src/HOL/Library/positivstellensatz.ML
 author wenzelm Wed Jun 01 10:45:35 2016 +0200 (2016-06-01) changeset 63201 f151704c08e4 parent 63198 c583ca33076a child 63205 97b721666890 permissions -rw-r--r--
tuned signature;
1 (*  Title:      HOL/Library/positivstellensatz.ML
2     Author:     Amine Chaieb, University of Cambridge
4 A generic arithmetic prover based on Positivstellensatz certificates
5 --- also implements Fourier-Motzkin elimination as a special case
6 Fourier-Motzkin elimination.
7 *)
9 (* A functor for finite mappings based on Tables *)
11 signature FUNC =
12 sig
13   include TABLE
14   val apply : 'a table -> key -> 'a
15   val applyd :'a table -> (key -> 'a) -> key -> 'a
16   val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a table -> 'a table -> 'a table
17   val dom : 'a table -> key list
18   val tryapplyd : 'a table -> key -> 'a -> 'a
19   val updatep : (key * 'a -> bool) -> key * 'a -> 'a table -> 'a table
20   val choose : 'a table -> key * 'a
21   val onefunc : key * 'a -> 'a table
22 end;
24 functor FuncFun(Key: KEY) : FUNC =
25 struct
27 structure Tab = Table(Key);
29 open Tab;
31 fun dom a = sort Key.ord (Tab.keys a);
32 fun applyd f d x = case Tab.lookup f x of
33    SOME y => y
34  | NONE => d x;
36 fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
37 fun tryapplyd f a d = applyd f (K d) a;
38 fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
39 fun combine f z a b =
40   let
41     fun h (k,v) t = case Tab.lookup t k of
42         NONE => Tab.update (k,v) t
43       | SOME v' => let val w = f v v'
44         in if z w then Tab.delete k t else Tab.update (k,w) t end;
45   in Tab.fold h a b end;
47 fun choose f =
48   (case Tab.min f of
49     SOME entry => entry
50   | NONE => error "FuncFun.choose : Completely empty function")
52 fun onefunc kv = update kv empty
54 end;
56 (* Some standard functors and utility functions for them *)
58 structure FuncUtil =
59 struct
61 structure Intfunc = FuncFun(type key = int val ord = int_ord);
62 structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
63 structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
64 structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
65 structure Termfunc = FuncFun(type key = term val ord = Term_Ord.fast_term_ord);
67 val cterm_ord = Term_Ord.fast_term_ord o apply2 Thm.term_of
69 structure Ctermfunc = FuncFun(type key = cterm val ord = cterm_ord);
71 type monomial = int Ctermfunc.table;
73 val monomial_ord = list_ord (prod_ord cterm_ord int_ord) o apply2 Ctermfunc.dest
75 structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
77 type poly = Rat.rat Monomialfunc.table;
79 (* The ordering so we can create canonical HOL polynomials.                  *)
81 fun dest_monomial mon = sort (cterm_ord o apply2 fst) (Ctermfunc.dest mon);
83 fun monomial_order (m1,m2) =
84   if Ctermfunc.is_empty m2 then LESS
85   else if Ctermfunc.is_empty m1 then GREATER
86   else
87     let
88       val mon1 = dest_monomial m1
89       val mon2 = dest_monomial m2
90       val deg1 = fold (Integer.add o snd) mon1 0
91       val deg2 = fold (Integer.add o snd) mon2 0
92     in if deg1 < deg2 then GREATER
93        else if deg1 > deg2 then LESS
94        else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
95     end;
97 end
99 (* positivstellensatz datatype and prover generation *)
101 signature REAL_ARITH =
102 sig
104   datatype positivstellensatz =
105     Axiom_eq of int
106   | Axiom_le of int
107   | Axiom_lt of int
108   | Rational_eq of Rat.rat
109   | Rational_le of Rat.rat
110   | Rational_lt of Rat.rat
111   | Square of FuncUtil.poly
112   | Eqmul of FuncUtil.poly * positivstellensatz
113   | Sum of positivstellensatz * positivstellensatz
114   | Product of positivstellensatz * positivstellensatz;
116   datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
118   datatype tree_choice = Left | Right
120   type prover = tree_choice list ->
121     (thm list * thm list * thm list -> positivstellensatz -> thm) ->
122       thm list * thm list * thm list -> thm * pss_tree
123   type cert_conv = cterm -> thm * pss_tree
125   val gen_gen_real_arith :
126     Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
127      conv * conv * conv * conv * conv * conv * prover -> cert_conv
128   val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
129     thm list * thm list * thm list -> thm * pss_tree
131   val gen_real_arith : Proof.context ->
132     (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
134   val gen_prover_real_arith : Proof.context -> prover -> cert_conv
136   val is_ratconst : cterm -> bool
137   val dest_ratconst : cterm -> Rat.rat
138   val cterm_of_rat : Rat.rat -> cterm
140 end
142 structure RealArith : REAL_ARITH =
143 struct
145 open Conv
146 (* ------------------------------------------------------------------------- *)
147 (* Data structure for Positivstellensatz refutations.                        *)
148 (* ------------------------------------------------------------------------- *)
150 datatype positivstellensatz =
151     Axiom_eq of int
152   | Axiom_le of int
153   | Axiom_lt of int
154   | Rational_eq of Rat.rat
155   | Rational_le of Rat.rat
156   | Rational_lt of Rat.rat
157   | Square of FuncUtil.poly
158   | Eqmul of FuncUtil.poly * positivstellensatz
159   | Sum of positivstellensatz * positivstellensatz
160   | Product of positivstellensatz * positivstellensatz;
161          (* Theorems used in the procedure *)
163 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
164 datatype tree_choice = Left | Right
165 type prover = tree_choice list ->
166   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
167     thm list * thm list * thm list -> thm * pss_tree
168 type cert_conv = cterm -> thm * pss_tree
171     (* Some useful derived rules *)
172 fun deduct_antisym_rule tha thb =
173     Thm.equal_intr (Thm.implies_intr (Thm.cprop_of thb) tha)
174      (Thm.implies_intr (Thm.cprop_of tha) thb);
176 fun prove_hyp tha thb =
177   if exists (curry op aconv (Thm.concl_of tha)) (Thm.hyps_of thb)  (* FIXME !? *)
178   then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb;
180 val pth = @{lemma "(((x::real) < y) == (y - x > 0))" and "((x <= y) == (y - x >= 0))" and
181      "((x = y) == (x - y = 0))" and "((~(x < y)) == (x - y >= 0))" and
182      "((~(x <= y)) == (x - y > 0))" and "((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
183   by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};
185 val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
187   @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 )" and "( x = 0 ==> y >= 0 ==> x + y >= 0)" and
188     "(x = 0 ==> y > 0 ==> x + y > 0)" and "(x >= 0 ==> y = 0 ==> x + y >= 0)" and
189     "(x >= 0 ==> y >= 0 ==> x + y >= 0)" and "(x >= 0 ==> y > 0 ==> x + y > 0)" and
190     "(x > 0 ==> y = 0 ==> x + y > 0)" and "(x > 0 ==> y >= 0 ==> x + y > 0)" and
191     "(x > 0 ==> y > 0 ==> x + y > 0)" by simp_all};
193 val pth_mul =
194   @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0)" and "(x = 0 ==> y >= 0 ==> x * y = 0)" and
195     "(x = 0 ==> y > 0 ==> x * y = 0)" and "(x >= 0 ==> y = 0 ==> x * y = 0)" and
196     "(x >= 0 ==> y >= 0 ==> x * y >= 0)" and "(x >= 0 ==> y > 0 ==> x * y >= 0)" and
197     "(x > 0 ==>  y = 0 ==> x * y = 0)" and "(x > 0 ==> y >= 0 ==> x * y >= 0)" and
198     "(x > 0 ==>  y > 0 ==> x * y > 0)"
199   by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
200     mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])};
202 val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
203 val pth_square = @{lemma "x * x >= (0::real)"  by simp};
205 val weak_dnf_simps =
206   List.take (@{thms simp_thms}, 34) @
207     @{lemma "((P & (Q | R)) = ((P&Q) | (P&R)))" and "((Q | R) & P) = ((Q&P) | (R&P))" and
208       "(P & Q) = (Q & P)" and "((P | Q) = (Q | P))" by blast+};
210 (*
211 val nnfD_simps =
212   @{lemma "((~(P & Q)) = (~P | ~Q))" and "((~(P | Q)) = (~P & ~Q) )" and
213     "((P --> Q) = (~P | Q) )" and "((P = Q) = ((P & Q) | (~P & ~ Q)))" and
214     "((~(P = Q)) = ((P & ~ Q) | (~P & Q)) )" and "((~ ~(P)) = P)" by blast+};
215 *)
217 val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
218 val prenex_simps =
219   map (fn th => th RS sym)
220     ([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @
221       @{thms "HOL.all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});
223 val real_abs_thms1 = @{lemma
224   "((-1 * \<bar>x::real\<bar> >= r) = (-1 * x >= r & 1 * x >= r))" and
225   "((-1 * \<bar>x\<bar> + a >= r) = (a + -1 * x >= r & a + 1 * x >= r))" and
226   "((a + -1 * \<bar>x\<bar> >= r) = (a + -1 * x >= r & a + 1 * x >= r))" and
227   "((a + -1 * \<bar>x\<bar> + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r))" and
228   "((a + b + -1 * \<bar>x\<bar> >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r))" and
229   "((a + b + -1 * \<bar>x\<bar> + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r))" and
230   "((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r))" and
231   "((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r))" and
232   "((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r))" and
233   "((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y  + b >= r))" and
234   "((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r))" and
235   "((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y  + c >= r))" and
236   "((1 * min x y >= r) = (1 * x >= r & 1 * y >= r))" and
237   "((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and
238   "((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and
239   "((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y  + b >= r))" and
240   "((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r))" and
241   "((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y  + c >= r))" and
242   "((min x y >= r) = (x >= r &  y >= r))" and
243   "((min x y + a >= r) = (a + x >= r & a + y >= r))" and
244   "((a + min x y >= r) = (a + x >= r & a + y >= r))" and
245   "((a + min x y + b >= r) = (a + x + b >= r & a + y  + b >= r))" and
246   "((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r))" and
247   "((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r))" and
248   "((-1 * \<bar>x\<bar> > r) = (-1 * x > r & 1 * x > r))" and
249   "((-1 * \<bar>x\<bar> + a > r) = (a + -1 * x > r & a + 1 * x > r))" and
250   "((a + -1 * \<bar>x\<bar> > r) = (a + -1 * x > r & a + 1 * x > r))" and
251   "((a + -1 * \<bar>x\<bar> + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r))" and
252   "((a + b + -1 * \<bar>x\<bar> > r) = (a + b + -1 * x > r & a + b + 1 * x > r))" and
253   "((a + b + -1 * \<bar>x\<bar> + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r))" and
254   "((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r))" and
255   "((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r))" and
256   "((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r))" and
257   "((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y  + b > r))" and
258   "((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r))" and
259   "((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y  + c > r))" and
260   "((min x y > r) = (x > r &  y > r))" and
261   "((min x y + a > r) = (a + x > r & a + y > r))" and
262   "((a + min x y > r) = (a + x > r & a + y > r))" and
263   "((a + min x y + b > r) = (a + x + b > r & a + y  + b > r))" and
264   "((a + b + min x y > r) = (a + b + x > r & a + b + y > r))" and
265   "((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
266   by auto};
268 val abs_split' = @{lemma "P \<bar>x::'a::linordered_idom\<bar> == (x >= 0 & P x | x < 0 & P (-x))"
269   by (atomize (full)) (auto split add: abs_split)};
271 val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
272   by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
274 val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
275   by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
278          (* Miscellaneous *)
279 fun literals_conv bops uops cv =
280   let
281     fun h t =
282       (case Thm.term_of t of
283         b\$_\$_ => if member (op aconv) bops b then binop_conv h t else cv t
284       | u\$_ => if member (op aconv) uops u then arg_conv h t else cv t
285       | _ => cv t)
286   in h end;
288 fun cterm_of_rat x =
289   let
290     val (a, b) = Rat.dest x
291   in
292     if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
293     else Thm.apply (Thm.apply @{cterm "op / :: real => _"}
294       (Numeral.mk_cnumber @{ctyp "real"} a))
295       (Numeral.mk_cnumber @{ctyp "real"} b)
296   end;
298 fun dest_ratconst t =
299   case Thm.term_of t of
300     Const(@{const_name divide}, _)\$a\$b => Rat.make(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
301   | _ => Rat.of_int (HOLogic.dest_number (Thm.term_of t) |> snd)
302 fun is_ratconst t = can dest_ratconst t
304 (*
305 fun find_term p t = if p t then t else
306  case t of
307   a\$b => (find_term p a handle TERM _ => find_term p b)
308  | Abs (_,_,t') => find_term p t'
309  | _ => raise TERM ("find_term",[t]);
310 *)
312 fun find_cterm p t =
313   if p t then t else
314   case Thm.term_of t of
315     _\$_ => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
316   | Abs (_,_,_) => find_cterm p (Thm.dest_abs NONE t |> snd)
317   | _ => raise CTERM ("find_cterm",[t]);
319 fun is_comb t = (case Thm.term_of t of _ \$ _ => true | _ => false);
321 fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
322   handle CTERM _ => false;
325 (* Map back polynomials to HOL.                         *)
327 fun cterm_of_varpow x k = if k = 1 then x else Thm.apply (Thm.apply @{cterm "op ^ :: real => _"} x)
328   (Numeral.mk_cnumber @{ctyp nat} k)
330 fun cterm_of_monomial m =
331   if FuncUtil.Ctermfunc.is_empty m then @{cterm "1::real"}
332   else
333     let
334       val m' = FuncUtil.dest_monomial m
335       val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' []
336     in foldr1 (fn (s, t) => Thm.apply (Thm.apply @{cterm "op * :: real => _"} s) t) vps
337     end
339 fun cterm_of_cmonomial (m,c) =
340   if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
341   else if c = Rat.one then cterm_of_monomial m
342   else Thm.apply (Thm.apply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
344 fun cterm_of_poly p =
345   if FuncUtil.Monomialfunc.is_empty p then @{cterm "0::real"}
346   else
347     let
348       val cms = map cterm_of_cmonomial
349         (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
350     in foldr1 (fn (t1, t2) => Thm.apply(Thm.apply @{cterm "op + :: real => _"} t1) t2) cms
351     end;
353 (* A general real arithmetic prover *)
355 fun gen_gen_real_arith ctxt (mk_numeric,
356        numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
358        absconv1,absconv2,prover) =
359   let
360     val pre_ss = put_simpset HOL_basic_ss ctxt addsimps
361       @{thms simp_thms ex_simps all_simps not_all not_ex ex_disj_distrib
362           all_conj_distrib if_bool_eq_disj}
363     val prenex_ss = put_simpset HOL_basic_ss ctxt addsimps prenex_simps
364     val skolemize_ss = put_simpset HOL_basic_ss ctxt addsimps [choice_iff]
365     val presimp_conv = Simplifier.rewrite pre_ss
366     val prenex_conv = Simplifier.rewrite prenex_ss
367     val skolemize_conv = Simplifier.rewrite skolemize_ss
368     val weak_dnf_ss = put_simpset HOL_basic_ss ctxt addsimps weak_dnf_simps
369     val weak_dnf_conv = Simplifier.rewrite weak_dnf_ss
370     fun eqT_elim th = Thm.equal_elim (Thm.symmetric th) @{thm TrueI}
371     fun oprconv cv ct =
372       let val g = Thm.dest_fun2 ct
373       in if g aconvc @{cterm "op <= :: real => _"}
374             orelse g aconvc @{cterm "op < :: real => _"}
375          then arg_conv cv ct else arg1_conv cv ct
376       end
378     fun real_ineq_conv th ct =
379       let
380         val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th
381           handle Pattern.MATCH => raise CTERM ("real_ineq_conv", [ct]))
382       in Thm.transitive th' (oprconv poly_conv (Thm.rhs_of th'))
383       end
384     val [real_lt_conv, real_le_conv, real_eq_conv,
385          real_not_lt_conv, real_not_le_conv, _] =
386          map real_ineq_conv pth
387     fun match_mp_rule ths ths' =
388       let
389         fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
390           | th::ths => (ths' MRS th handle THM _ => f ths ths')
391       in f ths ths' end
392     fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
393          (match_mp_rule pth_mul [th, th'])
396     fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
397        (Thm.instantiate' [] [SOME ct] (th RS pth_emul))
398     fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
399        (Thm.instantiate' [] [SOME t] pth_square)
401     fun hol_of_positivstellensatz(eqs,les,lts) proof =
402       let
403         fun translate prf =
404           case prf of
405             Axiom_eq n => nth eqs n
406           | Axiom_le n => nth les n
407           | Axiom_lt n => nth lts n
408           | Rational_eq x => eqT_elim(numeric_eq_conv(Thm.apply @{cterm Trueprop}
409                           (Thm.apply (Thm.apply @{cterm "op =::real => _"} (mk_numeric x))
410                                @{cterm "0::real"})))
411           | Rational_le x => eqT_elim(numeric_ge_conv(Thm.apply @{cterm Trueprop}
412                           (Thm.apply (Thm.apply @{cterm "op <=::real => _"}
413                                      @{cterm "0::real"}) (mk_numeric x))))
414           | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.apply @{cterm Trueprop}
415                       (Thm.apply (Thm.apply @{cterm "op <::real => _"} @{cterm "0::real"})
416                         (mk_numeric x))))
417           | Square pt => square_rule (cterm_of_poly pt)
418           | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
419           | Sum(p1,p2) => add_rule (translate p1) (translate p2)
420           | Product(p1,p2) => mul_rule (translate p1) (translate p2)
421       in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv])
422           (translate proof)
423       end
425     val init_conv = presimp_conv then_conv
426         nnf_conv ctxt then_conv skolemize_conv then_conv prenex_conv then_conv
427         weak_dnf_conv
429     val concl = Thm.dest_arg o Thm.cprop_of
430     fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
431     val is_req = is_binop @{cterm "op =:: real => _"}
432     val is_ge = is_binop @{cterm "op <=:: real => _"}
433     val is_gt = is_binop @{cterm "op <:: real => _"}
434     val is_conj = is_binop @{cterm HOL.conj}
435     val is_disj = is_binop @{cterm HOL.disj}
436     fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
437     fun disj_cases th th1 th2 =
438       let
439         val (p,q) = Thm.dest_binop (concl th)
440         val c = concl th1
441         val _ =
442           if c aconvc (concl th2) then ()
443           else error "disj_cases : conclusions not alpha convertible"
444       in Thm.implies_elim (Thm.implies_elim
445           (Thm.implies_elim (Thm.instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
446           (Thm.implies_intr (Thm.apply @{cterm Trueprop} p) th1))
447         (Thm.implies_intr (Thm.apply @{cterm Trueprop} q) th2)
448       end
449     fun overall cert_choice dun ths =
450       case ths of
451         [] =>
452         let
453           val (eq,ne) = List.partition (is_req o concl) dun
454           val (le,nl) = List.partition (is_ge o concl) ne
455           val lt = filter (is_gt o concl) nl
456         in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
457       | th::oths =>
458         let
459           val ct = concl th
460         in
461           if is_conj ct then
462             let
463               val (th1,th2) = conj_pair th
464             in overall cert_choice dun (th1::th2::oths) end
465           else if is_disj ct then
466             let
467               val (th1, cert1) =
468                 overall (Left::cert_choice) dun
469                   (Thm.assume (Thm.apply @{cterm Trueprop} (Thm.dest_arg1 ct))::oths)
470               val (th2, cert2) =
471                 overall (Right::cert_choice) dun
472                   (Thm.assume (Thm.apply @{cterm Trueprop} (Thm.dest_arg ct))::oths)
473             in (disj_cases th th1 th2, Branch (cert1, cert2)) end
474           else overall cert_choice (th::dun) oths
475         end
476     fun dest_binary b ct =
477         if is_binop b ct then Thm.dest_binop ct
478         else raise CTERM ("dest_binary",[b,ct])
479     val dest_eq = dest_binary @{cterm "op = :: real => _"}
480     val neq_th = nth pth 5
481     fun real_not_eq_conv ct =
482       let
483         val (l,r) = dest_eq (Thm.dest_arg ct)
484         val th = Thm.instantiate ([],[((("x", 0), @{typ real}),l),((("y", 0), @{typ real}),r)]) neq_th
485         val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
486         val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
487         val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
488         val th' = Drule.binop_cong_rule @{cterm HOL.disj}
489           (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
490           (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
491       in Thm.transitive th th'
492       end
493     fun equal_implies_1_rule PQ =
494       let
495         val P = Thm.lhs_of PQ
496       in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P))
497       end
498     (* FIXME!!! Copied from groebner.ml *)
499     val strip_exists =
500       let
501         fun h (acc, t) =
502           case Thm.term_of t of
503             Const(@{const_name Ex},_)\$Abs(_,_,_) =>
504               h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
505           | _ => (acc,t)
506       in fn t => h ([],t)
507       end
508     fun name_of x =
509       case Thm.term_of x of
510         Free(s,_) => s
511       | Var ((s,_),_) => s
512       | _ => "x"
514     fun mk_forall x th =
515       let
516         val T = Thm.typ_of_cterm x
517         val all = Thm.cterm_of ctxt (Const (@{const_name All}, (T --> @{typ bool}) --> @{typ bool}))
518       in Drule.arg_cong_rule all (Thm.abstract_rule (name_of x) x th) end
520     val specl = fold_rev (fn x => fn th => Thm.instantiate' [] [SOME x] (th RS spec));
522     fun ext T = Thm.cterm_of ctxt (Const (@{const_name Ex}, (T --> @{typ bool}) --> @{typ bool}))
523     fun mk_ex v t = Thm.apply (ext (Thm.typ_of_cterm v)) (Thm.lambda v t)
525     fun choose v th th' =
526       case Thm.concl_of th of
527         @{term Trueprop} \$ (Const(@{const_name Ex},_)\$_) =>
528         let
529           val p = (funpow 2 Thm.dest_arg o Thm.cprop_of) th
530           val T = (hd o Thm.dest_ctyp o Thm.ctyp_of_cterm) p
531           val th0 = fconv_rule (Thm.beta_conversion true)
532             (Thm.instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o Thm.cprop_of) th'] exE)
533           val pv = (Thm.rhs_of o Thm.beta_conversion true)
534             (Thm.apply @{cterm Trueprop} (Thm.apply p v))
535           val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
536         in Thm.implies_elim (Thm.implies_elim th0 th) th1  end
537       | _ => raise THM ("choose",0,[th, th'])
539     fun simple_choose v th =
540       choose v
541         (Thm.assume
542           ((Thm.apply @{cterm Trueprop} o mk_ex v) (Thm.dest_arg (hd (Thm.chyps_of th))))) th
544     val strip_forall =
545       let
546         fun h (acc, t) =
547           case Thm.term_of t of
548             Const(@{const_name All},_)\$Abs(_,_,_) =>
549               h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
550           | _ => (acc,t)
551       in fn t => h ([],t)
552       end
554     fun f ct =
555       let
556         val nnf_norm_conv' =
557           nnf_conv ctxt then_conv
558           literals_conv [@{term HOL.conj}, @{term HOL.disj}] []
559           (Conv.cache_conv
560             (first_conv [real_lt_conv, real_le_conv,
561                          real_eq_conv, real_not_lt_conv,
562                          real_not_le_conv, real_not_eq_conv, all_conv]))
563         fun absremover ct = (literals_conv [@{term HOL.conj}, @{term HOL.disj}] []
564                   (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv
565                   try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
566         val nct = Thm.apply @{cterm Trueprop} (Thm.apply @{cterm "Not"} ct)
567         val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
568         val tm0 = Thm.dest_arg (Thm.rhs_of th0)
569         val (th, certificates) =
570           if tm0 aconvc @{cterm False} then (equal_implies_1_rule th0, Trivial) else
571           let
572             val (evs,bod) = strip_exists tm0
573             val (avs,ibod) = strip_forall bod
574             val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
575             val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))]
576             val th3 =
577               fold simple_choose evs
578                 (prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.apply @{cterm Trueprop} bod))) th2)
579           in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
580           end
581       in (Thm.implies_elim (Thm.instantiate' [] [SOME ct] pth_final) th, certificates)
582       end
583   in f
584   end;
586 (* A linear arithmetic prover *)
587 local
588   val linear_add = FuncUtil.Ctermfunc.combine (curry op +) (fn z => z = Rat.zero)
589   fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c * x)
590   val one_tm = @{cterm "1::real"}
591   fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p Rat.zero)) orelse
592      ((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso
593        not(p(FuncUtil.Ctermfunc.apply e one_tm)))
595   fun linear_ineqs vars (les,lts) =
596     case find_first (contradictory (fn x => x > Rat.zero)) lts of
597       SOME r => r
598     | NONE =>
599       (case find_first (contradictory (fn x => x > Rat.zero)) les of
600          SOME r => r
601        | NONE =>
602          if null vars then error "linear_ineqs: no contradiction" else
603          let
604            val ineqs = les @ lts
605            fun blowup v =
606              length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero = Rat.zero) ineqs) +
607              length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero > Rat.zero) ineqs) *
608              length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero < Rat.zero) ineqs)
609            val v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
610              (map (fn v => (v,blowup v)) vars)))
611            fun addup (e1,p1) (e2,p2) acc =
612              let
613                val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v Rat.zero
614                val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v Rat.zero
615              in
616                if c1 * c2 >= Rat.zero then acc else
617                let
618                  val e1' = linear_cmul (Rat.abs c2) e1
619                  val e2' = linear_cmul (Rat.abs c1) e2
620                  val p1' = Product(Rational_lt(Rat.abs c2),p1)
621                  val p2' = Product(Rational_lt(Rat.abs c1),p2)
623                end
624              end
625            val (les0,les1) =
626              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero = Rat.zero) les
627            val (lts0,lts1) =
628              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero = Rat.zero) lts
629            val (lesp,lesn) =
630              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero > Rat.zero) les1
631            val (ltsp,ltsn) =
632              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero > Rat.zero) lts1
633            val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
634            val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
635                       (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
636          in linear_ineqs (remove (op aconvc) v vars) (les',lts')
637          end)
639   fun linear_eqs(eqs,les,lts) =
640     case find_first (contradictory (fn x => x = Rat.zero)) eqs of
641       SOME r => r
642     | NONE =>
643       (case eqs of
644          [] =>
645          let val vars = remove (op aconvc) one_tm
646              (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) [])
647          in linear_ineqs vars (les,lts) end
648        | (e,p)::es =>
649          if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
650          let
651            val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
652            fun xform (inp as (t,q)) =
653              let val d = FuncUtil.Ctermfunc.tryapplyd t x Rat.zero in
654                if d = Rat.zero then inp else
655                let
656                  val k = (Rat.neg d) * Rat.abs c / c
657                  val e' = linear_cmul k e
658                  val t' = linear_cmul (Rat.abs c) t
659                  val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
660                  val q' = Product(Rational_lt(Rat.abs c),q)
662                end
663              end
664          in linear_eqs(map xform es,map xform les,map xform lts)
665          end)
667   fun linear_prover (eq,le,lt) =
668     let
669       val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq
670       val les = map_index (fn (n, p) => (p,Axiom_le n)) le
671       val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt
672     in linear_eqs(eqs,les,lts)
673     end
675   fun lin_of_hol ct =
676     if ct aconvc @{cterm "0::real"} then FuncUtil.Ctermfunc.empty
677     else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
678     else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
679     else
680       let val (lop,r) = Thm.dest_comb ct
681       in
682         if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
683         else
684           let val (opr,l) = Thm.dest_comb lop
685           in
686             if opr aconvc @{cterm "op + :: real =>_"}
687             then linear_add (lin_of_hol l) (lin_of_hol r)
688             else if opr aconvc @{cterm "op * :: real =>_"}
689                     andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
690             else FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
691           end
692       end
694   fun is_alien ct =
695     case Thm.term_of ct of
696       Const(@{const_name "of_nat"}, _)\$ n => not (can HOLogic.dest_number n)
697     | Const(@{const_name "of_int"}, _)\$ n => not (can HOLogic.dest_number n)
698     | _ => false
699 in
700 fun real_linear_prover translator (eq,le,lt) =
701   let
702     val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o Thm.cprop_of
703     val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o Thm.cprop_of
704     val eq_pols = map lhs eq
705     val le_pols = map rhs le
706     val lt_pols = map rhs lt
707     val aliens = filter is_alien
708       (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom)
709                 (eq_pols @ le_pols @ lt_pols) [])
710     val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,Rat.one)) aliens
711     val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
712     val le' = le @ map (fn a => Thm.instantiate' [] [SOME (Thm.dest_arg a)] @{thm of_nat_0_le_iff}) aliens
713   in ((translator (eq,le',lt) proof), Trivial)
714   end
715 end;
717 (* A less general generic arithmetic prover dealing with abs,max and min*)
719 local
720   val absmaxmin_elim_ss1 =
721     simpset_of (put_simpset HOL_basic_ss @{context} addsimps real_abs_thms1)
722   fun absmaxmin_elim_conv1 ctxt =
723     Simplifier.rewrite (put_simpset absmaxmin_elim_ss1 ctxt)
725   val absmaxmin_elim_conv2 =
726     let
727       val pth_abs = Thm.instantiate' [SOME @{ctyp real}] [] abs_split'
728       val pth_max = Thm.instantiate' [SOME @{ctyp real}] [] max_split
729       val pth_min = Thm.instantiate' [SOME @{ctyp real}] [] min_split
730       val abs_tm = @{cterm "abs :: real => _"}
731       val p_v = (("P", 0), @{typ "real \<Rightarrow> bool"})
732       val x_v = (("x", 0), @{typ real})
733       val y_v = (("y", 0), @{typ real})
734       val is_max = is_binop @{cterm "max :: real => _"}
735       val is_min = is_binop @{cterm "min :: real => _"}
736       fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm
737       fun eliminate_construct p c tm =
738         let
739           val t = find_cterm p tm
740           val th0 = (Thm.symmetric o Thm.beta_conversion false) (Thm.apply (Thm.lambda t tm) t)
741           val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0
742         in fconv_rule(arg_conv(binop_conv (arg_conv (Thm.beta_conversion false))))
743                      (Thm.transitive th0 (c p ax))
744         end
746       val elim_abs = eliminate_construct is_abs
747         (fn p => fn ax =>
748           Thm.instantiate ([], [(p_v,p), (x_v, Thm.dest_arg ax)]) pth_abs)
749       val elim_max = eliminate_construct is_max
750         (fn p => fn ax =>
751           let val (ax,y) = Thm.dest_comb ax
752           in Thm.instantiate ([], [(p_v,p), (x_v, Thm.dest_arg ax), (y_v,y)])
753                              pth_max end)
754       val elim_min = eliminate_construct is_min
755         (fn p => fn ax =>
756           let val (ax,y) = Thm.dest_comb ax
757           in Thm.instantiate ([], [(p_v,p), (x_v, Thm.dest_arg ax), (y_v,y)])
758                              pth_min end)
759     in first_conv [elim_abs, elim_max, elim_min, all_conv]
760     end;
761 in
762 fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
763   gen_gen_real_arith ctxt
765      absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
766 end;
768 (* An instance for reals*)
770 fun gen_prover_real_arith ctxt prover =
771   let
772     fun simple_cterm_ord t u = Term_Ord.term_ord (Thm.term_of t, Thm.term_of u) = LESS
773     val {add, mul, neg, pow = _, sub = _, main} =
774         Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
775         (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
776         simple_cterm_ord
777   in gen_real_arith ctxt
778      (cterm_of_rat,
779       Numeral_Simprocs.field_comp_conv ctxt,
780       Numeral_Simprocs.field_comp_conv ctxt,
781       Numeral_Simprocs.field_comp_conv ctxt,
782       main ctxt, neg ctxt, add ctxt, mul ctxt, prover)
783   end;
785 end