src/HOL/Library/positivstellensatz.ML
author krauss
Fri May 07 15:36:03 2010 +0200 (2010-05-07)
changeset 39920 7479334d2c90
parent 39027 e4262f9e6a4e
child 44058 ae85c5d64913
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     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 val mon1 = dest_monomial m1 
    87       val mon2 = dest_monomial m2
    88       val deg1 = fold (Integer.add o snd) mon1 0
    89       val deg2 = fold (Integer.add o snd) mon2 0 
    90   in if deg1 < deg2 then GREATER else if deg1 > deg2 then LESS
    91      else list_ord (prod_ord cterm_ord int_ord) (mon1,mon2)
    92   end;
    93 
    94 end
    95 
    96 (* positivstellensatz datatype and prover generation *)
    97 
    98 signature REAL_ARITH = 
    99 sig
   100   
   101   datatype positivstellensatz =
   102    Axiom_eq of int
   103  | Axiom_le of int
   104  | Axiom_lt of int
   105  | Rational_eq of Rat.rat
   106  | Rational_le of Rat.rat
   107  | Rational_lt of Rat.rat
   108  | Square of FuncUtil.poly
   109  | Eqmul of FuncUtil.poly * positivstellensatz
   110  | Sum of positivstellensatz * positivstellensatz
   111  | Product of positivstellensatz * positivstellensatz;
   112 
   113 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
   114 
   115 datatype tree_choice = Left | Right
   116 
   117 type prover = tree_choice list -> 
   118   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   119   thm list * thm list * thm list -> thm * pss_tree
   120 type cert_conv = cterm -> thm * pss_tree
   121 
   122 val gen_gen_real_arith :
   123   Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
   124    conv * conv * conv * conv * conv * conv * prover -> cert_conv
   125 val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   126   thm list * thm list * thm list -> thm * pss_tree
   127 
   128 val gen_real_arith : Proof.context ->
   129   (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
   130 
   131 val gen_prover_real_arith : Proof.context -> prover -> cert_conv
   132 
   133 val is_ratconst : cterm -> bool
   134 val dest_ratconst : cterm -> Rat.rat
   135 val cterm_of_rat : Rat.rat -> cterm
   136 
   137 end
   138 
   139 structure RealArith : REAL_ARITH =
   140 struct
   141 
   142  open Conv
   143 (* ------------------------------------------------------------------------- *)
   144 (* Data structure for Positivstellensatz refutations.                        *)
   145 (* ------------------------------------------------------------------------- *)
   146 
   147 datatype positivstellensatz =
   148    Axiom_eq of int
   149  | Axiom_le of int
   150  | Axiom_lt of int
   151  | Rational_eq of Rat.rat
   152  | Rational_le of Rat.rat
   153  | Rational_lt of Rat.rat
   154  | Square of FuncUtil.poly
   155  | Eqmul of FuncUtil.poly * positivstellensatz
   156  | Sum of positivstellensatz * positivstellensatz
   157  | Product of positivstellensatz * positivstellensatz;
   158          (* Theorems used in the procedure *)
   159 
   160 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
   161 datatype tree_choice = Left | Right
   162 type prover = tree_choice list -> 
   163   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   164   thm list * thm list * thm list -> thm * pss_tree
   165 type cert_conv = cterm -> thm * pss_tree
   166 
   167 
   168     (* Some useful derived rules *)
   169 fun deduct_antisym_rule tha thb = 
   170     Thm.equal_intr (Thm.implies_intr (cprop_of thb) tha) 
   171      (Thm.implies_intr (cprop_of tha) thb);
   172 
   173 fun prove_hyp tha thb = 
   174   if exists (curry op aconv (concl_of tha)) (#hyps (rep_thm thb)) 
   175   then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb;
   176 
   177 val pth = @{lemma "(((x::real) < y) == (y - x > 0))" and "((x <= y) == (y - x >= 0))" and
   178      "((x = y) == (x - y = 0))" and "((~(x < y)) == (x - y >= 0))" and
   179      "((~(x <= y)) == (x - y > 0))" and "((~(x = y)) == (x - y > 0 | -(x - y) > 0))"
   180   by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};
   181 
   182 val pth_final = @{lemma "(~p ==> False) ==> p" by blast}
   183 val pth_add = 
   184   @{lemma "(x = (0::real) ==> y = 0 ==> x + y = 0 )" and "( x = 0 ==> y >= 0 ==> x + y >= 0)" and
   185     "(x = 0 ==> y > 0 ==> x + y > 0)" and "(x >= 0 ==> y = 0 ==> x + y >= 0)" and
   186     "(x >= 0 ==> 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)" by simp_all};
   189 
   190 val pth_mul = 
   191   @{lemma "(x = (0::real) ==> y = 0 ==> x * y = 0)" and "(x = 0 ==> y >= 0 ==> x * y = 0)" and
   192     "(x = 0 ==> y > 0 ==> x * y = 0)" and "(x >= 0 ==> y = 0 ==> x * y = 0)" and
   193     "(x >= 0 ==> 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)"
   196   by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
   197     mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])};
   198 
   199 val pth_emul = @{lemma "y = (0::real) ==> x * y = 0"  by simp};
   200 val pth_square = @{lemma "x * x >= (0::real)"  by simp};
   201 
   202 val weak_dnf_simps =
   203   List.take (simp_thms, 34) @
   204     @{lemma "((P & (Q | R)) = ((P&Q) | (P&R)))" and "((Q | R) & P) = ((Q&P) | (R&P))" and
   205       "(P & Q) = (Q & P)" and "((P | Q) = (Q | P))" by blast+};
   206 
   207 val nnfD_simps =
   208   @{lemma "((~(P & Q)) = (~P | ~Q))" and "((~(P | Q)) = (~P & ~Q) )" and
   209     "((P --> Q) = (~P | Q) )" and "((P = Q) = ((P & Q) | (~P & ~ Q)))" and
   210     "((~(P = Q)) = ((P & ~ Q) | (~P & Q)) )" and "((~ ~(P)) = P)" by blast+};
   211 
   212 val choice_iff = @{lemma "(ALL x. EX y. P x y) = (EX f. ALL x. P x (f x))" by metis};
   213 val prenex_simps =
   214   map (fn th => th RS sym)
   215     ([@{thm "all_conj_distrib"}, @{thm "ex_disj_distrib"}] @
   216       @{thms "HOL.all_simps"(1-4)} @ @{thms "ex_simps"(1-4)});
   217 
   218 val real_abs_thms1 = @{lemma
   219   "((-1 * abs(x::real) >= r) = (-1 * x >= r & 1 * x >= r))" and
   220   "((-1 * abs(x) + a >= r) = (a + -1 * x >= r & a + 1 * x >= r))" and
   221   "((a + -1 * abs(x) >= r) = (a + -1 * x >= r & a + 1 * x >= r))" and
   222   "((a + -1 * abs(x) + b >= r) = (a + -1 * x + b >= r & a + 1 * x + b >= r))" and
   223   "((a + b + -1 * abs(x) >= r) = (a + b + -1 * x >= r & a + b + 1 * x >= r))" and
   224   "((a + b + -1 * abs(x) + c >= r) = (a + b + -1 * x + c >= r & a + b + 1 * x + c >= r))" and
   225   "((-1 * max x y >= r) = (-1 * x >= r & -1 * y >= r))" and
   226   "((-1 * max x y + a >= r) = (a + -1 * x >= r & a + -1 * y >= r))" and
   227   "((a + -1 * max x y >= r) = (a + -1 * x >= r & a + -1 * y >= r))" and
   228   "((a + -1 * max x y + b >= r) = (a + -1 * x + b >= r & a + -1 * y  + b >= r))" and
   229   "((a + b + -1 * max x y >= r) = (a + b + -1 * x >= r & a + b + -1 * y >= r))" and
   230   "((a + b + -1 * max x y + c >= r) = (a + b + -1 * x + c >= r & a + b + -1 * y  + c >= r))" and
   231   "((1 * min x y >= r) = (1 * x >= r & 1 * y >= r))" and
   232   "((1 * min x y + a >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and
   233   "((a + 1 * min x y >= r) = (a + 1 * x >= r & a + 1 * y >= r))" and
   234   "((a + 1 * min x y + b >= r) = (a + 1 * x + b >= r & a + 1 * y  + b >= r))" and
   235   "((a + b + 1 * min x y >= r) = (a + b + 1 * x >= r & a + b + 1 * y >= r))" and
   236   "((a + b + 1 * min x y + c >= r) = (a + b + 1 * x + c >= r & a + b + 1 * y  + c >= r))" and
   237   "((min x y >= r) = (x >= r &  y >= r))" and
   238   "((min x y + a >= r) = (a + x >= r & a + y >= r))" and
   239   "((a + min x y >= r) = (a + x >= r & a + y >= r))" and
   240   "((a + min x y + b >= r) = (a + x + b >= r & a + y  + b >= r))" and
   241   "((a + b + min x y >= r) = (a + b + x >= r & a + b + y >= r))" and
   242   "((a + b + min x y + c >= r) = (a + b + x + c >= r & a + b + y + c >= r))" and
   243   "((-1 * abs(x) > r) = (-1 * x > r & 1 * x > r))" and
   244   "((-1 * abs(x) + a > r) = (a + -1 * x > r & a + 1 * x > r))" and
   245   "((a + -1 * abs(x) > r) = (a + -1 * x > r & a + 1 * x > r))" and
   246   "((a + -1 * abs(x) + b > r) = (a + -1 * x + b > r & a + 1 * x + b > r))" and
   247   "((a + b + -1 * abs(x) > r) = (a + b + -1 * x > r & a + b + 1 * x > r))" and
   248   "((a + b + -1 * abs(x) + c > r) = (a + b + -1 * x + c > r & a + b + 1 * x + c > r))" and
   249   "((-1 * max x y > r) = ((-1 * x > r) & -1 * y > r))" and
   250   "((-1 * max x y + a > r) = (a + -1 * x > r & a + -1 * y > r))" and
   251   "((a + -1 * max x y > r) = (a + -1 * x > r & a + -1 * y > r))" and
   252   "((a + -1 * max x y + b > r) = (a + -1 * x + b > r & a + -1 * y  + b > r))" and
   253   "((a + b + -1 * max x y > r) = (a + b + -1 * x > r & a + b + -1 * y > r))" and
   254   "((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r & a + b + -1 * y  + c > r))" and
   255   "((min x y > r) = (x > r &  y > r))" and
   256   "((min x y + a > r) = (a + x > r & a + y > r))" and
   257   "((a + min x y > r) = (a + x > r & a + y > r))" and
   258   "((a + min x y + b > r) = (a + x + b > r & a + y  + b > r))" and
   259   "((a + b + min x y > r) = (a + b + x > r & a + b + y > r))" and
   260   "((a + b + min x y + c > r) = (a + b + x + c > r & a + b + y + c > r))"
   261   by auto};
   262 
   263 val abs_split' = @{lemma "P (abs (x::'a::linordered_idom)) == (x >= 0 & P x | x < 0 & P (-x))"
   264   by (atomize (full)) (auto split add: abs_split)};
   265 
   266 val max_split = @{lemma "P (max x y) == ((x::'a::linorder) <= y & P y | x > y & P x)"
   267   by (atomize (full)) (cases "x <= y", auto simp add: max_def)};
   268 
   269 val min_split = @{lemma "P (min x y) == ((x::'a::linorder) <= y & P x | x > y & P y)"
   270   by (atomize (full)) (cases "x <= y", auto simp add: min_def)};
   271 
   272 
   273          (* Miscellaneous *)
   274 fun literals_conv bops uops cv = 
   275  let fun h t =
   276   case (term_of t) of 
   277    b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
   278  | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
   279  | _ => cv t
   280  in h end;
   281 
   282 fun cterm_of_rat x = 
   283 let val (a, b) = Rat.quotient_of_rat x
   284 in 
   285  if b = 1 then Numeral.mk_cnumber @{ctyp "real"} a
   286   else Thm.capply (Thm.capply @{cterm "op / :: real => _"} 
   287                    (Numeral.mk_cnumber @{ctyp "real"} a))
   288         (Numeral.mk_cnumber @{ctyp "real"} b)
   289 end;
   290 
   291   fun dest_ratconst t = case term_of t of
   292    Const(@{const_name divide}, _)$a$b => Rat.rat_of_quotient(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
   293  | _ => Rat.rat_of_int (HOLogic.dest_number (term_of t) |> snd)
   294  fun is_ratconst t = can dest_ratconst t
   295 
   296 fun find_term p t = if p t then t else 
   297  case t of
   298   a$b => (find_term p a handle TERM _ => find_term p b)
   299  | Abs (_,_,t') => find_term p t'
   300  | _ => raise TERM ("find_term",[t]);
   301 
   302 fun find_cterm p t = if p t then t else 
   303  case term_of t of
   304   a$b => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
   305  | Abs (_,_,t') => find_cterm p (Thm.dest_abs NONE t |> snd)
   306  | _ => raise CTERM ("find_cterm",[t]);
   307 
   308     (* Some conversions-related stuff which has been forbidden entrance into Pure/conv.ML*)
   309 fun instantiate_cterm' ty tms = Drule.cterm_rule (Drule.instantiate' ty tms)
   310 fun is_comb t = case (term_of t) of _$_ => true | _ => false;
   311 
   312 fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
   313   handle CTERM _ => false;
   314 
   315 
   316 (* Map back polynomials to HOL.                         *)
   317 
   318 fun cterm_of_varpow x k = if k = 1 then x else Thm.capply (Thm.capply @{cterm "op ^ :: real => _"} x) 
   319   (Numeral.mk_cnumber @{ctyp nat} k)
   320 
   321 fun cterm_of_monomial m = 
   322  if FuncUtil.Ctermfunc.is_empty m then @{cterm "1::real"} 
   323  else 
   324   let 
   325    val m' = FuncUtil.dest_monomial m
   326    val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' [] 
   327   in foldr1 (fn (s, t) => Thm.capply (Thm.capply @{cterm "op * :: real => _"} s) t) vps
   328   end
   329 
   330 fun cterm_of_cmonomial (m,c) = if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
   331     else if c = Rat.one then cterm_of_monomial m
   332     else Thm.capply (Thm.capply @{cterm "op *::real => _"} (cterm_of_rat c)) (cterm_of_monomial m);
   333 
   334 fun cterm_of_poly p = 
   335  if FuncUtil.Monomialfunc.is_empty p then @{cterm "0::real"} 
   336  else
   337   let 
   338    val cms = map cterm_of_cmonomial
   339      (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
   340   in foldr1 (fn (t1, t2) => Thm.capply(Thm.capply @{cterm "op + :: real => _"} t1) t2) cms
   341   end;
   342 
   343     (* A general real arithmetic prover *)
   344 
   345 fun gen_gen_real_arith ctxt (mk_numeric,
   346        numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
   347        poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
   348        absconv1,absconv2,prover) = 
   349 let
   350  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}]
   351  val prenex_ss = HOL_basic_ss addsimps prenex_simps
   352  val skolemize_ss = HOL_basic_ss addsimps [choice_iff]
   353  val presimp_conv = Simplifier.rewrite (Simplifier.context ctxt pre_ss)
   354  val prenex_conv = Simplifier.rewrite (Simplifier.context ctxt prenex_ss)
   355  val skolemize_conv = Simplifier.rewrite (Simplifier.context ctxt skolemize_ss)
   356  val weak_dnf_ss = HOL_basic_ss addsimps weak_dnf_simps
   357  val weak_dnf_conv = Simplifier.rewrite (Simplifier.context ctxt weak_dnf_ss)
   358  fun eqT_elim th = Thm.equal_elim (Thm.symmetric th) @{thm TrueI}
   359  fun oprconv cv ct = 
   360   let val g = Thm.dest_fun2 ct
   361   in if g aconvc @{cterm "op <= :: real => _"} 
   362        orelse g aconvc @{cterm "op < :: real => _"} 
   363      then arg_conv cv ct else arg1_conv cv ct
   364   end
   365 
   366  fun real_ineq_conv th ct =
   367   let
   368    val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th 
   369       handle Pattern.MATCH => raise CTERM ("real_ineq_conv", [ct]))
   370   in Thm.transitive th' (oprconv poly_conv (Thm.rhs_of th'))
   371   end 
   372   val [real_lt_conv, real_le_conv, real_eq_conv,
   373        real_not_lt_conv, real_not_le_conv, _] =
   374        map real_ineq_conv pth
   375   fun match_mp_rule ths ths' = 
   376    let
   377      fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
   378       | th::ths => (ths' MRS th handle THM _ => f ths ths')
   379    in f ths ths' end
   380   fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
   381          (match_mp_rule pth_mul [th, th'])
   382   fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
   383          (match_mp_rule pth_add [th, th'])
   384   fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv)) 
   385        (instantiate' [] [SOME ct] (th RS pth_emul)) 
   386   fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
   387        (instantiate' [] [SOME t] pth_square)
   388 
   389   fun hol_of_positivstellensatz(eqs,les,lts) proof =
   390    let 
   391     fun translate prf = case prf of
   392         Axiom_eq n => nth eqs n
   393       | Axiom_le n => nth les n
   394       | Axiom_lt n => nth lts n
   395       | Rational_eq x => eqT_elim(numeric_eq_conv(Thm.capply @{cterm Trueprop} 
   396                           (Thm.capply (Thm.capply @{cterm "op =::real => _"} (mk_numeric x)) 
   397                                @{cterm "0::real"})))
   398       | Rational_le x => eqT_elim(numeric_ge_conv(Thm.capply @{cterm Trueprop} 
   399                           (Thm.capply (Thm.capply @{cterm "op <=::real => _"} 
   400                                      @{cterm "0::real"}) (mk_numeric x))))
   401       | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.capply @{cterm Trueprop} 
   402                       (Thm.capply (Thm.capply @{cterm "op <::real => _"} @{cterm "0::real"})
   403                         (mk_numeric x))))
   404       | Square pt => square_rule (cterm_of_poly pt)
   405       | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
   406       | Sum(p1,p2) => add_rule (translate p1) (translate p2)
   407       | Product(p1,p2) => mul_rule (translate p1) (translate p2)
   408    in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv]) 
   409           (translate proof)
   410    end
   411   
   412   val init_conv = presimp_conv then_conv
   413       nnf_conv then_conv skolemize_conv then_conv prenex_conv then_conv
   414       weak_dnf_conv
   415 
   416   val concl = Thm.dest_arg o cprop_of
   417   fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
   418   val is_req = is_binop @{cterm "op =:: real => _"}
   419   val is_ge = is_binop @{cterm "op <=:: real => _"}
   420   val is_gt = is_binop @{cterm "op <:: real => _"}
   421   val is_conj = is_binop @{cterm HOL.conj}
   422   val is_disj = is_binop @{cterm HOL.disj}
   423   fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
   424   fun disj_cases th th1 th2 = 
   425    let val (p,q) = Thm.dest_binop (concl th)
   426        val c = concl th1
   427        val _ = if c aconvc (concl th2) then () else error "disj_cases : conclusions not alpha convertible"
   428    in Thm.implies_elim (Thm.implies_elim
   429           (Thm.implies_elim (instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
   430           (Thm.implies_intr (Thm.capply @{cterm Trueprop} p) th1))
   431         (Thm.implies_intr (Thm.capply @{cterm Trueprop} q) th2)
   432    end
   433  fun overall cert_choice dun ths = case ths of
   434   [] =>
   435    let 
   436     val (eq,ne) = List.partition (is_req o concl) dun
   437      val (le,nl) = List.partition (is_ge o concl) ne
   438      val lt = filter (is_gt o concl) nl 
   439     in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
   440  | th::oths =>
   441    let 
   442     val ct = concl th 
   443    in 
   444     if is_conj ct  then
   445      let 
   446       val (th1,th2) = conj_pair th in
   447       overall cert_choice dun (th1::th2::oths) end
   448     else if is_disj ct then
   449       let 
   450        val (th1, cert1) = overall (Left::cert_choice) dun (Thm.assume (Thm.capply @{cterm Trueprop} (Thm.dest_arg1 ct))::oths)
   451        val (th2, cert2) = overall (Right::cert_choice) dun (Thm.assume (Thm.capply @{cterm Trueprop} (Thm.dest_arg ct))::oths)
   452       in (disj_cases th th1 th2, Branch (cert1, cert2)) end
   453    else overall cert_choice (th::dun) oths
   454   end
   455   fun dest_binary b ct = if is_binop b ct then Thm.dest_binop ct 
   456                          else raise CTERM ("dest_binary",[b,ct])
   457   val dest_eq = dest_binary @{cterm "op = :: real => _"}
   458   val neq_th = nth pth 5
   459   fun real_not_eq_conv ct = 
   460    let 
   461     val (l,r) = dest_eq (Thm.dest_arg ct)
   462     val th = Thm.instantiate ([],[(@{cpat "?x::real"},l),(@{cpat "?y::real"},r)]) neq_th
   463     val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
   464     val th_x = Drule.arg_cong_rule @{cterm "uminus :: real => _"} th_p
   465     val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
   466     val th' = Drule.binop_cong_rule @{cterm HOL.disj} 
   467      (Drule.arg_cong_rule (Thm.capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_p)
   468      (Drule.arg_cong_rule (Thm.capply @{cterm "op <::real=>_"} @{cterm "0::real"}) th_n)
   469     in Thm.transitive th th' 
   470   end
   471  fun equal_implies_1_rule PQ = 
   472   let 
   473    val P = Thm.lhs_of PQ
   474   in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P))
   475   end
   476  (* FIXME!!! Copied from groebner.ml *)
   477  val strip_exists =
   478   let fun h (acc, t) =
   479    case (term_of t) of
   480     Const(@{const_name Ex},_)$Abs(x,T,p) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
   481   | _ => (acc,t)
   482   in fn t => h ([],t)
   483   end
   484   fun name_of x = case term_of x of
   485    Free(s,_) => s
   486  | Var ((s,_),_) => s
   487  | _ => "x"
   488 
   489   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)
   490 
   491   val specl = fold_rev (fn x => fn th => instantiate' [] [SOME x] (th RS spec));
   492 
   493  fun ext T = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat Ex}
   494  fun mk_ex v t = Thm.capply (ext (ctyp_of_term v)) (Thm.cabs v t)
   495 
   496  fun choose v th th' = case concl_of th of 
   497    @{term Trueprop} $ (Const(@{const_name Ex},_)$_) => 
   498     let
   499      val p = (funpow 2 Thm.dest_arg o cprop_of) th
   500      val T = (hd o Thm.dest_ctyp o ctyp_of_term) p
   501      val th0 = fconv_rule (Thm.beta_conversion true)
   502          (instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o cprop_of) th'] exE)
   503      val pv = (Thm.rhs_of o Thm.beta_conversion true) 
   504            (Thm.capply @{cterm Trueprop} (Thm.capply p v))
   505      val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
   506     in Thm.implies_elim (Thm.implies_elim th0 th) th1  end
   507  | _ => raise THM ("choose",0,[th, th'])
   508 
   509   fun simple_choose v th = 
   510      choose v (Thm.assume ((Thm.capply @{cterm Trueprop} o mk_ex v) ((Thm.dest_arg o hd o #hyps o Thm.crep_thm) th))) th
   511 
   512  val strip_forall =
   513   let fun h (acc, t) =
   514    case (term_of t) of
   515     Const(@{const_name All},_)$Abs(x,T,p) => h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
   516   | _ => (acc,t)
   517   in fn t => h ([],t)
   518   end
   519 
   520  fun f ct =
   521   let 
   522    val nnf_norm_conv' = 
   523      nnf_conv then_conv 
   524      literals_conv [@{term HOL.conj}, @{term HOL.disj}] [] 
   525      (Conv.cache_conv 
   526        (first_conv [real_lt_conv, real_le_conv, 
   527                     real_eq_conv, real_not_lt_conv, 
   528                     real_not_le_conv, real_not_eq_conv, all_conv]))
   529   fun absremover ct = (literals_conv [@{term HOL.conj}, @{term HOL.disj}] [] 
   530                   (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv 
   531         try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
   532   val nct = Thm.capply @{cterm Trueprop} (Thm.capply @{cterm "Not"} ct)
   533   val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
   534   val tm0 = Thm.dest_arg (Thm.rhs_of th0)
   535   val (th, certificates) = if tm0 aconvc @{cterm False} then (equal_implies_1_rule th0, Trivial) else
   536    let 
   537     val (evs,bod) = strip_exists tm0
   538     val (avs,ibod) = strip_forall bod
   539     val th1 = Drule.arg_cong_rule @{cterm Trueprop} (fold mk_forall avs (absremover ibod))
   540     val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))]
   541     val th3 = fold simple_choose evs (prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.capply @{cterm Trueprop} bod))) th2)
   542    in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
   543    end
   544   in (Thm.implies_elim (instantiate' [] [SOME ct] pth_final) th, certificates)
   545  end
   546 in f
   547 end;
   548 
   549 (* A linear arithmetic prover *)
   550 local
   551   val linear_add = FuncUtil.Ctermfunc.combine (curry op +/) (fn z => z =/ Rat.zero)
   552   fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c */ x)
   553   val one_tm = @{cterm "1::real"}
   554   fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p Rat.zero)) orelse
   555      ((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso
   556        not(p(FuncUtil.Ctermfunc.apply e one_tm)))
   557 
   558   fun linear_ineqs vars (les,lts) = 
   559    case find_first (contradictory (fn x => x >/ Rat.zero)) lts of
   560     SOME r => r
   561   | NONE => 
   562    (case find_first (contradictory (fn x => x >/ Rat.zero)) les of
   563      SOME r => r
   564    | NONE => 
   565      if null vars then error "linear_ineqs: no contradiction" else
   566      let 
   567       val ineqs = les @ lts
   568       fun blowup v =
   569        length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) ineqs) +
   570        length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) ineqs) *
   571        length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero </ Rat.zero) ineqs)
   572       val  v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
   573                  (map (fn v => (v,blowup v)) vars)))
   574       fun addup (e1,p1) (e2,p2) acc =
   575        let 
   576         val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v Rat.zero 
   577         val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v Rat.zero
   578        in if c1 */ c2 >=/ Rat.zero then acc else
   579         let 
   580          val e1' = linear_cmul (Rat.abs c2) e1
   581          val e2' = linear_cmul (Rat.abs c1) e2
   582          val p1' = Product(Rational_lt(Rat.abs c2),p1)
   583          val p2' = Product(Rational_lt(Rat.abs c1),p2)
   584         in (linear_add e1' e2',Sum(p1',p2'))::acc
   585         end
   586        end
   587       val (les0,les1) = 
   588          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) les
   589       val (lts0,lts1) = 
   590          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero =/ Rat.zero) lts
   591       val (lesp,lesn) = 
   592          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) les1
   593       val (ltsp,ltsn) = 
   594          List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v Rat.zero >/ Rat.zero) lts1
   595       val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
   596       val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
   597                       (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
   598      in linear_ineqs (remove (op aconvc) v vars) (les',lts')
   599      end)
   600 
   601   fun linear_eqs(eqs,les,lts) = 
   602    case find_first (contradictory (fn x => x =/ Rat.zero)) eqs of
   603     SOME r => r
   604   | NONE => (case eqs of 
   605     [] => 
   606      let val vars = remove (op aconvc) one_tm 
   607            (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) []) 
   608      in linear_ineqs vars (les,lts) end
   609    | (e,p)::es => 
   610      if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
   611      let 
   612       val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
   613       fun xform (inp as (t,q)) =
   614        let val d = FuncUtil.Ctermfunc.tryapplyd t x Rat.zero in
   615         if d =/ Rat.zero then inp else
   616         let 
   617          val k = (Rat.neg d) */ Rat.abs c // c
   618          val e' = linear_cmul k e
   619          val t' = linear_cmul (Rat.abs c) t
   620          val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
   621          val q' = Product(Rational_lt(Rat.abs c),q) 
   622         in (linear_add e' t',Sum(p',q')) 
   623         end 
   624       end
   625      in linear_eqs(map xform es,map xform les,map xform lts)
   626      end)
   627 
   628   fun linear_prover (eq,le,lt) = 
   629    let 
   630     val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq
   631     val les = map_index (fn (n, p) => (p,Axiom_le n)) le
   632     val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt
   633    in linear_eqs(eqs,les,lts)
   634    end 
   635   
   636   fun lin_of_hol ct = 
   637    if ct aconvc @{cterm "0::real"} then FuncUtil.Ctermfunc.empty
   638    else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
   639    else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
   640    else
   641     let val (lop,r) = Thm.dest_comb ct 
   642     in if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
   643        else
   644         let val (opr,l) = Thm.dest_comb lop 
   645         in if opr aconvc @{cterm "op + :: real =>_"} 
   646            then linear_add (lin_of_hol l) (lin_of_hol r)
   647            else if opr aconvc @{cterm "op * :: real =>_"} 
   648                    andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
   649            else FuncUtil.Ctermfunc.onefunc (ct, Rat.one)
   650         end
   651     end
   652 
   653   fun is_alien ct = case term_of ct of 
   654    Const(@{const_name "real"}, _)$ n => 
   655      if can HOLogic.dest_number n then false else true
   656   | _ => false
   657 in 
   658 fun real_linear_prover translator (eq,le,lt) = 
   659  let 
   660   val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o cprop_of
   661   val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o cprop_of
   662   val eq_pols = map lhs eq
   663   val le_pols = map rhs le
   664   val lt_pols = map rhs lt 
   665   val aliens =  filter is_alien
   666       (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom) 
   667           (eq_pols @ le_pols @ lt_pols) [])
   668   val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,Rat.one)) aliens
   669   val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
   670   val le' = le @ map (fn a => instantiate' [] [SOME (Thm.dest_arg a)] @{thm real_of_nat_ge_zero}) aliens 
   671  in ((translator (eq,le',lt) proof), Trivial)
   672  end
   673 end;
   674 
   675 (* A less general generic arithmetic prover dealing with abs,max and min*)
   676 
   677 local
   678  val absmaxmin_elim_ss1 = HOL_basic_ss addsimps real_abs_thms1
   679  fun absmaxmin_elim_conv1 ctxt = 
   680     Simplifier.rewrite (Simplifier.context ctxt absmaxmin_elim_ss1)
   681 
   682  val absmaxmin_elim_conv2 =
   683   let 
   684    val pth_abs = instantiate' [SOME @{ctyp real}] [] abs_split'
   685    val pth_max = instantiate' [SOME @{ctyp real}] [] max_split
   686    val pth_min = instantiate' [SOME @{ctyp real}] [] min_split
   687    val abs_tm = @{cterm "abs :: real => _"}
   688    val p_tm = @{cpat "?P :: real => bool"}
   689    val x_tm = @{cpat "?x :: real"}
   690    val y_tm = @{cpat "?y::real"}
   691    val is_max = is_binop @{cterm "max :: real => _"}
   692    val is_min = is_binop @{cterm "min :: real => _"} 
   693    fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm
   694    fun eliminate_construct p c tm =
   695     let 
   696      val t = find_cterm p tm
   697      val th0 = (Thm.symmetric o Thm.beta_conversion false) (Thm.capply (Thm.cabs t tm) t)
   698      val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0
   699     in fconv_rule(arg_conv(binop_conv (arg_conv (Thm.beta_conversion false))))
   700                (Thm.transitive th0 (c p ax))
   701    end
   702 
   703    val elim_abs = eliminate_construct is_abs
   704     (fn p => fn ax => 
   705        Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax)]) pth_abs)
   706    val elim_max = eliminate_construct is_max
   707     (fn p => fn ax => 
   708       let val (ax,y) = Thm.dest_comb ax 
   709       in  Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)]) 
   710       pth_max end)
   711    val elim_min = eliminate_construct is_min
   712     (fn p => fn ax => 
   713       let val (ax,y) = Thm.dest_comb ax 
   714       in  Thm.instantiate ([], [(p_tm,p), (x_tm, Thm.dest_arg ax), (y_tm,y)]) 
   715       pth_min end)
   716    in first_conv [elim_abs, elim_max, elim_min, all_conv]
   717   end;
   718 in fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
   719         gen_gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,
   720                        absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
   721 end;
   722 
   723 (* An instance for reals*) 
   724 
   725 fun gen_prover_real_arith ctxt prover = 
   726  let
   727   fun simple_cterm_ord t u = Term_Ord.term_ord (term_of t, term_of u) = LESS
   728   val {add,mul,neg,pow,sub,main} = 
   729      Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
   730       (the (Semiring_Normalizer.match ctxt @{cterm "(0::real) + 1"})) 
   731      simple_cterm_ord
   732 in gen_real_arith ctxt
   733    (cterm_of_rat, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv,
   734     main,neg,add,mul, prover)
   735 end;
   736 
   737 end