src/HOL/Library/positivstellensatz.ML
 author haftmann Thu Aug 19 16:08:59 2010 +0200 (2010-08-19) changeset 38558 32ad17fe2b9c parent 38549 d0385f2764d8 child 38795 848be46708dc permissions -rw-r--r--
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```     1 (*  Title:      HOL/Library/positivstellensatz.ML
```
```     2     Author:     Amine Chaieb, University of Cambridge
```
```     3
```
```     4 A generic arithmetic prover based on Positivstellensatz certificates
```
```     5 --- also implements Fourrier-Motzkin elimination as a special case
```
```     6 Fourrier-Motzkin elimination.
```
```     7 *)
```
```     8
```
```     9 (* A functor for finite mappings based on Tables *)
```
```    10
```
```    11 signature FUNC =
```
```    12 sig
```
```    13  include TABLE
```
```    14  val apply : 'a table -> key -> 'a
```
```    15  val applyd :'a table -> (key -> 'a) -> key -> 'a
```
```    16  val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a table -> 'a table -> 'a table
```
```    17  val dom : 'a table -> key list
```
```    18  val tryapplyd : 'a table -> key -> 'a -> 'a
```
```    19  val updatep : (key * 'a -> bool) -> key * 'a -> 'a table -> 'a table
```
```    20  val choose : 'a table -> key * 'a
```
```    21  val onefunc : key * 'a -> 'a table
```
```    22 end;
```
```    23
```
```    24 functor FuncFun(Key: KEY) : FUNC=
```
```    25 struct
```
```    26
```
```    27 structure Tab = Table(Key);
```
```    28
```
```    29 open Tab;
```
```    30
```
```    31 fun dom a = sort Key.ord (Tab.keys a);
```
```    32 fun applyd f d x = case Tab.lookup f x of
```
```    33    SOME y => y
```
```    34  | NONE => d x;
```
```    35
```
```    36 fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
```
```    37 fun tryapplyd f a d = applyd f (K d) a;
```
```    38 fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
```
```    39 fun combine f z a b =
```
```    40  let
```
```    41   fun h (k,v) t = case Tab.lookup t k of
```
```    42      NONE => Tab.update (k,v) t
```
```    43    | SOME v' => let val w = f v v'
```
```    44      in if z w then Tab.delete k t else Tab.update (k,w) t end;
```
```    45   in Tab.fold h a b end;
```
```    46
```
```    47 fun choose f = case Tab.min_key f of
```
```    48    SOME k => (k, the (Tab.lookup f k))
```
```    49  | NONE => error "FuncFun.choose : Completely empty function"
```
```    50
```
```    51 fun onefunc kv = update kv empty
```
```    52
```
```    53 end;
```
```    54
```
```    55 (* Some standard functors and utility functions for them *)
```
```    56
```
```    57 structure FuncUtil =
```
```    58 struct
```
```    59
```
```    60 structure Intfunc = FuncFun(type key = int val ord = int_ord);
```
```    61 structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
```
```    62 structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
```
```    63 structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
```
```    64 structure Termfunc = FuncFun(type key = term val ord = Term_Ord.fast_term_ord);
```
```    65
```
```    66 val cterm_ord = Term_Ord.fast_term_ord o pairself term_of
```
```    67
```
```    68 structure Ctermfunc = FuncFun(type key = cterm val ord = cterm_ord);
```
```    69
```
```    70 type monomial = int Ctermfunc.table;
```
```    71
```
```    72 val monomial_ord = list_ord (prod_ord cterm_ord int_ord) o pairself Ctermfunc.dest
```
```    73
```
```    74 structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
```
```    75
```
```    76 type poly = Rat.rat Monomialfunc.table;
```
```    77
```
```    78 (* The ordering so we can create canonical HOL polynomials.                  *)
```
```    79
```
```    80 fun dest_monomial mon = sort (cterm_ord o pairself fst) (Ctermfunc.dest mon);
```
```    81
```
```    82 fun monomial_order (m1,m2) =
```
```    83  if Ctermfunc.is_empty m2 then LESS
```
```    84  else if Ctermfunc.is_empty m1 then GREATER
```
```    85  else
```
```    86   let val mon1 = dest_monomial m1
```
```    87       val mon2 = dest_monomial m2
```
```    88       val deg1 = fold (Integer.add o snd) mon1 0
```
```    89       val deg2 = fold (Integer.add o snd) mon2 0
```
```    90   in if deg1 < deg2 then GREATER else if deg1 > deg2 then LESS
```
```    91      else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
```
```    92   end;
```
```    93
```
```    94 end
```
```    95
```
```    96 (* positivstellensatz datatype and prover generation *)
```
```    97
```
```    98 signature REAL_ARITH =
```
```    99 sig
```
```   100
```
```   101   datatype positivstellensatz =
```
```   102    Axiom_eq of int
```
```   103  | Axiom_le of int
```
```   104  | Axiom_lt of int
```
```   105  | Rational_eq of Rat.rat
```
```   106  | Rational_le of Rat.rat
```
```   107  | Rational_lt of Rat.rat
```
```   108  | Square of FuncUtil.poly
```
```   109  | Eqmul of FuncUtil.poly * positivstellensatz
```
```   110  | Sum of positivstellensatz * positivstellensatz
```
```   111  | Product of positivstellensatz * positivstellensatz;
```
```   112
```
```   113 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
```
```   114
```
```   115 datatype tree_choice = Left | Right
```
```   116
```
```   117 type prover = tree_choice list ->
```
```   118   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
```
```   119   thm list * thm list * thm list -> thm * pss_tree
```
```   120 type cert_conv = cterm -> thm * pss_tree
```
```   121
```
```   122 val gen_gen_real_arith :
```
```   123   Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
```
```   124    conv * conv * conv * conv * conv * conv * prover -> cert_conv
```
```   125 val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
```
```   126   thm list * thm list * thm list -> thm * pss_tree
```
```   127
```
```   128 val gen_real_arith : Proof.context ->
```
```   129   (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
```
```   130
```
```   131 val gen_prover_real_arith : Proof.context -> prover -> cert_conv
```
```   132
```
```   133 val is_ratconst : cterm -> bool
```
```   134 val dest_ratconst : cterm -> Rat.rat
```
```   135 val cterm_of_rat : Rat.rat -> cterm
```
```   136
```
```   137 end
```
```   138
```
```   139 structure RealArith : REAL_ARITH =
```
```   140 struct
```
```   141
```
```   142  open Conv
```
```   143 (* ------------------------------------------------------------------------- *)
```
```   144 (* Data structure for Positivstellensatz refutations.                        *)
```
```   145 (* ------------------------------------------------------------------------- *)
```
```   146
```
```   147 datatype positivstellensatz =
```
```   148    Axiom_eq of int
```
```   149  | Axiom_le of int
```
```   150  | Axiom_lt of int
```
```   151  | Rational_eq of Rat.rat
```
```   152  | Rational_le of Rat.rat
```
```   153  | Rational_lt of Rat.rat
```
```   154  | Square of FuncUtil.poly
```
```   155  | Eqmul of FuncUtil.poly * positivstellensatz
```
```   156  | Sum of positivstellensatz * positivstellensatz
```
```   157  | Product of positivstellensatz * positivstellensatz;
```
```   158          (* Theorems used in the procedure *)
```
```   159
```
```   160 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
```
```   161 datatype tree_choice = Left | Right
```
```   162 type prover = tree_choice list ->
```
```   163   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
```
```   164   thm list * thm list * thm list -> thm * pss_tree
```
```   165 type cert_conv = cterm -> thm * pss_tree
```
```   166
```
```   167 val my_eqs = Unsynchronized.ref ([] : thm list);
```
```   168 val my_les = Unsynchronized.ref ([] : thm list);
```
```   169 val my_lts = Unsynchronized.ref ([] : thm list);
```
```   170 val my_proof = Unsynchronized.ref (Axiom_eq 0);
```
```   171 val my_context = Unsynchronized.ref @{context};
```
```   172
```
```   173 val my_mk_numeric = Unsynchronized.ref ((K @{cterm True}) :Rat.rat -> cterm);
```
```   174 val my_numeric_eq_conv = Unsynchronized.ref no_conv;
```
```   175 val my_numeric_ge_conv = Unsynchronized.ref no_conv;
```
```   176 val my_numeric_gt_conv = Unsynchronized.ref no_conv;
```
```   177 val my_poly_conv = Unsynchronized.ref no_conv;
```
```   178 val my_poly_neg_conv = Unsynchronized.ref no_conv;
```
```   179 val my_poly_add_conv = Unsynchronized.ref no_conv;
```
```   180 val my_poly_mul_conv = Unsynchronized.ref no_conv;
```
```   181
```
```   182
```
```   183     (* Some useful derived rules *)
```
```   184 fun deduct_antisym_rule tha thb =
```
```   185     Thm.equal_intr (Thm.implies_intr (cprop_of thb) tha)
```
```   186      (Thm.implies_intr (cprop_of tha) thb);
```
```   187
```
```   188 fun prove_hyp tha thb =
```
```   189   if exists (curry op aconv (concl_of tha)) (#hyps (rep_thm thb))
```
```   190   then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb;
```
```   191
```
```   192 val pth = @{lemma "(((x::real) < y) == (y - x > 0))" and "((x <= y) == (y - x >= 0))" and
```
```   193      "((x = y) == (x - y = 0))" and "((~(x < y)) == (x - y >= 0))" and
```
```   194      "((~(x <= y)) == (x - y > 0))" and "((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
```
```   195   by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};
```
```   196
```
```   197 val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
```
```   198 val pth_add =
```
```   199   @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 )" and "( x = 0 ==> y >= 0 ==> x + y >= 0)" and
```
```   200     "(x = 0 ==> y > 0 ==> x + y > 0)" and "(x >= 0 ==> y = 0 ==> x + y >= 0)" and
```
```   201     "(x >= 0 ==> y >= 0 ==> x + y >= 0)" and "(x >= 0 ==> y > 0 ==> x + y > 0)" and
```
```   202     "(x > 0 ==> y = 0 ==> x + y > 0)" and "(x > 0 ==> y >= 0 ==> x + y > 0)" and
```
```   203     "(x > 0 ==> y > 0 ==> x + y > 0)" by simp_all};
```
```   204
```
```   205 val pth_mul =
```
```   206   @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0)" and "(x = 0 ==> y >= 0 ==> x * y = 0)" and
```
```   207     "(x = 0 ==> y > 0 ==> x * y = 0)" and "(x >= 0 ==> y = 0 ==> x * y = 0)" and
```
```   208     "(x >= 0 ==> y >= 0 ==> x * y >= 0)" and "(x >= 0 ==> y > 0 ==> x * y >= 0)" and
```
```   209     "(x > 0 ==>  y = 0 ==> x * y = 0)" and "(x > 0 ==> y >= 0 ==> x * y >= 0)" and
```
```   210     "(x > 0 ==>  y > 0 ==> x * y > 0)"
```
```   211   by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
```
```   212     mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])};
```
```   213
```
```   214 val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
```
```   215 val pth_square = @{lemma "x * x >= (0::real)"  by simp};
```
```   216
```
```   217 val weak_dnf_simps =
```
```   218   List.take (simp_thms, 34) @
```
```   219     @{lemma "((P & (Q | R)) = ((P&Q) | (P&R)))" and "((Q | R) & P) = ((Q&P) | (R&P))" and
```
```   220       "(P & Q) = (Q & P)" and "((P | Q) = (Q | P))" by blast+};
```
```   221
```
```   222 val nnfD_simps =
```
```   223   @{lemma "((~(P & Q)) = (~P | ~Q))" and "((~(P | Q)) = (~P & ~Q) )" and
```
```   224     "((P --> Q) = (~P | Q) )" and "((P = Q) = ((P & Q) | (~P & ~ Q)))" and
```
```   225     "((~(P = Q)) = ((P & ~ Q) | (~P & Q)) )" and "((~ ~(P)) = P)" by blast+};
```
```   226
```
```   227 val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
```
```   228 val prenex_simps =
```
```   229   map (fn th => th RS sym)
```
```   230     ([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @
```
```   231       @{thms "HOL.all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});
```
```   232
```
```   233 val real_abs_thms1 = @{lemma
```
```   234   "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r))" and
```
```   235   "((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r))" and
```
```   236   "((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r))" and
```
```   237   "((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r))" and
```
```   238   "((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r))" and
```
```   239   "((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r))" and
```
```   240   "((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r))" and
```
```   241   "((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r))" and
```
```   242   "((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r))" and
```
```   243   "((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y  + b >= r))" and
```
```   244   "((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r))" and
```
```   245   "((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y  + c >= r))" and
```
```   246   "((1 * min x y >= r) = (1 * x >= r & 1 * y >= r))" and
```
```   247   "((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and
```
```   248   "((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and
```
```   249   "((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y  + b >= r))" and
```
```   250   "((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r))" and
```
```   251   "((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y  + c >= r))" and
```
```   252   "((min x y >= r) = (x >= r &  y >= r))" and
```
```   253   "((min x y + a >= r) = (a + x >= r & a + y >= r))" and
```
```   254   "((a + min x y >= r) = (a + x >= r & a + y >= r))" and
```
```   255   "((a + min x y + b >= r) = (a + x + b >= r & a + y  + b >= r))" and
```
```   256   "((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r))" and
```
```   257   "((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r))" and
```
```   258   "((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r))" and
```
```   259   "((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r))" and
```
```   260   "((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r))" and
```
```   261   "((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r))" and
```
```   262   "((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r))" and
```
```   263   "((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r))" and
```
```   264   "((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r))" and
```
```   265   "((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r))" and
```
```   266   "((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r))" and
```
```   267   "((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y  + b > r))" and
```
```   268   "((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r))" and
```
```   269   "((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y  + c > r))" and
```
```   270   "((min x y > r) = (x > r &  y > r))" and
```
```   271   "((min x y + a > r) = (a + x > r & a + y > r))" and
```
```   272   "((a + min x y > r) = (a + x > r & a + y > r))" and
```
```   273   "((a + min x y + b > r) = (a + x + b > r & a + y  + b > r))" and
```
```   274   "((a + b + min x y > r) = (a + b + x > r & a + b + y > r))" and
```
```   275   "((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
```
```   276   by auto};
```
```   277
```
```   278 val abs_split' = @{lemma "P (abs (x::'a::linordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
```
```   279   by (atomize (full)) (auto split add: abs_split)};
```
```   280
```
```   281 val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
```
```   282   by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
```
```   283
```
```   284 val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
```
```   285   by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
```
```   286
```
```   287
```
```   288          (* Miscalineous *)
```
```   289 fun literals_conv bops uops cv =
```
```   290  let fun h t =
```
```   291   case (term_of t) of
```
```   292    b\$_\$_ => if member (op aconv) bops b then binop_conv h t else cv t
```
```   293  | u\$_ => if member (op aconv) uops u then arg_conv h t else cv t
```
```   294  | _ => cv t
```
```   295  in h end;
```
```   296
```
```   297 fun cterm_of_rat x =
```
```   298 let val (a, b) = Rat.quotient_of_rat x
```
```   299 in
```
```   300  if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
```
```   301   else Thm.capply (Thm.capply @{cterm "op / :: real => _"}
```
```   302                    (Numeral.mk_cnumber @{ctyp "real"} a))
```
```   303         (Numeral.mk_cnumber @{ctyp "real"} b)
```
```   304 end;
```
```   305
```
```   306   fun dest_ratconst t = case term_of t of
```
```   307    Const(@{const_name divide}, _)\$a\$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
```
```   308  | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
```
```   309  fun is_ratconst t = can dest_ratconst t
```
```   310
```
```   311 fun find_term p t = if p t then t else
```
```   312  case t of
```
```   313   a\$b => (find_term p a handle TERM _ => find_term p b)
```
```   314  | Abs (_,_,t') => find_term p t'
```
```   315  | _ => raise TERM ("find_term",[t]);
```
```   316
```
```   317 fun find_cterm p t = if p t then t else
```
```   318  case term_of t of
```
```   319   a\$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
```
```   320  | Abs (_,_,t') => find_cterm p (Thm.dest_abs NONE t |> snd)
```
```   321  | _ => raise CTERM ("find_cterm",[t]);
```
```   322
```
```   323     (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
```
```   324 fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
```
```   325 fun is_comb t = case (term_of t) of _\$_ => true | _ => false;
```
```   326
```
```   327 fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
```
```   328   handle CTERM _ => false;
```
```   329
```
```   330
```
```   331 (* Map back polynomials to HOL.                         *)
```
```   332
```
```   333 fun cterm_of_varpow x k = if k = 1 then x else Thm.capply (Thm.capply @{cterm "op ^ :: real => _"} x)
```
```   334   (Numeral.mk_cnumber @{ctyp nat} k)
```
```   335
```
```   336 fun cterm_of_monomial m =
```
```   337  if FuncUtil.Ctermfunc.is_empty m then @{cterm "1::real"}
```
```   338  else
```
```   339   let
```
```   340    val m' = FuncUtil.dest_monomial m
```
```   341    val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' []
```
```   342   in foldr1 (fn (s, t) => Thm.capply (Thm.capply @{cterm "op * :: real => _"} s) t) vps
```
```   343   end
```
```   344
```
```   345 fun cterm_of_cmonomial (m,c) = if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
```
```   346     else if c = Rat.one then cterm_of_monomial m
```
```   347     else Thm.capply (Thm.capply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
```
```   348
```
```   349 fun cterm_of_poly p =
```
```   350  if FuncUtil.Monomialfunc.is_empty p then @{cterm "0::real"}
```
```   351  else
```
```   352   let
```
```   353    val cms = map cterm_of_cmonomial
```
```   354      (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
```
```   355   in foldr1 (fn (t1, t2) => Thm.capply(Thm.capply @{cterm "op + :: real => _"} t1) t2) cms
```
```   356   end;
```
```   357
```
```   358     (* A general real arithmetic prover *)
```
```   359
```
```   360 fun gen_gen_real_arith ctxt (mk_numeric,
```
```   361        numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
```
```   362        poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
```
```   363        absconv1,absconv2,prover) =
```
```   364 let
```
```   365  val _ = my_context := ctxt
```
```   366  val _ = (my_mk_numeric := mk_numeric ; my_numeric_eq_conv := numeric_eq_conv ;
```
```   367           my_numeric_ge_conv := numeric_ge_conv; my_numeric_gt_conv := numeric_gt_conv ;
```
```   368           my_poly_conv := poly_conv; my_poly_neg_conv := poly_neg_conv;
```
```   369           my_poly_add_conv := poly_add_conv; my_poly_mul_conv := poly_mul_conv)
```
```   370  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}]
```
```   371  val prenex_ss = HOL_basic_ss addsimps prenex_simps
```
```   372  val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
```
```   373  val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
```
```   374  val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
```
```   375  val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
```
```   376  val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
```
```   377  val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
```
```   378  fun eqT_elim th = Thm.equal_elim (Thm.symmetric th) @{thm TrueI}
```
```   379  fun oprconv cv ct =
```
```   380   let val g = Thm.dest_fun2 ct
```
```   381   in if g aconvc @{cterm "op <= :: real => _"}
```
```   382        orelse g aconvc @{cterm "op < :: real => _"}
```
```   383      then arg_conv cv ct else arg1_conv cv ct
```
```   384   end
```
```   385
```
```   386  fun real_ineq_conv th ct =
```
```   387   let
```
```   388    val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th
```
```   389       handle Pattern.MATCH => raise CTERM ("real_ineq_conv", [ct]))
```
```   390   in Thm.transitive th' (oprconv poly_conv (Thm.rhs_of th'))
```
```   391   end
```
```   392   val [real_lt_conv, real_le_conv, real_eq_conv,
```
```   393        real_not_lt_conv, real_not_le_conv, _] =
```
```   394        map real_ineq_conv pth
```
```   395   fun match_mp_rule ths ths' =
```
```   396    let
```
```   397      fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
```
```   398       | th::ths => (ths' MRS th handle THM _ => f ths ths')
```
```   399    in f ths ths' end
```
```   400   fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
```
```   401          (match_mp_rule pth_mul [th, th'])
```
```   402   fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
```
```   403          (match_mp_rule pth_add [th, th'])
```
```   404   fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
```
```   405        (instantiate' [] [SOME ct] (th RS pth_emul))
```
```   406   fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
```
```   407        (instantiate' [] [SOME t] pth_square)
```
```   408
```
```   409   fun hol_of_positivstellensatz(eqs,les,lts) proof =
```
```   410    let
```
```   411     val _ = (my_eqs := eqs ; my_les := les ; my_lts := lts ; my_proof := proof)
```
```   412     fun translate prf = case prf of
```
```   413         Axiom_eq n => nth eqs n
```
```   414       | Axiom_le n => nth les n
```
```   415       | Axiom_lt n => nth lts n
```
```   416       | Rational_eq x => eqT_elim(numeric_eq_conv(Thm.capply @{cterm Trueprop}
```
```   417                           (Thm.capply (Thm.capply @{cterm "op =::real => _"} (mk_numeric x))
```
```   418                                @{cterm "0::real"})))
```
```   419       | Rational_le x => eqT_elim(numeric_ge_conv(Thm.capply @{cterm Trueprop}
```
```   420                           (Thm.capply (Thm.capply @{cterm "op <=::real => _"}
```
```   421                                      @{cterm "0::real"}) (mk_numeric x))))
```
```   422       | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.capply @{cterm Trueprop}
```
```   423                       (Thm.capply (Thm.capply @{cterm "op <::real => _"} @{cterm "0::real"})
```
```   424                         (mk_numeric x))))
```
```   425       | Square pt => square_rule (cterm_of_poly pt)
```
```   426       | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
```
```   427       | Sum(p1,p2) => add_rule (translate p1) (translate p2)
```
```   428       | Product(p1,p2) => mul_rule (translate p1) (translate p2)
```
```   429    in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv])
```
```   430           (translate proof)
```
```   431    end
```
```   432
```
```   433   val init_conv = presimp_conv then_conv
```
```   434       nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
```
```   435       weak_dnf_conv
```
```   436
```
```   437   val concl = Thm.dest_arg o cprop_of
```
```   438   fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
```
```   439   val is_req = is_binop @{cterm "op =:: real => _"}
```
```   440   val is_ge = is_binop @{cterm "op <=:: real => _"}
```
```   441   val is_gt = is_binop @{cterm "op <:: real => _"}
```
```   442   val is_conj = is_binop @{cterm "op &"}
```
```   443   val is_disj = is_binop @{cterm "op |"}
```
```   444   fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
```
```   445   fun disj_cases th th1 th2 =
```
```   446    let val (p,q) = Thm.dest_binop (concl th)
```
```   447        val c = concl th1
```
```   448        val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
```
```   449    in Thm.implies_elim (Thm.implies_elim
```
```   450           (Thm.implies_elim (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
```
```   451           (Thm.implies_intr (Thm.capply @{cterm Trueprop} p) th1))
```
```   452         (Thm.implies_intr (Thm.capply @{cterm Trueprop} q) th2)
```
```   453    end
```
```   454  fun overall cert_choice dun ths = case ths of
```
```   455   [] =>
```
```   456    let
```
```   457     val (eq,ne) = List.partition (is_req o concl) dun
```
```   458      val (le,nl) = List.partition (is_ge o concl) ne
```
```   459      val lt = filter (is_gt o concl) nl
```
```   460     in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
```
```   461  | th::oths =>
```
```   462    let
```
```   463     val ct = concl th
```
```   464    in
```
```   465     if is_conj ct  then
```
```   466      let
```
```   467       val (th1,th2) = conj_pair th in
```
```   468       overall cert_choice dun (th1::th2::oths) end
```
```   469     else if is_disj ct then
```
```   470       let
```
```   471        val (th1, cert1) = overall (Left::cert_choice) dun (Thm.assume (Thm.capply @{cterm Trueprop} (Thm.dest_arg1 ct))::oths)
```
```   472        val (th2, cert2) = overall (Right::cert_choice) dun (Thm.assume (Thm.capply @{cterm Trueprop} (Thm.dest_arg ct))::oths)
```
```   473       in (disj_cases th th1 th2, Branch (cert1, cert2)) end
```
```   474    else overall cert_choice (th::dun) oths
```
```   475   end
```
```   476   fun dest_binary b ct = if is_binop b ct then Thm.dest_binop ct
```
```   477                          else raise CTERM ("dest_binary",[b,ct])
```
```   478   val dest_eq = dest_binary @{cterm "op = :: real => _"}
```
```   479   val neq_th = nth pth 5
```
```   480   fun real_not_eq_conv ct =
```
```   481    let
```
```   482     val (l,r) = dest_eq (Thm.dest_arg ct)
```
```   483     val th = Thm.instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
```
```   484     val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
```
```   485     val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
```
```   486     val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
```
```   487     val th' = Drule.binop_cong_rule @{cterm "op |"}
```
```   488      (Drule.arg_cong_rule (Thm.capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
```
```   489      (Drule.arg_cong_rule (Thm.capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
```
```   490     in Thm.transitive th th'
```
```   491   end
```
```   492  fun equal_implies_1_rule PQ =
```
```   493   let
```
```   494    val P = Thm.lhs_of PQ
```
```   495   in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P))
```
```   496   end
```
```   497  (* FIXME!!! Copied from groebner.ml *)
```
```   498  val strip_exists =
```
```   499   let fun h (acc, t) =
```
```   500    case (term_of t) of
```
```   501     Const(@{const_name Ex},_)\$Abs(x,T,p) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
```
```   502   | _ => (acc,t)
```
```   503   in fn t => h ([],t)
```
```   504   end
```
```   505   fun name_of x = case term_of x of
```
```   506    Free(s,_) => s
```
```   507  | Var ((s,_),_) => s
```
```   508  | _ => "x"
```
```   509
```
```   510   fun mk_forall x th = Drule.arg_cong_rule (instantiate_cterm' [SOME (ctyp_of_term x)] [] @{cpat "All :: (?'a => bool) => _" }) (Thm.abstract_rule (name_of x) x th)
```
```   511
```
```   512   val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
```
```   513
```
```   514  fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
```
```   515  fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
```
```   516
```
```   517  fun choose v th th' = case concl_of th of
```
```   518    @{term Trueprop} \$ (Const(@{const_name Ex},_)\$_) =>
```
```   519     let
```
```   520      val p = (funpow 2 Thm.dest_arg o cprop_of) th
```
```   521      val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
```
```   522      val th0 = fconv_rule (Thm.beta_conversion true)
```
```   523          (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
```
```   524      val pv = (Thm.rhs_of o Thm.beta_conversion true)
```
```   525            (Thm.capply @{cterm Trueprop} (Thm.capply p v))
```
```   526      val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
```
```   527     in Thm.implies_elim (Thm.implies_elim th0 th) th1  end
```
```   528  | _ => raise THM ("choose",0,[th, th'])
```
```   529
```
```   530   fun simple_choose v th =
```
```   531      choose v (Thm.assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
```
```   532
```
```   533  val strip_forall =
```
```   534   let fun h (acc, t) =
```
```   535    case (term_of t) of
```
```   536     Const(@{const_name All},_)\$Abs(x,T,p) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
```
```   537   | _ => (acc,t)
```
```   538   in fn t => h ([],t)
```
```   539   end
```
```   540
```
```   541  fun f ct =
```
```   542   let
```
```   543    val nnf_norm_conv' =
```
```   544      nnf_conv then_conv
```
```   545      literals_conv [@{term "op &"}, @{term "op |"}] []
```
```   546      (Conv.cache_conv
```
```   547        (first_conv [real_lt_conv, real_le_conv,
```
```   548                     real_eq_conv, real_not_lt_conv,
```
```   549                     real_not_le_conv, real_not_eq_conv, all_conv]))
```
```   550   fun absremover ct = (literals_conv [@{term "op &"}, @{term "op |"}] []
```
```   551                   (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv
```
```   552         try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
```
```   553   val nct = Thm.capply @{cterm Trueprop} (Thm.capply @{cterm "Not"} ct)
```
```   554   val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
```
```   555   val tm0 = Thm.dest_arg (Thm.rhs_of th0)
```
```   556   val (th, certificates) = if tm0 aconvc @{cterm False} then (equal_implies_1_rule th0, Trivial) else
```
```   557    let
```
```   558     val (evs,bod) = strip_exists tm0
```
```   559     val (avs,ibod) = strip_forall bod
```
```   560     val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
```
```   561     val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))]
```
```   562     val th3 = fold simple_choose evs (prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.capply @{cterm Trueprop} bod))) th2)
```
```   563    in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
```
```   564    end
```
```   565   in (Thm.implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates)
```
```   566  end
```
```   567 in f
```
```   568 end;
```
```   569
```
```   570 (* A linear arithmetic prover *)
```
```   571 local
```
```   572   val linear_add = FuncUtil.Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
```
```   573   fun linear_cmul c = FuncUtil.Ctermfunc.map (fn x => c */ x)
```
```   574   val one_tm = @{cterm "1::real"}
```
```   575   fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p Rat.zero)) orelse
```
```   576      ((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso
```
```   577        not(p(FuncUtil.Ctermfunc.apply e one_tm)))
```
```   578
```
```   579   fun linear_ineqs vars (les,lts) =
```
```   580    case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
```
```   581     SOME r => r
```
```   582   | NONE =>
```
```   583    (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
```
```   584      SOME r => r
```
```   585    | NONE =>
```
```   586      if null vars then error "linear_ineqs: no contradiction" else
```
```   587      let
```
```   588       val ineqs = les @ lts
```
```   589       fun blowup v =
```
```   590        length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
```
```   591        length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
```
```   592        length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
```
```   593       val  v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
```
```   594                  (map (fn v => (v,blowup v)) vars)))
```
```   595       fun addup (e1,p1) (e2,p2) acc =
```
```   596        let
```
```   597         val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v Rat.zero
```
```   598         val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v Rat.zero
```
```   599        in if c1 */ c2 >=/ Rat.zero then acc else
```
```   600         let
```
```   601          val e1' = linear_cmul (Rat.abs c2) e1
```
```   602          val e2' = linear_cmul (Rat.abs c1) e2
```
```   603          val p1' = Product(Rational_lt(Rat.abs c2),p1)
```
```   604          val p2' = Product(Rational_lt(Rat.abs c1),p2)
```
```   605         in (linear_add e1' e2',Sum(p1',p2'))::acc
```
```   606         end
```
```   607        end
```
```   608       val (les0,les1) =
```
```   609          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
```
```   610       val (lts0,lts1) =
```
```   611          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
```
```   612       val (lesp,lesn) =
```
```   613          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
```
```   614       val (ltsp,ltsn) =
```
```   615          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
```
```   616       val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
```
```   617       val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
```
```   618                       (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
```
```   619      in linear_ineqs (remove (op aconvc) v vars) (les',lts')
```
```   620      end)
```
```   621
```
```   622   fun linear_eqs(eqs,les,lts) =
```
```   623    case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
```
```   624     SOME r => r
```
```   625   | NONE => (case eqs of
```
```   626     [] =>
```
```   627      let val vars = remove (op aconvc) one_tm
```
```   628            (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) [])
```
```   629      in linear_ineqs vars (les,lts) end
```
```   630    | (e,p)::es =>
```
```   631      if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
```
```   632      let
```
```   633       val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
```
```   634       fun xform (inp as (t,q)) =
```
```   635        let val d = FuncUtil.Ctermfunc.tryapplyd t x Rat.zero in
```
```   636         if d =/ Rat.zero then inp else
```
```   637         let
```
```   638          val k = (Rat.neg d) */ Rat.abs c // c
```
```   639          val e' = linear_cmul k e
```
```   640          val t' = linear_cmul (Rat.abs c) t
```
```   641          val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
```
```   642          val q' = Product(Rational_lt(Rat.abs c),q)
```
```   643         in (linear_add e' t',Sum(p',q'))
```
```   644         end
```
```   645       end
```
```   646      in linear_eqs(map xform es,map xform les,map xform lts)
```
```   647      end)
```
```   648
```
```   649   fun linear_prover (eq,le,lt) =
```
```   650    let
```
```   651     val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq
```
```   652     val les = map_index (fn (n, p) => (p,Axiom_le n)) le
```
```   653     val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt
```
```   654    in linear_eqs(eqs,les,lts)
```
```   655    end
```
```   656
```
```   657   fun lin_of_hol ct =
```
```   658    if ct aconvc @{cterm "0::real"} then FuncUtil.Ctermfunc.empty
```
```   659    else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
```
```   660    else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
```
```   661    else
```
```   662     let val (lop,r) = Thm.dest_comb ct
```
```   663     in if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
```
```   664        else
```
```   665         let val (opr,l) = Thm.dest_comb lop
```
```   666         in if opr aconvc @{cterm "op + :: real =>_"}
```
```   667            then linear_add (lin_of_hol l) (lin_of_hol r)
```
```   668            else if opr aconvc @{cterm "op * :: real =>_"}
```
```   669                    andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
```
```   670            else FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
```
```   671         end
```
```   672     end
```
```   673
```
```   674   fun is_alien ct = case term_of ct of
```
```   675    Const(@{const_name "real"}, _)\$ n =>
```
```   676      if can HOLogic.dest_number n then false else true
```
```   677   | _ => false
```
```   678 in
```
```   679 fun real_linear_prover translator (eq,le,lt) =
```
```   680  let
```
```   681   val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o cprop_of
```
```   682   val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o cprop_of
```
```   683   val eq_pols = map lhs eq
```
```   684   val le_pols = map rhs le
```
```   685   val lt_pols = map rhs lt
```
```   686   val aliens =  filter is_alien
```
```   687       (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom)
```
```   688           (eq_pols @ le_pols @ lt_pols) [])
```
```   689   val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,Rat.one)) aliens
```
```   690   val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
```
```   691   val le' = le @ map (fn a => instantiate' [] [SOME (Thm.dest_arg a)] @{thm real_of_nat_ge_zero}) aliens
```
```   692  in ((translator (eq,le',lt) proof), Trivial)
```
```   693  end
```
```   694 end;
```
```   695
```
```   696 (* A less general generic arithmetic prover dealing with abs,max and min*)
```
```   697
```
```   698 local
```
```   699  val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
```
```   700  fun absmaxmin_elim_conv1 ctxt =
```
```   701     Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
```
```   702
```
```   703  val absmaxmin_elim_conv2 =
```
```   704   let
```
```   705    val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
```
```   706    val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
```
```   707    val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
```
```   708    val abs_tm = @{cterm "abs :: real => _"}
```
```   709    val p_tm = @{cpat "?P :: real => bool"}
```
```   710    val x_tm = @{cpat "?x :: real"}
```
```   711    val y_tm = @{cpat "?y::real"}
```
```   712    val is_max = is_binop @{cterm "max :: real => _"}
```
```   713    val is_min = is_binop @{cterm "min :: real => _"}
```
```   714    fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm
```
```   715    fun eliminate_construct p c tm =
```
```   716     let
```
```   717      val t = find_cterm p tm
```
```   718      val th0 = (Thm.symmetric o Thm.beta_conversion false) (Thm.capply (Thm.cabs t tm) t)
```
```   719      val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0
```
```   720     in fconv_rule(arg_conv(binop_conv (arg_conv (Thm.beta_conversion false))))
```
```   721                (Thm.transitive th0 (c p ax))
```
```   722    end
```
```   723
```
```   724    val elim_abs = eliminate_construct is_abs
```
```   725     (fn p => fn ax =>
```
```   726        Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax)]) pth_abs)
```
```   727    val elim_max = eliminate_construct is_max
```
```   728     (fn p => fn ax =>
```
```   729       let val (ax,y) = Thm.dest_comb ax
```
```   730       in  Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)])
```
```   731       pth_max end)
```
```   732    val elim_min = eliminate_construct is_min
```
```   733     (fn p => fn ax =>
```
```   734       let val (ax,y) = Thm.dest_comb ax
```
```   735       in  Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)])
```
```   736       pth_min end)
```
```   737    in first_conv [elim_abs, elim_max, elim_min, all_conv]
```
```   738   end;
```
```   739 in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
```
```   740         gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,
```
```   741                        absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
```
```   742 end;
```
```   743
```
```   744 (* An instance for reals*)
```
```   745
```
```   746 fun gen_prover_real_arith ctxt prover =
```
```   747  let
```
```   748   fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS
```
```   749   val {add,mul,neg,pow,sub,main} =
```
```   750      Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
```
```   751       (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
```
```   752      simple_cterm_ord
```
```   753 in gen_real_arith ctxt
```
```   754    (cterm_of_rat, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv,
```
```   755     main,neg,add,mul, prover)
```
```   756 end;
```
```   757
```
```   758 end
```