src/HOL/Library/normarith.ML
author haftmann
Wed Apr 22 19:09:21 2009 +0200 (2009-04-22)
changeset 30960 fec1a04b7220
parent 30868 1040425c86a2
child 31118 541d43bee678
permissions -rw-r--r--
power operation defined generic
     1 (* A functor for finite mappings based on Tables *)
     2 signature FUNC = 
     3 sig
     4  type 'a T
     5  type key
     6  val apply : 'a T -> key -> 'a
     7  val applyd :'a T -> (key -> 'a) -> key -> 'a
     8  val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a T -> 'a T -> 'a T
     9  val defined : 'a T -> key -> bool
    10  val dom : 'a T -> key list
    11  val fold : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b
    12  val graph : 'a T -> (key * 'a) list
    13  val is_undefined : 'a T -> bool
    14  val mapf : ('a -> 'b) -> 'a T -> 'b T
    15  val tryapplyd : 'a T -> key -> 'a -> 'a
    16  val undefine :  key -> 'a T -> 'a T
    17  val undefined : 'a T
    18  val update : key * 'a -> 'a T -> 'a T
    19  val updatep : (key * 'a -> bool) -> key * 'a -> 'a T -> 'a T
    20  val choose : 'a T -> key * 'a
    21  val onefunc : key * 'a -> 'a T
    22  val get_first: (key*'a -> 'a option) -> 'a T -> 'a option
    23  val fns: 
    24    {key_ord: key*key -> order,
    25     apply : 'a T -> key -> 'a,
    26     applyd :'a T -> (key -> 'a) -> key -> 'a,
    27     combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a T -> 'a T -> 'a T,
    28     defined : 'a T -> key -> bool,
    29     dom : 'a T -> key list,
    30     fold : (key * 'a -> 'b -> 'b) -> 'a T -> 'b -> 'b,
    31     graph : 'a T -> (key * 'a) list,
    32     is_undefined : 'a T -> bool,
    33     mapf : ('a -> 'b) -> 'a T -> 'b T,
    34     tryapplyd : 'a T -> key -> 'a -> 'a,
    35     undefine :  key -> 'a T -> 'a T,
    36     undefined : 'a T,
    37     update : key * 'a -> 'a T -> 'a T,
    38     updatep : (key * 'a -> bool) -> key * 'a -> 'a T -> 'a T,
    39     choose : 'a T -> key * 'a,
    40     onefunc : key * 'a -> 'a T,
    41     get_first: (key*'a -> 'a option) -> 'a T -> 'a option}
    42 end;
    43 
    44 functor FuncFun(Key: KEY) : FUNC=
    45 struct
    46 
    47 type key = Key.key;
    48 structure Tab = TableFun(Key);
    49 type 'a T = 'a Tab.table;
    50 
    51 val undefined = Tab.empty;
    52 val is_undefined = Tab.is_empty;
    53 val mapf = Tab.map;
    54 val fold = Tab.fold;
    55 val graph = Tab.dest;
    56 val dom = Tab.keys;
    57 fun applyd f d x = case Tab.lookup f x of 
    58    SOME y => y
    59  | NONE => d x;
    60 
    61 fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
    62 fun tryapplyd f a d = applyd f (K d) a;
    63 val defined = Tab.defined;
    64 fun undefine x t = (Tab.delete x t handle UNDEF => t);
    65 val update = Tab.update;
    66 fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
    67 fun combine f z a b = 
    68  let
    69   fun h (k,v) t = case Tab.lookup t k of
    70      NONE => Tab.update (k,v) t
    71    | SOME v' => let val w = f v v'
    72      in if z w then Tab.delete k t else Tab.update (k,w) t end;
    73   in Tab.fold h a b end;
    74 
    75 fun choose f = case Tab.max_key f of 
    76    SOME k => (k,valOf (Tab.lookup f k))
    77  | NONE => error "FuncFun.choose : Completely undefined function"
    78 
    79 fun onefunc kv = update kv undefined
    80 
    81 local
    82 fun  find f (k,v) NONE = f (k,v)
    83    | find f (k,v) r = r
    84 in
    85 fun get_first f t = fold (find f) t NONE
    86 end
    87 
    88 val fns = 
    89    {key_ord = Key.ord,
    90     apply = apply,
    91     applyd = applyd,
    92     combine = combine,
    93     defined = defined,
    94     dom = dom,
    95     fold = fold,
    96     graph = graph,
    97     is_undefined = is_undefined,
    98     mapf = mapf,
    99     tryapplyd = tryapplyd,
   100     undefine = undefine,
   101     undefined = undefined,
   102     update = update,
   103     updatep = updatep,
   104     choose = choose,
   105     onefunc = onefunc,
   106     get_first = get_first}
   107 
   108 end;
   109 
   110 structure Intfunc = FuncFun(type key = int val ord = int_ord);
   111 structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
   112 structure Termfunc = FuncFun(type key = term val ord = TermOrd.fast_term_ord);
   113 structure Ctermfunc = FuncFun(type key = cterm val ord = (fn (s,t) => TermOrd.fast_term_ord(term_of s, term_of t)));
   114 structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
   115 
   116     (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
   117 structure Conv2 = 
   118 struct
   119  open Conv
   120 fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
   121 fun is_comb t = case (term_of t) of _$_ => true | _ => false;
   122 fun is_abs t = case (term_of t) of Abs _ => true | _ => false;
   123 
   124 fun end_itlist f l =
   125  case l of 
   126    []     => error "end_itlist"
   127  | [x]    => x
   128  | (h::t) => f h (end_itlist f t);
   129 
   130  fun absc cv ct = case term_of ct of 
   131  Abs (v,_, _) => 
   132   let val (x,t) = Thm.dest_abs (SOME v) ct
   133   in Thm.abstract_rule ((fst o dest_Free o term_of) x) x (cv t)
   134   end
   135  | _ => all_conv ct;
   136 
   137 fun cache_conv conv =
   138  let 
   139   val tab = ref Termtab.empty
   140   fun cconv t =  
   141     case Termtab.lookup (!tab) (term_of t) of
   142      SOME th => th
   143    | NONE => let val th = conv t
   144              in ((tab := Termtab.insert Thm.eq_thm (term_of t, th) (!tab)); th) end
   145  in cconv end;
   146 fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
   147   handle CTERM _ => false;
   148 
   149 local
   150  fun thenqc conv1 conv2 tm =
   151    case try conv1 tm of
   152     SOME th1 => (case try conv2 (Thm.rhs_of th1) of SOME th2 => Thm.transitive th1 th2 | NONE => th1)
   153   | NONE => conv2 tm
   154 
   155  fun thencqc conv1 conv2 tm =
   156     let val th1 = conv1 tm 
   157     in (case try conv2 (Thm.rhs_of th1) of SOME th2 => Thm.transitive th1 th2 | NONE => th1)
   158     end
   159  fun comb_qconv conv tm =
   160    let val (l,r) = Thm.dest_comb tm 
   161    in (case try conv l of 
   162         SOME th1 => (case try conv r of SOME th2 => Thm.combination th1 th2 
   163                                       | NONE => Drule.fun_cong_rule th1 r)
   164       | NONE => Drule.arg_cong_rule l (conv r))
   165    end
   166  fun repeatqc conv tm = thencqc conv (repeatqc conv) tm 
   167  fun sub_qconv conv tm =  if is_abs tm then absc conv tm else comb_qconv conv tm 
   168  fun once_depth_qconv conv tm =
   169       (conv else_conv (sub_qconv (once_depth_qconv conv))) tm
   170  fun depth_qconv conv tm =
   171     thenqc (sub_qconv (depth_qconv conv))
   172            (repeatqc conv) tm
   173  fun redepth_qconv conv tm =
   174     thenqc (sub_qconv (redepth_qconv conv))
   175            (thencqc conv (redepth_qconv conv)) tm
   176  fun top_depth_qconv conv tm =
   177     thenqc (repeatqc conv)
   178            (thencqc (sub_qconv (top_depth_qconv conv))
   179                     (thencqc conv (top_depth_qconv conv))) tm
   180  fun top_sweep_qconv conv tm =
   181     thenqc (repeatqc conv)
   182            (sub_qconv (top_sweep_qconv conv)) tm
   183 in 
   184 val (once_depth_conv, depth_conv, rdepth_conv, top_depth_conv, top_sweep_conv) = 
   185   (fn c => try_conv (once_depth_qconv c),
   186    fn c => try_conv (depth_qconv c),
   187    fn c => try_conv (redepth_qconv c),
   188    fn c => try_conv (top_depth_qconv c),
   189    fn c => try_conv (top_sweep_qconv c));
   190 end;
   191 end;
   192 
   193 
   194     (* Some useful derived rules *)
   195 fun deduct_antisym_rule tha thb = 
   196     equal_intr (implies_intr (cprop_of thb) tha) 
   197      (implies_intr (cprop_of tha) thb);
   198 
   199 fun prove_hyp tha thb = 
   200   if exists (curry op aconv (concl_of tha)) (#hyps (rep_thm thb)) 
   201   then equal_elim (symmetric (deduct_antisym_rule tha thb)) tha else thb;
   202 
   203 
   204 
   205 signature REAL_ARITH = 
   206 sig
   207   datatype positivstellensatz =
   208    Axiom_eq of int
   209  | Axiom_le of int
   210  | Axiom_lt of int
   211  | Rational_eq of Rat.rat
   212  | Rational_le of Rat.rat
   213  | Rational_lt of Rat.rat
   214  | Square of cterm
   215  | Eqmul of cterm * positivstellensatz
   216  | Sum of positivstellensatz * positivstellensatz
   217  | Product of positivstellensatz * positivstellensatz;
   218 
   219 val gen_gen_real_arith :
   220   Proof.context -> (Rat.rat -> Thm.cterm) * conv * conv * conv * 
   221    conv * conv * conv * conv * conv * conv * 
   222     ( (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   223         thm list * thm list * thm list -> thm) -> conv
   224 val real_linear_prover : 
   225   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   226    thm list * thm list * thm list -> thm
   227 
   228 val gen_real_arith : Proof.context ->
   229    (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv *
   230    ( (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   231        thm list * thm list * thm list -> thm) -> conv
   232 val gen_prover_real_arith : Proof.context ->
   233    ((thm list * thm list * thm list -> positivstellensatz -> thm) ->
   234      thm list * thm list * thm list -> thm) -> conv
   235 val real_arith : Proof.context -> conv
   236 end
   237 
   238 structure RealArith (* : REAL_ARITH *)=
   239 struct
   240 
   241  open Conv Thm Conv2;;
   242 (* ------------------------------------------------------------------------- *)
   243 (* Data structure for Positivstellensatz refutations.                        *)
   244 (* ------------------------------------------------------------------------- *)
   245 
   246 datatype positivstellensatz =
   247    Axiom_eq of int
   248  | Axiom_le of int
   249  | Axiom_lt of int
   250  | Rational_eq of Rat.rat
   251  | Rational_le of Rat.rat
   252  | Rational_lt of Rat.rat
   253  | Square of cterm
   254  | Eqmul of cterm * positivstellensatz
   255  | Sum of positivstellensatz * positivstellensatz
   256  | Product of positivstellensatz * positivstellensatz;
   257          (* Theorems used in the procedure *)
   258 
   259 fun conjunctions th = case try Conjunction.elim th of
   260    SOME (th1,th2) => (conjunctions th1) @ conjunctions th2
   261  | NONE => [th];
   262 
   263 val pth = @{lemma "(((x::real) < y) == (y - x > 0)) &&& ((x <= y) == (y - x >= 0)) 
   264      &&& ((x = y) == (x - y = 0)) &&& ((~(x < y)) == (x - y >= 0)) &&& ((~(x <= y)) == (x - y > 0))
   265      &&& ((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
   266   by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)} |> 
   267 conjunctions;
   268 
   269 val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
   270 val pth_add = 
   271  @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 ) &&& ( x = 0 ==> y >= 0 ==> x + y >= 0) 
   272     &&& (x = 0 ==> y > 0 ==> x + y > 0) &&& (x >= 0 ==> y = 0 ==> x + y >= 0) 
   273     &&& (x >= 0 ==> y >= 0 ==> x + y >= 0) &&& (x >= 0 ==> y > 0 ==> x + y > 0) 
   274     &&& (x > 0 ==> y = 0 ==> x + y > 0) &&& (x > 0 ==> y >= 0 ==> x + y > 0) 
   275     &&& (x > 0 ==> y > 0 ==> x + y > 0)"  by simp_all} |> conjunctions ;
   276 
   277 val pth_mul = 
   278   @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0) &&& (x = 0 ==> y >= 0 ==> x * y = 0) &&& 
   279            (x = 0 ==> y > 0 ==> x * y = 0) &&& (x >= 0 ==> y = 0 ==> x * y = 0) &&& 
   280            (x >= 0 ==> y >= 0 ==> x * y >= 0 ) &&& ( x >= 0 ==> y > 0 ==> x * y >= 0 ) &&&
   281            (x > 0 ==>  y = 0 ==> x * y = 0 ) &&& ( x > 0 ==> y >= 0 ==> x * y >= 0 ) &&&
   282            (x > 0 ==>  y > 0 ==> x * y > 0)"
   283   by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
   284     mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])} |> conjunctions;
   285 
   286 val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
   287 val pth_square = @{lemma "x * x >= (0::real)"  by simp};
   288 
   289 val weak_dnf_simps = List.take (simp_thms, 34) 
   290     @ 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+};
   291 
   292 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+}
   293 
   294 val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
   295 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)});
   296 
   297 val real_abs_thms1 = conjunctions @{lemma
   298   "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r)) &&&
   299   ((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
   300   ((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r)) &&&
   301   ((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r)) &&&
   302   ((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r)) &&&
   303   ((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r)) &&&
   304   ((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r)) &&&
   305   ((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
   306   ((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r)) &&&
   307   ((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y  + b >= r)) &&&
   308   ((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r)) &&&
   309   ((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y  + c >= r)) &&&
   310   ((1 * min x y >= r) = (1 * x >= r & 1 * y >= r)) &&&
   311   ((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
   312   ((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r)) &&&
   313   ((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y  + b >= r) )&&&
   314   ((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r)) &&&
   315   ((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y  + c >= r)) &&&
   316   ((min x y >= r) = (x >= r &  y >= r)) &&&
   317   ((min x y + a >= r) = (a + x >= r & a + y >= r)) &&&
   318   ((a + min x y >= r) = (a + x >= r & a + y >= r)) &&&
   319   ((a + min x y + b >= r) = (a + x + b >= r & a + y  + b >= r)) &&&
   320   ((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r) )&&&
   321   ((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r)) &&&
   322   ((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r)) &&&
   323   ((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
   324   ((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r)) &&&
   325   ((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r)) &&&
   326   ((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r)) &&&
   327   ((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r)) &&&
   328   ((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r)) &&&
   329   ((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
   330   ((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r)) &&&
   331   ((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y  + b > r)) &&&
   332   ((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r)) &&&
   333   ((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y  + c > r)) &&&
   334   ((min x y > r) = (x > r &  y > r)) &&&
   335   ((min x y + a > r) = (a + x > r & a + y > r)) &&&
   336   ((a + min x y > r) = (a + x > r & a + y > r)) &&&
   337   ((a + min x y + b > r) = (a + x + b > r & a + y  + b > r)) &&&
   338   ((a + b + min x y > r) = (a + b + x > r & a + b + y > r)) &&&
   339   ((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
   340   by auto};
   341 
   342 val abs_split' = @{lemma "P (abs (x::'a::ordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
   343   by (atomize (full)) (auto split add: abs_split)};
   344 
   345 val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
   346   by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
   347 
   348 val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
   349   by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
   350 
   351 
   352          (* Miscalineous *)
   353 fun literals_conv bops uops cv = 
   354  let fun h t =
   355   case (term_of t) of 
   356    b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
   357  | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
   358  | _ => cv t
   359  in h end;
   360 
   361 fun cterm_of_rat x = 
   362 let val (a, b) = Rat.quotient_of_rat x
   363 in 
   364  if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
   365   else Thm.capply (Thm.capply @{cterm "op / :: real => _"} 
   366                    (Numeral.mk_cnumber @{ctyp "real"} a))
   367         (Numeral.mk_cnumber @{ctyp "real"} b)
   368 end;
   369 
   370   fun dest_ratconst t = case term_of t of
   371    Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
   372  | Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd)
   373  | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
   374  fun is_ratconst t = can dest_ratconst t
   375 
   376 fun find_term p t = if p t then t else 
   377  case t of
   378   a$b => (find_term p a handle TERM _ => find_term p b)
   379  | Abs (_,_,t') => find_term p t'
   380  | _ => raise TERM ("find_term",[t]);
   381 
   382 fun find_cterm p t = if p t then t else 
   383  case term_of t of
   384   a$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
   385  | Abs (_,_,t') => find_cterm p (Thm.dest_abs NONE t |> snd)
   386  | _ => raise CTERM ("find_cterm",[t]);
   387 
   388 
   389     (* A general real arithmetic prover *)
   390 
   391 fun gen_gen_real_arith ctxt (mk_numeric,
   392        numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
   393        poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
   394        absconv1,absconv2,prover) = 
   395 let
   396  open Conv Thm; 
   397  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}]
   398  val prenex_ss = HOL_basic_ss addsimps prenex_simps
   399  val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
   400  val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
   401  val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
   402  val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
   403  val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
   404  val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
   405  fun eqT_elim th = equal_elim (symmetric th) @{thm TrueI}
   406  fun oprconv cv ct = 
   407   let val g = Thm.dest_fun2 ct
   408   in if g aconvc @{cterm "op <= :: real => _"} 
   409        orelse g aconvc @{cterm "op < :: real => _"} 
   410      then arg_conv cv ct else arg1_conv cv ct
   411   end
   412 
   413  fun real_ineq_conv th ct =
   414   let
   415    val th' = (instantiate (match (lhs_of th, ct)) th 
   416       handle MATCH => raise CTERM ("real_ineq_conv", [ct]))
   417   in transitive th' (oprconv poly_conv (Thm.rhs_of th'))
   418   end 
   419   val [real_lt_conv, real_le_conv, real_eq_conv,
   420        real_not_lt_conv, real_not_le_conv, _] =
   421        map real_ineq_conv pth
   422   fun match_mp_rule ths ths' = 
   423    let
   424      fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
   425       | th::ths => (ths' MRS th handle THM _ => f ths ths')
   426    in f ths ths' end
   427   fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
   428          (match_mp_rule pth_mul [th, th'])
   429   fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
   430          (match_mp_rule pth_add [th, th'])
   431   fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv)) 
   432        (instantiate' [] [SOME ct] (th RS pth_emul)) 
   433   fun square_rule t = fconv_rule (arg_conv (oprconv poly_mul_conv))
   434        (instantiate' [] [SOME t] pth_square)
   435 
   436   fun hol_of_positivstellensatz(eqs,les,lts) =
   437    let 
   438     fun translate prf = case prf of
   439         Axiom_eq n => nth eqs n
   440       | Axiom_le n => nth les n
   441       | Axiom_lt n => nth lts n
   442       | Rational_eq x => eqT_elim(numeric_eq_conv(capply @{cterm Trueprop} 
   443                           (capply (capply @{cterm "op =::real => _"} (mk_numeric x)) 
   444                                @{cterm "0::real"})))
   445       | Rational_le x => eqT_elim(numeric_ge_conv(capply @{cterm Trueprop} 
   446                           (capply (capply @{cterm "op <=::real => _"} 
   447                                      @{cterm "0::real"}) (mk_numeric x))))
   448       | Rational_lt x => eqT_elim(numeric_gt_conv(capply @{cterm Trueprop} 
   449                       (capply (capply @{cterm "op <::real => _"} @{cterm "0::real"})
   450                         (mk_numeric x))))
   451       | Square t => square_rule t
   452       | Eqmul(t,p) => emul_rule t (translate p)
   453       | Sum(p1,p2) => add_rule (translate p1) (translate p2)
   454       | Product(p1,p2) => mul_rule (translate p1) (translate p2)
   455    in fn prf => 
   456       fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv]) 
   457           (translate prf)
   458    end
   459   
   460   val init_conv = presimp_conv then_conv
   461       nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
   462       weak_dnf_conv
   463 
   464   val concl = dest_arg o cprop_of
   465   fun is_binop opr ct = (dest_fun2 ct aconvc opr handle CTERM _ => false)
   466   val is_req = is_binop @{cterm "op =:: real => _"}
   467   val is_ge = is_binop @{cterm "op <=:: real => _"}
   468   val is_gt = is_binop @{cterm "op <:: real => _"}
   469   val is_conj = is_binop @{cterm "op &"}
   470   val is_disj = is_binop @{cterm "op |"}
   471   fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
   472   fun disj_cases th th1 th2 = 
   473    let val (p,q) = dest_binop (concl th)
   474        val c = concl th1
   475        val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
   476    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)
   477    end
   478  fun overall dun ths = case ths of
   479   [] =>
   480    let 
   481     val (eq,ne) = List.partition (is_req o concl) dun
   482      val (le,nl) = List.partition (is_ge o concl) ne
   483      val lt = filter (is_gt o concl) nl 
   484     in prover hol_of_positivstellensatz (eq,le,lt) end
   485  | th::oths =>
   486    let 
   487     val ct = concl th 
   488    in 
   489     if is_conj ct  then
   490      let 
   491       val (th1,th2) = conj_pair th in
   492       overall dun (th1::th2::oths) end
   493     else if is_disj ct then
   494       let 
   495        val th1 = overall dun (assume (capply @{cterm Trueprop} (dest_arg1 ct))::oths)
   496        val th2 = overall dun (assume (capply @{cterm Trueprop} (dest_arg ct))::oths)
   497       in disj_cases th th1 th2 end
   498    else overall (th::dun) oths
   499   end
   500   fun dest_binary b ct = if is_binop b ct then dest_binop ct 
   501                          else raise CTERM ("dest_binary",[b,ct])
   502   val dest_eq = dest_binary @{cterm "op = :: real => _"}
   503   val neq_th = nth pth 5
   504   fun real_not_eq_conv ct = 
   505    let 
   506     val (l,r) = dest_eq (dest_arg ct)
   507     val th = instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
   508     val th_p = poly_conv(dest_arg(dest_arg1(Thm.rhs_of th)))
   509     val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
   510     val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
   511     val th' = Drule.binop_cong_rule @{cterm "op |"} 
   512      (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
   513      (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
   514     in transitive th th' 
   515   end
   516  fun equal_implies_1_rule PQ = 
   517   let 
   518    val P = lhs_of PQ
   519   in implies_intr P (equal_elim PQ (assume P))
   520   end
   521  (* FIXME!!! Copied from groebner.ml *)
   522  val strip_exists =
   523   let fun h (acc, t) =
   524    case (term_of t) of
   525     Const("Ex",_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
   526   | _ => (acc,t)
   527   in fn t => h ([],t)
   528   end
   529   fun name_of x = case term_of x of
   530    Free(s,_) => s
   531  | Var ((s,_),_) => s
   532  | _ => "x"
   533 
   534   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)
   535 
   536   val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
   537 
   538  fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
   539  fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
   540 
   541  fun choose v th th' = case concl_of th of 
   542    @{term Trueprop} $ (Const("Ex",_)$_) => 
   543     let
   544      val p = (funpow 2 Thm.dest_arg o cprop_of) th
   545      val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
   546      val th0 = fconv_rule (Thm.beta_conversion true)
   547          (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
   548      val pv = (Thm.rhs_of o Thm.beta_conversion true) 
   549            (Thm.capply @{cterm Trueprop} (Thm.capply p v))
   550      val th1 = forall_intr v (implies_intr pv th')
   551     in implies_elim (implies_elim th0 th) th1  end
   552  | _ => raise THM ("choose",0,[th, th'])
   553 
   554   fun simple_choose v th = 
   555      choose v (assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
   556 
   557  val strip_forall =
   558   let fun h (acc, t) =
   559    case (term_of t) of
   560     Const("All",_)$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
   561   | _ => (acc,t)
   562   in fn t => h ([],t)
   563   end
   564 
   565  fun f ct =
   566   let 
   567    val nnf_norm_conv' = 
   568      nnf_conv then_conv 
   569      literals_conv [@{term "op &"}, @{term "op |"}] [] 
   570      (cache_conv 
   571        (first_conv [real_lt_conv, real_le_conv, 
   572                     real_eq_conv, real_not_lt_conv, 
   573                     real_not_le_conv, real_not_eq_conv, all_conv]))
   574   fun absremover ct = (literals_conv [@{term "op &"}, @{term "op |"}] [] 
   575                   (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv 
   576         try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
   577   val nct = capply @{cterm Trueprop} (capply @{cterm "Not"} ct)
   578   val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
   579   val tm0 = dest_arg (Thm.rhs_of th0)
   580   val th = if tm0 aconvc @{cterm False} then equal_implies_1_rule th0 else
   581    let 
   582     val (evs,bod) = strip_exists tm0
   583     val (avs,ibod) = strip_forall bod
   584     val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
   585     val th2 = overall [] [specl avs (assume (Thm.rhs_of th1))]
   586     val th3 = fold simple_choose evs (prove_hyp (equal_elim th1 (assume (capply @{cterm Trueprop} bod))) th2)
   587    in  Drule.implies_intr_hyps (prove_hyp (equal_elim th0 (assume nct)) th3)
   588    end
   589   in implies_elim (instantiate' [] [SOME ct] pth_final) th
   590  end
   591 in f
   592 end;
   593 
   594 (* A linear arithmetic prover *)
   595 local
   596   val linear_add = Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
   597   fun linear_cmul c = Ctermfunc.mapf (fn x => c */ x)
   598   val one_tm = @{cterm "1::real"}
   599   fun contradictory p (e,_) = ((Ctermfunc.is_undefined e) andalso not(p Rat.zero)) orelse
   600      ((gen_eq_set (op aconvc) (Ctermfunc.dom e, [one_tm])) andalso not(p(Ctermfunc.apply e one_tm)))
   601 
   602   fun linear_ineqs vars (les,lts) = 
   603    case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
   604     SOME r => r
   605   | NONE => 
   606    (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
   607      SOME r => r
   608    | NONE => 
   609      if null vars then error "linear_ineqs: no contradiction" else
   610      let 
   611       val ineqs = les @ lts
   612       fun blowup v =
   613        length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
   614        length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
   615        length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
   616       val  v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
   617                  (map (fn v => (v,blowup v)) vars)))
   618       fun addup (e1,p1) (e2,p2) acc =
   619        let 
   620         val c1 = Ctermfunc.tryapplyd e1 v Rat.zero 
   621         val c2 = Ctermfunc.tryapplyd e2 v Rat.zero
   622        in if c1 */ c2 >=/ Rat.zero then acc else
   623         let 
   624          val e1' = linear_cmul (Rat.abs c2) e1
   625          val e2' = linear_cmul (Rat.abs c1) e2
   626          val p1' = Product(Rational_lt(Rat.abs c2),p1)
   627          val p2' = Product(Rational_lt(Rat.abs c1),p2)
   628         in (linear_add e1' e2',Sum(p1',p2'))::acc
   629         end
   630        end
   631       val (les0,les1) = 
   632          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
   633       val (lts0,lts1) = 
   634          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
   635       val (lesp,lesn) = 
   636          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
   637       val (ltsp,ltsn) = 
   638          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
   639       val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
   640       val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
   641                       (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
   642      in linear_ineqs (remove (op aconvc) v vars) (les',lts')
   643      end)
   644 
   645   fun linear_eqs(eqs,les,lts) = 
   646    case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
   647     SOME r => r
   648   | NONE => (case eqs of 
   649     [] => 
   650      let val vars = remove (op aconvc) one_tm 
   651            (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom o fst) (les@lts) []) 
   652      in linear_ineqs vars (les,lts) end
   653    | (e,p)::es => 
   654      if Ctermfunc.is_undefined e then linear_eqs (es,les,lts) else
   655      let 
   656       val (x,c) = Ctermfunc.choose (Ctermfunc.undefine one_tm e)
   657       fun xform (inp as (t,q)) =
   658        let val d = Ctermfunc.tryapplyd t x Rat.zero in
   659         if d =/ Rat.zero then inp else
   660         let 
   661          val k = (Rat.neg d) */ Rat.abs c // c
   662          val e' = linear_cmul k e
   663          val t' = linear_cmul (Rat.abs c) t
   664          val p' = Eqmul(cterm_of_rat k,p)
   665          val q' = Product(Rational_lt(Rat.abs c),q) 
   666         in (linear_add e' t',Sum(p',q')) 
   667         end 
   668       end
   669      in linear_eqs(map xform es,map xform les,map xform lts)
   670      end)
   671 
   672   fun linear_prover (eq,le,lt) = 
   673    let 
   674     val eqs = map2 (fn p => fn n => (p,Axiom_eq n)) eq (0 upto (length eq - 1))
   675     val les = map2 (fn p => fn n => (p,Axiom_le n)) le (0 upto (length le - 1))
   676     val lts = map2 (fn p => fn n => (p,Axiom_lt n)) lt (0 upto (length lt - 1))
   677    in linear_eqs(eqs,les,lts)
   678    end 
   679   
   680   fun lin_of_hol ct = 
   681    if ct aconvc @{cterm "0::real"} then Ctermfunc.undefined
   682    else if not (is_comb ct) then Ctermfunc.onefunc (ct, Rat.one)
   683    else if is_ratconst ct then Ctermfunc.onefunc (one_tm, dest_ratconst ct)
   684    else
   685     let val (lop,r) = Thm.dest_comb ct 
   686     in if not (is_comb lop) then Ctermfunc.onefunc (ct, Rat.one)
   687        else
   688         let val (opr,l) = Thm.dest_comb lop 
   689         in if opr aconvc @{cterm "op + :: real =>_"} 
   690            then linear_add (lin_of_hol l) (lin_of_hol r)
   691            else if opr aconvc @{cterm "op * :: real =>_"} 
   692                    andalso is_ratconst l then Ctermfunc.onefunc (r, dest_ratconst l)
   693            else Ctermfunc.onefunc (ct, Rat.one)
   694         end
   695     end
   696 
   697   fun is_alien ct = case term_of ct of 
   698    Const(@{const_name "real"}, _)$ n => 
   699      if can HOLogic.dest_number n then false else true
   700   | _ => false
   701  open Thm
   702 in 
   703 fun real_linear_prover translator (eq,le,lt) = 
   704  let 
   705   val lhs = lin_of_hol o dest_arg1 o dest_arg o cprop_of
   706   val rhs = lin_of_hol o dest_arg o dest_arg o cprop_of
   707   val eq_pols = map lhs eq
   708   val le_pols = map rhs le
   709   val lt_pols = map rhs lt 
   710   val aliens =  filter is_alien
   711       (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom) 
   712           (eq_pols @ le_pols @ lt_pols) [])
   713   val le_pols' = le_pols @ map (fn v => Ctermfunc.onefunc (v,Rat.one)) aliens
   714   val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
   715   val le' = le @ map (fn a => instantiate' [] [SOME (dest_arg a)] @{thm real_of_nat_ge_zero}) aliens 
   716  in (translator (eq,le',lt) proof) : thm
   717  end
   718 end;
   719 
   720 (* A less general generic arithmetic prover dealing with abs,max and min*)
   721 
   722 local
   723  val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
   724  fun absmaxmin_elim_conv1 ctxt = 
   725     Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
   726 
   727  val absmaxmin_elim_conv2 =
   728   let 
   729    val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
   730    val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
   731    val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
   732    val abs_tm = @{cterm "abs :: real => _"}
   733    val p_tm = @{cpat "?P :: real => bool"}
   734    val x_tm = @{cpat "?x :: real"}
   735    val y_tm = @{cpat "?y::real"}
   736    val is_max = is_binop @{cterm "max :: real => _"}
   737    val is_min = is_binop @{cterm "min :: real => _"} 
   738    fun is_abs t = is_comb t andalso dest_fun t aconvc abs_tm
   739    fun eliminate_construct p c tm =
   740     let 
   741      val t = find_cterm p tm
   742      val th0 = (symmetric o beta_conversion false) (capply (cabs t tm) t)
   743      val (p,ax) = (dest_comb o Thm.rhs_of) th0
   744     in fconv_rule(arg_conv(binop_conv (arg_conv (beta_conversion false))))
   745                (transitive th0 (c p ax))
   746    end
   747 
   748    val elim_abs = eliminate_construct is_abs
   749     (fn p => fn ax => 
   750        instantiate ([], [(p_tm,p), (x_tm, dest_arg ax)]) pth_abs)
   751    val elim_max = eliminate_construct is_max
   752     (fn p => fn ax => 
   753       let val (ax,y) = dest_comb ax 
   754       in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)]) 
   755       pth_max end)
   756    val elim_min = eliminate_construct is_min
   757     (fn p => fn ax => 
   758       let val (ax,y) = dest_comb ax 
   759       in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)]) 
   760       pth_min end)
   761    in first_conv [elim_abs, elim_max, elim_min, all_conv]
   762   end;
   763 in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
   764         gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,
   765                        absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
   766 end;
   767 
   768 (* An instance for reals*) 
   769 
   770 fun gen_prover_real_arith ctxt prover = 
   771  let
   772   fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS
   773   val {add,mul,neg,pow,sub,main} = 
   774      Normalizer.semiring_normalizers_ord_wrapper ctxt
   775       (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) 
   776      simple_cterm_ord
   777 in gen_real_arith ctxt
   778    (cterm_of_rat, field_comp_conv, field_comp_conv,field_comp_conv,
   779     main,neg,add,mul, prover)
   780 end;
   781 
   782 fun real_arith ctxt = gen_prover_real_arith ctxt real_linear_prover;
   783 end
   784 
   785   (* Now the norm procedure for euclidean spaces *)
   786 
   787 
   788 signature NORM_ARITH = 
   789 sig
   790  val norm_arith : Proof.context -> conv
   791  val norm_arith_tac : Proof.context -> int -> tactic
   792 end
   793 
   794 structure NormArith : NORM_ARITH = 
   795 struct
   796 
   797  open Conv Thm Conv2;
   798  val bool_eq = op = : bool *bool -> bool
   799   fun dest_ratconst t = case term_of t of
   800    Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
   801  | Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd)
   802  | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
   803  fun is_ratconst t = can dest_ratconst t
   804  fun augment_norm b t acc = case term_of t of 
   805      Const(@{const_name norm}, _) $ _ => insert (eq_pair bool_eq (op aconvc)) (b,dest_arg t) acc
   806    | _ => acc
   807  fun find_normedterms t acc = case term_of t of
   808     @{term "op + :: real => _"}$_$_ =>
   809             find_normedterms (dest_arg1 t) (find_normedterms (dest_arg t) acc)
   810       | @{term "op * :: real => _"}$_$n =>
   811             if not (is_ratconst (dest_arg1 t)) then acc else
   812             augment_norm (dest_ratconst (dest_arg1 t) >=/ Rat.zero) 
   813                       (dest_arg t) acc
   814       | _ => augment_norm true t acc 
   815 
   816  val cterm_lincomb_neg = Ctermfunc.mapf Rat.neg
   817  fun cterm_lincomb_cmul c t = 
   818     if c =/ Rat.zero then Ctermfunc.undefined else Ctermfunc.mapf (fn x => x */ c) t
   819  fun cterm_lincomb_add l r = Ctermfunc.combine (curry op +/) (fn x => x =/ Rat.zero) l r
   820  fun cterm_lincomb_sub l r = cterm_lincomb_add l (cterm_lincomb_neg r)
   821  fun cterm_lincomb_eq l r = Ctermfunc.is_undefined (cterm_lincomb_sub l r)
   822 
   823  val int_lincomb_neg = Intfunc.mapf Rat.neg
   824  fun int_lincomb_cmul c t = 
   825     if c =/ Rat.zero then Intfunc.undefined else Intfunc.mapf (fn x => x */ c) t
   826  fun int_lincomb_add l r = Intfunc.combine (curry op +/) (fn x => x =/ Rat.zero) l r
   827  fun int_lincomb_sub l r = int_lincomb_add l (int_lincomb_neg r)
   828  fun int_lincomb_eq l r = Intfunc.is_undefined (int_lincomb_sub l r)
   829 
   830 fun vector_lincomb t = case term_of t of 
   831    Const(@{const_name plus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_])) $ _ $ _ =>
   832     cterm_lincomb_add (vector_lincomb (dest_arg1 t)) (vector_lincomb (dest_arg t))
   833  | Const(@{const_name minus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_])) $ _ $ _ =>
   834     cterm_lincomb_sub (vector_lincomb (dest_arg1 t)) (vector_lincomb (dest_arg t))
   835  | Const(@{const_name vector_scalar_mult},Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))$_$_ =>
   836     cterm_lincomb_cmul (dest_ratconst (dest_arg1 t)) (vector_lincomb (dest_arg t))
   837  | Const(@{const_name uminus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))$_ =>
   838      cterm_lincomb_neg (vector_lincomb (dest_arg t))
   839  | Const(@{const_name vec},_)$_ => 
   840    let 
   841      val b = ((snd o HOLogic.dest_number o term_of o dest_arg) t = 0 
   842                handle TERM _=> false)
   843    in if b then Ctermfunc.onefunc (t,Rat.one)
   844       else Ctermfunc.undefined
   845    end
   846  | _ => Ctermfunc.onefunc (t,Rat.one)
   847 
   848  fun vector_lincombs ts =
   849   fold_rev 
   850    (fn t => fn fns => case AList.lookup (op aconvc) fns t of
   851      NONE => 
   852        let val f = vector_lincomb t 
   853        in case find_first (fn (_,f') => cterm_lincomb_eq f f') fns of
   854            SOME (_,f') => (t,f') :: fns
   855          | NONE => (t,f) :: fns 
   856        end
   857    | SOME _ => fns) ts []
   858 
   859 fun replacenegnorms cv t = case term_of t of 
   860   @{term "op + :: real => _"}$_$_ => binop_conv (replacenegnorms cv) t
   861 | @{term "op * :: real => _"}$_$_ => 
   862     if dest_ratconst (dest_arg1 t) </ Rat.zero then arg_conv cv t else reflexive t
   863 | _ => reflexive t
   864 fun flip v eq = 
   865   if Ctermfunc.defined eq v 
   866   then Ctermfunc.update (v, Rat.neg (Ctermfunc.apply eq v)) eq else eq
   867 fun allsubsets s = case s of 
   868   [] => [[]]
   869 |(a::t) => let val res = allsubsets t in
   870                map (cons a) res @ res end
   871 fun evaluate env lin =
   872  Intfunc.fold (fn (x,c) => fn s => s +/ c */ (Intfunc.apply env x)) 
   873    lin Rat.zero
   874 
   875 fun solve (vs,eqs) = case (vs,eqs) of
   876   ([],[]) => SOME (Intfunc.onefunc (0,Rat.one))
   877  |(_,eq::oeqs) => 
   878    (case filter (member (op =) vs) (Intfunc.dom eq) of (*FIXME use find_first here*)
   879      [] => NONE
   880     | v::_ => 
   881        if Intfunc.defined eq v 
   882        then 
   883         let 
   884          val c = Intfunc.apply eq v
   885          val vdef = int_lincomb_cmul (Rat.neg (Rat.inv c)) eq
   886          fun eliminate eqn = if not (Intfunc.defined eqn v) then eqn 
   887                              else int_lincomb_add (int_lincomb_cmul (Intfunc.apply eqn v) vdef) eqn
   888         in (case solve (vs \ v,map eliminate oeqs) of
   889             NONE => NONE
   890           | SOME soln => SOME (Intfunc.update (v, evaluate soln (Intfunc.undefine v vdef)) soln))
   891         end
   892        else NONE)
   893 
   894 fun combinations k l = if k = 0 then [[]] else
   895  case l of 
   896   [] => []
   897 | h::t => map (cons h) (combinations (k - 1) t) @ combinations k t
   898 
   899 
   900 fun forall2 p l1 l2 = case (l1,l2) of 
   901    ([],[]) => true
   902  | (h1::t1,h2::t2) => p h1 h2 andalso forall2 p t1 t2
   903  | _ => false;
   904 
   905 
   906 fun vertices vs eqs =
   907  let 
   908   fun vertex cmb = case solve(vs,cmb) of
   909     NONE => NONE
   910    | SOME soln => SOME (map (fn v => Intfunc.tryapplyd soln v Rat.zero) vs)
   911   val rawvs = map_filter vertex (combinations (length vs) eqs)
   912   val unset = filter (forall (fn c => c >=/ Rat.zero)) rawvs 
   913  in fold_rev (insert (uncurry (forall2 (curry op =/)))) unset [] 
   914  end 
   915 
   916 fun subsumes l m = forall2 (fn x => fn y => Rat.abs x <=/ Rat.abs y) l m 
   917 
   918 fun subsume todo dun = case todo of
   919  [] => dun
   920 |v::ovs => 
   921    let val dun' = if exists (fn w => subsumes w v) dun then dun
   922                   else v::(filter (fn w => not(subsumes v w)) dun) 
   923    in subsume ovs dun' 
   924    end;
   925 
   926 fun match_mp PQ P = P RS PQ;
   927 
   928 fun cterm_of_rat x = 
   929 let val (a, b) = Rat.quotient_of_rat x
   930 in 
   931  if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
   932   else Thm.capply (Thm.capply @{cterm "op / :: real => _"} 
   933                    (Numeral.mk_cnumber @{ctyp "real"} a))
   934         (Numeral.mk_cnumber @{ctyp "real"} b)
   935 end;
   936 
   937 fun norm_cmul_rule c th = instantiate' [] [SOME (cterm_of_rat c)] (th RS @{thm norm_cmul_rule_thm});
   938 
   939 fun norm_add_rule th1 th2 = [th1, th2] MRS @{thm norm_add_rule_thm};
   940 
   941   (* I think here the static context should be sufficient!! *)
   942 fun inequality_canon_rule ctxt = 
   943  let 
   944   (* FIXME : Should be computed statically!! *)
   945   val real_poly_conv = 
   946     Normalizer.semiring_normalize_wrapper ctxt
   947      (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
   948  in fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv arg_conv (field_comp_conv then_conv real_poly_conv)))
   949 end;
   950 
   951  fun absc cv ct = case term_of ct of 
   952  Abs (v,_, _) => 
   953   let val (x,t) = Thm.dest_abs (SOME v) ct
   954   in Thm.abstract_rule ((fst o dest_Free o term_of) x) x (cv t)
   955   end
   956  | _ => all_conv ct;
   957 
   958 fun sub_conv cv ct = (comb_conv cv else_conv absc cv) ct;
   959 fun botc1 conv ct = 
   960   ((sub_conv (botc1 conv)) then_conv (conv else_conv all_conv)) ct;
   961 
   962  fun rewrs_conv eqs ct = first_conv (map rewr_conv eqs) ct;
   963  val apply_pth1 = rewr_conv @{thm pth_1};
   964  val apply_pth2 = rewr_conv @{thm pth_2};
   965  val apply_pth3 = rewr_conv @{thm pth_3};
   966  val apply_pth4 = rewrs_conv @{thms pth_4};
   967  val apply_pth5 = rewr_conv @{thm pth_5};
   968  val apply_pth6 = rewr_conv @{thm pth_6};
   969  val apply_pth7 = rewrs_conv @{thms pth_7};
   970  val apply_pth8 = rewr_conv @{thm pth_8} then_conv arg1_conv field_comp_conv then_conv (try_conv (rewr_conv (mk_meta_eq @{thm vector_smult_lzero})));
   971  val apply_pth9 = rewrs_conv @{thms pth_9} then_conv arg1_conv (arg1_conv field_comp_conv);
   972  val apply_ptha = rewr_conv @{thm pth_a};
   973  val apply_pthb = rewrs_conv @{thms pth_b};
   974  val apply_pthc = rewrs_conv @{thms pth_c};
   975  val apply_pthd = try_conv (rewr_conv @{thm pth_d});
   976 
   977 fun headvector t = case t of 
   978   Const(@{const_name plus}, Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))$
   979    (Const(@{const_name vector_scalar_mult}, _)$l$v)$r => v
   980  | Const(@{const_name vector_scalar_mult}, _)$l$v => v
   981  | _ => error "headvector: non-canonical term"
   982 
   983 fun vector_cmul_conv ct =
   984    ((apply_pth5 then_conv arg1_conv field_comp_conv) else_conv
   985     (apply_pth6 then_conv binop_conv vector_cmul_conv)) ct
   986 
   987 fun vector_add_conv ct = apply_pth7 ct 
   988  handle CTERM _ => 
   989   (apply_pth8 ct 
   990    handle CTERM _ => 
   991     (case term_of ct of 
   992      Const(@{const_name plus},_)$lt$rt =>
   993       let 
   994        val l = headvector lt 
   995        val r = headvector rt
   996       in (case TermOrd.fast_term_ord (l,r) of
   997          LESS => (apply_pthb then_conv arg_conv vector_add_conv 
   998                   then_conv apply_pthd) ct
   999         | GREATER => (apply_pthc then_conv arg_conv vector_add_conv 
  1000                      then_conv apply_pthd) ct 
  1001         | EQUAL => (apply_pth9 then_conv 
  1002                 ((apply_ptha then_conv vector_add_conv) else_conv 
  1003               arg_conv vector_add_conv then_conv apply_pthd)) ct)
  1004       end
  1005      | _ => reflexive ct))
  1006 
  1007 fun vector_canon_conv ct = case term_of ct of
  1008  Const(@{const_name plus},_)$_$_ =>
  1009   let 
  1010    val ((p,l),r) = Thm.dest_comb ct |>> Thm.dest_comb
  1011    val lth = vector_canon_conv l 
  1012    val rth = vector_canon_conv r
  1013    val th = Drule.binop_cong_rule p lth rth 
  1014   in fconv_rule (arg_conv vector_add_conv) th end
  1015 
  1016 | Const(@{const_name vector_scalar_mult}, _)$_$_ =>
  1017   let 
  1018    val (p,r) = Thm.dest_comb ct
  1019    val rth = Drule.arg_cong_rule p (vector_canon_conv r) 
  1020   in fconv_rule (arg_conv (apply_pth4 else_conv vector_cmul_conv)) rth
  1021   end
  1022 
  1023 | Const(@{const_name minus},_)$_$_ => (apply_pth2 then_conv vector_canon_conv) ct
  1024 
  1025 | Const(@{const_name uminus},_)$_ => (apply_pth3 then_conv vector_canon_conv) ct
  1026 
  1027 | Const(@{const_name vec},_)$n => 
  1028   let val n = Thm.dest_arg ct
  1029   in if is_ratconst n andalso not (dest_ratconst n =/ Rat.zero) 
  1030      then reflexive ct else apply_pth1 ct
  1031   end
  1032 
  1033 | _ => apply_pth1 ct
  1034 
  1035 fun norm_canon_conv ct = case term_of ct of
  1036   Const(@{const_name norm},_)$_ => arg_conv vector_canon_conv ct
  1037  | _ => raise CTERM ("norm_canon_conv", [ct])
  1038 
  1039 fun fold_rev2 f [] [] z = z
  1040  | fold_rev2 f (x::xs) (y::ys) z = f x y (fold_rev2 f xs ys z)
  1041  | fold_rev2 f _ _ _ = raise UnequalLengths;
  1042 
  1043 fun int_flip v eq = 
  1044   if Intfunc.defined eq v 
  1045   then Intfunc.update (v, Rat.neg (Intfunc.apply eq v)) eq else eq;
  1046 
  1047 local
  1048  val pth_zero = @{thm "norm_0"}
  1049  val tv_n = (hd o tl o dest_ctyp o ctyp_of_term o dest_arg o dest_arg1 o dest_arg o cprop_of)
  1050              pth_zero
  1051  val concl = dest_arg o cprop_of 
  1052  fun real_vector_combo_prover ctxt translator (nubs,ges,gts) = 
  1053   let 
  1054    (* FIXME: Should be computed statically!!*)
  1055    val real_poly_conv = 
  1056       Normalizer.semiring_normalize_wrapper ctxt
  1057        (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
  1058    val sources = map (dest_arg o dest_arg1 o concl) nubs
  1059    val rawdests = fold_rev (find_normedterms o dest_arg o concl) (ges @ gts) [] 
  1060    val _ = if not (forall fst rawdests) then error "real_vector_combo_prover: Sanity check" 
  1061            else ()
  1062    val dests = distinct (op aconvc) (map snd rawdests)
  1063    val srcfuns = map vector_lincomb sources
  1064    val destfuns = map vector_lincomb dests 
  1065    val vvs = fold_rev (curry (gen_union op aconvc) o Ctermfunc.dom) (srcfuns @ destfuns) []
  1066    val n = length srcfuns
  1067    val nvs = 1 upto n
  1068    val srccombs = srcfuns ~~ nvs
  1069    fun consider d =
  1070     let 
  1071      fun coefficients x =
  1072       let 
  1073        val inp = if Ctermfunc.defined d x then Intfunc.onefunc (0, Rat.neg(Ctermfunc.apply d x))
  1074                       else Intfunc.undefined 
  1075       in fold_rev (fn (f,v) => fn g => if Ctermfunc.defined f x then Intfunc.update (v, Ctermfunc.apply f x) g else g) srccombs inp 
  1076       end
  1077      val equations = map coefficients vvs
  1078      val inequalities = map (fn n => Intfunc.onefunc (n,Rat.one)) nvs
  1079      fun plausiblevertices f =
  1080       let 
  1081        val flippedequations = map (fold_rev int_flip f) equations
  1082        val constraints = flippedequations @ inequalities
  1083        val rawverts = vertices nvs constraints
  1084        fun check_solution v =
  1085         let 
  1086           val f = fold_rev2 (curry Intfunc.update) nvs v (Intfunc.onefunc (0, Rat.one))
  1087         in forall (fn e => evaluate f e =/ Rat.zero) flippedequations
  1088         end
  1089        val goodverts = filter check_solution rawverts
  1090        val signfixups = map (fn n => if n mem_int  f then ~1 else 1) nvs 
  1091       in map (map2 (fn s => fn c => Rat.rat_of_int s */ c) signfixups) goodverts
  1092       end
  1093      val allverts = fold_rev append (map plausiblevertices (allsubsets nvs)) [] 
  1094     in subsume allverts []
  1095     end
  1096    fun compute_ineq v =
  1097     let 
  1098      val ths = map_filter (fn (v,t) => if v =/ Rat.zero then NONE 
  1099                                      else SOME(norm_cmul_rule v t))
  1100                             (v ~~ nubs) 
  1101     in inequality_canon_rule ctxt (end_itlist norm_add_rule ths)
  1102     end
  1103    val ges' = map_filter (try compute_ineq) (fold_rev (append o consider) destfuns []) @
  1104                  map (inequality_canon_rule ctxt) nubs @ ges
  1105    val zerodests = filter
  1106         (fn t => null (Ctermfunc.dom (vector_lincomb t))) (map snd rawdests)
  1107 
  1108   in RealArith.real_linear_prover translator
  1109         (map (fn t => instantiate ([(tv_n,(hd o tl o dest_ctyp o ctyp_of_term) t)],[]) pth_zero)
  1110             zerodests,
  1111         map (fconv_rule (once_depth_conv (norm_canon_conv) then_conv
  1112                        arg_conv (arg_conv real_poly_conv))) ges',
  1113         map (fconv_rule (once_depth_conv (norm_canon_conv) then_conv 
  1114                        arg_conv (arg_conv real_poly_conv))) gts)
  1115   end
  1116 in val real_vector_combo_prover = real_vector_combo_prover
  1117 end;
  1118 
  1119 local
  1120  val pth = @{thm norm_imp_pos_and_ge}
  1121  val norm_mp = match_mp pth
  1122  val concl = dest_arg o cprop_of
  1123  fun conjunct1 th = th RS @{thm conjunct1}
  1124  fun conjunct2 th = th RS @{thm conjunct2}
  1125  fun C f x y = f y x
  1126 fun real_vector_ineq_prover ctxt translator (ges,gts) = 
  1127  let 
  1128 (*   val _ = error "real_vector_ineq_prover: pause" *)
  1129   val ntms = fold_rev find_normedterms (map (dest_arg o concl) (ges @ gts)) []
  1130   val lctab = vector_lincombs (map snd (filter (not o fst) ntms))
  1131   val (fxns, ctxt') = Variable.variant_fixes (replicate (length lctab) "x") ctxt
  1132   fun mk_norm t = capply (instantiate_cterm' [SOME (ctyp_of_term t)] [] @{cpat "norm :: (?'a :: norm) => real"}) t
  1133   fun mk_equals l r = capply (capply (instantiate_cterm' [SOME (ctyp_of_term l)] [] @{cpat "op == :: ?'a =>_"}) l) r
  1134   val asl = map2 (fn (t,_) => fn n => assume (mk_equals (mk_norm t) (cterm_of (ProofContext.theory_of ctxt') (Free(n,@{typ real}))))) lctab fxns
  1135   val replace_conv = try_conv (rewrs_conv asl)
  1136   val replace_rule = fconv_rule (funpow 2 arg_conv (replacenegnorms replace_conv))
  1137   val ges' =
  1138        fold_rev (fn th => fn ths => conjunct1(norm_mp th)::ths)
  1139               asl (map replace_rule ges)
  1140   val gts' = map replace_rule gts
  1141   val nubs = map (conjunct2 o norm_mp) asl
  1142   val th1 = real_vector_combo_prover ctxt' translator (nubs,ges',gts')
  1143   val shs = filter (member (fn (t,th) => t aconvc cprop_of th) asl) (#hyps (crep_thm th1)) 
  1144   val th11 = hd (Variable.export ctxt' ctxt [fold implies_intr shs th1])
  1145   val cps = map (swap o dest_equals) (cprems_of th11)
  1146   val th12 = instantiate ([], cps) th11
  1147   val th13 = fold (C implies_elim) (map (reflexive o snd) cps) th12;
  1148  in hd (Variable.export ctxt' ctxt [th13])
  1149  end 
  1150 in val real_vector_ineq_prover = real_vector_ineq_prover
  1151 end;
  1152 
  1153 local
  1154  val rawrule = fconv_rule (arg_conv (rewr_conv @{thm real_eq_0_iff_le_ge_0}))
  1155  fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
  1156  fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS;
  1157   (* FIXME: Lookup in the context every time!!! Fix this !!!*)
  1158  fun splitequation ctxt th acc =
  1159   let 
  1160    val real_poly_neg_conv = #neg
  1161        (Normalizer.semiring_normalizers_ord_wrapper ctxt
  1162         (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) simple_cterm_ord)
  1163    val (th1,th2) = conj_pair(rawrule th)
  1164   in th1::fconv_rule (arg_conv (arg_conv real_poly_neg_conv)) th2::acc
  1165   end
  1166 in fun real_vector_prover ctxt translator (eqs,ges,gts) =
  1167      real_vector_ineq_prover ctxt translator
  1168          (fold_rev (splitequation ctxt) eqs ges,gts)
  1169 end;
  1170 
  1171   fun init_conv ctxt = 
  1172    Simplifier.rewrite (Simplifier.context ctxt 
  1173      (HOL_basic_ss addsimps ([@{thm vec_0}, @{thm vec_1}, @{thm dist_def}, @{thm diff_0_right}, @{thm right_minus}, @{thm diff_self}, @{thm norm_0}] @ @{thms arithmetic_simps} @ @{thms norm_pths})))
  1174    then_conv field_comp_conv 
  1175    then_conv nnf_conv
  1176 
  1177  fun pure ctxt = RealArith.gen_prover_real_arith ctxt (real_vector_prover ctxt);
  1178  fun norm_arith ctxt ct = 
  1179   let 
  1180    val ctxt' = Variable.declare_term (term_of ct) ctxt
  1181    val th = init_conv ctxt' ct
  1182   in equal_elim (Drule.arg_cong_rule @{cterm Trueprop} (symmetric th)) 
  1183                 (pure ctxt' (rhs_of th))
  1184  end
  1185 
  1186  fun norm_arith_tac ctxt = 
  1187    clarify_tac HOL_cs THEN'
  1188    ObjectLogic.full_atomize_tac THEN'
  1189    CSUBGOAL ( fn (p,i) => rtac (norm_arith ctxt (Thm.dest_arg p )) i);
  1190 
  1191 end;