src/HOL/Library/normarith.ML
 author chaieb Mon Feb 09 17:21:46 2009 +0000 (2009-02-09) changeset 29847 af32126ee729 parent 29843 4bb780545478 child 30373 ffdd7a1f1ff0 permissions -rw-r--r--
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;
44 functor FuncFun(Key: KEY) : FUNC=
45 struct
47 type key = Key.key;
48 structure Tab = TableFun(Key);
49 type 'a T = 'a Tab.table;
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;
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;
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"
79 fun onefunc kv = update kv undefined
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
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}
108 end;
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);
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;
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);
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;
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;
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
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;
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);
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;
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;
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
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
238 structure RealArith (* : REAL_ARITH *)=
239 struct
241  open Conv Thm Conv2;;
242 (* ------------------------------------------------------------------------- *)
243 (* Data structure for Positivstellensatz refutations.                        *)
244 (* ------------------------------------------------------------------------- *)
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 *)
259 fun conjunctions th = case try Conjunction.elim th of
260    SOME (th1,th2) => (conjunctions th1) @ conjunctions th2
261  | NONE => [th];
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;
269 val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
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 ;
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;
286 val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
287 val pth_square = @{lemma "x * x >= (0::real)"  by simp};
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+};
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+}
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)});
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};
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)};
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)};
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)};
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;
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;
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  | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
373  fun is_ratconst t = can dest_ratconst t
375 fun find_term p t = if p t then t else
376  case t of
377   a\$b => (find_term p a handle TERM _ => find_term p b)
378  | Abs (_,_,t') => find_term p t'
379  | _ => raise TERM ("find_term",[t]);
381 fun find_cterm p t = if p t then t else
382  case term_of t of
383   a\$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
384  | Abs (_,_,t') => find_cterm p (Thm.dest_abs NONE t |> snd)
385  | _ => raise CTERM ("find_cterm",[t]);
388     (* A general real arithmetic prover *)
390 fun gen_gen_real_arith ctxt (mk_numeric,
391        numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
393        absconv1,absconv2,prover) =
394 let
395  open Conv Thm;
396  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}]
397  val prenex_ss = HOL_basic_ss addsimps prenex_simps
398  val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
399  val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
400  val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
401  val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
402  val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
403  val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
404  fun eqT_elim th = equal_elim (symmetric th) @{thm TrueI}
405  fun oprconv cv ct =
406   let val g = Thm.dest_fun2 ct
407   in if g aconvc @{cterm "op <= :: real => _"}
408        orelse g aconvc @{cterm "op < :: real => _"}
409      then arg_conv cv ct else arg1_conv cv ct
410   end
412  fun real_ineq_conv th ct =
413   let
414    val th' = (instantiate (match (lhs_of th, ct)) th
415       handle MATCH => raise CTERM ("real_ineq_conv", [ct]))
416   in transitive th' (oprconv poly_conv (Thm.rhs_of th'))
417   end
418   val [real_lt_conv, real_le_conv, real_eq_conv,
419        real_not_lt_conv, real_not_le_conv, _] =
420        map real_ineq_conv pth
421   fun match_mp_rule ths ths' =
422    let
423      fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
424       | th::ths => (ths' MRS th handle THM _ => f ths ths')
425    in f ths ths' end
426   fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
427          (match_mp_rule pth_mul [th, th'])
430   fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
431        (instantiate' [] [SOME ct] (th RS pth_emul))
432   fun square_rule t = fconv_rule (arg_conv (oprconv poly_mul_conv))
433        (instantiate' [] [SOME t] pth_square)
435   fun hol_of_positivstellensatz(eqs,les,lts) =
436    let
437     fun translate prf = case prf of
438         Axiom_eq n => nth eqs n
439       | Axiom_le n => nth les n
440       | Axiom_lt n => nth lts n
441       | Rational_eq x => eqT_elim(numeric_eq_conv(capply @{cterm Trueprop}
442                           (capply (capply @{cterm "op =::real => _"} (mk_numeric x))
443                                @{cterm "0::real"})))
444       | Rational_le x => eqT_elim(numeric_ge_conv(capply @{cterm Trueprop}
445                           (capply (capply @{cterm "op <=::real => _"}
446                                      @{cterm "0::real"}) (mk_numeric x))))
447       | Rational_lt x => eqT_elim(numeric_gt_conv(capply @{cterm Trueprop}
448                       (capply (capply @{cterm "op <::real => _"} @{cterm "0::real"})
449                         (mk_numeric x))))
450       | Square t => square_rule t
451       | Eqmul(t,p) => emul_rule t (translate p)
452       | Sum(p1,p2) => add_rule (translate p1) (translate p2)
453       | Product(p1,p2) => mul_rule (translate p1) (translate p2)
454    in fn prf =>
455       fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv])
456           (translate prf)
457    end
459   val init_conv = presimp_conv then_conv
460       nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
461       weak_dnf_conv
463   val concl = dest_arg o cprop_of
464   fun is_binop opr ct = (dest_fun2 ct aconvc opr handle CTERM _ => false)
465   val is_req = is_binop @{cterm "op =:: real => _"}
466   val is_ge = is_binop @{cterm "op <=:: real => _"}
467   val is_gt = is_binop @{cterm "op <:: real => _"}
468   val is_conj = is_binop @{cterm "op &"}
469   val is_disj = is_binop @{cterm "op |"}
470   fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
471   fun disj_cases th th1 th2 =
472    let val (p,q) = dest_binop (concl th)
473        val c = concl th1
474        val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
475    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)
476    end
477  fun overall dun ths = case ths of
478   [] =>
479    let
480     val (eq,ne) = List.partition (is_req o concl) dun
481      val (le,nl) = List.partition (is_ge o concl) ne
482      val lt = filter (is_gt o concl) nl
483     in prover hol_of_positivstellensatz (eq,le,lt) end
484  | th::oths =>
485    let
486     val ct = concl th
487    in
488     if is_conj ct  then
489      let
490       val (th1,th2) = conj_pair th in
491       overall dun (th1::th2::oths) end
492     else if is_disj ct then
493       let
494        val th1 = overall dun (assume (capply @{cterm Trueprop} (dest_arg1 ct))::oths)
495        val th2 = overall dun (assume (capply @{cterm Trueprop} (dest_arg ct))::oths)
496       in disj_cases th th1 th2 end
497    else overall (th::dun) oths
498   end
499   fun dest_binary b ct = if is_binop b ct then dest_binop ct
500                          else raise CTERM ("dest_binary",[b,ct])
501   val dest_eq = dest_binary @{cterm "op = :: real => _"}
502   val neq_th = nth pth 5
503   fun real_not_eq_conv ct =
504    let
505     val (l,r) = dest_eq (dest_arg ct)
506     val th = instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
507     val th_p = poly_conv(dest_arg(dest_arg1(Thm.rhs_of th)))
508     val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
509     val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
510     val th' = Drule.binop_cong_rule @{cterm "op |"}
511      (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
512      (Drule.arg_cong_rule (capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
513     in transitive th th'
514   end
515  fun equal_implies_1_rule PQ =
516   let
517    val P = lhs_of PQ
518   in implies_intr P (equal_elim PQ (assume P))
519   end
520  (* FIXME!!! Copied from groebner.ml *)
521  val strip_exists =
522   let fun h (acc, t) =
523    case (term_of t) of
524     Const("Ex",_)\$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
525   | _ => (acc,t)
526   in fn t => h ([],t)
527   end
528   fun name_of x = case term_of x of
529    Free(s,_) => s
530  | Var ((s,_),_) => s
531  | _ => "x"
533   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   val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
537  fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
538  fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
540  fun choose v th th' = case concl_of th of
541    @{term Trueprop} \$ (Const("Ex",_)\$_) =>
542     let
543      val p = (funpow 2 Thm.dest_arg o cprop_of) th
544      val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
545      val th0 = fconv_rule (Thm.beta_conversion true)
546          (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
547      val pv = (Thm.rhs_of o Thm.beta_conversion true)
548            (Thm.capply @{cterm Trueprop} (Thm.capply p v))
549      val th1 = forall_intr v (implies_intr pv th')
550     in implies_elim (implies_elim th0 th) th1  end
551  | _ => raise THM ("choose",0,[th, th'])
553   fun simple_choose v th =
554      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  val strip_forall =
557   let fun h (acc, t) =
558    case (term_of t) of
559     Const("All",_)\$Abs(x,T,p) => h (dest_abs NONE (dest_arg t) |>> (fn v => v::acc))
560   | _ => (acc,t)
561   in fn t => h ([],t)
562   end
564  fun f ct =
565   let
566    val nnf_norm_conv' =
567      nnf_conv then_conv
568      literals_conv [@{term "op &"}, @{term "op |"}] []
569      (cache_conv
570        (first_conv [real_lt_conv, real_le_conv,
571                     real_eq_conv, real_not_lt_conv,
572                     real_not_le_conv, real_not_eq_conv, all_conv]))
573   fun absremover ct = (literals_conv [@{term "op &"}, @{term "op |"}] []
574                   (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv
575         try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
576   val nct = capply @{cterm Trueprop} (capply @{cterm "Not"} ct)
577   val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
578   val tm0 = dest_arg (Thm.rhs_of th0)
579   val th = if tm0 aconvc @{cterm False} then equal_implies_1_rule th0 else
580    let
581     val (evs,bod) = strip_exists tm0
582     val (avs,ibod) = strip_forall bod
583     val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
584     val th2 = overall [] [specl avs (assume (Thm.rhs_of th1))]
585     val th3 = fold simple_choose evs (prove_hyp (equal_elim th1 (assume (capply @{cterm Trueprop} bod))) th2)
586    in  Drule.implies_intr_hyps (prove_hyp (equal_elim th0 (assume nct)) th3)
587    end
588   in implies_elim (instantiate' [] [SOME ct] pth_final) th
589  end
590 in f
591 end;
593 (* A linear arithmetic prover *)
594 local
595   val linear_add = Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
596   fun linear_cmul c = Ctermfunc.mapf (fn x => c */ x)
597   val one_tm = @{cterm "1::real"}
598   fun contradictory p (e,_) = ((Ctermfunc.is_undefined e) andalso not(p Rat.zero)) orelse
599      ((gen_eq_set (op aconvc) (Ctermfunc.dom e, [one_tm])) andalso not(p(Ctermfunc.apply e one_tm)))
601   fun linear_ineqs vars (les,lts) =
602    case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
603     SOME r => r
604   | NONE =>
605    (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
606      SOME r => r
607    | NONE =>
608      if null vars then error "linear_ineqs: no contradiction" else
609      let
610       val ineqs = les @ lts
611       fun blowup v =
612        length(filter (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
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       val  v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
616                  (map (fn v => (v,blowup v)) vars)))
617       fun addup (e1,p1) (e2,p2) acc =
618        let
619         val c1 = Ctermfunc.tryapplyd e1 v Rat.zero
620         val c2 = Ctermfunc.tryapplyd e2 v Rat.zero
621        in if c1 */ c2 >=/ Rat.zero then acc else
622         let
623          val e1' = linear_cmul (Rat.abs c2) e1
624          val e2' = linear_cmul (Rat.abs c1) e2
625          val p1' = Product(Rational_lt(Rat.abs c2),p1)
626          val p2' = Product(Rational_lt(Rat.abs c1),p2)
628         end
629        end
630       val (les0,les1) =
631          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
632       val (lts0,lts1) =
633          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
634       val (lesp,lesn) =
635          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
636       val (ltsp,ltsn) =
637          List.partition (fn (e,_) => Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
638       val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
639       val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
640                       (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
641      in linear_ineqs (remove (op aconvc) v vars) (les',lts')
642      end)
644   fun linear_eqs(eqs,les,lts) =
645    case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
646     SOME r => r
647   | NONE => (case eqs of
648     [] =>
649      let val vars = remove (op aconvc) one_tm
650            (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom o fst) (les@lts) [])
651      in linear_ineqs vars (les,lts) end
652    | (e,p)::es =>
653      if Ctermfunc.is_undefined e then linear_eqs (es,les,lts) else
654      let
655       val (x,c) = Ctermfunc.choose (Ctermfunc.undefine one_tm e)
656       fun xform (inp as (t,q)) =
657        let val d = Ctermfunc.tryapplyd t x Rat.zero in
658         if d =/ Rat.zero then inp else
659         let
660          val k = (Rat.neg d) */ Rat.abs c // c
661          val e' = linear_cmul k e
662          val t' = linear_cmul (Rat.abs c) t
663          val p' = Eqmul(cterm_of_rat k,p)
664          val q' = Product(Rational_lt(Rat.abs c),q)
666         end
667       end
668      in linear_eqs(map xform es,map xform les,map xform lts)
669      end)
671   fun linear_prover (eq,le,lt) =
672    let
673     val eqs = map2 (fn p => fn n => (p,Axiom_eq n)) eq (0 upto (length eq - 1))
674     val les = map2 (fn p => fn n => (p,Axiom_le n)) le (0 upto (length le - 1))
675     val lts = map2 (fn p => fn n => (p,Axiom_lt n)) lt (0 upto (length lt - 1))
676    in linear_eqs(eqs,les,lts)
677    end
679   fun lin_of_hol ct =
680    if ct aconvc @{cterm "0::real"} then Ctermfunc.undefined
681    else if not (is_comb ct) then Ctermfunc.onefunc (ct, Rat.one)
682    else if is_ratconst ct then Ctermfunc.onefunc (one_tm, dest_ratconst ct)
683    else
684     let val (lop,r) = Thm.dest_comb ct
685     in if not (is_comb lop) then Ctermfunc.onefunc (ct, Rat.one)
686        else
687         let val (opr,l) = Thm.dest_comb lop
688         in if opr aconvc @{cterm "op + :: real =>_"}
689            then linear_add (lin_of_hol l) (lin_of_hol r)
690            else if opr aconvc @{cterm "op * :: real =>_"}
691                    andalso is_ratconst l then Ctermfunc.onefunc (r, dest_ratconst l)
692            else Ctermfunc.onefunc (ct, Rat.one)
693         end
694     end
696   fun is_alien ct = case term_of ct of
697    Const(@{const_name "real"}, _)\$ n =>
698      if can HOLogic.dest_number n then false else true
699   | _ => false
700  open Thm
701 in
702 fun real_linear_prover translator (eq,le,lt) =
703  let
704   val lhs = lin_of_hol o dest_arg1 o dest_arg o cprop_of
705   val rhs = lin_of_hol o dest_arg o dest_arg o cprop_of
706   val eq_pols = map lhs eq
707   val le_pols = map rhs le
708   val lt_pols = map rhs lt
709   val aliens =  filter is_alien
710       (fold_rev (curry (gen_union (op aconvc)) o Ctermfunc.dom)
711           (eq_pols @ le_pols @ lt_pols) [])
712   val le_pols' = le_pols @ map (fn v => Ctermfunc.onefunc (v,Rat.one)) aliens
713   val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
714   val le' = le @ map (fn a => instantiate' [] [SOME (dest_arg a)] @{thm real_of_nat_ge_zero}) aliens
715  in (translator (eq,le',lt) proof) : thm
716  end
717 end;
719 (* A less general generic arithmetic prover dealing with abs,max and min*)
721 local
722  val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
723  fun absmaxmin_elim_conv1 ctxt =
724     Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
726  val absmaxmin_elim_conv2 =
727   let
728    val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
729    val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
730    val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
731    val abs_tm = @{cterm "abs :: real => _"}
732    val p_tm = @{cpat "?P :: real => bool"}
733    val x_tm = @{cpat "?x :: real"}
734    val y_tm = @{cpat "?y::real"}
735    val is_max = is_binop @{cterm "max :: real => _"}
736    val is_min = is_binop @{cterm "min :: real => _"}
737    fun is_abs t = is_comb t andalso dest_fun t aconvc abs_tm
738    fun eliminate_construct p c tm =
739     let
740      val t = find_cterm p tm
741      val th0 = (symmetric o beta_conversion false) (capply (cabs t tm) t)
742      val (p,ax) = (dest_comb o Thm.rhs_of) th0
743     in fconv_rule(arg_conv(binop_conv (arg_conv (beta_conversion false))))
744                (transitive th0 (c p ax))
745    end
747    val elim_abs = eliminate_construct is_abs
748     (fn p => fn ax =>
749        instantiate ([], [(p_tm,p), (x_tm, dest_arg ax)]) pth_abs)
750    val elim_max = eliminate_construct is_max
751     (fn p => fn ax =>
752       let val (ax,y) = dest_comb ax
753       in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)])
754       pth_max end)
755    val elim_min = eliminate_construct is_min
756     (fn p => fn ax =>
757       let val (ax,y) = dest_comb ax
758       in  instantiate ([], [(p_tm,p), (x_tm, dest_arg ax), (y_tm,y)])
759       pth_min end)
760    in first_conv [elim_abs, elim_max, elim_min, all_conv]
761   end;
762 in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
764                        absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
765 end;
767 (* An instance for reals*)
769 fun gen_prover_real_arith ctxt prover =
770  let
771   fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS
773      Normalizer.semiring_normalizers_ord_wrapper ctxt
774       (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
775      simple_cterm_ord
776 in gen_real_arith ctxt
777    (cterm_of_rat, field_comp_conv, field_comp_conv,field_comp_conv,
779 end;
781 fun real_arith ctxt = gen_prover_real_arith ctxt real_linear_prover;
782 end
784   (* Now the norm procedure for euclidean spaces *)
787 signature NORM_ARITH =
788 sig
789  val norm_arith : Proof.context -> conv
790  val norm_arith_tac : Proof.context -> int -> tactic
791 end
793 structure NormArith : NORM_ARITH =
794 struct
796  open Conv Thm Conv2;
797  val bool_eq = op = : bool *bool -> bool
798  fun dest_ratconst t = case term_of t of
799    Const(@{const_name divide}, _)\$a\$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
800  | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
801  fun is_ratconst t = can dest_ratconst t
802  fun augment_norm b t acc = case term_of t of
803      Const(@{const_name norm}, _) \$ _ => insert (eq_pair bool_eq (op aconvc)) (b,dest_arg t) acc
804    | _ => acc
805  fun find_normedterms t acc = case term_of t of
806     @{term "op + :: real => _"}\$_\$_ =>
807             find_normedterms (dest_arg1 t) (find_normedterms (dest_arg t) acc)
808       | @{term "op * :: real => _"}\$_\$n =>
809             if not (is_ratconst (dest_arg1 t)) then acc else
810             augment_norm (dest_ratconst (dest_arg1 t) >=/ Rat.zero)
811                       (dest_arg t) acc
812       | _ => augment_norm true t acc
814  val cterm_lincomb_neg = Ctermfunc.mapf Rat.neg
815  fun cterm_lincomb_cmul c t =
816     if c =/ Rat.zero then Ctermfunc.undefined else Ctermfunc.mapf (fn x => x */ c) t
817  fun cterm_lincomb_add l r = Ctermfunc.combine (curry op +/) (fn x => x =/ Rat.zero) l r
818  fun cterm_lincomb_sub l r = cterm_lincomb_add l (cterm_lincomb_neg r)
819  fun cterm_lincomb_eq l r = Ctermfunc.is_undefined (cterm_lincomb_sub l r)
821  val int_lincomb_neg = Intfunc.mapf Rat.neg
822  fun int_lincomb_cmul c t =
823     if c =/ Rat.zero then Intfunc.undefined else Intfunc.mapf (fn x => x */ c) t
824  fun int_lincomb_add l r = Intfunc.combine (curry op +/) (fn x => x =/ Rat.zero) l r
825  fun int_lincomb_sub l r = int_lincomb_add l (int_lincomb_neg r)
826  fun int_lincomb_eq l r = Intfunc.is_undefined (int_lincomb_sub l r)
828 fun vector_lincomb t = case term_of t of
829    Const(@{const_name plus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_])) \$ _ \$ _ =>
830     cterm_lincomb_add (vector_lincomb (dest_arg1 t)) (vector_lincomb (dest_arg t))
831  | Const(@{const_name minus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_])) \$ _ \$ _ =>
832     cterm_lincomb_sub (vector_lincomb (dest_arg1 t)) (vector_lincomb (dest_arg t))
833  | Const(@{const_name vector_scalar_mult},Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))\$_\$_ =>
834     cterm_lincomb_cmul (dest_ratconst (dest_arg1 t)) (vector_lincomb (dest_arg t))
835  | Const(@{const_name uminus},Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))\$_ =>
836      cterm_lincomb_neg (vector_lincomb (dest_arg t))
837  | Const(@{const_name vec},_)\$_ =>
838    let
839      val b = ((snd o HOLogic.dest_number o term_of o dest_arg) t = 0
840                handle TERM _=> false)
841    in if b then Ctermfunc.onefunc (t,Rat.one)
842       else Ctermfunc.undefined
843    end
844  | _ => Ctermfunc.onefunc (t,Rat.one)
846  fun vector_lincombs ts =
847   fold_rev
848    (fn t => fn fns => case AList.lookup (op aconvc) fns t of
849      NONE =>
850        let val f = vector_lincomb t
851        in case find_first (fn (_,f') => cterm_lincomb_eq f f') fns of
852            SOME (_,f') => (t,f') :: fns
853          | NONE => (t,f) :: fns
854        end
855    | SOME _ => fns) ts []
857 fun replacenegnorms cv t = case term_of t of
858   @{term "op + :: real => _"}\$_\$_ => binop_conv (replacenegnorms cv) t
859 | @{term "op * :: real => _"}\$_\$_ =>
860     if dest_ratconst (dest_arg1 t) </ Rat.zero then arg_conv cv t else reflexive t
861 | _ => reflexive t
862 fun flip v eq =
863   if Ctermfunc.defined eq v
864   then Ctermfunc.update (v, Rat.neg (Ctermfunc.apply eq v)) eq else eq
865 fun allsubsets s = case s of
866   [] => [[]]
867 |(a::t) => let val res = allsubsets t in
868                map (cons a) res @ res end
869 fun evaluate env lin =
870  Intfunc.fold (fn (x,c) => fn s => s +/ c */ (Intfunc.apply env x))
871    lin Rat.zero
873 fun solve (vs,eqs) = case (vs,eqs) of
874   ([],[]) => SOME (Intfunc.onefunc (0,Rat.one))
875  |(_,eq::oeqs) =>
876    (case vs inter (Intfunc.dom eq) of
877      [] => NONE
878     | v::_ =>
879        if Intfunc.defined eq v
880        then
881         let
882          val c = Intfunc.apply eq v
883          val vdef = int_lincomb_cmul (Rat.neg (Rat.inv c)) eq
884          fun eliminate eqn = if not (Intfunc.defined eqn v) then eqn
885                              else int_lincomb_add (int_lincomb_cmul (Intfunc.apply eqn v) vdef) eqn
886         in (case solve (vs \ v,map eliminate oeqs) of
887             NONE => NONE
888           | SOME soln => SOME (Intfunc.update (v, evaluate soln (Intfunc.undefine v vdef)) soln))
889         end
890        else NONE)
892 fun combinations k l = if k = 0 then [[]] else
893  case l of
894   [] => []
895 | h::t => map (cons h) (combinations (k - 1) t) @ combinations k t
898 fun forall2 p l1 l2 = case (l1,l2) of
899    ([],[]) => true
900  | (h1::t1,h2::t2) => p h1 h2 andalso forall2 p t1 t2
901  | _ => false;
904 fun vertices vs eqs =
905  let
906   fun vertex cmb = case solve(vs,cmb) of
907     NONE => NONE
908    | SOME soln => SOME (map (fn v => Intfunc.tryapplyd soln v Rat.zero) vs)
909   val rawvs = map_filter vertex (combinations (length vs) eqs)
910   val unset = filter (forall (fn c => c >=/ Rat.zero)) rawvs
911  in fold_rev (insert (uncurry (forall2 (curry op =/)))) unset []
912  end
914 fun subsumes l m = forall2 (fn x => fn y => Rat.abs x <=/ Rat.abs y) l m
916 fun subsume todo dun = case todo of
917  [] => dun
918 |v::ovs =>
919    let val dun' = if exists (fn w => subsumes w v) dun then dun
920                   else v::(filter (fn w => not(subsumes v w)) dun)
921    in subsume ovs dun'
922    end;
924 fun match_mp PQ P = P RS PQ;
926 fun cterm_of_rat x =
927 let val (a, b) = Rat.quotient_of_rat x
928 in
929  if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
930   else Thm.capply (Thm.capply @{cterm "op / :: real => _"}
931                    (Numeral.mk_cnumber @{ctyp "real"} a))
932         (Numeral.mk_cnumber @{ctyp "real"} b)
933 end;
935 fun norm_cmul_rule c th = instantiate' [] [SOME (cterm_of_rat c)] (th RS @{thm norm_cmul_rule_thm});
939   (* I think here the static context should be sufficient!! *)
940 fun inequality_canon_rule ctxt =
941  let
942   (* FIXME : Should be computed statically!! *)
943   val real_poly_conv =
944     Normalizer.semiring_normalize_wrapper ctxt
945      (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
946  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)))
947 end;
949  fun absc cv ct = case term_of ct of
950  Abs (v,_, _) =>
951   let val (x,t) = Thm.dest_abs (SOME v) ct
952   in Thm.abstract_rule ((fst o dest_Free o term_of) x) x (cv t)
953   end
954  | _ => all_conv ct;
956 fun sub_conv cv ct = (comb_conv cv else_conv absc cv) ct;
957 fun botc1 conv ct =
958   ((sub_conv (botc1 conv)) then_conv (conv else_conv all_conv)) ct;
960  fun rewrs_conv eqs ct = first_conv (map rewr_conv eqs) ct;
961  val apply_pth1 = rewr_conv @{thm pth_1};
962  val apply_pth2 = rewr_conv @{thm pth_2};
963  val apply_pth3 = rewr_conv @{thm pth_3};
964  val apply_pth4 = rewrs_conv @{thms pth_4};
965  val apply_pth5 = rewr_conv @{thm pth_5};
966  val apply_pth6 = rewr_conv @{thm pth_6};
967  val apply_pth7 = rewrs_conv @{thms pth_7};
968  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})));
969  val apply_pth9 = rewrs_conv @{thms pth_9} then_conv arg1_conv (arg1_conv field_comp_conv);
970  val apply_ptha = rewr_conv @{thm pth_a};
971  val apply_pthb = rewrs_conv @{thms pth_b};
972  val apply_pthc = rewrs_conv @{thms pth_c};
973  val apply_pthd = try_conv (rewr_conv @{thm pth_d});
975 fun headvector t = case t of
976   Const(@{const_name plus}, Type("fun",[Type("Finite_Cartesian_Product.^",_),_]))\$
977    (Const(@{const_name vector_scalar_mult}, _)\$l\$v)\$r => v
978  | Const(@{const_name vector_scalar_mult}, _)\$l\$v => v
979  | _ => error "headvector: non-canonical term"
981 fun vector_cmul_conv ct =
982    ((apply_pth5 then_conv arg1_conv field_comp_conv) else_conv
983     (apply_pth6 then_conv binop_conv vector_cmul_conv)) ct
985 fun vector_add_conv ct = apply_pth7 ct
986  handle CTERM _ =>
987   (apply_pth8 ct
988    handle CTERM _ =>
989     (case term_of ct of
990      Const(@{const_name plus},_)\$lt\$rt =>
991       let
992        val l = headvector lt
993        val r = headvector rt
994       in (case TermOrd.fast_term_ord (l,r) of
995          LESS => (apply_pthb then_conv arg_conv vector_add_conv
996                   then_conv apply_pthd) ct
997         | GREATER => (apply_pthc then_conv arg_conv vector_add_conv
998                      then_conv apply_pthd) ct
999         | EQUAL => (apply_pth9 then_conv
1001               arg_conv vector_add_conv then_conv apply_pthd)) ct)
1002       end
1003      | _ => reflexive ct))
1005 fun vector_canon_conv ct = case term_of ct of
1006  Const(@{const_name plus},_)\$_\$_ =>
1007   let
1008    val ((p,l),r) = Thm.dest_comb ct |>> Thm.dest_comb
1009    val lth = vector_canon_conv l
1010    val rth = vector_canon_conv r
1011    val th = Drule.binop_cong_rule p lth rth
1012   in fconv_rule (arg_conv vector_add_conv) th end
1014 | Const(@{const_name vector_scalar_mult}, _)\$_\$_ =>
1015   let
1016    val (p,r) = Thm.dest_comb ct
1017    val rth = Drule.arg_cong_rule p (vector_canon_conv r)
1018   in fconv_rule (arg_conv (apply_pth4 else_conv vector_cmul_conv)) rth
1019   end
1021 | Const(@{const_name minus},_)\$_\$_ => (apply_pth2 then_conv vector_canon_conv) ct
1023 | Const(@{const_name uminus},_)\$_ => (apply_pth3 then_conv vector_canon_conv) ct
1025 | Const(@{const_name vec},_)\$n =>
1026   let val n = Thm.dest_arg ct
1027   in if is_ratconst n andalso not (dest_ratconst n =/ Rat.zero)
1028      then reflexive ct else apply_pth1 ct
1029   end
1031 | _ => apply_pth1 ct
1033 fun norm_canon_conv ct = case term_of ct of
1034   Const(@{const_name norm},_)\$_ => arg_conv vector_canon_conv ct
1035  | _ => raise CTERM ("norm_canon_conv", [ct])
1037 fun fold_rev2 f [] [] z = z
1038  | fold_rev2 f (x::xs) (y::ys) z = f x y (fold_rev2 f xs ys z)
1039  | fold_rev2 f _ _ _ = raise UnequalLengths;
1041 fun int_flip v eq =
1042   if Intfunc.defined eq v
1043   then Intfunc.update (v, Rat.neg (Intfunc.apply eq v)) eq else eq;
1045 local
1046  val pth_zero = @{thm "norm_0"}
1047  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)
1048              pth_zero
1049  val concl = dest_arg o cprop_of
1050  fun real_vector_combo_prover ctxt translator (nubs,ges,gts) =
1051   let
1052    (* FIXME: Should be computed statically!!*)
1053    val real_poly_conv =
1054       Normalizer.semiring_normalize_wrapper ctxt
1055        (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"}))
1056    val sources = map (dest_arg o dest_arg1 o concl) nubs
1057    val rawdests = fold_rev (find_normedterms o dest_arg o concl) (ges @ gts) []
1058    val _ = if not (forall fst rawdests) then error "real_vector_combo_prover: Sanity check"
1059            else ()
1060    val dests = distinct (op aconvc) (map snd rawdests)
1061    val srcfuns = map vector_lincomb sources
1062    val destfuns = map vector_lincomb dests
1063    val vvs = fold_rev (curry (gen_union op aconvc) o Ctermfunc.dom) (srcfuns @ destfuns) []
1064    val n = length srcfuns
1065    val nvs = 1 upto n
1066    val srccombs = srcfuns ~~ nvs
1067    fun consider d =
1068     let
1069      fun coefficients x =
1070       let
1071        val inp = if Ctermfunc.defined d x then Intfunc.onefunc (0, Rat.neg(Ctermfunc.apply d x))
1072                       else Intfunc.undefined
1073       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
1074       end
1075      val equations = map coefficients vvs
1076      val inequalities = map (fn n => Intfunc.onefunc (n,Rat.one)) nvs
1077      fun plausiblevertices f =
1078       let
1079        val flippedequations = map (fold_rev int_flip f) equations
1080        val constraints = flippedequations @ inequalities
1081        val rawverts = vertices nvs constraints
1082        fun check_solution v =
1083         let
1084           val f = fold_rev2 (curry Intfunc.update) nvs v (Intfunc.onefunc (0, Rat.one))
1085         in forall (fn e => evaluate f e =/ Rat.zero) flippedequations
1086         end
1087        val goodverts = filter check_solution rawverts
1088        val signfixups = map (fn n => if n mem_int  f then ~1 else 1) nvs
1089       in map (map2 (fn s => fn c => Rat.rat_of_int s */ c) signfixups) goodverts
1090       end
1091      val allverts = fold_rev append (map plausiblevertices (allsubsets nvs)) []
1092     in subsume allverts []
1093     end
1094    fun compute_ineq v =
1095     let
1096      val ths = map_filter (fn (v,t) => if v =/ Rat.zero then NONE
1097                                      else SOME(norm_cmul_rule v t))
1098                             (v ~~ nubs)
1099     in inequality_canon_rule ctxt (end_itlist norm_add_rule ths)
1100     end
1101    val ges' = map_filter (try compute_ineq) (fold_rev (append o consider) destfuns []) @
1102                  map (inequality_canon_rule ctxt) nubs @ ges
1103    val zerodests = filter
1104         (fn t => null (Ctermfunc.dom (vector_lincomb t))) (map snd rawdests)
1106   in RealArith.real_linear_prover translator
1107         (map (fn t => instantiate ([(tv_n,(hd o tl o dest_ctyp o ctyp_of_term) t)],[]) pth_zero)
1108             zerodests,
1109         map (fconv_rule (once_depth_conv (norm_canon_conv) then_conv
1110                        arg_conv (arg_conv real_poly_conv))) ges',
1111         map (fconv_rule (once_depth_conv (norm_canon_conv) then_conv
1112                        arg_conv (arg_conv real_poly_conv))) gts)
1113   end
1114 in val real_vector_combo_prover = real_vector_combo_prover
1115 end;
1117 local
1118  val pth = @{thm norm_imp_pos_and_ge}
1119  val norm_mp = match_mp pth
1120  val concl = dest_arg o cprop_of
1121  fun conjunct1 th = th RS @{thm conjunct1}
1122  fun conjunct2 th = th RS @{thm conjunct2}
1123  fun C f x y = f y x
1124 fun real_vector_ineq_prover ctxt translator (ges,gts) =
1125  let
1126 (*   val _ = error "real_vector_ineq_prover: pause" *)
1127   val ntms = fold_rev find_normedterms (map (dest_arg o concl) (ges @ gts)) []
1128   val lctab = vector_lincombs (map snd (filter (not o fst) ntms))
1129   val (fxns, ctxt') = Variable.variant_fixes (replicate (length lctab) "x") ctxt
1130   fun mk_norm t = capply (instantiate_cterm' [SOME (ctyp_of_term t)] [] @{cpat "norm :: (?'a :: norm) => real"}) t
1131   fun mk_equals l r = capply (capply (instantiate_cterm' [SOME (ctyp_of_term l)] [] @{cpat "op == :: ?'a =>_"}) l) r
1132   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
1133   val replace_conv = try_conv (rewrs_conv asl)
1134   val replace_rule = fconv_rule (funpow 2 arg_conv (replacenegnorms replace_conv))
1135   val ges' =
1136        fold_rev (fn th => fn ths => conjunct1(norm_mp th)::ths)
1137               asl (map replace_rule ges)
1138   val gts' = map replace_rule gts
1139   val nubs = map (conjunct2 o norm_mp) asl
1140   val th1 = real_vector_combo_prover ctxt' translator (nubs,ges',gts')
1141   val shs = filter (member (fn (t,th) => t aconvc cprop_of th) asl) (#hyps (crep_thm th1))
1142   val th11 = hd (Variable.export ctxt' ctxt [fold implies_intr shs th1])
1143   val cps = map (swap o dest_equals) (cprems_of th11)
1144   val th12 = instantiate ([], cps) th11
1145   val th13 = fold (C implies_elim) (map (reflexive o snd) cps) th12;
1146  in hd (Variable.export ctxt' ctxt [th13])
1147  end
1148 in val real_vector_ineq_prover = real_vector_ineq_prover
1149 end;
1151 local
1152  val rawrule = fconv_rule (arg_conv (rewr_conv @{thm real_eq_0_iff_le_ge_0}))
1153  fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
1154  fun simple_cterm_ord t u = TermOrd.term_ord (term_of t, term_of u) = LESS;
1155   (* FIXME: Lookup in the context every time!!! Fix this !!!*)
1156  fun splitequation ctxt th acc =
1157   let
1158    val real_poly_neg_conv = #neg
1159        (Normalizer.semiring_normalizers_ord_wrapper ctxt
1160         (valOf (NormalizerData.match ctxt @{cterm "(0::real) + 1"})) simple_cterm_ord)
1161    val (th1,th2) = conj_pair(rawrule th)
1162   in th1::fconv_rule (arg_conv (arg_conv real_poly_neg_conv)) th2::acc
1163   end
1164 in fun real_vector_prover ctxt translator (eqs,ges,gts) =
1165      real_vector_ineq_prover ctxt translator
1166          (fold_rev (splitequation ctxt) eqs ges,gts)
1167 end;
1169   fun init_conv ctxt =
1170    Simplifier.rewrite (Simplifier.context ctxt
1171      (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})))
1172    then_conv field_comp_conv
1173    then_conv nnf_conv
1175  fun pure ctxt = RealArith.gen_prover_real_arith ctxt (real_vector_prover ctxt);
1176  fun norm_arith ctxt ct =
1177   let
1178    val ctxt' = Variable.declare_term (term_of ct) ctxt
1179    val th = init_conv ctxt' ct
1180   in equal_elim (Drule.arg_cong_rule @{cterm Trueprop} (symmetric th))
1181                 (pure ctxt' (rhs_of th))
1182  end
1184  fun norm_arith_tac ctxt =
1185    clarify_tac HOL_cs THEN'
1186    ObjectLogic.full_atomize_tac THEN'
1187    CSUBGOAL ( fn (p,i) => rtac (norm_arith ctxt (Thm.dest_arg p )) i);
1189 end;