src/HOL/Library/positivstellensatz.ML
 author wenzelm Thu Jul 09 22:01:41 2009 +0200 (2009-07-09) changeset 31971 8c1b845ed105 parent 31120 fc654c95c29e child 32402 5731300da417 permissions -rw-r--r--
renamed functor TableFun to Table, and GraphFun to Graph;
```     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     (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
```
```    85 structure Conv2 =
```
```    86 struct
```
```    87  open Conv
```
```    88 fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
```
```    89 fun is_comb t = case (term_of t) of _\$_ => true | _ => false;
```
```    90 fun is_abs t = case (term_of t) of Abs _ => true | _ => false;
```
```    91
```
```    92 fun end_itlist f l =
```
```    93  case l of
```
```    94    []     => error "end_itlist"
```
```    95  | [x]    => x
```
```    96  | (h::t) => f h (end_itlist f t);
```
```    97
```
```    98  fun absc cv ct = case term_of ct of
```
```    99  Abs (v,_, _) =>
```
```   100   let val (x,t) = Thm.dest_abs (SOME v) ct
```
```   101   in Thm.abstract_rule ((fst o dest_Free o term_of) x) x (cv t)
```
```   102   end
```
```   103  | _ => all_conv ct;
```
```   104
```
```   105 fun cache_conv conv =
```
```   106  let
```
```   107   val tab = ref Termtab.empty
```
```   108   fun cconv t =
```
```   109     case Termtab.lookup (!tab) (term_of t) of
```
```   110      SOME th => th
```
```   111    | NONE => let val th = conv t
```
```   112              in ((tab := Termtab.insert Thm.eq_thm (term_of t, th) (!tab)); th) end
```
```   113  in cconv end;
```
```   114 fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
```
```   115   handle CTERM _ => false;
```
```   116
```
```   117 local
```
```   118  fun thenqc conv1 conv2 tm =
```
```   119    case try conv1 tm of
```
```   120     SOME th1 => (case try conv2 (Thm.rhs_of th1) of SOME th2 => Thm.transitive th1 th2 | NONE => th1)
```
```   121   | NONE => conv2 tm
```
```   122
```
```   123  fun thencqc conv1 conv2 tm =
```
```   124     let val th1 = conv1 tm
```
```   125     in (case try conv2 (Thm.rhs_of th1) of SOME th2 => Thm.transitive th1 th2 | NONE => th1)
```
```   126     end
```
```   127  fun comb_qconv conv tm =
```
```   128    let val (l,r) = Thm.dest_comb tm
```
```   129    in (case try conv l of
```
```   130         SOME th1 => (case try conv r of SOME th2 => Thm.combination th1 th2
```
```   131                                       | NONE => Drule.fun_cong_rule th1 r)
```
```   132       | NONE => Drule.arg_cong_rule l (conv r))
```
```   133    end
```
```   134  fun repeatqc conv tm = thencqc conv (repeatqc conv) tm
```
```   135  fun sub_qconv conv tm =  if is_abs tm then absc conv tm else comb_qconv conv tm
```
```   136  fun once_depth_qconv conv tm =
```
```   137       (conv else_conv (sub_qconv (once_depth_qconv conv))) tm
```
```   138  fun depth_qconv conv tm =
```
```   139     thenqc (sub_qconv (depth_qconv conv))
```
```   140            (repeatqc conv) tm
```
```   141  fun redepth_qconv conv tm =
```
```   142     thenqc (sub_qconv (redepth_qconv conv))
```
```   143            (thencqc conv (redepth_qconv conv)) tm
```
```   144  fun top_depth_qconv conv tm =
```
```   145     thenqc (repeatqc conv)
```
```   146            (thencqc (sub_qconv (top_depth_qconv conv))
```
```   147                     (thencqc conv (top_depth_qconv conv))) tm
```
```   148  fun top_sweep_qconv conv tm =
```
```   149     thenqc (repeatqc conv)
```
```   150            (sub_qconv (top_sweep_qconv conv)) tm
```
```   151 in
```
```   152 val (once_depth_conv, depth_conv, rdepth_conv, top_depth_conv, top_sweep_conv) =
```
```   153   (fn c => try_conv (once_depth_qconv c),
```
```   154    fn c => try_conv (depth_qconv c),
```
```   155    fn c => try_conv (redepth_qconv c),
```
```   156    fn c => try_conv (top_depth_qconv c),
```
```   157    fn c => try_conv (top_sweep_qconv c));
```
```   158 end;
```
```   159 end;
```
```   160
```
```   161
```
```   162     (* Some useful derived rules *)
```
```   163 fun deduct_antisym_rule tha thb =
```
```   164     equal_intr (implies_intr (cprop_of thb) tha)
```
```   165      (implies_intr (cprop_of tha) thb);
```
```   166
```
```   167 fun prove_hyp tha thb =
```
```   168   if exists (curry op aconv (concl_of tha)) (#hyps (rep_thm thb))
```
```   169   then equal_elim (symmetric (deduct_antisym_rule tha thb)) tha else thb;
```
```   170
```
```   171
```
```   172
```
```   173 signature REAL_ARITH =
```
```   174 sig
```
```   175   datatype positivstellensatz =
```
```   176    Axiom_eq of int
```
```   177  | Axiom_le of int
```
```   178  | Axiom_lt of int
```
```   179  | Rational_eq of Rat.rat
```
```   180  | Rational_le of Rat.rat
```
```   181  | Rational_lt of Rat.rat
```
```   182  | Square of cterm
```
```   183  | Eqmul of cterm * positivstellensatz
```
```   184  | Sum of positivstellensatz * positivstellensatz
```
```   185  | Product of positivstellensatz * positivstellensatz;
```
```   186
```
```   187 val gen_gen_real_arith :
```
```   188   Proof.context -> (Rat.rat -> Thm.cterm) * conv * conv * conv *
```
```   189    conv * conv * conv * conv * conv * conv *
```
```   190     ( (thm list * thm list * thm list -> positivstellensatz -> thm) ->
```
```   191         thm list * thm list * thm list -> thm) -> conv
```
```   192 val real_linear_prover :
```
```   193   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
```
```   194    thm list * thm list * thm list -> thm
```
```   195
```
```   196 val gen_real_arith : Proof.context ->
```
```   197    (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv *
```
```   198    ( (thm list * thm list * thm list -> positivstellensatz -> thm) ->
```
```   199        thm list * thm list * thm list -> thm) -> conv
```
```   200 val gen_prover_real_arith : Proof.context ->
```
```   201    ((thm list * thm list * thm list -> positivstellensatz -> thm) ->
```
```   202      thm list * thm list * thm list -> thm) -> conv
```
```   203 val real_arith : Proof.context -> conv
```
```   204 end
```
```   205
```
```   206 structure RealArith (* : REAL_ARITH *)=
```
```   207 struct
```
```   208
```
```   209  open Conv Thm;;
```
```   210 (* ------------------------------------------------------------------------- *)
```
```   211 (* Data structure for Positivstellensatz refutations.                        *)
```
```   212 (* ------------------------------------------------------------------------- *)
```
```   213
```
```   214 datatype positivstellensatz =
```
```   215    Axiom_eq of int
```
```   216  | Axiom_le of int
```
```   217  | Axiom_lt of int
```
```   218  | Rational_eq of Rat.rat
```
```   219  | Rational_le of Rat.rat
```
```   220  | Rational_lt of Rat.rat
```
```   221  | Square of cterm
```
```   222  | Eqmul of cterm * positivstellensatz
```
```   223  | Sum of positivstellensatz * positivstellensatz
```
```   224  | Product of positivstellensatz * positivstellensatz;
```
```   225          (* Theorems used in the procedure *)
```
```   226
```
```   227
```
```   228 val my_eqs = ref ([] : thm list);
```
```   229 val my_les = ref ([] : thm list);
```
```   230 val my_lts = ref ([] : thm list);
```
```   231 val my_proof = ref (Axiom_eq 0);
```
```   232 val my_context = ref @{context};
```
```   233
```
```   234 val my_mk_numeric = ref ((K @{cterm True}) :Rat.rat -> cterm);
```
```   235 val my_numeric_eq_conv = ref no_conv;
```
```   236 val my_numeric_ge_conv = ref no_conv;
```
```   237 val my_numeric_gt_conv = ref no_conv;
```
```   238 val my_poly_conv = ref no_conv;
```
```   239 val my_poly_neg_conv = ref no_conv;
```
```   240 val my_poly_add_conv = ref no_conv;
```
```   241 val my_poly_mul_conv = ref no_conv;
```
```   242
```
```   243 fun conjunctions th = case try Conjunction.elim th of
```
```   244    SOME (th1,th2) => (conjunctions th1) @ conjunctions th2
```
```   245  | NONE => [th];
```
```   246
```
```   247 val pth = @{lemma "(((x::real) < y) == (y - x > 0)) &&& ((x <= y) == (y - x >= 0))
```
```   248      &&& ((x = y) == (x - y = 0)) &&& ((~(x < y)) == (x - y >= 0)) &&& ((~(x <= y)) == (x - y > 0))
```
```   249      &&& ((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
```
```   250   by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)} |>
```
```   251 conjunctions;
```
```   252
```
```   253 val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
```
```   254 val pth_add =
```
```   255  @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 ) &&& ( x = 0 ==> y >= 0 ==> x + y >= 0)
```
```   256     &&& (x = 0 ==> y > 0 ==> x + y > 0) &&& (x >= 0 ==> y = 0 ==> x + y >= 0)
```
```   257     &&& (x >= 0 ==> y >= 0 ==> x + y >= 0) &&& (x >= 0 ==> y > 0 ==> x + y > 0)
```
```   258     &&& (x > 0 ==> y = 0 ==> x + y > 0) &&& (x > 0 ==> y >= 0 ==> x + y > 0)
```
```   259     &&& (x > 0 ==> y > 0 ==> x + y > 0)"  by simp_all} |> conjunctions ;
```
```   260
```
```   261 val pth_mul =
```
```   262   @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0) &&& (x = 0 ==> y >= 0 ==> x * y = 0) &&&
```
```   263            (x = 0 ==> y > 0 ==> x * y = 0) &&& (x >= 0 ==> y = 0 ==> x * y = 0) &&&
```
```   264            (x >= 0 ==> y >= 0 ==> x * y >= 0 ) &&& ( x >= 0 ==> y > 0 ==> x * y >= 0 ) &&&
```
```   265            (x > 0 ==>  y = 0 ==> x * y = 0 ) &&& ( x > 0 ==> y >= 0 ==> x * y >= 0 ) &&&
```
```   266            (x > 0 ==>  y > 0 ==> x * y > 0)"
```
```   267   by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
```
```   268     mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])} |> conjunctions;
```
```   269
```
```   270 val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
```
```   271 val pth_square = @{lemma "x * x >= (0::real)"  by simp};
```
```   272
```
```   273 val weak_dnf_simps = List.take (simp_thms, 34)
```
```   274     @ 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+};
```
```   275
```
```   276 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+}
```
```   277
```
```   278 val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
```
```   279 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)});
```
```   280
```
```   281 val real_abs_thms1 = conjunctions @{lemma
```
```   282   "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r)) &&&
```
```   283   ((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
```
```   284   ((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
```
```   285   ((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r)) &&&
```
```   286   ((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r)) &&&
```
```   287   ((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r)) &&&
```
```   288   ((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r)) &&&
```
```   289   ((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
```
```   290   ((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
```
```   291   ((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y  + b >= r)) &&&
```
```   292   ((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r)) &&&
```
```   293   ((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y  + c >= r)) &&&
```
```   294   ((1 * min x y >= r) = (1 * x >= r & 1 * y >= r)) &&&
```
```   295   ((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
```
```   296   ((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
```
```   297   ((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y  + b >= r) )&&&
```
```   298   ((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r)) &&&
```
```   299   ((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y  + c >= r)) &&&
```
```   300   ((min x y >= r) = (x >= r &  y >= r)) &&&
```
```   301   ((min x y + a >= r) = (a + x >= r & a + y >= r)) &&&
```
```   302   ((a + min x y >= r) = (a + x >= r & a + y >= r)) &&&
```
```   303   ((a + min x y + b >= r) = (a + x + b >= r & a + y  + b >= r)) &&&
```
```   304   ((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r) )&&&
```
```   305   ((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r)) &&&
```
```   306   ((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r)) &&&
```
```   307   ((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
```
```   308   ((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
```
```   309   ((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r)) &&&
```
```   310   ((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r)) &&&
```
```   311   ((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r)) &&&
```
```   312   ((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r)) &&&
```
```   313   ((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
```
```   314   ((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
```
```   315   ((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y  + b > r)) &&&
```
```   316   ((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r)) &&&
```
```   317   ((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y  + c > r)) &&&
```
```   318   ((min x y > r) = (x > r &  y > r)) &&&
```
```   319   ((min x y + a > r) = (a + x > r & a + y > r)) &&&
```
```   320   ((a + min x y > r) = (a + x > r & a + y > r)) &&&
```
```   321   ((a + min x y + b > r) = (a + x + b > r & a + y  + b > r)) &&&
```
```   322   ((a + b + min x y > r) = (a + b + x > r & a + b + y > r)) &&&
```
```   323   ((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
```
```   324   by auto};
```
```   325
```
```   326 val abs_split' = @{lemma "P (abs (x::'a::ordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
```
```   327   by (atomize (full)) (auto split add: abs_split)};
```
```   328
```
```   329 val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
```
```   330   by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
```
```   331
```
```   332 val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
```
```   333   by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
```
```   334
```
```   335
```
```   336          (* Miscalineous *)
```
```   337 fun literals_conv bops uops cv =
```
```   338  let fun h t =
```
```   339   case (term_of t) of
```
```   340    b\$_\$_ => if member (op aconv) bops b then binop_conv h t else cv t
```
```   341  | u\$_ => if member (op aconv) uops u then arg_conv h t else cv t
```
```   342  | _ => cv t
```
```   343  in h end;
```
```   344
```
```   345 fun cterm_of_rat x =
```
```   346 let val (a, b) = Rat.quotient_of_rat x
```
```   347 in
```
```   348  if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
```
```   349   else Thm.capply (Thm.capply @{cterm "op / :: real => _"}
```
```   350                    (Numeral.mk_cnumber @{ctyp "real"} a))
```
```   351         (Numeral.mk_cnumber @{ctyp "real"} b)
```
```   352 end;
```
```   353
```
```   354   fun dest_ratconst t = case term_of t of
```
```   355    Const(@{const_name divide}, _)\$a\$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
```
```   356  | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
```
```   357  fun is_ratconst t = can dest_ratconst t
```
```   358
```
```   359 fun find_term p t = if p t then t else
```
```   360  case t of
```
```   361   a\$b => (find_term p a handle TERM _ => find_term p b)
```
```   362  | Abs (_,_,t') => find_term p t'
```
```   363  | _ => raise TERM ("find_term",[t]);
```
```   364
```
```   365 fun find_cterm p t = if p t then t else
```
```   366  case term_of t of
```
```   367   a\$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
```
```   368  | Abs (_,_,t') => find_cterm p (Thm.dest_abs NONE t |> snd)
```
```   369  | _ => raise CTERM ("find_cterm",[t]);
```
```   370
```
```   371
```
```   372     (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
```
```   373 fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
```
```   374 fun is_comb t = case (term_of t) of _\$_ => true | _ => false;
```
```   375
```
```   376 fun cache_conv conv =
```
```   377  let
```
```   378   val tab = ref Termtab.empty
```
```   379   fun cconv t =
```
```   380     case Termtab.lookup (!tab) (term_of t) of
```
```   381      SOME th => th
```
```   382    | NONE => let val th = conv t
```
```   383              in ((tab := Termtab.insert Thm.eq_thm (term_of t, th) (!tab)); th) end
```
```   384  in cconv end;
```
```   385 fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
```
```   386   handle CTERM _ => false;
```
```   387
```
```   388     (* A general real arithmetic prover *)
```
```   389
```
```   390 fun gen_gen_real_arith ctxt (mk_numeric,
```
```   391        numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
```
```   392        poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
```
```   393        absconv1,absconv2,prover) =
```
```   394 let
```
```   395  open Conv Thm;
```
```   396  val _ = my_context := ctxt
```
```   397  val _ = (my_mk_numeric := mk_numeric ; my_numeric_eq_conv := numeric_eq_conv ;
```
```   398           my_numeric_ge_conv := numeric_ge_conv; my_numeric_gt_conv := numeric_gt_conv ;
```
```   399           my_poly_conv := poly_conv; my_poly_neg_conv := poly_neg_conv;
```
```   400           my_poly_add_conv := poly_add_conv; my_poly_mul_conv := poly_mul_conv)
```
```   401  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}]
```
```   402  val prenex_ss = HOL_basic_ss addsimps prenex_simps
```
```   403  val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
```
```   404  val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
```
```   405  val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
```
```   406  val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
```
```   407  val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
```
```   408  val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
```
```   409  fun eqT_elim th = equal_elim (symmetric th) @{thm TrueI}
```
```   410  fun oprconv cv ct =
```
```   411   let val g = Thm.dest_fun2 ct
```
```   412   in if g aconvc @{cterm "op <= :: real => _"}
```
```   413        orelse g aconvc @{cterm "op < :: real => _"}
```
```   414      then arg_conv cv ct else arg1_conv cv ct
```
```   415   end
```
```   416
```
```   417  fun real_ineq_conv th ct =
```
```   418   let
```
```   419    val th' = (instantiate (match (lhs_of th, ct)) th
```
```   420       handle MATCH => raise CTERM ("real_ineq_conv", [ct]))
```
```   421   in transitive th' (oprconv poly_conv (Thm.rhs_of th'))
```
```   422   end
```
```   423   val [real_lt_conv, real_le_conv, real_eq_conv,
```
```   424        real_not_lt_conv, real_not_le_conv, _] =
```
```   425        map real_ineq_conv pth
```
```   426   fun match_mp_rule ths ths' =
```
```   427    let
```
```   428      fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
```
```   429       | th::ths => (ths' MRS th handle THM _ => f ths ths')
```
```   430    in f ths ths' end
```
```   431   fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
```
```   432          (match_mp_rule pth_mul [th, th'])
```
```   433   fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
```
```   434          (match_mp_rule pth_add [th, th'])
```
```   435   fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
```
```   436        (instantiate' [] [SOME ct] (th RS pth_emul))
```
```   437   fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
```
```   438        (instantiate' [] [SOME t] pth_square)
```
```   439
```
```   440   fun hol_of_positivstellensatz(eqs,les,lts) proof =
```
```   441    let
```
```   442     val _ = (my_eqs := eqs ; my_les := les ; my_lts := lts ; my_proof := proof)
```
```   443     fun translate prf = case prf of
```
```   444         Axiom_eq n => nth eqs n
```
```   445       | Axiom_le n => nth les n
```
```   446       | Axiom_lt n => nth lts n
```
```   447       | Rational_eq x => eqT_elim(numeric_eq_conv(capply @{cterm Trueprop}
```
```   448                           (capply (capply @{cterm "op =::real => _"} (mk_numeric x))
```
```   449                                @{cterm "0::real"})))
```
```   450       | Rational_le x => eqT_elim(numeric_ge_conv(capply @{cterm Trueprop}
```
```   451                           (capply (capply @{cterm "op <=::real => _"}
```
```   452                                      @{cterm "0::real"}) (mk_numeric x))))
```
```   453       | Rational_lt x => eqT_elim(numeric_gt_conv(capply @{cterm Trueprop}
```
```   454                       (capply (capply @{cterm "op <::real => _"} @{cterm "0::real"})
```
```   455                         (mk_numeric x))))
```
```   456       | Square t => square_rule t
```
```   457       | Eqmul(t,p) => emul_rule t (translate p)
```
```   458       | Sum(p1,p2) => add_rule (translate p1) (translate p2)
```
```   459       | Product(p1,p2) => mul_rule (translate p1) (translate p2)
```
```   460    in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv])
```
```   461           (translate proof)
```
```   462    end
```
```   463
```
```   464   val init_conv = presimp_conv then_conv
```
```   465       nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
```
```   466       weak_dnf_conv
```
```   467
```
```   468   val concl = dest_arg o cprop_of
```
```   469   fun is_binop opr ct = (dest_fun2 ct aconvc opr handle CTERM _ => false)
```
```   470   val is_req = is_binop @{cterm "op =:: real => _"}
```
```   471   val is_ge = is_binop @{cterm "op <=:: real => _"}
```
```   472   val is_gt = is_binop @{cterm "op <:: real => _"}
```
```   473   val is_conj = is_binop @{cterm "op &"}
```
```   474   val is_disj = is_binop @{cterm "op |"}
```
```   475   fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
```
```   476   fun disj_cases th th1 th2 =
```
```   477    let val (p,q) = dest_binop (concl th)
```
```   478        val c = concl th1
```
```   479        val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
```
```   480    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)
```
```   481    end
```
```   482  fun overall dun ths = case ths of
```
```   483   [] =>
```
```   484    let
```
```   485     val (eq,ne) = List.partition (is_req o concl) dun
```
```   486      val (le,nl) = List.partition (is_ge o concl) ne
```
```   487      val lt = filter (is_gt o concl) nl
```
```   488     in prover hol_of_positivstellensatz (eq,le,lt) end
```
```   489  | th::oths =>
```
```   490    let
```
```   491     val ct = concl th
```
```   492    in
```
```   493     if is_conj ct  then
```
```   494      let
```
```   495       val (th1,th2) = conj_pair th in
```
```   496       overall dun (th1::th2::oths) end
```
```   497     else if is_disj ct then
```
```   498       let
```
```   499        val th1 = overall dun (assume (capply @{cterm Trueprop} (dest_arg1 ct))::oths)
```
```   500        val th2 = overall dun (assume (capply @{cterm Trueprop} (dest_arg ct))::oths)
```
```   501       in disj_cases th th1 th2 end
```
```   502    else overall (th::dun) oths
```
```   503   end
```
```   504   fun dest_binary b ct = if is_binop b ct then dest_binop ct
```
```   505                          else raise CTERM ("dest_binary",[b,ct])
```
```   506   val dest_eq = dest_binary @{cterm "op = :: real => _"}
```
```   507   val neq_th = nth pth 5
```
```   508   fun real_not_eq_conv ct =
```
```   509    let
```
```   510     val (l,r) = dest_eq (dest_arg ct)
```
```   511     val th = instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
```
```   512     val th_p = poly_conv(dest_arg(dest_arg1(rhs_of th)))
```
```   513     val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
```
```   514     val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
```
```   515     val th' = Drule.binop_cong_rule @{cterm "op |"}
```
```   516      (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
```
```   517      (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
```
```   518     in transitive th th'
```
```   519   end
```
```   520  fun equal_implies_1_rule PQ =
```
```   521   let
```
```   522    val P = lhs_of PQ
```
```   523   in implies_intr P (equal_elim PQ (assume P))
```
```   524   end
```
```   525  (* FIXME!!! Copied from groebner.ml *)
```
```   526  val strip_exists =
```
```   527   let fun h (acc, t) =
```
```   528    case (term_of t) of
```
```   529     Const("Ex",_)\$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
```
```   530   | _ => (acc,t)
```
```   531   in fn t => h ([],t)
```
```   532   end
```
```   533   fun name_of x = case term_of x of
```
```   534    Free(s,_) => s
```
```   535  | Var ((s,_),_) => s
```
```   536  | _ => "x"
```
```   537
```
```   538   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)
```
```   539
```
```   540   val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
```
```   541
```
```   542  fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
```
```   543  fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
```
```   544
```
```   545  fun choose v th th' = case concl_of th of
```
```   546    @{term Trueprop} \$ (Const("Ex",_)\$_) =>
```
```   547     let
```
```   548      val p = (funpow 2 Thm.dest_arg o cprop_of) th
```
```   549      val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
```
```   550      val th0 = fconv_rule (Thm.beta_conversion true)
```
```   551          (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
```
```   552      val pv = (Thm.rhs_of o Thm.beta_conversion true)
```
```   553            (Thm.capply @{cterm Trueprop} (Thm.capply p v))
```
```   554      val th1 = forall_intr v (implies_intr pv th')
```
```   555     in implies_elim (implies_elim th0 th) th1  end
```
```   556  | _ => raise THM ("choose",0,[th, th'])
```
```   557
```
```   558   fun simple_choose v th =
```
```   559      choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
```
```   560
```
```   561  val strip_forall =
```
```   562   let fun h (acc, t) =
```
```   563    case (term_of t) of
```
```   564     Const("All",_)\$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
```
```   565   | _ => (acc,t)
```
```   566   in fn t => h ([],t)
```
```   567   end
```
```   568
```
```   569  fun f ct =
```
```   570   let
```
```   571    val nnf_norm_conv' =
```
```   572      nnf_conv then_conv
```
```   573      literals_conv [@{term "op &"}, @{term "op |"}] []
```
```   574      (cache_conv
```
```   575        (first_conv [real_lt_conv, real_le_conv,
```
```   576                     real_eq_conv, real_not_lt_conv,
```
```   577                     real_not_le_conv, real_not_eq_conv, all_conv]))
```
```   578   fun absremover ct = (literals_conv [@{term "op &"}, @{term "op |"}] []
```
```   579                   (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv
```
```   580         try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
```
```   581   val nct = capply @{cterm Trueprop} (capply @{cterm "Not"} ct)
```
```   582   val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
```
```   583   val tm0 = dest_arg (rhs_of th0)
```
```   584   val th = if tm0 aconvc @{cterm False} then equal_implies_1_rule th0 else
```
```   585    let
```
```   586     val (evs,bod) = strip_exists tm0
```
```   587     val (avs,ibod) = strip_forall bod
```
```   588     val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
```
```   589     val th2 = overall [] [specl avs (assume (rhs_of th1))]
```
```   590     val th3 = fold simple_choose evs (prove_hyp (equal_elim th1 (assume (capply @{cterm Trueprop} bod))) th2)
```
```   591    in  Drule.implies_intr_hyps (prove_hyp (equal_elim th0 (assume nct)) th3)
```
```   592    end
```
```   593   in implies_elim (instantiate' [] [SOME ct] pth_final) th
```
```   594  end
```
```   595 in f
```
```   596 end;
```
```   597
```
```   598 (* A linear arithmetic prover *)
```
```   599 local
```
```   600   val linear_add = Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
```
```   601   fun linear_cmul c = Ctermfunc.mapf (fn x => c */ x)
```
```   602   val one_tm = @{cterm "1::real"}
```
```   603   fun contradictory p (e,_) = ((Ctermfunc.is_undefined e) andalso not(p Rat.zero)) orelse
```
```   604      ((gen_eq_set (op aconvc) (Ctermfunc.dom e, [one_tm])) andalso not(p(Ctermfunc.apply e one_tm)))
```
```   605
```
```   606   fun linear_ineqs vars (les,lts) =
```
```   607    case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
```
```   608     SOME r => r
```
```   609   | NONE =>
```
```   610    (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
```
```   611      SOME r => r
```
```   612    | NONE =>
```
```   613      if null vars then error "linear_ineqs: no contradiction" else
```
```   614      let
```
```   615       val ineqs = les @ lts
```
```   616       fun blowup v =
```
```   617        length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
```
```   618        length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
```
```   619        length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
```
```   620       val  v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
```
```   621                  (map (fn v => (v,blowup v)) vars)))
```
```   622       fun addup (e1,p1) (e2,p2) acc =
```
```   623        let
```
```   624         val c1 = Ctermfunc.tryapplyd e1 v Rat.zero
```
```   625         val c2 = Ctermfunc.tryapplyd e2 v Rat.zero
```
```   626        in if c1 */ c2 >=/ Rat.zero then acc else
```
```   627         let
```
```   628          val e1' = linear_cmul (Rat.abs c2) e1
```
```   629          val e2' = linear_cmul (Rat.abs c1) e2
```
```   630          val p1' = Product(Rational_lt(Rat.abs c2),p1)
```
```   631          val p2' = Product(Rational_lt(Rat.abs c1),p2)
```
```   632         in (linear_add e1' e2',Sum(p1',p2'))::acc
```
```   633         end
```
```   634        end
```
```   635       val (les0,les1) =
```
```   636          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
```
```   637       val (lts0,lts1) =
```
```   638          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
```
```   639       val (lesp,lesn) =
```
```   640          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
```
```   641       val (ltsp,ltsn) =
```
```   642          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
```
```   643       val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
```
```   644       val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
```
```   645                       (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
```
```   646      in linear_ineqs (remove (op aconvc) v vars) (les',lts')
```
```   647      end)
```
```   648
```
```   649   fun linear_eqs(eqs,les,lts) =
```
```   650    case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
```
```   651     SOME r => r
```
```   652   | NONE => (case eqs of
```
```   653     [] =>
```
```   654      let val vars = remove (op aconvc) one_tm
```
```   655            (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom o fst) (les@lts) [])
```
```   656      in linear_ineqs vars (les,lts) end
```
```   657    | (e,p)::es =>
```
```   658      if Ctermfunc.is_undefined e then linear_eqs (es,les,lts) else
```
```   659      let
```
```   660       val (x,c) = Ctermfunc.choose (Ctermfunc.undefine one_tm e)
```
```   661       fun xform (inp as (t,q)) =
```
```   662        let val d = Ctermfunc.tryapplyd t x Rat.zero in
```
```   663         if d =/ Rat.zero then inp else
```
```   664         let
```
```   665          val k = (Rat.neg d) */ Rat.abs c // c
```
```   666          val e' = linear_cmul k e
```
```   667          val t' = linear_cmul (Rat.abs c) t
```
```   668          val p' = Eqmul(cterm_of_rat k,p)
```
```   669          val q' = Product(Rational_lt(Rat.abs c),q)
```
```   670         in (linear_add e' t',Sum(p',q'))
```
```   671         end
```
```   672       end
```
```   673      in linear_eqs(map xform es,map xform les,map xform lts)
```
```   674      end)
```
```   675
```
```   676   fun linear_prover (eq,le,lt) =
```
```   677    let
```
```   678     val eqs = map2 (fn p => fn n => (p,Axiom_eq n)) eq (0 upto (length eq - 1))
```
```   679     val les = map2 (fn p => fn n => (p,Axiom_le n)) le (0 upto (length le - 1))
```
```   680     val lts = map2 (fn p => fn n => (p,Axiom_lt n)) lt (0 upto (length lt - 1))
```
```   681    in linear_eqs(eqs,les,lts)
```
```   682    end
```
```   683
```
```   684   fun lin_of_hol ct =
```
```   685    if ct aconvc @{cterm "0::real"} then Ctermfunc.undefined
```
```   686    else if not (is_comb ct) then Ctermfunc.onefunc (ct, Rat.one)
```
```   687    else if is_ratconst ct then Ctermfunc.onefunc (one_tm, dest_ratconst ct)
```
```   688    else
```
```   689     let val (lop,r) = Thm.dest_comb ct
```
```   690     in if not (is_comb lop) then Ctermfunc.onefunc (ct, Rat.one)
```
```   691        else
```
```   692         let val (opr,l) = Thm.dest_comb lop
```
```   693         in if opr aconvc @{cterm "op + :: real =>_"}
```
```   694            then linear_add (lin_of_hol l) (lin_of_hol r)
```
```   695            else if opr aconvc @{cterm "op * :: real =>_"}
```
```   696                    andalso is_ratconst l then Ctermfunc.onefunc (r, dest_ratconst l)
```
```   697            else Ctermfunc.onefunc (ct, Rat.one)
```
```   698         end
```
```   699     end
```
```   700
```
```   701   fun is_alien ct = case term_of ct of
```
```   702    Const(@{const_name "real"}, _)\$ n =>
```
```   703      if can HOLogic.dest_number n then false else true
```
```   704   | _ => false
```
```   705  open Thm
```
```   706 in
```
```   707 fun real_linear_prover translator (eq,le,lt) =
```
```   708  let
```
```   709   val lhs = lin_of_hol o dest_arg1 o dest_arg o cprop_of
```
```   710   val rhs = lin_of_hol o dest_arg o dest_arg o cprop_of
```
```   711   val eq_pols = map lhs eq
```
```   712   val le_pols = map rhs le
```
```   713   val lt_pols = map rhs lt
```
```   714   val aliens =  filter is_alien
```
```   715       (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom)
```
```   716           (eq_pols @ le_pols @ lt_pols) [])
```
```   717   val le_pols' = le_pols @ map (fn v => Ctermfunc.onefunc (v,Rat.one)) aliens
```
```   718   val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
```
```   719   val le' = le @ map (fn a => instantiate' [] [SOME (dest_arg a)] @{thm real_of_nat_ge_zero}) aliens
```
```   720  in (translator (eq,le',lt) proof) : thm
```
```   721  end
```
```   722 end;
```
```   723
```
```   724 (* A less general generic arithmetic prover dealing with abs,max and min*)
```
```   725
```
```   726 local
```
```   727  val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
```
```   728  fun absmaxmin_elim_conv1 ctxt =
```
```   729     Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
```
```   730
```
```   731  val absmaxmin_elim_conv2 =
```
```   732   let
```
```   733    val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
```
```   734    val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
```
```   735    val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
```
```   736    val abs_tm = @{cterm "abs :: real => _"}
```
```   737    val p_tm = @{cpat "?P :: real => bool"}
```
```   738    val x_tm = @{cpat "?x :: real"}
```
```   739    val y_tm = @{cpat "?y::real"}
```
```   740    val is_max = is_binop @{cterm "max :: real => _"}
```
```   741    val is_min = is_binop @{cterm "min :: real => _"}
```
```   742    fun is_abs t = is_comb t andalso dest_fun t aconvc abs_tm
```
```   743    fun eliminate_construct p c tm =
```
```   744     let
```
```   745      val t = find_cterm p tm
```
```   746      val th0 = (symmetric o beta_conversion false) (capply (cabs t tm) t)
```
```   747      val (p,ax) = (dest_comb o rhs_of) th0
```
```   748     in fconv_rule(arg_conv(binop_conv (arg_conv (beta_conversion false))))
```
```   749                (transitive th0 (c p ax))
```
```   750    end
```
```   751
```
```   752    val elim_abs = eliminate_construct is_abs
```
```   753     (fn p => fn ax =>
```
```   754        instantiate ([], [(p_tm,p), (x_tm, dest_arg ax)]) pth_abs)
```
```   755    val elim_max = eliminate_construct is_max
```
```   756     (fn p => fn ax =>
```
```   757       let val (ax,y) = dest_comb ax
```
```   758       in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)])
```
```   759       pth_max end)
```
```   760    val elim_min = eliminate_construct is_min
```
```   761     (fn p => fn ax =>
```
```   762       let val (ax,y) = dest_comb ax
```
```   763       in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)])
```
```   764       pth_min end)
```
```   765    in first_conv [elim_abs, elim_max, elim_min, all_conv]
```
```   766   end;
```
```   767 in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
```
```   768         gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,
```
```   769                        absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
```
```   770 end;
```
```   771
```
```   772 (* An instance for reals*)
```
```   773
```
```   774 fun gen_prover_real_arith ctxt prover =
```
```   775  let
```
```   776   fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS
```
```   777   val {add,mul,neg,pow,sub,main} =
```
```   778      Normalizer.semiring_normalizers_ord_wrapper ctxt
```
```   779       (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
```
```   780      simple_cterm_ord
```
```   781 in gen_real_arith ctxt
```
```   782    (cterm_of_rat, field_comp_conv, field_comp_conv,field_comp_conv,
```
```   783     main,neg,add,mul, prover)
```
```   784 end;
```
```   785
```
```   786 fun real_arith ctxt = gen_prover_real_arith ctxt real_linear_prover;
```
```   787 end
```