src/HOL/Library/positivstellensatz.ML

explicit indication of Unsynchronized.ref;

(* Title:      Library/Sum_Of_Squares/positivstellensatz
   Author:     Amine Chaieb, University of Cambridge
   Description: A generic arithmetic prover based on Positivstellensatz certificates --- 
    also implements Fourrier-Motzkin elimination as a special case Fourrier-Motzkin elimination.
*)

(* A functor for finite mappings based on Tables *)

signature FUNC = 
sig
 type 'a T
 type key
 val apply : 'a T -> key -> 'a
 val applyd :'a T -> (key -> 'a) -> key -> 'a
 val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a T -> 'a T -> 'a T
 val defined : 'a T -> key -> bool
 val dom : 'a T -> key list
 val fold : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b
 val fold_rev : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b
 val graph : 'a T -> (key * 'a) list
 val is_undefined : 'a T -> bool
 val mapf : ('a -> 'b) -> 'a T -> 'b T
 val tryapplyd : 'a T -> key -> 'a -> 'a
 val undefine :  key -> 'a T -> 'a T
 val undefined : 'a T
 val update : key * 'a -> 'a T -> 'a T
 val updatep : (key * 'a -> bool) -> key * 'a -> 'a T -> 'a T
 val choose : 'a T -> key * 'a
 val onefunc : key * 'a -> 'a T
 val get_first: (key*'a -> 'a option) -> 'a T -> 'a option
end;

functor FuncFun(Key: KEY) : FUNC=
struct

type key = Key.key;
structure Tab = Table(Key);
type 'a T = 'a Tab.table;

val undefined = Tab.empty;
val is_undefined = Tab.is_empty;
val mapf = Tab.map;
val fold = Tab.fold;
val fold_rev = Tab.fold_rev;
val graph = Tab.dest;
fun dom a = sort Key.ord (Tab.keys a);
fun applyd f d x = case Tab.lookup f x of 
   SOME y => y
 | NONE => d x;

fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
fun tryapplyd f a d = applyd f (K d) a;
val defined = Tab.defined;
fun undefine x t = (Tab.delete x t handle UNDEF => t);
val update = Tab.update;
fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
fun combine f z a b = 
 let
  fun h (k,v) t = case Tab.lookup t k of
     NONE => Tab.update (k,v) t
   | SOME v' => let val w = f v v'
     in if z w then Tab.delete k t else Tab.update (k,w) t end;
  in Tab.fold h a b end;

fun choose f = case Tab.min_key f of 
   SOME k => (k,valOf (Tab.lookup f k))
 | NONE => error "FuncFun.choose : Completely undefined function"

fun onefunc kv = update kv undefined

local
fun  find f (k,v) NONE = f (k,v)
   | find f (k,v) r = r
in
fun get_first f t = fold (find f) t NONE
end
end;

(* Some standard functors and utility functions for them *)

structure FuncUtil =
struct

fun increasing f ord (x,y) = ord (f x, f y);

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 = TermOrd.fast_term_ord);

val cterm_ord = (fn (s,t) => TermOrd.fast_term_ord(term_of s, term_of t))

structure Ctermfunc = FuncFun(type key = cterm val ord = cterm_ord);

type monomial = int Ctermfunc.T;

fun monomial_ord (m1,m2) = list_ord (prod_ord cterm_ord int_ord) (Ctermfunc.graph m1, Ctermfunc.graph m2)

structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)

type poly = Rat.rat Monomialfunc.T;

(* The ordering so we can create canonical HOL polynomials.                  *)

fun dest_monomial mon = sort (increasing fst cterm_ord) (Ctermfunc.graph mon);

fun monomial_order (m1,m2) =
 if Ctermfunc.is_undefined m2 then LESS 
 else if Ctermfunc.is_undefined m1 then GREATER 
 else
  let val mon1 = dest_monomial m1 
      val mon2 = dest_monomial m2
      val deg1 = fold (curry op + o snd) mon1 0
      val deg2 = fold (curry op + o snd) mon2 0 
  in if deg1 < deg2 then GREATER else if deg1 > deg2 then LESS
     else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
  end;

end

(* positivstellensatz datatype and prover generation *)

signature REAL_ARITH = 
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 Thm FuncUtil;;
(* ------------------------------------------------------------------------- *)
(* 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

val my_eqs = Unsynchronized.ref ([] : thm list);
val my_les = Unsynchronized.ref ([] : thm list);
val my_lts = Unsynchronized.ref ([] : thm list);
val my_proof = Unsynchronized.ref (Axiom_eq 0);
val my_context = Unsynchronized.ref @{context};

val my_mk_numeric = Unsynchronized.ref ((K @{cterm True}) :Rat.rat -> cterm);
val my_numeric_eq_conv = Unsynchronized.ref no_conv;
val my_numeric_ge_conv = Unsynchronized.ref no_conv;
val my_numeric_gt_conv = Unsynchronized.ref no_conv;
val my_poly_conv = Unsynchronized.ref no_conv;
val my_poly_neg_conv = Unsynchronized.ref no_conv;
val my_poly_add_conv = Unsynchronized.ref no_conv;
val my_poly_mul_conv = Unsynchronized.ref no_conv;


    (* 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;

fun conjunctions th = case try Conjunction.elim th of
   SOME (th1,th2) => (conjunctions th1) @ conjunctions th2
 | NONE => [th];

val pth = @{lemma "(((x::real) < y) == (y - x > 0)) &&& ((x <= y) == (y - x >= 0)) 
     &&& ((x = y) == (x - y = 0)) &&& ((~(x < y)) == (x - y >= 0)) &&& ((~(x <= y)) == (x - y > 0))
     &&& ((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
  by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)} |> 
conjunctions;

val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
val pth_add = 
 @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 ) &&& ( x = 0 ==> y >= 0 ==> x + y >= 0) 
    &&& (x = 0 ==> y > 0 ==> x + y > 0) &&& (x >= 0 ==> y = 0 ==> x + y >= 0) 
    &&& (x >= 0 ==> y >= 0 ==> x + y >= 0) &&& (x >= 0 ==> y > 0 ==> x + y > 0) 
    &&& (x > 0 ==> y = 0 ==> x + y > 0) &&& (x > 0 ==> y >= 0 ==> x + y > 0) 
    &&& (x > 0 ==> y > 0 ==> x + y > 0)"  by simp_all} |> conjunctions ;

val pth_mul = 
  @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0) &&& (x = 0 ==> y >= 0 ==> x * y = 0) &&& 
           (x = 0 ==> y > 0 ==> x * y = 0) &&& (x >= 0 ==> y = 0 ==> x * y = 0) &&& 
           (x >= 0 ==> y >= 0 ==> x * y >= 0 ) &&& ( x >= 0 ==> y > 0 ==> x * y >= 0 ) &&&
           (x > 0 ==>  y = 0 ==> x * y = 0 ) &&& ( x > 0 ==> y >= 0 ==> x * y >= 0 ) &&&
           (x > 0 ==>  y > 0 ==> x * y > 0)"
  by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
    mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])} |> conjunctions;

val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
val pth_square = @{lemma "x * x >= (0::real)"  by simp};

val weak_dnf_simps = List.take (simp_thms, 34) 
    @ conjunctions @{lemma "((P & (Q | R)) = ((P&Q) | (P&R))) &&& ((Q | R) & P) = ((Q&P) | (R&P)) &&& (P & Q) = (Q & P) &&& ((P | Q) = (Q | P))" by blast+};

val nnfD_simps = conjunctions @{lemma "((~(P & Q)) = (~P | ~Q)) &&& ((~(P | Q)) = (~P & ~Q) ) &&& ((P --> Q) = (~P | Q) ) &&& ((P = Q) = ((P & Q) | (~P & ~ Q))) &&& ((~(P = Q)) = ((P & ~ Q) | (~P & Q)) ) &&& ((~ ~(P)) = P)" by blast+}

val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
val prenex_simps = map (fn th => th RS sym) ([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @ @{thms "all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});

val real_abs_thms1 = conjunctions @{lemma
  "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r)) &&&
  ((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
  ((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
  ((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r)) &&&
  ((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r)) &&&
  ((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r)) &&&
  ((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r)) &&&
  ((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
  ((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
  ((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y  + b >= r)) &&&
  ((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r)) &&&
  ((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y  + c >= r)) &&&
  ((1 * min x y >= r) = (1 * x >= r & 1 * y >= r)) &&&
  ((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
  ((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
  ((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y  + b >= r) )&&&
  ((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r)) &&&
  ((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y  + c >= r)) &&&
  ((min x y >= r) = (x >= r &  y >= r)) &&&
  ((min x y + a >= r) = (a + x >= r & a + y >= r)) &&&
  ((a + min x y >= r) = (a + x >= r & a + y >= r)) &&&
  ((a + min x y + b >= r) = (a + x + b >= r & a + y  + b >= r)) &&&
  ((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r) )&&&
  ((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r)) &&&
  ((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r)) &&&
  ((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
  ((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
  ((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r)) &&&
  ((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r)) &&&
  ((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r)) &&&
  ((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r)) &&&
  ((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
  ((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
  ((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y  + b > r)) &&&
  ((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r)) &&&
  ((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y  + c > r)) &&&
  ((min x y > r) = (x > r &  y > r)) &&&
  ((min x y + a > r) = (a + x > r & a + y > r)) &&&
  ((a + min x y > r) = (a + x > r & a + y > r)) &&&
  ((a + min x y + b > r) = (a + x + b > r & a + y  + b > r)) &&&
  ((a + b + min x y > r) = (a + b + x > r & a + b + y > r)) &&&
  ((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
  by auto};

val abs_split' = @{lemma "P (abs (x::'a::ordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
  by (atomize (full)) (auto split add: abs_split)};

val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
  by (atomize (full)) (cases "x <= y", auto simp add: max_def)};

val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
  by (atomize (full)) (cases "x <= y", auto simp add: min_def)};


         (* Miscalineous *)
fun literals_conv bops uops cv = 
 let fun h t =
  case (term_of t) of 
   b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
 | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
 | _ => cv t
 in h end;

fun cterm_of_rat x = 
let val (a, b) = Rat.quotient_of_rat x
in 
 if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
  else Thm.capply (Thm.capply @{cterm "op / :: real => _"} 
                   (Numeral.mk_cnumber @{ctyp "real"} a))
        (Numeral.mk_cnumber @{ctyp "real"} b)
end;

  fun dest_ratconst t = case term_of t of
   Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
 | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
 fun is_ratconst t = can dest_ratconst t

fun find_term p t = if p t then t else 
 case t of
  a$b => (find_term p a handle TERM _ => find_term p b)
 | Abs (_,_,t') => find_term p t'
 | _ => raise TERM ("find_term",[t]);

fun find_cterm p t = if p t then t else 
 case term_of t of
  a$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
 | Abs (_,_,t') => find_cterm p (Thm.dest_abs NONE t |> snd)
 | _ => raise CTERM ("find_cterm",[t]);

    (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
fun is_comb t = case (term_of t) of _$_ => true | _ => false;

fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
  handle CTERM _ => false;


(* Map back polynomials to HOL.                         *)

local
 open Thm Numeral
in

fun cterm_of_varpow x k = if k = 1 then x else capply (capply @{cterm "op ^ :: real => _"} x) 
  (mk_cnumber @{ctyp nat} k)

fun cterm_of_monomial m = 
 if Ctermfunc.is_undefined m then @{cterm "1::real"} 
 else 
  let 
   val m' = dest_monomial m
   val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' [] 
  in foldr1 (fn (s, t) => capply (capply @{cterm "op * :: real => _"} s) t) vps
  end

fun cterm_of_cmonomial (m,c) = if Ctermfunc.is_undefined m then cterm_of_rat c
    else if c = Rat.one then cterm_of_monomial m
    else capply (capply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);

fun cterm_of_poly p = 
 if Monomialfunc.is_undefined p then @{cterm "0::real"} 
 else
  let 
   val cms = map cterm_of_cmonomial
     (sort (prod_ord monomial_order (K EQUAL)) (Monomialfunc.graph p))
  in foldr1 (fn (t1, t2) => capply(capply @{cterm "op + :: real => _"} t1) t2) cms
  end;

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
 open Conv Thm;
 val _ = my_context := ctxt 
 val _ = (my_mk_numeric := mk_numeric ; my_numeric_eq_conv := numeric_eq_conv ; 
          my_numeric_ge_conv := numeric_ge_conv; my_numeric_gt_conv := numeric_gt_conv ;
          my_poly_conv := poly_conv; my_poly_neg_conv := poly_neg_conv; 
          my_poly_add_conv := poly_add_conv; my_poly_mul_conv := poly_mul_conv)
 val pre_ss = HOL_basic_ss addsimps simp_thms@ ex_simps@ all_simps@[@{thm not_all},@{thm not_ex},ex_disj_distrib, all_conj_distrib, @{thm if_bool_eq_disj}]
 val prenex_ss = HOL_basic_ss addsimps prenex_simps
 val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
 val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
 val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
 val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
 val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
 val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
 fun eqT_elim th = equal_elim (symmetric th) @{thm TrueI}
 fun oprconv cv ct = 
  let val g = Thm.dest_fun2 ct
  in if g aconvc @{cterm "op <= :: real => _"} 
       orelse g aconvc @{cterm "op < :: real => _"} 
     then arg_conv cv ct else arg1_conv cv ct
  end

 fun real_ineq_conv th ct =
  let
   val th' = (instantiate (match (lhs_of th, ct)) th 
      handle MATCH => raise CTERM ("real_ineq_conv", [ct]))
  in transitive th' (oprconv poly_conv (Thm.rhs_of th'))
  end 
  val [real_lt_conv, real_le_conv, real_eq_conv,
       real_not_lt_conv, real_not_le_conv, _] =
       map real_ineq_conv pth
  fun match_mp_rule ths ths' = 
   let
     fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
      | th::ths => (ths' MRS th handle THM _ => f ths ths')
   in f ths ths' end
  fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
         (match_mp_rule pth_mul [th, th'])
  fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
         (match_mp_rule pth_add [th, th'])
  fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv)) 
       (instantiate' [] [SOME ct] (th RS pth_emul)) 
  fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
       (instantiate' [] [SOME t] pth_square)

  fun hol_of_positivstellensatz(eqs,les,lts) proof =
   let 
    val _ = (my_eqs := eqs ; my_les := les ; my_lts := lts ; my_proof := proof)
    fun translate prf = case prf of
        Axiom_eq n => nth eqs n
      | Axiom_le n => nth les n
      | Axiom_lt n => nth lts n
      | Rational_eq x => eqT_elim(numeric_eq_conv(capply @{cterm Trueprop} 
                          (capply (capply @{cterm "op =::real => _"} (mk_numeric x)) 
                               @{cterm "0::real"})))
      | Rational_le x => eqT_elim(numeric_ge_conv(capply @{cterm Trueprop} 
                          (capply (capply @{cterm "op <=::real => _"} 
                                     @{cterm "0::real"}) (mk_numeric x))))
      | Rational_lt x => eqT_elim(numeric_gt_conv(capply @{cterm Trueprop} 
                      (capply (capply @{cterm "op <::real => _"} @{cterm "0::real"})
                        (mk_numeric x))))
      | Square pt => square_rule (cterm_of_poly pt)
      | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
      | Sum(p1,p2) => add_rule (translate p1) (translate p2)
      | Product(p1,p2) => mul_rule (translate p1) (translate p2)
   in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv]) 
          (translate proof)
   end
  
  val init_conv = presimp_conv then_conv
      nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
      weak_dnf_conv

  val concl = 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 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 (assume (capply @{cterm Trueprop} (dest_arg1 ct))::oths)
       val (th2, cert2) = overall (Right::cert_choice) dun (assume (capply @{cterm Trueprop} (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 dest_binop ct 
                         else raise CTERM ("dest_binary",[b,ct])
  val dest_eq = dest_binary @{cterm "op = :: real => _"}
  val neq_th = nth pth 5
  fun real_not_eq_conv ct = 
   let 
    val (l,r) = dest_eq (dest_arg ct)
    val th = instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
    val th_p = poly_conv(dest_arg(dest_arg1(rhs_of th)))
    val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
    val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
    val th' = Drule.binop_cong_rule @{cterm "op |"} 
     (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
     (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
    in transitive th th' 
  end
 fun equal_implies_1_rule PQ = 
  let 
   val P = lhs_of PQ
  in implies_intr P (equal_elim PQ (assume P))
  end
 (* FIXME!!! Copied from groebner.ml *)
 val strip_exists =
  let fun h (acc, t) =
   case (term_of t) of
    Const("Ex",_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
  | _ => (acc,t)
  in fn t => h ([],t)
  end
  fun name_of x = case term_of x of
   Free(s,_) => s
 | Var ((s,_),_) => s
 | _ => "x"

  fun mk_forall x th = Drule.arg_cong_rule (instantiate_cterm' [SOME (ctyp_of_term x)] [] @{cpat "All :: (?'a => bool) => _" }) (abstract_rule (name_of x) x th)

  val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));

 fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
 fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)

 fun choose v th th' = case concl_of th of 
   @{term Trueprop} $ (Const("Ex",_)$_) => 
    let
     val p = (funpow 2 Thm.dest_arg o cprop_of) th
     val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
     val th0 = fconv_rule (Thm.beta_conversion true)
         (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
     val pv = (Thm.rhs_of o Thm.beta_conversion true) 
           (Thm.capply @{cterm Trueprop} (Thm.capply p v))
     val th1 = forall_intr v (implies_intr pv th')
    in implies_elim (implies_elim th0 th) th1  end
 | _ => raise THM ("choose",0,[th, th'])

  fun simple_choose v th = 
     choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th

 val strip_forall =
  let fun h (acc, t) =
   case (term_of t) of
    Const("All",_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
  | _ => (acc,t)
  in fn t => h ([],t)
  end

 fun f ct =
  let 
   val nnf_norm_conv' = 
     nnf_conv then_conv 
     literals_conv [@{term "op &"}, @{term "op |"}] [] 
     (More_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 "op &"}, @{term "op |"}] [] 
                  (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv 
        try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
  val nct = capply @{cterm Trueprop} (capply @{cterm "Not"} ct)
  val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
  val tm0 = dest_arg (rhs_of th0)
  val (th, certificates) = if tm0 aconvc @{cterm False} then (equal_implies_1_rule th0, Trivial) else
   let 
    val (evs,bod) = strip_exists tm0
    val (avs,ibod) = strip_forall bod
    val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
    val (th2, certs) = overall [] [] [specl avs (assume (rhs_of th1))]
    val th3 = fold simple_choose evs (prove_hyp (equal_elim th1 (assume (capply @{cterm Trueprop} bod))) th2)
   in (Drule.implies_intr_hyps (prove_hyp (equal_elim th0 (assume nct)) th3), certs)
   end
  in (implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates)
 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(Monomialfunc.onefunc (Ctermfunc.undefined, 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), Trivial)
 end
end;

(* A less general generic arithmetic prover dealing with abs,max and min*)

local
 val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
 fun absmaxmin_elim_conv1 ctxt = 
    Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)

 val absmaxmin_elim_conv2 =
  let 
   val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
   val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
   val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
   val abs_tm = @{cterm "abs :: real => _"}
   val p_tm = @{cpat "?P :: real => bool"}
   val x_tm = @{cpat "?x :: real"}
   val y_tm = @{cpat "?y::real"}
   val is_max = is_binop @{cterm "max :: real => _"}
   val is_min = is_binop @{cterm "min :: real => _"} 
   fun is_abs t = is_comb t andalso dest_fun t aconvc abs_tm
   fun eliminate_construct p c tm =
    let 
     val t = find_cterm p tm
     val th0 = (symmetric o beta_conversion false) (capply (cabs t tm) t)
     val (p,ax) = (dest_comb o rhs_of) th0
    in fconv_rule(arg_conv(binop_conv (arg_conv (beta_conversion false))))
               (transitive th0 (c p ax))
   end

   val elim_abs = eliminate_construct is_abs
    (fn p => fn ax => 
       instantiate ([], [(p_tm,p), (x_tm, dest_arg ax)]) pth_abs)
   val elim_max = eliminate_construct is_max
    (fn p => fn ax => 
      let val (ax,y) = dest_comb ax 
      in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)]) 
      pth_max end)
   val elim_min = eliminate_construct is_min
    (fn p => fn ax => 
      let val (ax,y) = dest_comb ax 
      in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)]) 
      pth_min end)
   in first_conv [elim_abs, elim_max, elim_min, all_conv]
  end;
in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
        gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,
                       absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
end;

(* An instance for reals*) 

fun gen_prover_real_arith ctxt prover = 
 let
  fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS
  val {add,mul,neg,pow,sub,main} = 
     Normalizer.semiring_normalizers_ord_wrapper ctxt
      (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) 
     simple_cterm_ord
in gen_real_arith ctxt
   (cterm_of_rat, field_comp_conv, field_comp_conv,field_comp_conv,
    main,neg,add,mul, prover)
end;

end