src/HOL/Library/positivstellensatz.ML
author wenzelm
Wed Feb 15 23:19:30 2012 +0100 (2012-02-15)
changeset 46497 89ccf66aa73d
parent 45654 cf10bde35973
child 46594 f11f332b964f
permissions -rw-r--r--
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
     1 (*  Title:      HOL/Library/positivstellensatz.ML
     2     Author:     Amine Chaieb, University of Cambridge
     3 
     4 A generic arithmetic prover based on Positivstellensatz certificates
     5 --- also implements Fourrier-Motzkin elimination as a special case
     6 Fourrier-Motzkin elimination.
     7 *)
     8 
     9 (* A functor for finite mappings based on Tables *)
    10 
    11 signature FUNC = 
    12 sig
    13  include TABLE
    14  val apply : 'a table -> key -> 'a
    15  val applyd :'a table -> (key -> 'a) -> key -> 'a
    16  val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a table -> 'a table -> 'a table
    17  val dom : 'a table -> key list
    18  val tryapplyd : 'a table -> key -> 'a -> 'a
    19  val updatep : (key * 'a -> bool) -> key * 'a -> 'a table -> 'a table
    20  val choose : 'a table -> key * 'a
    21  val onefunc : key * 'a -> 'a table
    22 end;
    23 
    24 functor FuncFun(Key: KEY) : FUNC=
    25 struct
    26 
    27 structure Tab = Table(Key);
    28 
    29 open Tab;
    30 
    31 fun dom a = sort Key.ord (Tab.keys a);
    32 fun applyd f d x = case Tab.lookup f x of 
    33    SOME y => y
    34  | NONE => d x;
    35 
    36 fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
    37 fun tryapplyd f a d = applyd f (K d) a;
    38 fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
    39 fun combine f z a b = 
    40  let
    41   fun h (k,v) t = case Tab.lookup t k of
    42      NONE => Tab.update (k,v) t
    43    | SOME v' => let val w = f v v'
    44      in if z w then Tab.delete k t else Tab.update (k,w) t end;
    45   in Tab.fold h a b end;
    46 
    47 fun choose f = case Tab.min_key f of 
    48    SOME k => (k, the (Tab.lookup f k))
    49  | NONE => error "FuncFun.choose : Completely empty function"
    50 
    51 fun onefunc kv = update kv empty
    52 
    53 end;
    54 
    55 (* Some standard functors and utility functions for them *)
    56 
    57 structure FuncUtil =
    58 struct
    59 
    60 structure Intfunc = FuncFun(type key = int val ord = int_ord);
    61 structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
    62 structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
    63 structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
    64 structure Termfunc = FuncFun(type key = term val ord = Term_Ord.fast_term_ord);
    65 
    66 val cterm_ord = Term_Ord.fast_term_ord o pairself term_of
    67 
    68 structure Ctermfunc = FuncFun(type key = cterm val ord = cterm_ord);
    69 
    70 type monomial = int Ctermfunc.table;
    71 
    72 val monomial_ord = list_ord (prod_ord cterm_ord int_ord) o pairself Ctermfunc.dest
    73 
    74 structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
    75 
    76 type poly = Rat.rat Monomialfunc.table;
    77 
    78 (* The ordering so we can create canonical HOL polynomials.                  *)
    79 
    80 fun dest_monomial mon = sort (cterm_ord o pairself fst) (Ctermfunc.dest mon);
    81 
    82 fun monomial_order (m1,m2) =
    83  if Ctermfunc.is_empty m2 then LESS 
    84  else if Ctermfunc.is_empty m1 then GREATER 
    85  else
    86   let val mon1 = dest_monomial m1 
    87       val mon2 = dest_monomial m2
    88       val deg1 = fold (Integer.add o snd) mon1 0
    89       val deg2 = fold (Integer.add o snd) mon2 0 
    90   in if deg1 < deg2 then GREATER else if deg1 > deg2 then LESS
    91      else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
    92   end;
    93 
    94 end
    95 
    96 (* positivstellensatz datatype and prover generation *)
    97 
    98 signature REAL_ARITH = 
    99 sig
   100   
   101   datatype positivstellensatz =
   102    Axiom_eq of int
   103  | Axiom_le of int
   104  | Axiom_lt of int
   105  | Rational_eq of Rat.rat
   106  | Rational_le of Rat.rat
   107  | Rational_lt of Rat.rat
   108  | Square of FuncUtil.poly
   109  | Eqmul of FuncUtil.poly * positivstellensatz
   110  | Sum of positivstellensatz * positivstellensatz
   111  | Product of positivstellensatz * positivstellensatz;
   112 
   113 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
   114 
   115 datatype tree_choice = Left | Right
   116 
   117 type prover = tree_choice list -> 
   118   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   119   thm list * thm list * thm list -> thm * pss_tree
   120 type cert_conv = cterm -> thm * pss_tree
   121 
   122 val gen_gen_real_arith :
   123   Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
   124    conv * conv * conv * conv * conv * conv * prover -> cert_conv
   125 val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   126   thm list * thm list * thm list -> thm * pss_tree
   127 
   128 val gen_real_arith : Proof.context ->
   129   (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
   130 
   131 val gen_prover_real_arith : Proof.context -> prover -> cert_conv
   132 
   133 val is_ratconst : cterm -> bool
   134 val dest_ratconst : cterm -> Rat.rat
   135 val cterm_of_rat : Rat.rat -> cterm
   136 
   137 end
   138 
   139 structure RealArith : REAL_ARITH =
   140 struct
   141 
   142  open Conv
   143 (* ------------------------------------------------------------------------- *)
   144 (* Data structure for Positivstellensatz refutations.                        *)
   145 (* ------------------------------------------------------------------------- *)
   146 
   147 datatype positivstellensatz =
   148    Axiom_eq of int
   149  | Axiom_le of int
   150  | Axiom_lt of int
   151  | Rational_eq of Rat.rat
   152  | Rational_le of Rat.rat
   153  | Rational_lt of Rat.rat
   154  | Square of FuncUtil.poly
   155  | Eqmul of FuncUtil.poly * positivstellensatz
   156  | Sum of positivstellensatz * positivstellensatz
   157  | Product of positivstellensatz * positivstellensatz;
   158          (* Theorems used in the procedure *)
   159 
   160 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
   161 datatype tree_choice = Left | Right
   162 type prover = tree_choice list -> 
   163   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   164   thm list * thm list * thm list -> thm * pss_tree
   165 type cert_conv = cterm -> thm * pss_tree
   166 
   167 
   168     (* Some useful derived rules *)
   169 fun deduct_antisym_rule tha thb = 
   170     Thm.equal_intr (Thm.implies_intr (cprop_of thb) tha) 
   171      (Thm.implies_intr (cprop_of tha) thb);
   172 
   173 fun prove_hyp tha thb =
   174   if exists (curry op aconv (concl_of tha)) (Thm.hyps_of thb)  (* FIXME !? *)
   175   then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb;
   176 
   177 val pth = @{lemma "(((x::real) < y) == (y - x > 0))" and "((x <= y) == (y - x >= 0))" and
   178      "((x = y) == (x - y = 0))" and "((~(x < y)) == (x - y >= 0))" and
   179      "((~(x <= y)) == (x - y > 0))" and "((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
   180   by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};
   181 
   182 val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
   183 val pth_add = 
   184   @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 )" and "( x = 0 ==> y >= 0 ==> x + y >= 0)" and
   185     "(x = 0 ==> y > 0 ==> x + y > 0)" and "(x >= 0 ==> y = 0 ==> x + y >= 0)" and
   186     "(x >= 0 ==> y >= 0 ==> x + y >= 0)" and "(x >= 0 ==> y > 0 ==> x + y > 0)" and
   187     "(x > 0 ==> y = 0 ==> x + y > 0)" and "(x > 0 ==> y >= 0 ==> x + y > 0)" and
   188     "(x > 0 ==> y > 0 ==> x + y > 0)" by simp_all};
   189 
   190 val pth_mul = 
   191   @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0)" and "(x = 0 ==> y >= 0 ==> x * y = 0)" and
   192     "(x = 0 ==> y > 0 ==> x * y = 0)" and "(x >= 0 ==> y = 0 ==> x * y = 0)" and
   193     "(x >= 0 ==> y >= 0 ==> x * y >= 0)" and "(x >= 0 ==> y > 0 ==> x * y >= 0)" and
   194     "(x > 0 ==>  y = 0 ==> x * y = 0)" and "(x > 0 ==> y >= 0 ==> x * y >= 0)" and
   195     "(x > 0 ==>  y > 0 ==> x * y > 0)"
   196   by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
   197     mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])};
   198 
   199 val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
   200 val pth_square = @{lemma "x * x >= (0::real)"  by simp};
   201 
   202 val weak_dnf_simps =
   203   List.take (@{thms simp_thms}, 34) @
   204     @{lemma "((P & (Q | R)) = ((P&Q) | (P&R)))" and "((Q | R) & P) = ((Q&P) | (R&P))" and
   205       "(P & Q) = (Q & P)" and "((P | Q) = (Q | P))" by blast+};
   206 
   207 (*
   208 val nnfD_simps =
   209   @{lemma "((~(P & Q)) = (~P | ~Q))" and "((~(P | Q)) = (~P & ~Q) )" and
   210     "((P --> Q) = (~P | Q) )" and "((P = Q) = ((P & Q) | (~P & ~ Q)))" and
   211     "((~(P = Q)) = ((P & ~ Q) | (~P & Q)) )" and "((~ ~(P)) = P)" by blast+};
   212 *)
   213 
   214 val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
   215 val prenex_simps =
   216   map (fn th => th RS sym)
   217     ([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @
   218       @{thms "HOL.all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});
   219 
   220 val real_abs_thms1 = @{lemma
   221   "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r))" and
   222   "((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r))" and
   223   "((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r))" and
   224   "((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r))" and
   225   "((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r))" and
   226   "((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r))" and
   227   "((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r))" and
   228   "((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r))" and
   229   "((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r))" and
   230   "((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y  + b >= r))" and
   231   "((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r))" and
   232   "((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y  + c >= r))" and
   233   "((1 * min x y >= r) = (1 * x >= r & 1 * y >= r))" and
   234   "((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and
   235   "((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and
   236   "((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y  + b >= r))" and
   237   "((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r))" and
   238   "((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y  + c >= r))" and
   239   "((min x y >= r) = (x >= r &  y >= r))" and
   240   "((min x y + a >= r) = (a + x >= r & a + y >= r))" and
   241   "((a + min x y >= r) = (a + x >= r & a + y >= r))" and
   242   "((a + min x y + b >= r) = (a + x + b >= r & a + y  + b >= r))" and
   243   "((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r))" and
   244   "((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r))" and
   245   "((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r))" and
   246   "((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r))" and
   247   "((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r))" and
   248   "((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r))" and
   249   "((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r))" and
   250   "((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r))" and
   251   "((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r))" and
   252   "((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r))" and
   253   "((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r))" and
   254   "((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y  + b > r))" and
   255   "((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r))" and
   256   "((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y  + c > r))" and
   257   "((min x y > r) = (x > r &  y > r))" and
   258   "((min x y + a > r) = (a + x > r & a + y > r))" and
   259   "((a + min x y > r) = (a + x > r & a + y > r))" and
   260   "((a + min x y + b > r) = (a + x + b > r & a + y  + b > r))" and
   261   "((a + b + min x y > r) = (a + b + x > r & a + b + y > r))" and
   262   "((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
   263   by auto};
   264 
   265 val abs_split' = @{lemma "P (abs (x::'a::linordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
   266   by (atomize (full)) (auto split add: abs_split)};
   267 
   268 val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
   269   by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
   270 
   271 val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
   272   by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
   273 
   274 
   275          (* Miscellaneous *)
   276 fun literals_conv bops uops cv = 
   277  let fun h t =
   278   case (term_of t) of 
   279    b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
   280  | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
   281  | _ => cv t
   282  in h end;
   283 
   284 fun cterm_of_rat x = 
   285 let val (a, b) = Rat.quotient_of_rat x
   286 in 
   287  if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
   288   else Thm.apply (Thm.apply @{cterm "op / :: real => _"} 
   289                    (Numeral.mk_cnumber @{ctyp "real"} a))
   290         (Numeral.mk_cnumber @{ctyp "real"} b)
   291 end;
   292 
   293   fun dest_ratconst t = case term_of t of
   294    Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
   295  | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
   296  fun is_ratconst t = can dest_ratconst t
   297 
   298 (*
   299 fun find_term p t = if p t then t else 
   300  case t of
   301   a$b => (find_term p a handle TERM _ => find_term p b)
   302  | Abs (_,_,t') => find_term p t'
   303  | _ => raise TERM ("find_term",[t]);
   304 *)
   305 
   306 fun find_cterm p t = if p t then t else 
   307  case term_of t of
   308   _$_ => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
   309  | Abs (_,_,_) => find_cterm p (Thm.dest_abs NONE t |> snd)
   310  | _ => raise CTERM ("find_cterm",[t]);
   311 
   312     (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
   313 fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
   314 fun is_comb t = case (term_of t) of _$_ => true | _ => false;
   315 
   316 fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
   317   handle CTERM _ => false;
   318 
   319 
   320 (* Map back polynomials to HOL.                         *)
   321 
   322 fun cterm_of_varpow x k = if k = 1 then x else Thm.apply (Thm.apply @{cterm "op ^ :: real => _"} x) 
   323   (Numeral.mk_cnumber @{ctyp nat} k)
   324 
   325 fun cterm_of_monomial m = 
   326  if FuncUtil.Ctermfunc.is_empty m then @{cterm "1::real"} 
   327  else 
   328   let 
   329    val m' = FuncUtil.dest_monomial m
   330    val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' [] 
   331   in foldr1 (fn (s, t) => Thm.apply (Thm.apply @{cterm "op * :: real => _"} s) t) vps
   332   end
   333 
   334 fun cterm_of_cmonomial (m,c) = if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
   335     else if c = Rat.one then cterm_of_monomial m
   336     else Thm.apply (Thm.apply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
   337 
   338 fun cterm_of_poly p = 
   339  if FuncUtil.Monomialfunc.is_empty p then @{cterm "0::real"} 
   340  else
   341   let 
   342    val cms = map cterm_of_cmonomial
   343      (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
   344   in foldr1 (fn (t1, t2) => Thm.apply(Thm.apply @{cterm "op + :: real => _"} t1) t2) cms
   345   end;
   346 
   347     (* A general real arithmetic prover *)
   348 
   349 fun gen_gen_real_arith ctxt (mk_numeric,
   350        numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
   351        poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
   352        absconv1,absconv2,prover) = 
   353 let
   354  val pre_ss = HOL_basic_ss addsimps
   355   @{thms simp_thms ex_simps all_simps not_all not_ex ex_disj_distrib all_conj_distrib if_bool_eq_disj}
   356  val prenex_ss = HOL_basic_ss addsimps prenex_simps
   357  val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
   358  val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
   359  val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
   360  val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
   361  val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
   362  val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
   363  fun eqT_elim th = Thm.equal_elim (Thm.symmetric th) @{thm TrueI}
   364  fun oprconv cv ct = 
   365   let val g = Thm.dest_fun2 ct
   366   in if g aconvc @{cterm "op <= :: real => _"} 
   367        orelse g aconvc @{cterm "op < :: real => _"} 
   368      then arg_conv cv ct else arg1_conv cv ct
   369   end
   370 
   371  fun real_ineq_conv th ct =
   372   let
   373    val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th 
   374       handle Pattern.MATCH => raise CTERM ("real_ineq_conv", [ct]))
   375   in Thm.transitive th' (oprconv poly_conv (Thm.rhs_of th'))
   376   end 
   377   val [real_lt_conv, real_le_conv, real_eq_conv,
   378        real_not_lt_conv, real_not_le_conv, _] =
   379        map real_ineq_conv pth
   380   fun match_mp_rule ths ths' = 
   381    let
   382      fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
   383       | th::ths => (ths' MRS th handle THM _ => f ths ths')
   384    in f ths ths' end
   385   fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
   386          (match_mp_rule pth_mul [th, th'])
   387   fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
   388          (match_mp_rule pth_add [th, th'])
   389   fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv)) 
   390        (instantiate' [] [SOME ct] (th RS pth_emul)) 
   391   fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
   392        (instantiate' [] [SOME t] pth_square)
   393 
   394   fun hol_of_positivstellensatz(eqs,les,lts) proof =
   395    let 
   396     fun translate prf = case prf of
   397         Axiom_eq n => nth eqs n
   398       | Axiom_le n => nth les n
   399       | Axiom_lt n => nth lts n
   400       | Rational_eq x => eqT_elim(numeric_eq_conv(Thm.apply @{cterm Trueprop} 
   401                           (Thm.apply (Thm.apply @{cterm "op =::real => _"} (mk_numeric x)) 
   402                                @{cterm "0::real"})))
   403       | Rational_le x => eqT_elim(numeric_ge_conv(Thm.apply @{cterm Trueprop} 
   404                           (Thm.apply (Thm.apply @{cterm "op <=::real => _"} 
   405                                      @{cterm "0::real"}) (mk_numeric x))))
   406       | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.apply @{cterm Trueprop} 
   407                       (Thm.apply (Thm.apply @{cterm "op <::real => _"} @{cterm "0::real"})
   408                         (mk_numeric x))))
   409       | Square pt => square_rule (cterm_of_poly pt)
   410       | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
   411       | Sum(p1,p2) => add_rule (translate p1) (translate p2)
   412       | Product(p1,p2) => mul_rule (translate p1) (translate p2)
   413    in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv]) 
   414           (translate proof)
   415    end
   416   
   417   val init_conv = presimp_conv then_conv
   418       nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
   419       weak_dnf_conv
   420 
   421   val concl = Thm.dest_arg o cprop_of
   422   fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
   423   val is_req = is_binop @{cterm "op =:: real => _"}
   424   val is_ge = is_binop @{cterm "op <=:: real => _"}
   425   val is_gt = is_binop @{cterm "op <:: real => _"}
   426   val is_conj = is_binop @{cterm HOL.conj}
   427   val is_disj = is_binop @{cterm HOL.disj}
   428   fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
   429   fun disj_cases th th1 th2 = 
   430    let val (p,q) = Thm.dest_binop (concl th)
   431        val c = concl th1
   432        val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
   433    in Thm.implies_elim (Thm.implies_elim
   434           (Thm.implies_elim (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
   435           (Thm.implies_intr (Thm.apply @{cterm Trueprop} p) th1))
   436         (Thm.implies_intr (Thm.apply @{cterm Trueprop} q) th2)
   437    end
   438  fun overall cert_choice dun ths = case ths of
   439   [] =>
   440    let 
   441     val (eq,ne) = List.partition (is_req o concl) dun
   442      val (le,nl) = List.partition (is_ge o concl) ne
   443      val lt = filter (is_gt o concl) nl 
   444     in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
   445  | th::oths =>
   446    let 
   447     val ct = concl th 
   448    in 
   449     if is_conj ct  then
   450      let 
   451       val (th1,th2) = conj_pair th in
   452       overall cert_choice dun (th1::th2::oths) end
   453     else if is_disj ct then
   454       let 
   455        val (th1, cert1) = overall (Left::cert_choice) dun (Thm.assume (Thm.apply @{cterm Trueprop} (Thm.dest_arg1 ct))::oths)
   456        val (th2, cert2) = overall (Right::cert_choice) dun (Thm.assume (Thm.apply @{cterm Trueprop} (Thm.dest_arg ct))::oths)
   457       in (disj_cases th th1 th2, Branch (cert1, cert2)) end
   458    else overall cert_choice (th::dun) oths
   459   end
   460   fun dest_binary b ct = if is_binop b ct then Thm.dest_binop ct 
   461                          else raise CTERM ("dest_binary",[b,ct])
   462   val dest_eq = dest_binary @{cterm "op = :: real => _"}
   463   val neq_th = nth pth 5
   464   fun real_not_eq_conv ct = 
   465    let 
   466     val (l,r) = dest_eq (Thm.dest_arg ct)
   467     val th = Thm.instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
   468     val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
   469     val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
   470     val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
   471     val th' = Drule.binop_cong_rule @{cterm HOL.disj} 
   472      (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
   473      (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
   474     in Thm.transitive th th' 
   475   end
   476  fun equal_implies_1_rule PQ = 
   477   let 
   478    val P = Thm.lhs_of PQ
   479   in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P))
   480   end
   481  (* FIXME!!! Copied from groebner.ml *)
   482  val strip_exists =
   483   let fun h (acc, t) =
   484    case (term_of t) of
   485     Const(@{const_name Ex},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
   486   | _ => (acc,t)
   487   in fn t => h ([],t)
   488   end
   489   fun name_of x = case term_of x of
   490    Free(s,_) => s
   491  | Var ((s,_),_) => s
   492  | _ => "x"
   493 
   494   fun mk_forall x th = Drule.arg_cong_rule (instantiate_cterm' [SOME (ctyp_of_term x)] [] @{cpat "All :: (?'a => bool) => _" }) (Thm.abstract_rule (name_of x) x th)
   495 
   496   val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
   497 
   498  fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
   499  fun mk_ex v t = Thm.apply (ext (ctyp_of_term v)) (Thm.lambda v t)
   500 
   501  fun choose v th th' = case concl_of th of 
   502    @{term Trueprop} $ (Const(@{const_name Ex},_)$_) => 
   503     let
   504      val p = (funpow 2 Thm.dest_arg o cprop_of) th
   505      val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
   506      val th0 = fconv_rule (Thm.beta_conversion true)
   507          (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
   508      val pv = (Thm.rhs_of o Thm.beta_conversion true) 
   509            (Thm.apply @{cterm Trueprop} (Thm.apply p v))
   510      val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
   511     in Thm.implies_elim (Thm.implies_elim th0 th) th1  end
   512  | _ => raise THM ("choose",0,[th, th'])
   513 
   514   fun simple_choose v th = 
   515      choose v (Thm.assume ((Thm.apply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
   516 
   517  val strip_forall =
   518   let fun h (acc, t) =
   519    case (term_of t) of
   520     Const(@{const_name All},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
   521   | _ => (acc,t)
   522   in fn t => h ([],t)
   523   end
   524 
   525  fun f ct =
   526   let 
   527    val nnf_norm_conv' = 
   528      nnf_conv then_conv 
   529      literals_conv [@{term HOL.conj}, @{term HOL.disj}] [] 
   530      (Conv.cache_conv 
   531        (first_conv [real_lt_conv, real_le_conv, 
   532                     real_eq_conv, real_not_lt_conv, 
   533                     real_not_le_conv, real_not_eq_conv, all_conv]))
   534   fun absremover ct = (literals_conv [@{term HOL.conj}, @{term HOL.disj}] [] 
   535                   (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv 
   536         try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
   537   val nct = Thm.apply @{cterm Trueprop} (Thm.apply @{cterm "Not"} ct)
   538   val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
   539   val tm0 = Thm.dest_arg (Thm.rhs_of th0)
   540   val (th, certificates) = if tm0 aconvc @{cterm False} then (equal_implies_1_rule th0, Trivial) else
   541    let 
   542     val (evs,bod) = strip_exists tm0
   543     val (avs,ibod) = strip_forall bod
   544     val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
   545     val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))]
   546     val th3 = fold simple_choose evs (prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.apply @{cterm Trueprop} bod))) th2)
   547    in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
   548    end
   549   in (Thm.implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates)
   550  end
   551 in f
   552 end;
   553 
   554 (* A linear arithmetic prover *)
   555 local
   556   val linear_add = FuncUtil.Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
   557   fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c */ x)
   558   val one_tm = @{cterm "1::real"}
   559   fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p Rat.zero)) orelse
   560      ((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso
   561        not(p(FuncUtil.Ctermfunc.apply e one_tm)))
   562 
   563   fun linear_ineqs vars (les,lts) = 
   564    case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
   565     SOME r => r
   566   | NONE => 
   567    (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
   568      SOME r => r
   569    | NONE => 
   570      if null vars then error "linear_ineqs: no contradiction" else
   571      let 
   572       val ineqs = les @ lts
   573       fun blowup v =
   574        length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
   575        length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
   576        length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
   577       val  v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
   578                  (map (fn v => (v,blowup v)) vars)))
   579       fun addup (e1,p1) (e2,p2) acc =
   580        let 
   581         val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v Rat.zero 
   582         val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v Rat.zero
   583        in if c1 */ c2 >=/ Rat.zero then acc else
   584         let 
   585          val e1' = linear_cmul (Rat.abs c2) e1
   586          val e2' = linear_cmul (Rat.abs c1) e2
   587          val p1' = Product(Rational_lt(Rat.abs c2),p1)
   588          val p2' = Product(Rational_lt(Rat.abs c1),p2)
   589         in (linear_add e1' e2',Sum(p1',p2'))::acc
   590         end
   591        end
   592       val (les0,les1) = 
   593          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
   594       val (lts0,lts1) = 
   595          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
   596       val (lesp,lesn) = 
   597          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
   598       val (ltsp,ltsn) = 
   599          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
   600       val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
   601       val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
   602                       (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
   603      in linear_ineqs (remove (op aconvc) v vars) (les',lts')
   604      end)
   605 
   606   fun linear_eqs(eqs,les,lts) = 
   607    case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
   608     SOME r => r
   609   | NONE => (case eqs of 
   610     [] => 
   611      let val vars = remove (op aconvc) one_tm 
   612            (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) []) 
   613      in linear_ineqs vars (les,lts) end
   614    | (e,p)::es => 
   615      if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
   616      let 
   617       val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
   618       fun xform (inp as (t,q)) =
   619        let val d = FuncUtil.Ctermfunc.tryapplyd t x Rat.zero in
   620         if d =/ Rat.zero then inp else
   621         let 
   622          val k = (Rat.neg d) */ Rat.abs c // c
   623          val e' = linear_cmul k e
   624          val t' = linear_cmul (Rat.abs c) t
   625          val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
   626          val q' = Product(Rational_lt(Rat.abs c),q) 
   627         in (linear_add e' t',Sum(p',q')) 
   628         end 
   629       end
   630      in linear_eqs(map xform es,map xform les,map xform lts)
   631      end)
   632 
   633   fun linear_prover (eq,le,lt) = 
   634    let 
   635     val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq
   636     val les = map_index (fn (n, p) => (p,Axiom_le n)) le
   637     val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt
   638    in linear_eqs(eqs,les,lts)
   639    end 
   640   
   641   fun lin_of_hol ct = 
   642    if ct aconvc @{cterm "0::real"} then FuncUtil.Ctermfunc.empty
   643    else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
   644    else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
   645    else
   646     let val (lop,r) = Thm.dest_comb ct 
   647     in if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
   648        else
   649         let val (opr,l) = Thm.dest_comb lop 
   650         in if opr aconvc @{cterm "op + :: real =>_"} 
   651            then linear_add (lin_of_hol l) (lin_of_hol r)
   652            else if opr aconvc @{cterm "op * :: real =>_"} 
   653                    andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
   654            else FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
   655         end
   656     end
   657 
   658   fun is_alien ct = case term_of ct of 
   659    Const(@{const_name "real"}, _)$ n => 
   660      if can HOLogic.dest_number n then false else true
   661   | _ => false
   662 in 
   663 fun real_linear_prover translator (eq,le,lt) = 
   664  let 
   665   val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o cprop_of
   666   val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o cprop_of
   667   val eq_pols = map lhs eq
   668   val le_pols = map rhs le
   669   val lt_pols = map rhs lt 
   670   val aliens =  filter is_alien
   671       (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom) 
   672           (eq_pols @ le_pols @ lt_pols) [])
   673   val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,Rat.one)) aliens
   674   val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
   675   val le' = le @ map (fn a => instantiate' [] [SOME (Thm.dest_arg a)] @{thm real_of_nat_ge_zero}) aliens 
   676  in ((translator (eq,le',lt) proof), Trivial)
   677  end
   678 end;
   679 
   680 (* A less general generic arithmetic prover dealing with abs,max and min*)
   681 
   682 local
   683  val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
   684  fun absmaxmin_elim_conv1 ctxt = 
   685     Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
   686 
   687  val absmaxmin_elim_conv2 =
   688   let 
   689    val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
   690    val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
   691    val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
   692    val abs_tm = @{cterm "abs :: real => _"}
   693    val p_tm = @{cpat "?P :: real => bool"}
   694    val x_tm = @{cpat "?x :: real"}
   695    val y_tm = @{cpat "?y::real"}
   696    val is_max = is_binop @{cterm "max :: real => _"}
   697    val is_min = is_binop @{cterm "min :: real => _"} 
   698    fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm
   699    fun eliminate_construct p c tm =
   700     let 
   701      val t = find_cterm p tm
   702      val th0 = (Thm.symmetric o Thm.beta_conversion false) (Thm.apply (Thm.lambda t tm) t)
   703      val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0
   704     in fconv_rule(arg_conv(binop_conv (arg_conv (Thm.beta_conversion false))))
   705                (Thm.transitive th0 (c p ax))
   706    end
   707 
   708    val elim_abs = eliminate_construct is_abs
   709     (fn p => fn ax => 
   710        Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax)]) pth_abs)
   711    val elim_max = eliminate_construct is_max
   712     (fn p => fn ax => 
   713       let val (ax,y) = Thm.dest_comb ax 
   714       in  Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)]) 
   715       pth_max end)
   716    val elim_min = eliminate_construct is_min
   717     (fn p => fn ax => 
   718       let val (ax,y) = Thm.dest_comb ax 
   719       in  Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)]) 
   720       pth_min end)
   721    in first_conv [elim_abs, elim_max, elim_min, all_conv]
   722   end;
   723 in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
   724         gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,
   725                        absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
   726 end;
   727 
   728 (* An instance for reals*) 
   729 
   730 fun gen_prover_real_arith ctxt prover = 
   731  let
   732   fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS
   733   val {add, mul, neg, pow = _, sub = _, main} = 
   734      Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
   735       (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"})) 
   736      simple_cterm_ord
   737 in gen_real_arith ctxt
   738    (cterm_of_rat, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv,
   739     main,neg,add,mul, prover)
   740 end;
   741 
   742 end