# HG changeset patch
# User chaieb
# Date 1242145969 -3600
# Node ID 541d43bee6782e4ce7e515609c19197ea86bc03f
# Parent  527ba4a378436a60e795bff37f05acbe13299fd5
Isolated decision procedure for noms and the general arithmetic solver

diff -r 527ba4a37843 -r 541d43bee678 src/HOL/Library/Euclidean_Space.thy
--- a/src/HOL/Library/Euclidean_Space.thy	Tue May 12 17:32:49 2009 +0100
+++ b/src/HOL/Library/Euclidean_Space.thy	Tue May 12 17:32:49 2009 +0100
@@ -9,7 +9,7 @@
   Complex_Main "~~/src/HOL/Decision_Procs/Dense_Linear_Order"
   Finite_Cartesian_Product Glbs Infinite_Set Numeral_Type
   Inner_Product
-uses ("normarith.ML")
+uses "positivstellensatz.ML" ("normarith.ML")
 begin
 
 text{* Some common special cases.*}
diff -r 527ba4a37843 -r 541d43bee678 src/HOL/Library/normarith.ML
--- a/src/HOL/Library/normarith.ML	Tue May 12 17:32:49 2009 +0100
+++ b/src/HOL/Library/normarith.ML	Tue May 12 17:32:49 2009 +0100
@@ -1,786 +1,7 @@
-(* 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 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
- val fns: 
-   {key_ord: key*key -> order,
-    apply : 'a T -> key -> 'a,
-    applyd :'a T -> (key -> 'a) -> key -> 'a,
-    combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a T -> 'a T -> 'a T,
-    defined : 'a T -> key -> bool,
-    dom : 'a T -> key list,
-    fold : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b,
-    graph : 'a T -> (key * 'a) list,
-    is_undefined : 'a T -> bool,
-    mapf : ('a -> 'b) -> 'a T -> 'b T,
-    tryapplyd : 'a T -> key -> 'a -> 'a,
-    undefine :  key -> 'a T -> 'a T,
-    undefined : 'a T,
-    update : key * 'a -> 'a T -> 'a T,
-    updatep : (key * 'a -> bool) -> key * 'a -> 'a T -> 'a T,
-    choose : 'a T -> key * 'a,
-    onefunc : key * 'a -> 'a T,
-    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 graph = Tab.dest;
-val dom = Tab.keys;
-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.max_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
-
-val fns = 
-   {key_ord = Key.ord,
-    apply = apply,
-    applyd = applyd,
-    combine = combine,
-    defined = defined,
-    dom = dom,
-    fold = fold,
-    graph = graph,
-    is_undefined = is_undefined,
-    mapf = mapf,
-    tryapplyd = tryapplyd,
-    undefine = undefine,
-    undefined = undefined,
-    update = update,
-    updatep = updatep,
-    choose = choose,
-    onefunc = onefunc,
-    get_first = get_first}
-
-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 Conv2;;
-(* ------------------------------------------------------------------------- *)
-(* 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 *)
-
-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)
- | Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd)
- | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
- fun is_ratconst t = can dest_ratconst t
-
-fun find_term p t = if p t then t else 
- case t of
-  a$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]);
-
-
-    (* 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 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_mul_conv))
-       (instantiate' [] [SOME t] pth_square)
-
-  fun hol_of_positivstellensatz(eqs,les,lts) =
-   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(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 fn prf => 
-      fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv]) 
-          (translate prf)
-   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(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 "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 (Thm.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 (Thm.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 Thm.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
+(* Title:      Library/normarith.ML
+   Author:     Amine Chaieb, University of Cambridge
+   Description: A simple decision procedure for linear problems in euclidean space
+*)
 
   (* Now the norm procedure for euclidean spaces *)