src/HOL/Library/positivstellensatz.ML
 author haftmann Thu Oct 22 13:48:06 2009 +0200 (2009-10-22) changeset 33063 4d462963a7db parent 33042 ddf1f03a9ad9 child 33443 b9bbd0f3dcdb permissions -rw-r--r--
map_range (and map_index) combinator
```     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  include TABLE
```
```    12  val apply : 'a table -> key -> 'a
```
```    13  val applyd :'a table -> (key -> 'a) -> key -> 'a
```
```    14  val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a table -> 'a table -> 'a table
```
```    15  val dom : 'a table -> key list
```
```    16  val tryapplyd : 'a table -> key -> 'a -> 'a
```
```    17  val updatep : (key * 'a -> bool) -> key * 'a -> 'a table -> 'a table
```
```    18  val choose : 'a table -> key * 'a
```
```    19  val onefunc : key * 'a -> 'a table
```
```    20 end;
```
```    21
```
```    22 functor FuncFun(Key: KEY) : FUNC=
```
```    23 struct
```
```    24
```
```    25 structure Tab = Table(Key);
```
```    26
```
```    27 open Tab;
```
```    28
```
```    29 fun dom a = sort Key.ord (Tab.keys a);
```
```    30 fun applyd f d x = case Tab.lookup f x of
```
```    31    SOME y => y
```
```    32  | NONE => d x;
```
```    33
```
```    34 fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
```
```    35 fun tryapplyd f a d = applyd f (K d) a;
```
```    36 fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
```
```    37 fun combine f z a b =
```
```    38  let
```
```    39   fun h (k,v) t = case Tab.lookup t k of
```
```    40      NONE => Tab.update (k,v) t
```
```    41    | SOME v' => let val w = f v v'
```
```    42      in if z w then Tab.delete k t else Tab.update (k,w) t end;
```
```    43   in Tab.fold h a b end;
```
```    44
```
```    45 fun choose f = case Tab.min_key f of
```
```    46    SOME k => (k, the (Tab.lookup f k))
```
```    47  | NONE => error "FuncFun.choose : Completely empty function"
```
```    48
```
```    49 fun onefunc kv = update kv empty
```
```    50
```
```    51 end;
```
```    52
```
```    53 (* Some standard functors and utility functions for them *)
```
```    54
```
```    55 structure FuncUtil =
```
```    56 struct
```
```    57
```
```    58 structure Intfunc = FuncFun(type key = int val ord = int_ord);
```
```    59 structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
```
```    60 structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
```
```    61 structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
```
```    62 structure Termfunc = FuncFun(type key = term val ord = TermOrd.fast_term_ord);
```
```    63
```
```    64 val cterm_ord = TermOrd.fast_term_ord o pairself term_of
```
```    65
```
```    66 structure Ctermfunc = FuncFun(type key = cterm val ord = cterm_ord);
```
```    67
```
```    68 type monomial = int Ctermfunc.table;
```
```    69
```
```    70 val monomial_ord = list_ord (prod_ord cterm_ord int_ord) o pairself Ctermfunc.dest
```
```    71
```
```    72 structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
```
```    73
```
```    74 type poly = Rat.rat Monomialfunc.table;
```
```    75
```
```    76 (* The ordering so we can create canonical HOL polynomials.                  *)
```
```    77
```
```    78 fun dest_monomial mon = sort (cterm_ord o pairself fst) (Ctermfunc.dest mon);
```
```    79
```
```    80 fun monomial_order (m1,m2) =
```
```    81  if Ctermfunc.is_empty m2 then LESS
```
```    82  else if Ctermfunc.is_empty m1 then GREATER
```
```    83  else
```
```    84   let val mon1 = dest_monomial m1
```
```    85       val mon2 = dest_monomial m2
```
```    86       val deg1 = fold (Integer.add o snd) mon1 0
```
```    87       val deg2 = fold (Integer.add o snd) mon2 0
```
```    88   in if deg1 < deg2 then GREATER else if deg1 > deg2 then LESS
```
```    89      else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
```
```    90   end;
```
```    91
```
```    92 end
```
```    93
```
```    94 (* positivstellensatz datatype and prover generation *)
```
```    95
```
```    96 signature REAL_ARITH =
```
```    97 sig
```
```    98
```
```    99   datatype positivstellensatz =
```
```   100    Axiom_eq of int
```
```   101  | Axiom_le of int
```
```   102  | Axiom_lt of int
```
```   103  | Rational_eq of Rat.rat
```
```   104  | Rational_le of Rat.rat
```
```   105  | Rational_lt of Rat.rat
```
```   106  | Square of FuncUtil.poly
```
```   107  | Eqmul of FuncUtil.poly * positivstellensatz
```
```   108  | Sum of positivstellensatz * positivstellensatz
```
```   109  | Product of positivstellensatz * positivstellensatz;
```
```   110
```
```   111 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
```
```   112
```
```   113 datatype tree_choice = Left | Right
```
```   114
```
```   115 type prover = tree_choice list ->
```
```   116   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
```
```   117   thm list * thm list * thm list -> thm * pss_tree
```
```   118 type cert_conv = cterm -> thm * pss_tree
```
```   119
```
```   120 val gen_gen_real_arith :
```
```   121   Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
```
```   122    conv * conv * conv * conv * conv * conv * prover -> cert_conv
```
```   123 val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
```
```   124   thm list * thm list * thm list -> thm * pss_tree
```
```   125
```
```   126 val gen_real_arith : Proof.context ->
```
```   127   (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
```
```   128
```
```   129 val gen_prover_real_arith : Proof.context -> prover -> cert_conv
```
```   130
```
```   131 val is_ratconst : cterm -> bool
```
```   132 val dest_ratconst : cterm -> Rat.rat
```
```   133 val cterm_of_rat : Rat.rat -> cterm
```
```   134
```
```   135 end
```
```   136
```
```   137 structure RealArith : REAL_ARITH =
```
```   138 struct
```
```   139
```
```   140  open Conv
```
```   141 (* ------------------------------------------------------------------------- *)
```
```   142 (* Data structure for Positivstellensatz refutations.                        *)
```
```   143 (* ------------------------------------------------------------------------- *)
```
```   144
```
```   145 datatype positivstellensatz =
```
```   146    Axiom_eq of int
```
```   147  | Axiom_le of int
```
```   148  | Axiom_lt of int
```
```   149  | Rational_eq of Rat.rat
```
```   150  | Rational_le of Rat.rat
```
```   151  | Rational_lt of Rat.rat
```
```   152  | Square of FuncUtil.poly
```
```   153  | Eqmul of FuncUtil.poly * positivstellensatz
```
```   154  | Sum of positivstellensatz * positivstellensatz
```
```   155  | Product of positivstellensatz * positivstellensatz;
```
```   156          (* Theorems used in the procedure *)
```
```   157
```
```   158 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
```
```   159 datatype tree_choice = Left | Right
```
```   160 type prover = tree_choice list ->
```
```   161   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
```
```   162   thm list * thm list * thm list -> thm * pss_tree
```
```   163 type cert_conv = cterm -> thm * pss_tree
```
```   164
```
```   165 val my_eqs = Unsynchronized.ref ([] : thm list);
```
```   166 val my_les = Unsynchronized.ref ([] : thm list);
```
```   167 val my_lts = Unsynchronized.ref ([] : thm list);
```
```   168 val my_proof = Unsynchronized.ref (Axiom_eq 0);
```
```   169 val my_context = Unsynchronized.ref @{context};
```
```   170
```
```   171 val my_mk_numeric = Unsynchronized.ref ((K @{cterm True}) :Rat.rat -> cterm);
```
```   172 val my_numeric_eq_conv = Unsynchronized.ref no_conv;
```
```   173 val my_numeric_ge_conv = Unsynchronized.ref no_conv;
```
```   174 val my_numeric_gt_conv = Unsynchronized.ref no_conv;
```
```   175 val my_poly_conv = Unsynchronized.ref no_conv;
```
```   176 val my_poly_neg_conv = Unsynchronized.ref no_conv;
```
```   177 val my_poly_add_conv = Unsynchronized.ref no_conv;
```
```   178 val my_poly_mul_conv = Unsynchronized.ref no_conv;
```
```   179
```
```   180
```
```   181     (* Some useful derived rules *)
```
```   182 fun deduct_antisym_rule tha thb =
```
```   183     equal_intr (implies_intr (cprop_of thb) tha)
```
```   184      (implies_intr (cprop_of tha) thb);
```
```   185
```
```   186 fun prove_hyp tha thb =
```
```   187   if exists (curry op aconv (concl_of tha)) (#hyps (rep_thm thb))
```
```   188   then equal_elim (symmetric (deduct_antisym_rule tha thb)) tha else thb;
```
```   189
```
```   190 fun conjunctions th = case try Conjunction.elim th of
```
```   191    SOME (th1,th2) => (conjunctions th1) @ conjunctions th2
```
```   192  | NONE => [th];
```
```   193
```
```   194 val pth = @{lemma "(((x::real) < y) == (y - x > 0)) &&& ((x <= y) == (y - x >= 0))
```
```   195      &&& ((x = y) == (x - y = 0)) &&& ((~(x < y)) == (x - y >= 0)) &&& ((~(x <= y)) == (x - y > 0))
```
```   196      &&& ((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
```
```   197   by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)} |>
```
```   198 conjunctions;
```
```   199
```
```   200 val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
```
```   201 val pth_add =
```
```   202  @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 ) &&& ( x = 0 ==> y >= 0 ==> x + y >= 0)
```
```   203     &&& (x = 0 ==> y > 0 ==> x + y > 0) &&& (x >= 0 ==> y = 0 ==> x + y >= 0)
```
```   204     &&& (x >= 0 ==> y >= 0 ==> x + y >= 0) &&& (x >= 0 ==> y > 0 ==> x + y > 0)
```
```   205     &&& (x > 0 ==> y = 0 ==> x + y > 0) &&& (x > 0 ==> y >= 0 ==> x + y > 0)
```
```   206     &&& (x > 0 ==> y > 0 ==> x + y > 0)"  by simp_all} |> conjunctions ;
```
```   207
```
```   208 val pth_mul =
```
```   209   @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0) &&& (x = 0 ==> y >= 0 ==> x * y = 0) &&&
```
```   210            (x = 0 ==> y > 0 ==> x * y = 0) &&& (x >= 0 ==> y = 0 ==> x * y = 0) &&&
```
```   211            (x >= 0 ==> y >= 0 ==> x * y >= 0 ) &&& ( x >= 0 ==> y > 0 ==> x * y >= 0 ) &&&
```
```   212            (x > 0 ==>  y = 0 ==> x * y = 0 ) &&& ( x > 0 ==> y >= 0 ==> x * y >= 0 ) &&&
```
```   213            (x > 0 ==>  y > 0 ==> x * y > 0)"
```
```   214   by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
```
```   215     mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])} |> conjunctions;
```
```   216
```
```   217 val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
```
```   218 val pth_square = @{lemma "x * x >= (0::real)"  by simp};
```
```   219
```
```   220 val weak_dnf_simps = List.take (simp_thms, 34)
```
```   221     @ 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+};
```
```   222
```
```   223 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+}
```
```   224
```
```   225 val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
```
```   226 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)});
```
```   227
```
```   228 val real_abs_thms1 = conjunctions @{lemma
```
```   229   "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r)) &&&
```
```   230   ((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
```
```   231   ((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
```
```   232   ((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r)) &&&
```
```   233   ((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r)) &&&
```
```   234   ((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r)) &&&
```
```   235   ((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r)) &&&
```
```   236   ((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
```
```   237   ((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
```
```   238   ((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y  + b >= r)) &&&
```
```   239   ((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r)) &&&
```
```   240   ((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y  + c >= r)) &&&
```
```   241   ((1 * min x y >= r) = (1 * x >= r & 1 * y >= r)) &&&
```
```   242   ((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
```
```   243   ((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
```
```   244   ((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y  + b >= r) )&&&
```
```   245   ((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r)) &&&
```
```   246   ((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y  + c >= r)) &&&
```
```   247   ((min x y >= r) = (x >= r &  y >= r)) &&&
```
```   248   ((min x y + a >= r) = (a + x >= r & a + y >= r)) &&&
```
```   249   ((a + min x y >= r) = (a + x >= r & a + y >= r)) &&&
```
```   250   ((a + min x y + b >= r) = (a + x + b >= r & a + y  + b >= r)) &&&
```
```   251   ((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r) )&&&
```
```   252   ((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r)) &&&
```
```   253   ((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r)) &&&
```
```   254   ((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
```
```   255   ((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
```
```   256   ((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r)) &&&
```
```   257   ((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r)) &&&
```
```   258   ((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r)) &&&
```
```   259   ((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r)) &&&
```
```   260   ((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
```
```   261   ((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
```
```   262   ((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y  + b > r)) &&&
```
```   263   ((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r)) &&&
```
```   264   ((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y  + c > r)) &&&
```
```   265   ((min x y > r) = (x > r &  y > r)) &&&
```
```   266   ((min x y + a > r) = (a + x > r & a + y > r)) &&&
```
```   267   ((a + min x y > r) = (a + x > r & a + y > r)) &&&
```
```   268   ((a + min x y + b > r) = (a + x + b > r & a + y  + b > r)) &&&
```
```   269   ((a + b + min x y > r) = (a + b + x > r & a + b + y > r)) &&&
```
```   270   ((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
```
```   271   by auto};
```
```   272
```
```   273 val abs_split' = @{lemma "P (abs (x::'a::ordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
```
```   274   by (atomize (full)) (auto split add: abs_split)};
```
```   275
```
```   276 val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
```
```   277   by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
```
```   278
```
```   279 val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
```
```   280   by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
```
```   281
```
```   282
```
```   283          (* Miscalineous *)
```
```   284 fun literals_conv bops uops cv =
```
```   285  let fun h t =
```
```   286   case (term_of t) of
```
```   287    b\$_\$_ => if member (op aconv) bops b then binop_conv h t else cv t
```
```   288  | u\$_ => if member (op aconv) uops u then arg_conv h t else cv t
```
```   289  | _ => cv t
```
```   290  in h end;
```
```   291
```
```   292 fun cterm_of_rat x =
```
```   293 let val (a, b) = Rat.quotient_of_rat x
```
```   294 in
```
```   295  if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
```
```   296   else Thm.capply (Thm.capply @{cterm "op / :: real => _"}
```
```   297                    (Numeral.mk_cnumber @{ctyp "real"} a))
```
```   298         (Numeral.mk_cnumber @{ctyp "real"} b)
```
```   299 end;
```
```   300
```
```   301   fun dest_ratconst t = case term_of t of
```
```   302    Const(@{const_name divide}, _)\$a\$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
```
```   303  | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
```
```   304  fun is_ratconst t = can dest_ratconst t
```
```   305
```
```   306 fun find_term p t = if p t then t else
```
```   307  case t of
```
```   308   a\$b => (find_term p a handle TERM _ => find_term p b)
```
```   309  | Abs (_,_,t') => find_term p t'
```
```   310  | _ => raise TERM ("find_term",[t]);
```
```   311
```
```   312 fun find_cterm p t = if p t then t else
```
```   313  case term_of t of
```
```   314   a\$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
```
```   315  | Abs (_,_,t') => find_cterm p (Thm.dest_abs NONE t |> snd)
```
```   316  | _ => raise CTERM ("find_cterm",[t]);
```
```   317
```
```   318     (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
```
```   319 fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
```
```   320 fun is_comb t = case (term_of t) of _\$_ => true | _ => false;
```
```   321
```
```   322 fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
```
```   323   handle CTERM _ => false;
```
```   324
```
```   325
```
```   326 (* Map back polynomials to HOL.                         *)
```
```   327
```
```   328 fun cterm_of_varpow x k = if k = 1 then x else Thm.capply (Thm.capply @{cterm "op ^ :: real => _"} x)
```
```   329   (Numeral.mk_cnumber @{ctyp nat} k)
```
```   330
```
```   331 fun cterm_of_monomial m =
```
```   332  if FuncUtil.Ctermfunc.is_empty m then @{cterm "1::real"}
```
```   333  else
```
```   334   let
```
```   335    val m' = FuncUtil.dest_monomial m
```
```   336    val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' []
```
```   337   in foldr1 (fn (s, t) => Thm.capply (Thm.capply @{cterm "op * :: real => _"} s) t) vps
```
```   338   end
```
```   339
```
```   340 fun cterm_of_cmonomial (m,c) = if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
```
```   341     else if c = Rat.one then cterm_of_monomial m
```
```   342     else Thm.capply (Thm.capply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
```
```   343
```
```   344 fun cterm_of_poly p =
```
```   345  if FuncUtil.Monomialfunc.is_empty p then @{cterm "0::real"}
```
```   346  else
```
```   347   let
```
```   348    val cms = map cterm_of_cmonomial
```
```   349      (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
```
```   350   in foldr1 (fn (t1, t2) => Thm.capply(Thm.capply @{cterm "op + :: real => _"} t1) t2) cms
```
```   351   end;
```
```   352
```
```   353     (* A general real arithmetic prover *)
```
```   354
```
```   355 fun gen_gen_real_arith ctxt (mk_numeric,
```
```   356        numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
```
```   357        poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
```
```   358        absconv1,absconv2,prover) =
```
```   359 let
```
```   360  val _ = my_context := ctxt
```
```   361  val _ = (my_mk_numeric := mk_numeric ; my_numeric_eq_conv := numeric_eq_conv ;
```
```   362           my_numeric_ge_conv := numeric_ge_conv; my_numeric_gt_conv := numeric_gt_conv ;
```
```   363           my_poly_conv := poly_conv; my_poly_neg_conv := poly_neg_conv;
```
```   364           my_poly_add_conv := poly_add_conv; my_poly_mul_conv := poly_mul_conv)
```
```   365  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}]
```
```   366  val prenex_ss = HOL_basic_ss addsimps prenex_simps
```
```   367  val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
```
```   368  val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
```
```   369  val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
```
```   370  val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
```
```   371  val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
```
```   372  val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
```
```   373  fun eqT_elim th = equal_elim (symmetric th) @{thm TrueI}
```
```   374  fun oprconv cv ct =
```
```   375   let val g = Thm.dest_fun2 ct
```
```   376   in if g aconvc @{cterm "op <= :: real => _"}
```
```   377        orelse g aconvc @{cterm "op < :: real => _"}
```
```   378      then arg_conv cv ct else arg1_conv cv ct
```
```   379   end
```
```   380
```
```   381  fun real_ineq_conv th ct =
```
```   382   let
```
```   383    val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th
```
```   384       handle MATCH => raise CTERM ("real_ineq_conv", [ct]))
```
```   385   in transitive th' (oprconv poly_conv (Thm.rhs_of th'))
```
```   386   end
```
```   387   val [real_lt_conv, real_le_conv, real_eq_conv,
```
```   388        real_not_lt_conv, real_not_le_conv, _] =
```
```   389        map real_ineq_conv pth
```
```   390   fun match_mp_rule ths ths' =
```
```   391    let
```
```   392      fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
```
```   393       | th::ths => (ths' MRS th handle THM _ => f ths ths')
```
```   394    in f ths ths' end
```
```   395   fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
```
```   396          (match_mp_rule pth_mul [th, th'])
```
```   397   fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
```
```   398          (match_mp_rule pth_add [th, th'])
```
```   399   fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
```
```   400        (instantiate' [] [SOME ct] (th RS pth_emul))
```
```   401   fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
```
```   402        (instantiate' [] [SOME t] pth_square)
```
```   403
```
```   404   fun hol_of_positivstellensatz(eqs,les,lts) proof =
```
```   405    let
```
```   406     val _ = (my_eqs := eqs ; my_les := les ; my_lts := lts ; my_proof := proof)
```
```   407     fun translate prf = case prf of
```
```   408         Axiom_eq n => nth eqs n
```
```   409       | Axiom_le n => nth les n
```
```   410       | Axiom_lt n => nth lts n
```
```   411       | Rational_eq x => eqT_elim(numeric_eq_conv(Thm.capply @{cterm Trueprop}
```
```   412                           (Thm.capply (Thm.capply @{cterm "op =::real => _"} (mk_numeric x))
```
```   413                                @{cterm "0::real"})))
```
```   414       | Rational_le x => eqT_elim(numeric_ge_conv(Thm.capply @{cterm Trueprop}
```
```   415                           (Thm.capply (Thm.capply @{cterm "op <=::real => _"}
```
```   416                                      @{cterm "0::real"}) (mk_numeric x))))
```
```   417       | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.capply @{cterm Trueprop}
```
```   418                       (Thm.capply (Thm.capply @{cterm "op <::real => _"} @{cterm "0::real"})
```
```   419                         (mk_numeric x))))
```
```   420       | Square pt => square_rule (cterm_of_poly pt)
```
```   421       | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
```
```   422       | Sum(p1,p2) => add_rule (translate p1) (translate p2)
```
```   423       | Product(p1,p2) => mul_rule (translate p1) (translate p2)
```
```   424    in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv])
```
```   425           (translate proof)
```
```   426    end
```
```   427
```
```   428   val init_conv = presimp_conv then_conv
```
```   429       nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
```
```   430       weak_dnf_conv
```
```   431
```
```   432   val concl = Thm.dest_arg o cprop_of
```
```   433   fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
```
```   434   val is_req = is_binop @{cterm "op =:: real => _"}
```
```   435   val is_ge = is_binop @{cterm "op <=:: real => _"}
```
```   436   val is_gt = is_binop @{cterm "op <:: real => _"}
```
```   437   val is_conj = is_binop @{cterm "op &"}
```
```   438   val is_disj = is_binop @{cterm "op |"}
```
```   439   fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
```
```   440   fun disj_cases th th1 th2 =
```
```   441    let val (p,q) = Thm.dest_binop (concl th)
```
```   442        val c = concl th1
```
```   443        val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
```
```   444    in implies_elim (implies_elim
```
```   445           (implies_elim (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
```
```   446           (implies_intr (Thm.capply @{cterm Trueprop} p) th1))
```
```   447         (implies_intr (Thm.capply @{cterm Trueprop} q) th2)
```
```   448    end
```
```   449  fun overall cert_choice dun ths = case ths of
```
```   450   [] =>
```
```   451    let
```
```   452     val (eq,ne) = List.partition (is_req o concl) dun
```
```   453      val (le,nl) = List.partition (is_ge o concl) ne
```
```   454      val lt = filter (is_gt o concl) nl
```
```   455     in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
```
```   456  | th::oths =>
```
```   457    let
```
```   458     val ct = concl th
```
```   459    in
```
```   460     if is_conj ct  then
```
```   461      let
```
```   462       val (th1,th2) = conj_pair th in
```
```   463       overall cert_choice dun (th1::th2::oths) end
```
```   464     else if is_disj ct then
```
```   465       let
```
```   466        val (th1, cert1) = overall (Left::cert_choice) dun (assume (Thm.capply @{cterm Trueprop} (Thm.dest_arg1 ct))::oths)
```
```   467        val (th2, cert2) = overall (Right::cert_choice) dun (assume (Thm.capply @{cterm Trueprop} (Thm.dest_arg ct))::oths)
```
```   468       in (disj_cases th th1 th2, Branch (cert1, cert2)) end
```
```   469    else overall cert_choice (th::dun) oths
```
```   470   end
```
```   471   fun dest_binary b ct = if is_binop b ct then Thm.dest_binop ct
```
```   472                          else raise CTERM ("dest_binary",[b,ct])
```
```   473   val dest_eq = dest_binary @{cterm "op = :: real => _"}
```
```   474   val neq_th = nth pth 5
```
```   475   fun real_not_eq_conv ct =
```
```   476    let
```
```   477     val (l,r) = dest_eq (Thm.dest_arg ct)
```
```   478     val th = Thm.instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
```
```   479     val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
```
```   480     val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
```
```   481     val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
```
```   482     val th' = Drule.binop_cong_rule @{cterm "op |"}
```
```   483      (Drule.arg_cong_rule (Thm.capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
```
```   484      (Drule.arg_cong_rule (Thm.capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
```
```   485     in transitive th th'
```
```   486   end
```
```   487  fun equal_implies_1_rule PQ =
```
```   488   let
```
```   489    val P = Thm.lhs_of PQ
```
```   490   in implies_intr P (equal_elim PQ (assume P))
```
```   491   end
```
```   492  (* FIXME!!! Copied from groebner.ml *)
```
```   493  val strip_exists =
```
```   494   let fun h (acc, t) =
```
```   495    case (term_of t) of
```
```   496     Const("Ex",_)\$Abs(x,T,p) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
```
```   497   | _ => (acc,t)
```
```   498   in fn t => h ([],t)
```
```   499   end
```
```   500   fun name_of x = case term_of x of
```
```   501    Free(s,_) => s
```
```   502  | Var ((s,_),_) => s
```
```   503  | _ => "x"
```
```   504
```
```   505   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)
```
```   506
```
```   507   val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
```
```   508
```
```   509  fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
```
```   510  fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
```
```   511
```
```   512  fun choose v th th' = case concl_of th of
```
```   513    @{term Trueprop} \$ (Const("Ex",_)\$_) =>
```
```   514     let
```
```   515      val p = (funpow 2 Thm.dest_arg o cprop_of) th
```
```   516      val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
```
```   517      val th0 = fconv_rule (Thm.beta_conversion true)
```
```   518          (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
```
```   519      val pv = (Thm.rhs_of o Thm.beta_conversion true)
```
```   520            (Thm.capply @{cterm Trueprop} (Thm.capply p v))
```
```   521      val th1 = forall_intr v (implies_intr pv th')
```
```   522     in implies_elim (implies_elim th0 th) th1  end
```
```   523  | _ => raise THM ("choose",0,[th, th'])
```
```   524
```
```   525   fun simple_choose v th =
```
```   526      choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
```
```   527
```
```   528  val strip_forall =
```
```   529   let fun h (acc, t) =
```
```   530    case (term_of t) of
```
```   531     Const("All",_)\$Abs(x,T,p) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
```
```   532   | _ => (acc,t)
```
```   533   in fn t => h ([],t)
```
```   534   end
```
```   535
```
```   536  fun f ct =
```
```   537   let
```
```   538    val nnf_norm_conv' =
```
```   539      nnf_conv then_conv
```
```   540      literals_conv [@{term "op &"}, @{term "op |"}] []
```
```   541      (Conv.cache_conv
```
```   542        (first_conv [real_lt_conv, real_le_conv,
```
```   543                     real_eq_conv, real_not_lt_conv,
```
```   544                     real_not_le_conv, real_not_eq_conv, all_conv]))
```
```   545   fun absremover ct = (literals_conv [@{term "op &"}, @{term "op |"}] []
```
```   546                   (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv
```
```   547         try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
```
```   548   val nct = Thm.capply @{cterm Trueprop} (Thm.capply @{cterm "Not"} ct)
```
```   549   val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
```
```   550   val tm0 = Thm.dest_arg (Thm.rhs_of th0)
```
```   551   val (th, certificates) = if tm0 aconvc @{cterm False} then (equal_implies_1_rule th0, Trivial) else
```
```   552    let
```
```   553     val (evs,bod) = strip_exists tm0
```
```   554     val (avs,ibod) = strip_forall bod
```
```   555     val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
```
```   556     val (th2, certs) = overall [] [] [specl avs (assume (Thm.rhs_of th1))]
```
```   557     val th3 = fold simple_choose evs (prove_hyp (equal_elim th1 (assume (Thm.capply @{cterm Trueprop} bod))) th2)
```
```   558    in (Drule.implies_intr_hyps (prove_hyp (equal_elim th0 (assume nct)) th3), certs)
```
```   559    end
```
```   560   in (implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates)
```
```   561  end
```
```   562 in f
```
```   563 end;
```
```   564
```
```   565 (* A linear arithmetic prover *)
```
```   566 local
```
```   567   val linear_add = FuncUtil.Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
```
```   568   fun linear_cmul c = FuncUtil.Ctermfunc.map (fn x => c */ x)
```
```   569   val one_tm = @{cterm "1::real"}
```
```   570   fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p Rat.zero)) orelse
```
```   571      ((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso
```
```   572        not(p(FuncUtil.Ctermfunc.apply e one_tm)))
```
```   573
```
```   574   fun linear_ineqs vars (les,lts) =
```
```   575    case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
```
```   576     SOME r => r
```
```   577   | NONE =>
```
```   578    (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
```
```   579      SOME r => r
```
```   580    | NONE =>
```
```   581      if null vars then error "linear_ineqs: no contradiction" else
```
```   582      let
```
```   583       val ineqs = les @ lts
```
```   584       fun blowup v =
```
```   585        length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
```
```   586        length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
```
```   587        length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
```
```   588       val  v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
```
```   589                  (map (fn v => (v,blowup v)) vars)))
```
```   590       fun addup (e1,p1) (e2,p2) acc =
```
```   591        let
```
```   592         val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v Rat.zero
```
```   593         val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v Rat.zero
```
```   594        in if c1 */ c2 >=/ Rat.zero then acc else
```
```   595         let
```
```   596          val e1' = linear_cmul (Rat.abs c2) e1
```
```   597          val e2' = linear_cmul (Rat.abs c1) e2
```
```   598          val p1' = Product(Rational_lt(Rat.abs c2),p1)
```
```   599          val p2' = Product(Rational_lt(Rat.abs c1),p2)
```
```   600         in (linear_add e1' e2',Sum(p1',p2'))::acc
```
```   601         end
```
```   602        end
```
```   603       val (les0,les1) =
```
```   604          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
```
```   605       val (lts0,lts1) =
```
```   606          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
```
```   607       val (lesp,lesn) =
```
```   608          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
```
```   609       val (ltsp,ltsn) =
```
```   610          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
```
```   611       val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
```
```   612       val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
```
```   613                       (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
```
```   614      in linear_ineqs (remove (op aconvc) v vars) (les',lts')
```
```   615      end)
```
```   616
```
```   617   fun linear_eqs(eqs,les,lts) =
```
```   618    case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
```
```   619     SOME r => r
```
```   620   | NONE => (case eqs of
```
```   621     [] =>
```
```   622      let val vars = remove (op aconvc) one_tm
```
```   623            (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) [])
```
```   624      in linear_ineqs vars (les,lts) end
```
```   625    | (e,p)::es =>
```
```   626      if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
```
```   627      let
```
```   628       val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
```
```   629       fun xform (inp as (t,q)) =
```
```   630        let val d = FuncUtil.Ctermfunc.tryapplyd t x Rat.zero in
```
```   631         if d =/ Rat.zero then inp else
```
```   632         let
```
```   633          val k = (Rat.neg d) */ Rat.abs c // c
```
```   634          val e' = linear_cmul k e
```
```   635          val t' = linear_cmul (Rat.abs c) t
```
```   636          val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
```
```   637          val q' = Product(Rational_lt(Rat.abs c),q)
```
```   638         in (linear_add e' t',Sum(p',q'))
```
```   639         end
```
```   640       end
```
```   641      in linear_eqs(map xform es,map xform les,map xform lts)
```
```   642      end)
```
```   643
```
```   644   fun linear_prover (eq,le,lt) =
```
```   645    let
```
```   646     val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq
```
```   647     val les = map_index (fn (n, p) => (p,Axiom_le n)) le
```
```   648     val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt
```
```   649    in linear_eqs(eqs,les,lts)
```
```   650    end
```
```   651
```
```   652   fun lin_of_hol ct =
```
```   653    if ct aconvc @{cterm "0::real"} then FuncUtil.Ctermfunc.empty
```
```   654    else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
```
```   655    else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
```
```   656    else
```
```   657     let val (lop,r) = Thm.dest_comb ct
```
```   658     in if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
```
```   659        else
```
```   660         let val (opr,l) = Thm.dest_comb lop
```
```   661         in if opr aconvc @{cterm "op + :: real =>_"}
```
```   662            then linear_add (lin_of_hol l) (lin_of_hol r)
```
```   663            else if opr aconvc @{cterm "op * :: real =>_"}
```
```   664                    andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
```
```   665            else FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
```
```   666         end
```
```   667     end
```
```   668
```
```   669   fun is_alien ct = case term_of ct of
```
```   670    Const(@{const_name "real"}, _)\$ n =>
```
```   671      if can HOLogic.dest_number n then false else true
```
```   672   | _ => false
```
```   673 in
```
```   674 fun real_linear_prover translator (eq,le,lt) =
```
```   675  let
```
```   676   val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o cprop_of
```
```   677   val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o cprop_of
```
```   678   val eq_pols = map lhs eq
```
```   679   val le_pols = map rhs le
```
```   680   val lt_pols = map rhs lt
```
```   681   val aliens =  filter is_alien
```
```   682       (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom)
```
```   683           (eq_pols @ le_pols @ lt_pols) [])
```
```   684   val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,Rat.one)) aliens
```
```   685   val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
```
```   686   val le' = le @ map (fn a => instantiate' [] [SOME (Thm.dest_arg a)] @{thm real_of_nat_ge_zero}) aliens
```
```   687  in ((translator (eq,le',lt) proof), Trivial)
```
```   688  end
```
```   689 end;
```
```   690
```
```   691 (* A less general generic arithmetic prover dealing with abs,max and min*)
```
```   692
```
```   693 local
```
```   694  val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
```
```   695  fun absmaxmin_elim_conv1 ctxt =
```
```   696     Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
```
```   697
```
```   698  val absmaxmin_elim_conv2 =
```
```   699   let
```
```   700    val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
```
```   701    val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
```
```   702    val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
```
```   703    val abs_tm = @{cterm "abs :: real => _"}
```
```   704    val p_tm = @{cpat "?P :: real => bool"}
```
```   705    val x_tm = @{cpat "?x :: real"}
```
```   706    val y_tm = @{cpat "?y::real"}
```
```   707    val is_max = is_binop @{cterm "max :: real => _"}
```
```   708    val is_min = is_binop @{cterm "min :: real => _"}
```
```   709    fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm
```
```   710    fun eliminate_construct p c tm =
```
```   711     let
```
```   712      val t = find_cterm p tm
```
```   713      val th0 = (symmetric o beta_conversion false) (Thm.capply (Thm.cabs t tm) t)
```
```   714      val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0
```
```   715     in fconv_rule(arg_conv(binop_conv (arg_conv (beta_conversion false))))
```
```   716                (transitive th0 (c p ax))
```
```   717    end
```
```   718
```
```   719    val elim_abs = eliminate_construct is_abs
```
```   720     (fn p => fn ax =>
```
```   721        Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax)]) pth_abs)
```
```   722    val elim_max = eliminate_construct is_max
```
```   723     (fn p => fn ax =>
```
```   724       let val (ax,y) = Thm.dest_comb ax
```
```   725       in  Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)])
```
```   726       pth_max end)
```
```   727    val elim_min = eliminate_construct is_min
```
```   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_min end)
```
```   732    in first_conv [elim_abs, elim_max, elim_min, all_conv]
```
```   733   end;
```
```   734 in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
```
```   735         gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,
```
```   736                        absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
```
```   737 end;
```
```   738
```
```   739 (* An instance for reals*)
```
```   740
```
```   741 fun gen_prover_real_arith ctxt prover =
```
```   742  let
```
```   743   fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS
```
```   744   val {add,mul,neg,pow,sub,main} =
```
```   745      Normalizer.semiring_normalizers_ord_wrapper ctxt
```
```   746       (the (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
```
```   747      simple_cterm_ord
```
```   748 in gen_real_arith ctxt
```
```   749    (cterm_of_rat, field_comp_conv, field_comp_conv,field_comp_conv,
```
```   750     main,neg,add,mul, prover)
```
```   751 end;
```
```   752
```
```   753 end
```