src/HOL/Library/positivstellensatz.ML
author wenzelm
Thu Apr 18 17:07:01 2013 +0200 (2013-04-18)
changeset 51717 9e7d1c139569
parent 46594 f11f332b964f
child 52049 156e12d5cb92
permissions -rw-r--r--
simplifier uses proper Proof.context instead of historic type simpset;
     1 (*  Title:      HOL/Library/positivstellensatz.ML
     2     Author:     Amine Chaieb, University of Cambridge
     3 
     4 A generic arithmetic prover based on Positivstellensatz certificates
     5 --- also implements Fourrier-Motzkin elimination as a special case
     6 Fourrier-Motzkin elimination.
     7 *)
     8 
     9 (* A functor for finite mappings based on Tables *)
    10 
    11 signature FUNC =
    12 sig
    13   include TABLE
    14   val apply : 'a table -> key -> 'a
    15   val applyd :'a table -> (key -> 'a) -> key -> 'a
    16   val combine : ('a -> 'a -> 'a) -> ('a -> bool) -> 'a table -> 'a table -> 'a table
    17   val dom : 'a table -> key list
    18   val tryapplyd : 'a table -> key -> 'a -> 'a
    19   val updatep : (key * 'a -> bool) -> key * 'a -> 'a table -> 'a table
    20   val choose : 'a table -> key * 'a
    21   val onefunc : key * 'a -> 'a table
    22 end;
    23 
    24 functor FuncFun(Key: KEY) : FUNC =
    25 struct
    26 
    27 structure Tab = Table(Key);
    28 
    29 open Tab;
    30 
    31 fun dom a = sort Key.ord (Tab.keys a);
    32 fun applyd f d x = case Tab.lookup f x of
    33    SOME y => y
    34  | NONE => d x;
    35 
    36 fun apply f x = applyd f (fn _ => raise Tab.UNDEF x) x;
    37 fun tryapplyd f a d = applyd f (K d) a;
    38 fun updatep p (k,v) t = if p (k, v) then t else update (k,v) t
    39 fun combine f z a b =
    40   let
    41     fun h (k,v) t = case Tab.lookup t k of
    42         NONE => Tab.update (k,v) t
    43       | SOME v' => let val w = f v v'
    44         in if z w then Tab.delete k t else Tab.update (k,w) t end;
    45   in Tab.fold h a b end;
    46 
    47 fun choose f = case Tab.min_key f of
    48     SOME k => (k, the (Tab.lookup f k))
    49   | NONE => error "FuncFun.choose : Completely empty function"
    50 
    51 fun onefunc kv = update kv empty
    52 
    53 end;
    54 
    55 (* Some standard functors and utility functions for them *)
    56 
    57 structure FuncUtil =
    58 struct
    59 
    60 structure Intfunc = FuncFun(type key = int val ord = int_ord);
    61 structure Ratfunc = FuncFun(type key = Rat.rat val ord = Rat.ord);
    62 structure Intpairfunc = FuncFun(type key = int*int val ord = prod_ord int_ord int_ord);
    63 structure Symfunc = FuncFun(type key = string val ord = fast_string_ord);
    64 structure Termfunc = FuncFun(type key = term val ord = Term_Ord.fast_term_ord);
    65 
    66 val cterm_ord = Term_Ord.fast_term_ord o pairself term_of
    67 
    68 structure Ctermfunc = FuncFun(type key = cterm val ord = cterm_ord);
    69 
    70 type monomial = int Ctermfunc.table;
    71 
    72 val monomial_ord = list_ord (prod_ord cterm_ord int_ord) o pairself Ctermfunc.dest
    73 
    74 structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
    75 
    76 type poly = Rat.rat Monomialfunc.table;
    77 
    78 (* The ordering so we can create canonical HOL polynomials.                  *)
    79 
    80 fun dest_monomial mon = sort (cterm_ord o pairself fst) (Ctermfunc.dest mon);
    81 
    82 fun monomial_order (m1,m2) =
    83   if Ctermfunc.is_empty m2 then LESS
    84   else if Ctermfunc.is_empty m1 then GREATER
    85   else
    86     let
    87       val mon1 = dest_monomial m1
    88       val mon2 = dest_monomial m2
    89       val deg1 = fold (Integer.add o snd) mon1 0
    90       val deg2 = fold (Integer.add o snd) mon2 0
    91     in if deg1 < deg2 then GREATER
    92        else if deg1 > deg2 then LESS
    93        else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
    94     end;
    95 
    96 end
    97 
    98 (* positivstellensatz datatype and prover generation *)
    99 
   100 signature REAL_ARITH =
   101 sig
   102 
   103   datatype positivstellensatz =
   104     Axiom_eq of int
   105   | Axiom_le of int
   106   | Axiom_lt of int
   107   | Rational_eq of Rat.rat
   108   | Rational_le of Rat.rat
   109   | Rational_lt of Rat.rat
   110   | Square of FuncUtil.poly
   111   | Eqmul of FuncUtil.poly * positivstellensatz
   112   | Sum of positivstellensatz * positivstellensatz
   113   | Product of positivstellensatz * positivstellensatz;
   114 
   115   datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
   116 
   117   datatype tree_choice = Left | Right
   118 
   119   type prover = tree_choice list ->
   120     (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   121       thm list * thm list * thm list -> thm * pss_tree
   122   type cert_conv = cterm -> thm * pss_tree
   123 
   124   val gen_gen_real_arith :
   125     Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
   126      conv * conv * conv * conv * conv * conv * prover -> cert_conv
   127   val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   128     thm list * thm list * thm list -> thm * pss_tree
   129 
   130   val gen_real_arith : Proof.context ->
   131     (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
   132 
   133   val gen_prover_real_arith : Proof.context -> prover -> cert_conv
   134 
   135   val is_ratconst : cterm -> bool
   136   val dest_ratconst : cterm -> Rat.rat
   137   val cterm_of_rat : Rat.rat -> cterm
   138 
   139 end
   140 
   141 structure RealArith : REAL_ARITH =
   142 struct
   143 
   144 open Conv
   145 (* ------------------------------------------------------------------------- *)
   146 (* Data structure for Positivstellensatz refutations.                        *)
   147 (* ------------------------------------------------------------------------- *)
   148 
   149 datatype positivstellensatz =
   150     Axiom_eq of int
   151   | Axiom_le of int
   152   | Axiom_lt of int
   153   | Rational_eq of Rat.rat
   154   | Rational_le of Rat.rat
   155   | Rational_lt of Rat.rat
   156   | Square of FuncUtil.poly
   157   | Eqmul of FuncUtil.poly * positivstellensatz
   158   | Sum of positivstellensatz * positivstellensatz
   159   | Product of positivstellensatz * positivstellensatz;
   160          (* Theorems used in the procedure *)
   161 
   162 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
   163 datatype tree_choice = Left | Right
   164 type prover = tree_choice list ->
   165   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   166     thm list * thm list * thm list -> thm * pss_tree
   167 type cert_conv = cterm -> thm * pss_tree
   168 
   169 
   170     (* Some useful derived rules *)
   171 fun deduct_antisym_rule tha thb =
   172     Thm.equal_intr (Thm.implies_intr (cprop_of thb) tha)
   173      (Thm.implies_intr (cprop_of tha) thb);
   174 
   175 fun prove_hyp tha thb =
   176   if exists (curry op aconv (concl_of tha)) (Thm.hyps_of thb)  (* FIXME !? *)
   177   then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb;
   178 
   179 val pth = @{lemma "(((x::real) < y) == (y - x > 0))" and "((x <= y) == (y - x >= 0))" and
   180      "((x = y) == (x - y = 0))" and "((~(x < y)) == (x - y >= 0))" and
   181      "((~(x <= y)) == (x - y > 0))" and "((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
   182   by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};
   183 
   184 val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
   185 val pth_add =
   186   @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 )" and "( x = 0 ==> y >= 0 ==> x + y >= 0)" and
   187     "(x = 0 ==> 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)" by simp_all};
   191 
   192 val pth_mul =
   193   @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0)" and "(x = 0 ==> y >= 0 ==> x * y = 0)" and
   194     "(x = 0 ==> 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)"
   198   by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
   199     mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])};
   200 
   201 val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
   202 val pth_square = @{lemma "x * x >= (0::real)"  by simp};
   203 
   204 val weak_dnf_simps =
   205   List.take (@{thms simp_thms}, 34) @
   206     @{lemma "((P & (Q | R)) = ((P&Q) | (P&R)))" and "((Q | R) & P) = ((Q&P) | (R&P))" and
   207       "(P & Q) = (Q & P)" and "((P | Q) = (Q | P))" by blast+};
   208 
   209 (*
   210 val nnfD_simps =
   211   @{lemma "((~(P & Q)) = (~P | ~Q))" and "((~(P | Q)) = (~P & ~Q) )" and
   212     "((P --> Q) = (~P | Q) )" and "((P = Q) = ((P & Q) | (~P & ~ Q)))" and
   213     "((~(P = Q)) = ((P & ~ Q) | (~P & Q)) )" and "((~ ~(P)) = P)" by blast+};
   214 *)
   215 
   216 val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
   217 val prenex_simps =
   218   map (fn th => th RS sym)
   219     ([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @
   220       @{thms "HOL.all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});
   221 
   222 val real_abs_thms1 = @{lemma
   223   "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r))" and
   224   "((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r))" and
   225   "((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r))" and
   226   "((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r))" and
   227   "((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r))" and
   228   "((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r))" and
   229   "((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r))" and
   230   "((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r))" and
   231   "((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r))" and
   232   "((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y  + b >= r))" and
   233   "((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r))" and
   234   "((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y  + c >= r))" and
   235   "((1 * min x y >= r) = (1 * x >= r & 1 * y >= r))" and
   236   "((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and
   237   "((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and
   238   "((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y  + b >= r))" and
   239   "((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r))" and
   240   "((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y  + c >= r))" and
   241   "((min x y >= r) = (x >= r &  y >= r))" and
   242   "((min x y + a >= r) = (a + x >= r & a + y >= r))" and
   243   "((a + min x y >= r) = (a + x >= r & a + y >= r))" and
   244   "((a + min x y + b >= r) = (a + x + b >= r & a + y  + b >= r))" and
   245   "((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r))" and
   246   "((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r))" and
   247   "((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r))" and
   248   "((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r))" and
   249   "((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r))" and
   250   "((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r))" and
   251   "((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r))" and
   252   "((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r))" and
   253   "((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r))" and
   254   "((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r))" and
   255   "((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r))" and
   256   "((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y  + b > r))" and
   257   "((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r))" and
   258   "((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y  + c > r))" and
   259   "((min x y > r) = (x > r &  y > r))" and
   260   "((min x y + a > r) = (a + x > r & a + y > r))" and
   261   "((a + min x y > r) = (a + x > r & a + y > r))" and
   262   "((a + min x y + b > r) = (a + x + b > r & a + y  + b > r))" and
   263   "((a + b + min x y > r) = (a + b + x > r & a + b + y > r))" and
   264   "((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
   265   by auto};
   266 
   267 val abs_split' = @{lemma "P (abs (x::'a::linordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
   268   by (atomize (full)) (auto split add: abs_split)};
   269 
   270 val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
   271   by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
   272 
   273 val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
   274   by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
   275 
   276 
   277          (* Miscellaneous *)
   278 fun literals_conv bops uops cv =
   279   let
   280     fun h t =
   281       case (term_of t) of
   282         b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
   283       | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
   284       | _ => cv t
   285   in h end;
   286 
   287 fun cterm_of_rat x =
   288   let
   289     val (a, b) = Rat.quotient_of_rat x
   290   in
   291     if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
   292     else Thm.apply (Thm.apply @{cterm "op / :: real => _"}
   293       (Numeral.mk_cnumber @{ctyp "real"} a))
   294       (Numeral.mk_cnumber @{ctyp "real"} b)
   295   end;
   296 
   297 fun dest_ratconst t =
   298   case term_of t of
   299     Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
   300   | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
   301 fun is_ratconst t = can dest_ratconst t
   302 
   303 (*
   304 fun find_term p t = if p t then t else
   305  case t of
   306   a$b => (find_term p a handle TERM _ => find_term p b)
   307  | Abs (_,_,t') => find_term p t'
   308  | _ => raise TERM ("find_term",[t]);
   309 *)
   310 
   311 fun find_cterm p t =
   312   if p t then t else
   313   case term_of t of
   314     _$_ => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
   315   | Abs (_,_,_) => find_cterm p (Thm.dest_abs NONE t |> snd)
   316   | _ => raise CTERM ("find_cterm",[t]);
   317 
   318     (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
   319 fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
   320 fun is_comb t = case (term_of t) of _$_ => true | _ => false;
   321 
   322 fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
   323   handle CTERM _ => false;
   324 
   325 
   326 (* Map back polynomials to HOL.                         *)
   327 
   328 fun cterm_of_varpow x k = if k = 1 then x else Thm.apply (Thm.apply @{cterm "op ^ :: real => _"} x)
   329   (Numeral.mk_cnumber @{ctyp nat} k)
   330 
   331 fun cterm_of_monomial m =
   332   if FuncUtil.Ctermfunc.is_empty m then @{cterm "1::real"}
   333   else
   334     let
   335       val m' = FuncUtil.dest_monomial m
   336       val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' []
   337     in foldr1 (fn (s, t) => Thm.apply (Thm.apply @{cterm "op * :: real => _"} s) t) vps
   338     end
   339 
   340 fun cterm_of_cmonomial (m,c) =
   341   if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
   342   else if c = Rat.one then cterm_of_monomial m
   343   else Thm.apply (Thm.apply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
   344 
   345 fun cterm_of_poly p =
   346   if FuncUtil.Monomialfunc.is_empty p then @{cterm "0::real"}
   347   else
   348     let
   349       val cms = map cterm_of_cmonomial
   350         (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
   351     in foldr1 (fn (t1, t2) => Thm.apply(Thm.apply @{cterm "op + :: real => _"} t1) t2) cms
   352     end;
   353 
   354 (* A general real arithmetic prover *)
   355 
   356 fun gen_gen_real_arith ctxt (mk_numeric,
   357        numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
   358        poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
   359        absconv1,absconv2,prover) =
   360   let
   361     val pre_ss = put_simpset HOL_basic_ss ctxt addsimps
   362       @{thms simp_thms ex_simps all_simps not_all not_ex ex_disj_distrib 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
   377 
   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'])
   394     fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
   395          (match_mp_rule pth_add [th, th'])
   396     fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
   397        (instantiate' [] [SOME ct] (th RS pth_emul))
   398     fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
   399        (instantiate' [] [SOME t] pth_square)
   400 
   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
   424 
   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
   428 
   429     val concl = Thm.dest_arg o 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 _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
   442       in Thm.implies_elim (Thm.implies_elim
   443           (Thm.implies_elim (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
   444           (Thm.implies_intr (Thm.apply @{cterm Trueprop} p) th1))
   445         (Thm.implies_intr (Thm.apply @{cterm Trueprop} q) th2)
   446       end
   447     fun overall cert_choice dun ths =
   448       case ths of
   449         [] =>
   450         let
   451           val (eq,ne) = List.partition (is_req o concl) dun
   452           val (le,nl) = List.partition (is_ge o concl) ne
   453           val lt = filter (is_gt o concl) nl
   454         in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
   455       | th::oths =>
   456         let
   457           val ct = concl th
   458         in
   459           if is_conj ct then
   460             let
   461               val (th1,th2) = conj_pair th
   462             in overall cert_choice dun (th1::th2::oths) end
   463           else if is_disj ct then
   464             let
   465               val (th1, cert1) = overall (Left::cert_choice) dun (Thm.assume (Thm.apply @{cterm Trueprop} (Thm.dest_arg1 ct))::oths)
   466               val (th2, cert2) = overall (Right::cert_choice) dun (Thm.assume (Thm.apply @{cterm Trueprop} (Thm.dest_arg ct))::oths)
   467             in (disj_cases th th1 th2, Branch (cert1, cert2)) end
   468           else overall cert_choice (th::dun) oths
   469         end
   470     fun dest_binary b ct =
   471         if is_binop b ct then Thm.dest_binop ct
   472         else raise CTERM ("dest_binary",[b,ct])
   473     val dest_eq = dest_binary @{cterm "op = :: real => _"}
   474     val neq_th = nth pth 5
   475     fun real_not_eq_conv ct =
   476       let
   477         val (l,r) = dest_eq (Thm.dest_arg ct)
   478         val th = Thm.instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
   479         val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
   480         val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
   481         val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
   482         val th' = Drule.binop_cong_rule @{cterm HOL.disj}
   483           (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
   484           (Drule.arg_cong_rule (Thm.apply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
   485       in Thm.transitive th th'
   486       end
   487     fun equal_implies_1_rule PQ =
   488       let
   489         val P = Thm.lhs_of PQ
   490       in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P))
   491       end
   492     (* FIXME!!! Copied from groebner.ml *)
   493     val strip_exists =
   494       let
   495         fun h (acc, t) =
   496           case (term_of t) of
   497             Const(@{const_name Ex},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
   498           | _ => (acc,t)
   499       in fn t => h ([],t)
   500       end
   501     fun name_of x =
   502       case term_of x of
   503         Free(s,_) => s
   504       | Var ((s,_),_) => s
   505       | _ => "x"
   506 
   507     fun mk_forall x th = Drule.arg_cong_rule (instantiate_cterm' [SOME (ctyp_of_term x)] [] @{cpat "All :: (?'a => bool) => _" }) (Thm.abstract_rule (name_of x) x th)
   508 
   509     val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
   510 
   511     fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
   512     fun mk_ex v t = Thm.apply (ext (ctyp_of_term v)) (Thm.lambda v t)
   513 
   514     fun choose v th th' =
   515       case concl_of th of
   516         @{term Trueprop} $ (Const(@{const_name Ex},_)$_) =>
   517         let
   518           val p = (funpow 2 Thm.dest_arg o cprop_of) th
   519           val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
   520           val th0 = fconv_rule (Thm.beta_conversion true)
   521             (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
   522           val pv = (Thm.rhs_of o Thm.beta_conversion true)
   523             (Thm.apply @{cterm Trueprop} (Thm.apply p v))
   524           val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
   525         in Thm.implies_elim (Thm.implies_elim th0 th) th1  end
   526       | _ => raise THM ("choose",0,[th, th'])
   527 
   528     fun simple_choose v th =
   529       choose v (Thm.assume ((Thm.apply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
   530 
   531     val strip_forall =
   532       let
   533         fun h (acc, t) =
   534           case (term_of t) of
   535             Const(@{const_name All},_)$Abs(_,_,_) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
   536           | _ => (acc,t)
   537       in fn t => h ([],t)
   538       end
   539 
   540     fun f ct =
   541       let
   542         val nnf_norm_conv' =
   543           nnf_conv ctxt then_conv
   544           literals_conv [@{term HOL.conj}, @{term HOL.disj}] []
   545           (Conv.cache_conv
   546             (first_conv [real_lt_conv, real_le_conv,
   547                          real_eq_conv, real_not_lt_conv,
   548                          real_not_le_conv, real_not_eq_conv, all_conv]))
   549         fun absremover ct = (literals_conv [@{term HOL.conj}, @{term HOL.disj}] []
   550                   (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv
   551                   try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
   552         val nct = Thm.apply @{cterm Trueprop} (Thm.apply @{cterm "Not"} ct)
   553         val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
   554         val tm0 = Thm.dest_arg (Thm.rhs_of th0)
   555         val (th, certificates) =
   556           if tm0 aconvc @{cterm False} then (equal_implies_1_rule th0, Trivial) else
   557           let
   558             val (evs,bod) = strip_exists tm0
   559             val (avs,ibod) = strip_forall bod
   560             val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
   561             val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))]
   562             val th3 = fold simple_choose evs (prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.apply @{cterm Trueprop} bod))) th2)
   563           in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
   564           end
   565       in (Thm.implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates)
   566       end
   567   in f
   568   end;
   569 
   570 (* A linear arithmetic prover *)
   571 local
   572   val linear_add = FuncUtil.Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
   573   fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c */ x)
   574   val one_tm = @{cterm "1::real"}
   575   fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p Rat.zero)) orelse
   576      ((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso
   577        not(p(FuncUtil.Ctermfunc.apply e one_tm)))
   578 
   579   fun linear_ineqs vars (les,lts) =
   580     case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
   581       SOME r => r
   582     | NONE =>
   583       (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
   584          SOME r => r
   585        | NONE =>
   586          if null vars then error "linear_ineqs: no contradiction" else
   587          let
   588            val ineqs = les @ lts
   589            fun blowup v =
   590              length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
   591              length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
   592              length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
   593            val v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
   594              (map (fn v => (v,blowup v)) vars)))
   595            fun addup (e1,p1) (e2,p2) acc =
   596              let
   597                val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v Rat.zero
   598                val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v Rat.zero
   599              in
   600                if c1 */ c2 >=/ Rat.zero then acc else
   601                let
   602                  val e1' = linear_cmul (Rat.abs c2) e1
   603                  val e2' = linear_cmul (Rat.abs c1) e2
   604                  val p1' = Product(Rational_lt(Rat.abs c2),p1)
   605                  val p2' = Product(Rational_lt(Rat.abs c1),p2)
   606                in (linear_add e1' e2',Sum(p1',p2'))::acc
   607                end
   608              end
   609            val (les0,les1) =
   610              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
   611            val (lts0,lts1) =
   612              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
   613            val (lesp,lesn) =
   614              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
   615            val (ltsp,ltsn) =
   616              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
   617            val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
   618            val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
   619                       (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
   620          in linear_ineqs (remove (op aconvc) v vars) (les',lts')
   621          end)
   622 
   623   fun linear_eqs(eqs,les,lts) =
   624     case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
   625       SOME r => r
   626     | NONE =>
   627       (case eqs of
   628          [] =>
   629          let val vars = remove (op aconvc) one_tm
   630              (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) [])
   631          in linear_ineqs vars (les,lts) end
   632        | (e,p)::es =>
   633          if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
   634          let
   635            val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
   636            fun xform (inp as (t,q)) =
   637              let val d = FuncUtil.Ctermfunc.tryapplyd t x Rat.zero in
   638                if d =/ Rat.zero then inp else
   639                let
   640                  val k = (Rat.neg d) */ Rat.abs c // c
   641                  val e' = linear_cmul k e
   642                  val t' = linear_cmul (Rat.abs c) t
   643                  val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
   644                  val q' = Product(Rational_lt(Rat.abs c),q)
   645                in (linear_add e' t',Sum(p',q'))
   646                end
   647              end
   648          in linear_eqs(map xform es,map xform les,map xform lts)
   649          end)
   650 
   651   fun linear_prover (eq,le,lt) =
   652     let
   653       val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq
   654       val les = map_index (fn (n, p) => (p,Axiom_le n)) le
   655       val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt
   656     in linear_eqs(eqs,les,lts)
   657     end
   658 
   659   fun lin_of_hol ct =
   660     if ct aconvc @{cterm "0::real"} then FuncUtil.Ctermfunc.empty
   661     else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
   662     else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
   663     else
   664       let val (lop,r) = Thm.dest_comb ct
   665       in
   666         if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
   667         else
   668           let val (opr,l) = Thm.dest_comb lop
   669           in
   670             if opr aconvc @{cterm "op + :: real =>_"}
   671             then linear_add (lin_of_hol l) (lin_of_hol r)
   672             else if opr aconvc @{cterm "op * :: real =>_"}
   673                     andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
   674             else FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
   675           end
   676       end
   677 
   678   fun is_alien ct =
   679       case term_of ct of
   680         Const(@{const_name "real"}, _)$ n =>
   681         if can HOLogic.dest_number n then false else true
   682       | _ => false
   683 in
   684 fun real_linear_prover translator (eq,le,lt) =
   685   let
   686     val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o cprop_of
   687     val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o cprop_of
   688     val eq_pols = map lhs eq
   689     val le_pols = map rhs le
   690     val lt_pols = map rhs lt
   691     val aliens = filter is_alien
   692       (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom)
   693                 (eq_pols @ le_pols @ lt_pols) [])
   694     val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,Rat.one)) aliens
   695     val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
   696     val le' = le @ map (fn a => instantiate' [] [SOME (Thm.dest_arg a)] @{thm real_of_nat_ge_zero}) aliens
   697   in ((translator (eq,le',lt) proof), Trivial)
   698   end
   699 end;
   700 
   701 (* A less general generic arithmetic prover dealing with abs,max and min*)
   702 
   703 local
   704   val absmaxmin_elim_ss1 =
   705     simpset_of (put_simpset HOL_basic_ss @{context} addsimps real_abs_thms1)
   706   fun absmaxmin_elim_conv1 ctxt =
   707     Simplifier.rewrite (put_simpset absmaxmin_elim_ss1 ctxt)
   708 
   709   val absmaxmin_elim_conv2 =
   710     let
   711       val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
   712       val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
   713       val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
   714       val abs_tm = @{cterm "abs :: real => _"}
   715       val p_tm = @{cpat "?P :: real => bool"}
   716       val x_tm = @{cpat "?x :: real"}
   717       val y_tm = @{cpat "?y::real"}
   718       val is_max = is_binop @{cterm "max :: real => _"}
   719       val is_min = is_binop @{cterm "min :: real => _"}
   720       fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm
   721       fun eliminate_construct p c tm =
   722         let
   723           val t = find_cterm p tm
   724           val th0 = (Thm.symmetric o Thm.beta_conversion false) (Thm.apply (Thm.lambda t tm) t)
   725           val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0
   726         in fconv_rule(arg_conv(binop_conv (arg_conv (Thm.beta_conversion false))))
   727                      (Thm.transitive th0 (c p ax))
   728         end
   729 
   730       val elim_abs = eliminate_construct is_abs
   731         (fn p => fn ax =>
   732           Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax)]) pth_abs)
   733       val elim_max = eliminate_construct is_max
   734         (fn p => fn ax =>
   735           let val (ax,y) = Thm.dest_comb ax
   736           in Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)])
   737                              pth_max end)
   738       val elim_min = eliminate_construct is_min
   739         (fn p => fn ax =>
   740           let val (ax,y) = Thm.dest_comb ax
   741           in Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)])
   742                              pth_min end)
   743     in first_conv [elim_abs, elim_max, elim_min, all_conv]
   744     end;
   745 in
   746 fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
   747   gen_gen_real_arith ctxt
   748     (mkconst,eq,ge,gt,norm,neg,add,mul,
   749      absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
   750 end;
   751 
   752 (* An instance for reals*)
   753 
   754 fun gen_prover_real_arith ctxt prover =
   755   let
   756     fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS
   757     val {add, mul, neg, pow = _, sub = _, main} =
   758         Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
   759         (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"}))
   760         simple_cterm_ord
   761   in gen_real_arith ctxt
   762      (cterm_of_rat,
   763       Numeral_Simprocs.field_comp_conv ctxt,
   764       Numeral_Simprocs.field_comp_conv ctxt,
   765       Numeral_Simprocs.field_comp_conv ctxt,
   766       main ctxt, neg ctxt, add ctxt, mul ctxt, prover)
   767   end;
   768 
   769 end