# HG changeset patch
# User chaieb
# Date 1242145970 -3600
# Node ID fc654c95c29e362b6240af01069a0e46891e76dd
# Parent  2532bb2d65c7cddbd03d2fdfd4fbe9bdd61fac8b
A generic arithmetic prover based on Positivstellensatz certificates --- also implements Fourrier-Motzkin elimination as a special case Fourrier-Motzkin elimination

diff -r 2532bb2d65c7 -r fc654c95c29e src/HOL/Library/positivstellensatz.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/positivstellensatz.ML	Tue May 12 17:32:50 2009 +0100
@@ -0,0 +1,787 @@
+(* Title:      Library/positivstellensatz
+   Author:     Amine Chaieb, University of Cambridge
+   Description: A generic arithmetic prover based on Positivstellensatz certificates --- 
+    also implements Fourrier-Motzkin elimination as a special case Fourrier-Motzkin elimination.
+*)
+
+(* A functor for finite mappings based on Tables *)
+signature FUNC = 
+sig
+ type 'a T
+ type key
+ val apply : 'a T -> key -> 'a
+ val applyd :'a T -> (key -> 'a) -> key -> 'a
+ val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a T -> 'a T -> 'a T
+ val defined : 'a T -> key -> bool
+ val dom : 'a T -> key list
+ val fold : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b
+ val fold_rev : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b
+ val graph : 'a T -> (key * 'a) list
+ val is_undefined : 'a T -> bool
+ val mapf : ('a -> 'b) -> 'a T -> 'b T
+ val tryapplyd : 'a T -> key -> 'a -> 'a
+ val undefine :  key -> 'a T -> 'a T
+ val undefined : 'a T
+ val update : key * 'a -> 'a T -> 'a T
+ val updatep : (key * 'a -> bool) -> key * 'a -> 'a T -> 'a T
+ val choose : 'a T -> key * 'a
+ val onefunc : key * 'a -> 'a T
+ val get_first: (key*'a -> 'a option) -> 'a T -> 'a option
+end;
+
+functor FuncFun(Key: KEY) : FUNC=
+struct
+
+type key = Key.key;
+structure Tab = TableFun(Key);
+type 'a T = 'a Tab.table;
+
+val undefined = Tab.empty;
+val is_undefined = Tab.is_empty;
+val mapf = Tab.map;
+val fold = Tab.fold;
+val fold_rev = Tab.fold_rev;
+val graph = Tab.dest;
+fun dom a = sort Key.ord (Tab.keys a);
+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;
+val defined = Tab.defined;
+fun undefine x t = (Tab.delete x t handle UNDEF => t);
+val update = Tab.update;
+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;
+  in Tab.fold h a b end;
+
+fun choose f = case Tab.min_key f of 
+   SOME k => (k,valOf (Tab.lookup f k))
+ | NONE => error "FuncFun.choose : Completely undefined function"
+
+fun onefunc kv = update kv undefined
+
+local
+fun  find f (k,v) NONE = f (k,v)
+   | find f (k,v) r = r
+in
+fun get_first f t = fold (find f) t NONE
+end
+end;
+
+structure Intfunc = FuncFun(type key = int val ord = int_ord);
+structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
+structure Termfunc = FuncFun(type key = term val ord = TermOrd.fast_term_ord);
+structure Ctermfunc = FuncFun(type key = cterm val ord = (fn (s,t) => TermOrd.fast_term_ord(term_of s, term_of t)));
+
+structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
+    (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
+structure Conv2 = 
+struct
+ open Conv
+fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
+fun is_comb t = case (term_of t) of _$_ => true | _ => false;
+fun is_abs t = case (term_of t) of Abs _ => true | _ => false;
+
+fun end_itlist f l =
+ case l of 
+   []     => error "end_itlist"
+ | [x]    => x
+ | (h::t) => f h (end_itlist f t);
+
+ fun absc cv ct = case term_of ct of 
+ Abs (v,_, _) => 
+  let val (x,t) = Thm.dest_abs (SOME v) ct
+  in Thm.abstract_rule ((fst o dest_Free o term_of) x) x (cv t)
+  end
+ | _ => all_conv ct;
+
+fun cache_conv conv =
+ let 
+  val tab = ref Termtab.empty
+  fun cconv t =  
+    case Termtab.lookup (!tab) (term_of t) of
+     SOME th => th
+   | NONE => let val th = conv t
+             in ((tab := Termtab.insert Thm.eq_thm (term_of t, th) (!tab)); th) end
+ in cconv end;
+fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
+  handle CTERM _ => false;
+
+local
+ fun thenqc conv1 conv2 tm =
+   case try conv1 tm of
+    SOME th1 => (case try conv2 (Thm.rhs_of th1) of SOME th2 => Thm.transitive th1 th2 | NONE => th1)
+  | NONE => conv2 tm
+
+ fun thencqc conv1 conv2 tm =
+    let val th1 = conv1 tm 
+    in (case try conv2 (Thm.rhs_of th1) of SOME th2 => Thm.transitive th1 th2 | NONE => th1)
+    end
+ fun comb_qconv conv tm =
+   let val (l,r) = Thm.dest_comb tm 
+   in (case try conv l of 
+        SOME th1 => (case try conv r of SOME th2 => Thm.combination th1 th2 
+                                      | NONE => Drule.fun_cong_rule th1 r)
+      | NONE => Drule.arg_cong_rule l (conv r))
+   end
+ fun repeatqc conv tm = thencqc conv (repeatqc conv) tm 
+ fun sub_qconv conv tm =  if is_abs tm then absc conv tm else comb_qconv conv tm 
+ fun once_depth_qconv conv tm =
+      (conv else_conv (sub_qconv (once_depth_qconv conv))) tm
+ fun depth_qconv conv tm =
+    thenqc (sub_qconv (depth_qconv conv))
+           (repeatqc conv) tm
+ fun redepth_qconv conv tm =
+    thenqc (sub_qconv (redepth_qconv conv))
+           (thencqc conv (redepth_qconv conv)) tm
+ fun top_depth_qconv conv tm =
+    thenqc (repeatqc conv)
+           (thencqc (sub_qconv (top_depth_qconv conv))
+                    (thencqc conv (top_depth_qconv conv))) tm
+ fun top_sweep_qconv conv tm =
+    thenqc (repeatqc conv)
+           (sub_qconv (top_sweep_qconv conv)) tm
+in 
+val (once_depth_conv, depth_conv, rdepth_conv, top_depth_conv, top_sweep_conv) = 
+  (fn c => try_conv (once_depth_qconv c),
+   fn c => try_conv (depth_qconv c),
+   fn c => try_conv (redepth_qconv c),
+   fn c => try_conv (top_depth_qconv c),
+   fn c => try_conv (top_sweep_qconv c));
+end;
+end;
+
+
+    (* Some useful derived rules *)
+fun deduct_antisym_rule tha thb = 
+    equal_intr (implies_intr (cprop_of thb) tha) 
+     (implies_intr (cprop_of tha) thb);
+
+fun prove_hyp tha thb = 
+  if exists (curry op aconv (concl_of tha)) (#hyps (rep_thm thb)) 
+  then equal_elim (symmetric (deduct_antisym_rule tha thb)) tha else thb;
+
+
+
+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 cterm
+ | Eqmul of cterm * positivstellensatz
+ | Sum of positivstellensatz * positivstellensatz
+ | Product of positivstellensatz * positivstellensatz;
+
+val gen_gen_real_arith :
+  Proof.context -> (Rat.rat -> Thm.cterm) * conv * conv * conv * 
+   conv * conv * conv * conv * conv * conv * 
+    ( (thm list * thm list * thm list -> positivstellensatz -> thm) ->
+        thm list * thm list * thm list -> thm) -> conv
+val real_linear_prover : 
+  (thm list * thm list * thm list -> positivstellensatz -> thm) ->
+   thm list * thm list * thm list -> thm
+
+val gen_real_arith : Proof.context ->
+   (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv *
+   ( (thm list * thm list * thm list -> positivstellensatz -> thm) ->
+       thm list * thm list * thm list -> thm) -> conv
+val gen_prover_real_arith : Proof.context ->
+   ((thm list * thm list * thm list -> positivstellensatz -> thm) ->
+     thm list * thm list * thm list -> thm) -> conv
+val real_arith : Proof.context -> conv
+end
+
+structure RealArith (* : REAL_ARITH *)=
+struct
+
+ open Conv Thm;;
+(* ------------------------------------------------------------------------- *)
+(* 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 cterm
+ | Eqmul of cterm * positivstellensatz
+ | Sum of positivstellensatz * positivstellensatz
+ | Product of positivstellensatz * positivstellensatz;
+         (* Theorems used in the procedure *)
+
+
+val my_eqs = ref ([] : thm list);
+val my_les = ref ([] : thm list);
+val my_lts = ref ([] : thm list);
+val my_proof = ref (Axiom_eq 0);
+val my_context = ref @{context};
+
+val my_mk_numeric = ref ((K @{cterm True}) :Rat.rat -> cterm);
+val my_numeric_eq_conv = ref no_conv;
+val my_numeric_ge_conv = ref no_conv;
+val my_numeric_gt_conv = ref no_conv;
+val my_poly_conv = ref no_conv;
+val my_poly_neg_conv = ref no_conv;
+val my_poly_add_conv = ref no_conv;
+val my_poly_mul_conv = ref no_conv;
+
+fun conjunctions th = case try Conjunction.elim th of
+   SOME (th1,th2) => (conjunctions th1) @ conjunctions th2
+ | NONE => [th];
+
+val pth = @{lemma "(((x::real) < y) == (y - x > 0)) &&& ((x <= y) == (y - x >= 0)) 
+     &&& ((x = y) == (x - y = 0)) &&& ((~(x < y)) == (x - y >= 0)) &&& ((~(x <= y)) == (x - y > 0))
+     &&& ((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
+  by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)} |> 
+conjunctions;
+
+val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
+val pth_add = 
+ @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 ) &&& ( x = 0 ==> y >= 0 ==> x + y >= 0) 
+    &&& (x = 0 ==> y > 0 ==> x + y > 0) &&& (x >= 0 ==> y = 0 ==> x + y >= 0) 
+    &&& (x >= 0 ==> y >= 0 ==> x + y >= 0) &&& (x >= 0 ==> y > 0 ==> x + y > 0) 
+    &&& (x > 0 ==> y = 0 ==> x + y > 0) &&& (x > 0 ==> y >= 0 ==> x + y > 0) 
+    &&& (x > 0 ==> y > 0 ==> x + y > 0)"  by simp_all} |> conjunctions ;
+
+val pth_mul = 
+  @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0) &&& (x = 0 ==> y >= 0 ==> x * y = 0) &&& 
+           (x = 0 ==> y > 0 ==> x * y = 0) &&& (x >= 0 ==> y = 0 ==> x * y = 0) &&& 
+           (x >= 0 ==> y >= 0 ==> x * y >= 0 ) &&& ( x >= 0 ==> y > 0 ==> x * y >= 0 ) &&&
+           (x > 0 ==>  y = 0 ==> x * y = 0 ) &&& ( x > 0 ==> y >= 0 ==> x * y >= 0 ) &&&
+           (x > 0 ==>  y > 0 ==> x * y > 0)"
+  by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
+    mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])} |> conjunctions;
+
+val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
+val pth_square = @{lemma "x * x >= (0::real)"  by simp};
+
+val weak_dnf_simps = List.take (simp_thms, 34) 
+    @ conjunctions @{lemma "((P & (Q | R)) = ((P&Q) | (P&R))) &&& ((Q | R) & P) = ((Q&P) | (R&P)) &&& (P & Q) = (Q & P) &&& ((P | Q) = (Q | P))" by blast+};
+
+val nnfD_simps = conjunctions @{lemma "((~(P & Q)) = (~P | ~Q)) &&& ((~(P | Q)) = (~P & ~Q) ) &&& ((P --> Q) = (~P | Q) ) &&& ((P = Q) = ((P & Q) | (~P & ~ Q))) &&& ((~(P = Q)) = ((P & ~ Q) | (~P & Q)) ) &&& ((~ ~(P)) = P)" by blast+}
+
+val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
+val prenex_simps = map (fn th => th RS sym) ([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @ @{thms "all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});
+
+val real_abs_thms1 = conjunctions @{lemma
+  "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r)) &&&
+  ((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
+  ((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
+  ((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r)) &&&
+  ((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r)) &&&
+  ((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r)) &&&
+  ((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r)) &&&
+  ((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
+  ((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
+  ((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y  + b >= r)) &&&
+  ((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r)) &&&
+  ((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y  + c >= r)) &&&
+  ((1 * min x y >= r) = (1 * x >= r & 1 * y >= r)) &&&
+  ((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
+  ((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
+  ((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y  + b >= r) )&&&
+  ((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r)) &&&
+  ((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y  + c >= r)) &&&
+  ((min x y >= r) = (x >= r &  y >= r)) &&&
+  ((min x y + a >= r) = (a + x >= r & a + y >= r)) &&&
+  ((a + min x y >= r) = (a + x >= r & a + y >= r)) &&&
+  ((a + min x y + b >= r) = (a + x + b >= r & a + y  + b >= r)) &&&
+  ((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r) )&&&
+  ((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r)) &&&
+  ((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r)) &&&
+  ((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
+  ((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
+  ((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r)) &&&
+  ((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r)) &&&
+  ((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r)) &&&
+  ((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r)) &&&
+  ((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
+  ((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
+  ((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y  + b > r)) &&&
+  ((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r)) &&&
+  ((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y  + c > r)) &&&
+  ((min x y > r) = (x > r &  y > r)) &&&
+  ((min x y + a > r) = (a + x > r & a + y > r)) &&&
+  ((a + min x y > r) = (a + x > r & a + y > r)) &&&
+  ((a + min x y + b > r) = (a + x + b > r & a + y  + b > r)) &&&
+  ((a + b + min x y > r) = (a + b + x > r & a + b + y > r)) &&&
+  ((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
+  by auto};
+
+val abs_split' = @{lemma "P (abs (x::'a::ordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
+  by (atomize (full)) (auto split add: abs_split)};
+
+val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
+  by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
+
+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)};
+
+
+         (* Miscalineous *)
+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.capply (Thm.capply @{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 
+ case t of
+  a$b => (find_term p a handle TERM _ => find_term p b)
+ | Abs (_,_,t') => find_term p t'
+ | _ => raise TERM ("find_term",[t]);
+
+fun find_cterm p t = if p t then t else 
+ case term_of t of
+  a$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
+ | Abs (_,_,t') => 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;
+
+fun cache_conv conv =
+ let 
+  val tab = ref Termtab.empty
+  fun cconv t =  
+    case Termtab.lookup (!tab) (term_of t) of
+     SOME th => th
+   | NONE => let val th = conv t
+             in ((tab := Termtab.insert Thm.eq_thm (term_of t, th) (!tab)); th) end
+ in cconv end;
+fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
+  handle CTERM _ => false;
+
+    (* 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
+ open Conv Thm;
+ val _ = my_context := ctxt 
+ val _ = (my_mk_numeric := mk_numeric ; my_numeric_eq_conv := numeric_eq_conv ; 
+          my_numeric_ge_conv := numeric_ge_conv; my_numeric_gt_conv := numeric_gt_conv ;
+          my_poly_conv := poly_conv; my_poly_neg_conv := poly_neg_conv; 
+          my_poly_add_conv := poly_add_conv; my_poly_mul_conv := poly_mul_conv)
+ val pre_ss = HOL_basic_ss addsimps simp_thms@ ex_simps@ all_simps@[@{thm not_all},@{thm not_ex},ex_disj_distrib, all_conj_distrib, @{thm 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 = equal_elim (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' = (instantiate (match (lhs_of th, ct)) th 
+      handle MATCH => raise CTERM ("real_ineq_conv", [ct]))
+  in 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))
+         (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))
+       (instantiate' [] [SOME t] pth_square)
+
+  fun hol_of_positivstellensatz(eqs,les,lts) proof =
+   let 
+    val _ = (my_eqs := eqs ; my_les := les ; my_lts := lts ; my_proof := proof)
+    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(capply @{cterm Trueprop} 
+                          (capply (capply @{cterm "op =::real => _"} (mk_numeric x)) 
+                               @{cterm "0::real"})))
+      | Rational_le x => eqT_elim(numeric_ge_conv(capply @{cterm Trueprop} 
+                          (capply (capply @{cterm "op <=::real => _"} 
+                                     @{cterm "0::real"}) (mk_numeric x))))
+      | Rational_lt x => eqT_elim(numeric_gt_conv(capply @{cterm Trueprop} 
+                      (capply (capply @{cterm "op <::real => _"} @{cterm "0::real"})
+                        (mk_numeric x))))
+      | Square t => square_rule t
+      | Eqmul(t,p) => emul_rule t (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 = dest_arg o cprop_of
+  fun is_binop opr ct = (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 "op &"}
+  val is_disj = is_binop @{cterm "op |"}
+  fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
+  fun disj_cases th th1 th2 = 
+   let val (p,q) = dest_binop (concl th)
+       val c = concl th1
+       val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
+   in implies_elim (implies_elim (implies_elim (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th) (implies_intr (capply @{cterm Trueprop} p) th1)) (implies_intr (capply @{cterm Trueprop} q) th2)
+   end
+ fun overall 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 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 dun (th1::th2::oths) end
+    else if is_disj ct then
+      let 
+       val th1 = overall dun (assume (capply @{cterm Trueprop} (dest_arg1 ct))::oths)
+       val th2 = overall dun (assume (capply @{cterm Trueprop} (dest_arg ct))::oths)
+      in disj_cases th th1 th2 end
+   else overall (th::dun) oths
+  end
+  fun dest_binary b ct = if is_binop b ct then 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 (dest_arg ct)
+    val th = instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
+    val th_p = poly_conv(dest_arg(dest_arg1(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 "op |"} 
+     (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
+     (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
+    in transitive th th' 
+  end
+ fun equal_implies_1_rule PQ = 
+  let 
+   val P = lhs_of PQ
+  in implies_intr P (equal_elim PQ (assume P))
+  end
+ (* FIXME!!! Copied from groebner.ml *)
+ val strip_exists =
+  let fun h (acc, t) =
+   case (term_of t) of
+    Const("Ex",_)$Abs(x,T,p) => h (dest_abs NONE (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) => _" }) (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.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
+
+ fun choose v th th' = case concl_of th of 
+   @{term Trueprop} $ (Const("Ex",_)$_) => 
+    let
+     val p = (funpow 2 Thm.dest_arg o cprop_of) th
+     val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
+     val th0 = fconv_rule (Thm.beta_conversion true)
+         (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
+     val pv = (Thm.rhs_of o Thm.beta_conversion true) 
+           (Thm.capply @{cterm Trueprop} (Thm.capply p v))
+     val th1 = forall_intr v (implies_intr pv th')
+    in implies_elim (implies_elim th0 th) th1  end
+ | _ => raise THM ("choose",0,[th, th'])
+
+  fun simple_choose v th = 
+     choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
+
+ val strip_forall =
+  let fun h (acc, t) =
+   case (term_of t) of
+    Const("All",_)$Abs(x,T,p) => h (dest_abs NONE (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 "op &"}, @{term "op |"}] [] 
+     (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 "op &"}, @{term "op |"}] [] 
+                  (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 = capply @{cterm Trueprop} (capply @{cterm "Not"} ct)
+  val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
+  val tm0 = dest_arg (rhs_of th0)
+  val th = if tm0 aconvc @{cterm False} then equal_implies_1_rule th0 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 = overall [] [specl avs (assume (rhs_of th1))]
+    val th3 = fold simple_choose evs (prove_hyp (equal_elim th1 (assume (capply @{cterm Trueprop} bod))) th2)
+   in  Drule.implies_intr_hyps (prove_hyp (equal_elim th0 (assume nct)) th3)
+   end
+  in implies_elim (instantiate' [] [SOME ct] pth_final) th
+ end
+in f
+end;
+
+(* A linear arithmetic prover *)
+local
+  val linear_add = Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
+  fun linear_cmul c = Ctermfunc.mapf (fn x => c */ x)
+  val one_tm = @{cterm "1::real"}
+  fun contradictory p (e,_) = ((Ctermfunc.is_undefined e) andalso not(p Rat.zero)) orelse
+     ((gen_eq_set (op aconvc) (Ctermfunc.dom e, [one_tm])) andalso not(p(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,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
+       length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
+       length(filter (fn (e,_) => 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 = Ctermfunc.tryapplyd e1 v Rat.zero 
+        val c2 = 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,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
+      val (lts0,lts1) = 
+         List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
+      val (lesp,lesn) = 
+         List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
+      val (ltsp,ltsn) = 
+         List.partition (fn (e,_) => 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 (curry (gen_union (op aconvc)) o Ctermfunc.dom o fst) (les@lts) []) 
+     in linear_ineqs vars (les,lts) end
+   | (e,p)::es => 
+     if Ctermfunc.is_undefined e then linear_eqs (es,les,lts) else
+     let 
+      val (x,c) = Ctermfunc.choose (Ctermfunc.undefine one_tm e)
+      fun xform (inp as (t,q)) =
+       let val d = 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(cterm_of_rat 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 = map2 (fn p => fn n => (p,Axiom_eq n)) eq (0 upto (length eq - 1))
+    val les = map2 (fn p => fn n => (p,Axiom_le n)) le (0 upto (length le - 1))
+    val lts = map2 (fn p => fn n => (p,Axiom_lt n)) lt (0 upto (length lt - 1))
+   in linear_eqs(eqs,les,lts)
+   end 
+  
+  fun lin_of_hol ct = 
+   if ct aconvc @{cterm "0::real"} then Ctermfunc.undefined
+   else if not (is_comb ct) then Ctermfunc.onefunc (ct, Rat.one)
+   else if is_ratconst ct then Ctermfunc.onefunc (one_tm, dest_ratconst ct)
+   else
+    let val (lop,r) = Thm.dest_comb ct 
+    in if not (is_comb lop) then 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 Ctermfunc.onefunc (r, dest_ratconst l)
+           else 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
+ open Thm
+in 
+fun real_linear_prover translator (eq,le,lt) = 
+ let 
+  val lhs = lin_of_hol o dest_arg1 o dest_arg o cprop_of
+  val rhs = lin_of_hol o dest_arg o 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 (curry (gen_union (op aconvc)) o Ctermfunc.dom) 
+          (eq_pols @ le_pols @ lt_pols) [])
+  val le_pols' = le_pols @ map (fn v => Ctermfunc.onefunc (v,Rat.one)) aliens
+  val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
+  val le' = le @ map (fn a => instantiate' [] [SOME (dest_arg a)] @{thm real_of_nat_ge_zero}) aliens 
+ in (translator (eq,le',lt) proof) : thm
+ 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 dest_fun t aconvc abs_tm
+   fun eliminate_construct p c tm =
+    let 
+     val t = find_cterm p tm
+     val th0 = (symmetric o beta_conversion false) (capply (cabs t tm) t)
+     val (p,ax) = (dest_comb o rhs_of) th0
+    in fconv_rule(arg_conv(binop_conv (arg_conv (beta_conversion false))))
+               (transitive th0 (c p ax))
+   end
+
+   val elim_abs = eliminate_construct is_abs
+    (fn p => fn ax => 
+       instantiate ([], [(p_tm,p), (x_tm, dest_arg ax)]) pth_abs)
+   val elim_max = eliminate_construct is_max
+    (fn p => fn ax => 
+      let val (ax,y) = dest_comb ax 
+      in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)]) 
+      pth_max end)
+   val elim_min = eliminate_construct is_min
+    (fn p => fn ax => 
+      let val (ax,y) = dest_comb ax 
+      in  instantiate ([], [(p_tm,p), (x_tm, 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;
+
+(* An instance for reals*) 
+
+fun gen_prover_real_arith ctxt prover = 
+ let
+  fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS
+  val {add,mul,neg,pow,sub,main} = 
+     Normalizer.semiring_normalizers_ord_wrapper ctxt
+      (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) 
+     simple_cterm_ord
+in gen_real_arith ctxt
+   (cterm_of_rat, field_comp_conv, field_comp_conv,field_comp_conv,
+    main,neg,add,mul, prover)
+end;
+
+fun real_arith ctxt = gen_prover_real_arith ctxt real_linear_prover;
+end