src/HOL/Library/positivstellensatz.ML
author wenzelm
Thu Feb 01 17:27:13 2018 +0100 (21 months ago)
changeset 67564 d615e9ca77dc
parent 67562 2427d3e72b6e
child 69064 5840724b1d71
permissions -rw-r--r--
tuned;
     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 Fourier-Motzkin elimination as a special case
     6 Fourier-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 =
    48   (case Tab.min f of
    49     SOME entry => entry
    50   | NONE => error "FuncFun.choose : Completely empty function")
    51 
    52 fun onefunc kv = update kv empty
    53 
    54 end;
    55 
    56 (* Some standard functors and utility functions for them *)
    57 
    58 structure FuncUtil =
    59 struct
    60 
    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);
    66 structure Ctermfunc = FuncFun(type key = cterm val ord = Thm.fast_term_ord);
    67 
    68 type monomial = int Ctermfunc.table;
    69 val monomial_ord = list_ord (prod_ord Thm.fast_term_ord int_ord) o apply2 Ctermfunc.dest
    70 structure Monomialfunc = FuncFun(type key = monomial val ord = monomial_ord)
    71 
    72 type poly = Rat.rat Monomialfunc.table;
    73 
    74 (* The ordering so we can create canonical HOL polynomials.                  *)
    75 
    76 fun dest_monomial mon = sort (Thm.fast_term_ord o apply2 fst) (Ctermfunc.dest mon);
    77 
    78 fun monomial_order (m1,m2) =
    79   if Ctermfunc.is_empty m2 then LESS
    80   else if Ctermfunc.is_empty m1 then GREATER
    81   else
    82     let
    83       val mon1 = dest_monomial m1
    84       val mon2 = dest_monomial m2
    85       val deg1 = fold (Integer.add o snd) mon1 0
    86       val deg2 = fold (Integer.add o snd) mon2 0
    87     in if deg1 < deg2 then GREATER
    88        else if deg1 > deg2 then LESS
    89        else list_ord (prod_ord Thm.fast_term_ord int_ord) (mon1,mon2)
    90     end;
    91 
    92 end
    93 
    94 (* positivstellensatz datatype and prover generation *)
    95 
    96 signature REAL_ARITH =
    97 sig
    98 
    99   datatype positivstellensatz =
   100     Axiom_eq of int
   101   | Axiom_le of int
   102   | Axiom_lt of int
   103   | Rational_eq of Rat.rat
   104   | Rational_le of Rat.rat
   105   | Rational_lt of Rat.rat
   106   | Square of FuncUtil.poly
   107   | Eqmul of FuncUtil.poly * positivstellensatz
   108   | Sum of positivstellensatz * positivstellensatz
   109   | Product of positivstellensatz * positivstellensatz;
   110 
   111   datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
   112 
   113   datatype tree_choice = Left | Right
   114 
   115   type prover = tree_choice list ->
   116     (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   117       thm list * thm list * thm list -> thm * pss_tree
   118   type cert_conv = cterm -> thm * pss_tree
   119 
   120   val gen_gen_real_arith :
   121     Proof.context -> (Rat.rat -> cterm) * conv * conv * conv *
   122      conv * conv * conv * conv * conv * conv * prover -> cert_conv
   123   val real_linear_prover : (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   124     thm list * thm list * thm list -> thm * pss_tree
   125 
   126   val gen_real_arith : Proof.context ->
   127     (Rat.rat -> cterm) * conv * conv * conv * conv * conv * conv * conv * prover -> cert_conv
   128 
   129   val gen_prover_real_arith : Proof.context -> prover -> cert_conv
   130 
   131   val is_ratconst : cterm -> bool
   132   val dest_ratconst : cterm -> Rat.rat
   133   val cterm_of_rat : Rat.rat -> cterm
   134 
   135 end
   136 
   137 structure RealArith : REAL_ARITH =
   138 struct
   139 
   140 open Conv
   141 (* ------------------------------------------------------------------------- *)
   142 (* Data structure for Positivstellensatz refutations.                        *)
   143 (* ------------------------------------------------------------------------- *)
   144 
   145 datatype positivstellensatz =
   146     Axiom_eq of int
   147   | Axiom_le of int
   148   | Axiom_lt of int
   149   | Rational_eq of Rat.rat
   150   | Rational_le of Rat.rat
   151   | Rational_lt of Rat.rat
   152   | Square of FuncUtil.poly
   153   | Eqmul of FuncUtil.poly * positivstellensatz
   154   | Sum of positivstellensatz * positivstellensatz
   155   | Product of positivstellensatz * positivstellensatz;
   156          (* Theorems used in the procedure *)
   157 
   158 datatype pss_tree = Trivial | Cert of positivstellensatz | Branch of pss_tree * pss_tree
   159 datatype tree_choice = Left | Right
   160 type prover = tree_choice list ->
   161   (thm list * thm list * thm list -> positivstellensatz -> thm) ->
   162     thm list * thm list * thm list -> thm * pss_tree
   163 type cert_conv = cterm -> thm * pss_tree
   164 
   165 
   166     (* Some useful derived rules *)
   167 fun deduct_antisym_rule tha thb =
   168     Thm.equal_intr (Thm.implies_intr (Thm.cprop_of thb) tha)
   169      (Thm.implies_intr (Thm.cprop_of tha) thb);
   170 
   171 fun prove_hyp tha thb =
   172   if exists (curry op aconv (Thm.concl_of tha)) (Thm.hyps_of thb)  (* FIXME !? *)
   173   then Thm.equal_elim (Thm.symmetric (deduct_antisym_rule tha thb)) tha else thb;
   174 
   175 val pth = @{lemma "(((x::real) < y) \<equiv> (y - x > 0))" and "((x \<le> y) \<equiv> (y - x \<ge> 0))" and
   176      "((x = y) \<equiv> (x - y = 0))" and "((\<not>(x < y)) \<equiv> (x - y \<ge> 0))" and
   177      "((\<not>(x \<le> y)) \<equiv> (x - y > 0))" and "((\<not>(x = y)) \<equiv> (x - y > 0 \<or> -(x - y) > 0))"
   178   by (atomize (full), auto simp add: less_diff_eq le_diff_eq not_less)};
   179 
   180 val pth_final = @{lemma "(\<not>p \<Longrightarrow> False) \<Longrightarrow> p" by blast}
   181 val pth_add =
   182   @{lemma "(x = (0::real) \<Longrightarrow> y = 0 \<Longrightarrow> x + y = 0 )" and "( x = 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y \<ge> 0)" and
   183     "(x = 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" and "(x \<ge> 0 \<Longrightarrow> y = 0 \<Longrightarrow> x + y \<ge> 0)" and
   184     "(x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y \<ge> 0)" and "(x \<ge> 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" and
   185     "(x > 0 \<Longrightarrow> y = 0 \<Longrightarrow> x + y > 0)" and "(x > 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x + y > 0)" and
   186     "(x > 0 \<Longrightarrow> y > 0 \<Longrightarrow> x + y > 0)" by simp_all};
   187 
   188 val pth_mul =
   189   @{lemma "(x = (0::real) \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and "(x = 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y = 0)" and
   190     "(x = 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y = 0)" and "(x \<ge> 0 \<Longrightarrow> y = 0 \<Longrightarrow> x * y = 0)" and
   191     "(x \<ge> 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y \<ge> 0)" and "(x \<ge> 0 \<Longrightarrow> y > 0 \<Longrightarrow> x * y \<ge> 0)" and
   192     "(x > 0 \<Longrightarrow>  y = 0 \<Longrightarrow> x * y = 0)" and "(x > 0 \<Longrightarrow> y \<ge> 0 \<Longrightarrow> x * y \<ge> 0)" and
   193     "(x > 0 \<Longrightarrow>  y > 0 \<Longrightarrow> x * y > 0)"
   194   by (auto intro: mult_mono[where a="0::real" and b="x" and d="y" and c="0", simplified]
   195     mult_strict_mono[where b="x" and d="y" and a="0" and c="0", simplified])};
   196 
   197 val pth_emul = @{lemma "y = (0::real) \<Longrightarrow> x * y = 0"  by simp};
   198 val pth_square = @{lemma "x * x \<ge> (0::real)"  by simp};
   199 
   200 val weak_dnf_simps =
   201   List.take (@{thms simp_thms}, 34) @
   202     @{lemma "((P \<and> (Q \<or> R)) = ((P\<and>Q) \<or> (P\<and>R)))" and "((Q \<or> R) \<and> P) = ((Q\<and>P) \<or> (R\<and>P))" and
   203       "(P \<and> Q) = (Q \<and> P)" and "((P \<or> Q) = (Q \<or> P))" by blast+};
   204 
   205 (*
   206 val nnfD_simps =
   207   @{lemma "((~(P & Q)) = (~P | ~Q))" and "((~(P | Q)) = (~P & ~Q) )" and
   208     "((P --> Q) = (~P | Q) )" and "((P = Q) = ((P & Q) | (~P & ~ Q)))" and
   209     "((~(P = Q)) = ((P & ~ Q) | (~P & Q)) )" and "((~ ~(P)) = P)" by blast+};
   210 *)
   211 
   212 val choice_iff = @{lemma "(\<forall>x. \<exists>y. P x y) = (\<exists>f. \<forall>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 * \<bar>x::real\<bar> \<ge> r) = (-1 * x \<ge> r \<and> 1 * x \<ge> r))" and
   220   "((-1 * \<bar>x\<bar> + a \<ge> r) = (a + -1 * x \<ge> r \<and> a + 1 * x \<ge> r))" and
   221   "((a + -1 * \<bar>x\<bar> \<ge> r) = (a + -1 * x \<ge> r \<and> a + 1 * x \<ge> r))" and
   222   "((a + -1 * \<bar>x\<bar> + b \<ge> r) = (a + -1 * x + b \<ge> r \<and> a + 1 * x + b \<ge> r))" and
   223   "((a + b + -1 * \<bar>x\<bar> \<ge> r) = (a + b + -1 * x \<ge> r \<and> a + b + 1 * x \<ge> r))" and
   224   "((a + b + -1 * \<bar>x\<bar> + c \<ge> r) = (a + b + -1 * x + c \<ge> r \<and> a + b + 1 * x + c \<ge> r))" and
   225   "((-1 * max x y \<ge> r) = (-1 * x \<ge> r \<and> -1 * y \<ge> r))" and
   226   "((-1 * max x y + a \<ge> r) = (a + -1 * x \<ge> r \<and> a + -1 * y \<ge> r))" and
   227   "((a + -1 * max x y \<ge> r) = (a + -1 * x \<ge> r \<and> a + -1 * y \<ge> r))" and
   228   "((a + -1 * max x y + b \<ge> r) = (a + -1 * x + b \<ge> r \<and> a + -1 * y  + b \<ge> r))" and
   229   "((a + b + -1 * max x y \<ge> r) = (a + b + -1 * x \<ge> r \<and> a + b + -1 * y \<ge> r))" and
   230   "((a + b + -1 * max x y + c \<ge> r) = (a + b + -1 * x + c \<ge> r \<and> a + b + -1 * y  + c \<ge> r))" and
   231   "((1 * min x y \<ge> r) = (1 * x \<ge> r \<and> 1 * y \<ge> r))" and
   232   "((1 * min x y + a \<ge> r) = (a + 1 * x \<ge> r \<and> a + 1 * y \<ge> r))" and
   233   "((a + 1 * min x y \<ge> r) = (a + 1 * x \<ge> r \<and> a + 1 * y \<ge> r))" and
   234   "((a + 1 * min x y + b \<ge> r) = (a + 1 * x + b \<ge> r \<and> a + 1 * y  + b \<ge> r))" and
   235   "((a + b + 1 * min x y \<ge> r) = (a + b + 1 * x \<ge> r \<and> a + b + 1 * y \<ge> r))" and
   236   "((a + b + 1 * min x y + c \<ge> r) = (a + b + 1 * x + c \<ge> r \<and> a + b + 1 * y  + c \<ge> r))" and
   237   "((min x y \<ge> r) = (x \<ge> r \<and> y \<ge> r))" and
   238   "((min x y + a \<ge> r) = (a + x \<ge> r \<and> a + y \<ge> r))" and
   239   "((a + min x y \<ge> r) = (a + x \<ge> r \<and> a + y \<ge> r))" and
   240   "((a + min x y + b \<ge> r) = (a + x + b \<ge> r \<and> a + y  + b \<ge> r))" and
   241   "((a + b + min x y \<ge> r) = (a + b + x \<ge> r \<and> a + b + y \<ge> r))" and
   242   "((a + b + min x y + c \<ge> r) = (a + b + x + c \<ge> r \<and> a + b + y + c \<ge> r))" and
   243   "((-1 * \<bar>x\<bar> > r) = (-1 * x > r \<and> 1 * x > r))" and
   244   "((-1 * \<bar>x\<bar> + a > r) = (a + -1 * x > r \<and> a + 1 * x > r))" and
   245   "((a + -1 * \<bar>x\<bar> > r) = (a + -1 * x > r \<and> a + 1 * x > r))" and
   246   "((a + -1 * \<bar>x\<bar> + b > r) = (a + -1 * x + b > r \<and> a + 1 * x + b > r))" and
   247   "((a + b + -1 * \<bar>x\<bar> > r) = (a + b + -1 * x > r \<and> a + b + 1 * x > r))" and
   248   "((a + b + -1 * \<bar>x\<bar> + c > r) = (a + b + -1 * x + c > r \<and> a + b + 1 * x + c > r))" and
   249   "((-1 * max x y > r) = ((-1 * x > r) \<and> -1 * y > r))" and
   250   "((-1 * max x y + a > r) = (a + -1 * x > r \<and> a + -1 * y > r))" and
   251   "((a + -1 * max x y > r) = (a + -1 * x > r \<and> a + -1 * y > r))" and
   252   "((a + -1 * max x y + b > r) = (a + -1 * x + b > r \<and> a + -1 * y  + b > r))" and
   253   "((a + b + -1 * max x y > r) = (a + b + -1 * x > r \<and> a + b + -1 * y > r))" and
   254   "((a + b + -1 * max x y + c > r) = (a + b + -1 * x + c > r \<and> a + b + -1 * y  + c > r))" and
   255   "((min x y > r) = (x > r \<and> y > r))" and
   256   "((min x y + a > r) = (a + x > r \<and> a + y > r))" and
   257   "((a + min x y > r) = (a + x > r \<and> a + y > r))" and
   258   "((a + min x y + b > r) = (a + x + b > r \<and> a + y  + b > r))" and
   259   "((a + b + min x y > r) = (a + b + x > r \<and> a + b + y > r))" and
   260   "((a + b + min x y + c > r) = (a + b + x + c > r \<and> a + b + y + c > r))"
   261   by auto};
   262 
   263 val abs_split' = @{lemma "P \<bar>x::'a::linordered_idom\<bar> == (x \<ge> 0 \<and> P x \<or> x < 0 \<and> P (-x))"
   264   by (atomize (full)) (auto split: abs_split)};
   265 
   266 val max_split = @{lemma "P (max x y) \<equiv> ((x::'a::linorder) \<le> y \<and> P y \<or> x > y \<and> P x)"
   267   by (atomize (full)) (cases "x \<le> y", auto simp add: max_def)};
   268 
   269 val min_split = @{lemma "P (min x y) \<equiv> ((x::'a::linorder) \<le> y \<and> P x \<or> x > y \<and> P y)"
   270   by (atomize (full)) (cases "x \<le> y", auto simp add: min_def)};
   271 
   272 
   273          (* Miscellaneous *)
   274 fun literals_conv bops uops cv =
   275   let
   276     fun h t =
   277       (case Thm.term_of t of
   278         b$_$_ => if member (op aconv) bops b then binop_conv h t else cv t
   279       | u$_ => if member (op aconv) uops u then arg_conv h t else cv t
   280       | _ => cv t)
   281   in h end;
   282 
   283 fun cterm_of_rat x =
   284   let
   285     val (a, b) = Rat.dest x
   286   in
   287     if b = 1 then Numeral.mk_cnumber \<^ctyp>\<open>real\<close> a
   288     else Thm.apply (Thm.apply \<^cterm>\<open>(/) :: real \<Rightarrow> _\<close>
   289       (Numeral.mk_cnumber \<^ctyp>\<open>real\<close> a))
   290       (Numeral.mk_cnumber \<^ctyp>\<open>real\<close> b)
   291   end;
   292 
   293 fun dest_ratconst t =
   294   case Thm.term_of t of
   295     Const(\<^const_name>\<open>divide\<close>, _)$a$b => Rat.make(HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
   296   | _ => Rat.of_int (HOLogic.dest_number (Thm.term_of t) |> snd)
   297 fun is_ratconst t = can dest_ratconst t
   298 
   299 (*
   300 fun find_term p t = if p t then t else
   301  case t of
   302   a$b => (find_term p a handle TERM _ => find_term p b)
   303  | Abs (_,_,t') => find_term p t'
   304  | _ => raise TERM ("find_term",[t]);
   305 *)
   306 
   307 fun find_cterm p t =
   308   if p t then t else
   309   case Thm.term_of t of
   310     _$_ => (find_cterm p (Thm.dest_fun t) handle CTERM _ => find_cterm p (Thm.dest_arg t))
   311   | Abs (_,_,_) => find_cterm p (Thm.dest_abs NONE t |> snd)
   312   | _ => raise CTERM ("find_cterm",[t]);
   313 
   314 fun is_comb t = (case Thm.term_of t of _ $ _ => true | _ => false);
   315 
   316 fun is_binop ct ct' = ct aconvc (Thm.dest_fun (Thm.dest_fun ct'))
   317   handle CTERM _ => false;
   318 
   319 
   320 (* Map back polynomials to HOL.                         *)
   321 
   322 fun cterm_of_varpow x k = if k = 1 then x else Thm.apply (Thm.apply \<^cterm>\<open>(^) :: real \<Rightarrow> _\<close> x)
   323   (Numeral.mk_cnumber \<^ctyp>\<open>nat\<close> k)
   324 
   325 fun cterm_of_monomial m =
   326   if FuncUtil.Ctermfunc.is_empty m then \<^cterm>\<open>1::real\<close>
   327   else
   328     let
   329       val m' = FuncUtil.dest_monomial m
   330       val vps = fold_rev (fn (x,k) => cons (cterm_of_varpow x k)) m' []
   331     in foldr1 (fn (s, t) => Thm.apply (Thm.apply \<^cterm>\<open>( * ) :: real \<Rightarrow> _\<close> s) t) vps
   332     end
   333 
   334 fun cterm_of_cmonomial (m,c) =
   335   if FuncUtil.Ctermfunc.is_empty m then cterm_of_rat c
   336   else if c = @1 then cterm_of_monomial m
   337   else Thm.apply (Thm.apply \<^cterm>\<open>( * )::real \<Rightarrow> _\<close> (cterm_of_rat c)) (cterm_of_monomial m);
   338 
   339 fun cterm_of_poly p =
   340   if FuncUtil.Monomialfunc.is_empty p then \<^cterm>\<open>0::real\<close>
   341   else
   342     let
   343       val cms = map cterm_of_cmonomial
   344         (sort (prod_ord FuncUtil.monomial_order (K EQUAL)) (FuncUtil.Monomialfunc.dest p))
   345     in foldr1 (fn (t1, t2) => Thm.apply(Thm.apply \<^cterm>\<open>(+) :: real \<Rightarrow> _\<close> t1) t2) cms
   346     end;
   347 
   348 (* A general real arithmetic prover *)
   349 
   350 fun gen_gen_real_arith ctxt (mk_numeric,
   351        numeric_eq_conv,numeric_ge_conv,numeric_gt_conv,
   352        poly_conv,poly_neg_conv,poly_add_conv,poly_mul_conv,
   353        absconv1,absconv2,prover) =
   354   let
   355     val pre_ss = put_simpset HOL_basic_ss ctxt addsimps
   356       @{thms simp_thms ex_simps all_simps not_all not_ex ex_disj_distrib
   357           all_conj_distrib if_bool_eq_disj}
   358     val prenex_ss = put_simpset HOL_basic_ss ctxt addsimps prenex_simps
   359     val skolemize_ss = put_simpset HOL_basic_ss ctxt addsimps [choice_iff]
   360     val presimp_conv = Simplifier.rewrite pre_ss
   361     val prenex_conv = Simplifier.rewrite prenex_ss
   362     val skolemize_conv = Simplifier.rewrite skolemize_ss
   363     val weak_dnf_ss = put_simpset HOL_basic_ss ctxt addsimps weak_dnf_simps
   364     val weak_dnf_conv = Simplifier.rewrite weak_dnf_ss
   365     fun eqT_elim th = Thm.equal_elim (Thm.symmetric th) @{thm TrueI}
   366     fun oprconv cv ct =
   367       let val g = Thm.dest_fun2 ct
   368       in if g aconvc \<^cterm>\<open>(\<le>) :: real \<Rightarrow> _\<close>
   369             orelse g aconvc \<^cterm>\<open>(<) :: real \<Rightarrow> _\<close>
   370          then arg_conv cv ct else arg1_conv cv ct
   371       end
   372 
   373     fun real_ineq_conv th ct =
   374       let
   375         val th' = (Thm.instantiate (Thm.match (Thm.lhs_of th, ct)) th
   376           handle Pattern.MATCH => raise CTERM ("real_ineq_conv", [ct]))
   377       in Thm.transitive th' (oprconv poly_conv (Thm.rhs_of th'))
   378       end
   379     val [real_lt_conv, real_le_conv, real_eq_conv,
   380          real_not_lt_conv, real_not_le_conv, _] =
   381          map real_ineq_conv pth
   382     fun match_mp_rule ths ths' =
   383       let
   384         fun f ths ths' = case ths of [] => raise THM("match_mp_rule",0,ths)
   385           | th::ths => (ths' MRS th handle THM _ => f ths ths')
   386       in f ths ths' end
   387     fun mul_rule th th' = fconv_rule (arg_conv (oprconv poly_mul_conv))
   388          (match_mp_rule pth_mul [th, th'])
   389     fun add_rule th th' = fconv_rule (arg_conv (oprconv poly_add_conv))
   390          (match_mp_rule pth_add [th, th'])
   391     fun emul_rule ct th = fconv_rule (arg_conv (oprconv poly_mul_conv))
   392        (Thm.instantiate' [] [SOME ct] (th RS pth_emul))
   393     fun square_rule t = fconv_rule (arg_conv (oprconv poly_conv))
   394        (Thm.instantiate' [] [SOME t] pth_square)
   395 
   396     fun hol_of_positivstellensatz(eqs,les,lts) proof =
   397       let
   398         fun translate prf =
   399           case prf of
   400             Axiom_eq n => nth eqs n
   401           | Axiom_le n => nth les n
   402           | Axiom_lt n => nth lts n
   403           | Rational_eq x => eqT_elim(numeric_eq_conv(Thm.apply \<^cterm>\<open>Trueprop\<close>
   404                           (Thm.apply (Thm.apply \<^cterm>\<open>(=)::real \<Rightarrow> _\<close> (mk_numeric x))
   405                                \<^cterm>\<open>0::real\<close>)))
   406           | Rational_le x => eqT_elim(numeric_ge_conv(Thm.apply \<^cterm>\<open>Trueprop\<close>
   407                           (Thm.apply (Thm.apply \<^cterm>\<open>(\<le>)::real \<Rightarrow> _\<close>
   408                                      \<^cterm>\<open>0::real\<close>) (mk_numeric x))))
   409           | Rational_lt x => eqT_elim(numeric_gt_conv(Thm.apply \<^cterm>\<open>Trueprop\<close>
   410                       (Thm.apply (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>)
   411                         (mk_numeric x))))
   412           | Square pt => square_rule (cterm_of_poly pt)
   413           | Eqmul(pt,p) => emul_rule (cterm_of_poly pt) (translate p)
   414           | Sum(p1,p2) => add_rule (translate p1) (translate p2)
   415           | Product(p1,p2) => mul_rule (translate p1) (translate p2)
   416       in fconv_rule (first_conv [numeric_ge_conv, numeric_gt_conv, numeric_eq_conv, all_conv])
   417           (translate proof)
   418       end
   419 
   420     val init_conv = presimp_conv then_conv
   421         nnf_conv ctxt then_conv skolemize_conv then_conv prenex_conv then_conv
   422         weak_dnf_conv
   423 
   424     val concl = Thm.dest_arg o Thm.cprop_of
   425     fun is_binop opr ct = (Thm.dest_fun2 ct aconvc opr handle CTERM _ => false)
   426     val is_req = is_binop \<^cterm>\<open>(=):: real \<Rightarrow> _\<close>
   427     val is_ge = is_binop \<^cterm>\<open>(\<le>):: real \<Rightarrow> _\<close>
   428     val is_gt = is_binop \<^cterm>\<open>(<):: real \<Rightarrow> _\<close>
   429     val is_conj = is_binop \<^cterm>\<open>HOL.conj\<close>
   430     val is_disj = is_binop \<^cterm>\<open>HOL.disj\<close>
   431     fun conj_pair th = (th RS @{thm conjunct1}, th RS @{thm conjunct2})
   432     fun disj_cases th th1 th2 =
   433       let
   434         val (p,q) = Thm.dest_binop (concl th)
   435         val c = concl th1
   436         val _ =
   437           if c aconvc (concl th2) then ()
   438           else error "disj_cases : conclusions not alpha convertible"
   439       in Thm.implies_elim (Thm.implies_elim
   440           (Thm.implies_elim (Thm.instantiate' [] (map SOME [p,q,c]) @{thm disjE}) th)
   441           (Thm.implies_intr (Thm.apply \<^cterm>\<open>Trueprop\<close> p) th1))
   442         (Thm.implies_intr (Thm.apply \<^cterm>\<open>Trueprop\<close> q) th2)
   443       end
   444     fun overall cert_choice dun ths =
   445       case ths of
   446         [] =>
   447         let
   448           val (eq,ne) = List.partition (is_req o concl) dun
   449           val (le,nl) = List.partition (is_ge o concl) ne
   450           val lt = filter (is_gt o concl) nl
   451         in prover (rev cert_choice) hol_of_positivstellensatz (eq,le,lt) end
   452       | th::oths =>
   453         let
   454           val ct = concl th
   455         in
   456           if is_conj ct then
   457             let
   458               val (th1,th2) = conj_pair th
   459             in overall cert_choice dun (th1::th2::oths) end
   460           else if is_disj ct then
   461             let
   462               val (th1, cert1) =
   463                 overall (Left::cert_choice) dun
   464                   (Thm.assume (Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.dest_arg1 ct))::oths)
   465               val (th2, cert2) =
   466                 overall (Right::cert_choice) dun
   467                   (Thm.assume (Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.dest_arg ct))::oths)
   468             in (disj_cases th th1 th2, Branch (cert1, cert2)) end
   469           else overall cert_choice (th::dun) oths
   470         end
   471     fun dest_binary b ct =
   472         if is_binop b ct then Thm.dest_binop ct
   473         else raise CTERM ("dest_binary",[b,ct])
   474     val dest_eq = dest_binary \<^cterm>\<open>(=) :: real \<Rightarrow> _\<close>
   475     val neq_th = nth pth 5
   476     fun real_not_eq_conv ct =
   477       let
   478         val (l,r) = dest_eq (Thm.dest_arg ct)
   479         val th = Thm.instantiate ([],[((("x", 0), \<^typ>\<open>real\<close>),l),((("y", 0), \<^typ>\<open>real\<close>),r)]) neq_th
   480         val th_p = poly_conv(Thm.dest_arg(Thm.dest_arg1(Thm.rhs_of th)))
   481         val th_x = Drule.arg_cong_rule \<^cterm>\<open>uminus :: real \<Rightarrow> _\<close> th_p
   482         val th_n = fconv_rule (arg_conv poly_neg_conv) th_x
   483         val th' = Drule.binop_cong_rule \<^cterm>\<open>HOL.disj\<close>
   484           (Drule.arg_cong_rule (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>) th_p)
   485           (Drule.arg_cong_rule (Thm.apply \<^cterm>\<open>(<)::real \<Rightarrow> _\<close> \<^cterm>\<open>0::real\<close>) th_n)
   486       in Thm.transitive th th'
   487       end
   488     fun equal_implies_1_rule PQ =
   489       let
   490         val P = Thm.lhs_of PQ
   491       in Thm.implies_intr P (Thm.equal_elim PQ (Thm.assume P))
   492       end
   493     (*FIXME!!! Copied from groebner.ml*)
   494     val strip_exists =
   495       let
   496         fun h (acc, t) =
   497           case Thm.term_of t of
   498             Const(\<^const_name>\<open>Ex\<close>,_)$Abs(_,_,_) =>
   499               h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
   500           | _ => (acc,t)
   501       in fn t => h ([],t)
   502       end
   503     fun name_of x =
   504       case Thm.term_of x of
   505         Free(s,_) => s
   506       | Var ((s,_),_) => s
   507       | _ => "x"
   508 
   509     fun mk_forall x th =
   510       let
   511         val T = Thm.typ_of_cterm x
   512         val all = Thm.cterm_of ctxt (Const (\<^const_name>\<open>All\<close>, (T --> \<^typ>\<open>bool\<close>) --> \<^typ>\<open>bool\<close>))
   513       in Drule.arg_cong_rule all (Thm.abstract_rule (name_of x) x th) end
   514 
   515     val specl = fold_rev (fn x => fn th => Thm.instantiate' [] [SOME x] (th RS spec));
   516 
   517     fun ext T = Thm.cterm_of ctxt (Const (\<^const_name>\<open>Ex\<close>, (T --> \<^typ>\<open>bool\<close>) --> \<^typ>\<open>bool\<close>))
   518     fun mk_ex v t = Thm.apply (ext (Thm.typ_of_cterm v)) (Thm.lambda v t)
   519 
   520     fun choose v th th' =
   521       case Thm.concl_of th of
   522         \<^term>\<open>Trueprop\<close> $ (Const(\<^const_name>\<open>Ex\<close>,_)$_) =>
   523         let
   524           val p = (funpow 2 Thm.dest_arg o Thm.cprop_of) th
   525           val T = (hd o Thm.dest_ctyp o Thm.ctyp_of_cterm) p
   526           val th0 = fconv_rule (Thm.beta_conversion true)
   527             (Thm.instantiate' [SOME T] [SOME p, (SOME o Thm.dest_arg o Thm.cprop_of) th'] exE)
   528           val pv = (Thm.rhs_of o Thm.beta_conversion true)
   529             (Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.apply p v))
   530           val th1 = Thm.forall_intr v (Thm.implies_intr pv th')
   531         in Thm.implies_elim (Thm.implies_elim th0 th) th1  end
   532       | _ => raise THM ("choose",0,[th, th'])
   533 
   534     fun simple_choose v th =
   535       choose v
   536         (Thm.assume
   537           ((Thm.apply \<^cterm>\<open>Trueprop\<close> o mk_ex v) (Thm.dest_arg (hd (Thm.chyps_of th))))) th
   538 
   539     val strip_forall =
   540       let
   541         fun h (acc, t) =
   542           case Thm.term_of t of
   543             Const(\<^const_name>\<open>All\<close>,_)$Abs(_,_,_) =>
   544               h (Thm.dest_abs NONE (Thm.dest_arg t) |>> (fn v => v::acc))
   545           | _ => (acc,t)
   546       in fn t => h ([],t)
   547       end
   548 
   549     fun f ct =
   550       let
   551         val nnf_norm_conv' =
   552           nnf_conv ctxt then_conv
   553           literals_conv [\<^term>\<open>HOL.conj\<close>, \<^term>\<open>HOL.disj\<close>] []
   554           (Conv.cache_conv
   555             (first_conv [real_lt_conv, real_le_conv,
   556                          real_eq_conv, real_not_lt_conv,
   557                          real_not_le_conv, real_not_eq_conv, all_conv]))
   558         fun absremover ct = (literals_conv [\<^term>\<open>HOL.conj\<close>, \<^term>\<open>HOL.disj\<close>] []
   559                   (try_conv (absconv1 then_conv binop_conv (arg_conv poly_conv))) then_conv
   560                   try_conv (absconv2 then_conv nnf_norm_conv' then_conv binop_conv absremover)) ct
   561         val nct = Thm.apply \<^cterm>\<open>Trueprop\<close> (Thm.apply \<^cterm>\<open>Not\<close> ct)
   562         val th0 = (init_conv then_conv arg_conv nnf_norm_conv') nct
   563         val tm0 = Thm.dest_arg (Thm.rhs_of th0)
   564         val (th, certificates) =
   565           if tm0 aconvc \<^cterm>\<open>False\<close> then (equal_implies_1_rule th0, Trivial) else
   566           let
   567             val (evs,bod) = strip_exists tm0
   568             val (avs,ibod) = strip_forall bod
   569             val th1 = Drule.arg_cong_rule \<^cterm>\<open>Trueprop\<close> (fold mk_forall avs (absremover ibod))
   570             val (th2, certs) = overall [] [] [specl avs (Thm.assume (Thm.rhs_of th1))]
   571             val th3 =
   572               fold simple_choose evs
   573                 (prove_hyp (Thm.equal_elim th1 (Thm.assume (Thm.apply \<^cterm>\<open>Trueprop\<close> bod))) th2)
   574           in (Drule.implies_intr_hyps (prove_hyp (Thm.equal_elim th0 (Thm.assume nct)) th3), certs)
   575           end
   576       in (Thm.implies_elim (Thm.instantiate' [] [SOME ct] pth_final) th, certificates)
   577       end
   578   in f
   579   end;
   580 
   581 (* A linear arithmetic prover *)
   582 local
   583   val linear_add = FuncUtil.Ctermfunc.combine (curry op +) (fn z => z = @0)
   584   fun linear_cmul c = FuncUtil.Ctermfunc.map (fn _ => fn x => c * x)
   585   val one_tm = \<^cterm>\<open>1::real\<close>
   586   fun contradictory p (e,_) = ((FuncUtil.Ctermfunc.is_empty e) andalso not(p @0)) orelse
   587      ((eq_set (op aconvc) (FuncUtil.Ctermfunc.dom e, [one_tm])) andalso
   588        not(p(FuncUtil.Ctermfunc.apply e one_tm)))
   589 
   590   fun linear_ineqs vars (les,lts) =
   591     case find_first (contradictory (fn x => x > @0)) lts of
   592       SOME r => r
   593     | NONE =>
   594       (case find_first (contradictory (fn x => x > @0)) les of
   595          SOME r => r
   596        | NONE =>
   597          if null vars then error "linear_ineqs: no contradiction" else
   598          let
   599            val ineqs = les @ lts
   600            fun blowup v =
   601              length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) ineqs) +
   602              length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) ineqs) *
   603              length(filter (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 < @0) ineqs)
   604            val v = fst(hd(sort (fn ((_,i),(_,j)) => int_ord (i,j))
   605              (map (fn v => (v,blowup v)) vars)))
   606            fun addup (e1,p1) (e2,p2) acc =
   607              let
   608                val c1 = FuncUtil.Ctermfunc.tryapplyd e1 v @0
   609                val c2 = FuncUtil.Ctermfunc.tryapplyd e2 v @0
   610              in
   611                if c1 * c2 >= @0 then acc else
   612                let
   613                  val e1' = linear_cmul (abs c2) e1
   614                  val e2' = linear_cmul (abs c1) e2
   615                  val p1' = Product(Rational_lt (abs c2), p1)
   616                  val p2' = Product(Rational_lt (abs c1), p2)
   617                in (linear_add e1' e2',Sum(p1',p2'))::acc
   618                end
   619              end
   620            val (les0,les1) =
   621              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) les
   622            val (lts0,lts1) =
   623              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 = @0) lts
   624            val (lesp,lesn) =
   625              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) les1
   626            val (ltsp,ltsn) =
   627              List.partition (fn (e,_) => FuncUtil.Ctermfunc.tryapplyd e v @0 > @0) lts1
   628            val les' = fold_rev (fn ep1 => fold_rev (addup ep1) lesp) lesn les0
   629            val lts' = fold_rev (fn ep1 => fold_rev (addup ep1) (lesp@ltsp)) ltsn
   630                       (fold_rev (fn ep1 => fold_rev (addup ep1) (lesn@ltsn)) ltsp lts0)
   631          in linear_ineqs (remove (op aconvc) v vars) (les',lts')
   632          end)
   633 
   634   fun linear_eqs(eqs,les,lts) =
   635     case find_first (contradictory (fn x => x = @0)) eqs of
   636       SOME r => r
   637     | NONE =>
   638       (case eqs of
   639          [] =>
   640          let val vars = remove (op aconvc) one_tm
   641              (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom o fst) (les@lts) [])
   642          in linear_ineqs vars (les,lts) end
   643        | (e,p)::es =>
   644          if FuncUtil.Ctermfunc.is_empty e then linear_eqs (es,les,lts) else
   645          let
   646            val (x,c) = FuncUtil.Ctermfunc.choose (FuncUtil.Ctermfunc.delete_safe one_tm e)
   647            fun xform (inp as (t,q)) =
   648              let val d = FuncUtil.Ctermfunc.tryapplyd t x @0 in
   649                if d = @0 then inp else
   650                let
   651                  val k = ~ d * abs c / c
   652                  val e' = linear_cmul k e
   653                  val t' = linear_cmul (abs c) t
   654                  val p' = Eqmul(FuncUtil.Monomialfunc.onefunc (FuncUtil.Ctermfunc.empty, k),p)
   655                  val q' = Product(Rational_lt (abs c), q)
   656                in (linear_add e' t',Sum(p',q'))
   657                end
   658              end
   659          in linear_eqs(map xform es,map xform les,map xform lts)
   660          end)
   661 
   662   fun linear_prover (eq,le,lt) =
   663     let
   664       val eqs = map_index (fn (n, p) => (p,Axiom_eq n)) eq
   665       val les = map_index (fn (n, p) => (p,Axiom_le n)) le
   666       val lts = map_index (fn (n, p) => (p,Axiom_lt n)) lt
   667     in linear_eqs(eqs,les,lts)
   668     end
   669 
   670   fun lin_of_hol ct =
   671     if ct aconvc \<^cterm>\<open>0::real\<close> then FuncUtil.Ctermfunc.empty
   672     else if not (is_comb ct) then FuncUtil.Ctermfunc.onefunc (ct, @1)
   673     else if is_ratconst ct then FuncUtil.Ctermfunc.onefunc (one_tm, dest_ratconst ct)
   674     else
   675       let val (lop,r) = Thm.dest_comb ct
   676       in
   677         if not (is_comb lop) then FuncUtil.Ctermfunc.onefunc (ct, @1)
   678         else
   679           let val (opr,l) = Thm.dest_comb lop
   680           in
   681             if opr aconvc \<^cterm>\<open>(+) :: real \<Rightarrow> _\<close>
   682             then linear_add (lin_of_hol l) (lin_of_hol r)
   683             else if opr aconvc \<^cterm>\<open>( * ) :: real \<Rightarrow> _\<close>
   684                     andalso is_ratconst l then FuncUtil.Ctermfunc.onefunc (r, dest_ratconst l)
   685             else FuncUtil.Ctermfunc.onefunc (ct, @1)
   686           end
   687       end
   688 
   689   fun is_alien ct =
   690     case Thm.term_of ct of
   691       Const(\<^const_name>\<open>of_nat\<close>, _)$ n => not (can HOLogic.dest_number n)
   692     | Const(\<^const_name>\<open>of_int\<close>, _)$ n => not (can HOLogic.dest_number n)
   693     | _ => false
   694 in
   695 fun real_linear_prover translator (eq,le,lt) =
   696   let
   697     val lhs = lin_of_hol o Thm.dest_arg1 o Thm.dest_arg o Thm.cprop_of
   698     val rhs = lin_of_hol o Thm.dest_arg o Thm.dest_arg o Thm.cprop_of
   699     val eq_pols = map lhs eq
   700     val le_pols = map rhs le
   701     val lt_pols = map rhs lt
   702     val aliens = filter is_alien
   703       (fold_rev (union (op aconvc) o FuncUtil.Ctermfunc.dom)
   704                 (eq_pols @ le_pols @ lt_pols) [])
   705     val le_pols' = le_pols @ map (fn v => FuncUtil.Ctermfunc.onefunc (v,@1)) aliens
   706     val (_,proof) = linear_prover (eq_pols,le_pols',lt_pols)
   707     val le' = le @ map (fn a => Thm.instantiate' [] [SOME (Thm.dest_arg a)] @{thm of_nat_0_le_iff}) aliens
   708   in ((translator (eq,le',lt) proof), Trivial)
   709   end
   710 end;
   711 
   712 (* A less general generic arithmetic prover dealing with abs,max and min*)
   713 
   714 local
   715   val absmaxmin_elim_ss1 =
   716     simpset_of (put_simpset HOL_basic_ss @{context} addsimps real_abs_thms1)
   717   fun absmaxmin_elim_conv1 ctxt =
   718     Simplifier.rewrite (put_simpset absmaxmin_elim_ss1 ctxt)
   719 
   720   val absmaxmin_elim_conv2 =
   721     let
   722       val pth_abs = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] abs_split'
   723       val pth_max = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] max_split
   724       val pth_min = Thm.instantiate' [SOME \<^ctyp>\<open>real\<close>] [] min_split
   725       val abs_tm = \<^cterm>\<open>abs :: real \<Rightarrow> _\<close>
   726       val p_v = (("P", 0), \<^typ>\<open>real \<Rightarrow> bool\<close>)
   727       val x_v = (("x", 0), \<^typ>\<open>real\<close>)
   728       val y_v = (("y", 0), \<^typ>\<open>real\<close>)
   729       val is_max = is_binop \<^cterm>\<open>max :: real \<Rightarrow> _\<close>
   730       val is_min = is_binop \<^cterm>\<open>min :: real \<Rightarrow> _\<close>
   731       fun is_abs t = is_comb t andalso Thm.dest_fun t aconvc abs_tm
   732       fun eliminate_construct p c tm =
   733         let
   734           val t = find_cterm p tm
   735           val th0 = (Thm.symmetric o Thm.beta_conversion false) (Thm.apply (Thm.lambda t tm) t)
   736           val (p,ax) = (Thm.dest_comb o Thm.rhs_of) th0
   737         in fconv_rule(arg_conv(binop_conv (arg_conv (Thm.beta_conversion false))))
   738                      (Thm.transitive th0 (c p ax))
   739         end
   740 
   741       val elim_abs = eliminate_construct is_abs
   742         (fn p => fn ax =>
   743           Thm.instantiate ([], [(p_v,p), (x_v, Thm.dest_arg ax)]) pth_abs)
   744       val elim_max = eliminate_construct is_max
   745         (fn p => fn ax =>
   746           let val (ax,y) = Thm.dest_comb ax
   747           in Thm.instantiate ([], [(p_v,p), (x_v, Thm.dest_arg ax), (y_v,y)])
   748                              pth_max end)
   749       val elim_min = eliminate_construct is_min
   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_min end)
   754     in first_conv [elim_abs, elim_max, elim_min, all_conv]
   755     end;
   756 in
   757 fun gen_real_arith ctxt (mkconst,eq,ge,gt,norm,neg,add,mul,prover) =
   758   gen_gen_real_arith ctxt
   759     (mkconst,eq,ge,gt,norm,neg,add,mul,
   760      absmaxmin_elim_conv1 ctxt,absmaxmin_elim_conv2,prover)
   761 end;
   762 
   763 (* An instance for reals*)
   764 
   765 fun gen_prover_real_arith ctxt prover =
   766   let
   767     val {add, mul, neg, pow = _, sub = _, main} =
   768         Semiring_Normalizer.semiring_normalizers_ord_wrapper ctxt
   769         (the (Semiring_Normalizer.match ctxt \<^cterm>\<open>(0::real) + 1\<close>))
   770         Thm.term_ord
   771   in gen_real_arith ctxt
   772      (cterm_of_rat,
   773       Numeral_Simprocs.field_comp_conv ctxt,
   774       Numeral_Simprocs.field_comp_conv ctxt,
   775       Numeral_Simprocs.field_comp_conv ctxt,
   776       main ctxt, neg ctxt, add ctxt, mul ctxt, prover)
   777   end;
   778 
   779 end