src/HOL/Library/positivstellensatz.ML
author boehmes
Wed Aug 26 11:40:28 2009 +0200 (2009-08-26)
changeset 32402 5731300da417
parent 31971 8c1b845ed105
child 32645 1cc5b24f5a01
permissions -rw-r--r--
added further conversions and conversionals
     1 (* Title:      Library/positivstellensatz
     2    Author:     Amine Chaieb, University of Cambridge
     3    Description: A generic arithmetic prover based on Positivstellensatz certificates --- 
     4     also implements Fourrier-Motzkin elimination as a special case Fourrier-Motzkin elimination.
     5 *)
     6 
     7 (* A functor for finite mappings based on Tables *)
     8 signature FUNC = 
     9 sig
    10  type 'a T
    11  type key
    12  val apply : 'a T -> key -> 'a
    13  val applyd :'a T -> (key -> 'a) -> key -> 'a
    14  val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a T -> 'a T -> 'a T
    15  val defined : 'a T -> key -> bool
    16  val dom : 'a T -> key list
    17  val fold : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b
    18  val fold_rev : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b
    19  val graph : 'a T -> (key * 'a) list
    20  val is_undefined : 'a T -> bool
    21  val mapf : ('a -> 'b) -> 'a T -> 'b T
    22  val tryapplyd : 'a T -> key -> 'a -> 'a
    23  val undefine :  key -> 'a T -> 'a T
    24  val undefined : 'a T
    25  val update : key * 'a -> 'a T -> 'a T
    26  val updatep : (key * 'a -> bool) -> key * 'a -> 'a T -> 'a T
    27  val choose : 'a T -> key * 'a
    28  val onefunc : key * 'a -> 'a T
    29  val get_first: (key*'a -> 'a option) -> 'a T -> 'a option
    30 end;
    31 
    32 functor FuncFun(Key: KEY) : FUNC=
    33 struct
    34 
    35 type key = Key.key;
    36 structure Tab = Table(Key);
    37 type 'a T = 'a Tab.table;
    38 
    39 val undefined = Tab.empty;
    40 val is_undefined = Tab.is_empty;
    41 val mapf = Tab.map;
    42 val fold = Tab.fold;
    43 val fold_rev = Tab.fold_rev;
    44 val graph = Tab.dest;
    45 fun dom a = sort Key.ord (Tab.keys a);
    46 fun applyd f d x = case Tab.lookup f x of 
    47    SOME y => y
    48  | NONE => d x;
    49 
    50 fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
    51 fun tryapplyd f a d = applyd f (K d) a;
    52 val defined = Tab.defined;
    53 fun undefine x t = (Tab.delete x t handle UNDEF => t);
    54 val update = Tab.update;
    55 fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
    56 fun combine f z a b = 
    57  let
    58   fun h (k,v) t = case Tab.lookup t k of
    59      NONE => Tab.update (k,v) t
    60    | SOME v' => let val w = f v v'
    61      in if z w then Tab.delete k t else Tab.update (k,w) t end;
    62   in Tab.fold h a b end;
    63 
    64 fun choose f = case Tab.min_key f of 
    65    SOME k => (k,valOf (Tab.lookup f k))
    66  | NONE => error "FuncFun.choose : Completely undefined function"
    67 
    68 fun onefunc kv = update kv undefined
    69 
    70 local
    71 fun  find f (k,v) NONE = f (k,v)
    72    | find f (k,v) r = r
    73 in
    74 fun get_first f t = fold (find f) t NONE
    75 end
    76 end;
    77 
    78 structure Intfunc = FuncFun(type key = int val ord = int_ord);
    79 structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
    80 structure Termfunc = FuncFun(type key = term val ord = TermOrd.fast_term_ord);
    81 structure Ctermfunc = FuncFun(type key = cterm val ord = (fn (s,t) => TermOrd.fast_term_ord(term_of s, term_of t)));
    82 
    83 structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
    84 
    85 
    86     (* Some useful derived rules *)
    87 fun deduct_antisym_rule tha thb = 
    88     equal_intr (implies_intr (cprop_of thb) tha) 
    89      (implies_intr (cprop_of tha) thb);
    90 
    91 fun prove_hyp tha thb = 
    92   if exists (curry op aconv (concl_of tha)) (#hyps (rep_thm thb)) 
    93   then equal_elim (symmetric (deduct_antisym_rule tha thb)) tha else thb;
    94 
    95 
    96 
    97 signature REAL_ARITH = 
    98 sig
    99   datatype positivstellensatz =
   100    Axiom_eq of int
   101  | Axiom_le of int
   102  | Axiom_lt of int
   103  | Rational_eq of Rat.rat
   104  | Rational_le of Rat.rat
   105  | Rational_lt of Rat.rat
   106  | Square of cterm
   107  | Eqmul of cterm * positivstellensatz
   108  | Sum of positivstellensatz * positivstellensatz
   109  | Product of positivstellensatz * positivstellensatz;
   110 
   111 val gen_gen_real_arith :
   112   Proof.context -> (Rat.rat -> Thm.cterm) * conv * conv * conv * 
   113    conv * conv * conv * conv * conv * conv * 
   114     ( (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   115         thm list * thm list * thm list -> thm) -> conv
   116 val real_linear_prover : 
   117   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   118    thm list * thm list * thm list -> thm
   119 
   120 val gen_real_arith : Proof.context ->
   121    (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv *
   122    ( (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   123        thm list * thm list * thm list -> thm) -> conv
   124 val gen_prover_real_arith : Proof.context ->
   125    ((thm list * thm list * thm list -> positivstellensatz -> thm) ->
   126      thm list * thm list * thm list -> thm) -> conv
   127 val real_arith : Proof.context -> conv
   128 end
   129 
   130 structure RealArith (* : REAL_ARITH *)=
   131 struct
   132 
   133  open Conv Thm;;
   134 (* ------------------------------------------------------------------------- *)
   135 (* Data structure for Positivstellensatz refutations.                        *)
   136 (* ------------------------------------------------------------------------- *)
   137 
   138 datatype positivstellensatz =
   139    Axiom_eq of int
   140  | Axiom_le of int
   141  | Axiom_lt of int
   142  | Rational_eq of Rat.rat
   143  | Rational_le of Rat.rat
   144  | Rational_lt of Rat.rat
   145  | Square of cterm
   146  | Eqmul of cterm * positivstellensatz
   147  | Sum of positivstellensatz * positivstellensatz
   148  | Product of positivstellensatz * positivstellensatz;
   149          (* Theorems used in the procedure *)
   150 
   151 
   152 val my_eqs = ref ([] : thm list);
   153 val my_les = ref ([] : thm list);
   154 val my_lts = ref ([] : thm list);
   155 val my_proof = ref (Axiom_eq 0);
   156 val my_context = ref @{context};
   157 
   158 val my_mk_numeric = ref ((K @{cterm True}) :Rat.rat -> cterm);
   159 val my_numeric_eq_conv = ref no_conv;
   160 val my_numeric_ge_conv = ref no_conv;
   161 val my_numeric_gt_conv = ref no_conv;
   162 val my_poly_conv = ref no_conv;
   163 val my_poly_neg_conv = ref no_conv;
   164 val my_poly_add_conv = ref no_conv;
   165 val my_poly_mul_conv = ref no_conv;
   166 
   167 fun conjunctions th = case try Conjunction.elim th of
   168    SOME (th1,th2) => (conjunctions th1) @ conjunctions th2
   169  | NONE => [th];
   170 
   171 val pth = @{lemma "(((x::real) < y) == (y - x > 0)) &&& ((x <= y) == (y - x >= 0)) 
   172      &&& ((x = y) == (x - y = 0)) &&& ((~(x < y)) == (x - y >= 0)) &&& ((~(x <= y)) == (x - y > 0))
   173      &&& ((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
   174   by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)} |> 
   175 conjunctions;
   176 
   177 val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
   178 val pth_add = 
   179  @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 ) &&& ( x = 0 ==> y >= 0 ==> x + y >= 0) 
   180     &&& (x = 0 ==> y > 0 ==> x + y > 0) &&& (x >= 0 ==> y = 0 ==> x + y >= 0) 
   181     &&& (x >= 0 ==> y >= 0 ==> x + y >= 0) &&& (x >= 0 ==> y > 0 ==> x + y > 0) 
   182     &&& (x > 0 ==> y = 0 ==> x + y > 0) &&& (x > 0 ==> y >= 0 ==> x + y > 0) 
   183     &&& (x > 0 ==> y > 0 ==> x + y > 0)"  by simp_all} |> conjunctions ;
   184 
   185 val pth_mul = 
   186   @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0) &&& (x = 0 ==> y >= 0 ==> x * y = 0) &&& 
   187            (x = 0 ==> y > 0 ==> x * y = 0) &&& (x >= 0 ==> y = 0 ==> x * y = 0) &&& 
   188            (x >= 0 ==> y >= 0 ==> x * y >= 0 ) &&& ( x >= 0 ==> y > 0 ==> x * y >= 0 ) &&&
   189            (x > 0 ==>  y = 0 ==> x * y = 0 ) &&& ( x > 0 ==> y >= 0 ==> x * y >= 0 ) &&&
   190            (x > 0 ==>  y > 0 ==> x * y > 0)"
   191   by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
   192     mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])} |> conjunctions;
   193 
   194 val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
   195 val pth_square = @{lemma "x * x >= (0::real)"  by simp};
   196 
   197 val weak_dnf_simps = List.take (simp_thms, 34) 
   198     @ 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+};
   199 
   200 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+}
   201 
   202 val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
   203 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)});
   204 
   205 val real_abs_thms1 = conjunctions @{lemma
   206   "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r)) &&&
   207   ((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
   208   ((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
   209   ((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r)) &&&
   210   ((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r)) &&&
   211   ((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r)) &&&
   212   ((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r)) &&&
   213   ((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
   214   ((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
   215   ((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y  + b >= r)) &&&
   216   ((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r)) &&&
   217   ((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y  + c >= r)) &&&
   218   ((1 * min x y >= r) = (1 * x >= r & 1 * y >= r)) &&&
   219   ((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
   220   ((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
   221   ((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y  + b >= r) )&&&
   222   ((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r)) &&&
   223   ((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y  + c >= r)) &&&
   224   ((min x y >= r) = (x >= r &  y >= r)) &&&
   225   ((min x y + a >= r) = (a + x >= r & a + y >= r)) &&&
   226   ((a + min x y >= r) = (a + x >= r & a + y >= r)) &&&
   227   ((a + min x y + b >= r) = (a + x + b >= r & a + y  + b >= r)) &&&
   228   ((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r) )&&&
   229   ((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r)) &&&
   230   ((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r)) &&&
   231   ((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
   232   ((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
   233   ((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r)) &&&
   234   ((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r)) &&&
   235   ((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r)) &&&
   236   ((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r)) &&&
   237   ((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
   238   ((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
   239   ((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y  + b > r)) &&&
   240   ((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r)) &&&
   241   ((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y  + c > r)) &&&
   242   ((min x y > r) = (x > r &  y > r)) &&&
   243   ((min x y + a > r) = (a + x > r & a + y > r)) &&&
   244   ((a + min x y > r) = (a + x > r & a + y > r)) &&&
   245   ((a + min x y + b > r) = (a + x + b > r & a + y  + b > r)) &&&
   246   ((a + b + min x y > r) = (a + b + x > r & a + b + y > r)) &&&
   247   ((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
   248   by auto};
   249 
   250 val abs_split' = @{lemma "P (abs (x::'a::ordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
   251   by (atomize (full)) (auto split add: abs_split)};
   252 
   253 val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
   254   by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
   255 
   256 val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
   257   by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
   258 
   259 
   260          (* Miscalineous *)
   261 fun literals_conv bops uops cv = 
   262  let fun h t =
   263   case (term_of t) of 
   264    b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
   265  | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
   266  | _ => cv t
   267  in h end;
   268 
   269 fun cterm_of_rat x = 
   270 let val (a, b) = Rat.quotient_of_rat x
   271 in 
   272  if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
   273   else Thm.capply (Thm.capply @{cterm "op / :: real => _"} 
   274                    (Numeral.mk_cnumber @{ctyp "real"} a))
   275         (Numeral.mk_cnumber @{ctyp "real"} b)
   276 end;
   277 
   278   fun dest_ratconst t = case term_of t of
   279    Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
   280  | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
   281  fun is_ratconst t = can dest_ratconst t
   282 
   283 fun find_term p t = if p t then t else 
   284  case t of
   285   a$b => (find_term p a handle TERM _ => find_term p b)
   286  | Abs (_,_,t') => find_term p t'
   287  | _ => raise TERM ("find_term",[t]);
   288 
   289 fun find_cterm p t = if p t then t else 
   290  case term_of t of
   291   a$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
   292  | Abs (_,_,t') => find_cterm p (Thm.dest_abs NONE t |> snd)
   293  | _ => raise CTERM ("find_cterm",[t]);
   294 
   295 
   296     (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
   297 fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
   298 fun is_comb t = case (term_of t) of _$_ => true | _ => false;
   299 
   300 fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
   301   handle CTERM _ => false;
   302 
   303     (* A general real arithmetic prover *)
   304 
   305 fun gen_gen_real_arith ctxt (mk_numeric,
   306        numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
   307        poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
   308        absconv1,absconv2,prover) = 
   309 let
   310  open Conv Thm;
   311  val _ = my_context := ctxt 
   312  val _ = (my_mk_numeric := mk_numeric ; my_numeric_eq_conv := numeric_eq_conv ; 
   313           my_numeric_ge_conv := numeric_ge_conv; my_numeric_gt_conv := numeric_gt_conv ;
   314           my_poly_conv := poly_conv; my_poly_neg_conv := poly_neg_conv; 
   315           my_poly_add_conv := poly_add_conv; my_poly_mul_conv := poly_mul_conv)
   316  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}]
   317  val prenex_ss = HOL_basic_ss addsimps prenex_simps
   318  val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
   319  val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
   320  val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
   321  val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
   322  val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
   323  val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
   324  fun eqT_elim th = equal_elim (symmetric th) @{thm TrueI}
   325  fun oprconv cv ct = 
   326   let val g = Thm.dest_fun2 ct
   327   in if g aconvc @{cterm "op <= :: real => _"} 
   328        orelse g aconvc @{cterm "op < :: real => _"} 
   329      then arg_conv cv ct else arg1_conv cv ct
   330   end
   331 
   332  fun real_ineq_conv th ct =
   333   let
   334    val th' = (instantiate (match (lhs_of th, ct)) th 
   335       handle MATCH => raise CTERM ("real_ineq_conv", [ct]))
   336   in transitive th' (oprconv poly_conv (Thm.rhs_of th'))
   337   end 
   338   val [real_lt_conv, real_le_conv, real_eq_conv,
   339        real_not_lt_conv, real_not_le_conv, _] =
   340        map real_ineq_conv pth
   341   fun match_mp_rule ths ths' = 
   342    let
   343      fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
   344       | th::ths => (ths' MRS th handle THM _ => f ths ths')
   345    in f ths ths' end
   346   fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
   347          (match_mp_rule pth_mul [th, th'])
   348   fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
   349          (match_mp_rule pth_add [th, th'])
   350   fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv)) 
   351        (instantiate' [] [SOME ct] (th RS pth_emul)) 
   352   fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
   353        (instantiate' [] [SOME t] pth_square)
   354 
   355   fun hol_of_positivstellensatz(eqs,les,lts) proof =
   356    let 
   357     val _ = (my_eqs := eqs ; my_les := les ; my_lts := lts ; my_proof := proof)
   358     fun translate prf = case prf of
   359         Axiom_eq n => nth eqs n
   360       | Axiom_le n => nth les n
   361       | Axiom_lt n => nth lts n
   362       | Rational_eq x => eqT_elim(numeric_eq_conv(capply @{cterm Trueprop} 
   363                           (capply (capply @{cterm "op =::real => _"} (mk_numeric x)) 
   364                                @{cterm "0::real"})))
   365       | Rational_le x => eqT_elim(numeric_ge_conv(capply @{cterm Trueprop} 
   366                           (capply (capply @{cterm "op <=::real => _"} 
   367                                      @{cterm "0::real"}) (mk_numeric x))))
   368       | Rational_lt x => eqT_elim(numeric_gt_conv(capply @{cterm Trueprop} 
   369                       (capply (capply @{cterm "op <::real => _"} @{cterm "0::real"})
   370                         (mk_numeric x))))
   371       | Square t => square_rule t
   372       | Eqmul(t,p) => emul_rule t (translate p)
   373       | Sum(p1,p2) => add_rule (translate p1) (translate p2)
   374       | Product(p1,p2) => mul_rule (translate p1) (translate p2)
   375    in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv]) 
   376           (translate proof)
   377    end
   378   
   379   val init_conv = presimp_conv then_conv
   380       nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
   381       weak_dnf_conv
   382 
   383   val concl = dest_arg o cprop_of
   384   fun is_binop opr ct = (dest_fun2 ct aconvc opr handle CTERM _ => false)
   385   val is_req = is_binop @{cterm "op =:: real => _"}
   386   val is_ge = is_binop @{cterm "op <=:: real => _"}
   387   val is_gt = is_binop @{cterm "op <:: real => _"}
   388   val is_conj = is_binop @{cterm "op &"}
   389   val is_disj = is_binop @{cterm "op |"}
   390   fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
   391   fun disj_cases th th1 th2 = 
   392    let val (p,q) = dest_binop (concl th)
   393        val c = concl th1
   394        val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
   395    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)
   396    end
   397  fun overall dun ths = case ths of
   398   [] =>
   399    let 
   400     val (eq,ne) = List.partition (is_req o concl) dun
   401      val (le,nl) = List.partition (is_ge o concl) ne
   402      val lt = filter (is_gt o concl) nl 
   403     in prover hol_of_positivstellensatz (eq,le,lt) end
   404  | th::oths =>
   405    let 
   406     val ct = concl th 
   407    in 
   408     if is_conj ct  then
   409      let 
   410       val (th1,th2) = conj_pair th in
   411       overall dun (th1::th2::oths) end
   412     else if is_disj ct then
   413       let 
   414        val th1 = overall dun (assume (capply @{cterm Trueprop} (dest_arg1 ct))::oths)
   415        val th2 = overall dun (assume (capply @{cterm Trueprop} (dest_arg ct))::oths)
   416       in disj_cases th th1 th2 end
   417    else overall (th::dun) oths
   418   end
   419   fun dest_binary b ct = if is_binop b ct then dest_binop ct 
   420                          else raise CTERM ("dest_binary",[b,ct])
   421   val dest_eq = dest_binary @{cterm "op = :: real => _"}
   422   val neq_th = nth pth 5
   423   fun real_not_eq_conv ct = 
   424    let 
   425     val (l,r) = dest_eq (dest_arg ct)
   426     val th = instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
   427     val th_p = poly_conv(dest_arg(dest_arg1(rhs_of th)))
   428     val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
   429     val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
   430     val th' = Drule.binop_cong_rule @{cterm "op |"} 
   431      (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
   432      (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
   433     in transitive th th' 
   434   end
   435  fun equal_implies_1_rule PQ = 
   436   let 
   437    val P = lhs_of PQ
   438   in implies_intr P (equal_elim PQ (assume P))
   439   end
   440  (* FIXME!!! Copied from groebner.ml *)
   441  val strip_exists =
   442   let fun h (acc, t) =
   443    case (term_of t) of
   444     Const("Ex",_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
   445   | _ => (acc,t)
   446   in fn t => h ([],t)
   447   end
   448   fun name_of x = case term_of x of
   449    Free(s,_) => s
   450  | Var ((s,_),_) => s
   451  | _ => "x"
   452 
   453   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)
   454 
   455   val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
   456 
   457  fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
   458  fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
   459 
   460  fun choose v th th' = case concl_of th of 
   461    @{term Trueprop} $ (Const("Ex",_)$_) => 
   462     let
   463      val p = (funpow 2 Thm.dest_arg o cprop_of) th
   464      val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
   465      val th0 = fconv_rule (Thm.beta_conversion true)
   466          (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
   467      val pv = (Thm.rhs_of o Thm.beta_conversion true) 
   468            (Thm.capply @{cterm Trueprop} (Thm.capply p v))
   469      val th1 = forall_intr v (implies_intr pv th')
   470     in implies_elim (implies_elim th0 th) th1  end
   471  | _ => raise THM ("choose",0,[th, th'])
   472 
   473   fun simple_choose v th = 
   474      choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
   475 
   476  val strip_forall =
   477   let fun h (acc, t) =
   478    case (term_of t) of
   479     Const("All",_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
   480   | _ => (acc,t)
   481   in fn t => h ([],t)
   482   end
   483 
   484  fun f ct =
   485   let 
   486    val nnf_norm_conv' = 
   487      nnf_conv then_conv 
   488      literals_conv [@{term "op &"}, @{term "op |"}] [] 
   489      (More_Conv.cache_conv 
   490        (first_conv [real_lt_conv, real_le_conv, 
   491                     real_eq_conv, real_not_lt_conv, 
   492                     real_not_le_conv, real_not_eq_conv, all_conv]))
   493   fun absremover ct = (literals_conv [@{term "op &"}, @{term "op |"}] [] 
   494                   (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv 
   495         try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
   496   val nct = capply @{cterm Trueprop} (capply @{cterm "Not"} ct)
   497   val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
   498   val tm0 = dest_arg (rhs_of th0)
   499   val th = if tm0 aconvc @{cterm False} then equal_implies_1_rule th0 else
   500    let 
   501     val (evs,bod) = strip_exists tm0
   502     val (avs,ibod) = strip_forall bod
   503     val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
   504     val th2 = overall [] [specl avs (assume (rhs_of th1))]
   505     val th3 = fold simple_choose evs (prove_hyp (equal_elim th1 (assume (capply @{cterm Trueprop} bod))) th2)
   506    in  Drule.implies_intr_hyps (prove_hyp (equal_elim th0 (assume nct)) th3)
   507    end
   508   in implies_elim (instantiate' [] [SOME ct] pth_final) th
   509  end
   510 in f
   511 end;
   512 
   513 (* A linear arithmetic prover *)
   514 local
   515   val linear_add = Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
   516   fun linear_cmul c = Ctermfunc.mapf (fn x => c */ x)
   517   val one_tm = @{cterm "1::real"}
   518   fun contradictory p (e,_) = ((Ctermfunc.is_undefined e) andalso not(p Rat.zero)) orelse
   519      ((gen_eq_set (op aconvc) (Ctermfunc.dom e, [one_tm])) andalso not(p(Ctermfunc.apply e one_tm)))
   520 
   521   fun linear_ineqs vars (les,lts) = 
   522    case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
   523     SOME r => r
   524   | NONE => 
   525    (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
   526      SOME r => r
   527    | NONE => 
   528      if null vars then error "linear_ineqs: no contradiction" else
   529      let 
   530       val ineqs = les @ lts
   531       fun blowup v =
   532        length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
   533        length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
   534        length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
   535       val  v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
   536                  (map (fn v => (v,blowup v)) vars)))
   537       fun addup (e1,p1) (e2,p2) acc =
   538        let 
   539         val c1 = Ctermfunc.tryapplyd e1 v Rat.zero 
   540         val c2 = Ctermfunc.tryapplyd e2 v Rat.zero
   541        in if c1 */ c2 >=/ Rat.zero then acc else
   542         let 
   543          val e1' = linear_cmul (Rat.abs c2) e1
   544          val e2' = linear_cmul (Rat.abs c1) e2
   545          val p1' = Product(Rational_lt(Rat.abs c2),p1)
   546          val p2' = Product(Rational_lt(Rat.abs c1),p2)
   547         in (linear_add e1' e2',Sum(p1',p2'))::acc
   548         end
   549        end
   550       val (les0,les1) = 
   551          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
   552       val (lts0,lts1) = 
   553          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
   554       val (lesp,lesn) = 
   555          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
   556       val (ltsp,ltsn) = 
   557          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
   558       val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
   559       val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
   560                       (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
   561      in linear_ineqs (remove (op aconvc) v vars) (les',lts')
   562      end)
   563 
   564   fun linear_eqs(eqs,les,lts) = 
   565    case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
   566     SOME r => r
   567   | NONE => (case eqs of 
   568     [] => 
   569      let val vars = remove (op aconvc) one_tm 
   570            (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom o fst) (les@lts) []) 
   571      in linear_ineqs vars (les,lts) end
   572    | (e,p)::es => 
   573      if Ctermfunc.is_undefined e then linear_eqs (es,les,lts) else
   574      let 
   575       val (x,c) = Ctermfunc.choose (Ctermfunc.undefine one_tm e)
   576       fun xform (inp as (t,q)) =
   577        let val d = Ctermfunc.tryapplyd t x Rat.zero in
   578         if d =/ Rat.zero then inp else
   579         let 
   580          val k = (Rat.neg d) */ Rat.abs c // c
   581          val e' = linear_cmul k e
   582          val t' = linear_cmul (Rat.abs c) t
   583          val p' = Eqmul(cterm_of_rat k,p)
   584          val q' = Product(Rational_lt(Rat.abs c),q) 
   585         in (linear_add e' t',Sum(p',q')) 
   586         end 
   587       end
   588      in linear_eqs(map xform es,map xform les,map xform lts)
   589      end)
   590 
   591   fun linear_prover (eq,le,lt) = 
   592    let 
   593     val eqs = map2 (fn p => fn n => (p,Axiom_eq n)) eq (0 upto (length eq - 1))
   594     val les = map2 (fn p => fn n => (p,Axiom_le n)) le (0 upto (length le - 1))
   595     val lts = map2 (fn p => fn n => (p,Axiom_lt n)) lt (0 upto (length lt - 1))
   596    in linear_eqs(eqs,les,lts)
   597    end 
   598   
   599   fun lin_of_hol ct = 
   600    if ct aconvc @{cterm "0::real"} then Ctermfunc.undefined
   601    else if not (is_comb ct) then Ctermfunc.onefunc (ct, Rat.one)
   602    else if is_ratconst ct then Ctermfunc.onefunc (one_tm, dest_ratconst ct)
   603    else
   604     let val (lop,r) = Thm.dest_comb ct 
   605     in if not (is_comb lop) then Ctermfunc.onefunc (ct, Rat.one)
   606        else
   607         let val (opr,l) = Thm.dest_comb lop 
   608         in if opr aconvc @{cterm "op + :: real =>_"} 
   609            then linear_add (lin_of_hol l) (lin_of_hol r)
   610            else if opr aconvc @{cterm "op * :: real =>_"} 
   611                    andalso is_ratconst l then Ctermfunc.onefunc (r, dest_ratconst l)
   612            else Ctermfunc.onefunc (ct, Rat.one)
   613         end
   614     end
   615 
   616   fun is_alien ct = case term_of ct of 
   617    Const(@{const_name "real"}, _)$ n => 
   618      if can HOLogic.dest_number n then false else true
   619   | _ => false
   620  open Thm
   621 in 
   622 fun real_linear_prover translator (eq,le,lt) = 
   623  let 
   624   val lhs = lin_of_hol o dest_arg1 o dest_arg o cprop_of
   625   val rhs = lin_of_hol o dest_arg o dest_arg o cprop_of
   626   val eq_pols = map lhs eq
   627   val le_pols = map rhs le
   628   val lt_pols = map rhs lt 
   629   val aliens =  filter is_alien
   630       (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom) 
   631           (eq_pols @ le_pols @ lt_pols) [])
   632   val le_pols' = le_pols @ map (fn v => Ctermfunc.onefunc (v,Rat.one)) aliens
   633   val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
   634   val le' = le @ map (fn a => instantiate' [] [SOME (dest_arg a)] @{thm real_of_nat_ge_zero}) aliens 
   635  in (translator (eq,le',lt) proof) : thm
   636  end
   637 end;
   638 
   639 (* A less general generic arithmetic prover dealing with abs,max and min*)
   640 
   641 local
   642  val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
   643  fun absmaxmin_elim_conv1 ctxt = 
   644     Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
   645 
   646  val absmaxmin_elim_conv2 =
   647   let 
   648    val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
   649    val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
   650    val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
   651    val abs_tm = @{cterm "abs :: real => _"}
   652    val p_tm = @{cpat "?P :: real => bool"}
   653    val x_tm = @{cpat "?x :: real"}
   654    val y_tm = @{cpat "?y::real"}
   655    val is_max = is_binop @{cterm "max :: real => _"}
   656    val is_min = is_binop @{cterm "min :: real => _"} 
   657    fun is_abs t = is_comb t andalso dest_fun t aconvc abs_tm
   658    fun eliminate_construct p c tm =
   659     let 
   660      val t = find_cterm p tm
   661      val th0 = (symmetric o beta_conversion false) (capply (cabs t tm) t)
   662      val (p,ax) = (dest_comb o rhs_of) th0
   663     in fconv_rule(arg_conv(binop_conv (arg_conv (beta_conversion false))))
   664                (transitive th0 (c p ax))
   665    end
   666 
   667    val elim_abs = eliminate_construct is_abs
   668     (fn p => fn ax => 
   669        instantiate ([], [(p_tm,p), (x_tm, dest_arg ax)]) pth_abs)
   670    val elim_max = eliminate_construct is_max
   671     (fn p => fn ax => 
   672       let val (ax,y) = dest_comb ax 
   673       in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)]) 
   674       pth_max end)
   675    val elim_min = eliminate_construct is_min
   676     (fn p => fn ax => 
   677       let val (ax,y) = dest_comb ax 
   678       in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)]) 
   679       pth_min end)
   680    in first_conv [elim_abs, elim_max, elim_min, all_conv]
   681   end;
   682 in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
   683         gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,
   684                        absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
   685 end;
   686 
   687 (* An instance for reals*) 
   688 
   689 fun gen_prover_real_arith ctxt prover = 
   690  let
   691   fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS
   692   val {add,mul,neg,pow,sub,main} = 
   693      Normalizer.semiring_normalizers_ord_wrapper ctxt
   694       (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) 
   695      simple_cterm_ord
   696 in gen_real_arith ctxt
   697    (cterm_of_rat, field_comp_conv, field_comp_conv,field_comp_conv,
   698     main,neg,add,mul, prover)
   699 end;
   700 
   701 fun real_arith ctxt = gen_prover_real_arith ctxt real_linear_prover;
   702 end