src/HOL/Library/positivstellensatz.ML
 author Philipp Meyer Mon Sep 21 15:05:26 2009 +0200 (2009-09-21) changeset 32645 1cc5b24f5a01 parent 32402 5731300da417 child 32646 962b4354ed90 permissions -rw-r--r--
sos method generates and uses proof certificates
```     1 (* Title:      Library/Sum_Of_Squares/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
```
```     9 signature FUNC =
```
```    10 sig
```
```    11  type 'a T
```
```    12  type key
```
```    13  val apply : 'a T -> key -> 'a
```
```    14  val applyd :'a T -> (key -> 'a) -> key -> 'a
```
```    15  val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a T -> 'a T -> 'a T
```
```    16  val defined : 'a T -> key -> bool
```
```    17  val dom : 'a T -> key list
```
```    18  val fold : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b
```
```    19  val fold_rev : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b
```
```    20  val graph : 'a T -> (key * 'a) list
```
```    21  val is_undefined : 'a T -> bool
```
```    22  val mapf : ('a -> 'b) -> 'a T -> 'b T
```
```    23  val tryapplyd : 'a T -> key -> 'a -> 'a
```
```    24  val undefine :  key -> 'a T -> 'a T
```
```    25  val undefined : 'a T
```
```    26  val update : key * 'a -> 'a T -> 'a T
```
```    27  val updatep : (key * 'a -> bool) -> key * 'a -> 'a T -> 'a T
```
```    28  val choose : 'a T -> key * 'a
```
```    29  val onefunc : key * 'a -> 'a T
```
```    30  val get_first: (key*'a -> 'a option) -> 'a T -> 'a option
```
```    31 end;
```
```    32
```
```    33 functor FuncFun(Key: KEY) : FUNC=
```
```    34 struct
```
```    35
```
```    36 type key = Key.key;
```
```    37 structure Tab = Table(Key);
```
```    38 type 'a T = 'a Tab.table;
```
```    39
```
```    40 val undefined = Tab.empty;
```
```    41 val is_undefined = Tab.is_empty;
```
```    42 val mapf = Tab.map;
```
```    43 val fold = Tab.fold;
```
```    44 val fold_rev = Tab.fold_rev;
```
```    45 val graph = Tab.dest;
```
```    46 fun dom a = sort Key.ord (Tab.keys a);
```
```    47 fun applyd f d x = case Tab.lookup f x of
```
```    48    SOME y => y
```
```    49  | NONE => d x;
```
```    50
```
```    51 fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
```
```    52 fun tryapplyd f a d = applyd f (K d) a;
```
```    53 val defined = Tab.defined;
```
```    54 fun undefine x t = (Tab.delete x t handle UNDEF => t);
```
```    55 val update = Tab.update;
```
```    56 fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
```
```    57 fun combine f z a b =
```
```    58  let
```
```    59   fun h (k,v) t = case Tab.lookup t k of
```
```    60      NONE => Tab.update (k,v) t
```
```    61    | SOME v' => let val w = f v v'
```
```    62      in if z w then Tab.delete k t else Tab.update (k,w) t end;
```
```    63   in Tab.fold h a b end;
```
```    64
```
```    65 fun choose f = case Tab.min_key f of
```
```    66    SOME k => (k,valOf (Tab.lookup f k))
```
```    67  | NONE => error "FuncFun.choose : Completely undefined function"
```
```    68
```
```    69 fun onefunc kv = update kv undefined
```
```    70
```
```    71 local
```
```    72 fun  find f (k,v) NONE = f (k,v)
```
```    73    | find f (k,v) r = r
```
```    74 in
```
```    75 fun get_first f t = fold (find f) t NONE
```
```    76 end
```
```    77 end;
```
```    78
```
```    79 (* Some standard functors and utility functions for them *)
```
```    80
```
```    81 structure FuncUtil =
```
```    82 struct
```
```    83
```
```    84 fun increasing f ord (x,y) = ord (f x, f y);
```
```    85
```
```    86 structure Intfunc = FuncFun(type key = int val ord = int_ord);
```
```    87 structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
```
```    88 structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
```
```    89 structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
```
```    90 structure Termfunc = FuncFun(type key = term val ord = TermOrd.fast_term_ord);
```
```    91
```
```    92 val cterm_ord = (fn (s,t) => TermOrd.fast_term_ord(term_of s, term_of t))
```
```    93
```
```    94 structure Ctermfunc = FuncFun(type key = cterm val ord = cterm_ord);
```
```    95
```
```    96 type monomial = int Ctermfunc.T;
```
```    97
```
```    98 fun monomial_ord (m1,m2) = list_ord (prod_ord cterm_ord int_ord) (Ctermfunc.graph m1, Ctermfunc.graph m2)
```
```    99
```
```   100 structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
```
```   101
```
```   102 type poly = Rat.rat Monomialfunc.T;
```
```   103
```
```   104 (* The ordering so we can create canonical HOL polynomials.                  *)
```
```   105
```
```   106 fun dest_monomial mon = sort (increasing fst cterm_ord) (Ctermfunc.graph mon);
```
```   107
```
```   108 fun monomial_order (m1,m2) =
```
```   109  if Ctermfunc.is_undefined m2 then LESS
```
```   110  else if Ctermfunc.is_undefined m1 then GREATER
```
```   111  else
```
```   112   let val mon1 = dest_monomial m1
```
```   113       val mon2 = dest_monomial m2
```
```   114       val deg1 = fold (curry op + o snd) mon1 0
```
```   115       val deg2 = fold (curry op + o snd) mon2 0
```
```   116   in if deg1 < deg2 then GREATER else if deg1 > deg2 then LESS
```
```   117      else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
```
```   118   end;
```
```   119
```
```   120 end
```
```   121
```
```   122 (* positivstellensatz datatype and prover generation *)
```
```   123
```
```   124 signature REAL_ARITH =
```
```   125 sig
```
```   126
```
```   127   datatype positivstellensatz =
```
```   128    Axiom_eq of int
```
```   129  | Axiom_le of int
```
```   130  | Axiom_lt of int
```
```   131  | Rational_eq of Rat.rat
```
```   132  | Rational_le of Rat.rat
```
```   133  | Rational_lt of Rat.rat
```
```   134  | Square of FuncUtil.poly
```
```   135  | Eqmul of FuncUtil.poly * positivstellensatz
```
```   136  | Sum of positivstellensatz * positivstellensatz
```
```   137  | Product of positivstellensatz * positivstellensatz;
```
```   138
```
```   139 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
```
```   140
```
```   141 datatype tree_choice = Left | Right
```
```   142
```
```   143 type prover = tree_choice list ->
```
```   144   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
```
```   145   thm list * thm list * thm list -> thm * pss_tree
```
```   146 type cert_conv = cterm -> thm * pss_tree
```
```   147
```
```   148 val gen_gen_real_arith :
```
```   149   Proof.context -> (Rat.rat -> Thm.cterm) * conv * conv * conv *
```
```   150    conv * conv * conv * conv * conv * conv * prover -> cert_conv
```
```   151 val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
```
```   152   thm list * thm list * thm list -> thm * pss_tree
```
```   153
```
```   154 val gen_real_arith : Proof.context ->
```
```   155   (Rat.rat -> Thm.cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
```
```   156
```
```   157 val gen_prover_real_arith : Proof.context -> prover -> cert_conv
```
```   158
```
```   159 val is_ratconst : Thm.cterm -> bool
```
```   160 val dest_ratconst : Thm.cterm -> Rat.rat
```
```   161 val cterm_of_rat : Rat.rat -> Thm.cterm
```
```   162
```
```   163 end
```
```   164
```
```   165 structure RealArith : REAL_ARITH =
```
```   166 struct
```
```   167
```
```   168  open Conv Thm FuncUtil;;
```
```   169 (* ------------------------------------------------------------------------- *)
```
```   170 (* Data structure for Positivstellensatz refutations.                        *)
```
```   171 (* ------------------------------------------------------------------------- *)
```
```   172
```
```   173 datatype positivstellensatz =
```
```   174    Axiom_eq of int
```
```   175  | Axiom_le of int
```
```   176  | Axiom_lt of int
```
```   177  | Rational_eq of Rat.rat
```
```   178  | Rational_le of Rat.rat
```
```   179  | Rational_lt of Rat.rat
```
```   180  | Square of FuncUtil.poly
```
```   181  | Eqmul of FuncUtil.poly * positivstellensatz
```
```   182  | Sum of positivstellensatz * positivstellensatz
```
```   183  | Product of positivstellensatz * positivstellensatz;
```
```   184          (* Theorems used in the procedure *)
```
```   185
```
```   186 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
```
```   187 datatype tree_choice = Left | Right
```
```   188 type prover = tree_choice list ->
```
```   189   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
```
```   190   thm list * thm list * thm list -> thm * pss_tree
```
```   191 type cert_conv = cterm -> thm * pss_tree
```
```   192
```
```   193 val my_eqs = ref ([] : thm list);
```
```   194 val my_les = ref ([] : thm list);
```
```   195 val my_lts = ref ([] : thm list);
```
```   196 val my_proof = ref (Axiom_eq 0);
```
```   197 val my_context = ref @{context};
```
```   198
```
```   199 val my_mk_numeric = ref ((K @{cterm True}) :Rat.rat -> cterm);
```
```   200 val my_numeric_eq_conv = ref no_conv;
```
```   201 val my_numeric_ge_conv = ref no_conv;
```
```   202 val my_numeric_gt_conv = ref no_conv;
```
```   203 val my_poly_conv = ref no_conv;
```
```   204 val my_poly_neg_conv = ref no_conv;
```
```   205 val my_poly_add_conv = ref no_conv;
```
```   206 val my_poly_mul_conv = ref no_conv;
```
```   207
```
```   208
```
```   209     (* Some useful derived rules *)
```
```   210 fun deduct_antisym_rule tha thb =
```
```   211     equal_intr (implies_intr (cprop_of thb) tha)
```
```   212      (implies_intr (cprop_of tha) thb);
```
```   213
```
```   214 fun prove_hyp tha thb =
```
```   215   if exists (curry op aconv (concl_of tha)) (#hyps (rep_thm thb))
```
```   216   then equal_elim (symmetric (deduct_antisym_rule tha thb)) tha else thb;
```
```   217
```
```   218 fun conjunctions th = case try Conjunction.elim th of
```
```   219    SOME (th1,th2) => (conjunctions th1) @ conjunctions th2
```
```   220  | NONE => [th];
```
```   221
```
```   222 val pth = @{lemma "(((x::real) < y) == (y - x > 0)) &&& ((x <= y) == (y - x >= 0))
```
```   223      &&& ((x = y) == (x - y = 0)) &&& ((~(x < y)) == (x - y >= 0)) &&& ((~(x <= y)) == (x - y > 0))
```
```   224      &&& ((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
```
```   225   by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)} |>
```
```   226 conjunctions;
```
```   227
```
```   228 val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
```
```   229 val pth_add =
```
```   230  @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 ) &&& ( x = 0 ==> y >= 0 ==> x + y >= 0)
```
```   231     &&& (x = 0 ==> y > 0 ==> x + y > 0) &&& (x >= 0 ==> y = 0 ==> x + y >= 0)
```
```   232     &&& (x >= 0 ==> y >= 0 ==> x + y >= 0) &&& (x >= 0 ==> y > 0 ==> x + y > 0)
```
```   233     &&& (x > 0 ==> y = 0 ==> x + y > 0) &&& (x > 0 ==> y >= 0 ==> x + y > 0)
```
```   234     &&& (x > 0 ==> y > 0 ==> x + y > 0)"  by simp_all} |> conjunctions ;
```
```   235
```
```   236 val pth_mul =
```
```   237   @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0) &&& (x = 0 ==> y >= 0 ==> x * y = 0) &&&
```
```   238            (x = 0 ==> y > 0 ==> x * y = 0) &&& (x >= 0 ==> y = 0 ==> x * y = 0) &&&
```
```   239            (x >= 0 ==> y >= 0 ==> x * y >= 0 ) &&& ( x >= 0 ==> y > 0 ==> x * y >= 0 ) &&&
```
```   240            (x > 0 ==>  y = 0 ==> x * y = 0 ) &&& ( x > 0 ==> y >= 0 ==> x * y >= 0 ) &&&
```
```   241            (x > 0 ==>  y > 0 ==> x * y > 0)"
```
```   242   by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
```
```   243     mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])} |> conjunctions;
```
```   244
```
```   245 val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
```
```   246 val pth_square = @{lemma "x * x >= (0::real)"  by simp};
```
```   247
```
```   248 val weak_dnf_simps = List.take (simp_thms, 34)
```
```   249     @ 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+};
```
```   250
```
```   251 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+}
```
```   252
```
```   253 val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
```
```   254 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)});
```
```   255
```
```   256 val real_abs_thms1 = conjunctions @{lemma
```
```   257   "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r)) &&&
```
```   258   ((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
```
```   259   ((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
```
```   260   ((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r)) &&&
```
```   261   ((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r)) &&&
```
```   262   ((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r)) &&&
```
```   263   ((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r)) &&&
```
```   264   ((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
```
```   265   ((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
```
```   266   ((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y  + b >= r)) &&&
```
```   267   ((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r)) &&&
```
```   268   ((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y  + c >= r)) &&&
```
```   269   ((1 * min x y >= r) = (1 * x >= r & 1 * y >= r)) &&&
```
```   270   ((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
```
```   271   ((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
```
```   272   ((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y  + b >= r) )&&&
```
```   273   ((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r)) &&&
```
```   274   ((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y  + c >= r)) &&&
```
```   275   ((min x y >= r) = (x >= r &  y >= r)) &&&
```
```   276   ((min x y + a >= r) = (a + x >= r & a + y >= r)) &&&
```
```   277   ((a + min x y >= r) = (a + x >= r & a + y >= r)) &&&
```
```   278   ((a + min x y + b >= r) = (a + x + b >= r & a + y  + b >= r)) &&&
```
```   279   ((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r) )&&&
```
```   280   ((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r)) &&&
```
```   281   ((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r)) &&&
```
```   282   ((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
```
```   283   ((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
```
```   284   ((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r)) &&&
```
```   285   ((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r)) &&&
```
```   286   ((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r)) &&&
```
```   287   ((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r)) &&&
```
```   288   ((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
```
```   289   ((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
```
```   290   ((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y  + b > r)) &&&
```
```   291   ((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r)) &&&
```
```   292   ((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y  + c > r)) &&&
```
```   293   ((min x y > r) = (x > r &  y > r)) &&&
```
```   294   ((min x y + a > r) = (a + x > r & a + y > r)) &&&
```
```   295   ((a + min x y > r) = (a + x > r & a + y > r)) &&&
```
```   296   ((a + min x y + b > r) = (a + x + b > r & a + y  + b > r)) &&&
```
```   297   ((a + b + min x y > r) = (a + b + x > r & a + b + y > r)) &&&
```
```   298   ((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
```
```   299   by auto};
```
```   300
```
```   301 val abs_split' = @{lemma "P (abs (x::'a::ordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
```
```   302   by (atomize (full)) (auto split add: abs_split)};
```
```   303
```
```   304 val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
```
```   305   by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
```
```   306
```
```   307 val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
```
```   308   by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
```
```   309
```
```   310
```
```   311          (* Miscalineous *)
```
```   312 fun literals_conv bops uops cv =
```
```   313  let fun h t =
```
```   314   case (term_of t) of
```
```   315    b\$_\$_ => if member (op aconv) bops b then binop_conv h t else cv t
```
```   316  | u\$_ => if member (op aconv) uops u then arg_conv h t else cv t
```
```   317  | _ => cv t
```
```   318  in h end;
```
```   319
```
```   320 fun cterm_of_rat x =
```
```   321 let val (a, b) = Rat.quotient_of_rat x
```
```   322 in
```
```   323  if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
```
```   324   else Thm.capply (Thm.capply @{cterm "op / :: real => _"}
```
```   325                    (Numeral.mk_cnumber @{ctyp "real"} a))
```
```   326         (Numeral.mk_cnumber @{ctyp "real"} b)
```
```   327 end;
```
```   328
```
```   329   fun dest_ratconst t = case term_of t of
```
```   330    Const(@{const_name divide}, _)\$a\$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
```
```   331  | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
```
```   332  fun is_ratconst t = can dest_ratconst t
```
```   333
```
```   334 fun find_term p t = if p t then t else
```
```   335  case t of
```
```   336   a\$b => (find_term p a handle TERM _ => find_term p b)
```
```   337  | Abs (_,_,t') => find_term p t'
```
```   338  | _ => raise TERM ("find_term",[t]);
```
```   339
```
```   340 fun find_cterm p t = if p t then t else
```
```   341  case term_of t of
```
```   342   a\$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
```
```   343  | Abs (_,_,t') => find_cterm p (Thm.dest_abs NONE t |> snd)
```
```   344  | _ => raise CTERM ("find_cterm",[t]);
```
```   345
```
```   346     (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
```
```   347 fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
```
```   348 fun is_comb t = case (term_of t) of _\$_ => true | _ => false;
```
```   349
```
```   350 fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
```
```   351   handle CTERM _ => false;
```
```   352
```
```   353
```
```   354 (* Map back polynomials to HOL.                         *)
```
```   355
```
```   356 local
```
```   357  open Thm Numeral
```
```   358 in
```
```   359
```
```   360 fun cterm_of_varpow x k = if k = 1 then x else capply (capply @{cterm "op ^ :: real => _"} x)
```
```   361   (mk_cnumber @{ctyp nat} k)
```
```   362
```
```   363 fun cterm_of_monomial m =
```
```   364  if Ctermfunc.is_undefined m then @{cterm "1::real"}
```
```   365  else
```
```   366   let
```
```   367    val m' = dest_monomial m
```
```   368    val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' []
```
```   369   in foldr1 (fn (s, t) => capply (capply @{cterm "op * :: real => _"} s) t) vps
```
```   370   end
```
```   371
```
```   372 fun cterm_of_cmonomial (m,c) = if Ctermfunc.is_undefined m then cterm_of_rat c
```
```   373     else if c = Rat.one then cterm_of_monomial m
```
```   374     else capply (capply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
```
```   375
```
```   376 fun cterm_of_poly p =
```
```   377  if Monomialfunc.is_undefined p then @{cterm "0::real"}
```
```   378  else
```
```   379   let
```
```   380    val cms = map cterm_of_cmonomial
```
```   381      (sort (prod_ord monomial_order (K EQUAL)) (Monomialfunc.graph p))
```
```   382   in foldr1 (fn (t1, t2) => capply(capply @{cterm "op + :: real => _"} t1) t2) cms
```
```   383   end;
```
```   384
```
```   385 end;
```
```   386     (* A general real arithmetic prover *)
```
```   387
```
```   388 fun gen_gen_real_arith ctxt (mk_numeric,
```
```   389        numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
```
```   390        poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
```
```   391        absconv1,absconv2,prover) =
```
```   392 let
```
```   393  open Conv Thm;
```
```   394  val _ = my_context := ctxt
```
```   395  val _ = (my_mk_numeric := mk_numeric ; my_numeric_eq_conv := numeric_eq_conv ;
```
```   396           my_numeric_ge_conv := numeric_ge_conv; my_numeric_gt_conv := numeric_gt_conv ;
```
```   397           my_poly_conv := poly_conv; my_poly_neg_conv := poly_neg_conv;
```
```   398           my_poly_add_conv := poly_add_conv; my_poly_mul_conv := poly_mul_conv)
```
```   399  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}]
```
```   400  val prenex_ss = HOL_basic_ss addsimps prenex_simps
```
```   401  val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
```
```   402  val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
```
```   403  val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
```
```   404  val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
```
```   405  val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
```
```   406  val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
```
```   407  fun eqT_elim th = equal_elim (symmetric th) @{thm TrueI}
```
```   408  fun oprconv cv ct =
```
```   409   let val g = Thm.dest_fun2 ct
```
```   410   in if g aconvc @{cterm "op <= :: real => _"}
```
```   411        orelse g aconvc @{cterm "op < :: real => _"}
```
```   412      then arg_conv cv ct else arg1_conv cv ct
```
```   413   end
```
```   414
```
```   415  fun real_ineq_conv th ct =
```
```   416   let
```
```   417    val th' = (instantiate (match (lhs_of th, ct)) th
```
```   418       handle MATCH => raise CTERM ("real_ineq_conv", [ct]))
```
```   419   in transitive th' (oprconv poly_conv (Thm.rhs_of th'))
```
```   420   end
```
```   421   val [real_lt_conv, real_le_conv, real_eq_conv,
```
```   422        real_not_lt_conv, real_not_le_conv, _] =
```
```   423        map real_ineq_conv pth
```
```   424   fun match_mp_rule ths ths' =
```
```   425    let
```
```   426      fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
```
```   427       | th::ths => (ths' MRS th handle THM _ => f ths ths')
```
```   428    in f ths ths' end
```
```   429   fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
```
```   430          (match_mp_rule pth_mul [th, th'])
```
```   431   fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
```
```   432          (match_mp_rule pth_add [th, th'])
```
```   433   fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
```
```   434        (instantiate' [] [SOME ct] (th RS pth_emul))
```
```   435   fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
```
```   436        (instantiate' [] [SOME t] pth_square)
```
```   437
```
```   438   fun hol_of_positivstellensatz(eqs,les,lts) proof =
```
```   439    let
```
```   440     val _ = (my_eqs := eqs ; my_les := les ; my_lts := lts ; my_proof := proof)
```
```   441     fun translate prf = case prf of
```
```   442         Axiom_eq n => nth eqs n
```
```   443       | Axiom_le n => nth les n
```
```   444       | Axiom_lt n => nth lts n
```
```   445       | Rational_eq x => eqT_elim(numeric_eq_conv(capply @{cterm Trueprop}
```
```   446                           (capply (capply @{cterm "op =::real => _"} (mk_numeric x))
```
```   447                                @{cterm "0::real"})))
```
```   448       | Rational_le x => eqT_elim(numeric_ge_conv(capply @{cterm Trueprop}
```
```   449                           (capply (capply @{cterm "op <=::real => _"}
```
```   450                                      @{cterm "0::real"}) (mk_numeric x))))
```
```   451       | Rational_lt x => eqT_elim(numeric_gt_conv(capply @{cterm Trueprop}
```
```   452                       (capply (capply @{cterm "op <::real => _"} @{cterm "0::real"})
```
```   453                         (mk_numeric x))))
```
```   454       | Square pt => square_rule (cterm_of_poly pt)
```
```   455       | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
```
```   456       | Sum(p1,p2) => add_rule (translate p1) (translate p2)
```
```   457       | Product(p1,p2) => mul_rule (translate p1) (translate p2)
```
```   458    in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv])
```
```   459           (translate proof)
```
```   460    end
```
```   461
```
```   462   val init_conv = presimp_conv then_conv
```
```   463       nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
```
```   464       weak_dnf_conv
```
```   465
```
```   466   val concl = dest_arg o cprop_of
```
```   467   fun is_binop opr ct = (dest_fun2 ct aconvc opr handle CTERM _ => false)
```
```   468   val is_req = is_binop @{cterm "op =:: real => _"}
```
```   469   val is_ge = is_binop @{cterm "op <=:: real => _"}
```
```   470   val is_gt = is_binop @{cterm "op <:: real => _"}
```
```   471   val is_conj = is_binop @{cterm "op &"}
```
```   472   val is_disj = is_binop @{cterm "op |"}
```
```   473   fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
```
```   474   fun disj_cases th th1 th2 =
```
```   475    let val (p,q) = dest_binop (concl th)
```
```   476        val c = concl th1
```
```   477        val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
```
```   478    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)
```
```   479    end
```
```   480  fun overall cert_choice dun ths = case ths of
```
```   481   [] =>
```
```   482    let
```
```   483     val (eq,ne) = List.partition (is_req o concl) dun
```
```   484      val (le,nl) = List.partition (is_ge o concl) ne
```
```   485      val lt = filter (is_gt o concl) nl
```
```   486     in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
```
```   487  | th::oths =>
```
```   488    let
```
```   489     val ct = concl th
```
```   490    in
```
```   491     if is_conj ct  then
```
```   492      let
```
```   493       val (th1,th2) = conj_pair th in
```
```   494       overall cert_choice dun (th1::th2::oths) end
```
```   495     else if is_disj ct then
```
```   496       let
```
```   497        val (th1, cert1) = overall (Left::cert_choice) dun (assume (capply @{cterm Trueprop} (dest_arg1 ct))::oths)
```
```   498        val (th2, cert2) = overall (Right::cert_choice) dun (assume (capply @{cterm Trueprop} (dest_arg ct))::oths)
```
```   499       in (disj_cases th th1 th2, Branch (cert1, cert2)) end
```
```   500    else overall cert_choice (th::dun) oths
```
```   501   end
```
```   502   fun dest_binary b ct = if is_binop b ct then dest_binop ct
```
```   503                          else raise CTERM ("dest_binary",[b,ct])
```
```   504   val dest_eq = dest_binary @{cterm "op = :: real => _"}
```
```   505   val neq_th = nth pth 5
```
```   506   fun real_not_eq_conv ct =
```
```   507    let
```
```   508     val (l,r) = dest_eq (dest_arg ct)
```
```   509     val th = instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
```
```   510     val th_p = poly_conv(dest_arg(dest_arg1(rhs_of th)))
```
```   511     val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
```
```   512     val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
```
```   513     val th' = Drule.binop_cong_rule @{cterm "op |"}
```
```   514      (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
```
```   515      (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
```
```   516     in transitive th th'
```
```   517   end
```
```   518  fun equal_implies_1_rule PQ =
```
```   519   let
```
```   520    val P = lhs_of PQ
```
```   521   in implies_intr P (equal_elim PQ (assume P))
```
```   522   end
```
```   523  (* FIXME!!! Copied from groebner.ml *)
```
```   524  val strip_exists =
```
```   525   let fun h (acc, t) =
```
```   526    case (term_of t) of
```
```   527     Const("Ex",_)\$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
```
```   528   | _ => (acc,t)
```
```   529   in fn t => h ([],t)
```
```   530   end
```
```   531   fun name_of x = case term_of x of
```
```   532    Free(s,_) => s
```
```   533  | Var ((s,_),_) => s
```
```   534  | _ => "x"
```
```   535
```
```   536   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)
```
```   537
```
```   538   val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
```
```   539
```
```   540  fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
```
```   541  fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
```
```   542
```
```   543  fun choose v th th' = case concl_of th of
```
```   544    @{term Trueprop} \$ (Const("Ex",_)\$_) =>
```
```   545     let
```
```   546      val p = (funpow 2 Thm.dest_arg o cprop_of) th
```
```   547      val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
```
```   548      val th0 = fconv_rule (Thm.beta_conversion true)
```
```   549          (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
```
```   550      val pv = (Thm.rhs_of o Thm.beta_conversion true)
```
```   551            (Thm.capply @{cterm Trueprop} (Thm.capply p v))
```
```   552      val th1 = forall_intr v (implies_intr pv th')
```
```   553     in implies_elim (implies_elim th0 th) th1  end
```
```   554  | _ => raise THM ("choose",0,[th, th'])
```
```   555
```
```   556   fun simple_choose v th =
```
```   557      choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
```
```   558
```
```   559  val strip_forall =
```
```   560   let fun h (acc, t) =
```
```   561    case (term_of t) of
```
```   562     Const("All",_)\$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
```
```   563   | _ => (acc,t)
```
```   564   in fn t => h ([],t)
```
```   565   end
```
```   566
```
```   567  fun f ct =
```
```   568   let
```
```   569    val nnf_norm_conv' =
```
```   570      nnf_conv then_conv
```
```   571      literals_conv [@{term "op &"}, @{term "op |"}] []
```
```   572      (More_Conv.cache_conv
```
```   573        (first_conv [real_lt_conv, real_le_conv,
```
```   574                     real_eq_conv, real_not_lt_conv,
```
```   575                     real_not_le_conv, real_not_eq_conv, all_conv]))
```
```   576   fun absremover ct = (literals_conv [@{term "op &"}, @{term "op |"}] []
```
```   577                   (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv
```
```   578         try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
```
```   579   val nct = capply @{cterm Trueprop} (capply @{cterm "Not"} ct)
```
```   580   val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
```
```   581   val tm0 = dest_arg (rhs_of th0)
```
```   582   val (th, certificates) = if tm0 aconvc @{cterm False} then (equal_implies_1_rule th0, Trivial) else
```
```   583    let
```
```   584     val (evs,bod) = strip_exists tm0
```
```   585     val (avs,ibod) = strip_forall bod
```
```   586     val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
```
```   587     val (th2, certs) = overall [] [] [specl avs (assume (rhs_of th1))]
```
```   588     val th3 = fold simple_choose evs (prove_hyp (equal_elim th1 (assume (capply @{cterm Trueprop} bod))) th2)
```
```   589    in (Drule.implies_intr_hyps (prove_hyp (equal_elim th0 (assume nct)) th3), certs)
```
```   590    end
```
```   591   in (implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates)
```
```   592  end
```
```   593 in f
```
```   594 end;
```
```   595
```
```   596 (* A linear arithmetic prover *)
```
```   597 local
```
```   598   val linear_add = Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
```
```   599   fun linear_cmul c = Ctermfunc.mapf (fn x => c */ x)
```
```   600   val one_tm = @{cterm "1::real"}
```
```   601   fun contradictory p (e,_) = ((Ctermfunc.is_undefined e) andalso not(p Rat.zero)) orelse
```
```   602      ((gen_eq_set (op aconvc) (Ctermfunc.dom e, [one_tm])) andalso not(p(Ctermfunc.apply e one_tm)))
```
```   603
```
```   604   fun linear_ineqs vars (les,lts) =
```
```   605    case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
```
```   606     SOME r => r
```
```   607   | NONE =>
```
```   608    (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
```
```   609      SOME r => r
```
```   610    | NONE =>
```
```   611      if null vars then error "linear_ineqs: no contradiction" else
```
```   612      let
```
```   613       val ineqs = les @ lts
```
```   614       fun blowup v =
```
```   615        length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
```
```   616        length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
```
```   617        length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
```
```   618       val  v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
```
```   619                  (map (fn v => (v,blowup v)) vars)))
```
```   620       fun addup (e1,p1) (e2,p2) acc =
```
```   621        let
```
```   622         val c1 = Ctermfunc.tryapplyd e1 v Rat.zero
```
```   623         val c2 = Ctermfunc.tryapplyd e2 v Rat.zero
```
```   624        in if c1 */ c2 >=/ Rat.zero then acc else
```
```   625         let
```
```   626          val e1' = linear_cmul (Rat.abs c2) e1
```
```   627          val e2' = linear_cmul (Rat.abs c1) e2
```
```   628          val p1' = Product(Rational_lt(Rat.abs c2),p1)
```
```   629          val p2' = Product(Rational_lt(Rat.abs c1),p2)
```
```   630         in (linear_add e1' e2',Sum(p1',p2'))::acc
```
```   631         end
```
```   632        end
```
```   633       val (les0,les1) =
```
```   634          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
```
```   635       val (lts0,lts1) =
```
```   636          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
```
```   637       val (lesp,lesn) =
```
```   638          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
```
```   639       val (ltsp,ltsn) =
```
```   640          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
```
```   641       val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
```
```   642       val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
```
```   643                       (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
```
```   644      in linear_ineqs (remove (op aconvc) v vars) (les',lts')
```
```   645      end)
```
```   646
```
```   647   fun linear_eqs(eqs,les,lts) =
```
```   648    case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
```
```   649     SOME r => r
```
```   650   | NONE => (case eqs of
```
```   651     [] =>
```
```   652      let val vars = remove (op aconvc) one_tm
```
```   653            (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom o fst) (les@lts) [])
```
```   654      in linear_ineqs vars (les,lts) end
```
```   655    | (e,p)::es =>
```
```   656      if Ctermfunc.is_undefined e then linear_eqs (es,les,lts) else
```
```   657      let
```
```   658       val (x,c) = Ctermfunc.choose (Ctermfunc.undefine one_tm e)
```
```   659       fun xform (inp as (t,q)) =
```
```   660        let val d = Ctermfunc.tryapplyd t x Rat.zero in
```
```   661         if d =/ Rat.zero then inp else
```
```   662         let
```
```   663          val k = (Rat.neg d) */ Rat.abs c // c
```
```   664          val e' = linear_cmul k e
```
```   665          val t' = linear_cmul (Rat.abs c) t
```
```   666          val p' = Eqmul(Monomialfunc.onefunc (Ctermfunc.undefined, k),p)
```
```   667          val q' = Product(Rational_lt(Rat.abs c),q)
```
```   668         in (linear_add e' t',Sum(p',q'))
```
```   669         end
```
```   670       end
```
```   671      in linear_eqs(map xform es,map xform les,map xform lts)
```
```   672      end)
```
```   673
```
```   674   fun linear_prover (eq,le,lt) =
```
```   675    let
```
```   676     val eqs = map2 (fn p => fn n => (p,Axiom_eq n)) eq (0 upto (length eq - 1))
```
```   677     val les = map2 (fn p => fn n => (p,Axiom_le n)) le (0 upto (length le - 1))
```
```   678     val lts = map2 (fn p => fn n => (p,Axiom_lt n)) lt (0 upto (length lt - 1))
```
```   679    in linear_eqs(eqs,les,lts)
```
```   680    end
```
```   681
```
```   682   fun lin_of_hol ct =
```
```   683    if ct aconvc @{cterm "0::real"} then Ctermfunc.undefined
```
```   684    else if not (is_comb ct) then Ctermfunc.onefunc (ct, Rat.one)
```
```   685    else if is_ratconst ct then Ctermfunc.onefunc (one_tm, dest_ratconst ct)
```
```   686    else
```
```   687     let val (lop,r) = Thm.dest_comb ct
```
```   688     in if not (is_comb lop) then Ctermfunc.onefunc (ct, Rat.one)
```
```   689        else
```
```   690         let val (opr,l) = Thm.dest_comb lop
```
```   691         in if opr aconvc @{cterm "op + :: real =>_"}
```
```   692            then linear_add (lin_of_hol l) (lin_of_hol r)
```
```   693            else if opr aconvc @{cterm "op * :: real =>_"}
```
```   694                    andalso is_ratconst l then Ctermfunc.onefunc (r, dest_ratconst l)
```
```   695            else Ctermfunc.onefunc (ct, Rat.one)
```
```   696         end
```
```   697     end
```
```   698
```
```   699   fun is_alien ct = case term_of ct of
```
```   700    Const(@{const_name "real"}, _)\$ n =>
```
```   701      if can HOLogic.dest_number n then false else true
```
```   702   | _ => false
```
```   703  open Thm
```
```   704 in
```
```   705 fun real_linear_prover translator (eq,le,lt) =
```
```   706  let
```
```   707   val lhs = lin_of_hol o dest_arg1 o dest_arg o cprop_of
```
```   708   val rhs = lin_of_hol o dest_arg o dest_arg o cprop_of
```
```   709   val eq_pols = map lhs eq
```
```   710   val le_pols = map rhs le
```
```   711   val lt_pols = map rhs lt
```
```   712   val aliens =  filter is_alien
```
```   713       (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom)
```
```   714           (eq_pols @ le_pols @ lt_pols) [])
```
```   715   val le_pols' = le_pols @ map (fn v => Ctermfunc.onefunc (v,Rat.one)) aliens
```
```   716   val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
```
```   717   val le' = le @ map (fn a => instantiate' [] [SOME (dest_arg a)] @{thm real_of_nat_ge_zero}) aliens
```
```   718  in ((translator (eq,le',lt) proof), Trivial)
```
```   719  end
```
```   720 end;
```
```   721
```
```   722 (* A less general generic arithmetic prover dealing with abs,max and min*)
```
```   723
```
```   724 local
```
```   725  val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
```
```   726  fun absmaxmin_elim_conv1 ctxt =
```
```   727     Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
```
```   728
```
```   729  val absmaxmin_elim_conv2 =
```
```   730   let
```
```   731    val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
```
```   732    val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
```
```   733    val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
```
```   734    val abs_tm = @{cterm "abs :: real => _"}
```
```   735    val p_tm = @{cpat "?P :: real => bool"}
```
```   736    val x_tm = @{cpat "?x :: real"}
```
```   737    val y_tm = @{cpat "?y::real"}
```
```   738    val is_max = is_binop @{cterm "max :: real => _"}
```
```   739    val is_min = is_binop @{cterm "min :: real => _"}
```
```   740    fun is_abs t = is_comb t andalso dest_fun t aconvc abs_tm
```
```   741    fun eliminate_construct p c tm =
```
```   742     let
```
```   743      val t = find_cterm p tm
```
```   744      val th0 = (symmetric o beta_conversion false) (capply (cabs t tm) t)
```
```   745      val (p,ax) = (dest_comb o rhs_of) th0
```
```   746     in fconv_rule(arg_conv(binop_conv (arg_conv (beta_conversion false))))
```
```   747                (transitive th0 (c p ax))
```
```   748    end
```
```   749
```
```   750    val elim_abs = eliminate_construct is_abs
```
```   751     (fn p => fn ax =>
```
```   752        instantiate ([], [(p_tm,p), (x_tm, dest_arg ax)]) pth_abs)
```
```   753    val elim_max = eliminate_construct is_max
```
```   754     (fn p => fn ax =>
```
```   755       let val (ax,y) = dest_comb ax
```
```   756       in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)])
```
```   757       pth_max end)
```
```   758    val elim_min = eliminate_construct is_min
```
```   759     (fn p => fn ax =>
```
```   760       let val (ax,y) = dest_comb ax
```
```   761       in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)])
```
```   762       pth_min end)
```
```   763    in first_conv [elim_abs, elim_max, elim_min, all_conv]
```
```   764   end;
```
```   765 in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
```
```   766         gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,
```
```   767                        absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
```
```   768 end;
```
```   769
```
```   770 (* An instance for reals*)
```
```   771
```
```   772 fun gen_prover_real_arith ctxt prover =
```
```   773  let
```
```   774   fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS
```
```   775   val {add,mul,neg,pow,sub,main} =
```
```   776      Normalizer.semiring_normalizers_ord_wrapper ctxt
```
```   777       (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
```
```   778      simple_cterm_ord
```
```   779 in gen_real_arith ctxt
```
```   780    (cterm_of_rat, field_comp_conv, field_comp_conv,field_comp_conv,
```
```   781     main,neg,add,mul, prover)
```
```   782 end;
```
```   783
```
```   784 end
```