eq_fm T F = false
+  | eq_fm F T = false
+  | eq_fm T (Lt num) = false
+  | eq_fm (Lt num) T = false
+  | eq_fm T (Le num) = false
+  | eq_fm (Le num) T = false
+  | eq_fm T (Gt num) = false
+  | eq_fm (Gt num) T = false
+  | eq_fm T (Ge num) = false
+  | eq_fm (Ge num) T = false
+  | eq_fm T (Eq num) = false
+  | eq_fm (Eq num) T = false
+  | eq_fm T (NEq num) = false
+  | eq_fm (NEq num) T = false
+  | eq_fm T (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, num)) T = false
+  | eq_fm T (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, num)) T = false
+  | eq_fm T (Not fm) = false
+  | eq_fm (Not fm) T = false
+  | eq_fm T (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) T = false
+  | eq_fm T (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) T = false
+  | eq_fm T (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) T = false
+  | eq_fm T (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) T = false
+  | eq_fm T (E fm) = false
+  | eq_fm (E fm) T = false
+  | eq_fm T (A fm) = false
+  | eq_fm (A fm) T = false
+  | eq_fm T (Closed nat) = false
+  | eq_fm (Closed nat) T = false
+  | eq_fm T (NClosed nat) = false
+  | eq_fm (NClosed nat) T = false
+  | eq_fm F (Lt num) = false
+  | eq_fm (Lt num) F = false
+  | eq_fm F (Le num) = false
+  | eq_fm (Le num) F = false
+  | eq_fm F (Gt num) = false
+  | eq_fm (Gt num) F = false
+  | eq_fm F (Ge num) = false
+  | eq_fm (Ge num) F = false
+  | eq_fm F (Eq num) = false
+  | eq_fm (Eq num) F = false
+  | eq_fm F (NEq num) = false
+  | eq_fm (NEq num) F = false
+  | eq_fm F (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, num)) F = false
+  | eq_fm F (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, num)) F = false
+  | eq_fm F (Not fm) = false
+  | eq_fm (Not fm) F = false
+  | eq_fm F (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) F = false
+  | eq_fm F (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) F = false
+  | eq_fm F (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) F = false
+  | eq_fm F (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) F = false
+  | eq_fm F (E fm) = false
+  | eq_fm (E fm) F = false
+  | eq_fm F (A fm) = false
+  | eq_fm (A fm) F = false
+  | eq_fm F (Closed nat) = false
+  | eq_fm (Closed nat) F = false
+  | eq_fm F (NClosed nat) = false
+  | eq_fm (NClosed nat) F = false
+  | eq_fm (Lt numa) (Le num) = false
+  | eq_fm (Le numa) (Lt num) = false
+  | eq_fm (Lt numa) (Gt num) = false
+  | eq_fm (Gt numa) (Lt num) = false
+  | eq_fm (Lt numa) (Ge num) = false
+  | eq_fm (Ge numa) (Lt num) = false
+  | eq_fm (Lt numa) (Eq num) = false
+  | eq_fm (Eq numa) (Lt num) = false
+  | eq_fm (Lt numa) (NEq num) = false
+  | eq_fm (NEq numa) (Lt num) = false
+  | eq_fm (Lt numa) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, numa)) (Lt num) = false
+  | eq_fm (Lt numa) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, numa)) (Lt num) = false
+  | eq_fm (Lt num) (Not fm) = false
+  | eq_fm (Not fm) (Lt num) = false
+  | eq_fm (Lt num) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) (Lt num) = false
+  | eq_fm (Lt num) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) (Lt num) = false
+  | eq_fm (Lt num) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) (Lt num) = false
+  | eq_fm (Lt num) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) (Lt num) = false
+  | eq_fm (Lt num) (E fm) = false
+  | eq_fm (E fm) (Lt num) = false
+  | eq_fm (Lt num) (A fm) = false
+  | eq_fm (A fm) (Lt num) = false
+  | eq_fm (Lt num) (Closed nat) = false
+  | eq_fm (Closed nat) (Lt num) = false
+  | eq_fm (Lt num) (NClosed nat) = false
+  | eq_fm (NClosed nat) (Lt num) = false
+  | eq_fm (Le numa) (Gt num) = false
+  | eq_fm (Gt numa) (Le num) = false
+  | eq_fm (Le numa) (Ge num) = false
+  | eq_fm (Ge numa) (Le num) = false
+  | eq_fm (Le numa) (Eq num) = false
+  | eq_fm (Eq numa) (Le num) = false
+  | eq_fm (Le numa) (NEq num) = false
+  | eq_fm (NEq numa) (Le num) = false
+  | eq_fm (Le numa) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, numa)) (Le num) = false
+  | eq_fm (Le numa) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, numa)) (Le num) = false
+  | eq_fm (Le num) (Not fm) = false
+  | eq_fm (Not fm) (Le num) = false
+  | eq_fm (Le num) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) (Le num) = false
+  | eq_fm (Le num) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) (Le num) = false
+  | eq_fm (Le num) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) (Le num) = false
+  | eq_fm (Le num) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) (Le num) = false
+  | eq_fm (Le num) (E fm) = false
+  | eq_fm (E fm) (Le num) = false
+  | eq_fm (Le num) (A fm) = false
+  | eq_fm (A fm) (Le num) = false
+  | eq_fm (Le num) (Closed nat) = false
+  | eq_fm (Closed nat) (Le num) = false
+  | eq_fm (Le num) (NClosed nat) = false
+  | eq_fm (NClosed nat) (Le num) = false
+  | eq_fm (Gt numa) (Ge num) = false
+  | eq_fm (Ge numa) (Gt num) = false
+  | eq_fm (Gt numa) (Eq num) = false
+  | eq_fm (Eq numa) (Gt num) = false
+  | eq_fm (Gt numa) (NEq num) = false
+  | eq_fm (NEq numa) (Gt num) = false
+  | eq_fm (Gt numa) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, numa)) (Gt num) = false
+  | eq_fm (Gt numa) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, numa)) (Gt num) = false
+  | eq_fm (Gt num) (Not fm) = false
+  | eq_fm (Not fm) (Gt num) = false
+  | eq_fm (Gt num) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) (Gt num) = false
+  | eq_fm (Gt num) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) (Gt num) = false
+  | eq_fm (Gt num) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) (Gt num) = false
+  | eq_fm (Gt num) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) (Gt num) = false
+  | eq_fm (Gt num) (E fm) = false
+  | eq_fm (E fm) (Gt num) = false
+  | eq_fm (Gt num) (A fm) = false
+  | eq_fm (A fm) (Gt num) = false
+  | eq_fm (Gt num) (Closed nat) = false
+  | eq_fm (Closed nat) (Gt num) = false
+  | eq_fm (Gt num) (NClosed nat) = false
+  | eq_fm (NClosed nat) (Gt num) = false
+  | eq_fm (Ge numa) (Eq num) = false
+  | eq_fm (Eq numa) (Ge num) = false
+  | eq_fm (Ge numa) (NEq num) = false
+  | eq_fm (NEq numa) (Ge num) = false
+  | eq_fm (Ge numa) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, numa)) (Ge num) = false
+  | eq_fm (Ge numa) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, numa)) (Ge num) = false
+  | eq_fm (Ge num) (Not fm) = false
+  | eq_fm (Not fm) (Ge num) = false
+  | eq_fm (Ge num) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) (Ge num) = false
+  | eq_fm (Ge num) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) (Ge num) = false
+  | eq_fm (Ge num) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) (Ge num) = false
+  | eq_fm (Ge num) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) (Ge num) = false
+  | eq_fm (Ge num) (E fm) = false
+  | eq_fm (E fm) (Ge num) = false
+  | eq_fm (Ge num) (A fm) = false
+  | eq_fm (A fm) (Ge num) = false
+  | eq_fm (Ge num) (Closed nat) = false
+  | eq_fm (Closed nat) (Ge num) = false
+  | eq_fm (Ge num) (NClosed nat) = false
+  | eq_fm (NClosed nat) (Ge num) = false
+  | eq_fm (Eq numa) (NEq num) = false
+  | eq_fm (NEq numa) (Eq num) = false
+  | eq_fm (Eq numa) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, numa)) (Eq num) = false
+  | eq_fm (Eq numa) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, numa)) (Eq num) = false
+  | eq_fm (Eq num) (Not fm) = false
+  | eq_fm (Not fm) (Eq num) = false
+  | eq_fm (Eq num) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) (Eq num) = false
+  | eq_fm (Eq num) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) (Eq num) = false
+  | eq_fm (Eq num) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) (Eq num) = false
+  | eq_fm (Eq num) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) (Eq num) = false
+  | eq_fm (Eq num) (E fm) = false
+  | eq_fm (E fm) (Eq num) = false
+  | eq_fm (Eq num) (A fm) = false
+  | eq_fm (A fm) (Eq num) = false
+  | eq_fm (Eq num) (Closed nat) = false
+  | eq_fm (Closed nat) (Eq num) = false
+  | eq_fm (Eq num) (NClosed nat) = false
+  | eq_fm (NClosed nat) (Eq num) = false
+  | eq_fm (NEq numa) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, numa)) (NEq num) = false
+  | eq_fm (NEq numa) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, numa)) (NEq num) = false
+  | eq_fm (NEq num) (Not fm) = false
+  | eq_fm (Not fm) (NEq num) = false
+  | eq_fm (NEq num) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) (NEq num) = false
+  | eq_fm (NEq num) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) (NEq num) = false
+  | eq_fm (NEq num) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) (NEq num) = false
+  | eq_fm (NEq num) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) (NEq num) = false
+  | eq_fm (NEq num) (E fm) = false
+  | eq_fm (E fm) (NEq num) = false
+  | eq_fm (NEq num) (A fm) = false
+  | eq_fm (A fm) (NEq num) = false
+  | eq_fm (NEq num) (Closed nat) = false
+  | eq_fm (Closed nat) (NEq num) = false
+  | eq_fm (NEq num) (NClosed nat) = false
+  | eq_fm (NClosed nat) (NEq num) = false
+  | eq_fm (Dvd (intaa, numa)) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (intaa, numa)) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, num)) (Not fm) = false
+  | eq_fm (Not fm) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, num)) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, num)) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, num)) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, num)) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, num)) (E fm) = false
+  | eq_fm (E fm) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, num)) (A fm) = false
+  | eq_fm (A fm) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, num)) (Closed nat) = false
+  | eq_fm (Closed nat) (Dvd (inta, num)) = false
+  | eq_fm (Dvd (inta, num)) (NClosed nat) = false
+  | eq_fm (NClosed nat) (Dvd (inta, num)) = false
+  | eq_fm (NDvd (inta, num)) (Not fm) = false
+  | eq_fm (Not fm) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, num)) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, num)) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, num)) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, num)) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, num)) (E fm) = false
+  | eq_fm (E fm) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, num)) (A fm) = false
+  | eq_fm (A fm) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, num)) (Closed nat) = false
+  | eq_fm (Closed nat) (NDvd (inta, num)) = false
+  | eq_fm (NDvd (inta, num)) (NClosed nat) = false
+  | eq_fm (NClosed nat) (NDvd (inta, num)) = false
+  | eq_fm (Not fm) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) (Not fm) = false
+  | eq_fm (Not fm) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) (Not fm) = false
+  | eq_fm (Not fm) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) (Not fm) = false
+  | eq_fm (Not fm) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) (Not fm) = false
+  | eq_fm (Not fma) (E fm) = false
+  | eq_fm (E fma) (Not fm) = false
+  | eq_fm (Not fma) (A fm) = false
+  | eq_fm (A fma) (Not fm) = false
+  | eq_fm (Not fm) (Closed nat) = false
+  | eq_fm (Closed nat) (Not fm) = false
+  | eq_fm (Not fm) (NClosed nat) = false
+  | eq_fm (NClosed nat) (Not fm) = false
+  | eq_fm (And (fm1a, fm2a)) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1a, fm2a)) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1a, fm2a)) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1a, fm2a)) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1a, fm2a)) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1a, fm2a)) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) (E fm) = false
+  | eq_fm (E fm) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) (A fm) = false
+  | eq_fm (A fm) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) (Closed nat) = false
+  | eq_fm (Closed nat) (And (fm1, fm2)) = false
+  | eq_fm (And (fm1, fm2)) (NClosed nat) = false
+  | eq_fm (NClosed nat) (And (fm1, fm2)) = false
+  | eq_fm (Or (fm1a, fm2a)) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1a, fm2a)) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1a, fm2a)) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1a, fm2a)) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) (E fm) = false
+  | eq_fm (E fm) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) (A fm) = false
+  | eq_fm (A fm) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) (Closed nat) = false
+  | eq_fm (Closed nat) (Or (fm1, fm2)) = false
+  | eq_fm (Or (fm1, fm2)) (NClosed nat) = false
+  | eq_fm (NClosed nat) (Or (fm1, fm2)) = false
+  | eq_fm (Imp (fm1a, fm2a)) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1a, fm2a)) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) (E fm) = false
+  | eq_fm (E fm) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) (A fm) = false
+  | eq_fm (A fm) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) (Closed nat) = false
+  | eq_fm (Closed nat) (Imp (fm1, fm2)) = false
+  | eq_fm (Imp (fm1, fm2)) (NClosed nat) = false
+  | eq_fm (NClosed nat) (Imp (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) (E fm) = false
+  | eq_fm (E fm) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) (A fm) = false
+  | eq_fm (A fm) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) (Closed nat) = false
+  | eq_fm (Closed nat) (Iff (fm1, fm2)) = false
+  | eq_fm (Iff (fm1, fm2)) (NClosed nat) = false
+  | eq_fm (NClosed nat) (Iff (fm1, fm2)) = false
+  | eq_fm (E fma) (A fm) = false
+  | eq_fm (A fma) (E fm) = false
+  | eq_fm (E fm) (Closed nat) = false
+  | eq_fm (Closed nat) (E fm) = false
+  | eq_fm (E fm) (NClosed nat) = false
+  | eq_fm (NClosed nat) (E fm) = false
+  | eq_fm (A fm) (Closed nat) = false
+  | eq_fm (Closed nat) (A fm) = false
+  | eq_fm (A fm) (NClosed nat) = false
+  | eq_fm (NClosed nat) (A fm) = false
+  | eq_fm (Closed nata) (NClosed nat) = false
+  | eq_fm (NClosed nata) (Closed nat) = 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 _ => Or (f p, q)
+                  | Le _ => Or (f p, q) | Gt _ => Or (f p, q)
+                  | Ge _ => Or (f p, q) | Eq _ => Or (f p, q)
+                  | NEq _ => Or (f p, q) | Dvd (_, _) => Or (f p, q)
+                  | NDvd (_, _) => Or (f p, q) | Not _ => Or (f p, q)
+                  | And (_, _) => Or (f p, q) | Or (_, _) => Or (f p, q)
+                  | Imp (_, _) => Or (f p, q) | Iff (_, _) => Or (f p, q)
+                  | E _ => Or (f p, q) | A _ => Or (f p, q)
+                  | Closed _ => Or (f p, q) | NClosed _ => Or (f p, q))));
+
+fun foldr f [] a = a
+  | foldr f (x :: xs) a = f x (foldr f xs a);
+
+fun evaldjf f ps = foldr (djf f) ps F;
+
+fun dj f p = evaldjf f (disjuncts p);
+
+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 minus_nat n m = IntInf.max (0, (IntInf.- (n, m)));
+
+fun decrnum (Bound n) = Bound (minus_nat n (1 : IntInf.int))
+  | 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 (Cn (n, i, a)) = Cn (minus_nat n (1 : IntInf.int), i, decrnum a)
+  | decrnum (C u) = C u;
+
+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 concat_map f [] = []
+  | concat_map f (x :: xs) = append (f x) (concat_map f xs);
+
+fun numsubst0 t (C c) = C c
+  | numsubst0 t (Bound n) =
+    (if ((n : IntInf.int) = (0 : IntInf.int)) then t else Bound n)
+  | 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)
+  | numsubst0 t (Cn (v, i, a)) =
+    (if ((v : IntInf.int) = (0 : IntInf.int))
+      then Add (Mul (i, t), numsubst0 t a)
+      else Cn (suc (minus_nat v (1 : IntInf.int)), 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 T = T
+  | minusinf F = F
+  | minusinf (Lt (C bo)) = Lt (C bo)
+  | minusinf (Lt (Bound bp)) = Lt (Bound bp)
+  | minusinf (Lt (Neg bt)) = Lt (Neg bt)
+  | minusinf (Lt (Add (bu, bv))) = Lt (Add (bu, bv))
+  | minusinf (Lt (Sub (bw, bx))) = Lt (Sub (bw, bx))
+  | minusinf (Lt (Mul (by, bz))) = Lt (Mul (by, bz))
+  | minusinf (Le (C co)) = Le (C co)
+  | minusinf (Le (Bound cp)) = Le (Bound cp)
+  | minusinf (Le (Neg ct)) = Le (Neg ct)
+  | minusinf (Le (Add (cu, cv))) = Le (Add (cu, cv))
+  | minusinf (Le (Sub (cw, cx))) = Le (Sub (cw, cx))
+  | minusinf (Le (Mul (cy, cz))) = Le (Mul (cy, cz))
+  | minusinf (Gt (C doa)) = Gt (C doa)
+  | minusinf (Gt (Bound dp)) = Gt (Bound dp)
+  | minusinf (Gt (Neg dt)) = Gt (Neg dt)
+  | minusinf (Gt (Add (du, dv))) = Gt (Add (du, dv))
+  | minusinf (Gt (Sub (dw, dx))) = Gt (Sub (dw, dx))
+  | minusinf (Gt (Mul (dy, dz))) = Gt (Mul (dy, dz))
+  | minusinf (Ge (C eo)) = Ge (C eo)
+  | minusinf (Ge (Bound ep)) = Ge (Bound ep)
+  | minusinf (Ge (Neg et)) = Ge (Neg et)
+  | minusinf (Ge (Add (eu, ev))) = Ge (Add (eu, ev))
+  | minusinf (Ge (Sub (ew, ex))) = Ge (Sub (ew, ex))
+  | minusinf (Ge (Mul (ey, ez))) = Ge (Mul (ey, ez))
+  | minusinf (Eq (C fo)) = Eq (C fo)
+  | minusinf (Eq (Bound fp)) = Eq (Bound fp)
+  | minusinf (Eq (Neg ft)) = Eq (Neg ft)
+  | minusinf (Eq (Add (fu, fv))) = Eq (Add (fu, fv))
+  | minusinf (Eq (Sub (fw, fx))) = Eq (Sub (fw, fx))
+  | minusinf (Eq (Mul (fy, fz))) = Eq (Mul (fy, fz))
+  | minusinf (NEq (C go)) = NEq (C go)
+  | minusinf (NEq (Bound gp)) = NEq (Bound gp)
+  | minusinf (NEq (Neg gt)) = NEq (Neg gt)
+  | minusinf (NEq (Add (gu, gv))) = NEq (Add (gu, gv))
+  | minusinf (NEq (Sub (gw, gx))) = NEq (Sub (gw, gx))
+  | minusinf (NEq (Mul (gy, gz))) = NEq (Mul (gy, gz))
+  | 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
+  | minusinf (Lt (Cn (cm, c, e))) =
+    (if ((cm : IntInf.int) = (0 : IntInf.int)) then T
+      else Lt (Cn (suc (minus_nat cm (1 : IntInf.int)), c, e)))
+  | minusinf (Le (Cn (dm, c, e))) =
+    (if ((dm : IntInf.int) = (0 : IntInf.int)) then T
+      else Le (Cn (suc (minus_nat dm (1 : IntInf.int)), c, e)))
+  | minusinf (Gt (Cn (em, c, e))) =
+    (if ((em : IntInf.int) = (0 : IntInf.int)) then F
+      else Gt (Cn (suc (minus_nat em (1 : IntInf.int)), c, e)))
+  | minusinf (Ge (Cn (fm, c, e))) =
+    (if ((fm : IntInf.int) = (0 : IntInf.int)) then F
+      else Ge (Cn (suc (minus_nat fm (1 : IntInf.int)), c, e)))
+  | minusinf (Eq (Cn (gm, c, e))) =
+    (if ((gm : IntInf.int) = (0 : IntInf.int)) then F
+      else Eq (Cn (suc (minus_nat gm (1 : IntInf.int)), c, e)))
+  | minusinf (NEq (Cn (hm, c, e))) =
+    (if ((hm : IntInf.int) = (0 : IntInf.int)) then T
+      else NEq (Cn (suc (minus_nat hm (1 : IntInf.int)), c, e)));
+
+val eq_int = {eq = (fn a => fn b => ((a : IntInf.int) = b))} : IntInf.int eq;
+
+val zero_int : IntInf.int = (0 : IntInf.int);
+
+type 'a zero = {zero : 'a};
+val zero = #zero : 'a zero -> 'a;
+
+val zero_inta = {zero = zero_int} : IntInf.int zero;
+
+type 'a times = {times : 'a -> 'a -> 'a};
+val times = #times : 'a times -> 'a -> 'a -> 'a;
+
+type 'a no_zero_divisors =
+  {times_no_zero_divisors : 'a times, zero_no_zero_divisors : 'a zero};
+val times_no_zero_divisors = #times_no_zero_divisors :
+  'a no_zero_divisors -> 'a times;
+val zero_no_zero_divisors = #zero_no_zero_divisors :
+  'a no_zero_divisors -> 'a zero;
+
+val times_int = {times = (fn a => fn b => IntInf.* (a, b))} : IntInf.int times;
+
+val no_zero_divisors_int =
+  {times_no_zero_divisors = times_int, zero_no_zero_divisors = zero_inta} :
+  IntInf.int no_zero_divisors;
+
+type 'a one = {one : 'a};
+val one = #one : 'a one -> 'a;
+
+type 'a zero_neq_one = {one_zero_neq_one : 'a one, zero_zero_neq_one : 'a zero};
+val one_zero_neq_one = #one_zero_neq_one : 'a zero_neq_one -> 'a one;
+val zero_zero_neq_one = #zero_zero_neq_one : 'a zero_neq_one -> 'a zero;
+
+type 'a semigroup_mult = {times_semigroup_mult : 'a times};
+val times_semigroup_mult = #times_semigroup_mult :
+  'a semigroup_mult -> 'a times;
+
+type 'a plus = {plus : 'a -> 'a -> 'a};
+val plus = #plus : 'a plus -> 'a -> 'a -> 'a;
+
+type 'a semigroup_add = {plus_semigroup_add : 'a plus};
+val plus_semigroup_add = #plus_semigroup_add : 'a semigroup_add -> 'a plus;
+
+type 'a ab_semigroup_add = {semigroup_add_ab_semigroup_add : 'a semigroup_add};
+val semigroup_add_ab_semigroup_add = #semigroup_add_ab_semigroup_add :
+  'a ab_semigroup_add -> 'a semigroup_add;
+
+type 'a semiring =
+  {ab_semigroup_add_semiring : 'a ab_semigroup_add,
+    semigroup_mult_semiring : 'a semigroup_mult};
+val ab_semigroup_add_semiring = #ab_semigroup_add_semiring :
+  'a semiring -> 'a ab_semigroup_add;
+val semigroup_mult_semiring = #semigroup_mult_semiring :
+  'a semiring -> 'a semigroup_mult;
+
+type 'a mult_zero = {times_mult_zero : 'a times, zero_mult_zero : 'a zero};
+val times_mult_zero = #times_mult_zero : 'a mult_zero -> 'a times;
+val zero_mult_zero = #zero_mult_zero : 'a mult_zero -> 'a zero;
+
+type 'a monoid_add =
+  {semigroup_add_monoid_add : 'a semigroup_add, zero_monoid_add : 'a zero};
+val semigroup_add_monoid_add = #semigroup_add_monoid_add :
+  'a monoid_add -> 'a semigroup_add;
+val zero_monoid_add = #zero_monoid_add : 'a monoid_add -> 'a zero;
+
+type 'a comm_monoid_add =
+  {ab_semigroup_add_comm_monoid_add : 'a ab_semigroup_add,
+    monoid_add_comm_monoid_add : 'a monoid_add};
+val ab_semigroup_add_comm_monoid_add = #ab_semigroup_add_comm_monoid_add :
+  'a comm_monoid_add -> 'a ab_semigroup_add;
+val monoid_add_comm_monoid_add = #monoid_add_comm_monoid_add :
+  'a comm_monoid_add -> 'a monoid_add;
+
+type 'a semiring_0 =
+  {comm_monoid_add_semiring_0 : 'a comm_monoid_add,
+    mult_zero_semiring_0 : 'a mult_zero, semiring_semiring_0 : 'a semiring};
+val comm_monoid_add_semiring_0 = #comm_monoid_add_semiring_0 :
+  'a semiring_0 -> 'a comm_monoid_add;
+val mult_zero_semiring_0 = #mult_zero_semiring_0 :
+  'a semiring_0 -> 'a mult_zero;
+val semiring_semiring_0 = #semiring_semiring_0 : 'a semiring_0 -> 'a semiring;
+
+type 'a power = {one_power : 'a one, times_power : 'a times};
+val one_power = #one_power : 'a power -> 'a one;
+val times_power = #times_power : 'a power -> 'a times;
+
+type 'a monoid_mult =
+  {semigroup_mult_monoid_mult : 'a semigroup_mult,
+    power_monoid_mult : 'a power};
+val semigroup_mult_monoid_mult = #semigroup_mult_monoid_mult :
+  'a monoid_mult -> 'a semigroup_mult;
+val power_monoid_mult = #power_monoid_mult : 'a monoid_mult -> 'a power;
+
+type 'a semiring_1 =
+  {monoid_mult_semiring_1 : 'a monoid_mult,
+    semiring_0_semiring_1 : 'a semiring_0,
+    zero_neq_one_semiring_1 : 'a zero_neq_one};
+val monoid_mult_semiring_1 = #monoid_mult_semiring_1 :
+  'a semiring_1 -> 'a monoid_mult;
+val semiring_0_semiring_1 = #semiring_0_semiring_1 :
+  'a semiring_1 -> 'a semiring_0;
+val zero_neq_one_semiring_1 = #zero_neq_one_semiring_1 :
+  'a semiring_1 -> 'a zero_neq_one;
+
+type 'a cancel_semigroup_add =
+  {semigroup_add_cancel_semigroup_add : 'a semigroup_add};
+val semigroup_add_cancel_semigroup_add = #semigroup_add_cancel_semigroup_add :
+  'a cancel_semigroup_add -> 'a semigroup_add;
+
+type 'a cancel_ab_semigroup_add =
+  {ab_semigroup_add_cancel_ab_semigroup_add : 'a ab_semigroup_add,
+    cancel_semigroup_add_cancel_ab_semigroup_add : 'a cancel_semigroup_add};
+val ab_semigroup_add_cancel_ab_semigroup_add =
+  #ab_semigroup_add_cancel_ab_semigroup_add :
+  'a cancel_ab_semigroup_add -> 'a ab_semigroup_add;
+val cancel_semigroup_add_cancel_ab_semigroup_add =
+  #cancel_semigroup_add_cancel_ab_semigroup_add :
+  'a cancel_ab_semigroup_add -> 'a cancel_semigroup_add;
+
+type 'a cancel_comm_monoid_add =
+  {cancel_ab_semigroup_add_cancel_comm_monoid_add : 'a cancel_ab_semigroup_add,
+    comm_monoid_add_cancel_comm_monoid_add : 'a comm_monoid_add};
+val cancel_ab_semigroup_add_cancel_comm_monoid_add =
+  #cancel_ab_semigroup_add_cancel_comm_monoid_add :
+  'a cancel_comm_monoid_add -> 'a cancel_ab_semigroup_add;
+val comm_monoid_add_cancel_comm_monoid_add =
+  #comm_monoid_add_cancel_comm_monoid_add :
+  'a cancel_comm_monoid_add -> 'a comm_monoid_add;
+
+type 'a semiring_0_cancel =
+  {cancel_comm_monoid_add_semiring_0_cancel : 'a cancel_comm_monoid_add,
+    semiring_0_semiring_0_cancel : 'a semiring_0};
+val cancel_comm_monoid_add_semiring_0_cancel =
+  #cancel_comm_monoid_add_semiring_0_cancel :
+  'a semiring_0_cancel -> 'a cancel_comm_monoid_add;
+val semiring_0_semiring_0_cancel = #semiring_0_semiring_0_cancel :
+  'a semiring_0_cancel -> 'a semiring_0;
+
+type 'a semiring_1_cancel =
+  {semiring_0_cancel_semiring_1_cancel : 'a semiring_0_cancel,
+    semiring_1_semiring_1_cancel : 'a semiring_1};
+val semiring_0_cancel_semiring_1_cancel = #semiring_0_cancel_semiring_1_cancel :
+  'a semiring_1_cancel -> 'a semiring_0_cancel;
+val semiring_1_semiring_1_cancel = #semiring_1_semiring_1_cancel :
+  'a semiring_1_cancel -> 'a semiring_1;
+
+type 'a dvd = {times_dvd : 'a times};
+val times_dvd = #times_dvd : 'a dvd -> 'a times;
+
+type 'a ab_semigroup_mult =
+  {semigroup_mult_ab_semigroup_mult : 'a semigroup_mult};
+val semigroup_mult_ab_semigroup_mult = #semigroup_mult_ab_semigroup_mult :
+  'a ab_semigroup_mult -> 'a semigroup_mult;
+
+type 'a comm_semiring =
+  {ab_semigroup_mult_comm_semiring : 'a ab_semigroup_mult,
+    semiring_comm_semiring : 'a semiring};
+val ab_semigroup_mult_comm_semiring = #ab_semigroup_mult_comm_semiring :
+  'a comm_semiring -> 'a ab_semigroup_mult;
+val semiring_comm_semiring = #semiring_comm_semiring :
+  'a comm_semiring -> 'a semiring;
+
+type 'a comm_semiring_0 =
+  {comm_semiring_comm_semiring_0 : 'a comm_semiring,
+    semiring_0_comm_semiring_0 : 'a semiring_0};
+val comm_semiring_comm_semiring_0 = #comm_semiring_comm_semiring_0 :
+  'a comm_semiring_0 -> 'a comm_semiring;
+val semiring_0_comm_semiring_0 = #semiring_0_comm_semiring_0 :
+  'a comm_semiring_0 -> 'a semiring_0;
+
+type 'a comm_monoid_mult =
+  {ab_semigroup_mult_comm_monoid_mult : 'a ab_semigroup_mult,
+    monoid_mult_comm_monoid_mult : 'a monoid_mult};
+val ab_semigroup_mult_comm_monoid_mult = #ab_semigroup_mult_comm_monoid_mult :
+  'a comm_monoid_mult -> 'a ab_semigroup_mult;
+val monoid_mult_comm_monoid_mult = #monoid_mult_comm_monoid_mult :
+  'a comm_monoid_mult -> 'a monoid_mult;
+
+type 'a comm_semiring_1 =
+  {comm_monoid_mult_comm_semiring_1 : 'a comm_monoid_mult,
+    comm_semiring_0_comm_semiring_1 : 'a comm_semiring_0,
+    dvd_comm_semiring_1 : 'a dvd, semiring_1_comm_semiring_1 : 'a semiring_1};
+val comm_monoid_mult_comm_semiring_1 = #comm_monoid_mult_comm_semiring_1 :
+  'a comm_semiring_1 -> 'a comm_monoid_mult;
+val comm_semiring_0_comm_semiring_1 = #comm_semiring_0_comm_semiring_1 :
+  'a comm_semiring_1 -> 'a comm_semiring_0;
+val dvd_comm_semiring_1 = #dvd_comm_semiring_1 : 'a comm_semiring_1 -> 'a dvd;
+val semiring_1_comm_semiring_1 = #semiring_1_comm_semiring_1 :
+  'a comm_semiring_1 -> 'a semiring_1;
+
+type 'a comm_semiring_0_cancel =
+  {comm_semiring_0_comm_semiring_0_cancel : 'a comm_semiring_0,
+    semiring_0_cancel_comm_semiring_0_cancel : 'a semiring_0_cancel};
+val comm_semiring_0_comm_semiring_0_cancel =
+  #comm_semiring_0_comm_semiring_0_cancel :
+  'a comm_semiring_0_cancel -> 'a comm_semiring_0;
+val semiring_0_cancel_comm_semiring_0_cancel =
+  #semiring_0_cancel_comm_semiring_0_cancel :
+  'a comm_semiring_0_cancel -> 'a semiring_0_cancel;
+
+type 'a comm_semiring_1_cancel =
+  {comm_semiring_0_cancel_comm_semiring_1_cancel : 'a comm_semiring_0_cancel,
+    comm_semiring_1_comm_semiring_1_cancel : 'a comm_semiring_1,
+    semiring_1_cancel_comm_semiring_1_cancel : 'a semiring_1_cancel};
+val comm_semiring_0_cancel_comm_semiring_1_cancel =
+  #comm_semiring_0_cancel_comm_semiring_1_cancel :
+  'a comm_semiring_1_cancel -> 'a comm_semiring_0_cancel;
+val comm_semiring_1_comm_semiring_1_cancel =
+  #comm_semiring_1_comm_semiring_1_cancel :
+  'a comm_semiring_1_cancel -> 'a comm_semiring_1;
+val semiring_1_cancel_comm_semiring_1_cancel =
+  #semiring_1_cancel_comm_semiring_1_cancel :
+  'a comm_semiring_1_cancel -> 'a semiring_1_cancel;
+
+type 'a diva = {dvd_div : 'a dvd, diva : 'a -> 'a -> 'a, moda : 'a -> 'a -> 'a};
+val dvd_div = #dvd_div : 'a diva -> 'a dvd;
+val diva = #diva : 'a diva -> 'a -> 'a -> 'a;
+val moda = #moda : 'a diva -> 'a -> 'a -> 'a;
+
+type 'a semiring_div =
+  {div_semiring_div : 'a diva,
+    comm_semiring_1_cancel_semiring_div : 'a comm_semiring_1_cancel,
+    no_zero_divisors_semiring_div : 'a no_zero_divisors};
+val div_semiring_div = #div_semiring_div : 'a semiring_div -> 'a diva;
+val comm_semiring_1_cancel_semiring_div = #comm_semiring_1_cancel_semiring_div :
+  'a semiring_div -> 'a comm_semiring_1_cancel;
+val no_zero_divisors_semiring_div = #no_zero_divisors_semiring_div :
+  'a semiring_div -> 'a no_zero_divisors;
+
+val one_int : IntInf.int = (1 : IntInf.int);
+
+val one_inta = {one = one_int} : IntInf.int one;
+
+val zero_neq_one_int =
+  {one_zero_neq_one = one_inta, zero_zero_neq_one = zero_inta} :
+  IntInf.int zero_neq_one;
+
+val semigroup_mult_int = {times_semigroup_mult = times_int} :
+  IntInf.int semigroup_mult;
+
+val plus_int = {plus = (fn a => fn b => IntInf.+ (a, b))} : IntInf.int plus;
+
+val semigroup_add_int = {plus_semigroup_add = plus_int} :
+  IntInf.int semigroup_add;
+
+val ab_semigroup_add_int = {semigroup_add_ab_semigroup_add = semigroup_add_int}
+  : IntInf.int ab_semigroup_add;
+
+val semiring_int =
+  {ab_semigroup_add_semiring = ab_semigroup_add_int,
+    semigroup_mult_semiring = semigroup_mult_int}
+  : IntInf.int semiring;
+
+val mult_zero_int = {times_mult_zero = times_int, zero_mult_zero = zero_inta} :
+  IntInf.int mult_zero;
+
+val monoid_add_int =
+  {semigroup_add_monoid_add = semigroup_add_int, zero_monoid_add = zero_inta} :
+  IntInf.int monoid_add;
+
+val comm_monoid_add_int =
+  {ab_semigroup_add_comm_monoid_add = ab_semigroup_add_int,
+    monoid_add_comm_monoid_add = monoid_add_int}
+  : IntInf.int comm_monoid_add;
+
+val semiring_0_int =
+  {comm_monoid_add_semiring_0 = comm_monoid_add_int,
+    mult_zero_semiring_0 = mult_zero_int, semiring_semiring_0 = semiring_int}
+  : IntInf.int semiring_0;
+
+val power_int = {one_power = one_inta, times_power = times_int} :
+  IntInf.int power;
+
+val monoid_mult_int =
+  {semigroup_mult_monoid_mult = semigroup_mult_int,
+    power_monoid_mult = power_int}
+  : IntInf.int monoid_mult;
+
+val semiring_1_int =
+  {monoid_mult_semiring_1 = monoid_mult_int,
+    semiring_0_semiring_1 = semiring_0_int,
+    zero_neq_one_semiring_1 = zero_neq_one_int}
+  : IntInf.int semiring_1;
+
+val cancel_semigroup_add_int =
+  {semigroup_add_cancel_semigroup_add = semigroup_add_int} :
+  IntInf.int cancel_semigroup_add;
+
+val cancel_ab_semigroup_add_int =
+  {ab_semigroup_add_cancel_ab_semigroup_add = ab_semigroup_add_int,
+    cancel_semigroup_add_cancel_ab_semigroup_add = cancel_semigroup_add_int}
+  : IntInf.int cancel_ab_semigroup_add;
+
+val cancel_comm_monoid_add_int =
+  {cancel_ab_semigroup_add_cancel_comm_monoid_add = cancel_ab_semigroup_add_int,
+    comm_monoid_add_cancel_comm_monoid_add = comm_monoid_add_int}
+  : IntInf.int cancel_comm_monoid_add;
+
+val semiring_0_cancel_int =
+  {cancel_comm_monoid_add_semiring_0_cancel = cancel_comm_monoid_add_int,
+    semiring_0_semiring_0_cancel = semiring_0_int}
+  : IntInf.int semiring_0_cancel;
+
+val semiring_1_cancel_int =
+  {semiring_0_cancel_semiring_1_cancel = semiring_0_cancel_int,
+    semiring_1_semiring_1_cancel = semiring_1_int}
+  : IntInf.int semiring_1_cancel;
+
+val dvd_int = {times_dvd = times_int} : IntInf.int dvd;
+
+val ab_semigroup_mult_int =
+  {semigroup_mult_ab_semigroup_mult = semigroup_mult_int} :
+  IntInf.int ab_semigroup_mult;
+
+val comm_semiring_int =
+  {ab_semigroup_mult_comm_semiring = ab_semigroup_mult_int,
+    semiring_comm_semiring = semiring_int}
+  : IntInf.int comm_semiring;
+
+val comm_semiring_0_int =
+  {comm_semiring_comm_semiring_0 = comm_semiring_int,
+    semiring_0_comm_semiring_0 = semiring_0_int}
+  : IntInf.int comm_semiring_0;
+
+val comm_monoid_mult_int =
+  {ab_semigroup_mult_comm_monoid_mult = ab_semigroup_mult_int,
+    monoid_mult_comm_monoid_mult = monoid_mult_int}
+  : IntInf.int comm_monoid_mult;
+
+val comm_semiring_1_int =
+  {comm_monoid_mult_comm_semiring_1 = comm_monoid_mult_int,
+    comm_semiring_0_comm_semiring_1 = comm_semiring_0_int,
+    dvd_comm_semiring_1 = dvd_int, semiring_1_comm_semiring_1 = semiring_1_int}
+  : IntInf.int comm_semiring_1;
+
+val comm_semiring_0_cancel_int =
+  {comm_semiring_0_comm_semiring_0_cancel = comm_semiring_0_int,
+    semiring_0_cancel_comm_semiring_0_cancel = semiring_0_cancel_int}
+  : IntInf.int comm_semiring_0_cancel;
+
+val comm_semiring_1_cancel_int =
+  {comm_semiring_0_cancel_comm_semiring_1_cancel = comm_semiring_0_cancel_int,
+    comm_semiring_1_comm_semiring_1_cancel = comm_semiring_1_int,
+    semiring_1_cancel_comm_semiring_1_cancel = semiring_1_cancel_int}
+  : IntInf.int comm_semiring_1_cancel;
+
+fun abs_int i = (if IntInf.< (i, (0 : IntInf.int)) then IntInf.~ i else i);
+
+fun split f (a, b) = f a b;
+
+fun sgn_int i =
+  (if ((i : IntInf.int) = (0 : IntInf.int)) then (0 : IntInf.int)
+    else (if IntInf.< ((0 : IntInf.int), i) then (1 : IntInf.int)
+           else IntInf.~ (1 : IntInf.int)));
+
+fun apsnd f (x, y) = (x, f y);
+
+fun divmod_int k l =
+  (if ((k : IntInf.int) = (0 : IntInf.int))
+    then ((0 : IntInf.int), (0 : IntInf.int))
+    else (if ((l : IntInf.int) = (0 : IntInf.int)) then ((0 : IntInf.int), k)
+           else apsnd (fn a => IntInf.* (sgn_int l, a))
+                  (if (((sgn_int k) : IntInf.int) = (sgn_int l))
+                    then IntInf.divMod (IntInf.abs k, IntInf.abs l)
+                    else let
+                           val (r, s) =
+                             IntInf.divMod (IntInf.abs k, IntInf.abs l);
+                         in
+                           (if ((s : IntInf.int) = (0 : IntInf.int))
+                             then (IntInf.~ r, (0 : IntInf.int))
+                             else (IntInf.- (IntInf.~ r, (1 : IntInf.int)),
+                                    IntInf.- (abs_int l, s)))
+                         end)));
+
+fun snd (a, b) = b;
+
+fun mod_int a b = snd (divmod_int a b);
+
+fun fst (a, b) = a;
+
+fun div_int a b = fst (divmod_int a b);
+
+val div_inta = {dvd_div = dvd_int, diva = div_int, moda = mod_int} :
+  IntInf.int diva;
+
+val semiring_div_int =
+  {div_semiring_div = div_inta,
+    comm_semiring_1_cancel_semiring_div = comm_semiring_1_cancel_int,
+    no_zero_divisors_semiring_div = no_zero_divisors_int}
+  : IntInf.int semiring_div;
+
+fun dvd (A1_, A2_) a b =
+  eqa A2_ (moda (div_semiring_div A1_) b a)
+    (zero ((zero_no_zero_divisors o no_zero_divisors_semiring_div) A1_));
+
+fun num_case f1 f2 f3 f4 f5 f6 f7 (Mul (inta, num)) = f7 inta num
+  | num_case f1 f2 f3 f4 f5 f6 f7 (Sub (num1, num2)) = f6 num1 num2
+  | num_case f1 f2 f3 f4 f5 f6 f7 (Add (num1, num2)) = f5 num1 num2
+  | num_case f1 f2 f3 f4 f5 f6 f7 (Neg num) = f4 num
+  | num_case f1 f2 f3 f4 f5 f6 f7 (Cn (nat, inta, num)) = f3 nat inta num
+  | num_case f1 f2 f3 f4 f5 f6 f7 (Bound nat) = f2 nat
+  | num_case f1 f2 f3 f4 f5 f6 f7 (C inta) = f1 inta;
+
+fun nummul i (C j) = C (IntInf.* (i, j))
+  | nummul i (Cn (n, c, t)) = Cn (n, IntInf.* (c, i), nummul i t)
+  | nummul i (Bound v) = Mul (i, Bound v)
+  | nummul i (Neg v) = Mul (i, Neg v)
+  | nummul i (Add (v, va)) = Mul (i, Add (v, va))
+  | nummul i (Sub (v, va)) = Mul (i, Sub (v, va))
+  | nummul i (Mul (v, va)) = Mul (i, Mul (v, va));
+
+fun numneg t = nummul (IntInf.~ (1 : IntInf.int)) t;
+
+fun numadd (Cn (n1, c1, r1), Cn (n2, c2, 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 Cn (n1, c, numadd (r1, r2)))
+         end
+    else (if IntInf.<= (n1, n2)
+           then Cn (n1, c1, numadd (r1, Add (Mul (c2, Bound n2), r2)))
+           else Cn (n2, c2, numadd (Add (Mul (c1, Bound n1), r1), r2))))
+  | numadd (Cn (n1, c1, r1), C dd) = Cn (n1, c1, numadd (r1, C dd))
+  | numadd (Cn (n1, c1, r1), Bound de) = Cn (n1, c1, numadd (r1, Bound de))
+  | numadd (Cn (n1, c1, r1), Neg di) = Cn (n1, c1, numadd (r1, Neg di))
+  | numadd (Cn (n1, c1, r1), Add (dj, dk)) =
+    Cn (n1, c1, numadd (r1, Add (dj, dk)))
+  | numadd (Cn (n1, c1, r1), Sub (dl, dm)) =
+    Cn (n1, c1, numadd (r1, Sub (dl, dm)))
+  | numadd (Cn (n1, c1, r1), Mul (dn, doa)) =
+    Cn (n1, c1, numadd (r1, Mul (dn, doa)))
+  | numadd (C w, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (C w, r2))
+  | numadd (Bound x, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (Bound x, r2))
+  | numadd (Neg ac, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (Neg ac, r2))
+  | numadd (Add (ad, ae), Cn (n2, c2, r2)) =
+    Cn (n2, c2, numadd (Add (ad, ae), r2))
+  | numadd (Sub (af, ag), Cn (n2, c2, r2)) =
+    Cn (n2, c2, numadd (Sub (af, ag), r2))
+  | numadd (Mul (ah, ai), Cn (n2, c2, r2)) =
+    Cn (n2, c2, numadd (Mul (ah, ai), r2))
+  | numadd (C b1, C b2) = C (IntInf.+ (b1, b2))
+  | numadd (C aj, Bound bi) = Add (C aj, Bound bi)
+  | numadd (C aj, Neg bm) = Add (C aj, Neg bm)
+  | numadd (C aj, Add (bn, bo)) = Add (C aj, Add (bn, bo))
+  | numadd (C aj, Sub (bp, bq)) = Add (C aj, Sub (bp, bq))
+  | numadd (C aj, Mul (br, bs)) = Add (C aj, Mul (br, bs))
+  | numadd (Bound ak, C cf) = Add (Bound ak, C cf)
+  | numadd (Bound ak, Bound cg) = Add (Bound ak, Bound cg)
+  | numadd (Bound ak, Neg ck) = Add (Bound ak, Neg ck)
+  | numadd (Bound ak, Add (cl, cm)) = Add (Bound ak, Add (cl, cm))
+  | numadd (Bound ak, Sub (cn, co)) = Add (Bound ak, Sub (cn, co))
+  | numadd (Bound ak, Mul (cp, cq)) = Add (Bound ak, Mul (cp, cq))
+  | numadd (Neg ao, C en) = Add (Neg ao, C en)
+  | numadd (Neg ao, Bound eo) = Add (Neg ao, Bound eo)
+  | numadd (Neg ao, Neg es) = Add (Neg ao, Neg es)
+  | numadd (Neg ao, Add (et, eu)) = Add (Neg ao, Add (et, eu))
+  | numadd (Neg ao, Sub (ev, ew)) = Add (Neg ao, Sub (ev, ew))
+  | numadd (Neg ao, Mul (ex, ey)) = Add (Neg ao, Mul (ex, ey))
+  | numadd (Add (ap, aq), C fl) = Add (Add (ap, aq), C fl)
+  | numadd (Add (ap, aq), Bound fm) = Add (Add (ap, aq), Bound fm)
+  | numadd (Add (ap, aq), Neg fq) = Add (Add (ap, aq), Neg fq)
+  | numadd (Add (ap, aq), Add (fr, fs)) = Add (Add (ap, aq), Add (fr, fs))
+  | numadd (Add (ap, aq), Sub (ft, fu)) = Add (Add (ap, aq), Sub (ft, fu))
+  | numadd (Add (ap, aq), Mul (fv, fw)) = Add (Add (ap, aq), Mul (fv, fw))
+  | numadd (Sub (ar, asa), C gj) = Add (Sub (ar, asa), C gj)
+  | numadd (Sub (ar, asa), Bound gk) = Add (Sub (ar, asa), Bound gk)
+  | numadd (Sub (ar, asa), Neg go) = Add (Sub (ar, asa), Neg go)
+  | numadd (Sub (ar, asa), Add (gp, gq)) = Add (Sub (ar, asa), Add (gp, gq))
+  | numadd (Sub (ar, asa), Sub (gr, gs)) = Add (Sub (ar, asa), Sub (gr, gs))
+  | numadd (Sub (ar, asa), Mul (gt, gu)) = Add (Sub (ar, asa), Mul (gt, gu))
+  | numadd (Mul (at, au), C hh) = Add (Mul (at, au), C hh)
+  | numadd (Mul (at, au), Bound hi) = Add (Mul (at, au), Bound hi)
+  | numadd (Mul (at, au), Neg hm) = Add (Mul (at, au), Neg hm)
+  | numadd (Mul (at, au), Add (hn, ho)) = Add (Mul (at, au), Add (hn, ho))
+  | numadd (Mul (at, au), Sub (hp, hq)) = Add (Mul (at, au), Sub (hp, hq))
+  | numadd (Mul (at, au), Mul (hr, hs)) = Add (Mul (at, au), Mul (hr, hs));
+
+fun numsub s t =
+  (if eq_num s t then C (0 : IntInf.int) else numadd (s, numneg t));
+
+fun simpnum (C j) = C j
+  | simpnum (Bound n) = Cn (n, (1 : IntInf.int), C (0 : IntInf.int))
+  | 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 : IntInf.int) = (0 : IntInf.int)) then C (0 : IntInf.int)
+      else nummul i (simpnum t))
+  | simpnum (Cn (v, va, vb)) = Cn (v, va, vb);
+
+fun nota (Not p) = p
+  | nota T = F
+  | nota F = T
+  | nota (Lt v) = Not (Lt v)
+  | nota (Le v) = Not (Le v)
+  | nota (Gt v) = Not (Gt v)
+  | nota (Ge v) = Not (Ge v)
+  | nota (Eq v) = Not (Eq v)
+  | nota (NEq v) = Not (NEq v)
+  | nota (Dvd (v, va)) = Not (Dvd (v, va))
+  | nota (NDvd (v, va)) = Not (NDvd (v, va))
+  | nota (And (v, va)) = Not (And (v, va))
+  | nota (Or (v, va)) = Not (Or (v, va))
+  | nota (Imp (v, va)) = Not (Imp (v, va))
+  | nota (Iff (v, va)) = Not (Iff (v, va))
+  | nota (E v) = Not (E v)
+  | nota (A v) = Not (A v)
+  | nota (Closed v) = Not (Closed v)
+  | nota (NClosed v) = Not (NClosed v);
+
+fun iffa 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 Iff (p, q)))))));
+
+fun impa 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 Imp (p, q))));
+
+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 simpfm (And (p, q)) = conj (simpfm p) (simpfm q)
+  | simpfm (Or (p, q)) = disj (simpfm p) (simpfm q)
+  | simpfm (Imp (p, q)) = impa (simpfm p) (simpfm q)
+  | simpfm (Iff (p, q)) = iffa (simpfm p) (simpfm q)
+  | simpfm (Not p) = nota (simpfm p)
+  | simpfm (Lt a) =
+    let
+      val aa = simpnum a;
+    in
+      (case aa of C v => (if IntInf.< (v, (0 : IntInf.int)) then T else F)
+        | Bound _ => Lt aa | Cn (_, _, _) => Lt aa | Neg _ => Lt aa
+        | Add (_, _) => Lt aa | Sub (_, _) => Lt aa | Mul (_, _) => Lt aa)
+    end
+  | simpfm (Le a) =
+    let
+      val aa = simpnum a;
+    in
+      (case aa of C v => (if IntInf.<= (v, (0 : IntInf.int)) then T else F)
+        | Bound _ => Le aa | Cn (_, _, _) => Le aa | Neg _ => Le aa
+        | Add (_, _) => Le aa | Sub (_, _) => Le aa | Mul (_, _) => Le aa)
+    end
+  | simpfm (Gt a) =
+    let
+      val aa = simpnum a;
+    in
+      (case aa of C v => (if IntInf.< ((0 : IntInf.int), v) then T else F)
+        | Bound _ => Gt aa | Cn (_, _, _) => Gt aa | Neg _ => Gt aa
+        | Add (_, _) => Gt aa | Sub (_, _) => Gt aa | Mul (_, _) => Gt aa)
+    end
+  | simpfm (Ge a) =
+    let
+      val aa = simpnum a;
+    in
+      (case aa of C v => (if IntInf.<= ((0 : IntInf.int), v) then T else F)
+        | Bound _ => Ge aa | Cn (_, _, _) => Ge aa | Neg _ => Ge aa
+        | Add (_, _) => Ge aa | Sub (_, _) => Ge aa | Mul (_, _) => Ge aa)
+    end
+  | simpfm (Eq a) =
+    let
+      val aa = simpnum a;
+    in
+      (case aa
+        of C v => (if ((v : IntInf.int) = (0 : IntInf.int)) then T else F)
+        | Bound _ => Eq aa | Cn (_, _, _) => Eq aa | Neg _ => Eq aa
+        | Add (_, _) => Eq aa | Sub (_, _) => Eq aa | Mul (_, _) => Eq aa)
+    end
+  | simpfm (NEq a) =
+    let
+      val aa = simpnum a;
+    in
+      (case aa
+        of C v => (if not ((v : IntInf.int) = (0 : IntInf.int)) then T else F)
+        | Bound _ => NEq aa | Cn (_, _, _) => NEq aa | Neg _ => NEq aa
+        | Add (_, _) => NEq aa | Sub (_, _) => NEq aa | Mul (_, _) => NEq aa)
+    end
+  | simpfm (Dvd (i, a)) =
+    (if ((i : IntInf.int) = (0 : IntInf.int)) then simpfm (Eq a)
+      else (if (((abs_int i) : IntInf.int) = (1 : IntInf.int)) then T
+             else let
+                    val aa = simpnum a;
+                  in
+                    (case aa
+                      of C v =>
+                        (if dvd (semiring_div_int, eq_int) i v then T else F)
+                      | Bound _ => Dvd (i, aa) | Cn (_, _, _) => Dvd (i, aa)
+                      | Neg _ => Dvd (i, aa) | Add (_, _) => Dvd (i, aa)
+                      | Sub (_, _) => Dvd (i, aa) | Mul (_, _) => Dvd (i, aa))
+                  end))
+  | simpfm (NDvd (i, a)) =
+    (if ((i : IntInf.int) = (0 : IntInf.int)) then simpfm (NEq a)
+      else (if (((abs_int i) : IntInf.int) = (1 : IntInf.int)) then F
+             else let
+                    val aa = simpnum a;
+                  in
+                    (case aa
+                      of C v =>
+                        (if not (dvd (semiring_div_int, eq_int) i v) then T
+                          else F)
+                      | Bound _ => NDvd (i, aa) | Cn (_, _, _) => NDvd (i, aa)
+                      | Neg _ => NDvd (i, aa) | Add (_, _) => NDvd (i, aa)
+                      | Sub (_, _) => NDvd (i, aa) | Mul (_, _) => NDvd (i, aa))
+                  end))
+  | simpfm T = T
+  | simpfm F = F
+  | simpfm (E v) = E v
+  | simpfm (A v) = A v
+  | simpfm (Closed v) = Closed v
+  | simpfm (NClosed v) = NClosed v;
+
+fun iupt i j =
+  (if IntInf.< (j, i) then []
+    else i :: iupt (IntInf.+ (i, (1 : IntInf.int))) j);
+
+fun mirror (And (p, q)) = And (mirror p, mirror q)
+  | mirror (Or (p, q)) = Or (mirror p, mirror q)
+  | mirror T = T
+  | mirror F = F
+  | mirror (Lt (C bo)) = Lt (C bo)
+  | mirror (Lt (Bound bp)) = Lt (Bound bp)
+  | mirror (Lt (Neg bt)) = Lt (Neg bt)
+  | mirror (Lt (Add (bu, bv))) = Lt (Add (bu, bv))
+  | mirror (Lt (Sub (bw, bx))) = Lt (Sub (bw, bx))
+  | mirror (Lt (Mul (by, bz))) = Lt (Mul (by, bz))
+  | mirror (Le (C co)) = Le (C co)
+  | mirror (Le (Bound cp)) = Le (Bound cp)
+  | mirror (Le (Neg ct)) = Le (Neg ct)
+  | mirror (Le (Add (cu, cv))) = Le (Add (cu, cv))
+  | mirror (Le (Sub (cw, cx))) = Le (Sub (cw, cx))
+  | mirror (Le (Mul (cy, cz))) = Le (Mul (cy, cz))
+  | mirror (Gt (C doa)) = Gt (C doa)
+  | mirror (Gt (Bound dp)) = Gt (Bound dp)
+  | mirror (Gt (Neg dt)) = Gt (Neg dt)
+  | mirror (Gt (Add (du, dv))) = Gt (Add (du, dv))
+  | mirror (Gt (Sub (dw, dx))) = Gt (Sub (dw, dx))
+  | mirror (Gt (Mul (dy, dz))) = Gt (Mul (dy, dz))
+  | mirror (Ge (C eo)) = Ge (C eo)
+  | mirror (Ge (Bound ep)) = Ge (Bound ep)
+  | mirror (Ge (Neg et)) = Ge (Neg et)
+  | mirror (Ge (Add (eu, ev))) = Ge (Add (eu, ev))
+  | mirror (Ge (Sub (ew, ex))) = Ge (Sub (ew, ex))
+  | mirror (Ge (Mul (ey, ez))) = Ge (Mul (ey, ez))
+  | mirror (Eq (C fo)) = Eq (C fo)
+  | mirror (Eq (Bound fp)) = Eq (Bound fp)
+  | mirror (Eq (Neg ft)) = Eq (Neg ft)
+  | mirror (Eq (Add (fu, fv))) = Eq (Add (fu, fv))
+  | mirror (Eq (Sub (fw, fx))) = Eq (Sub (fw, fx))
+  | mirror (Eq (Mul (fy, fz))) = Eq (Mul (fy, fz))
+  | mirror (NEq (C go)) = NEq (C go)
+  | mirror (NEq (Bound gp)) = NEq (Bound gp)
+  | mirror (NEq (Neg gt)) = NEq (Neg gt)
+  | mirror (NEq (Add (gu, gv))) = NEq (Add (gu, gv))
+  | mirror (NEq (Sub (gw, gx))) = NEq (Sub (gw, gx))
+  | mirror (NEq (Mul (gy, gz))) = NEq (Mul (gy, gz))
+  | mirror (Dvd (aa, C ho)) = Dvd (aa, C ho)
+  | mirror (Dvd (aa, Bound hp)) = Dvd (aa, Bound hp)
+  | mirror (Dvd (aa, Neg ht)) = Dvd (aa, Neg ht)
+  | mirror (Dvd (aa, Add (hu, hv))) = Dvd (aa, Add (hu, hv))
+  | mirror (Dvd (aa, Sub (hw, hx))) = Dvd (aa, Sub (hw, hx))
+  | mirror (Dvd (aa, Mul (hy, hz))) = Dvd (aa, Mul (hy, hz))
+  | mirror (NDvd (ac, C io)) = NDvd (ac, C io)
+  | mirror (NDvd (ac, Bound ip)) = NDvd (ac, Bound ip)
+  | mirror (NDvd (ac, Neg it)) = NDvd (ac, Neg it)
+  | mirror (NDvd (ac, Add (iu, iv))) = NDvd (ac, Add (iu, iv))
+  | mirror (NDvd (ac, Sub (iw, ix))) = NDvd (ac, Sub (iw, ix))
+  | mirror (NDvd (ac, Mul (iy, iz))) = NDvd (ac, Mul (iy, iz))
+  | 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
+  | mirror (Lt (Cn (cm, c, e))) =
+    (if ((cm : IntInf.int) = (0 : IntInf.int))
+      then Gt (Cn ((0 : IntInf.int), c, Neg e))
+      else Lt (Cn (suc (minus_nat cm (1 : IntInf.int)), c, e)))
+  | mirror (Le (Cn (dm, c, e))) =
+    (if ((dm : IntInf.int) = (0 : IntInf.int))
+      then Ge (Cn ((0 : IntInf.int), c, Neg e))
+      else Le (Cn (suc (minus_nat dm (1 : IntInf.int)), c, e)))
+  | mirror (Gt (Cn (em, c, e))) =
+    (if ((em : IntInf.int) = (0 : IntInf.int))
+      then Lt (Cn ((0 : IntInf.int), c, Neg e))
+      else Gt (Cn (suc (minus_nat em (1 : IntInf.int)), c, e)))
+  | mirror (Ge (Cn (fm, c, e))) =
+    (if ((fm : IntInf.int) = (0 : IntInf.int))
+      then Le (Cn ((0 : IntInf.int), c, Neg e))
+      else Ge (Cn (suc (minus_nat fm (1 : IntInf.int)), c, e)))
+  | mirror (Eq (Cn (gm, c, e))) =
+    (if ((gm : IntInf.int) = (0 : IntInf.int))
+      then Eq (Cn ((0 : IntInf.int), c, Neg e))
+      else Eq (Cn (suc (minus_nat gm (1 : IntInf.int)), c, e)))
+  | mirror (NEq (Cn (hm, c, e))) =
+    (if ((hm : IntInf.int) = (0 : IntInf.int))
+      then NEq (Cn ((0 : IntInf.int), c, Neg e))
+      else NEq (Cn (suc (minus_nat hm (1 : IntInf.int)), c, e)))
+  | mirror (Dvd (i, Cn (im, c, e))) =
+    (if ((im : IntInf.int) = (0 : IntInf.int))
+      then Dvd (i, Cn ((0 : IntInf.int), c, Neg e))
+      else Dvd (i, Cn (suc (minus_nat im (1 : IntInf.int)), c, e)))
+  | mirror (NDvd (i, Cn (jm, c, e))) =
+    (if ((jm : IntInf.int) = (0 : IntInf.int))
+      then NDvd (i, Cn ((0 : IntInf.int), c, Neg e))
+      else NDvd (i, Cn (suc (minus_nat jm (1 : IntInf.int)), c, e)));
+
+fun size_list [] = (0 : IntInf.int)
+  | size_list (a :: lista) = IntInf.+ (size_list lista, suc (0 : IntInf.int));
+
+fun alpha (And (p, q)) = append (alpha p) (alpha q)
+  | alpha (Or (p, q)) = append (alpha p) (alpha q)
+  | alpha T = []
+  | alpha F = []
+  | alpha (Lt (C bo)) = []
+  | alpha (Lt (Bound bp)) = []
+  | alpha (Lt (Neg bt)) = []
+  | alpha (Lt (Add (bu, bv))) = []
+  | alpha (Lt (Sub (bw, bx))) = []
+  | alpha (Lt (Mul (by, bz))) = []
+  | alpha (Le (C co)) = []
+  | alpha (Le (Bound cp)) = []
+  | alpha (Le (Neg ct)) = []
+  | alpha (Le (Add (cu, cv))) = []
+  | alpha (Le (Sub (cw, cx))) = []
+  | alpha (Le (Mul (cy, cz))) = []
+  | alpha (Gt (C doa)) = []
+  | alpha (Gt (Bound dp)) = []
+  | alpha (Gt (Neg dt)) = []
+  | alpha (Gt (Add (du, dv))) = []
+  | alpha (Gt (Sub (dw, dx))) = []
+  | alpha (Gt (Mul (dy, dz))) = []
+  | alpha (Ge (C eo)) = []
+  | alpha (Ge (Bound ep)) = []
+  | alpha (Ge (Neg et)) = []
+  | alpha (Ge (Add (eu, ev))) = []
+  | alpha (Ge (Sub (ew, ex))) = []
+  | alpha (Ge (Mul (ey, ez))) = []
+  | alpha (Eq (C fo)) = []
+  | alpha (Eq (Bound fp)) = []
+  | alpha (Eq (Neg ft)) = []
+  | alpha (Eq (Add (fu, fv))) = []
+  | alpha (Eq (Sub (fw, fx))) = []
+  | alpha (Eq (Mul (fy, fz))) = []
+  | alpha (NEq (C go)) = []
+  | alpha (NEq (Bound gp)) = []
+  | alpha (NEq (Neg gt)) = []
+  | alpha (NEq (Add (gu, gv))) = []
+  | alpha (NEq (Sub (gw, gx))) = []
+  | alpha (NEq (Mul (gy, gz))) = []
+  | 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) = []
+  | alpha (Lt (Cn (cm, c, e))) =
+    (if ((cm : IntInf.int) = (0 : IntInf.int)) then [e] else [])
+  | alpha (Le (Cn (dm, c, e))) =
+    (if ((dm : IntInf.int) = (0 : IntInf.int))
+      then [Add (C (~1 : IntInf.int), e)] else [])
+  | alpha (Gt (Cn (em, c, e))) =
+    (if ((em : IntInf.int) = (0 : IntInf.int)) then [] else [])
+  | alpha (Ge (Cn (fm, c, e))) =
+    (if ((fm : IntInf.int) = (0 : IntInf.int)) then [] else [])
+  | alpha (Eq (Cn (gm, c, e))) =
+    (if ((gm : IntInf.int) = (0 : IntInf.int))
+      then [Add (C (~1 : IntInf.int), e)] else [])
+  | alpha (NEq (Cn (hm, c, e))) =
+    (if ((hm : IntInf.int) = (0 : IntInf.int)) then [e] else []);
+
+fun beta (And (p, q)) = append (beta p) (beta q)
+  | beta (Or (p, q)) = append (beta p) (beta q)
+  | beta T = []
+  | beta F = []
+  | beta (Lt (C bo)) = []
+  | beta (Lt (Bound bp)) = []
+  | beta (Lt (Neg bt)) = []
+  | beta (Lt (Add (bu, bv))) = []
+  | beta (Lt (Sub (bw, bx))) = []
+  | beta (Lt (Mul (by, bz))) = []
+  | beta (Le (C co)) = []
+  | beta (Le (Bound cp)) = []
+  | beta (Le (Neg ct)) = []
+  | beta (Le (Add (cu, cv))) = []
+  | beta (Le (Sub (cw, cx))) = []
+  | beta (Le (Mul (cy, cz))) = []
+  | beta (Gt (C doa)) = []
+  | beta (Gt (Bound dp)) = []
+  | beta (Gt (Neg dt)) = []
+  | beta (Gt (Add (du, dv))) = []
+  | beta (Gt (Sub (dw, dx))) = []
+  | beta (Gt (Mul (dy, dz))) = []
+  | beta (Ge (C eo)) = []
+  | beta (Ge (Bound ep)) = []
+  | beta (Ge (Neg et)) = []
+  | beta (Ge (Add (eu, ev))) = []
+  | beta (Ge (Sub (ew, ex))) = []
+  | beta (Ge (Mul (ey, ez))) = []
+  | beta (Eq (C fo)) = []
+  | beta (Eq (Bound fp)) = []
+  | beta (Eq (Neg ft)) = []
+  | beta (Eq (Add (fu, fv))) = []
+  | beta (Eq (Sub (fw, fx))) = []
+  | beta (Eq (Mul (fy, fz))) = []
+  | beta (NEq (C go)) = []
+  | beta (NEq (Bound gp)) = []
+  | beta (NEq (Neg gt)) = []
+  | beta (NEq (Add (gu, gv))) = []
+  | beta (NEq (Sub (gw, gx))) = []
+  | beta (NEq (Mul (gy, gz))) = []
+  | 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) = []
+  | beta (Lt (Cn (cm, c, e))) =
+    (if ((cm : IntInf.int) = (0 : IntInf.int)) then [] else [])
+  | beta (Le (Cn (dm, c, e))) =
+    (if ((dm : IntInf.int) = (0 : IntInf.int)) then [] else [])
+  | beta (Gt (Cn (em, c, e))) =
+    (if ((em : IntInf.int) = (0 : IntInf.int)) then [Neg e] else [])
+  | beta (Ge (Cn (fm, c, e))) =
+    (if ((fm : IntInf.int) = (0 : IntInf.int))
+      then [Sub (C (~1 : IntInf.int), e)] else [])
+  | beta (Eq (Cn (gm, c, e))) =
+    (if ((gm : IntInf.int) = (0 : IntInf.int))
+      then [Sub (C (~1 : IntInf.int), e)] else [])
+  | beta (NEq (Cn (hm, c, e))) =
+    (if ((hm : IntInf.int) = (0 : IntInf.int)) then [Neg e] else []);
+
+val eq_numa = {eq = eq_num} : num eq;
+
+fun member A_ x [] = false
+  | member A_ x (y :: ys) = eqa A_ x y orelse member A_ x ys;
+
+fun remdups A_ [] = []
+  | remdups A_ (x :: xs) =
+    (if member A_ x xs then remdups A_ xs else x :: remdups A_ xs);
+
+fun gcd_int k l =
+  abs_int
+    (if ((l : IntInf.int) = (0 : IntInf.int)) then k
+      else gcd_int l (mod_int (abs_int k) (abs_int l)));
+
+fun lcm_int a b = div_int (IntInf.* (abs_int a, abs_int b)) (gcd_int a b);
+
+fun delta (And (p, q)) = lcm_int (delta p) (delta q)
+  | delta (Or (p, q)) = lcm_int (delta p) (delta q)
+  | delta T = (1 : IntInf.int)
+  | delta F = (1 : IntInf.int)
+  | delta (Lt u) = (1 : IntInf.int)
+  | delta (Le v) = (1 : IntInf.int)
+  | delta (Gt w) = (1 : IntInf.int)
+  | delta (Ge x) = (1 : IntInf.int)
+  | delta (Eq y) = (1 : IntInf.int)
+  | delta (NEq z) = (1 : IntInf.int)
+  | delta (Dvd (aa, C bo)) = (1 : IntInf.int)
+  | delta (Dvd (aa, Bound bp)) = (1 : IntInf.int)
+  | delta (Dvd (aa, Neg bt)) = (1 : IntInf.int)
+  | delta (Dvd (aa, Add (bu, bv))) = (1 : IntInf.int)
+  | delta (Dvd (aa, Sub (bw, bx))) = (1 : IntInf.int)
+  | delta (Dvd (aa, Mul (by, bz))) = (1 : IntInf.int)
+  | delta (NDvd (ac, C co)) = (1 : IntInf.int)
+  | delta (NDvd (ac, Bound cp)) = (1 : IntInf.int)
+  | delta (NDvd (ac, Neg ct)) = (1 : IntInf.int)
+  | delta (NDvd (ac, Add (cu, cv))) = (1 : IntInf.int)
+  | delta (NDvd (ac, Sub (cw, cx))) = (1 : IntInf.int)
+  | delta (NDvd (ac, Mul (cy, cz))) = (1 : IntInf.int)
+  | delta (Not ae) = (1 : IntInf.int)
+  | delta (Imp (aj, ak)) = (1 : IntInf.int)
+  | delta (Iff (al, am)) = (1 : IntInf.int)
+  | delta (E an) = (1 : IntInf.int)
+  | delta (A ao) = (1 : IntInf.int)
+  | delta (Closed ap) = (1 : IntInf.int)
+  | delta (NClosed aq) = (1 : IntInf.int)
+  | delta (Dvd (i, Cn (cm, c, e))) =
+    (if ((cm : IntInf.int) = (0 : IntInf.int)) then i else (1 : IntInf.int))
+  | delta (NDvd (i, Cn (dm, c, e))) =
+    (if ((dm : IntInf.int) = (0 : IntInf.int)) then i else (1 : IntInf.int));
+
+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 T = (fn _ => T)
+  | a_beta F = (fn _ => F)
+  | a_beta (Lt (C bo)) = (fn _ => Lt (C bo))
+  | a_beta (Lt (Bound bp)) = (fn _ => Lt (Bound bp))
+  | a_beta (Lt (Neg bt)) = (fn _ => Lt (Neg bt))
+  | a_beta (Lt (Add (bu, bv))) = (fn _ => Lt (Add (bu, bv)))
+  | a_beta (Lt (Sub (bw, bx))) = (fn _ => Lt (Sub (bw, bx)))
+  | a_beta (Lt (Mul (by, bz))) = (fn _ => Lt (Mul (by, bz)))
+  | a_beta (Le (C co)) = (fn _ => Le (C co))
+  | a_beta (Le (Bound cp)) = (fn _ => Le (Bound cp))
+  | a_beta (Le (Neg ct)) = (fn _ => Le (Neg ct))
+  | a_beta (Le (Add (cu, cv))) = (fn _ => Le (Add (cu, cv)))
+  | a_beta (Le (Sub (cw, cx))) = (fn _ => Le (Sub (cw, cx)))
+  | a_beta (Le (Mul (cy, cz))) = (fn _ => Le (Mul (cy, cz)))
+  | a_beta (Gt (C doa)) = (fn _ => Gt (C doa))
+  | a_beta (Gt (Bound dp)) = (fn _ => Gt (Bound dp))
+  | a_beta (Gt (Neg dt)) = (fn _ => Gt (Neg dt))
+  | a_beta (Gt (Add (du, dv))) = (fn _ => Gt (Add (du, dv)))
+  | a_beta (Gt (Sub (dw, dx))) = (fn _ => Gt (Sub (dw, dx)))
+  | a_beta (Gt (Mul (dy, dz))) = (fn _ => Gt (Mul (dy, dz)))
+  | a_beta (Ge (C eo)) = (fn _ => Ge (C eo))
+  | a_beta (Ge (Bound ep)) = (fn _ => Ge (Bound ep))
+  | a_beta (Ge (Neg et)) = (fn _ => Ge (Neg et))
+  | a_beta (Ge (Add (eu, ev))) = (fn _ => Ge (Add (eu, ev)))
+  | a_beta (Ge (Sub (ew, ex))) = (fn _ => Ge (Sub (ew, ex)))
+  | a_beta (Ge (Mul (ey, ez))) = (fn _ => Ge (Mul (ey, ez)))
+  | a_beta (Eq (C fo)) = (fn _ => Eq (C fo))
+  | a_beta (Eq (Bound fp)) = (fn _ => Eq (Bound fp))
+  | a_beta (Eq (Neg ft)) = (fn _ => Eq (Neg ft))
+  | a_beta (Eq (Add (fu, fv))) = (fn _ => Eq (Add (fu, fv)))
+  | a_beta (Eq (Sub (fw, fx))) = (fn _ => Eq (Sub (fw, fx)))
+  | a_beta (Eq (Mul (fy, fz))) = (fn _ => Eq (Mul (fy, fz)))
+  | a_beta (NEq (C go)) = (fn _ => NEq (C go))
+  | a_beta (NEq (Bound gp)) = (fn _ => NEq (Bound gp))
+  | a_beta (NEq (Neg gt)) = (fn _ => NEq (Neg gt))
+  | a_beta (NEq (Add (gu, gv))) = (fn _ => NEq (Add (gu, gv)))
+  | a_beta (NEq (Sub (gw, gx))) = (fn _ => NEq (Sub (gw, gx)))
+  | a_beta (NEq (Mul (gy, gz))) = (fn _ => NEq (Mul (gy, gz)))
+  | a_beta (Dvd (aa, C ho)) = (fn _ => Dvd (aa, C ho))
+  | a_beta (Dvd (aa, Bound hp)) = (fn _ => Dvd (aa, Bound hp))
+  | a_beta (Dvd (aa, Neg ht)) = (fn _ => Dvd (aa, Neg ht))
+  | a_beta (Dvd (aa, Add (hu, hv))) = (fn _ => Dvd (aa, Add (hu, hv)))
+  | a_beta (Dvd (aa, Sub (hw, hx))) = (fn _ => Dvd (aa, Sub (hw, hx)))
+  | a_beta (Dvd (aa, Mul (hy, hz))) = (fn _ => Dvd (aa, Mul (hy, hz)))
+  | a_beta (NDvd (ac, C io)) = (fn _ => NDvd (ac, C io))
+  | a_beta (NDvd (ac, Bound ip)) = (fn _ => NDvd (ac, Bound ip))
+  | a_beta (NDvd (ac, Neg it)) = (fn _ => NDvd (ac, Neg it))
+  | a_beta (NDvd (ac, Add (iu, iv))) = (fn _ => NDvd (ac, Add (iu, iv)))
+  | a_beta (NDvd (ac, Sub (iw, ix))) = (fn _ => NDvd (ac, Sub (iw, ix)))
+  | a_beta (NDvd (ac, Mul (iy, iz))) = (fn _ => NDvd (ac, Mul (iy, iz)))
+  | a_beta (Not ae) = (fn _ => Not ae)
+  | a_beta (Imp (aj, ak)) = (fn _ => Imp (aj, ak))
+  | a_beta (Iff (al, am)) = (fn _ => Iff (al, am))
+  | a_beta (E an) = (fn _ => E an)
+  | a_beta (A ao) = (fn _ => A ao)
+  | a_beta (Closed ap) = (fn _ => Closed ap)
+  | a_beta (NClosed aq) = (fn _ => NClosed aq)
+  | a_beta (Lt (Cn (cm, c, e))) =
+    (if ((cm : IntInf.int) = (0 : IntInf.int))
+      then (fn k =>
+             Lt (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e))))
+      else (fn _ => Lt (Cn (suc (minus_nat cm (1 : IntInf.int)), c, e))))
+  | a_beta (Le (Cn (dm, c, e))) =
+    (if ((dm : IntInf.int) = (0 : IntInf.int))
+      then (fn k =>
+             Le (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e))))
+      else (fn _ => Le (Cn (suc (minus_nat dm (1 : IntInf.int)), c, e))))
+  | a_beta (Gt (Cn (em, c, e))) =
+    (if ((em : IntInf.int) = (0 : IntInf.int))
+      then (fn k =>
+             Gt (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e))))
+      else (fn _ => Gt (Cn (suc (minus_nat em (1 : IntInf.int)), c, e))))
+  | a_beta (Ge (Cn (fm, c, e))) =
+    (if ((fm : IntInf.int) = (0 : IntInf.int))
+      then (fn k =>
+             Ge (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e))))
+      else (fn _ => Ge (Cn (suc (minus_nat fm (1 : IntInf.int)), c, e))))
+  | a_beta (Eq (Cn (gm, c, e))) =
+    (if ((gm : IntInf.int) = (0 : IntInf.int))
+      then (fn k =>
+             Eq (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e))))
+      else (fn _ => Eq (Cn (suc (minus_nat gm (1 : IntInf.int)), c, e))))
+  | a_beta (NEq (Cn (hm, c, e))) =
+    (if ((hm : IntInf.int) = (0 : IntInf.int))
+      then (fn k =>
+             NEq (Cn ((0 : IntInf.int), (1 : IntInf.int),
+                       Mul (div_int k c, e))))
+      else (fn _ => NEq (Cn (suc (minus_nat hm (1 : IntInf.int)), c, e))))
+  | a_beta (Dvd (i, Cn (im, c, e))) =
+    (if ((im : IntInf.int) = (0 : IntInf.int))
+      then (fn k =>
+             Dvd (IntInf.* (div_int k c, i),
+                   Cn ((0 : IntInf.int), (1 : IntInf.int),
+                        Mul (div_int k c, e))))
+      else (fn _ => Dvd (i, Cn (suc (minus_nat im (1 : IntInf.int)), c, e))))
+  | a_beta (NDvd (i, Cn (jm, c, e))) =
+    (if ((jm : IntInf.int) = (0 : IntInf.int))
+      then (fn k =>
+             NDvd (IntInf.* (div_int k c, i),
+                    Cn ((0 : IntInf.int), (1 : IntInf.int),
+                         Mul (div_int k c, e))))
+      else (fn _ => NDvd (i, Cn (suc (minus_nat jm (1 : IntInf.int)), c, e))));
+
+fun zeta (And (p, q)) = lcm_int (zeta p) (zeta q)
+  | zeta (Or (p, q)) = lcm_int (zeta p) (zeta q)
+  | zeta T = (1 : IntInf.int)
+  | zeta F = (1 : IntInf.int)
+  | zeta (Lt (C bo)) = (1 : IntInf.int)
+  | zeta (Lt (Bound bp)) = (1 : IntInf.int)
+  | zeta (Lt (Neg bt)) = (1 : IntInf.int)
+  | zeta (Lt (Add (bu, bv))) = (1 : IntInf.int)
+  | zeta (Lt (Sub (bw, bx))) = (1 : IntInf.int)
+  | zeta (Lt (Mul (by, bz))) = (1 : IntInf.int)
+  | zeta (Le (C co)) = (1 : IntInf.int)
+  | zeta (Le (Bound cp)) = (1 : IntInf.int)
+  | zeta (Le (Neg ct)) = (1 : IntInf.int)
+  | zeta (Le (Add (cu, cv))) = (1 : IntInf.int)
+  | zeta (Le (Sub (cw, cx))) = (1 : IntInf.int)
+  | zeta (Le (Mul (cy, cz))) = (1 : IntInf.int)
+  | zeta (Gt (C doa)) = (1 : IntInf.int)
+  | zeta (Gt (Bound dp)) = (1 : IntInf.int)
+  | zeta (Gt (Neg dt)) = (1 : IntInf.int)
+  | zeta (Gt (Add (du, dv))) = (1 : IntInf.int)
+  | zeta (Gt (Sub (dw, dx))) = (1 : IntInf.int)
+  | zeta (Gt (Mul (dy, dz))) = (1 : IntInf.int)
+  | zeta (Ge (C eo)) = (1 : IntInf.int)
+  | zeta (Ge (Bound ep)) = (1 : IntInf.int)
+  | zeta (Ge (Neg et)) = (1 : IntInf.int)
+  | zeta (Ge (Add (eu, ev))) = (1 : IntInf.int)
+  | zeta (Ge (Sub (ew, ex))) = (1 : IntInf.int)
+  | zeta (Ge (Mul (ey, ez))) = (1 : IntInf.int)
+  | zeta (Eq (C fo)) = (1 : IntInf.int)
+  | zeta (Eq (Bound fp)) = (1 : IntInf.int)
+  | zeta (Eq (Neg ft)) = (1 : IntInf.int)
+  | zeta (Eq (Add (fu, fv))) = (1 : IntInf.int)
+  | zeta (Eq (Sub (fw, fx))) = (1 : IntInf.int)
+  | zeta (Eq (Mul (fy, fz))) = (1 : IntInf.int)
+  | zeta (NEq (C go)) = (1 : IntInf.int)
+  | zeta (NEq (Bound gp)) = (1 : IntInf.int)
+  | zeta (NEq (Neg gt)) = (1 : IntInf.int)
+  | zeta (NEq (Add (gu, gv))) = (1 : IntInf.int)
+  | zeta (NEq (Sub (gw, gx))) = (1 : IntInf.int)
+  | zeta (NEq (Mul (gy, gz))) = (1 : IntInf.int)
+  | zeta (Dvd (aa, C ho)) = (1 : IntInf.int)
+  | zeta (Dvd (aa, Bound hp)) = (1 : IntInf.int)
+  | zeta (Dvd (aa, Neg ht)) = (1 : IntInf.int)
+  | zeta (Dvd (aa, Add (hu, hv))) = (1 : IntInf.int)
+  | zeta (Dvd (aa, Sub (hw, hx))) = (1 : IntInf.int)
+  | zeta (Dvd (aa, Mul (hy, hz))) = (1 : IntInf.int)
+  | zeta (NDvd (ac, C io)) = (1 : IntInf.int)
+  | zeta (NDvd (ac, Bound ip)) = (1 : IntInf.int)
+  | zeta (NDvd (ac, Neg it)) = (1 : IntInf.int)
+  | zeta (NDvd (ac, Add (iu, iv))) = (1 : IntInf.int)
+  | zeta (NDvd (ac, Sub (iw, ix))) = (1 : IntInf.int)
+  | zeta (NDvd (ac, Mul (iy, iz))) = (1 : IntInf.int)
+  | zeta (Not ae) = (1 : IntInf.int)
+  | zeta (Imp (aj, ak)) = (1 : IntInf.int)
+  | zeta (Iff (al, am)) = (1 : IntInf.int)
+  | zeta (E an) = (1 : IntInf.int)
+  | zeta (A ao) = (1 : IntInf.int)
+  | zeta (Closed ap) = (1 : IntInf.int)
+  | zeta (NClosed aq) = (1 : IntInf.int)
+  | zeta (Lt (Cn (cm, c, e))) =
+    (if ((cm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int))
+  | zeta (Le (Cn (dm, c, e))) =
+    (if ((dm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int))
+  | zeta (Gt (Cn (em, c, e))) =
+    (if ((em : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int))
+  | zeta (Ge (Cn (fm, c, e))) =
+    (if ((fm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int))
+  | zeta (Eq (Cn (gm, c, e))) =
+    (if ((gm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int))
+  | zeta (NEq (Cn (hm, c, e))) =
+    (if ((hm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int))
+  | zeta (Dvd (i, Cn (im, c, e))) =
+    (if ((im : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int))
+  | zeta (NDvd (i, Cn (jm, c, e))) =
+    (if ((jm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int));
+
+fun zsplit0 (C c) = ((0 : IntInf.int), C c)
+  | zsplit0 (Bound n) =
+    (if ((n : IntInf.int) = (0 : IntInf.int))
+      then ((1 : IntInf.int), C (0 : IntInf.int))
+      else ((0 : IntInf.int), Bound n))
+  | zsplit0 (Cn (n, i, a)) =
+    let
+      val (ia, aa) = zsplit0 a;
+    in
+      (if ((n : IntInf.int) = (0 : IntInf.int)) then (IntInf.+ (i, ia), aa)
+        else (ia, Cn (n, i, aa)))
+    end
+  | zsplit0 (Neg a) =
+    let
+      val (i, aa) = zsplit0 a;
+    in
+      (IntInf.~ i, Neg aa)
+    end
+  | zsplit0 (Add (a, b)) =
+    let
+      val (ia, aa) = zsplit0 a;
+      val (ib, ba) = zsplit0 b;
+    in
+      (IntInf.+ (ia, ib), Add (aa, ba))
+    end
+  | zsplit0 (Sub (a, b)) =
+    let
+      val (ia, aa) = zsplit0 a;
+      val (ib, ba) = zsplit0 b;
+    in
+      (IntInf.- (ia, ib), Sub (aa, ba))
+    end
+  | zsplit0 (Mul (i, a)) =
+    let
+      val (ia, aa) = zsplit0 a;
+    in
+      (IntInf.* (i, ia), Mul (i, aa))
+    end;
+
+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 (c, r) = zsplit0 a;
+    in
+      (if ((c : IntInf.int) = (0 : IntInf.int)) then Lt r
+        else (if IntInf.< ((0 : IntInf.int), c)
+               then Lt (Cn ((0 : IntInf.int), c, r))
+               else Gt (Cn ((0 : IntInf.int), 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 (Cn ((0 : IntInf.int), c, r))
+               else Ge (Cn ((0 : IntInf.int), 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 (Cn ((0 : IntInf.int), c, r))
+               else Lt (Cn ((0 : IntInf.int), 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 (Cn ((0 : IntInf.int), c, r))
+               else Le (Cn ((0 : IntInf.int), 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 (Cn ((0 : IntInf.int), c, r))
+               else Eq (Cn ((0 : IntInf.int), 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 (Cn ((0 : IntInf.int), c, r))
+               else NEq (Cn ((0 : IntInf.int), 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 (abs_int i, r)
+               else (if IntInf.< ((0 : IntInf.int), c)
+                      then Dvd (abs_int i, Cn ((0 : IntInf.int), c, r))
+                      else Dvd (abs_int i,
+                                 Cn ((0 : IntInf.int), IntInf.~ c, Neg r))))
+           end)
+  | 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 (abs_int i, r)
+               else (if IntInf.< ((0 : IntInf.int), c)
+                      then NDvd (abs_int i, Cn ((0 : IntInf.int), c, r))
+                      else NDvd (abs_int i,
+                                  Cn ((0 : IntInf.int), IntInf.~ c, Neg r))))
+           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 unita p =
+  let
+    val pa = zlfm p;
+    val l = zeta pa;
+    val q =
+      And (Dvd (l, Cn ((0 : IntInf.int), (1 : IntInf.int), C (0 : IntInf.int))),
+            a_beta pa l);
+    val d = delta q;
+    val b = remdups eq_numa (map simpnum (beta q));
+    val a = remdups eq_numa (map simpnum (alpha q));
+  in
+    (if IntInf.<= (size_list b, size_list a) then (q, (b, d))
+      else (mirror q, (a, d)))
+  end;
+
+fun cooper p =
+  let
+    val (q, (b, d)) = unita p;
+    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 (ba, j) => simpfm (subst0 (Add (ba, C j)) q))
+                 (concat_map (fn ba => map (fn a => (ba, a)) js) b);
+           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))
+  | 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;
+
+fun qelim (E p) = (fn qe => dj qe (qelim p qe))
+  | qelim (A p) = (fn qe => nota (qe (qelim (Not p) qe)))
+  | qelim (Not p) = (fn qe => nota (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 => impa (qelim p qe) (qelim q qe))
+  | qelim (Iff (p, q)) = (fn qe => iffa (qelim p qe) (qelim q qe))
+  | qelim T = (fn _ => simpfm T)
+  | qelim F = (fn _ => simpfm F)
+  | qelim (Lt u) = (fn _ => simpfm (Lt u))
+  | qelim (Le v) = (fn _ => simpfm (Le v))
+  | qelim (Gt w) = (fn _ => simpfm (Gt w))
+  | qelim (Ge x) = (fn _ => simpfm (Ge x))
+  | qelim (Eq y) = (fn _ => simpfm (Eq y))
+  | qelim (NEq z) = (fn _ => simpfm (NEq z))
+  | qelim (Dvd (aa, ab)) = (fn _ => simpfm (Dvd (aa, ab)))
+  | qelim (NDvd (ac, ad)) = (fn _ => simpfm (NDvd (ac, ad)))
+  | qelim (Closed ap) = (fn _ => simpfm (Closed ap))
+  | qelim (NClosed aq) = (fn _ => simpfm (NClosed aq));
+
+fun pa p = qelim (prep p) cooper;
+
+end; (*struct Cooper_Procedure*)
diff -r cb074cec7a30 -r 3981db162131 src/HOL/Tools/Qelim/generated_cooper.ML
--- a/src/HOL/Tools/Qelim/generated_cooper.ML	Mon May 10 12:25:49 2010 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,2274 +0,0 @@
-(* Generated from Cooper.thy; DO NOT EDIT! *)
-
-structure Generated_Cooper : sig
-  type 'a eq
-  val eq : 'a eq -> 'a -> 'a -> bool
-  val eqa : 'a eq -> 'a -> 'a -> bool
-  val leta : 'a -> ('a -> 'b) -> 'b
-  val suc : IntInf.int -> IntInf.int
-  datatype num = C of IntInf.int | Bound of IntInf.int |
-    Cn of IntInf.int * 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
-    | 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 IntInf.int |
-    NClosed of IntInf.int
-  val map : ('a -> 'b) -> 'a list -> 'b list
-  val append : 'a list -> 'a list -> 'a list
-  val disjuncts : fm -> fm list
-  val fm_case :
-    'a -> 'a -> (num -> 'a) ->
-                  (num -> 'a) ->
-                    (num -> 'a) ->
-                      (num -> 'a) ->
-                        (num -> 'a) ->
-                          (num -> 'a) ->
-                            (IntInf.int -> num -> 'a) ->
-                              (IntInf.int -> num -> 'a) ->
-                                (fm -> 'a) ->
-                                  (fm -> fm -> 'a) ->
-                                    (fm -> fm -> 'a) ->
-                                      (fm -> fm -> 'a) ->
-(fm -> fm -> 'a) ->
-  (fm -> 'a) ->
-    (fm -> 'a) -> (IntInf.int -> 'a) -> (IntInf.int -> 'a) -> fm -> 'a
-  val eq_num : num -> num -> bool
-  val eq_fm : fm -> fm -> bool
-  val djf : ('a -> fm) -> 'a -> fm -> fm
-  val foldr : ('a -> 'b -> 'b) -> 'a list -> 'b -> 'b
-  val evaldjf : ('a -> fm) -> 'a list -> fm
-  val dj : (fm -> fm) -> fm -> fm
-  val disj : fm -> fm -> fm
-  val minus_nat : IntInf.int -> IntInf.int -> IntInf.int
-  val decrnum : num -> num
-  val decr : fm -> fm
-  val concat_map : ('a -> 'b list) -> 'a list -> 'b list
-  val numsubst0 : num -> num -> num
-  val subst0 : num -> fm -> fm
-  val minusinf : fm -> fm
-  val eq_int : IntInf.int eq
-  val zero_int : IntInf.int
-  type 'a zero
-  val zero : 'a zero -> 'a
-  val zero_inta : IntInf.int zero
-  type 'a times
-  val times : 'a times -> 'a -> 'a -> 'a
-  type 'a no_zero_divisors
-  val times_no_zero_divisors : 'a no_zero_divisors -> 'a times
-  val zero_no_zero_divisors : 'a no_zero_divisors -> 'a zero
-  val times_int : IntInf.int times
-  val no_zero_divisors_int : IntInf.int no_zero_divisors
-  type 'a one
-  val one : 'a one -> 'a
-  type 'a zero_neq_one
-  val one_zero_neq_one : 'a zero_neq_one -> 'a one
-  val zero_zero_neq_one : 'a zero_neq_one -> 'a zero
-  type 'a semigroup_mult
-  val times_semigroup_mult : 'a semigroup_mult -> 'a times
-  type 'a plus
-  val plus : 'a plus -> 'a -> 'a -> 'a
-  type 'a semigroup_add
-  val plus_semigroup_add : 'a semigroup_add -> 'a plus
-  type 'a ab_semigroup_add
-  val semigroup_add_ab_semigroup_add : 'a ab_semigroup_add -> 'a semigroup_add
-  type 'a semiring
-  val ab_semigroup_add_semiring : 'a semiring -> 'a ab_semigroup_add
-  val semigroup_mult_semiring : 'a semiring -> 'a semigroup_mult
-  type 'a mult_zero
-  val times_mult_zero : 'a mult_zero -> 'a times
-  val zero_mult_zero : 'a mult_zero -> 'a zero
-  type 'a monoid_add
-  val semigroup_add_monoid_add : 'a monoid_add -> 'a semigroup_add
-  val zero_monoid_add : 'a monoid_add -> 'a zero
-  type 'a comm_monoid_add
-  val ab_semigroup_add_comm_monoid_add :
-    'a comm_monoid_add -> 'a ab_semigroup_add
-  val monoid_add_comm_monoid_add : 'a comm_monoid_add -> 'a monoid_add
-  type 'a semiring_0
-  val comm_monoid_add_semiring_0 : 'a semiring_0 -> 'a comm_monoid_add
-  val mult_zero_semiring_0 : 'a semiring_0 -> 'a mult_zero
-  val semiring_semiring_0 : 'a semiring_0 -> 'a semiring
-  type 'a power
-  val one_power : 'a power -> 'a one
-  val times_power : 'a power -> 'a times
-  type 'a monoid_mult
-  val semigroup_mult_monoid_mult : 'a monoid_mult -> 'a semigroup_mult
-  val power_monoid_mult : 'a monoid_mult -> 'a power
-  type 'a semiring_1
-  val monoid_mult_semiring_1 : 'a semiring_1 -> 'a monoid_mult
-  val semiring_0_semiring_1 : 'a semiring_1 -> 'a semiring_0
-  val zero_neq_one_semiring_1 : 'a semiring_1 -> 'a zero_neq_one
-  type 'a cancel_semigroup_add
-  val semigroup_add_cancel_semigroup_add :
-    'a cancel_semigroup_add -> 'a semigroup_add
-  type 'a cancel_ab_semigroup_add
-  val ab_semigroup_add_cancel_ab_semigroup_add :
-    'a cancel_ab_semigroup_add -> 'a ab_semigroup_add
-  val cancel_semigroup_add_cancel_ab_semigroup_add :
-    'a cancel_ab_semigroup_add -> 'a cancel_semigroup_add
-  type 'a cancel_comm_monoid_add
-  val cancel_ab_semigroup_add_cancel_comm_monoid_add :
-    'a cancel_comm_monoid_add -> 'a cancel_ab_semigroup_add
-  val comm_monoid_add_cancel_comm_monoid_add :
-    'a cancel_comm_monoid_add -> 'a comm_monoid_add
-  type 'a semiring_0_cancel
-  val cancel_comm_monoid_add_semiring_0_cancel :
-    'a semiring_0_cancel -> 'a cancel_comm_monoid_add
-  val semiring_0_semiring_0_cancel : 'a semiring_0_cancel -> 'a semiring_0
-  type 'a semiring_1_cancel
-  val semiring_0_cancel_semiring_1_cancel :
-    'a semiring_1_cancel -> 'a semiring_0_cancel
-  val semiring_1_semiring_1_cancel : 'a semiring_1_cancel -> 'a semiring_1
-  type 'a dvd
-  val times_dvd : 'a dvd -> 'a times
-  type 'a ab_semigroup_mult
-  val semigroup_mult_ab_semigroup_mult :
-    'a ab_semigroup_mult -> 'a semigroup_mult
-  type 'a comm_semiring
-  val ab_semigroup_mult_comm_semiring : 'a comm_semiring -> 'a ab_semigroup_mult
-  val semiring_comm_semiring : 'a comm_semiring -> 'a semiring
-  type 'a comm_semiring_0
-  val comm_semiring_comm_semiring_0 : 'a comm_semiring_0 -> 'a comm_semiring
-  val semiring_0_comm_semiring_0 : 'a comm_semiring_0 -> 'a semiring_0
-  type 'a comm_monoid_mult
-  val ab_semigroup_mult_comm_monoid_mult :
-    'a comm_monoid_mult -> 'a ab_semigroup_mult
-  val monoid_mult_comm_monoid_mult : 'a comm_monoid_mult -> 'a monoid_mult
-  type 'a comm_semiring_1
-  val comm_monoid_mult_comm_semiring_1 :
-    'a comm_semiring_1 -> 'a comm_monoid_mult
-  val comm_semiring_0_comm_semiring_1 : 'a comm_semiring_1 -> 'a comm_semiring_0
-  val dvd_comm_semiring_1 : 'a comm_semiring_1 -> 'a dvd
-  val semiring_1_comm_semiring_1 : 'a comm_semiring_1 -> 'a semiring_1
-  type 'a comm_semiring_0_cancel
-  val comm_semiring_0_comm_semiring_0_cancel :
-    'a comm_semiring_0_cancel -> 'a comm_semiring_0
-  val semiring_0_cancel_comm_semiring_0_cancel :
-    'a comm_semiring_0_cancel -> 'a semiring_0_cancel
-  type 'a comm_semiring_1_cancel
-  val comm_semiring_0_cancel_comm_semiring_1_cancel :
-    'a comm_semiring_1_cancel -> 'a comm_semiring_0_cancel
-  val comm_semiring_1_comm_semiring_1_cancel :
-    'a comm_semiring_1_cancel -> 'a comm_semiring_1
-  val semiring_1_cancel_comm_semiring_1_cancel :
-    'a comm_semiring_1_cancel -> 'a semiring_1_cancel
-  type 'a diva
-  val dvd_div : 'a diva -> 'a dvd
-  val diva : 'a diva -> 'a -> 'a -> 'a
-  val moda : 'a diva -> 'a -> 'a -> 'a
-  type 'a semiring_div
-  val div_semiring_div : 'a semiring_div -> 'a diva
-  val comm_semiring_1_cancel_semiring_div :
-    'a semiring_div -> 'a comm_semiring_1_cancel
-  val no_zero_divisors_semiring_div : 'a semiring_div -> 'a no_zero_divisors
-  val one_int : IntInf.int
-  val one_inta : IntInf.int one
-  val zero_neq_one_int : IntInf.int zero_neq_one
-  val semigroup_mult_int : IntInf.int semigroup_mult
-  val plus_int : IntInf.int plus
-  val semigroup_add_int : IntInf.int semigroup_add
-  val ab_semigroup_add_int : IntInf.int ab_semigroup_add
-  val semiring_int : IntInf.int semiring
-  val mult_zero_int : IntInf.int mult_zero
-  val monoid_add_int : IntInf.int monoid_add
-  val comm_monoid_add_int : IntInf.int comm_monoid_add
-  val semiring_0_int : IntInf.int semiring_0
-  val power_int : IntInf.int power
-  val monoid_mult_int : IntInf.int monoid_mult
-  val semiring_1_int : IntInf.int semiring_1
-  val cancel_semigroup_add_int : IntInf.int cancel_semigroup_add
-  val cancel_ab_semigroup_add_int : IntInf.int cancel_ab_semigroup_add
-  val cancel_comm_monoid_add_int : IntInf.int cancel_comm_monoid_add
-  val semiring_0_cancel_int : IntInf.int semiring_0_cancel
-  val semiring_1_cancel_int : IntInf.int semiring_1_cancel
-  val dvd_int : IntInf.int dvd
-  val ab_semigroup_mult_int : IntInf.int ab_semigroup_mult
-  val comm_semiring_int : IntInf.int comm_semiring
-  val comm_semiring_0_int : IntInf.int comm_semiring_0
-  val comm_monoid_mult_int : IntInf.int comm_monoid_mult
-  val comm_semiring_1_int : IntInf.int comm_semiring_1
-  val comm_semiring_0_cancel_int : IntInf.int comm_semiring_0_cancel
-  val comm_semiring_1_cancel_int : IntInf.int comm_semiring_1_cancel
-  val abs_int : IntInf.int -> IntInf.int
-  val split : ('a -> 'b -> 'c) -> 'a * 'b -> 'c
-  val sgn_int : IntInf.int -> IntInf.int
-  val apsnd : ('a -> 'b) -> 'c * 'a -> 'c * 'b
-  val divmod_int : IntInf.int -> IntInf.int -> IntInf.int * IntInf.int
-  val snd : 'a * 'b -> 'b
-  val mod_int : IntInf.int -> IntInf.int -> IntInf.int
-  val fst : 'a * 'b -> 'a
-  val div_int : IntInf.int -> IntInf.int -> IntInf.int
-  val div_inta : IntInf.int diva
-  val semiring_div_int : IntInf.int semiring_div
-  val dvd : 'a semiring_div * 'a eq -> 'a -> 'a -> bool
-  val num_case :
-    (IntInf.int -> 'a) ->
-      (IntInf.int -> 'a) ->
-        (IntInf.int -> IntInf.int -> num -> 'a) ->
-          (num -> 'a) ->
-            (num -> num -> 'a) ->
-              (num -> num -> 'a) -> (IntInf.int -> num -> 'a) -> num -> 'a
-  val nummul : IntInf.int -> num -> num
-  val numneg : num -> num
-  val numadd : num * num -> num
-  val numsub : num -> num -> num
-  val simpnum : num -> num
-  val nota : fm -> fm
-  val iffa : fm -> fm -> fm
-  val impa : fm -> fm -> fm
-  val conj : fm -> fm -> fm
-  val simpfm : fm -> fm
-  val iupt : IntInf.int -> IntInf.int -> IntInf.int list
-  val mirror : fm -> fm
-  val size_list : 'a list -> IntInf.int
-  val alpha : fm -> num list
-  val beta : fm -> num list
-  val eq_numa : num eq
-  val member : 'a eq -> 'a -> 'a list -> bool
-  val remdups : 'a eq -> 'a list -> 'a list
-  val gcd_int : IntInf.int -> IntInf.int -> IntInf.int
-  val lcm_int : IntInf.int -> IntInf.int -> IntInf.int
-  val delta : fm -> IntInf.int
-  val a_beta : fm -> IntInf.int -> fm
-  val zeta : fm -> IntInf.int
-  val zsplit0 : num -> IntInf.int * num
-  val zlfm : fm -> fm
-  val unita : fm -> fm * (num list * IntInf.int)
-  val cooper : fm -> fm
-  val prep : fm -> fm
-  val qelim : fm -> (fm -> fm) -> fm
-  val pa : fm -> fm
-end = struct
-
-type 'a eq = {eq : 'a -> 'a -> bool};
-val eq = #eq : 'a eq -> 'a -> 'a -> bool;
-
-fun eqa A_ a b = eq A_ a b;
-
-fun leta s f = f s;
-
-fun suc n = IntInf.+ (n, (1 : IntInf.int));
-
-datatype num = C of IntInf.int | Bound of IntInf.int |
-  Cn of IntInf.int * 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 | 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 IntInf.int | NClosed of IntInf.int;
-
-fun map f [] = []
-  | map f (x :: xs) = f x :: map f xs;
-
-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 fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-  (NClosed nat) = f19 nat
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (Closed nat) = f18 nat
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (A fm) = f17 fm
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (E fm) = f16 fm
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (Iff (fm1, fm2)) = f15 fm1 fm2
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (Imp (fm1, fm2)) = f14 fm1 fm2
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (Or (fm1, fm2)) = f13 fm1 fm2
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (And (fm1, fm2)) = f12 fm1 fm2
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (Not fm) = f11 fm
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (NDvd (inta, num)) = f10 inta num
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (Dvd (inta, num)) = f9 inta num
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (NEq num) = f8 num
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (Eq num) = f7 num
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (Ge num) = f6 num
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (Gt num) = f5 num
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (Le num) = f4 num
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19
-    (Lt num) = f3 num
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 F
-    = f2
-  | fm_case f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 f15 f16 f17 f18 f19 T
-    = f1;
-
-fun eq_num (C intaa) (C inta) = ((intaa : IntInf.int) = inta)
-  | eq_num (Bound nata) (Bound nat) = ((nata : IntInf.int) = nat)
-  | eq_num (Cn (nata, intaa, numa)) (Cn (nat, inta, num)) =
-    ((nata : IntInf.int) = nat) andalso
-      (((intaa : IntInf.int) = inta) andalso eq_num numa num)
-  | eq_num (Neg numa) (Neg num) = eq_num numa num
-  | eq_num (Add (num1a, num2a)) (Add (num1, num2)) =
-    eq_num num1a num1 andalso eq_num num2a num2
-  | eq_num (Sub (num1a, num2a)) (Sub (num1, num2)) =
-    eq_num num1a num1 andalso eq_num num2a num2
-  | eq_num (Mul (intaa, numa)) (Mul (inta, num)) =
-    ((intaa : IntInf.int) = inta) andalso eq_num numa num
-  | eq_num (C inta) (Bound nat) = false
-  | eq_num (Bound nat) (C inta) = false
-  | eq_num (C intaa) (Cn (nat, inta, num)) = false
-  | eq_num (Cn (nat, intaa, num)) (C inta) = false
-  | eq_num (C inta) (Neg num) = false
-  | eq_num (Neg num) (C inta) = false
-  | eq_num (C inta) (Add (num1, num2)) = false
-  | eq_num (Add (num1, num2)) (C inta) = false
-  | eq_num (C inta) (Sub (num1, num2)) = false
-  | eq_num (Sub (num1, num2)) (C inta) = false
-  | eq_num (C intaa) (Mul (inta, num)) = false
-  | eq_num (Mul (intaa, num)) (C inta) = false
-  | eq_num (Bound nata) (Cn (nat, inta, num)) = false
-  | eq_num (Cn (nata, inta, num)) (Bound nat) = false
-  | eq_num (Bound nat) (Neg num) = false
-  | eq_num (Neg num) (Bound nat) = false
-  | eq_num (Bound nat) (Add (num1, num2)) = false
-  | eq_num (Add (num1, num2)) (Bound nat) = false
-  | eq_num (Bound nat) (Sub (num1, num2)) = false
-  | eq_num (Sub (num1, num2)) (Bound nat) = false
-  | eq_num (Bound nat) (Mul (inta, num)) = false
-  | eq_num (Mul (inta, num)) (Bound nat) = false
-  | eq_num (Cn (nat, inta, numa)) (Neg num) = false
-  | eq_num (Neg numa) (Cn (nat, inta, num)) = false
-  | eq_num (Cn (nat, inta, num)) (Add (num1, num2)) = false
-  | eq_num (Add (num1, num2)) (Cn (nat, inta, num)) = false
-  | eq_num (Cn (nat, inta, num)) (Sub (num1, num2)) = false
-  | eq_num (Sub (num1, num2)) (Cn (nat, inta, num)) = false
-  | eq_num (Cn (nat, intaa, numa)) (Mul (inta, num)) = false
-  | eq_num (Mul (intaa, numa)) (Cn (nat, inta, num)) = false
-  | eq_num (Neg num) (Add (num1, num2)) = false
-  | eq_num (Add (num1, num2)) (Neg num) = false
-  | eq_num (Neg num) (Sub (num1, num2)) = false
-  | eq_num (Sub (num1, num2)) (Neg num) = false
-  | eq_num (Neg numa) (Mul (inta, num)) = false
-  | eq_num (Mul (inta, numa)) (Neg num) = false
-  | eq_num (Add (num1a, num2a)) (Sub (num1, num2)) = false
-  | eq_num (Sub (num1a, num2a)) (Add (num1, num2)) = false
-  | eq_num (Add (num1, num2)) (Mul (inta, num)) = false
-  | eq_num (Mul (inta, num)) (Add (num1, num2)) = false
-  | eq_num (Sub (num1, num2)) (Mul (inta, num)) = false
-  | eq_num (Mul (inta, num)) (Sub (num1, num2)) = false;
-
-fun eq_fm T T = true
-  | eq_fm F F = true
-  | eq_fm (Lt numa) (Lt num) = eq_num numa num
-  | eq_fm (Le numa) (Le num) = eq_num numa num
-  | eq_fm (Gt numa) (Gt num) = eq_num numa num
-  | eq_fm (Ge numa) (Ge num) = eq_num numa num
-  | eq_fm (Eq numa) (Eq num) = eq_num numa num
-  | eq_fm (NEq numa) (NEq num) = eq_num numa num
-  | eq_fm (Dvd (intaa, numa)) (Dvd (inta, num)) =
-    ((intaa : IntInf.int) = inta) andalso eq_num numa num
-  | eq_fm (NDvd (intaa, numa)) (NDvd (inta, num)) =
-    ((intaa : IntInf.int) = inta) andalso eq_num numa num
-  | eq_fm (Not fma) (Not fm) = eq_fm fma fm
-  | eq_fm (And (fm1a, fm2a)) (And (fm1, fm2)) =
-    eq_fm fm1a fm1 andalso eq_fm fm2a fm2
-  | eq_fm (Or (fm1a, fm2a)) (Or (fm1, fm2)) =
-    eq_fm fm1a fm1 andalso eq_fm fm2a fm2
-  | eq_fm (Imp (fm1a, fm2a)) (Imp (fm1, fm2)) =
-    eq_fm fm1a fm1 andalso eq_fm fm2a fm2
-  | eq_fm (Iff (fm1a, fm2a)) (Iff (fm1, fm2)) =
-    eq_fm fm1a fm1 andalso eq_fm fm2a fm2
-  | eq_fm (E fma) (E fm) = eq_fm fma fm
-  | eq_fm (A fma) (A fm) = eq_fm fma fm
-  | eq_fm (Closed nata) (Closed nat) = ((nata : IntInf.int) = nat)
-  | eq_fm (NClosed nata) (NClosed nat) = ((nata : IntInf.int) = nat)
-  | eq_fm T F = false
-  | eq_fm F T = false
-  | eq_fm T (Lt num) = false
-  | eq_fm (Lt num) T = false
-  | eq_fm T (Le num) = false
-  | eq_fm (Le num) T = false
-  | eq_fm T (Gt num) = false
-  | eq_fm (Gt num) T = false
-  | eq_fm T (Ge num) = false
-  | eq_fm (Ge num) T = false
-  | eq_fm T (Eq num) = false
-  | eq_fm (Eq num) T = false
-  | eq_fm T (NEq num) = false
-  | eq_fm (NEq num) T = false
-  | eq_fm T (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, num)) T = false
-  | eq_fm T (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, num)) T = false
-  | eq_fm T (Not fm) = false
-  | eq_fm (Not fm) T = false
-  | eq_fm T (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) T = false
-  | eq_fm T (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) T = false
-  | eq_fm T (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) T = false
-  | eq_fm T (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) T = false
-  | eq_fm T (E fm) = false
-  | eq_fm (E fm) T = false
-  | eq_fm T (A fm) = false
-  | eq_fm (A fm) T = false
-  | eq_fm T (Closed nat) = false
-  | eq_fm (Closed nat) T = false
-  | eq_fm T (NClosed nat) = false
-  | eq_fm (NClosed nat) T = false
-  | eq_fm F (Lt num) = false
-  | eq_fm (Lt num) F = false
-  | eq_fm F (Le num) = false
-  | eq_fm (Le num) F = false
-  | eq_fm F (Gt num) = false
-  | eq_fm (Gt num) F = false
-  | eq_fm F (Ge num) = false
-  | eq_fm (Ge num) F = false
-  | eq_fm F (Eq num) = false
-  | eq_fm (Eq num) F = false
-  | eq_fm F (NEq num) = false
-  | eq_fm (NEq num) F = false
-  | eq_fm F (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, num)) F = false
-  | eq_fm F (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, num)) F = false
-  | eq_fm F (Not fm) = false
-  | eq_fm (Not fm) F = false
-  | eq_fm F (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) F = false
-  | eq_fm F (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) F = false
-  | eq_fm F (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) F = false
-  | eq_fm F (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) F = false
-  | eq_fm F (E fm) = false
-  | eq_fm (E fm) F = false
-  | eq_fm F (A fm) = false
-  | eq_fm (A fm) F = false
-  | eq_fm F (Closed nat) = false
-  | eq_fm (Closed nat) F = false
-  | eq_fm F (NClosed nat) = false
-  | eq_fm (NClosed nat) F = false
-  | eq_fm (Lt numa) (Le num) = false
-  | eq_fm (Le numa) (Lt num) = false
-  | eq_fm (Lt numa) (Gt num) = false
-  | eq_fm (Gt numa) (Lt num) = false
-  | eq_fm (Lt numa) (Ge num) = false
-  | eq_fm (Ge numa) (Lt num) = false
-  | eq_fm (Lt numa) (Eq num) = false
-  | eq_fm (Eq numa) (Lt num) = false
-  | eq_fm (Lt numa) (NEq num) = false
-  | eq_fm (NEq numa) (Lt num) = false
-  | eq_fm (Lt numa) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, numa)) (Lt num) = false
-  | eq_fm (Lt numa) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, numa)) (Lt num) = false
-  | eq_fm (Lt num) (Not fm) = false
-  | eq_fm (Not fm) (Lt num) = false
-  | eq_fm (Lt num) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) (Lt num) = false
-  | eq_fm (Lt num) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) (Lt num) = false
-  | eq_fm (Lt num) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) (Lt num) = false
-  | eq_fm (Lt num) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) (Lt num) = false
-  | eq_fm (Lt num) (E fm) = false
-  | eq_fm (E fm) (Lt num) = false
-  | eq_fm (Lt num) (A fm) = false
-  | eq_fm (A fm) (Lt num) = false
-  | eq_fm (Lt num) (Closed nat) = false
-  | eq_fm (Closed nat) (Lt num) = false
-  | eq_fm (Lt num) (NClosed nat) = false
-  | eq_fm (NClosed nat) (Lt num) = false
-  | eq_fm (Le numa) (Gt num) = false
-  | eq_fm (Gt numa) (Le num) = false
-  | eq_fm (Le numa) (Ge num) = false
-  | eq_fm (Ge numa) (Le num) = false
-  | eq_fm (Le numa) (Eq num) = false
-  | eq_fm (Eq numa) (Le num) = false
-  | eq_fm (Le numa) (NEq num) = false
-  | eq_fm (NEq numa) (Le num) = false
-  | eq_fm (Le numa) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, numa)) (Le num) = false
-  | eq_fm (Le numa) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, numa)) (Le num) = false
-  | eq_fm (Le num) (Not fm) = false
-  | eq_fm (Not fm) (Le num) = false
-  | eq_fm (Le num) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) (Le num) = false
-  | eq_fm (Le num) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) (Le num) = false
-  | eq_fm (Le num) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) (Le num) = false
-  | eq_fm (Le num) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) (Le num) = false
-  | eq_fm (Le num) (E fm) = false
-  | eq_fm (E fm) (Le num) = false
-  | eq_fm (Le num) (A fm) = false
-  | eq_fm (A fm) (Le num) = false
-  | eq_fm (Le num) (Closed nat) = false
-  | eq_fm (Closed nat) (Le num) = false
-  | eq_fm (Le num) (NClosed nat) = false
-  | eq_fm (NClosed nat) (Le num) = false
-  | eq_fm (Gt numa) (Ge num) = false
-  | eq_fm (Ge numa) (Gt num) = false
-  | eq_fm (Gt numa) (Eq num) = false
-  | eq_fm (Eq numa) (Gt num) = false
-  | eq_fm (Gt numa) (NEq num) = false
-  | eq_fm (NEq numa) (Gt num) = false
-  | eq_fm (Gt numa) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, numa)) (Gt num) = false
-  | eq_fm (Gt numa) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, numa)) (Gt num) = false
-  | eq_fm (Gt num) (Not fm) = false
-  | eq_fm (Not fm) (Gt num) = false
-  | eq_fm (Gt num) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) (Gt num) = false
-  | eq_fm (Gt num) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) (Gt num) = false
-  | eq_fm (Gt num) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) (Gt num) = false
-  | eq_fm (Gt num) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) (Gt num) = false
-  | eq_fm (Gt num) (E fm) = false
-  | eq_fm (E fm) (Gt num) = false
-  | eq_fm (Gt num) (A fm) = false
-  | eq_fm (A fm) (Gt num) = false
-  | eq_fm (Gt num) (Closed nat) = false
-  | eq_fm (Closed nat) (Gt num) = false
-  | eq_fm (Gt num) (NClosed nat) = false
-  | eq_fm (NClosed nat) (Gt num) = false
-  | eq_fm (Ge numa) (Eq num) = false
-  | eq_fm (Eq numa) (Ge num) = false
-  | eq_fm (Ge numa) (NEq num) = false
-  | eq_fm (NEq numa) (Ge num) = false
-  | eq_fm (Ge numa) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, numa)) (Ge num) = false
-  | eq_fm (Ge numa) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, numa)) (Ge num) = false
-  | eq_fm (Ge num) (Not fm) = false
-  | eq_fm (Not fm) (Ge num) = false
-  | eq_fm (Ge num) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) (Ge num) = false
-  | eq_fm (Ge num) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) (Ge num) = false
-  | eq_fm (Ge num) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) (Ge num) = false
-  | eq_fm (Ge num) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) (Ge num) = false
-  | eq_fm (Ge num) (E fm) = false
-  | eq_fm (E fm) (Ge num) = false
-  | eq_fm (Ge num) (A fm) = false
-  | eq_fm (A fm) (Ge num) = false
-  | eq_fm (Ge num) (Closed nat) = false
-  | eq_fm (Closed nat) (Ge num) = false
-  | eq_fm (Ge num) (NClosed nat) = false
-  | eq_fm (NClosed nat) (Ge num) = false
-  | eq_fm (Eq numa) (NEq num) = false
-  | eq_fm (NEq numa) (Eq num) = false
-  | eq_fm (Eq numa) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, numa)) (Eq num) = false
-  | eq_fm (Eq numa) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, numa)) (Eq num) = false
-  | eq_fm (Eq num) (Not fm) = false
-  | eq_fm (Not fm) (Eq num) = false
-  | eq_fm (Eq num) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) (Eq num) = false
-  | eq_fm (Eq num) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) (Eq num) = false
-  | eq_fm (Eq num) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) (Eq num) = false
-  | eq_fm (Eq num) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) (Eq num) = false
-  | eq_fm (Eq num) (E fm) = false
-  | eq_fm (E fm) (Eq num) = false
-  | eq_fm (Eq num) (A fm) = false
-  | eq_fm (A fm) (Eq num) = false
-  | eq_fm (Eq num) (Closed nat) = false
-  | eq_fm (Closed nat) (Eq num) = false
-  | eq_fm (Eq num) (NClosed nat) = false
-  | eq_fm (NClosed nat) (Eq num) = false
-  | eq_fm (NEq numa) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, numa)) (NEq num) = false
-  | eq_fm (NEq numa) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, numa)) (NEq num) = false
-  | eq_fm (NEq num) (Not fm) = false
-  | eq_fm (Not fm) (NEq num) = false
-  | eq_fm (NEq num) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) (NEq num) = false
-  | eq_fm (NEq num) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) (NEq num) = false
-  | eq_fm (NEq num) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) (NEq num) = false
-  | eq_fm (NEq num) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) (NEq num) = false
-  | eq_fm (NEq num) (E fm) = false
-  | eq_fm (E fm) (NEq num) = false
-  | eq_fm (NEq num) (A fm) = false
-  | eq_fm (A fm) (NEq num) = false
-  | eq_fm (NEq num) (Closed nat) = false
-  | eq_fm (Closed nat) (NEq num) = false
-  | eq_fm (NEq num) (NClosed nat) = false
-  | eq_fm (NClosed nat) (NEq num) = false
-  | eq_fm (Dvd (intaa, numa)) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (intaa, numa)) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, num)) (Not fm) = false
-  | eq_fm (Not fm) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, num)) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, num)) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, num)) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, num)) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, num)) (E fm) = false
-  | eq_fm (E fm) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, num)) (A fm) = false
-  | eq_fm (A fm) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, num)) (Closed nat) = false
-  | eq_fm (Closed nat) (Dvd (inta, num)) = false
-  | eq_fm (Dvd (inta, num)) (NClosed nat) = false
-  | eq_fm (NClosed nat) (Dvd (inta, num)) = false
-  | eq_fm (NDvd (inta, num)) (Not fm) = false
-  | eq_fm (Not fm) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, num)) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, num)) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, num)) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, num)) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, num)) (E fm) = false
-  | eq_fm (E fm) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, num)) (A fm) = false
-  | eq_fm (A fm) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, num)) (Closed nat) = false
-  | eq_fm (Closed nat) (NDvd (inta, num)) = false
-  | eq_fm (NDvd (inta, num)) (NClosed nat) = false
-  | eq_fm (NClosed nat) (NDvd (inta, num)) = false
-  | eq_fm (Not fm) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) (Not fm) = false
-  | eq_fm (Not fm) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) (Not fm) = false
-  | eq_fm (Not fm) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) (Not fm) = false
-  | eq_fm (Not fm) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) (Not fm) = false
-  | eq_fm (Not fma) (E fm) = false
-  | eq_fm (E fma) (Not fm) = false
-  | eq_fm (Not fma) (A fm) = false
-  | eq_fm (A fma) (Not fm) = false
-  | eq_fm (Not fm) (Closed nat) = false
-  | eq_fm (Closed nat) (Not fm) = false
-  | eq_fm (Not fm) (NClosed nat) = false
-  | eq_fm (NClosed nat) (Not fm) = false
-  | eq_fm (And (fm1a, fm2a)) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1a, fm2a)) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1a, fm2a)) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1a, fm2a)) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1a, fm2a)) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1a, fm2a)) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) (E fm) = false
-  | eq_fm (E fm) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) (A fm) = false
-  | eq_fm (A fm) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) (Closed nat) = false
-  | eq_fm (Closed nat) (And (fm1, fm2)) = false
-  | eq_fm (And (fm1, fm2)) (NClosed nat) = false
-  | eq_fm (NClosed nat) (And (fm1, fm2)) = false
-  | eq_fm (Or (fm1a, fm2a)) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1a, fm2a)) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1a, fm2a)) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1a, fm2a)) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) (E fm) = false
-  | eq_fm (E fm) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) (A fm) = false
-  | eq_fm (A fm) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) (Closed nat) = false
-  | eq_fm (Closed nat) (Or (fm1, fm2)) = false
-  | eq_fm (Or (fm1, fm2)) (NClosed nat) = false
-  | eq_fm (NClosed nat) (Or (fm1, fm2)) = false
-  | eq_fm (Imp (fm1a, fm2a)) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1a, fm2a)) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) (E fm) = false
-  | eq_fm (E fm) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) (A fm) = false
-  | eq_fm (A fm) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) (Closed nat) = false
-  | eq_fm (Closed nat) (Imp (fm1, fm2)) = false
-  | eq_fm (Imp (fm1, fm2)) (NClosed nat) = false
-  | eq_fm (NClosed nat) (Imp (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) (E fm) = false
-  | eq_fm (E fm) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) (A fm) = false
-  | eq_fm (A fm) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) (Closed nat) = false
-  | eq_fm (Closed nat) (Iff (fm1, fm2)) = false
-  | eq_fm (Iff (fm1, fm2)) (NClosed nat) = false
-  | eq_fm (NClosed nat) (Iff (fm1, fm2)) = false
-  | eq_fm (E fma) (A fm) = false
-  | eq_fm (A fma) (E fm) = false
-  | eq_fm (E fm) (Closed nat) = false
-  | eq_fm (Closed nat) (E fm) = false
-  | eq_fm (E fm) (NClosed nat) = false
-  | eq_fm (NClosed nat) (E fm) = false
-  | eq_fm (A fm) (Closed nat) = false
-  | eq_fm (Closed nat) (A fm) = false
-  | eq_fm (A fm) (NClosed nat) = false
-  | eq_fm (NClosed nat) (A fm) = false
-  | eq_fm (Closed nata) (NClosed nat) = false
-  | eq_fm (NClosed nata) (Closed nat) = 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 _ => Or (f p, q)
-                  | Le _ => Or (f p, q) | Gt _ => Or (f p, q)
-                  | Ge _ => Or (f p, q) | Eq _ => Or (f p, q)
-                  | NEq _ => Or (f p, q) | Dvd (_, _) => Or (f p, q)
-                  | NDvd (_, _) => Or (f p, q) | Not _ => Or (f p, q)
-                  | And (_, _) => Or (f p, q) | Or (_, _) => Or (f p, q)
-                  | Imp (_, _) => Or (f p, q) | Iff (_, _) => Or (f p, q)
-                  | E _ => Or (f p, q) | A _ => Or (f p, q)
-                  | Closed _ => Or (f p, q) | NClosed _ => Or (f p, q))));
-
-fun foldr f [] a = a
-  | foldr f (x :: xs) a = f x (foldr f xs a);
-
-fun evaldjf f ps = foldr (djf f) ps F;
-
-fun dj f p = evaldjf f (disjuncts p);
-
-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 minus_nat n m = IntInf.max (0, (IntInf.- (n, m)));
-
-fun decrnum (Bound n) = Bound (minus_nat n (1 : IntInf.int))
-  | 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 (Cn (n, i, a)) = Cn (minus_nat n (1 : IntInf.int), i, decrnum a)
-  | decrnum (C u) = C u;
-
-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 concat_map f [] = []
-  | concat_map f (x :: xs) = append (f x) (concat_map f xs);
-
-fun numsubst0 t (C c) = C c
-  | numsubst0 t (Bound n) =
-    (if ((n : IntInf.int) = (0 : IntInf.int)) then t else Bound n)
-  | 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)
-  | numsubst0 t (Cn (v, i, a)) =
-    (if ((v : IntInf.int) = (0 : IntInf.int))
-      then Add (Mul (i, t), numsubst0 t a)
-      else Cn (suc (minus_nat v (1 : IntInf.int)), 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 T = T
-  | minusinf F = F
-  | minusinf (Lt (C bo)) = Lt (C bo)
-  | minusinf (Lt (Bound bp)) = Lt (Bound bp)
-  | minusinf (Lt (Neg bt)) = Lt (Neg bt)
-  | minusinf (Lt (Add (bu, bv))) = Lt (Add (bu, bv))
-  | minusinf (Lt (Sub (bw, bx))) = Lt (Sub (bw, bx))
-  | minusinf (Lt (Mul (by, bz))) = Lt (Mul (by, bz))
-  | minusinf (Le (C co)) = Le (C co)
-  | minusinf (Le (Bound cp)) = Le (Bound cp)
-  | minusinf (Le (Neg ct)) = Le (Neg ct)
-  | minusinf (Le (Add (cu, cv))) = Le (Add (cu, cv))
-  | minusinf (Le (Sub (cw, cx))) = Le (Sub (cw, cx))
-  | minusinf (Le (Mul (cy, cz))) = Le (Mul (cy, cz))
-  | minusinf (Gt (C doa)) = Gt (C doa)
-  | minusinf (Gt (Bound dp)) = Gt (Bound dp)
-  | minusinf (Gt (Neg dt)) = Gt (Neg dt)
-  | minusinf (Gt (Add (du, dv))) = Gt (Add (du, dv))
-  | minusinf (Gt (Sub (dw, dx))) = Gt (Sub (dw, dx))
-  | minusinf (Gt (Mul (dy, dz))) = Gt (Mul (dy, dz))
-  | minusinf (Ge (C eo)) = Ge (C eo)
-  | minusinf (Ge (Bound ep)) = Ge (Bound ep)
-  | minusinf (Ge (Neg et)) = Ge (Neg et)
-  | minusinf (Ge (Add (eu, ev))) = Ge (Add (eu, ev))
-  | minusinf (Ge (Sub (ew, ex))) = Ge (Sub (ew, ex))
-  | minusinf (Ge (Mul (ey, ez))) = Ge (Mul (ey, ez))
-  | minusinf (Eq (C fo)) = Eq (C fo)
-  | minusinf (Eq (Bound fp)) = Eq (Bound fp)
-  | minusinf (Eq (Neg ft)) = Eq (Neg ft)
-  | minusinf (Eq (Add (fu, fv))) = Eq (Add (fu, fv))
-  | minusinf (Eq (Sub (fw, fx))) = Eq (Sub (fw, fx))
-  | minusinf (Eq (Mul (fy, fz))) = Eq (Mul (fy, fz))
-  | minusinf (NEq (C go)) = NEq (C go)
-  | minusinf (NEq (Bound gp)) = NEq (Bound gp)
-  | minusinf (NEq (Neg gt)) = NEq (Neg gt)
-  | minusinf (NEq (Add (gu, gv))) = NEq (Add (gu, gv))
-  | minusinf (NEq (Sub (gw, gx))) = NEq (Sub (gw, gx))
-  | minusinf (NEq (Mul (gy, gz))) = NEq (Mul (gy, gz))
-  | 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
-  | minusinf (Lt (Cn (cm, c, e))) =
-    (if ((cm : IntInf.int) = (0 : IntInf.int)) then T
-      else Lt (Cn (suc (minus_nat cm (1 : IntInf.int)), c, e)))
-  | minusinf (Le (Cn (dm, c, e))) =
-    (if ((dm : IntInf.int) = (0 : IntInf.int)) then T
-      else Le (Cn (suc (minus_nat dm (1 : IntInf.int)), c, e)))
-  | minusinf (Gt (Cn (em, c, e))) =
-    (if ((em : IntInf.int) = (0 : IntInf.int)) then F
-      else Gt (Cn (suc (minus_nat em (1 : IntInf.int)), c, e)))
-  | minusinf (Ge (Cn (fm, c, e))) =
-    (if ((fm : IntInf.int) = (0 : IntInf.int)) then F
-      else Ge (Cn (suc (minus_nat fm (1 : IntInf.int)), c, e)))
-  | minusinf (Eq (Cn (gm, c, e))) =
-    (if ((gm : IntInf.int) = (0 : IntInf.int)) then F
-      else Eq (Cn (suc (minus_nat gm (1 : IntInf.int)), c, e)))
-  | minusinf (NEq (Cn (hm, c, e))) =
-    (if ((hm : IntInf.int) = (0 : IntInf.int)) then T
-      else NEq (Cn (suc (minus_nat hm (1 : IntInf.int)), c, e)));
-
-val eq_int = {eq = (fn a => fn b => ((a : IntInf.int) = b))} : IntInf.int eq;
-
-val zero_int : IntInf.int = (0 : IntInf.int);
-
-type 'a zero = {zero : 'a};
-val zero = #zero : 'a zero -> 'a;
-
-val zero_inta = {zero = zero_int} : IntInf.int zero;
-
-type 'a times = {times : 'a -> 'a -> 'a};
-val times = #times : 'a times -> 'a -> 'a -> 'a;
-
-type 'a no_zero_divisors =
-  {times_no_zero_divisors : 'a times, zero_no_zero_divisors : 'a zero};
-val times_no_zero_divisors = #times_no_zero_divisors :
-  'a no_zero_divisors -> 'a times;
-val zero_no_zero_divisors = #zero_no_zero_divisors :
-  'a no_zero_divisors -> 'a zero;
-
-val times_int = {times = (fn a => fn b => IntInf.* (a, b))} : IntInf.int times;
-
-val no_zero_divisors_int =
-  {times_no_zero_divisors = times_int, zero_no_zero_divisors = zero_inta} :
-  IntInf.int no_zero_divisors;
-
-type 'a one = {one : 'a};
-val one = #one : 'a one -> 'a;
-
-type 'a zero_neq_one = {one_zero_neq_one : 'a one, zero_zero_neq_one : 'a zero};
-val one_zero_neq_one = #one_zero_neq_one : 'a zero_neq_one -> 'a one;
-val zero_zero_neq_one = #zero_zero_neq_one : 'a zero_neq_one -> 'a zero;
-
-type 'a semigroup_mult = {times_semigroup_mult : 'a times};
-val times_semigroup_mult = #times_semigroup_mult :
-  'a semigroup_mult -> 'a times;
-
-type 'a plus = {plus : 'a -> 'a -> 'a};
-val plus = #plus : 'a plus -> 'a -> 'a -> 'a;
-
-type 'a semigroup_add = {plus_semigroup_add : 'a plus};
-val plus_semigroup_add = #plus_semigroup_add : 'a semigroup_add -> 'a plus;
-
-type 'a ab_semigroup_add = {semigroup_add_ab_semigroup_add : 'a semigroup_add};
-val semigroup_add_ab_semigroup_add = #semigroup_add_ab_semigroup_add :
-  'a ab_semigroup_add -> 'a semigroup_add;
-
-type 'a semiring =
-  {ab_semigroup_add_semiring : 'a ab_semigroup_add,
-    semigroup_mult_semiring : 'a semigroup_mult};
-val ab_semigroup_add_semiring = #ab_semigroup_add_semiring :
-  'a semiring -> 'a ab_semigroup_add;
-val semigroup_mult_semiring = #semigroup_mult_semiring :
-  'a semiring -> 'a semigroup_mult;
-
-type 'a mult_zero = {times_mult_zero : 'a times, zero_mult_zero : 'a zero};
-val times_mult_zero = #times_mult_zero : 'a mult_zero -> 'a times;
-val zero_mult_zero = #zero_mult_zero : 'a mult_zero -> 'a zero;
-
-type 'a monoid_add =
-  {semigroup_add_monoid_add : 'a semigroup_add, zero_monoid_add : 'a zero};
-val semigroup_add_monoid_add = #semigroup_add_monoid_add :
-  'a monoid_add -> 'a semigroup_add;
-val zero_monoid_add = #zero_monoid_add : 'a monoid_add -> 'a zero;
-
-type 'a comm_monoid_add =
-  {ab_semigroup_add_comm_monoid_add : 'a ab_semigroup_add,
-    monoid_add_comm_monoid_add : 'a monoid_add};
-val ab_semigroup_add_comm_monoid_add = #ab_semigroup_add_comm_monoid_add :
-  'a comm_monoid_add -> 'a ab_semigroup_add;
-val monoid_add_comm_monoid_add = #monoid_add_comm_monoid_add :
-  'a comm_monoid_add -> 'a monoid_add;
-
-type 'a semiring_0 =
-  {comm_monoid_add_semiring_0 : 'a comm_monoid_add,
-    mult_zero_semiring_0 : 'a mult_zero, semiring_semiring_0 : 'a semiring};
-val comm_monoid_add_semiring_0 = #comm_monoid_add_semiring_0 :
-  'a semiring_0 -> 'a comm_monoid_add;
-val mult_zero_semiring_0 = #mult_zero_semiring_0 :
-  'a semiring_0 -> 'a mult_zero;
-val semiring_semiring_0 = #semiring_semiring_0 : 'a semiring_0 -> 'a semiring;
-
-type 'a power = {one_power : 'a one, times_power : 'a times};
-val one_power = #one_power : 'a power -> 'a one;
-val times_power = #times_power : 'a power -> 'a times;
-
-type 'a monoid_mult =
-  {semigroup_mult_monoid_mult : 'a semigroup_mult,
-    power_monoid_mult : 'a power};
-val semigroup_mult_monoid_mult = #semigroup_mult_monoid_mult :
-  'a monoid_mult -> 'a semigroup_mult;
-val power_monoid_mult = #power_monoid_mult : 'a monoid_mult -> 'a power;
-
-type 'a semiring_1 =
-  {monoid_mult_semiring_1 : 'a monoid_mult,
-    semiring_0_semiring_1 : 'a semiring_0,
-    zero_neq_one_semiring_1 : 'a zero_neq_one};
-val monoid_mult_semiring_1 = #monoid_mult_semiring_1 :
-  'a semiring_1 -> 'a monoid_mult;
-val semiring_0_semiring_1 = #semiring_0_semiring_1 :
-  'a semiring_1 -> 'a semiring_0;
-val zero_neq_one_semiring_1 = #zero_neq_one_semiring_1 :
-  'a semiring_1 -> 'a zero_neq_one;
-
-type 'a cancel_semigroup_add =
-  {semigroup_add_cancel_semigroup_add : 'a semigroup_add};
-val semigroup_add_cancel_semigroup_add = #semigroup_add_cancel_semigroup_add :
-  'a cancel_semigroup_add -> 'a semigroup_add;
-
-type 'a cancel_ab_semigroup_add =
-  {ab_semigroup_add_cancel_ab_semigroup_add : 'a ab_semigroup_add,
-    cancel_semigroup_add_cancel_ab_semigroup_add : 'a cancel_semigroup_add};
-val ab_semigroup_add_cancel_ab_semigroup_add =
-  #ab_semigroup_add_cancel_ab_semigroup_add :
-  'a cancel_ab_semigroup_add -> 'a ab_semigroup_add;
-val cancel_semigroup_add_cancel_ab_semigroup_add =
-  #cancel_semigroup_add_cancel_ab_semigroup_add :
-  'a cancel_ab_semigroup_add -> 'a cancel_semigroup_add;
-
-type 'a cancel_comm_monoid_add =
-  {cancel_ab_semigroup_add_cancel_comm_monoid_add : 'a cancel_ab_semigroup_add,
-    comm_monoid_add_cancel_comm_monoid_add : 'a comm_monoid_add};
-val cancel_ab_semigroup_add_cancel_comm_monoid_add =
-  #cancel_ab_semigroup_add_cancel_comm_monoid_add :
-  'a cancel_comm_monoid_add -> 'a cancel_ab_semigroup_add;
-val comm_monoid_add_cancel_comm_monoid_add =
-  #comm_monoid_add_cancel_comm_monoid_add :
-  'a cancel_comm_monoid_add -> 'a comm_monoid_add;
-
-type 'a semiring_0_cancel =
-  {cancel_comm_monoid_add_semiring_0_cancel : 'a cancel_comm_monoid_add,
-    semiring_0_semiring_0_cancel : 'a semiring_0};
-val cancel_comm_monoid_add_semiring_0_cancel =
-  #cancel_comm_monoid_add_semiring_0_cancel :
-  'a semiring_0_cancel -> 'a cancel_comm_monoid_add;
-val semiring_0_semiring_0_cancel = #semiring_0_semiring_0_cancel :
-  'a semiring_0_cancel -> 'a semiring_0;
-
-type 'a semiring_1_cancel =
-  {semiring_0_cancel_semiring_1_cancel : 'a semiring_0_cancel,
-    semiring_1_semiring_1_cancel : 'a semiring_1};
-val semiring_0_cancel_semiring_1_cancel = #semiring_0_cancel_semiring_1_cancel :
-  'a semiring_1_cancel -> 'a semiring_0_cancel;
-val semiring_1_semiring_1_cancel = #semiring_1_semiring_1_cancel :
-  'a semiring_1_cancel -> 'a semiring_1;
-
-type 'a dvd = {times_dvd : 'a times};
-val times_dvd = #times_dvd : 'a dvd -> 'a times;
-
-type 'a ab_semigroup_mult =
-  {semigroup_mult_ab_semigroup_mult : 'a semigroup_mult};
-val semigroup_mult_ab_semigroup_mult = #semigroup_mult_ab_semigroup_mult :
-  'a ab_semigroup_mult -> 'a semigroup_mult;
-
-type 'a comm_semiring =
-  {ab_semigroup_mult_comm_semiring : 'a ab_semigroup_mult,
-    semiring_comm_semiring : 'a semiring};
-val ab_semigroup_mult_comm_semiring = #ab_semigroup_mult_comm_semiring :
-  'a comm_semiring -> 'a ab_semigroup_mult;
-val semiring_comm_semiring = #semiring_comm_semiring :
-  'a comm_semiring -> 'a semiring;
-
-type 'a comm_semiring_0 =
-  {comm_semiring_comm_semiring_0 : 'a comm_semiring,
-    semiring_0_comm_semiring_0 : 'a semiring_0};
-val comm_semiring_comm_semiring_0 = #comm_semiring_comm_semiring_0 :
-  'a comm_semiring_0 -> 'a comm_semiring;
-val semiring_0_comm_semiring_0 = #semiring_0_comm_semiring_0 :
-  'a comm_semiring_0 -> 'a semiring_0;
-
-type 'a comm_monoid_mult =
-  {ab_semigroup_mult_comm_monoid_mult : 'a ab_semigroup_mult,
-    monoid_mult_comm_monoid_mult : 'a monoid_mult};
-val ab_semigroup_mult_comm_monoid_mult = #ab_semigroup_mult_comm_monoid_mult :
-  'a comm_monoid_mult -> 'a ab_semigroup_mult;
-val monoid_mult_comm_monoid_mult = #monoid_mult_comm_monoid_mult :
-  'a comm_monoid_mult -> 'a monoid_mult;
-
-type 'a comm_semiring_1 =
-  {comm_monoid_mult_comm_semiring_1 : 'a comm_monoid_mult,
-    comm_semiring_0_comm_semiring_1 : 'a comm_semiring_0,
-    dvd_comm_semiring_1 : 'a dvd, semiring_1_comm_semiring_1 : 'a semiring_1};
-val comm_monoid_mult_comm_semiring_1 = #comm_monoid_mult_comm_semiring_1 :
-  'a comm_semiring_1 -> 'a comm_monoid_mult;
-val comm_semiring_0_comm_semiring_1 = #comm_semiring_0_comm_semiring_1 :
-  'a comm_semiring_1 -> 'a comm_semiring_0;
-val dvd_comm_semiring_1 = #dvd_comm_semiring_1 : 'a comm_semiring_1 -> 'a dvd;
-val semiring_1_comm_semiring_1 = #semiring_1_comm_semiring_1 :
-  'a comm_semiring_1 -> 'a semiring_1;
-
-type 'a comm_semiring_0_cancel =
-  {comm_semiring_0_comm_semiring_0_cancel : 'a comm_semiring_0,
-    semiring_0_cancel_comm_semiring_0_cancel : 'a semiring_0_cancel};
-val comm_semiring_0_comm_semiring_0_cancel =
-  #comm_semiring_0_comm_semiring_0_cancel :
-  'a comm_semiring_0_cancel -> 'a comm_semiring_0;
-val semiring_0_cancel_comm_semiring_0_cancel =
-  #semiring_0_cancel_comm_semiring_0_cancel :
-  'a comm_semiring_0_cancel -> 'a semiring_0_cancel;
-
-type 'a comm_semiring_1_cancel =
-  {comm_semiring_0_cancel_comm_semiring_1_cancel : 'a comm_semiring_0_cancel,
-    comm_semiring_1_comm_semiring_1_cancel : 'a comm_semiring_1,
-    semiring_1_cancel_comm_semiring_1_cancel : 'a semiring_1_cancel};
-val comm_semiring_0_cancel_comm_semiring_1_cancel =
-  #comm_semiring_0_cancel_comm_semiring_1_cancel :
-  'a comm_semiring_1_cancel -> 'a comm_semiring_0_cancel;
-val comm_semiring_1_comm_semiring_1_cancel =
-  #comm_semiring_1_comm_semiring_1_cancel :
-  'a comm_semiring_1_cancel -> 'a comm_semiring_1;
-val semiring_1_cancel_comm_semiring_1_cancel =
-  #semiring_1_cancel_comm_semiring_1_cancel :
-  'a comm_semiring_1_cancel -> 'a semiring_1_cancel;
-
-type 'a diva = {dvd_div : 'a dvd, diva : 'a -> 'a -> 'a, moda : 'a -> 'a -> 'a};
-val dvd_div = #dvd_div : 'a diva -> 'a dvd;
-val diva = #diva : 'a diva -> 'a -> 'a -> 'a;
-val moda = #moda : 'a diva -> 'a -> 'a -> 'a;
-
-type 'a semiring_div =
-  {div_semiring_div : 'a diva,
-    comm_semiring_1_cancel_semiring_div : 'a comm_semiring_1_cancel,
-    no_zero_divisors_semiring_div : 'a no_zero_divisors};
-val div_semiring_div = #div_semiring_div : 'a semiring_div -> 'a diva;
-val comm_semiring_1_cancel_semiring_div = #comm_semiring_1_cancel_semiring_div :
-  'a semiring_div -> 'a comm_semiring_1_cancel;
-val no_zero_divisors_semiring_div = #no_zero_divisors_semiring_div :
-  'a semiring_div -> 'a no_zero_divisors;
-
-val one_int : IntInf.int = (1 : IntInf.int);
-
-val one_inta = {one = one_int} : IntInf.int one;
-
-val zero_neq_one_int =
-  {one_zero_neq_one = one_inta, zero_zero_neq_one = zero_inta} :
-  IntInf.int zero_neq_one;
-
-val semigroup_mult_int = {times_semigroup_mult = times_int} :
-  IntInf.int semigroup_mult;
-
-val plus_int = {plus = (fn a => fn b => IntInf.+ (a, b))} : IntInf.int plus;
-
-val semigroup_add_int = {plus_semigroup_add = plus_int} :
-  IntInf.int semigroup_add;
-
-val ab_semigroup_add_int = {semigroup_add_ab_semigroup_add = semigroup_add_int}
-  : IntInf.int ab_semigroup_add;
-
-val semiring_int =
-  {ab_semigroup_add_semiring = ab_semigroup_add_int,
-    semigroup_mult_semiring = semigroup_mult_int}
-  : IntInf.int semiring;
-
-val mult_zero_int = {times_mult_zero = times_int, zero_mult_zero = zero_inta} :
-  IntInf.int mult_zero;
-
-val monoid_add_int =
-  {semigroup_add_monoid_add = semigroup_add_int, zero_monoid_add = zero_inta} :
-  IntInf.int monoid_add;
-
-val comm_monoid_add_int =
-  {ab_semigroup_add_comm_monoid_add = ab_semigroup_add_int,
-    monoid_add_comm_monoid_add = monoid_add_int}
-  : IntInf.int comm_monoid_add;
-
-val semiring_0_int =
-  {comm_monoid_add_semiring_0 = comm_monoid_add_int,
-    mult_zero_semiring_0 = mult_zero_int, semiring_semiring_0 = semiring_int}
-  : IntInf.int semiring_0;
-
-val power_int = {one_power = one_inta, times_power = times_int} :
-  IntInf.int power;
-
-val monoid_mult_int =
-  {semigroup_mult_monoid_mult = semigroup_mult_int,
-    power_monoid_mult = power_int}
-  : IntInf.int monoid_mult;
-
-val semiring_1_int =
-  {monoid_mult_semiring_1 = monoid_mult_int,
-    semiring_0_semiring_1 = semiring_0_int,
-    zero_neq_one_semiring_1 = zero_neq_one_int}
-  : IntInf.int semiring_1;
-
-val cancel_semigroup_add_int =
-  {semigroup_add_cancel_semigroup_add = semigroup_add_int} :
-  IntInf.int cancel_semigroup_add;
-
-val cancel_ab_semigroup_add_int =
-  {ab_semigroup_add_cancel_ab_semigroup_add = ab_semigroup_add_int,
-    cancel_semigroup_add_cancel_ab_semigroup_add = cancel_semigroup_add_int}
-  : IntInf.int cancel_ab_semigroup_add;
-
-val cancel_comm_monoid_add_int =
-  {cancel_ab_semigroup_add_cancel_comm_monoid_add = cancel_ab_semigroup_add_int,
-    comm_monoid_add_cancel_comm_monoid_add = comm_monoid_add_int}
-  : IntInf.int cancel_comm_monoid_add;
-
-val semiring_0_cancel_int =
-  {cancel_comm_monoid_add_semiring_0_cancel = cancel_comm_monoid_add_int,
-    semiring_0_semiring_0_cancel = semiring_0_int}
-  : IntInf.int semiring_0_cancel;
-
-val semiring_1_cancel_int =
-  {semiring_0_cancel_semiring_1_cancel = semiring_0_cancel_int,
-    semiring_1_semiring_1_cancel = semiring_1_int}
-  : IntInf.int semiring_1_cancel;
-
-val dvd_int = {times_dvd = times_int} : IntInf.int dvd;
-
-val ab_semigroup_mult_int =
-  {semigroup_mult_ab_semigroup_mult = semigroup_mult_int} :
-  IntInf.int ab_semigroup_mult;
-
-val comm_semiring_int =
-  {ab_semigroup_mult_comm_semiring = ab_semigroup_mult_int,
-    semiring_comm_semiring = semiring_int}
-  : IntInf.int comm_semiring;
-
-val comm_semiring_0_int =
-  {comm_semiring_comm_semiring_0 = comm_semiring_int,
-    semiring_0_comm_semiring_0 = semiring_0_int}
-  : IntInf.int comm_semiring_0;
-
-val comm_monoid_mult_int =
-  {ab_semigroup_mult_comm_monoid_mult = ab_semigroup_mult_int,
-    monoid_mult_comm_monoid_mult = monoid_mult_int}
-  : IntInf.int comm_monoid_mult;
-
-val comm_semiring_1_int =
-  {comm_monoid_mult_comm_semiring_1 = comm_monoid_mult_int,
-    comm_semiring_0_comm_semiring_1 = comm_semiring_0_int,
-    dvd_comm_semiring_1 = dvd_int, semiring_1_comm_semiring_1 = semiring_1_int}
-  : IntInf.int comm_semiring_1;
-
-val comm_semiring_0_cancel_int =
-  {comm_semiring_0_comm_semiring_0_cancel = comm_semiring_0_int,
-    semiring_0_cancel_comm_semiring_0_cancel = semiring_0_cancel_int}
-  : IntInf.int comm_semiring_0_cancel;
-
-val comm_semiring_1_cancel_int =
-  {comm_semiring_0_cancel_comm_semiring_1_cancel = comm_semiring_0_cancel_int,
-    comm_semiring_1_comm_semiring_1_cancel = comm_semiring_1_int,
-    semiring_1_cancel_comm_semiring_1_cancel = semiring_1_cancel_int}
-  : IntInf.int comm_semiring_1_cancel;
-
-fun abs_int i = (if IntInf.< (i, (0 : IntInf.int)) then IntInf.~ i else i);
-
-fun split f (a, b) = f a b;
-
-fun sgn_int i =
-  (if ((i : IntInf.int) = (0 : IntInf.int)) then (0 : IntInf.int)
-    else (if IntInf.< ((0 : IntInf.int), i) then (1 : IntInf.int)
-           else IntInf.~ (1 : IntInf.int)));
-
-fun apsnd f (x, y) = (x, f y);
-
-fun divmod_int k l =
-  (if ((k : IntInf.int) = (0 : IntInf.int))
-    then ((0 : IntInf.int), (0 : IntInf.int))
-    else (if ((l : IntInf.int) = (0 : IntInf.int)) then ((0 : IntInf.int), k)
-           else apsnd (fn a => IntInf.* (sgn_int l, a))
-                  (if (((sgn_int k) : IntInf.int) = (sgn_int l))
-                    then IntInf.divMod (IntInf.abs k, IntInf.abs l)
-                    else let
-                           val (r, s) =
-                             IntInf.divMod (IntInf.abs k, IntInf.abs l);
-                         in
-                           (if ((s : IntInf.int) = (0 : IntInf.int))
-                             then (IntInf.~ r, (0 : IntInf.int))
-                             else (IntInf.- (IntInf.~ r, (1 : IntInf.int)),
-                                    IntInf.- (abs_int l, s)))
-                         end)));
-
-fun snd (a, b) = b;
-
-fun mod_int a b = snd (divmod_int a b);
-
-fun fst (a, b) = a;
-
-fun div_int a b = fst (divmod_int a b);
-
-val div_inta = {dvd_div = dvd_int, diva = div_int, moda = mod_int} :
-  IntInf.int diva;
-
-val semiring_div_int =
-  {div_semiring_div = div_inta,
-    comm_semiring_1_cancel_semiring_div = comm_semiring_1_cancel_int,
-    no_zero_divisors_semiring_div = no_zero_divisors_int}
-  : IntInf.int semiring_div;
-
-fun dvd (A1_, A2_) a b =
-  eqa A2_ (moda (div_semiring_div A1_) b a)
-    (zero ((zero_no_zero_divisors o no_zero_divisors_semiring_div) A1_));
-
-fun num_case f1 f2 f3 f4 f5 f6 f7 (Mul (inta, num)) = f7 inta num
-  | num_case f1 f2 f3 f4 f5 f6 f7 (Sub (num1, num2)) = f6 num1 num2
-  | num_case f1 f2 f3 f4 f5 f6 f7 (Add (num1, num2)) = f5 num1 num2
-  | num_case f1 f2 f3 f4 f5 f6 f7 (Neg num) = f4 num
-  | num_case f1 f2 f3 f4 f5 f6 f7 (Cn (nat, inta, num)) = f3 nat inta num
-  | num_case f1 f2 f3 f4 f5 f6 f7 (Bound nat) = f2 nat
-  | num_case f1 f2 f3 f4 f5 f6 f7 (C inta) = f1 inta;
-
-fun nummul i (C j) = C (IntInf.* (i, j))
-  | nummul i (Cn (n, c, t)) = Cn (n, IntInf.* (c, i), nummul i t)
-  | nummul i (Bound v) = Mul (i, Bound v)
-  | nummul i (Neg v) = Mul (i, Neg v)
-  | nummul i (Add (v, va)) = Mul (i, Add (v, va))
-  | nummul i (Sub (v, va)) = Mul (i, Sub (v, va))
-  | nummul i (Mul (v, va)) = Mul (i, Mul (v, va));
-
-fun numneg t = nummul (IntInf.~ (1 : IntInf.int)) t;
-
-fun numadd (Cn (n1, c1, r1), Cn (n2, c2, 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 Cn (n1, c, numadd (r1, r2)))
-         end
-    else (if IntInf.<= (n1, n2)
-           then Cn (n1, c1, numadd (r1, Add (Mul (c2, Bound n2), r2)))
-           else Cn (n2, c2, numadd (Add (Mul (c1, Bound n1), r1), r2))))
-  | numadd (Cn (n1, c1, r1), C dd) = Cn (n1, c1, numadd (r1, C dd))
-  | numadd (Cn (n1, c1, r1), Bound de) = Cn (n1, c1, numadd (r1, Bound de))
-  | numadd (Cn (n1, c1, r1), Neg di) = Cn (n1, c1, numadd (r1, Neg di))
-  | numadd (Cn (n1, c1, r1), Add (dj, dk)) =
-    Cn (n1, c1, numadd (r1, Add (dj, dk)))
-  | numadd (Cn (n1, c1, r1), Sub (dl, dm)) =
-    Cn (n1, c1, numadd (r1, Sub (dl, dm)))
-  | numadd (Cn (n1, c1, r1), Mul (dn, doa)) =
-    Cn (n1, c1, numadd (r1, Mul (dn, doa)))
-  | numadd (C w, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (C w, r2))
-  | numadd (Bound x, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (Bound x, r2))
-  | numadd (Neg ac, Cn (n2, c2, r2)) = Cn (n2, c2, numadd (Neg ac, r2))
-  | numadd (Add (ad, ae), Cn (n2, c2, r2)) =
-    Cn (n2, c2, numadd (Add (ad, ae), r2))
-  | numadd (Sub (af, ag), Cn (n2, c2, r2)) =
-    Cn (n2, c2, numadd (Sub (af, ag), r2))
-  | numadd (Mul (ah, ai), Cn (n2, c2, r2)) =
-    Cn (n2, c2, numadd (Mul (ah, ai), r2))
-  | numadd (C b1, C b2) = C (IntInf.+ (b1, b2))
-  | numadd (C aj, Bound bi) = Add (C aj, Bound bi)
-  | numadd (C aj, Neg bm) = Add (C aj, Neg bm)
-  | numadd (C aj, Add (bn, bo)) = Add (C aj, Add (bn, bo))
-  | numadd (C aj, Sub (bp, bq)) = Add (C aj, Sub (bp, bq))
-  | numadd (C aj, Mul (br, bs)) = Add (C aj, Mul (br, bs))
-  | numadd (Bound ak, C cf) = Add (Bound ak, C cf)
-  | numadd (Bound ak, Bound cg) = Add (Bound ak, Bound cg)
-  | numadd (Bound ak, Neg ck) = Add (Bound ak, Neg ck)
-  | numadd (Bound ak, Add (cl, cm)) = Add (Bound ak, Add (cl, cm))
-  | numadd (Bound ak, Sub (cn, co)) = Add (Bound ak, Sub (cn, co))
-  | numadd (Bound ak, Mul (cp, cq)) = Add (Bound ak, Mul (cp, cq))
-  | numadd (Neg ao, C en) = Add (Neg ao, C en)
-  | numadd (Neg ao, Bound eo) = Add (Neg ao, Bound eo)
-  | numadd (Neg ao, Neg es) = Add (Neg ao, Neg es)
-  | numadd (Neg ao, Add (et, eu)) = Add (Neg ao, Add (et, eu))
-  | numadd (Neg ao, Sub (ev, ew)) = Add (Neg ao, Sub (ev, ew))
-  | numadd (Neg ao, Mul (ex, ey)) = Add (Neg ao, Mul (ex, ey))
-  | numadd (Add (ap, aq), C fl) = Add (Add (ap, aq), C fl)
-  | numadd (Add (ap, aq), Bound fm) = Add (Add (ap, aq), Bound fm)
-  | numadd (Add (ap, aq), Neg fq) = Add (Add (ap, aq), Neg fq)
-  | numadd (Add (ap, aq), Add (fr, fs)) = Add (Add (ap, aq), Add (fr, fs))
-  | numadd (Add (ap, aq), Sub (ft, fu)) = Add (Add (ap, aq), Sub (ft, fu))
-  | numadd (Add (ap, aq), Mul (fv, fw)) = Add (Add (ap, aq), Mul (fv, fw))
-  | numadd (Sub (ar, asa), C gj) = Add (Sub (ar, asa), C gj)
-  | numadd (Sub (ar, asa), Bound gk) = Add (Sub (ar, asa), Bound gk)
-  | numadd (Sub (ar, asa), Neg go) = Add (Sub (ar, asa), Neg go)
-  | numadd (Sub (ar, asa), Add (gp, gq)) = Add (Sub (ar, asa), Add (gp, gq))
-  | numadd (Sub (ar, asa), Sub (gr, gs)) = Add (Sub (ar, asa), Sub (gr, gs))
-  | numadd (Sub (ar, asa), Mul (gt, gu)) = Add (Sub (ar, asa), Mul (gt, gu))
-  | numadd (Mul (at, au), C hh) = Add (Mul (at, au), C hh)
-  | numadd (Mul (at, au), Bound hi) = Add (Mul (at, au), Bound hi)
-  | numadd (Mul (at, au), Neg hm) = Add (Mul (at, au), Neg hm)
-  | numadd (Mul (at, au), Add (hn, ho)) = Add (Mul (at, au), Add (hn, ho))
-  | numadd (Mul (at, au), Sub (hp, hq)) = Add (Mul (at, au), Sub (hp, hq))
-  | numadd (Mul (at, au), Mul (hr, hs)) = Add (Mul (at, au), Mul (hr, hs));
-
-fun numsub s t =
-  (if eq_num s t then C (0 : IntInf.int) else numadd (s, numneg t));
-
-fun simpnum (C j) = C j
-  | simpnum (Bound n) = Cn (n, (1 : IntInf.int), C (0 : IntInf.int))
-  | 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 : IntInf.int) = (0 : IntInf.int)) then C (0 : IntInf.int)
-      else nummul i (simpnum t))
-  | simpnum (Cn (v, va, vb)) = Cn (v, va, vb);
-
-fun nota (Not p) = p
-  | nota T = F
-  | nota F = T
-  | nota (Lt v) = Not (Lt v)
-  | nota (Le v) = Not (Le v)
-  | nota (Gt v) = Not (Gt v)
-  | nota (Ge v) = Not (Ge v)
-  | nota (Eq v) = Not (Eq v)
-  | nota (NEq v) = Not (NEq v)
-  | nota (Dvd (v, va)) = Not (Dvd (v, va))
-  | nota (NDvd (v, va)) = Not (NDvd (v, va))
-  | nota (And (v, va)) = Not (And (v, va))
-  | nota (Or (v, va)) = Not (Or (v, va))
-  | nota (Imp (v, va)) = Not (Imp (v, va))
-  | nota (Iff (v, va)) = Not (Iff (v, va))
-  | nota (E v) = Not (E v)
-  | nota (A v) = Not (A v)
-  | nota (Closed v) = Not (Closed v)
-  | nota (NClosed v) = Not (NClosed v);
-
-fun iffa 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 Iff (p, q)))))));
-
-fun impa 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 Imp (p, q))));
-
-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 simpfm (And (p, q)) = conj (simpfm p) (simpfm q)
-  | simpfm (Or (p, q)) = disj (simpfm p) (simpfm q)
-  | simpfm (Imp (p, q)) = impa (simpfm p) (simpfm q)
-  | simpfm (Iff (p, q)) = iffa (simpfm p) (simpfm q)
-  | simpfm (Not p) = nota (simpfm p)
-  | simpfm (Lt a) =
-    let
-      val aa = simpnum a;
-    in
-      (case aa of C v => (if IntInf.< (v, (0 : IntInf.int)) then T else F)
-        | Bound _ => Lt aa | Cn (_, _, _) => Lt aa | Neg _ => Lt aa
-        | Add (_, _) => Lt aa | Sub (_, _) => Lt aa | Mul (_, _) => Lt aa)
-    end
-  | simpfm (Le a) =
-    let
-      val aa = simpnum a;
-    in
-      (case aa of C v => (if IntInf.<= (v, (0 : IntInf.int)) then T else F)
-        | Bound _ => Le aa | Cn (_, _, _) => Le aa | Neg _ => Le aa
-        | Add (_, _) => Le aa | Sub (_, _) => Le aa | Mul (_, _) => Le aa)
-    end
-  | simpfm (Gt a) =
-    let
-      val aa = simpnum a;
-    in
-      (case aa of C v => (if IntInf.< ((0 : IntInf.int), v) then T else F)
-        | Bound _ => Gt aa | Cn (_, _, _) => Gt aa | Neg _ => Gt aa
-        | Add (_, _) => Gt aa | Sub (_, _) => Gt aa | Mul (_, _) => Gt aa)
-    end
-  | simpfm (Ge a) =
-    let
-      val aa = simpnum a;
-    in
-      (case aa of C v => (if IntInf.<= ((0 : IntInf.int), v) then T else F)
-        | Bound _ => Ge aa | Cn (_, _, _) => Ge aa | Neg _ => Ge aa
-        | Add (_, _) => Ge aa | Sub (_, _) => Ge aa | Mul (_, _) => Ge aa)
-    end
-  | simpfm (Eq a) =
-    let
-      val aa = simpnum a;
-    in
-      (case aa
-        of C v => (if ((v : IntInf.int) = (0 : IntInf.int)) then T else F)
-        | Bound _ => Eq aa | Cn (_, _, _) => Eq aa | Neg _ => Eq aa
-        | Add (_, _) => Eq aa | Sub (_, _) => Eq aa | Mul (_, _) => Eq aa)
-    end
-  | simpfm (NEq a) =
-    let
-      val aa = simpnum a;
-    in
-      (case aa
-        of C v => (if not ((v : IntInf.int) = (0 : IntInf.int)) then T else F)
-        | Bound _ => NEq aa | Cn (_, _, _) => NEq aa | Neg _ => NEq aa
-        | Add (_, _) => NEq aa | Sub (_, _) => NEq aa | Mul (_, _) => NEq aa)
-    end
-  | simpfm (Dvd (i, a)) =
-    (if ((i : IntInf.int) = (0 : IntInf.int)) then simpfm (Eq a)
-      else (if (((abs_int i) : IntInf.int) = (1 : IntInf.int)) then T
-             else let
-                    val aa = simpnum a;
-                  in
-                    (case aa
-                      of C v =>
-                        (if dvd (semiring_div_int, eq_int) i v then T else F)
-                      | Bound _ => Dvd (i, aa) | Cn (_, _, _) => Dvd (i, aa)
-                      | Neg _ => Dvd (i, aa) | Add (_, _) => Dvd (i, aa)
-                      | Sub (_, _) => Dvd (i, aa) | Mul (_, _) => Dvd (i, aa))
-                  end))
-  | simpfm (NDvd (i, a)) =
-    (if ((i : IntInf.int) = (0 : IntInf.int)) then simpfm (NEq a)
-      else (if (((abs_int i) : IntInf.int) = (1 : IntInf.int)) then F
-             else let
-                    val aa = simpnum a;
-                  in
-                    (case aa
-                      of C v =>
-                        (if not (dvd (semiring_div_int, eq_int) i v) then T
-                          else F)
-                      | Bound _ => NDvd (i, aa) | Cn (_, _, _) => NDvd (i, aa)
-                      | Neg _ => NDvd (i, aa) | Add (_, _) => NDvd (i, aa)
-                      | Sub (_, _) => NDvd (i, aa) | Mul (_, _) => NDvd (i, aa))
-                  end))
-  | simpfm T = T
-  | simpfm F = F
-  | simpfm (E v) = E v
-  | simpfm (A v) = A v
-  | simpfm (Closed v) = Closed v
-  | simpfm (NClosed v) = NClosed v;
-
-fun iupt i j =
-  (if IntInf.< (j, i) then []
-    else i :: iupt (IntInf.+ (i, (1 : IntInf.int))) j);
-
-fun mirror (And (p, q)) = And (mirror p, mirror q)
-  | mirror (Or (p, q)) = Or (mirror p, mirror q)
-  | mirror T = T
-  | mirror F = F
-  | mirror (Lt (C bo)) = Lt (C bo)
-  | mirror (Lt (Bound bp)) = Lt (Bound bp)
-  | mirror (Lt (Neg bt)) = Lt (Neg bt)
-  | mirror (Lt (Add (bu, bv))) = Lt (Add (bu, bv))
-  | mirror (Lt (Sub (bw, bx))) = Lt (Sub (bw, bx))
-  | mirror (Lt (Mul (by, bz))) = Lt (Mul (by, bz))
-  | mirror (Le (C co)) = Le (C co)
-  | mirror (Le (Bound cp)) = Le (Bound cp)
-  | mirror (Le (Neg ct)) = Le (Neg ct)
-  | mirror (Le (Add (cu, cv))) = Le (Add (cu, cv))
-  | mirror (Le (Sub (cw, cx))) = Le (Sub (cw, cx))
-  | mirror (Le (Mul (cy, cz))) = Le (Mul (cy, cz))
-  | mirror (Gt (C doa)) = Gt (C doa)
-  | mirror (Gt (Bound dp)) = Gt (Bound dp)
-  | mirror (Gt (Neg dt)) = Gt (Neg dt)
-  | mirror (Gt (Add (du, dv))) = Gt (Add (du, dv))
-  | mirror (Gt (Sub (dw, dx))) = Gt (Sub (dw, dx))
-  | mirror (Gt (Mul (dy, dz))) = Gt (Mul (dy, dz))
-  | mirror (Ge (C eo)) = Ge (C eo)
-  | mirror (Ge (Bound ep)) = Ge (Bound ep)
-  | mirror (Ge (Neg et)) = Ge (Neg et)
-  | mirror (Ge (Add (eu, ev))) = Ge (Add (eu, ev))
-  | mirror (Ge (Sub (ew, ex))) = Ge (Sub (ew, ex))
-  | mirror (Ge (Mul (ey, ez))) = Ge (Mul (ey, ez))
-  | mirror (Eq (C fo)) = Eq (C fo)
-  | mirror (Eq (Bound fp)) = Eq (Bound fp)
-  | mirror (Eq (Neg ft)) = Eq (Neg ft)
-  | mirror (Eq (Add (fu, fv))) = Eq (Add (fu, fv))
-  | mirror (Eq (Sub (fw, fx))) = Eq (Sub (fw, fx))
-  | mirror (Eq (Mul (fy, fz))) = Eq (Mul (fy, fz))
-  | mirror (NEq (C go)) = NEq (C go)
-  | mirror (NEq (Bound gp)) = NEq (Bound gp)
-  | mirror (NEq (Neg gt)) = NEq (Neg gt)
-  | mirror (NEq (Add (gu, gv))) = NEq (Add (gu, gv))
-  | mirror (NEq (Sub (gw, gx))) = NEq (Sub (gw, gx))
-  | mirror (NEq (Mul (gy, gz))) = NEq (Mul (gy, gz))
-  | mirror (Dvd (aa, C ho)) = Dvd (aa, C ho)
-  | mirror (Dvd (aa, Bound hp)) = Dvd (aa, Bound hp)
-  | mirror (Dvd (aa, Neg ht)) = Dvd (aa, Neg ht)
-  | mirror (Dvd (aa, Add (hu, hv))) = Dvd (aa, Add (hu, hv))
-  | mirror (Dvd (aa, Sub (hw, hx))) = Dvd (aa, Sub (hw, hx))
-  | mirror (Dvd (aa, Mul (hy, hz))) = Dvd (aa, Mul (hy, hz))
-  | mirror (NDvd (ac, C io)) = NDvd (ac, C io)
-  | mirror (NDvd (ac, Bound ip)) = NDvd (ac, Bound ip)
-  | mirror (NDvd (ac, Neg it)) = NDvd (ac, Neg it)
-  | mirror (NDvd (ac, Add (iu, iv))) = NDvd (ac, Add (iu, iv))
-  | mirror (NDvd (ac, Sub (iw, ix))) = NDvd (ac, Sub (iw, ix))
-  | mirror (NDvd (ac, Mul (iy, iz))) = NDvd (ac, Mul (iy, iz))
-  | 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
-  | mirror (Lt (Cn (cm, c, e))) =
-    (if ((cm : IntInf.int) = (0 : IntInf.int))
-      then Gt (Cn ((0 : IntInf.int), c, Neg e))
-      else Lt (Cn (suc (minus_nat cm (1 : IntInf.int)), c, e)))
-  | mirror (Le (Cn (dm, c, e))) =
-    (if ((dm : IntInf.int) = (0 : IntInf.int))
-      then Ge (Cn ((0 : IntInf.int), c, Neg e))
-      else Le (Cn (suc (minus_nat dm (1 : IntInf.int)), c, e)))
-  | mirror (Gt (Cn (em, c, e))) =
-    (if ((em : IntInf.int) = (0 : IntInf.int))
-      then Lt (Cn ((0 : IntInf.int), c, Neg e))
-      else Gt (Cn (suc (minus_nat em (1 : IntInf.int)), c, e)))
-  | mirror (Ge (Cn (fm, c, e))) =
-    (if ((fm : IntInf.int) = (0 : IntInf.int))
-      then Le (Cn ((0 : IntInf.int), c, Neg e))
-      else Ge (Cn (suc (minus_nat fm (1 : IntInf.int)), c, e)))
-  | mirror (Eq (Cn (gm, c, e))) =
-    (if ((gm : IntInf.int) = (0 : IntInf.int))
-      then Eq (Cn ((0 : IntInf.int), c, Neg e))
-      else Eq (Cn (suc (minus_nat gm (1 : IntInf.int)), c, e)))
-  | mirror (NEq (Cn (hm, c, e))) =
-    (if ((hm : IntInf.int) = (0 : IntInf.int))
-      then NEq (Cn ((0 : IntInf.int), c, Neg e))
-      else NEq (Cn (suc (minus_nat hm (1 : IntInf.int)), c, e)))
-  | mirror (Dvd (i, Cn (im, c, e))) =
-    (if ((im : IntInf.int) = (0 : IntInf.int))
-      then Dvd (i, Cn ((0 : IntInf.int), c, Neg e))
-      else Dvd (i, Cn (suc (minus_nat im (1 : IntInf.int)), c, e)))
-  | mirror (NDvd (i, Cn (jm, c, e))) =
-    (if ((jm : IntInf.int) = (0 : IntInf.int))
-      then NDvd (i, Cn ((0 : IntInf.int), c, Neg e))
-      else NDvd (i, Cn (suc (minus_nat jm (1 : IntInf.int)), c, e)));
-
-fun size_list [] = (0 : IntInf.int)
-  | size_list (a :: lista) = IntInf.+ (size_list lista, suc (0 : IntInf.int));
-
-fun alpha (And (p, q)) = append (alpha p) (alpha q)
-  | alpha (Or (p, q)) = append (alpha p) (alpha q)
-  | alpha T = []
-  | alpha F = []
-  | alpha (Lt (C bo)) = []
-  | alpha (Lt (Bound bp)) = []
-  | alpha (Lt (Neg bt)) = []
-  | alpha (Lt (Add (bu, bv))) = []
-  | alpha (Lt (Sub (bw, bx))) = []
-  | alpha (Lt (Mul (by, bz))) = []
-  | alpha (Le (C co)) = []
-  | alpha (Le (Bound cp)) = []
-  | alpha (Le (Neg ct)) = []
-  | alpha (Le (Add (cu, cv))) = []
-  | alpha (Le (Sub (cw, cx))) = []
-  | alpha (Le (Mul (cy, cz))) = []
-  | alpha (Gt (C doa)) = []
-  | alpha (Gt (Bound dp)) = []
-  | alpha (Gt (Neg dt)) = []
-  | alpha (Gt (Add (du, dv))) = []
-  | alpha (Gt (Sub (dw, dx))) = []
-  | alpha (Gt (Mul (dy, dz))) = []
-  | alpha (Ge (C eo)) = []
-  | alpha (Ge (Bound ep)) = []
-  | alpha (Ge (Neg et)) = []
-  | alpha (Ge (Add (eu, ev))) = []
-  | alpha (Ge (Sub (ew, ex))) = []
-  | alpha (Ge (Mul (ey, ez))) = []
-  | alpha (Eq (C fo)) = []
-  | alpha (Eq (Bound fp)) = []
-  | alpha (Eq (Neg ft)) = []
-  | alpha (Eq (Add (fu, fv))) = []
-  | alpha (Eq (Sub (fw, fx))) = []
-  | alpha (Eq (Mul (fy, fz))) = []
-  | alpha (NEq (C go)) = []
-  | alpha (NEq (Bound gp)) = []
-  | alpha (NEq (Neg gt)) = []
-  | alpha (NEq (Add (gu, gv))) = []
-  | alpha (NEq (Sub (gw, gx))) = []
-  | alpha (NEq (Mul (gy, gz))) = []
-  | 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) = []
-  | alpha (Lt (Cn (cm, c, e))) =
-    (if ((cm : IntInf.int) = (0 : IntInf.int)) then [e] else [])
-  | alpha (Le (Cn (dm, c, e))) =
-    (if ((dm : IntInf.int) = (0 : IntInf.int))
-      then [Add (C (~1 : IntInf.int), e)] else [])
-  | alpha (Gt (Cn (em, c, e))) =
-    (if ((em : IntInf.int) = (0 : IntInf.int)) then [] else [])
-  | alpha (Ge (Cn (fm, c, e))) =
-    (if ((fm : IntInf.int) = (0 : IntInf.int)) then [] else [])
-  | alpha (Eq (Cn (gm, c, e))) =
-    (if ((gm : IntInf.int) = (0 : IntInf.int))
-      then [Add (C (~1 : IntInf.int), e)] else [])
-  | alpha (NEq (Cn (hm, c, e))) =
-    (if ((hm : IntInf.int) = (0 : IntInf.int)) then [e] else []);
-
-fun beta (And (p, q)) = append (beta p) (beta q)
-  | beta (Or (p, q)) = append (beta p) (beta q)
-  | beta T = []
-  | beta F = []
-  | beta (Lt (C bo)) = []
-  | beta (Lt (Bound bp)) = []
-  | beta (Lt (Neg bt)) = []
-  | beta (Lt (Add (bu, bv))) = []
-  | beta (Lt (Sub (bw, bx))) = []
-  | beta (Lt (Mul (by, bz))) = []
-  | beta (Le (C co)) = []
-  | beta (Le (Bound cp)) = []
-  | beta (Le (Neg ct)) = []
-  | beta (Le (Add (cu, cv))) = []
-  | beta (Le (Sub (cw, cx))) = []
-  | beta (Le (Mul (cy, cz))) = []
-  | beta (Gt (C doa)) = []
-  | beta (Gt (Bound dp)) = []
-  | beta (Gt (Neg dt)) = []
-  | beta (Gt (Add (du, dv))) = []
-  | beta (Gt (Sub (dw, dx))) = []
-  | beta (Gt (Mul (dy, dz))) = []
-  | beta (Ge (C eo)) = []
-  | beta (Ge (Bound ep)) = []
-  | beta (Ge (Neg et)) = []
-  | beta (Ge (Add (eu, ev))) = []
-  | beta (Ge (Sub (ew, ex))) = []
-  | beta (Ge (Mul (ey, ez))) = []
-  | beta (Eq (C fo)) = []
-  | beta (Eq (Bound fp)) = []
-  | beta (Eq (Neg ft)) = []
-  | beta (Eq (Add (fu, fv))) = []
-  | beta (Eq (Sub (fw, fx))) = []
-  | beta (Eq (Mul (fy, fz))) = []
-  | beta (NEq (C go)) = []
-  | beta (NEq (Bound gp)) = []
-  | beta (NEq (Neg gt)) = []
-  | beta (NEq (Add (gu, gv))) = []
-  | beta (NEq (Sub (gw, gx))) = []
-  | beta (NEq (Mul (gy, gz))) = []
-  | 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) = []
-  | beta (Lt (Cn (cm, c, e))) =
-    (if ((cm : IntInf.int) = (0 : IntInf.int)) then [] else [])
-  | beta (Le (Cn (dm, c, e))) =
-    (if ((dm : IntInf.int) = (0 : IntInf.int)) then [] else [])
-  | beta (Gt (Cn (em, c, e))) =
-    (if ((em : IntInf.int) = (0 : IntInf.int)) then [Neg e] else [])
-  | beta (Ge (Cn (fm, c, e))) =
-    (if ((fm : IntInf.int) = (0 : IntInf.int))
-      then [Sub (C (~1 : IntInf.int), e)] else [])
-  | beta (Eq (Cn (gm, c, e))) =
-    (if ((gm : IntInf.int) = (0 : IntInf.int))
-      then [Sub (C (~1 : IntInf.int), e)] else [])
-  | beta (NEq (Cn (hm, c, e))) =
-    (if ((hm : IntInf.int) = (0 : IntInf.int)) then [Neg e] else []);
-
-val eq_numa = {eq = eq_num} : num eq;
-
-fun member A_ x [] = false
-  | member A_ x (y :: ys) = eqa A_ x y orelse member A_ x ys;
-
-fun remdups A_ [] = []
-  | remdups A_ (x :: xs) =
-    (if member A_ x xs then remdups A_ xs else x :: remdups A_ xs);
-
-fun gcd_int k l =
-  abs_int
-    (if ((l : IntInf.int) = (0 : IntInf.int)) then k
-      else gcd_int l (mod_int (abs_int k) (abs_int l)));
-
-fun lcm_int a b = div_int (IntInf.* (abs_int a, abs_int b)) (gcd_int a b);
-
-fun delta (And (p, q)) = lcm_int (delta p) (delta q)
-  | delta (Or (p, q)) = lcm_int (delta p) (delta q)
-  | delta T = (1 : IntInf.int)
-  | delta F = (1 : IntInf.int)
-  | delta (Lt u) = (1 : IntInf.int)
-  | delta (Le v) = (1 : IntInf.int)
-  | delta (Gt w) = (1 : IntInf.int)
-  | delta (Ge x) = (1 : IntInf.int)
-  | delta (Eq y) = (1 : IntInf.int)
-  | delta (NEq z) = (1 : IntInf.int)
-  | delta (Dvd (aa, C bo)) = (1 : IntInf.int)
-  | delta (Dvd (aa, Bound bp)) = (1 : IntInf.int)
-  | delta (Dvd (aa, Neg bt)) = (1 : IntInf.int)
-  | delta (Dvd (aa, Add (bu, bv))) = (1 : IntInf.int)
-  | delta (Dvd (aa, Sub (bw, bx))) = (1 : IntInf.int)
-  | delta (Dvd (aa, Mul (by, bz))) = (1 : IntInf.int)
-  | delta (NDvd (ac, C co)) = (1 : IntInf.int)
-  | delta (NDvd (ac, Bound cp)) = (1 : IntInf.int)
-  | delta (NDvd (ac, Neg ct)) = (1 : IntInf.int)
-  | delta (NDvd (ac, Add (cu, cv))) = (1 : IntInf.int)
-  | delta (NDvd (ac, Sub (cw, cx))) = (1 : IntInf.int)
-  | delta (NDvd (ac, Mul (cy, cz))) = (1 : IntInf.int)
-  | delta (Not ae) = (1 : IntInf.int)
-  | delta (Imp (aj, ak)) = (1 : IntInf.int)
-  | delta (Iff (al, am)) = (1 : IntInf.int)
-  | delta (E an) = (1 : IntInf.int)
-  | delta (A ao) = (1 : IntInf.int)
-  | delta (Closed ap) = (1 : IntInf.int)
-  | delta (NClosed aq) = (1 : IntInf.int)
-  | delta (Dvd (i, Cn (cm, c, e))) =
-    (if ((cm : IntInf.int) = (0 : IntInf.int)) then i else (1 : IntInf.int))
-  | delta (NDvd (i, Cn (dm, c, e))) =
-    (if ((dm : IntInf.int) = (0 : IntInf.int)) then i else (1 : IntInf.int));
-
-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 T = (fn _ => T)
-  | a_beta F = (fn _ => F)
-  | a_beta (Lt (C bo)) = (fn _ => Lt (C bo))
-  | a_beta (Lt (Bound bp)) = (fn _ => Lt (Bound bp))
-  | a_beta (Lt (Neg bt)) = (fn _ => Lt (Neg bt))
-  | a_beta (Lt (Add (bu, bv))) = (fn _ => Lt (Add (bu, bv)))
-  | a_beta (Lt (Sub (bw, bx))) = (fn _ => Lt (Sub (bw, bx)))
-  | a_beta (Lt (Mul (by, bz))) = (fn _ => Lt (Mul (by, bz)))
-  | a_beta (Le (C co)) = (fn _ => Le (C co))
-  | a_beta (Le (Bound cp)) = (fn _ => Le (Bound cp))
-  | a_beta (Le (Neg ct)) = (fn _ => Le (Neg ct))
-  | a_beta (Le (Add (cu, cv))) = (fn _ => Le (Add (cu, cv)))
-  | a_beta (Le (Sub (cw, cx))) = (fn _ => Le (Sub (cw, cx)))
-  | a_beta (Le (Mul (cy, cz))) = (fn _ => Le (Mul (cy, cz)))
-  | a_beta (Gt (C doa)) = (fn _ => Gt (C doa))
-  | a_beta (Gt (Bound dp)) = (fn _ => Gt (Bound dp))
-  | a_beta (Gt (Neg dt)) = (fn _ => Gt (Neg dt))
-  | a_beta (Gt (Add (du, dv))) = (fn _ => Gt (Add (du, dv)))
-  | a_beta (Gt (Sub (dw, dx))) = (fn _ => Gt (Sub (dw, dx)))
-  | a_beta (Gt (Mul (dy, dz))) = (fn _ => Gt (Mul (dy, dz)))
-  | a_beta (Ge (C eo)) = (fn _ => Ge (C eo))
-  | a_beta (Ge (Bound ep)) = (fn _ => Ge (Bound ep))
-  | a_beta (Ge (Neg et)) = (fn _ => Ge (Neg et))
-  | a_beta (Ge (Add (eu, ev))) = (fn _ => Ge (Add (eu, ev)))
-  | a_beta (Ge (Sub (ew, ex))) = (fn _ => Ge (Sub (ew, ex)))
-  | a_beta (Ge (Mul (ey, ez))) = (fn _ => Ge (Mul (ey, ez)))
-  | a_beta (Eq (C fo)) = (fn _ => Eq (C fo))
-  | a_beta (Eq (Bound fp)) = (fn _ => Eq (Bound fp))
-  | a_beta (Eq (Neg ft)) = (fn _ => Eq (Neg ft))
-  | a_beta (Eq (Add (fu, fv))) = (fn _ => Eq (Add (fu, fv)))
-  | a_beta (Eq (Sub (fw, fx))) = (fn _ => Eq (Sub (fw, fx)))
-  | a_beta (Eq (Mul (fy, fz))) = (fn _ => Eq (Mul (fy, fz)))
-  | a_beta (NEq (C go)) = (fn _ => NEq (C go))
-  | a_beta (NEq (Bound gp)) = (fn _ => NEq (Bound gp))
-  | a_beta (NEq (Neg gt)) = (fn _ => NEq (Neg gt))
-  | a_beta (NEq (Add (gu, gv))) = (fn _ => NEq (Add (gu, gv)))
-  | a_beta (NEq (Sub (gw, gx))) = (fn _ => NEq (Sub (gw, gx)))
-  | a_beta (NEq (Mul (gy, gz))) = (fn _ => NEq (Mul (gy, gz)))
-  | a_beta (Dvd (aa, C ho)) = (fn _ => Dvd (aa, C ho))
-  | a_beta (Dvd (aa, Bound hp)) = (fn _ => Dvd (aa, Bound hp))
-  | a_beta (Dvd (aa, Neg ht)) = (fn _ => Dvd (aa, Neg ht))
-  | a_beta (Dvd (aa, Add (hu, hv))) = (fn _ => Dvd (aa, Add (hu, hv)))
-  | a_beta (Dvd (aa, Sub (hw, hx))) = (fn _ => Dvd (aa, Sub (hw, hx)))
-  | a_beta (Dvd (aa, Mul (hy, hz))) = (fn _ => Dvd (aa, Mul (hy, hz)))
-  | a_beta (NDvd (ac, C io)) = (fn _ => NDvd (ac, C io))
-  | a_beta (NDvd (ac, Bound ip)) = (fn _ => NDvd (ac, Bound ip))
-  | a_beta (NDvd (ac, Neg it)) = (fn _ => NDvd (ac, Neg it))
-  | a_beta (NDvd (ac, Add (iu, iv))) = (fn _ => NDvd (ac, Add (iu, iv)))
-  | a_beta (NDvd (ac, Sub (iw, ix))) = (fn _ => NDvd (ac, Sub (iw, ix)))
-  | a_beta (NDvd (ac, Mul (iy, iz))) = (fn _ => NDvd (ac, Mul (iy, iz)))
-  | a_beta (Not ae) = (fn _ => Not ae)
-  | a_beta (Imp (aj, ak)) = (fn _ => Imp (aj, ak))
-  | a_beta (Iff (al, am)) = (fn _ => Iff (al, am))
-  | a_beta (E an) = (fn _ => E an)
-  | a_beta (A ao) = (fn _ => A ao)
-  | a_beta (Closed ap) = (fn _ => Closed ap)
-  | a_beta (NClosed aq) = (fn _ => NClosed aq)
-  | a_beta (Lt (Cn (cm, c, e))) =
-    (if ((cm : IntInf.int) = (0 : IntInf.int))
-      then (fn k =>
-             Lt (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e))))
-      else (fn _ => Lt (Cn (suc (minus_nat cm (1 : IntInf.int)), c, e))))
-  | a_beta (Le (Cn (dm, c, e))) =
-    (if ((dm : IntInf.int) = (0 : IntInf.int))
-      then (fn k =>
-             Le (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e))))
-      else (fn _ => Le (Cn (suc (minus_nat dm (1 : IntInf.int)), c, e))))
-  | a_beta (Gt (Cn (em, c, e))) =
-    (if ((em : IntInf.int) = (0 : IntInf.int))
-      then (fn k =>
-             Gt (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e))))
-      else (fn _ => Gt (Cn (suc (minus_nat em (1 : IntInf.int)), c, e))))
-  | a_beta (Ge (Cn (fm, c, e))) =
-    (if ((fm : IntInf.int) = (0 : IntInf.int))
-      then (fn k =>
-             Ge (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e))))
-      else (fn _ => Ge (Cn (suc (minus_nat fm (1 : IntInf.int)), c, e))))
-  | a_beta (Eq (Cn (gm, c, e))) =
-    (if ((gm : IntInf.int) = (0 : IntInf.int))
-      then (fn k =>
-             Eq (Cn ((0 : IntInf.int), (1 : IntInf.int), Mul (div_int k c, e))))
-      else (fn _ => Eq (Cn (suc (minus_nat gm (1 : IntInf.int)), c, e))))
-  | a_beta (NEq (Cn (hm, c, e))) =
-    (if ((hm : IntInf.int) = (0 : IntInf.int))
-      then (fn k =>
-             NEq (Cn ((0 : IntInf.int), (1 : IntInf.int),
-                       Mul (div_int k c, e))))
-      else (fn _ => NEq (Cn (suc (minus_nat hm (1 : IntInf.int)), c, e))))
-  | a_beta (Dvd (i, Cn (im, c, e))) =
-    (if ((im : IntInf.int) = (0 : IntInf.int))
-      then (fn k =>
-             Dvd (IntInf.* (div_int k c, i),
-                   Cn ((0 : IntInf.int), (1 : IntInf.int),
-                        Mul (div_int k c, e))))
-      else (fn _ => Dvd (i, Cn (suc (minus_nat im (1 : IntInf.int)), c, e))))
-  | a_beta (NDvd (i, Cn (jm, c, e))) =
-    (if ((jm : IntInf.int) = (0 : IntInf.int))
-      then (fn k =>
-             NDvd (IntInf.* (div_int k c, i),
-                    Cn ((0 : IntInf.int), (1 : IntInf.int),
-                         Mul (div_int k c, e))))
-      else (fn _ => NDvd (i, Cn (suc (minus_nat jm (1 : IntInf.int)), c, e))));
-
-fun zeta (And (p, q)) = lcm_int (zeta p) (zeta q)
-  | zeta (Or (p, q)) = lcm_int (zeta p) (zeta q)
-  | zeta T = (1 : IntInf.int)
-  | zeta F = (1 : IntInf.int)
-  | zeta (Lt (C bo)) = (1 : IntInf.int)
-  | zeta (Lt (Bound bp)) = (1 : IntInf.int)
-  | zeta (Lt (Neg bt)) = (1 : IntInf.int)
-  | zeta (Lt (Add (bu, bv))) = (1 : IntInf.int)
-  | zeta (Lt (Sub (bw, bx))) = (1 : IntInf.int)
-  | zeta (Lt (Mul (by, bz))) = (1 : IntInf.int)
-  | zeta (Le (C co)) = (1 : IntInf.int)
-  | zeta (Le (Bound cp)) = (1 : IntInf.int)
-  | zeta (Le (Neg ct)) = (1 : IntInf.int)
-  | zeta (Le (Add (cu, cv))) = (1 : IntInf.int)
-  | zeta (Le (Sub (cw, cx))) = (1 : IntInf.int)
-  | zeta (Le (Mul (cy, cz))) = (1 : IntInf.int)
-  | zeta (Gt (C doa)) = (1 : IntInf.int)
-  | zeta (Gt (Bound dp)) = (1 : IntInf.int)
-  | zeta (Gt (Neg dt)) = (1 : IntInf.int)
-  | zeta (Gt (Add (du, dv))) = (1 : IntInf.int)
-  | zeta (Gt (Sub (dw, dx))) = (1 : IntInf.int)
-  | zeta (Gt (Mul (dy, dz))) = (1 : IntInf.int)
-  | zeta (Ge (C eo)) = (1 : IntInf.int)
-  | zeta (Ge (Bound ep)) = (1 : IntInf.int)
-  | zeta (Ge (Neg et)) = (1 : IntInf.int)
-  | zeta (Ge (Add (eu, ev))) = (1 : IntInf.int)
-  | zeta (Ge (Sub (ew, ex))) = (1 : IntInf.int)
-  | zeta (Ge (Mul (ey, ez))) = (1 : IntInf.int)
-  | zeta (Eq (C fo)) = (1 : IntInf.int)
-  | zeta (Eq (Bound fp)) = (1 : IntInf.int)
-  | zeta (Eq (Neg ft)) = (1 : IntInf.int)
-  | zeta (Eq (Add (fu, fv))) = (1 : IntInf.int)
-  | zeta (Eq (Sub (fw, fx))) = (1 : IntInf.int)
-  | zeta (Eq (Mul (fy, fz))) = (1 : IntInf.int)
-  | zeta (NEq (C go)) = (1 : IntInf.int)
-  | zeta (NEq (Bound gp)) = (1 : IntInf.int)
-  | zeta (NEq (Neg gt)) = (1 : IntInf.int)
-  | zeta (NEq (Add (gu, gv))) = (1 : IntInf.int)
-  | zeta (NEq (Sub (gw, gx))) = (1 : IntInf.int)
-  | zeta (NEq (Mul (gy, gz))) = (1 : IntInf.int)
-  | zeta (Dvd (aa, C ho)) = (1 : IntInf.int)
-  | zeta (Dvd (aa, Bound hp)) = (1 : IntInf.int)
-  | zeta (Dvd (aa, Neg ht)) = (1 : IntInf.int)
-  | zeta (Dvd (aa, Add (hu, hv))) = (1 : IntInf.int)
-  | zeta (Dvd (aa, Sub (hw, hx))) = (1 : IntInf.int)
-  | zeta (Dvd (aa, Mul (hy, hz))) = (1 : IntInf.int)
-  | zeta (NDvd (ac, C io)) = (1 : IntInf.int)
-  | zeta (NDvd (ac, Bound ip)) = (1 : IntInf.int)
-  | zeta (NDvd (ac, Neg it)) = (1 : IntInf.int)
-  | zeta (NDvd (ac, Add (iu, iv))) = (1 : IntInf.int)
-  | zeta (NDvd (ac, Sub (iw, ix))) = (1 : IntInf.int)
-  | zeta (NDvd (ac, Mul (iy, iz))) = (1 : IntInf.int)
-  | zeta (Not ae) = (1 : IntInf.int)
-  | zeta (Imp (aj, ak)) = (1 : IntInf.int)
-  | zeta (Iff (al, am)) = (1 : IntInf.int)
-  | zeta (E an) = (1 : IntInf.int)
-  | zeta (A ao) = (1 : IntInf.int)
-  | zeta (Closed ap) = (1 : IntInf.int)
-  | zeta (NClosed aq) = (1 : IntInf.int)
-  | zeta (Lt (Cn (cm, c, e))) =
-    (if ((cm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int))
-  | zeta (Le (Cn (dm, c, e))) =
-    (if ((dm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int))
-  | zeta (Gt (Cn (em, c, e))) =
-    (if ((em : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int))
-  | zeta (Ge (Cn (fm, c, e))) =
-    (if ((fm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int))
-  | zeta (Eq (Cn (gm, c, e))) =
-    (if ((gm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int))
-  | zeta (NEq (Cn (hm, c, e))) =
-    (if ((hm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int))
-  | zeta (Dvd (i, Cn (im, c, e))) =
-    (if ((im : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int))
-  | zeta (NDvd (i, Cn (jm, c, e))) =
-    (if ((jm : IntInf.int) = (0 : IntInf.int)) then c else (1 : IntInf.int));
-
-fun zsplit0 (C c) = ((0 : IntInf.int), C c)
-  | zsplit0 (Bound n) =
-    (if ((n : IntInf.int) = (0 : IntInf.int))
-      then ((1 : IntInf.int), C (0 : IntInf.int))
-      else ((0 : IntInf.int), Bound n))
-  | zsplit0 (Cn (n, i, a)) =
-    let
-      val (ia, aa) = zsplit0 a;
-    in
-      (if ((n : IntInf.int) = (0 : IntInf.int)) then (IntInf.+ (i, ia), aa)
-        else (ia, Cn (n, i, aa)))
-    end
-  | zsplit0 (Neg a) =
-    let
-      val (i, aa) = zsplit0 a;
-    in
-      (IntInf.~ i, Neg aa)
-    end
-  | zsplit0 (Add (a, b)) =
-    let
-      val (ia, aa) = zsplit0 a;
-      val (ib, ba) = zsplit0 b;
-    in
-      (IntInf.+ (ia, ib), Add (aa, ba))
-    end
-  | zsplit0 (Sub (a, b)) =
-    let
-      val (ia, aa) = zsplit0 a;
-      val (ib, ba) = zsplit0 b;
-    in
-      (IntInf.- (ia, ib), Sub (aa, ba))
-    end
-  | zsplit0 (Mul (i, a)) =
-    let
-      val (ia, aa) = zsplit0 a;
-    in
-      (IntInf.* (i, ia), Mul (i, aa))
-    end;
-
-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 (c, r) = zsplit0 a;
-    in
-      (if ((c : IntInf.int) = (0 : IntInf.int)) then Lt r
-        else (if IntInf.< ((0 : IntInf.int), c)
-               then Lt (Cn ((0 : IntInf.int), c, r))
-               else Gt (Cn ((0 : IntInf.int), 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 (Cn ((0 : IntInf.int), c, r))
-               else Ge (Cn ((0 : IntInf.int), 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 (Cn ((0 : IntInf.int), c, r))
-               else Lt (Cn ((0 : IntInf.int), 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 (Cn ((0 : IntInf.int), c, r))
-               else Le (Cn ((0 : IntInf.int), 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 (Cn ((0 : IntInf.int), c, r))
-               else Eq (Cn ((0 : IntInf.int), 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 (Cn ((0 : IntInf.int), c, r))
-               else NEq (Cn ((0 : IntInf.int), 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 (abs_int i, r)
-               else (if IntInf.< ((0 : IntInf.int), c)
-                      then Dvd (abs_int i, Cn ((0 : IntInf.int), c, r))
-                      else Dvd (abs_int i,
-                                 Cn ((0 : IntInf.int), IntInf.~ c, Neg r))))
-           end)
-  | 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 (abs_int i, r)
-               else (if IntInf.< ((0 : IntInf.int), c)
-                      then NDvd (abs_int i, Cn ((0 : IntInf.int), c, r))
-                      else NDvd (abs_int i,
-                                  Cn ((0 : IntInf.int), IntInf.~ c, Neg r))))
-           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 unita p =
-  let
-    val pa = zlfm p;
-    val l = zeta pa;
-    val q =
-      And (Dvd (l, Cn ((0 : IntInf.int), (1 : IntInf.int), C (0 : IntInf.int))),
-            a_beta pa l);
-    val d = delta q;
-    val b = remdups eq_numa (map simpnum (beta q));
-    val a = remdups eq_numa (map simpnum (alpha q));
-  in
-    (if IntInf.<= (size_list b, size_list a) then (q, (b, d))
-      else (mirror q, (a, d)))
-  end;
-
-fun cooper p =
-  let
-    val (q, (b, d)) = unita p;
-    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 (ba, j) => simpfm (subst0 (Add (ba, C j)) q))
-                 (concat_map (fn ba => map (fn a => (ba, a)) js) b);
-           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))
-  | 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;
-
-fun qelim (E p) = (fn qe => dj qe (qelim p qe))
-  | qelim (A p) = (fn qe => nota (qe (qelim (Not p) qe)))
-  | qelim (Not p) = (fn qe => nota (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 => impa (qelim p qe) (qelim q qe))
-  | qelim (Iff (p, q)) = (fn qe => iffa (qelim p qe) (qelim q qe))
-  | qelim T = (fn _ => simpfm T)
-  | qelim F = (fn _ => simpfm F)
-  | qelim (Lt u) = (fn _ => simpfm (Lt u))
-  | qelim (Le v) = (fn _ => simpfm (Le v))
-  | qelim (Gt w) = (fn _ => simpfm (Gt w))
-  | qelim (Ge x) = (fn _ => simpfm (Ge x))
-  | qelim (Eq y) = (fn _ => simpfm (Eq y))
-  | qelim (NEq z) = (fn _ => simpfm (NEq z))
-  | qelim (Dvd (aa, ab)) = (fn _ => simpfm (Dvd (aa, ab)))
-  | qelim (NDvd (ac, ad)) = (fn _ => simpfm (NDvd (ac, ad)))
-  | qelim (Closed ap) = (fn _ => simpfm (Closed ap))
-  | qelim (NClosed aq) = (fn _ => simpfm (NClosed aq));
-
-fun pa p = qelim (prep p) cooper;
-
-end; (*struct Generated_Cooper*)