# HG changeset patch
# User haftmann
# Date 1560501267 0
# Node ID 0f7edf0853df05078ac10b69490eb89c0b493c40
# Parent  315489d836d8d3d47af6a11cf872ab3290a9e4c8
tuned file system structure

diff -r 315489d836d8 -r 0f7edf0853df src/HOL/Library/Sum_of_Squares.thy
--- a/src/HOL/Library/Sum_of_Squares.thy	Fri Jun 14 08:34:27 2019 +0000
+++ b/src/HOL/Library/Sum_of_Squares.thy	Fri Jun 14 08:34:27 2019 +0000
@@ -10,9 +10,9 @@
 imports Complex_Main
 begin
 
-ML_file \<open>positivstellensatz.ML\<close>
+ML_file \<open>Sum_of_Squares/positivstellensatz.ML\<close>
+ML_file \<open>Sum_of_Squares/positivstellensatz_tools.ML\<close>
 ML_file \<open>Sum_of_Squares/sum_of_squares.ML\<close>
-ML_file \<open>Sum_of_Squares/positivstellensatz_tools.ML\<close>
 ML_file \<open>Sum_of_Squares/sos_wrapper.ML\<close>
 
 end
diff -r 315489d836d8 -r 0f7edf0853df src/HOL/Library/Sum_of_Squares/positivstellensatz.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Sum_of_Squares/positivstellensatz.ML	Fri Jun 14 08:34:27 2019 +0000
@@ -0,0 +1,779 @@
+(*  Title:      HOL/Library/positivstellensatz.ML
+    Author:     Amine Chaieb, University of Cambridge
+
+A generic arithmetic prover based on Positivstellensatz certificates
+--- also implements Fourier-Motzkin elimination as a special case
+Fourier-Motzkin elimination.
+*)
+
+(* A functor for finite mappings based on Tables *)
+
+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
+end;
+
+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
+   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;
+  in Tab.fold h a b end;
+
+fun choose f =
+  (case Tab.min f of
+    SOME entry => entry
+  | NONE => error "FuncFun.choose : Completely empty function")
+
+fun onefunc kv = update kv empty
+
+end;
+
+(* Some standard functors and utility functions for them *)
+
+structure FuncUtil =
+struct
+
+structure Intfunc = FuncFun(type key = int val ord = int_ord);
+structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
+structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
+structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
+structure Termfunc = FuncFun(type key = term val ord = Term_Ord.fast_term_ord);
+structure Ctermfunc = FuncFun(type key = cterm val ord = Thm.fast_term_ord);
+
+type monomial = int Ctermfunc.table;
+val monomial_ord = list_ord (prod_ord Thm.fast_term_ord int_ord) o apply2 Ctermfunc.dest
+structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
+
+type poly = Rat.rat Monomialfunc.table;
+
+(* The ordering so we can create canonical HOL polynomials.                  *)
+
+fun dest_monomial mon = sort (Thm.fast_term_ord o apply2 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
+      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 Thm.fast_term_ord int_ord) (mon1,mon2)
+    end;
+
+end
+
+(* positivstellensatz datatype and prover generation *)
+
+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
+
+end
+
+structure RealArith : REAL_ARITH =
+struct
+
+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;
+         (* 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 ->
+  (thm list * thm list * thm list -> positivstellensatz -> thm) ->
+    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 (Thm.cprop_of thb) tha)
+     (Thm.implies_intr (Thm.cprop_of tha) thb);
+
+fun prove_hyp tha thb =
+  if exists (curry op aconv (Thm.concl_of tha)) (Thm.hyps_of thb)  (* FIXME !? *)
+  then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb;
+
+val pth = @{lemma "(((x::real) < y) \<equiv> (y - x > 0))" and "((x \<le> y) \<equiv> (y - x \<ge> 0))" and
+     "((x = y) \<equiv> (x - y = 0))" and "((\<not>(x < y)) \<equiv> (x - y \<ge> 0))" and
+     "((\<not>(x \<le> y)) \<equiv> (x - y > 0))" and "((\<not>(x = y)) \<equiv> (x - y > 0 \<or> -(x - y) > 0))"
+  by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};
+
+val pth_final = @{lemma "(\<not>p \<Longrightarrow> False) \<Longrightarrow> p" by blast}
+val pth_add =
+  @{lemma "(x = (0::real) \<Longrightarrow> y = 0 \<Longrightarrow> x + y = 0 )" and "( x = 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y \<ge> 0)" and
+    "(x = 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" and "(x \<ge> 0 \<Longrightarrow> y = 0 \<Longrightarrow> x + y \<ge> 0)" and
+    "(x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y \<ge> 0)" and "(x \<ge> 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" and
+    "(x > 0 \<Longrightarrow> y = 0 \<Longrightarrow> x + y > 0)" and "(x > 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y > 0)" and
+    "(x > 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" by simp_all};
+
+val pth_mul =
+  @{lemma "(x = (0::real) \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and "(x = 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y = 0)" and
+    "(x = 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y = 0)" and "(x \<ge> 0 \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and
+    "(x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y \<ge> 0)" and "(x \<ge> 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y \<ge> 0)" and
+    "(x > 0 \<Longrightarrow>  y = 0 \<Longrightarrow> x * y = 0)" and "(x > 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y \<ge> 0)" and
+    "(x > 0 \<Longrightarrow>  y > 0 \<Longrightarrow> 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])};
+
+val pth_emul = @{lemma "y = (0::real) \<Longrightarrow> x * y = 0"  by simp};
+val pth_square = @{lemma "x * x \<ge> (0::real)"  by simp};
+
+val weak_dnf_simps =
+  List.take (@{thms simp_thms}, 34) @
+    @{lemma "((P \<and> (Q \<or> R)) = ((P\<and>Q) \<or> (P\<and>R)))" and "((Q \<or> R) \<and> P) = ((Q\<and>P) \<or> (R\<and>P))" and
+      "(P \<and> Q) = (Q \<and> P)" and "((P \<or> Q) = (Q \<or> P))" by blast+};
+
+(*
+val nnfD_simps =
+  @{lemma "((~(P & Q)) = (~P | ~Q))" and "((~(P | Q)) = (~P & ~Q) )" and
+    "((P --> Q) = (~P | Q) )" and "((P = Q) = ((P & Q) | (~P & ~ Q)))" and
+    "((~(P = Q)) = ((P & ~ Q) | (~P & Q)) )" and "((~ ~(P)) = P)" by blast+};
+*)
+
+val choice_iff = @{lemma "(\<forall>x. \<exists>y. P x y) = (\<exists>f. \<forall>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 "HOL.all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});
+
+val real_abs_thms1 = @{lemma
+  "((-1 * \<bar>x::real\<bar> \<ge> r) = (-1 * x \<ge> r \<and> 1 * x \<ge> r))" and
+  "((-1 * \<bar>x\<bar> + a \<ge> r) = (a + -1 * x \<ge> r \<and> a + 1 * x \<ge> r))" and
+  "((a + -1 * \<bar>x\<bar> \<ge> r) = (a + -1 * x \<ge> r \<and> a + 1 * x \<ge> r))" and
+  "((a + -1 * \<bar>x\<bar> + b \<ge> r) = (a + -1 * x + b \<ge> r \<and> a + 1 * x + b \<ge> r))" and
+  "((a + b + -1 * \<bar>x\<bar> \<ge> r) = (a + b + -1 * x \<ge> r \<and> a + b + 1 * x \<ge> r))" and
+  "((a + b + -1 * \<bar>x\<bar> + c \<ge> r) = (a + b + -1 * x + c \<ge> r \<and> a + b + 1 * x + c \<ge> r))" and
+  "((-1 * max x y \<ge> r) = (-1 * x \<ge> r \<and> -1 * y \<ge> r))" and
+  "((-1 * max x y + a \<ge> r) = (a + -1 * x \<ge> r \<and> a + -1 * y \<ge> r))" and
+  "((a + -1 * max x y \<ge> r) = (a + -1 * x \<ge> r \<and> a + -1 * y \<ge> r))" and
+  "((a + -1 * max x y + b \<ge> r) = (a + -1 * x + b \<ge> r \<and> a + -1 * y  + b \<ge> r))" and
+  "((a + b + -1 * max x y \<ge> r) = (a + b + -1 * x \<ge> r \<and> a + b + -1 * y \<ge> r))" and
+  "((a + b + -1 * max x y + c \<ge> r) = (a + b + -1 * x + c \<ge> r \<and> a + b + -1 * y  + c \<ge> r))" and
+  "((1 * min x y \<ge> r) = (1 * x \<ge> r \<and> 1 * y \<ge> r))" and
+  "((1 * min x y + a \<ge> r) = (a + 1 * x \<ge> r \<and> a + 1 * y \<ge> r))" and
+  "((a + 1 * min x y \<ge> r) = (a + 1 * x \<ge> r \<and> a + 1 * y \<ge> r))" and
+  "((a + 1 * min x y + b \<ge> r) = (a + 1 * x + b \<ge> r \<and> a + 1 * y  + b \<ge> r))" and
+  "((a + b + 1 * min x y \<ge> r) = (a + b + 1 * x \<ge> r \<and> a + b + 1 * y \<ge> r))" and
+  "((a + b + 1 * min x y + c \<ge> r) = (a + b + 1 * x + c \<ge> r \<and> a + b + 1 * y  + c \<ge> r))" and
+  "((min x y \<ge> r) = (x \<ge> r \<and> y \<ge> r))" and
+  "((min x y + a \<ge> r) = (a + x \<ge> r \<and> a + y \<ge> r))" and
+  "((a + min x y \<ge> r) = (a + x \<ge> r \<and> a + y \<ge> r))" and
+  "((a + min x y + b \<ge> r) = (a + x + b \<ge> r \<and> a + y  + b \<ge> r))" and
+  "((a + b + min x y \<ge> r) = (a + b + x \<ge> r \<and> a + b + y \<ge> r))" and
+  "((a + b + min x y + c \<ge> r) = (a + b + x + c \<ge> r \<and> a + b + y + c \<ge> r))" and
+  "((-1 * \<bar>x\<bar> > r) = (-1 * x > r \<and> 1 * x > r))" and
+  "((-1 * \<bar>x\<bar> + a > r) = (a + -1 * x > r \<and> a + 1 * x > r))" and
+  "((a + -1 * \<bar>x\<bar> > r) = (a + -1 * x > r \<and> a + 1 * x > r))" and
+  "((a + -1 * \<bar>x\<bar> + b > r) = (a + -1 * x + b > r \<and> a + 1 * x + b > r))" and
+  "((a + b + -1 * \<bar>x\<bar> > r) = (a + b + -1 * x > r \<and> a + b + 1 * x > r))" and
+  "((a + b + -1 * \<bar>x\<bar> + c > r) = (a + b + -1 * x + c > r \<and> a + b + 1 * x + c > r))" and
+  "((-1 * max x y > r) = ((-1 * x > r) \<and> -1 * y > r))" and
+  "((-1 * max x y + a > r) = (a + -1 * x > r \<and> a + -1 * y > r))" and
+  "((a + -1 * max x y > r) = (a + -1 * x > r \<and> a + -1 * y > r))" and
+  "((a + -1 * max x y + b > r) = (a + -1 * x + b > r \<and> a + -1 * y  + b > r))" and
+  "((a + b + -1 * max x y > r) = (a + b + -1 * x > r \<and> a + b + -1 * y > r))" and
+  "((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r \<and> a + b + -1 * y  + c > r))" and
+  "((min x y > r) = (x > r \<and> y > r))" and
+  "((min x y + a > r) = (a + x > r \<and> a + y > r))" and
+  "((a + min x y > r) = (a + x > r \<and> a + y > r))" and
+  "((a + min x y + b > r) = (a + x + b > r \<and> a + y  + b > r))" and
+  "((a + b + min x y > r) = (a + b + x > r \<and> a + b + y > r))" and
+  "((a + b + min x y + c > r) = (a + b + x + c > r \<and> a + b + y + c > r))"
+  by auto};
+
+val abs_split' = @{lemma "P \<bar>x::'a::linordered_idom\<bar> == (x \<ge> 0 \<and> P x \<or> x < 0 \<and> P (-x))"
+  by (atomize (full)) (auto split: abs_split)};
+
+val max_split = @{lemma "P (max x y) \<equiv> ((x::'a::linorder) \<le> y \<and> P y \<or> x > y \<and> P x)"
+  by (atomize (full)) (cases "x \<le> y", auto simp add: max_def)};
+
+val min_split = @{lemma "P (min x y) \<equiv> ((x::'a::linorder) \<le> y \<and> P x \<or> x > y \<and> P y)"
+  by (atomize (full)) (cases "x \<le> y", auto simp add: min_def)};
+
+
+         (* Miscellaneous *)
+fun literals_conv bops uops cv =
+  let
+    fun h t =
+      (case Thm.term_of t of
+        b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
+      | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
+      | _ => cv t)
+  in h end;
+
+fun cterm_of_rat x =
+  let
+    val (a, b) = Rat.dest x
+  in
+    if b = 1 then Numeral.mk_cnumber \<^ctyp>\<open>real\<close> a
+    else Thm.apply (Thm.apply \<^cterm>\<open>(/) :: real \<Rightarrow> _\<close>
+      (Numeral.mk_cnumber \<^ctyp>\<open>real\<close> a))
+      (Numeral.mk_cnumber \<^ctyp>\<open>real\<close> b)
+  end;
+
+fun dest_ratconst t =
+  case Thm.term_of t of
+    Const(\<^const_name>\<open>divide\<close>, _)$a$b => Rat.make(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
+  | _ => Rat.of_int (HOLogic.dest_number (Thm.term_of t) |> snd)
+fun is_ratconst t = can dest_ratconst t
+
+(*
+fun find_term p t = if p t then t else
+ case t of
+  a$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 Thm.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 is_comb t = (case Thm.term_of t of _ $ _ => true | _ => false);
+
+fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
+  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>\<open>(^) :: real \<Rightarrow> _\<close> x)
+  (Numeral.mk_cnumber \<^ctyp>\<open>nat\<close> k)
+
+fun cterm_of_monomial m =
+  if FuncUtil.Ctermfunc.is_empty m then \<^cterm>\<open>1::real\<close>
+  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>\<open>(*) :: real \<Rightarrow> _\<close> s) t) vps
+    end
+
+fun cterm_of_cmonomial (m,c) =
+  if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
+  else if c = @1 then cterm_of_monomial m
+  else Thm.apply (Thm.apply \<^cterm>\<open>(*)::real \<Rightarrow> _\<close> (cterm_of_rat c)) (cterm_of_monomial m);
+
+fun cterm_of_poly p =
+  if FuncUtil.Monomialfunc.is_empty p then \<^cterm>\<open>0::real\<close>
+  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>\<open>(+) :: real \<Rightarrow> _\<close> 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 = put_simpset HOL_basic_ss ctxt 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 = put_simpset HOL_basic_ss ctxt addsimps prenex_simps
+    val skolemize_ss = put_simpset HOL_basic_ss ctxt addsimps [choice_iff]
+    val presimp_conv = Simplifier.rewrite pre_ss
+    val prenex_conv = Simplifier.rewrite prenex_ss
+    val skolemize_conv = Simplifier.rewrite skolemize_ss
+    val weak_dnf_ss = put_simpset HOL_basic_ss ctxt addsimps weak_dnf_simps
+    val weak_dnf_conv = Simplifier.rewrite 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>\<open>(\<le>) :: real \<Rightarrow> _\<close>
+            orelse g aconvc \<^cterm>\<open>(<) :: real \<Rightarrow> _\<close>
+         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))
+         (match_mp_rule pth_add [th, th'])
+    fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
+       (Thm.instantiate' [] [SOME ct] (th RS pth_emul))
+    fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
+       (Thm.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>\<open>Trueprop\<close>
+                          (Thm.apply (Thm.apply \<^cterm>\<open>(=)::real \<Rightarrow> _\<close> (mk_numeric x))
+                               \<^cterm>\<open>0::real\<close>)))
+          | Rational_le x => eqT_elim(numeric_ge_conv(Thm.apply \<^cterm>\<open>Trueprop\<close>
+                          (Thm.apply (Thm.apply \<^cterm>\<open>(\<le>)::real \<Rightarrow> _\<close>
+                                     \<^cterm>\<open>0::real\<close>) (mk_numeric x))))
+          | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.apply \<^cterm>\<open>Trueprop\<close>
+                      (Thm.apply (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>)
+                        (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])
+          (translate proof)
+      end
+
+    val init_conv = presimp_conv then_conv
+        nnf_conv ctxt then_conv skolemize_conv then_conv prenex_conv then_conv
+        weak_dnf_conv
+
+    val concl = Thm.dest_arg o Thm.cprop_of
+    fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
+    val is_req = is_binop \<^cterm>\<open>(=):: real \<Rightarrow> _\<close>
+    val is_ge = is_binop \<^cterm>\<open>(\<le>):: real \<Rightarrow> _\<close>
+    val is_gt = is_binop \<^cterm>\<open>(<):: real \<Rightarrow> _\<close>
+    val is_conj = is_binop \<^cterm>\<open>HOL.conj\<close>
+    val is_disj = is_binop \<^cterm>\<open>HOL.disj\<close>
+    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 (Thm.instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
+          (Thm.implies_intr (Thm.apply \<^cterm>\<open>Trueprop\<close> p) th1))
+        (Thm.implies_intr (Thm.apply \<^cterm>\<open>Trueprop\<close> 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>\<open>Trueprop\<close> (Thm.dest_arg1 ct))::oths)
+              val (th2, cert2) =
+                overall (Right::cert_choice) dun
+                  (Thm.assume (Thm.apply \<^cterm>\<open>Trueprop\<close> (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>\<open>(=) :: real \<Rightarrow> _\<close>
+    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 ([],[((("x", 0), \<^typ>\<open>real\<close>),l),((("y", 0), \<^typ>\<open>real\<close>),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>\<open>uminus :: real \<Rightarrow> _\<close> th_p
+        val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
+        val th' = Drule.binop_cong_rule \<^cterm>\<open>HOL.disj\<close>
+          (Drule.arg_cong_rule (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>) th_p)
+          (Drule.arg_cong_rule (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>) 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 Thm.term_of t of
+            Const(\<^const_name>\<open>Ex\<close>,_)$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 Thm.term_of x of
+        Free(s,_) => s
+      | Var ((s,_),_) => s
+      | _ => "x"
+
+    fun mk_forall x th =
+      let
+        val T = Thm.typ_of_cterm x
+        val all = Thm.cterm_of ctxt (Const (\<^const_name>\<open>All\<close>, (T --> \<^typ>\<open>bool\<close>) --> \<^typ>\<open>bool\<close>))
+      in Drule.arg_cong_rule all (Thm.abstract_rule (name_of x) x th) end
+
+    val specl = fold_rev (fn x => fn th => Thm.instantiate' [] [SOME x] (th RS spec));
+
+    fun ext T = Thm.cterm_of ctxt (Const (\<^const_name>\<open>Ex\<close>, (T --> \<^typ>\<open>bool\<close>) --> \<^typ>\<open>bool\<close>))
+    fun mk_ex v t = Thm.apply (ext (Thm.typ_of_cterm v)) (Thm.lambda v t)
+
+    fun choose v th th' =
+      case Thm.concl_of th of
+        \<^term>\<open>Trueprop\<close> $ (Const(\<^const_name>\<open>Ex\<close>,_)$_) =>
+        let
+          val p = (funpow 2 Thm.dest_arg o Thm.cprop_of) th
+          val T = Thm.dest_ctyp0 (Thm.ctyp_of_cterm p)
+          val th0 = fconv_rule (Thm.beta_conversion true)
+            (Thm.instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o Thm.cprop_of) th'] exE)
+          val pv = (Thm.rhs_of o Thm.beta_conversion true)
+            (Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.apply p v))
+          val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
+        in Thm.implies_elim (Thm.implies_elim th0 th) th1  end
+      | _ => raise THM ("choose",0,[th, th'])
+
+    fun simple_choose v th =
+      choose v
+        (Thm.assume
+          ((Thm.apply \<^cterm>\<open>Trueprop\<close> o mk_ex v) (Thm.dest_arg (hd (Thm.chyps_of th))))) th
+
+    val strip_forall =
+      let
+        fun h (acc, t) =
+          case Thm.term_of t of
+            Const(\<^const_name>\<open>All\<close>,_)$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 ctxt then_conv
+          literals_conv [\<^term>\<open>HOL.conj\<close>, \<^term>\<open>HOL.disj\<close>] []
+          (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>\<open>HOL.conj\<close>, \<^term>\<open>HOL.disj\<close>] []
+                  (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>\<open>Trueprop\<close> (Thm.apply \<^cterm>\<open>Not\<close> 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>\<open>False\<close> 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>\<open>Trueprop\<close> (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>\<open>Trueprop\<close> bod))) th2)
+          in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
+          end
+      in (Thm.implies_elim (Thm.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 = @0)
+  fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c * x)
+  val one_tm = \<^cterm>\<open>1::real\<close>
+  fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p @0)) 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 > @0)) lts of
+      SOME r => r
+    | NONE =>
+      (case find_first (contradictory (fn x => x > @0)) 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 @0 = @0) ineqs) +
+             length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) ineqs) *
+             length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 < @0) 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 @0
+               val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v @0
+             in
+               if c1 * c2 >= @0 then acc else
+               let
+                 val e1' = linear_cmul (abs c2) e1
+                 val e2' = linear_cmul (abs c1) e2
+                 val p1' = Product(Rational_lt (abs c2), p1)
+                 val p2' = Product(Rational_lt (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 @0 = @0) les
+           val (lts0,lts1) =
+             List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) lts
+           val (lesp,lesn) =
+             List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) les1
+           val (ltsp,ltsn) =
+             List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) 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 = @0)) 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 @0 in
+               if d = @0 then inp else
+               let
+                 val k = ~ d * abs c / c
+                 val e' = linear_cmul k e
+                 val t' = linear_cmul (abs c) t
+                 val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
+                 val q' = Product(Rational_lt (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>\<open>0::real\<close> then FuncUtil.Ctermfunc.empty
+    else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, @1)
+    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, @1)
+        else
+          let val (opr,l) = Thm.dest_comb lop
+          in
+            if opr aconvc \<^cterm>\<open>(+) :: real \<Rightarrow> _\<close>
+            then linear_add (lin_of_hol l) (lin_of_hol r)
+            else if opr aconvc \<^cterm>\<open>(*) :: real \<Rightarrow> _\<close>
+                    andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
+            else FuncUtil.Ctermfunc.onefunc (ct, @1)
+          end
+      end
+
+  fun is_alien ct =
+    case Thm.term_of ct of
+      Const(\<^const_name>\<open>of_nat\<close>, _)$ n => not (can HOLogic.dest_number n)
+    | Const(\<^const_name>\<open>of_int\<close>, _)$ n => not (can HOLogic.dest_number n)
+    | _ => 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 Thm.cprop_of
+    val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o Thm.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,@1)) aliens
+    val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
+    val le' = le @ map (fn a => Thm.instantiate' [] [SOME (Thm.dest_arg a)] @{thm of_nat_0_le_iff}) 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 =
+    simpset_of (put_simpset HOL_basic_ss \<^context> addsimps real_abs_thms1)
+  fun absmaxmin_elim_conv1 ctxt =
+    Simplifier.rewrite (put_simpset absmaxmin_elim_ss1 ctxt)
+
+  val absmaxmin_elim_conv2 =
+    let
+      val pth_abs = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] abs_split'
+      val pth_max = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] max_split
+      val pth_min = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] min_split
+      val abs_tm = \<^cterm>\<open>abs :: real \<Rightarrow> _\<close>
+      val p_v = (("P", 0), \<^typ>\<open>real \<Rightarrow> bool\<close>)
+      val x_v = (("x", 0), \<^typ>\<open>real\<close>)
+      val y_v = (("y", 0), \<^typ>\<open>real\<close>)
+      val is_max = is_binop \<^cterm>\<open>max :: real \<Rightarrow> _\<close>
+      val is_min = is_binop \<^cterm>\<open>min :: real \<Rightarrow> _\<close>
+      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_v,p), (x_v, 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_v,p), (x_v, Thm.dest_arg ax), (y_v,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_v,p), (x_v, Thm.dest_arg ax), (y_v,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
+    val {add, mul, neg, pow = _, sub = _, main} =
+        Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
+        (the (Semiring_Normalizer.match ctxt \<^cterm>\<open>(0::real) + 1\<close>))
+        Thm.term_ord
+  in gen_real_arith ctxt
+     (cterm_of_rat,
+      Numeral_Simprocs.field_comp_conv ctxt,
+      Numeral_Simprocs.field_comp_conv ctxt,
+      Numeral_Simprocs.field_comp_conv ctxt,
+      main ctxt, neg ctxt, add ctxt, mul ctxt, prover)
+  end;
+
+end
diff -r 315489d836d8 -r 0f7edf0853df src/HOL/Library/positivstellensatz.ML
--- a/src/HOL/Library/positivstellensatz.ML	Fri Jun 14 08:34:27 2019 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,779 +0,0 @@
-(*  Title:      HOL/Library/positivstellensatz.ML
-    Author:     Amine Chaieb, University of Cambridge
-
-A generic arithmetic prover based on Positivstellensatz certificates
---- also implements Fourier-Motzkin elimination as a special case
-Fourier-Motzkin elimination.
-*)
-
-(* A functor for finite mappings based on Tables *)
-
-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
-end;
-
-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
-   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;
-  in Tab.fold h a b end;
-
-fun choose f =
-  (case Tab.min f of
-    SOME entry => entry
-  | NONE => error "FuncFun.choose : Completely empty function")
-
-fun onefunc kv = update kv empty
-
-end;
-
-(* Some standard functors and utility functions for them *)
-
-structure FuncUtil =
-struct
-
-structure Intfunc = FuncFun(type key = int val ord = int_ord);
-structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
-structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
-structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
-structure Termfunc = FuncFun(type key = term val ord = Term_Ord.fast_term_ord);
-structure Ctermfunc = FuncFun(type key = cterm val ord = Thm.fast_term_ord);
-
-type monomial = int Ctermfunc.table;
-val monomial_ord = list_ord (prod_ord Thm.fast_term_ord int_ord) o apply2 Ctermfunc.dest
-structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
-
-type poly = Rat.rat Monomialfunc.table;
-
-(* The ordering so we can create canonical HOL polynomials.                  *)
-
-fun dest_monomial mon = sort (Thm.fast_term_ord o apply2 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
-      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 Thm.fast_term_ord int_ord) (mon1,mon2)
-    end;
-
-end
-
-(* positivstellensatz datatype and prover generation *)
-
-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
-
-end
-
-structure RealArith : REAL_ARITH =
-struct
-
-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;
-         (* 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 ->
-  (thm list * thm list * thm list -> positivstellensatz -> thm) ->
-    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 (Thm.cprop_of thb) tha)
-     (Thm.implies_intr (Thm.cprop_of tha) thb);
-
-fun prove_hyp tha thb =
-  if exists (curry op aconv (Thm.concl_of tha)) (Thm.hyps_of thb)  (* FIXME !? *)
-  then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb;
-
-val pth = @{lemma "(((x::real) < y) \<equiv> (y - x > 0))" and "((x \<le> y) \<equiv> (y - x \<ge> 0))" and
-     "((x = y) \<equiv> (x - y = 0))" and "((\<not>(x < y)) \<equiv> (x - y \<ge> 0))" and
-     "((\<not>(x \<le> y)) \<equiv> (x - y > 0))" and "((\<not>(x = y)) \<equiv> (x - y > 0 \<or> -(x - y) > 0))"
-  by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};
-
-val pth_final = @{lemma "(\<not>p \<Longrightarrow> False) \<Longrightarrow> p" by blast}
-val pth_add =
-  @{lemma "(x = (0::real) \<Longrightarrow> y = 0 \<Longrightarrow> x + y = 0 )" and "( x = 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y \<ge> 0)" and
-    "(x = 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" and "(x \<ge> 0 \<Longrightarrow> y = 0 \<Longrightarrow> x + y \<ge> 0)" and
-    "(x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y \<ge> 0)" and "(x \<ge> 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" and
-    "(x > 0 \<Longrightarrow> y = 0 \<Longrightarrow> x + y > 0)" and "(x > 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y > 0)" and
-    "(x > 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" by simp_all};
-
-val pth_mul =
-  @{lemma "(x = (0::real) \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and "(x = 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y = 0)" and
-    "(x = 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y = 0)" and "(x \<ge> 0 \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and
-    "(x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y \<ge> 0)" and "(x \<ge> 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y \<ge> 0)" and
-    "(x > 0 \<Longrightarrow>  y = 0 \<Longrightarrow> x * y = 0)" and "(x > 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y \<ge> 0)" and
-    "(x > 0 \<Longrightarrow>  y > 0 \<Longrightarrow> 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])};
-
-val pth_emul = @{lemma "y = (0::real) \<Longrightarrow> x * y = 0"  by simp};
-val pth_square = @{lemma "x * x \<ge> (0::real)"  by simp};
-
-val weak_dnf_simps =
-  List.take (@{thms simp_thms}, 34) @
-    @{lemma "((P \<and> (Q \<or> R)) = ((P\<and>Q) \<or> (P\<and>R)))" and "((Q \<or> R) \<and> P) = ((Q\<and>P) \<or> (R\<and>P))" and
-      "(P \<and> Q) = (Q \<and> P)" and "((P \<or> Q) = (Q \<or> P))" by blast+};
-
-(*
-val nnfD_simps =
-  @{lemma "((~(P & Q)) = (~P | ~Q))" and "((~(P | Q)) = (~P & ~Q) )" and
-    "((P --> Q) = (~P | Q) )" and "((P = Q) = ((P & Q) | (~P & ~ Q)))" and
-    "((~(P = Q)) = ((P & ~ Q) | (~P & Q)) )" and "((~ ~(P)) = P)" by blast+};
-*)
-
-val choice_iff = @{lemma "(\<forall>x. \<exists>y. P x y) = (\<exists>f. \<forall>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 "HOL.all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});
-
-val real_abs_thms1 = @{lemma
-  "((-1 * \<bar>x::real\<bar> \<ge> r) = (-1 * x \<ge> r \<and> 1 * x \<ge> r))" and
-  "((-1 * \<bar>x\<bar> + a \<ge> r) = (a + -1 * x \<ge> r \<and> a + 1 * x \<ge> r))" and
-  "((a + -1 * \<bar>x\<bar> \<ge> r) = (a + -1 * x \<ge> r \<and> a + 1 * x \<ge> r))" and
-  "((a + -1 * \<bar>x\<bar> + b \<ge> r) = (a + -1 * x + b \<ge> r \<and> a + 1 * x + b \<ge> r))" and
-  "((a + b + -1 * \<bar>x\<bar> \<ge> r) = (a + b + -1 * x \<ge> r \<and> a + b + 1 * x \<ge> r))" and
-  "((a + b + -1 * \<bar>x\<bar> + c \<ge> r) = (a + b + -1 * x + c \<ge> r \<and> a + b + 1 * x + c \<ge> r))" and
-  "((-1 * max x y \<ge> r) = (-1 * x \<ge> r \<and> -1 * y \<ge> r))" and
-  "((-1 * max x y + a \<ge> r) = (a + -1 * x \<ge> r \<and> a + -1 * y \<ge> r))" and
-  "((a + -1 * max x y \<ge> r) = (a + -1 * x \<ge> r \<and> a + -1 * y \<ge> r))" and
-  "((a + -1 * max x y + b \<ge> r) = (a + -1 * x + b \<ge> r \<and> a + -1 * y  + b \<ge> r))" and
-  "((a + b + -1 * max x y \<ge> r) = (a + b + -1 * x \<ge> r \<and> a + b + -1 * y \<ge> r))" and
-  "((a + b + -1 * max x y + c \<ge> r) = (a + b + -1 * x + c \<ge> r \<and> a + b + -1 * y  + c \<ge> r))" and
-  "((1 * min x y \<ge> r) = (1 * x \<ge> r \<and> 1 * y \<ge> r))" and
-  "((1 * min x y + a \<ge> r) = (a + 1 * x \<ge> r \<and> a + 1 * y \<ge> r))" and
-  "((a + 1 * min x y \<ge> r) = (a + 1 * x \<ge> r \<and> a + 1 * y \<ge> r))" and
-  "((a + 1 * min x y + b \<ge> r) = (a + 1 * x + b \<ge> r \<and> a + 1 * y  + b \<ge> r))" and
-  "((a + b + 1 * min x y \<ge> r) = (a + b + 1 * x \<ge> r \<and> a + b + 1 * y \<ge> r))" and
-  "((a + b + 1 * min x y + c \<ge> r) = (a + b + 1 * x + c \<ge> r \<and> a + b + 1 * y  + c \<ge> r))" and
-  "((min x y \<ge> r) = (x \<ge> r \<and> y \<ge> r))" and
-  "((min x y + a \<ge> r) = (a + x \<ge> r \<and> a + y \<ge> r))" and
-  "((a + min x y \<ge> r) = (a + x \<ge> r \<and> a + y \<ge> r))" and
-  "((a + min x y + b \<ge> r) = (a + x + b \<ge> r \<and> a + y  + b \<ge> r))" and
-  "((a + b + min x y \<ge> r) = (a + b + x \<ge> r \<and> a + b + y \<ge> r))" and
-  "((a + b + min x y + c \<ge> r) = (a + b + x + c \<ge> r \<and> a + b + y + c \<ge> r))" and
-  "((-1 * \<bar>x\<bar> > r) = (-1 * x > r \<and> 1 * x > r))" and
-  "((-1 * \<bar>x\<bar> + a > r) = (a + -1 * x > r \<and> a + 1 * x > r))" and
-  "((a + -1 * \<bar>x\<bar> > r) = (a + -1 * x > r \<and> a + 1 * x > r))" and
-  "((a + -1 * \<bar>x\<bar> + b > r) = (a + -1 * x + b > r \<and> a + 1 * x + b > r))" and
-  "((a + b + -1 * \<bar>x\<bar> > r) = (a + b + -1 * x > r \<and> a + b + 1 * x > r))" and
-  "((a + b + -1 * \<bar>x\<bar> + c > r) = (a + b + -1 * x + c > r \<and> a + b + 1 * x + c > r))" and
-  "((-1 * max x y > r) = ((-1 * x > r) \<and> -1 * y > r))" and
-  "((-1 * max x y + a > r) = (a + -1 * x > r \<and> a + -1 * y > r))" and
-  "((a + -1 * max x y > r) = (a + -1 * x > r \<and> a + -1 * y > r))" and
-  "((a + -1 * max x y + b > r) = (a + -1 * x + b > r \<and> a + -1 * y  + b > r))" and
-  "((a + b + -1 * max x y > r) = (a + b + -1 * x > r \<and> a + b + -1 * y > r))" and
-  "((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r \<and> a + b + -1 * y  + c > r))" and
-  "((min x y > r) = (x > r \<and> y > r))" and
-  "((min x y + a > r) = (a + x > r \<and> a + y > r))" and
-  "((a + min x y > r) = (a + x > r \<and> a + y > r))" and
-  "((a + min x y + b > r) = (a + x + b > r \<and> a + y  + b > r))" and
-  "((a + b + min x y > r) = (a + b + x > r \<and> a + b + y > r))" and
-  "((a + b + min x y + c > r) = (a + b + x + c > r \<and> a + b + y + c > r))"
-  by auto};
-
-val abs_split' = @{lemma "P \<bar>x::'a::linordered_idom\<bar> == (x \<ge> 0 \<and> P x \<or> x < 0 \<and> P (-x))"
-  by (atomize (full)) (auto split: abs_split)};
-
-val max_split = @{lemma "P (max x y) \<equiv> ((x::'a::linorder) \<le> y \<and> P y \<or> x > y \<and> P x)"
-  by (atomize (full)) (cases "x \<le> y", auto simp add: max_def)};
-
-val min_split = @{lemma "P (min x y) \<equiv> ((x::'a::linorder) \<le> y \<and> P x \<or> x > y \<and> P y)"
-  by (atomize (full)) (cases "x \<le> y", auto simp add: min_def)};
-
-
-         (* Miscellaneous *)
-fun literals_conv bops uops cv =
-  let
-    fun h t =
-      (case Thm.term_of t of
-        b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
-      | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
-      | _ => cv t)
-  in h end;
-
-fun cterm_of_rat x =
-  let
-    val (a, b) = Rat.dest x
-  in
-    if b = 1 then Numeral.mk_cnumber \<^ctyp>\<open>real\<close> a
-    else Thm.apply (Thm.apply \<^cterm>\<open>(/) :: real \<Rightarrow> _\<close>
-      (Numeral.mk_cnumber \<^ctyp>\<open>real\<close> a))
-      (Numeral.mk_cnumber \<^ctyp>\<open>real\<close> b)
-  end;
-
-fun dest_ratconst t =
-  case Thm.term_of t of
-    Const(\<^const_name>\<open>divide\<close>, _)$a$b => Rat.make(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
-  | _ => Rat.of_int (HOLogic.dest_number (Thm.term_of t) |> snd)
-fun is_ratconst t = can dest_ratconst t
-
-(*
-fun find_term p t = if p t then t else
- case t of
-  a$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 Thm.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 is_comb t = (case Thm.term_of t of _ $ _ => true | _ => false);
-
-fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
-  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>\<open>(^) :: real \<Rightarrow> _\<close> x)
-  (Numeral.mk_cnumber \<^ctyp>\<open>nat\<close> k)
-
-fun cterm_of_monomial m =
-  if FuncUtil.Ctermfunc.is_empty m then \<^cterm>\<open>1::real\<close>
-  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>\<open>(*) :: real \<Rightarrow> _\<close> s) t) vps
-    end
-
-fun cterm_of_cmonomial (m,c) =
-  if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
-  else if c = @1 then cterm_of_monomial m
-  else Thm.apply (Thm.apply \<^cterm>\<open>(*)::real \<Rightarrow> _\<close> (cterm_of_rat c)) (cterm_of_monomial m);
-
-fun cterm_of_poly p =
-  if FuncUtil.Monomialfunc.is_empty p then \<^cterm>\<open>0::real\<close>
-  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>\<open>(+) :: real \<Rightarrow> _\<close> 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 = put_simpset HOL_basic_ss ctxt 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 = put_simpset HOL_basic_ss ctxt addsimps prenex_simps
-    val skolemize_ss = put_simpset HOL_basic_ss ctxt addsimps [choice_iff]
-    val presimp_conv = Simplifier.rewrite pre_ss
-    val prenex_conv = Simplifier.rewrite prenex_ss
-    val skolemize_conv = Simplifier.rewrite skolemize_ss
-    val weak_dnf_ss = put_simpset HOL_basic_ss ctxt addsimps weak_dnf_simps
-    val weak_dnf_conv = Simplifier.rewrite 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>\<open>(\<le>) :: real \<Rightarrow> _\<close>
-            orelse g aconvc \<^cterm>\<open>(<) :: real \<Rightarrow> _\<close>
-         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))
-         (match_mp_rule pth_add [th, th'])
-    fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
-       (Thm.instantiate' [] [SOME ct] (th RS pth_emul))
-    fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
-       (Thm.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>\<open>Trueprop\<close>
-                          (Thm.apply (Thm.apply \<^cterm>\<open>(=)::real \<Rightarrow> _\<close> (mk_numeric x))
-                               \<^cterm>\<open>0::real\<close>)))
-          | Rational_le x => eqT_elim(numeric_ge_conv(Thm.apply \<^cterm>\<open>Trueprop\<close>
-                          (Thm.apply (Thm.apply \<^cterm>\<open>(\<le>)::real \<Rightarrow> _\<close>
-                                     \<^cterm>\<open>0::real\<close>) (mk_numeric x))))
-          | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.apply \<^cterm>\<open>Trueprop\<close>
-                      (Thm.apply (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>)
-                        (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])
-          (translate proof)
-      end
-
-    val init_conv = presimp_conv then_conv
-        nnf_conv ctxt then_conv skolemize_conv then_conv prenex_conv then_conv
-        weak_dnf_conv
-
-    val concl = Thm.dest_arg o Thm.cprop_of
-    fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
-    val is_req = is_binop \<^cterm>\<open>(=):: real \<Rightarrow> _\<close>
-    val is_ge = is_binop \<^cterm>\<open>(\<le>):: real \<Rightarrow> _\<close>
-    val is_gt = is_binop \<^cterm>\<open>(<):: real \<Rightarrow> _\<close>
-    val is_conj = is_binop \<^cterm>\<open>HOL.conj\<close>
-    val is_disj = is_binop \<^cterm>\<open>HOL.disj\<close>
-    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 (Thm.instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
-          (Thm.implies_intr (Thm.apply \<^cterm>\<open>Trueprop\<close> p) th1))
-        (Thm.implies_intr (Thm.apply \<^cterm>\<open>Trueprop\<close> 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>\<open>Trueprop\<close> (Thm.dest_arg1 ct))::oths)
-              val (th2, cert2) =
-                overall (Right::cert_choice) dun
-                  (Thm.assume (Thm.apply \<^cterm>\<open>Trueprop\<close> (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>\<open>(=) :: real \<Rightarrow> _\<close>
-    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 ([],[((("x", 0), \<^typ>\<open>real\<close>),l),((("y", 0), \<^typ>\<open>real\<close>),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>\<open>uminus :: real \<Rightarrow> _\<close> th_p
-        val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
-        val th' = Drule.binop_cong_rule \<^cterm>\<open>HOL.disj\<close>
-          (Drule.arg_cong_rule (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>) th_p)
-          (Drule.arg_cong_rule (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>) 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 Thm.term_of t of
-            Const(\<^const_name>\<open>Ex\<close>,_)$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 Thm.term_of x of
-        Free(s,_) => s
-      | Var ((s,_),_) => s
-      | _ => "x"
-
-    fun mk_forall x th =
-      let
-        val T = Thm.typ_of_cterm x
-        val all = Thm.cterm_of ctxt (Const (\<^const_name>\<open>All\<close>, (T --> \<^typ>\<open>bool\<close>) --> \<^typ>\<open>bool\<close>))
-      in Drule.arg_cong_rule all (Thm.abstract_rule (name_of x) x th) end
-
-    val specl = fold_rev (fn x => fn th => Thm.instantiate' [] [SOME x] (th RS spec));
-
-    fun ext T = Thm.cterm_of ctxt (Const (\<^const_name>\<open>Ex\<close>, (T --> \<^typ>\<open>bool\<close>) --> \<^typ>\<open>bool\<close>))
-    fun mk_ex v t = Thm.apply (ext (Thm.typ_of_cterm v)) (Thm.lambda v t)
-
-    fun choose v th th' =
-      case Thm.concl_of th of
-        \<^term>\<open>Trueprop\<close> $ (Const(\<^const_name>\<open>Ex\<close>,_)$_) =>
-        let
-          val p = (funpow 2 Thm.dest_arg o Thm.cprop_of) th
-          val T = Thm.dest_ctyp0 (Thm.ctyp_of_cterm p)
-          val th0 = fconv_rule (Thm.beta_conversion true)
-            (Thm.instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o Thm.cprop_of) th'] exE)
-          val pv = (Thm.rhs_of o Thm.beta_conversion true)
-            (Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.apply p v))
-          val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
-        in Thm.implies_elim (Thm.implies_elim th0 th) th1  end
-      | _ => raise THM ("choose",0,[th, th'])
-
-    fun simple_choose v th =
-      choose v
-        (Thm.assume
-          ((Thm.apply \<^cterm>\<open>Trueprop\<close> o mk_ex v) (Thm.dest_arg (hd (Thm.chyps_of th))))) th
-
-    val strip_forall =
-      let
-        fun h (acc, t) =
-          case Thm.term_of t of
-            Const(\<^const_name>\<open>All\<close>,_)$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 ctxt then_conv
-          literals_conv [\<^term>\<open>HOL.conj\<close>, \<^term>\<open>HOL.disj\<close>] []
-          (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>\<open>HOL.conj\<close>, \<^term>\<open>HOL.disj\<close>] []
-                  (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>\<open>Trueprop\<close> (Thm.apply \<^cterm>\<open>Not\<close> 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>\<open>False\<close> 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>\<open>Trueprop\<close> (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>\<open>Trueprop\<close> bod))) th2)
-          in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
-          end
-      in (Thm.implies_elim (Thm.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 = @0)
-  fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c * x)
-  val one_tm = \<^cterm>\<open>1::real\<close>
-  fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p @0)) 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 > @0)) lts of
-      SOME r => r
-    | NONE =>
-      (case find_first (contradictory (fn x => x > @0)) 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 @0 = @0) ineqs) +
-             length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) ineqs) *
-             length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 < @0) 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 @0
-               val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v @0
-             in
-               if c1 * c2 >= @0 then acc else
-               let
-                 val e1' = linear_cmul (abs c2) e1
-                 val e2' = linear_cmul (abs c1) e2
-                 val p1' = Product(Rational_lt (abs c2), p1)
-                 val p2' = Product(Rational_lt (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 @0 = @0) les
-           val (lts0,lts1) =
-             List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) lts
-           val (lesp,lesn) =
-             List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) les1
-           val (ltsp,ltsn) =
-             List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) 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 = @0)) 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 @0 in
-               if d = @0 then inp else
-               let
-                 val k = ~ d * abs c / c
-                 val e' = linear_cmul k e
-                 val t' = linear_cmul (abs c) t
-                 val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
-                 val q' = Product(Rational_lt (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>\<open>0::real\<close> then FuncUtil.Ctermfunc.empty
-    else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, @1)
-    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, @1)
-        else
-          let val (opr,l) = Thm.dest_comb lop
-          in
-            if opr aconvc \<^cterm>\<open>(+) :: real \<Rightarrow> _\<close>
-            then linear_add (lin_of_hol l) (lin_of_hol r)
-            else if opr aconvc \<^cterm>\<open>(*) :: real \<Rightarrow> _\<close>
-                    andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
-            else FuncUtil.Ctermfunc.onefunc (ct, @1)
-          end
-      end
-
-  fun is_alien ct =
-    case Thm.term_of ct of
-      Const(\<^const_name>\<open>of_nat\<close>, _)$ n => not (can HOLogic.dest_number n)
-    | Const(\<^const_name>\<open>of_int\<close>, _)$ n => not (can HOLogic.dest_number n)
-    | _ => 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 Thm.cprop_of
-    val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o Thm.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,@1)) aliens
-    val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
-    val le' = le @ map (fn a => Thm.instantiate' [] [SOME (Thm.dest_arg a)] @{thm of_nat_0_le_iff}) 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 =
-    simpset_of (put_simpset HOL_basic_ss \<^context> addsimps real_abs_thms1)
-  fun absmaxmin_elim_conv1 ctxt =
-    Simplifier.rewrite (put_simpset absmaxmin_elim_ss1 ctxt)
-
-  val absmaxmin_elim_conv2 =
-    let
-      val pth_abs = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] abs_split'
-      val pth_max = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] max_split
-      val pth_min = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] min_split
-      val abs_tm = \<^cterm>\<open>abs :: real \<Rightarrow> _\<close>
-      val p_v = (("P", 0), \<^typ>\<open>real \<Rightarrow> bool\<close>)
-      val x_v = (("x", 0), \<^typ>\<open>real\<close>)
-      val y_v = (("y", 0), \<^typ>\<open>real\<close>)
-      val is_max = is_binop \<^cterm>\<open>max :: real \<Rightarrow> _\<close>
-      val is_min = is_binop \<^cterm>\<open>min :: real \<Rightarrow> _\<close>
-      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_v,p), (x_v, 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_v,p), (x_v, Thm.dest_arg ax), (y_v,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_v,p), (x_v, Thm.dest_arg ax), (y_v,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
-    val {add, mul, neg, pow = _, sub = _, main} =
-        Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
-        (the (Semiring_Normalizer.match ctxt \<^cterm>\<open>(0::real) + 1\<close>))
-        Thm.term_ord
-  in gen_real_arith ctxt
-     (cterm_of_rat,
-      Numeral_Simprocs.field_comp_conv ctxt,
-      Numeral_Simprocs.field_comp_conv ctxt,
-      Numeral_Simprocs.field_comp_conv ctxt,
-      main ctxt, neg ctxt, add ctxt, mul ctxt, prover)
-  end;
-
-end