src/HOL/Library/positivstellensatz.ML

 Fourrier-Motzkin elimination.
 *)
 
 (* A functor for finite mappings based on Tables *)
 
-signature FUNC = 
+signature FUNC =
 sig
- include TABLE
- val apply : 'a table -> key -> 'a
- val applyd :'a table -> (key -> 'a) -> key -> 'a
- val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a table -> 'a table -> 'a table
- val dom : 'a table -> key list
- val tryapplyd : 'a table -> key -> 'a -> 'a
- val updatep : (key * 'a -> bool) -> key * 'a -> 'a table -> 'a table
- val choose : 'a table -> key * 'a
- val onefunc : key * 'a -> 'a table
+  include TABLE
+  val apply : 'a table -> key -> 'a
+  val applyd :'a table -> (key -> 'a) -> key -> 'a
+  val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a table -> 'a table -> 'a table
+  val dom : 'a table -> key list
+  val tryapplyd : 'a table -> key -> 'a -> 'a
+  val updatep : (key * 'a -> bool) -> key * 'a -> 'a table -> 'a table
+  val choose : 'a table -> key * 'a
+  val onefunc : key * 'a -> 'a table
 end;
 
-functor FuncFun(Key: KEY) : FUNC=
+functor FuncFun(Key: KEY) : FUNC =
 struct
 
 structure Tab = Table(Key);
 
 open Tab;
 
 fun dom a = sort Key.ord (Tab.keys a);
-fun applyd f d x = case Tab.lookup f x of 
+fun applyd f d x = case Tab.lookup f x of
    SOME y => y
  | NONE => d x;
 
 fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
 fun tryapplyd f a d = applyd f (K d) a;
 fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
-fun combine f z a b = 
- let
-  fun h (k,v) t = case Tab.lookup t k of
-     NONE => Tab.update (k,v) t
-   | SOME v' => let val w = f v v'
-     in if z w then Tab.delete k t else Tab.update (k,w) t end;
+fun combine f z a b =
+  let
+    fun h (k,v) t = case Tab.lookup t k of
+        NONE => Tab.update (k,v) t
+      | SOME v' => let val w = f v v'
+        in if z w then Tab.delete k t else Tab.update (k,w) t end;
   in Tab.fold h a b end;
 
-fun choose f = case Tab.min_key f of 
-   SOME k => (k, the (Tab.lookup f k))
- | NONE => error "FuncFun.choose : Completely empty function"
+fun choose f = case Tab.min_key f of
+    SOME k => (k, the (Tab.lookup f k))
+  | NONE => error "FuncFun.choose : Completely empty function"
 
 fun onefunc kv = update kv empty
 
 end;
 

 (* The ordering so we can create canonical HOL polynomials.                  *)
 
 fun dest_monomial mon = sort (cterm_ord o pairself fst) (Ctermfunc.dest mon);
 
 fun monomial_order (m1,m2) =
- if Ctermfunc.is_empty m2 then LESS 
- else if Ctermfunc.is_empty m1 then GREATER 
- else
-  let val mon1 = dest_monomial m1 
+  if Ctermfunc.is_empty m2 then LESS
+  else if Ctermfunc.is_empty m1 then GREATER
+  else
+    let
+      val mon1 = dest_monomial m1
       val mon2 = dest_monomial m2
       val deg1 = fold (Integer.add o snd) mon1 0
-      val deg2 = fold (Integer.add o snd) mon2 0 
-  in if deg1 < deg2 then GREATER else if deg1 > deg2 then LESS
-     else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
-  end;
+      val deg2 = fold (Integer.add o snd) mon2 0
+    in if deg1 < deg2 then GREATER
+       else if deg1 > deg2 then LESS
+       else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
+    end;
 
 end
 
 (* positivstellensatz datatype and prover generation *)
 
-signature REAL_ARITH = 
+signature REAL_ARITH =
 sig
-  
+
   datatype positivstellensatz =
-   Axiom_eq of int
- | Axiom_le of int
- | Axiom_lt of int
- | Rational_eq of Rat.rat
- | Rational_le of Rat.rat
- | Rational_lt of Rat.rat
- | Square of FuncUtil.poly
- | Eqmul of FuncUtil.poly * positivstellensatz
- | Sum of positivstellensatz * positivstellensatz
- | Product of positivstellensatz * positivstellensatz;
-
-datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
-
-datatype tree_choice = Left | Right
-
-type prover = tree_choice list -> 
-  (thm list * thm list * thm list -> positivstellensatz -> thm) ->
-  thm list * thm list * thm list -> thm * pss_tree
-type cert_conv = cterm -> thm * pss_tree
-
-val gen_gen_real_arith :
-  Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
-   conv * conv * conv * conv * conv * conv * prover -> cert_conv
-val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
-  thm list * thm list * thm list -> thm * pss_tree
-
-val gen_real_arith : Proof.context ->
-  (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
-
-val gen_prover_real_arith : Proof.context -> prover -> cert_conv
-
-val is_ratconst : cterm -> bool
-val dest_ratconst : cterm -> Rat.rat
-val cterm_of_rat : Rat.rat -> cterm
+    Axiom_eq of int
+  | Axiom_le of int
+  | Axiom_lt of int
+  | Rational_eq of Rat.rat
+  | Rational_le of Rat.rat
+  | Rational_lt of Rat.rat
+  | Square of FuncUtil.poly
+  | Eqmul of FuncUtil.poly * positivstellensatz
+  | Sum of positivstellensatz * positivstellensatz
+  | Product of positivstellensatz * positivstellensatz;
+
+  datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
+
+  datatype tree_choice = Left | Right
+
+  type prover = tree_choice list ->
+    (thm list * thm list * thm list -> positivstellensatz -> thm) ->
+      thm list * thm list * thm list -> thm * pss_tree
+  type cert_conv = cterm -> thm * pss_tree
+
+  val gen_gen_real_arith :
+    Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
+     conv * conv * conv * conv * conv * conv * prover -> cert_conv
+  val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
+    thm list * thm list * thm list -> thm * pss_tree
+
+  val gen_real_arith : Proof.context ->
+    (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
+
+  val gen_prover_real_arith : Proof.context -> prover -> cert_conv
+
+  val is_ratconst : cterm -> bool
+  val dest_ratconst : cterm -> Rat.rat
+  val cterm_of_rat : Rat.rat -> cterm
 
 end
 
 structure RealArith : REAL_ARITH =
 struct
 
- open Conv
+open Conv
 (* ------------------------------------------------------------------------- *)
 (* Data structure for Positivstellensatz refutations.                        *)
 (* ------------------------------------------------------------------------- *)
 
 datatype positivstellensatz =
-   Axiom_eq of int
- | Axiom_le of int
- | Axiom_lt of int
- | Rational_eq of Rat.rat
- | Rational_le of Rat.rat
- | Rational_lt of Rat.rat
- | Square of FuncUtil.poly
- | Eqmul of FuncUtil.poly * positivstellensatz
- | Sum of positivstellensatz * positivstellensatz
- | Product of positivstellensatz * positivstellensatz;
+    Axiom_eq of int
+  | Axiom_le of int
+  | Axiom_lt of int
+  | Rational_eq of Rat.rat
+  | Rational_le of Rat.rat
+  | Rational_lt of Rat.rat
+  | Square of FuncUtil.poly
+  | Eqmul of FuncUtil.poly * positivstellensatz
+  | Sum of positivstellensatz * positivstellensatz
+  | Product of positivstellensatz * positivstellensatz;
          (* Theorems used in the procedure *)
 
 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
 datatype tree_choice = Left | Right
-type prover = tree_choice list -> 
+type prover = tree_choice list ->
   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
-  thm list * thm list * thm list -> thm * pss_tree
+    thm list * thm list * thm list -> thm * pss_tree
 type cert_conv = cterm -> thm * pss_tree
 
 
     (* Some useful derived rules *)
-fun deduct_antisym_rule tha thb = 
-    Thm.equal_intr (Thm.implies_intr (cprop_of thb) tha) 
+fun deduct_antisym_rule tha thb =
+    Thm.equal_intr (Thm.implies_intr (cprop_of thb) tha)
      (Thm.implies_intr (cprop_of tha) thb);
 
 fun prove_hyp tha thb =
   if exists (curry op aconv (concl_of tha)) (Thm.hyps_of thb)  (* FIXME !? *)
   then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb;

      "((x = y) == (x - y = 0))" and "((~(x < y)) == (x - y >= 0))" and
      "((~(x <= y)) == (x - y > 0))" and "((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
   by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};
 
 val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
-val pth_add = 
+val pth_add =
   @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 )" and "( x = 0 ==> y >= 0 ==> x + y >= 0)" and
     "(x = 0 ==> y > 0 ==> x + y > 0)" and "(x >= 0 ==> y = 0 ==> x + y >= 0)" and
     "(x >= 0 ==> y >= 0 ==> x + y >= 0)" and "(x >= 0 ==> y > 0 ==> x + y > 0)" and
     "(x > 0 ==> y = 0 ==> x + y > 0)" and "(x > 0 ==> y >= 0 ==> x + y > 0)" and
     "(x > 0 ==> y > 0 ==> x + y > 0)" by simp_all};
 
-val pth_mul = 
+val pth_mul =
   @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0)" and "(x = 0 ==> y >= 0 ==> x * y = 0)" and
     "(x = 0 ==> y > 0 ==> x * y = 0)" and "(x >= 0 ==> y = 0 ==> x * y = 0)" and
     "(x >= 0 ==> y >= 0 ==> x * y >= 0)" and "(x >= 0 ==> y > 0 ==> x * y >= 0)" and
     "(x > 0 ==>  y = 0 ==> x * y = 0)" and "(x > 0 ==> y >= 0 ==> x * y >= 0)" and
     "(x > 0 ==>  y > 0 ==> x * y > 0)"

 val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
   by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
 
 
          (* Miscellaneous *)
-fun literals_conv bops uops cv = 
- let fun h t =
-  case (term_of t) of 
-   b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
- | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
- | _ => cv t
- in h end;
-
-fun cterm_of_rat x = 
-let val (a, b) = Rat.quotient_of_rat x
-in 
- if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
-  else Thm.apply (Thm.apply @{cterm "op / :: real => _"} 
-                   (Numeral.mk_cnumber @{ctyp "real"} a))
-        (Numeral.mk_cnumber @{ctyp "real"} b)
-end;
-
-  fun dest_ratconst t = case term_of t of
-   Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
- | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
- fun is_ratconst t = can dest_ratconst t
+fun literals_conv bops uops cv =
+  let
+    fun h t =
+      case (term_of t) of
+        b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
+      | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
+      | _ => cv t
+  in h end;
+
+fun cterm_of_rat x =
+  let
+    val (a, b) = Rat.quotient_of_rat x
+  in
+    if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
+    else Thm.apply (Thm.apply @{cterm "op / :: real => _"}
+      (Numeral.mk_cnumber @{ctyp "real"} a))
+      (Numeral.mk_cnumber @{ctyp "real"} b)
+  end;
+
+fun dest_ratconst t =
+  case term_of t of
+    Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
+  | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
+fun is_ratconst t = can dest_ratconst t
 
 (*
-fun find_term p t = if p t then t else 
+fun find_term p t = if p t then t else
  case t of
   a$b => (find_term p a handle TERM _ => find_term p b)
  | Abs (_,_,t') => find_term p t'
  | _ => raise TERM ("find_term",[t]);
 *)
 
-fun find_cterm p t = if p t then t else 
- case term_of t of
-  _$_ => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
- | Abs (_,_,_) => find_cterm p (Thm.dest_abs NONE t |> snd)
- | _ => raise CTERM ("find_cterm",[t]);
+fun find_cterm p t =
+  if p t then t else
+  case term_of t of
+    _$_ => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
+  | Abs (_,_,_) => find_cterm p (Thm.dest_abs NONE t |> snd)
+  | _ => raise CTERM ("find_cterm",[t]);
 
     (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
 fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
 fun is_comb t = case (term_of t) of _$_ => true | _ => false;
 

   handle CTERM _ => false;
 
 
 (* Map back polynomials to HOL.                         *)
 
-fun cterm_of_varpow x k = if k = 1 then x else Thm.apply (Thm.apply @{cterm "op ^ :: real => _"} x) 
+fun cterm_of_varpow x k = if k = 1 then x else Thm.apply (Thm.apply @{cterm "op ^ :: real => _"} x)
   (Numeral.mk_cnumber @{ctyp nat} k)
 
-fun cterm_of_monomial m = 
- if FuncUtil.Ctermfunc.is_empty m then @{cterm "1::real"} 
- else 
-  let 
-   val m' = FuncUtil.dest_monomial m
-   val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' [] 
-  in foldr1 (fn (s, t) => Thm.apply (Thm.apply @{cterm "op * :: real => _"} s) t) vps
-  end
-
-fun cterm_of_cmonomial (m,c) = if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
-    else if c = Rat.one then cterm_of_monomial m
-    else Thm.apply (Thm.apply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
-
-fun cterm_of_poly p = 
- if FuncUtil.Monomialfunc.is_empty p then @{cterm "0::real"} 
- else
-  let 
-   val cms = map cterm_of_cmonomial
-     (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
-  in foldr1 (fn (t1, t2) => Thm.apply(Thm.apply @{cterm "op + :: real => _"} t1) t2) cms
-  end;
-
-    (* A general real arithmetic prover *)
+fun cterm_of_monomial m =
+  if FuncUtil.Ctermfunc.is_empty m then @{cterm "1::real"}
+  else
+    let
+      val m' = FuncUtil.dest_monomial m
+      val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' []
+    in foldr1 (fn (s, t) => Thm.apply (Thm.apply @{cterm "op * :: real => _"} s) t) vps
+    end
+
+fun cterm_of_cmonomial (m,c) =
+  if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
+  else if c = Rat.one then cterm_of_monomial m
+  else Thm.apply (Thm.apply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
+
+fun cterm_of_poly p =
+  if FuncUtil.Monomialfunc.is_empty p then @{cterm "0::real"}
+  else
+    let
+      val cms = map cterm_of_cmonomial
+        (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
+    in foldr1 (fn (t1, t2) => Thm.apply(Thm.apply @{cterm "op + :: real => _"} t1) t2) cms
+    end;
+
+(* A general real arithmetic prover *)
 
 fun gen_gen_real_arith ctxt (mk_numeric,
        numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
        poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
-       absconv1,absconv2,prover) = 
-let
- val pre_ss = HOL_basic_ss addsimps
-  @{thms simp_thms ex_simps all_simps not_all not_ex ex_disj_distrib all_conj_distrib if_bool_eq_disj}
- val prenex_ss = HOL_basic_ss addsimps prenex_simps
- val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
- val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
- val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
- val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
- val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
- val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
- fun eqT_elim th = Thm.equal_elim (Thm.symmetric th) @{thm TrueI}
- fun oprconv cv ct = 
-  let val g = Thm.dest_fun2 ct
-  in if g aconvc @{cterm "op <= :: real => _"} 
-       orelse g aconvc @{cterm "op < :: real => _"} 
-     then arg_conv cv ct else arg1_conv cv ct
-  end
-
- fun real_ineq_conv th ct =
+       absconv1,absconv2,prover) =
   let
-   val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th 
-      handle Pattern.MATCH => raise CTERM ("real_ineq_conv", [ct]))
-  in Thm.transitive th' (oprconv poly_conv (Thm.rhs_of th'))
-  end 
-  val [real_lt_conv, real_le_conv, real_eq_conv,
-       real_not_lt_conv, real_not_le_conv, _] =
-       map real_ineq_conv pth
-  fun match_mp_rule ths ths' = 
-   let
-     fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
-      | th::ths => (ths' MRS th handle THM _ => f ths ths')
-   in f ths ths' end
-  fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
+    val pre_ss = HOL_basic_ss addsimps
+      @{thms simp_thms ex_simps all_simps not_all not_ex ex_disj_distrib all_conj_distrib if_bool_eq_disj}
+    val prenex_ss = HOL_basic_ss addsimps prenex_simps
+    val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
+    val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
+    val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
+    val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
+    val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
+    val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
+    fun eqT_elim th = Thm.equal_elim (Thm.symmetric th) @{thm TrueI}
+    fun oprconv cv ct =
+      let val g = Thm.dest_fun2 ct
+      in if g aconvc @{cterm "op <= :: real => _"}
+            orelse g aconvc @{cterm "op < :: real => _"}
+         then arg_conv cv ct else arg1_conv cv ct
+      end
+
+    fun real_ineq_conv th ct =
+      let
+        val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th
+          handle Pattern.MATCH => raise CTERM ("real_ineq_conv", [ct]))
+      in Thm.transitive th' (oprconv poly_conv (Thm.rhs_of th'))
+      end
+    val [real_lt_conv, real_le_conv, real_eq_conv,
+         real_not_lt_conv, real_not_le_conv, _] =
+         map real_ineq_conv pth
+    fun match_mp_rule ths ths' =
+      let
+        fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
+          | th::ths => (ths' MRS th handle THM _ => f ths ths')
+      in f ths ths' end
+    fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
          (match_mp_rule pth_mul [th, th'])
-  fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
+    fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
          (match_mp_rule pth_add [th, th'])
-  fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv)) 
-       (instantiate' [] [SOME ct] (th RS pth_emul)) 
-  fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
+    fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
+       (instantiate' [] [SOME ct] (th RS pth_emul))
+    fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
        (instantiate' [] [SOME t] pth_square)
 
-  fun hol_of_positivstellensatz(eqs,les,lts) proof =
-   let 
-    fun translate prf = case prf of
-        Axiom_eq n => nth eqs n
-      | Axiom_le n => nth les n
-      | Axiom_lt n => nth lts n
-      | Rational_eq x => eqT_elim(numeric_eq_conv(Thm.apply @{cterm Trueprop} 
-                          (Thm.apply (Thm.apply @{cterm "op =::real => _"} (mk_numeric x)) 
+    fun hol_of_positivstellensatz(eqs,les,lts) proof =
+      let
+        fun translate prf =
+          case prf of
+            Axiom_eq n => nth eqs n
+          | Axiom_le n => nth les n
+          | Axiom_lt n => nth lts n
+          | Rational_eq x => eqT_elim(numeric_eq_conv(Thm.apply @{cterm Trueprop}
+                          (Thm.apply (Thm.apply @{cterm "op =::real => _"} (mk_numeric x))
                                @{cterm "0::real"})))
-      | Rational_le x => eqT_elim(numeric_ge_conv(Thm.apply @{cterm Trueprop} 
-                          (Thm.apply (Thm.apply @{cterm "op <=::real => _"} 
+          | Rational_le x => eqT_elim(numeric_ge_conv(Thm.apply @{cterm Trueprop}
+                          (Thm.apply (Thm.apply @{cterm "op <=::real => _"}
                                      @{cterm "0::real"}) (mk_numeric x))))
-      | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.apply @{cterm Trueprop} 
+          | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.apply @{cterm Trueprop}
                       (Thm.apply (Thm.apply @{cterm "op <::real => _"} @{cterm "0::real"})
                         (mk_numeric x))))
-      | Square pt => square_rule (cterm_of_poly pt)
-      | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
-      | Sum(p1,p2) => add_rule (translate p1) (translate p2)
-      | Product(p1,p2) => mul_rule (translate p1) (translate p2)
-   in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv]) 
+          | Square pt => square_rule (cterm_of_poly pt)
+          | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
+          | Sum(p1,p2) => add_rule (translate p1) (translate p2)
+          | Product(p1,p2) => mul_rule (translate p1) (translate p2)
+      in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv])
           (translate proof)
-   end
-  
-  val init_conv = presimp_conv then_conv
-      nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
-      weak_dnf_conv
-
-  val concl = Thm.dest_arg o cprop_of
-  fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
-  val is_req = is_binop @{cterm "op =:: real => _"}
-  val is_ge = is_binop @{cterm "op <=:: real => _"}
-  val is_gt = is_binop @{cterm "op <:: real => _"}
-  val is_conj = is_binop @{cterm HOL.conj}
-  val is_disj = is_binop @{cterm HOL.disj}
-  fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
-  fun disj_cases th th1 th2 = 
-   let val (p,q) = Thm.dest_binop (concl th)
-       val c = concl th1
-       val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
-   in Thm.implies_elim (Thm.implies_elim
+      end
+
+    val init_conv = presimp_conv then_conv
+        nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
+        weak_dnf_conv
+
+    val concl = Thm.dest_arg o cprop_of
+    fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
+    val is_req = is_binop @{cterm "op =:: real => _"}
+    val is_ge = is_binop @{cterm "op <=:: real => _"}
+    val is_gt = is_binop @{cterm "op <:: real => _"}
+    val is_conj = is_binop @{cterm HOL.conj}
+    val is_disj = is_binop @{cterm HOL.disj}
+    fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
+    fun disj_cases th th1 th2 =
+      let
+        val (p,q) = Thm.dest_binop (concl th)
+        val c = concl th1
+        val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
+      in Thm.implies_elim (Thm.implies_elim
           (Thm.implies_elim (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
           (Thm.implies_intr (Thm.apply @{cterm Trueprop} p) th1))
         (Thm.implies_intr (Thm.apply @{cterm Trueprop} q) th2)
-   end
- fun overall cert_choice dun ths = case ths of
-  [] =>
-   let 
-    val (eq,ne) = List.partition (is_req o concl) dun
-     val (le,nl) = List.partition (is_ge o concl) ne
-     val lt = filter (is_gt o concl) nl 
-    in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
- | th::oths =>
-   let 
-    val ct = concl th 
-   in 
-    if is_conj ct  then
-     let 
-      val (th1,th2) = conj_pair th in
-      overall cert_choice dun (th1::th2::oths) end
-    else if is_disj ct then
-      let 
-       val (th1, cert1) = overall (Left::cert_choice) dun (Thm.assume (Thm.apply @{cterm Trueprop} (Thm.dest_arg1 ct))::oths)
-       val (th2, cert2) = overall (Right::cert_choice) dun (Thm.assume (Thm.apply @{cterm Trueprop} (Thm.dest_arg ct))::oths)
-      in (disj_cases th th1 th2, Branch (cert1, cert2)) end
-   else overall cert_choice (th::dun) oths
-  end
-  fun dest_binary b ct = if is_binop b ct then Thm.dest_binop ct 
-                         else raise CTERM ("dest_binary",[b,ct])
-  val dest_eq = dest_binary @{cterm "op = :: real => _"}
-  val neq_th = nth pth 5
-  fun real_not_eq_conv ct = 
-   let 
-    val (l,r) = dest_eq (Thm.dest_arg ct)
-    val th = Thm.instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
-    val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
-    val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
-    val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
-    val th' = Drule.binop_cong_rule @{cterm HOL.disj} 
-     (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
-     (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
-    in Thm.transitive th th' 
-  end
- fun equal_implies_1_rule PQ = 
-  let 
-   val P = Thm.lhs_of PQ
-  in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P))
-  end
- (* FIXME!!! Copied from groebner.ml *)
- val strip_exists =
-  let fun h (acc, t) =
-   case (term_of t) of
-    Const(@{const_name Ex},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
-  | _ => (acc,t)
-  in fn t => h ([],t)
-  end
-  fun name_of x = case term_of x of
-   Free(s,_) => s
- | Var ((s,_),_) => s
- | _ => "x"
-
-  fun mk_forall x th = Drule.arg_cong_rule (instantiate_cterm' [SOME (ctyp_of_term x)] [] @{cpat "All :: (?'a => bool) => _" }) (Thm.abstract_rule (name_of x) x th)
-
-  val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
-
- fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
- fun mk_ex v t = Thm.apply (ext (ctyp_of_term v)) (Thm.lambda v t)
-
- fun choose v th th' = case concl_of th of 
-   @{term Trueprop} $ (Const(@{const_name Ex},_)$_) => 
-    let
-     val p = (funpow 2 Thm.dest_arg o cprop_of) th
-     val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
-     val th0 = fconv_rule (Thm.beta_conversion true)
-         (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
-     val pv = (Thm.rhs_of o Thm.beta_conversion true) 
-           (Thm.apply @{cterm Trueprop} (Thm.apply p v))
-     val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
-    in Thm.implies_elim (Thm.implies_elim th0 th) th1  end
- | _ => raise THM ("choose",0,[th, th'])
-
-  fun simple_choose v th = 
-     choose v (Thm.assume ((Thm.apply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
-
- val strip_forall =
-  let fun h (acc, t) =
-   case (term_of t) of
-    Const(@{const_name All},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
-  | _ => (acc,t)
-  in fn t => h ([],t)
-  end
-
- fun f ct =
-  let 
-   val nnf_norm_conv' = 
-     nnf_conv then_conv 
-     literals_conv [@{term HOL.conj}, @{term HOL.disj}] [] 
-     (Conv.cache_conv 
-       (first_conv [real_lt_conv, real_le_conv, 
-                    real_eq_conv, real_not_lt_conv, 
-                    real_not_le_conv, real_not_eq_conv, all_conv]))
-  fun absremover ct = (literals_conv [@{term HOL.conj}, @{term HOL.disj}] [] 
-                  (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv 
-        try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
-  val nct = Thm.apply @{cterm Trueprop} (Thm.apply @{cterm "Not"} ct)
-  val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
-  val tm0 = Thm.dest_arg (Thm.rhs_of th0)
-  val (th, certificates) = if tm0 aconvc @{cterm False} then (equal_implies_1_rule th0, Trivial) else
-   let 
-    val (evs,bod) = strip_exists tm0
-    val (avs,ibod) = strip_forall bod
-    val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
-    val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))]
-    val th3 = fold simple_choose evs (prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.apply @{cterm Trueprop} bod))) th2)
-   in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
-   end
-  in (Thm.implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates)
- end
-in f
-end;
+      end
+    fun overall cert_choice dun ths =
+      case ths of
+        [] =>
+        let
+          val (eq,ne) = List.partition (is_req o concl) dun
+          val (le,nl) = List.partition (is_ge o concl) ne
+          val lt = filter (is_gt o concl) nl
+        in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
+      | th::oths =>
+        let
+          val ct = concl th
+        in
+          if is_conj ct then
+            let
+              val (th1,th2) = conj_pair th
+            in overall cert_choice dun (th1::th2::oths) end
+          else if is_disj ct then
+            let
+              val (th1, cert1) = overall (Left::cert_choice) dun (Thm.assume (Thm.apply @{cterm Trueprop} (Thm.dest_arg1 ct))::oths)
+              val (th2, cert2) = overall (Right::cert_choice) dun (Thm.assume (Thm.apply @{cterm Trueprop} (Thm.dest_arg ct))::oths)
+            in (disj_cases th th1 th2, Branch (cert1, cert2)) end
+          else overall cert_choice (th::dun) oths
+        end
+    fun dest_binary b ct =
+        if is_binop b ct then Thm.dest_binop ct
+        else raise CTERM ("dest_binary",[b,ct])
+    val dest_eq = dest_binary @{cterm "op = :: real => _"}
+    val neq_th = nth pth 5
+    fun real_not_eq_conv ct =
+      let
+        val (l,r) = dest_eq (Thm.dest_arg ct)
+        val th = Thm.instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
+        val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
+        val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
+        val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
+        val th' = Drule.binop_cong_rule @{cterm HOL.disj}
+          (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
+          (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
+      in Thm.transitive th th'
+      end
+    fun equal_implies_1_rule PQ =
+      let
+        val P = Thm.lhs_of PQ
+      in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P))
+      end
+    (* FIXME!!! Copied from groebner.ml *)
+    val strip_exists =
+      let
+        fun h (acc, t) =
+          case (term_of t) of
+            Const(@{const_name Ex},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
+          | _ => (acc,t)
+      in fn t => h ([],t)
+      end
+    fun name_of x =
+      case term_of x of
+        Free(s,_) => s
+      | Var ((s,_),_) => s
+      | _ => "x"
+
+    fun mk_forall x th = Drule.arg_cong_rule (instantiate_cterm' [SOME (ctyp_of_term x)] [] @{cpat "All :: (?'a => bool) => _" }) (Thm.abstract_rule (name_of x) x th)
+
+    val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
+
+    fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
+    fun mk_ex v t = Thm.apply (ext (ctyp_of_term v)) (Thm.lambda v t)
+
+    fun choose v th th' =
+      case concl_of th of
+        @{term Trueprop} $ (Const(@{const_name Ex},_)$_) =>
+        let
+          val p = (funpow 2 Thm.dest_arg o cprop_of) th
+          val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
+          val th0 = fconv_rule (Thm.beta_conversion true)
+            (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
+          val pv = (Thm.rhs_of o Thm.beta_conversion true)
+            (Thm.apply @{cterm Trueprop} (Thm.apply p v))
+          val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
+        in Thm.implies_elim (Thm.implies_elim th0 th) th1  end
+      | _ => raise THM ("choose",0,[th, th'])
+
+    fun simple_choose v th =
+      choose v (Thm.assume ((Thm.apply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
+
+    val strip_forall =
+      let
+        fun h (acc, t) =
+          case (term_of t) of
+            Const(@{const_name All},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
+          | _ => (acc,t)
+      in fn t => h ([],t)
+      end
+
+    fun f ct =
+      let
+        val nnf_norm_conv' =
+          nnf_conv then_conv
+          literals_conv [@{term HOL.conj}, @{term HOL.disj}] []
+          (Conv.cache_conv
+            (first_conv [real_lt_conv, real_le_conv,
+                         real_eq_conv, real_not_lt_conv,
+                         real_not_le_conv, real_not_eq_conv, all_conv]))
+        fun absremover ct = (literals_conv [@{term HOL.conj}, @{term HOL.disj}] []
+                  (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv
+                  try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
+        val nct = Thm.apply @{cterm Trueprop} (Thm.apply @{cterm "Not"} ct)
+        val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
+        val tm0 = Thm.dest_arg (Thm.rhs_of th0)
+        val (th, certificates) =
+          if tm0 aconvc @{cterm False} then (equal_implies_1_rule th0, Trivial) else
+          let
+            val (evs,bod) = strip_exists tm0
+            val (avs,ibod) = strip_forall bod
+            val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
+            val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))]
+            val th3 = fold simple_choose evs (prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.apply @{cterm Trueprop} bod))) th2)
+          in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
+          end
+      in (Thm.implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates)
+      end
+  in f
+  end;
 
 (* A linear arithmetic prover *)
 local
   val linear_add = FuncUtil.Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
   fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c */ x)
   val one_tm = @{cterm "1::real"}
   fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p Rat.zero)) orelse
      ((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso
        not(p(FuncUtil.Ctermfunc.apply e one_tm)))
 
-  fun linear_ineqs vars (les,lts) = 
-   case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
-    SOME r => r
-  | NONE => 
-   (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
-     SOME r => r
-   | NONE => 
-     if null vars then error "linear_ineqs: no contradiction" else
-     let 
-      val ineqs = les @ lts
-      fun blowup v =
-       length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
-       length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
-       length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
-      val  v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
-                 (map (fn v => (v,blowup v)) vars)))
-      fun addup (e1,p1) (e2,p2) acc =
-       let 
-        val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v Rat.zero 
-        val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v Rat.zero
-       in if c1 */ c2 >=/ Rat.zero then acc else
-        let 
-         val e1' = linear_cmul (Rat.abs c2) e1
-         val e2' = linear_cmul (Rat.abs c1) e2
-         val p1' = Product(Rational_lt(Rat.abs c2),p1)
-         val p2' = Product(Rational_lt(Rat.abs c1),p2)
-        in (linear_add e1' e2',Sum(p1',p2'))::acc
+  fun linear_ineqs vars (les,lts) =
+    case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
+      SOME r => r
+    | NONE =>
+      (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
+         SOME r => r
+       | NONE =>
+         if null vars then error "linear_ineqs: no contradiction" else
+         let
+           val ineqs = les @ lts
+           fun blowup v =
+             length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
+             length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
+             length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
+           val v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
+             (map (fn v => (v,blowup v)) vars)))
+           fun addup (e1,p1) (e2,p2) acc =
+             let
+               val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v Rat.zero
+               val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v Rat.zero
+             in
+               if c1 */ c2 >=/ Rat.zero then acc else
+               let
+                 val e1' = linear_cmul (Rat.abs c2) e1
+                 val e2' = linear_cmul (Rat.abs c1) e2
+                 val p1' = Product(Rational_lt(Rat.abs c2),p1)
+                 val p2' = Product(Rational_lt(Rat.abs c1),p2)
+               in (linear_add e1' e2',Sum(p1',p2'))::acc
+               end
+             end
+           val (les0,les1) =
+             List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
+           val (lts0,lts1) =
+             List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
+           val (lesp,lesn) =
+             List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
+           val (ltsp,ltsn) =
+             List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
+           val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
+           val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
+                      (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
+         in linear_ineqs (remove (op aconvc) v vars) (les',lts')
+         end)
+
+  fun linear_eqs(eqs,les,lts) =
+    case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
+      SOME r => r
+    | NONE =>
+      (case eqs of
+         [] =>
+         let val vars = remove (op aconvc) one_tm
+             (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) [])
+         in linear_ineqs vars (les,lts) end
+       | (e,p)::es =>
+         if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
+         let
+           val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
+           fun xform (inp as (t,q)) =
+             let val d = FuncUtil.Ctermfunc.tryapplyd t x Rat.zero in
+               if d =/ Rat.zero then inp else
+               let
+                 val k = (Rat.neg d) */ Rat.abs c // c
+                 val e' = linear_cmul k e
+                 val t' = linear_cmul (Rat.abs c) t
+                 val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
+                 val q' = Product(Rational_lt(Rat.abs c),q)
+               in (linear_add e' t',Sum(p',q'))
+               end
+             end
+         in linear_eqs(map xform es,map xform les,map xform lts)
+         end)
+
+  fun linear_prover (eq,le,lt) =
+    let
+      val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq
+      val les = map_index (fn (n, p) => (p,Axiom_le n)) le
+      val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt
+    in linear_eqs(eqs,les,lts)
+    end
+
+  fun lin_of_hol ct =
+    if ct aconvc @{cterm "0::real"} then FuncUtil.Ctermfunc.empty
+    else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
+    else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
+    else
+      let val (lop,r) = Thm.dest_comb ct
+      in
+        if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
+        else
+          let val (opr,l) = Thm.dest_comb lop
+          in
+            if opr aconvc @{cterm "op + :: real =>_"}
+            then linear_add (lin_of_hol l) (lin_of_hol r)
+            else if opr aconvc @{cterm "op * :: real =>_"}
+                    andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
+            else FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
+          end
+      end
+
+  fun is_alien ct =
+      case term_of ct of
+        Const(@{const_name "real"}, _)$ n =>
+        if can HOLogic.dest_number n then false else true
+      | _ => false
+in
+fun real_linear_prover translator (eq,le,lt) =
+  let
+    val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o cprop_of
+    val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o cprop_of
+    val eq_pols = map lhs eq
+    val le_pols = map rhs le
+    val lt_pols = map rhs lt
+    val aliens = filter is_alien
+      (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom)
+                (eq_pols @ le_pols @ lt_pols) [])
+    val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,Rat.one)) aliens
+    val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
+    val le' = le @ map (fn a => instantiate' [] [SOME (Thm.dest_arg a)] @{thm real_of_nat_ge_zero}) aliens
+  in ((translator (eq,le',lt) proof), Trivial)
+  end
+end;
+
+(* A less general generic arithmetic prover dealing with abs,max and min*)
+
+local
+  val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
+  fun absmaxmin_elim_conv1 ctxt =
+    Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
+
+  val absmaxmin_elim_conv2 =
+    let
+      val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
+      val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
+      val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
+      val abs_tm = @{cterm "abs :: real => _"}
+      val p_tm = @{cpat "?P :: real => bool"}
+      val x_tm = @{cpat "?x :: real"}
+      val y_tm = @{cpat "?y::real"}
+      val is_max = is_binop @{cterm "max :: real => _"}
+      val is_min = is_binop @{cterm "min :: real => _"}
+      fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm
+      fun eliminate_construct p c tm =
+        let
+          val t = find_cterm p tm
+          val th0 = (Thm.symmetric o Thm.beta_conversion false) (Thm.apply (Thm.lambda t tm) t)
+          val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0
+        in fconv_rule(arg_conv(binop_conv (arg_conv (Thm.beta_conversion false))))
+                     (Thm.transitive th0 (c p ax))
         end
-       end
-      val (les0,les1) = 
-         List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
-      val (lts0,lts1) = 
-         List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
-      val (lesp,lesn) = 
-         List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
-      val (ltsp,ltsn) = 
-         List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
-      val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
-      val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
-                      (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
-     in linear_ineqs (remove (op aconvc) v vars) (les',lts')
-     end)
-
-  fun linear_eqs(eqs,les,lts) = 
-   case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
-    SOME r => r
-  | NONE => (case eqs of 
-    [] => 
-     let val vars = remove (op aconvc) one_tm 
-           (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) []) 
-     in linear_ineqs vars (les,lts) end
-   | (e,p)::es => 
-     if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
-     let 
-      val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
-      fun xform (inp as (t,q)) =
-       let val d = FuncUtil.Ctermfunc.tryapplyd t x Rat.zero in
-        if d =/ Rat.zero then inp else
-        let 
-         val k = (Rat.neg d) */ Rat.abs c // c
-         val e' = linear_cmul k e
-         val t' = linear_cmul (Rat.abs c) t
-         val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
-         val q' = Product(Rational_lt(Rat.abs c),q) 
-        in (linear_add e' t',Sum(p',q')) 
-        end 
-      end
-     in linear_eqs(map xform es,map xform les,map xform lts)
-     end)
-
-  fun linear_prover (eq,le,lt) = 
-   let 
-    val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq
-    val les = map_index (fn (n, p) => (p,Axiom_le n)) le
-    val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt
-   in linear_eqs(eqs,les,lts)
-   end 
-  
-  fun lin_of_hol ct = 
-   if ct aconvc @{cterm "0::real"} then FuncUtil.Ctermfunc.empty
-   else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
-   else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
-   else
-    let val (lop,r) = Thm.dest_comb ct 
-    in if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
-       else
-        let val (opr,l) = Thm.dest_comb lop 
-        in if opr aconvc @{cterm "op + :: real =>_"} 
-           then linear_add (lin_of_hol l) (lin_of_hol r)
-           else if opr aconvc @{cterm "op * :: real =>_"} 
-                   andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
-           else FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
-        end
-    end
-
-  fun is_alien ct = case term_of ct of 
-   Const(@{const_name "real"}, _)$ n => 
-     if can HOLogic.dest_number n then false else true
-  | _ => false
-in 
-fun real_linear_prover translator (eq,le,lt) = 
- let 
-  val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o cprop_of
-  val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o cprop_of
-  val eq_pols = map lhs eq
-  val le_pols = map rhs le
-  val lt_pols = map rhs lt 
-  val aliens =  filter is_alien
-      (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom) 
-          (eq_pols @ le_pols @ lt_pols) [])
-  val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,Rat.one)) aliens
-  val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
-  val le' = le @ map (fn a => instantiate' [] [SOME (Thm.dest_arg a)] @{thm real_of_nat_ge_zero}) aliens 
- in ((translator (eq,le',lt) proof), Trivial)
- end
+
+      val elim_abs = eliminate_construct is_abs
+        (fn p => fn ax =>
+          Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax)]) pth_abs)
+      val elim_max = eliminate_construct is_max
+        (fn p => fn ax =>
+          let val (ax,y) = Thm.dest_comb ax
+          in Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)])
+                             pth_max end)
+      val elim_min = eliminate_construct is_min
+        (fn p => fn ax =>
+          let val (ax,y) = Thm.dest_comb ax
+          in Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)])
+                             pth_min end)
+    in first_conv [elim_abs, elim_max, elim_min, all_conv]
+    end;
+in
+fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
+  gen_gen_real_arith ctxt
+    (mkconst,eq,ge,gt,norm,neg,add,mul,
+     absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
 end;
 
-(* A less general generic arithmetic prover dealing with abs,max and min*)
-
-local
- val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
- fun absmaxmin_elim_conv1 ctxt = 
-    Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
-
- val absmaxmin_elim_conv2 =
-  let 
-   val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
-   val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
-   val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
-   val abs_tm = @{cterm "abs :: real => _"}
-   val p_tm = @{cpat "?P :: real => bool"}
-   val x_tm = @{cpat "?x :: real"}
-   val y_tm = @{cpat "?y::real"}
-   val is_max = is_binop @{cterm "max :: real => _"}
-   val is_min = is_binop @{cterm "min :: real => _"} 
-   fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm
-   fun eliminate_construct p c tm =
-    let 
-     val t = find_cterm p tm
-     val th0 = (Thm.symmetric o Thm.beta_conversion false) (Thm.apply (Thm.lambda t tm) t)
-     val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0
-    in fconv_rule(arg_conv(binop_conv (arg_conv (Thm.beta_conversion false))))
-               (Thm.transitive th0 (c p ax))
-   end
-
-   val elim_abs = eliminate_construct is_abs
-    (fn p => fn ax => 
-       Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax)]) pth_abs)
-   val elim_max = eliminate_construct is_max
-    (fn p => fn ax => 
-      let val (ax,y) = Thm.dest_comb ax 
-      in  Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)]) 
-      pth_max end)
-   val elim_min = eliminate_construct is_min
-    (fn p => fn ax => 
-      let val (ax,y) = Thm.dest_comb ax 
-      in  Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)]) 
-      pth_min end)
-   in first_conv [elim_abs, elim_max, elim_min, all_conv]
+(* An instance for reals*)
+
+fun gen_prover_real_arith ctxt prover =
+  let
+    fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS
+    val {add, mul, neg, pow = _, sub = _, main} =
+        Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
+        (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
+        simple_cterm_ord
+  in gen_real_arith ctxt
+     (cterm_of_rat, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv,
+      main,neg,add,mul, prover)
   end;
-in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
-        gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,
-                       absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
-end;
-
-(* An instance for reals*) 
-
-fun gen_prover_real_arith ctxt prover = 
- let
-  fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS
-  val {add, mul, neg, pow = _, sub = _, main} = 
-     Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
-      (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"})) 
-     simple_cterm_ord
-in gen_real_arith ctxt
-   (cterm_of_rat, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv,
-    main,neg,add,mul, prover)
-end;
 
 end