(* equational systems, minimal -- for use in Biegelinie

    author: Walther Neuper

    050826,

    (c) due to copyright terms

 *)

 theory EqSystem imports Integrate Rational Root begin

 consts

   occur'_exactly'_in :: 

    "[real list, real list, 'a] => bool" ("_ from _ occur'_exactly'_in _")

   (*descriptions in the related problems*)

   solveForVars       :: "real list => toreall"

   solution           :: "bool list => toreall"

   (*the CAS-command, eg. "solveSystem [x+y=1,y=2] [x,y]"*)

   solveSystem        :: "[bool list, real list] => bool list"

   (*Script-names*)

   SolveSystemScript  :: "[bool list, real list,     bool list]  

 						 => bool list"

                   ("((Script SolveSystemScript (_ _ =))// (_))" 9)

 axioms 

 (*stated as axioms, todo: prove as theorems

   'bdv' is a constant handled on the meta-level 

    specifically as a 'bound variable'            *)

   commute_0_equality:  "(0 = a) = (a = 0)"

   (*WN0510 see simliar rules 'isolate_' 'separate_' (by RL)

     [bdv_1,bdv_2,bdv_3,bdv_4] work also for 2 and 3 bdvs, ugly !*)

   separate_bdvs_add:   

     "[| [] from [bdv_1,bdv_2,bdv_3,bdv_4] occur_exactly_in a |] 

 		      			     ==> (a + b = c) = (b = c + -1*a)"

   separate_bdvs0:

     "[| some_of [bdv_1,bdv_2,bdv_3,bdv_4] occur_in b; Not (b=!=0)  |] 

 		      			     ==> (a = b) = (a + -1*b = 0)"

   separate_bdvs_add1:  

     "[| some_of [bdv_1,bdv_2,bdv_3,bdv_4] occur_in c |] 

 		      			     ==> (a = b + c) = (a + -1*c = b)"

   separate_bdvs_add2:

     "[| Not (some_of [bdv_1,bdv_2,bdv_3,bdv_4] occur_in a) |] 

 		      			     ==> (a + b = c) = (b = -1*a + c)"

   separate_bdvs_mult:  

     "[| [] from [bdv_1,bdv_2,bdv_3,bdv_4] occur_exactly_in a; Not (a=!=0) |] 

 		      			     ==>(a * b = c) = (b = c / a)"

   (*requires rew_ord for termination, eg. ord_simplify_Integral;

     works for lists of any length, interestingly !?!*)

   order_system_NxN:     "[a,b] = [b,a]"

 ML {*

 val thy = @{theory};

 (** eval functions **)

 (*certain variables of a given list occur _all_ in a term

   args: all: ..variables, which are under consideration (eg. the bound vars)

         vs:  variables which must be in t, 

              and none of the others in all must be in t

         t: the term under consideration

  *)

 fun occur_exactly_in vs all t =

     let fun occurs_in' a b = occurs_in b a

     in foldl and_ (true, map (occurs_in' t) vs)

        andalso not (foldl or_ (false, map (occurs_in' t) 

                                           (subtract op = vs all)))

     end;

 (*("occur_exactly_in", ("EqSystem.occur'_exactly'_in", 

 			eval_occur_exactly_in "#eval_occur_exactly_in_"))*)

 fun eval_occur_exactly_in _ "EqSystem.occur'_exactly'_in"

 			  (p as (Const ("EqSystem.occur'_exactly'_in",_) 

 				       $ vs $ all $ t)) _ =

     if occur_exactly_in (isalist2list vs) (isalist2list all) t

     then SOME ((term2str p) ^ " = True",

 	       Trueprop $ (mk_equality (p, HOLogic.true_const)))

     else SOME ((term2str p) ^ " = False",

 	       Trueprop $ (mk_equality (p, HOLogic.false_const)))

   | eval_occur_exactly_in _ _ _ _ = NONE;

 calclist':= 

 overwritel (!calclist', 

 	    [("occur_exactly_in", 

 	      ("EqSystem.occur'_exactly'_in", 

 	       eval_occur_exactly_in "#eval_occur_exactly_in_"))

     ]);

 (** rewrite order 'ord_simplify_System' **)

 (* order wrt. several linear (i.e. without exponents) variables "c","c_2",..

    which leaves the monomials containing c, c_2,... at the end of an Integral

    and puts the c, c_2,... rightmost within a monomial.

    WN050906 this is a quick and dirty adaption of ord_make_polynomial_in,

    which was most adequate, because it uses size_of_term*)

 (**)

 local (*. for simplify_System .*)

 (**)

 open Term;  (* for type order = EQUAL | LESS | GREATER *)

 fun pr_ord EQUAL = "EQUAL"

   | pr_ord LESS  = "LESS"

   | pr_ord GREATER = "GREATER";

 fun dest_hd' (Const (a, T)) = (((a, 0), T), 0)

   | dest_hd' (Free (ccc, T)) =

     (case explode ccc of

 	"c"::[] => ((("|||||||||||||||||||||", 0), T), 1)(*greatest string WN*)

       | "c"::"_"::_ => ((("|||||||||||||||||||||", 0), T), 1)

       | _ => (((ccc, 0), T), 1))

   | dest_hd' (Var v) = (v, 2)

   | dest_hd' (Bound i) = ((("", i), dummyT), 3)

   | dest_hd' (Abs (_, T, _)) = ((("", 0), T), 4);

 fun size_of_term' (Free (ccc, _)) =

     (case explode ccc of (*WN0510 hack for the bound variables*)

 	"c"::[] => 1000

       | "c"::"_"::is => 1000 * ((str2int o implode) is)

       | _ => 1)

   | size_of_term' (Abs (_,_,body)) = 1 + size_of_term' body

   | size_of_term' (f$t) = size_of_term' f  +  size_of_term' t

   | size_of_term' _ = 1;

 fun term_ord' pr thy (Abs (_, T, t), Abs(_, U, u)) =       (* ~ term.ML *)

       (case term_ord' pr thy (t, u) of EQUAL => Term_Ord.typ_ord (T, U) | ord => ord)

   | term_ord' pr thy (t, u) =

       (if pr then 

 	 let

 	   val (f, ts) = strip_comb t and (g, us) = strip_comb u;

 	   val _=writeln("t= f@ts= \""^

 	      ((Syntax.string_of_term (thy2ctxt thy)) f)^"\" @ \"["^

 	      (commas(map(Syntax.string_of_term (thy2ctxt thy)) ts))^"]\"");

 	   val _=writeln("u= g@us= \""^

 	      ((Syntax.string_of_term (thy2ctxt thy)) g)^"\" @ \"["^

 	      (commas(map(Syntax.string_of_term (thy2ctxt thy)) us))^"]\"");

 	   val _=writeln("size_of_term(t,u)= ("^

 	      (string_of_int(size_of_term' t))^", "^

 	      (string_of_int(size_of_term' u))^")");

 	   val _=writeln("hd_ord(f,g)      = "^((pr_ord o hd_ord)(f,g)));

 	   val _=writeln("terms_ord(ts,us) = "^

 			   ((pr_ord o terms_ord str false)(ts,us)));

 	   val _=writeln("-------");

 	 in () end

        else ();

 	 case int_ord (size_of_term' t, size_of_term' u) of

 	   EQUAL =>

 	     let val (f, ts) = strip_comb t and (g, us) = strip_comb u in

 	       (case hd_ord (f, g) of EQUAL => (terms_ord str pr) (ts, us) 

 	     | ord => ord)

 	     end

 	 | ord => ord)

 and hd_ord (f, g) =                                        (* ~ term.ML *)

   prod_ord (prod_ord Term_Ord.indexname_ord Term_Ord.typ_ord) int_ord (dest_hd' f, 

 						     dest_hd' g)

 and terms_ord str pr (ts, us) = 

     list_ord (term_ord' pr (assoc_thy "Isac"))(ts, us);

 (**)

 in

 (**)

 (*WN0510 for preliminary use in eval_order_system, see case-study mat-eng.tex

 fun ord_simplify_System_rev (pr:bool) thy subst tu = 

     (term_ord' pr thy (Library.swap tu) = LESS);*)

 (*for the rls's*)

 fun ord_simplify_System (pr:bool) thy subst tu = 

     (term_ord' pr thy tu = LESS);

 (**)

 end;

 (**)

 rew_ord' := overwritel (!rew_ord',

 [("ord_simplify_System", ord_simplify_System false thy)

  ]);

 *}

 ML {*

 (** rulesets **)

 (*.adapted from 'order_add_mult_in' by just replacing the rew_ord.*)

 val order_add_mult_System = 

   Rls{id = "order_add_mult_System", preconds = [], 

       rew_ord = ("ord_simplify_System",

 		 ord_simplify_System false (theory "Integrate")),

       erls = e_rls,srls = Erls, calc = [],

       rules = [Thm ("real_mult_commute",num_str @{thm real_mult_commute}),

 	       (* z * w = w * z *)

 	       Thm ("real_mult_left_commute",num_str @{thm real_mult_left_commute}),

 	       (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)

 	       Thm ("real_mult_assoc",num_str @{thm real_mult_assoc}),		

 	       (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)

 	       Thm ("add_commute",num_str @{thm add_commute}),	

 	       (*z + w = w + z*)

 	       Thm ("add_left_commute",num_str @{thm add_left_commute}),

 	       (*x + (y + z) = y + (x + z)*)

 	       Thm ("add_assoc",num_str @{thm add_assoc})	               

 	       (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)

 	       ], 

       scr = EmptyScr}:rls;

 *}

 ML {*

 (*.adapted from 'norm_Rational' by

   #1 using 'ord_simplify_System' in 'order_add_mult_System'

   #2 NOT using common_nominator_p                          .*)

 val norm_System_noadd_fractions = 

   Rls {id = "norm_System_noadd_fractions", preconds = [], 

        rew_ord = ("dummy_ord",dummy_ord), 

        erls = norm_rat_erls, srls = Erls, calc = [],

        rules = [(*sequence given by operator precedence*)

 		Rls_ discard_minus,

 		Rls_ powers,

 		Rls_ rat_mult_divide,

 		Rls_ expand,

 		Rls_ reduce_0_1_2,

 		Rls_ (*order_add_mult #1*) order_add_mult_System,

 		Rls_ collect_numerals,

 		(*Rls_ add_fractions_p, #2*)

 		Rls_ cancel_p

 		],

        scr = Script ((term_of o the o (parse thy)) 

 			 "empty_script")

        }:rls;

 *}

 ML {*

 (*.adapted from 'norm_Rational' by

   *1* using 'ord_simplify_System' in 'order_add_mult_System'.*)

 val norm_System = 

   Rls {id = "norm_System", preconds = [], 

        rew_ord = ("dummy_ord",dummy_ord), 

        erls = norm_rat_erls, srls = Erls, calc = [],

        rules = [(*sequence given by operator precedence*)

 		Rls_ discard_minus,

 		Rls_ powers,

 		Rls_ rat_mult_divide,

 		Rls_ expand,

 		Rls_ reduce_0_1_2,

 		Rls_ (*order_add_mult *1*) order_add_mult_System,

 		Rls_ collect_numerals,

 		Rls_ add_fractions_p,

 		Rls_ cancel_p

 		],

        scr = Script ((term_of o the o (parse thy)) 

 			 "empty_script")

        }:rls;

 *}

 ML {*

 (*.simplify an equational system BEFORE solving it such that parentheses are

    ( ((u0*v0)*w0) + ( ((u1*v1)*w1) * c + ... +((u4*v4)*w4) * c_4 ) )

 ATTENTION: works ONLY for bound variables c, c_1, c_2, c_3, c_4 :ATTENTION

    This is a copy from 'make_ratpoly_in' with respective reductions:

    *0* expand the term, ie. distribute * and / over +

    *1* ord_simplify_System instead of termlessI

    *2* no add_fractions_p (= common_nominator_p_rls !)

    *3* discard_parentheses only for (.*(.*.))

    analoguous to simplify_Integral                                       .*)

 val simplify_System_parenthesized = 

   Seq {id = "simplify_System_parenthesized", preconds = []:term list, 

        rew_ord = ("dummy_ord", dummy_ord),

       erls = Atools_erls, srls = Erls, calc = [],

       rules = [Thm ("left_distrib",num_str @{thm left_distrib}),

  	       (*"(?z1.0 + ?z2.0) * ?w = ?z1.0 * ?w + ?z2.0 * ?w"*)

 	       Thm ("add_divide_distrib",num_str @{thm add_divide_distrib}),

  	       (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)

 	       (*^^^^^ *0* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^*)

 	       Rls_ norm_Rational_noadd_fractions(**2**),

 	       Rls_ (*order_add_mult_in*) norm_System_noadd_fractions (**1**),

 	       Thm ("sym_real_mult_assoc",

                      num_str (@{thm real_mult_assoc} RS @{thm sym}))

 	       (*Rls_ discard_parentheses *3**),

 	       Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)

 	       Rls_ separate_bdv2,

 	       Calc ("Rings.inverse_class.divide"  ,eval_cancel "#divide_e")

 	       ],

       scr = EmptyScr}:rls;      

 *}

 ML {*

 (*.simplify an equational system AFTER solving it;

    This is a copy of 'make_ratpoly_in' with the differences

    *1* ord_simplify_System instead of termlessI           .*)

 (*TODO.WN051031 ^^^^^^^^^^ should be in EACH rls contained *)

 val simplify_System = 

   Seq {id = "simplify_System", preconds = []:term list, 

        rew_ord = ("dummy_ord", dummy_ord),

       erls = Atools_erls, srls = Erls, calc = [],

       rules = [Rls_ norm_Rational,

 	       Rls_ (*order_add_mult_in*) norm_System (**1**),

 	       Rls_ discard_parentheses,

 	       Rls_ collect_bdv, (*from make_polynomial_in WN051031 welldone?*)

 	       Rls_ separate_bdv2,

 	       Calc ("Rings.inverse_class.divide"  ,eval_cancel "#divide_e")

 	       ],

       scr = EmptyScr}:rls;      

 (*

 val simplify_System = 

     append_rls "simplify_System" simplify_System_parenthesized

 	       [Thm ("sym_add_assoc",

                       num_str (@{thm add_assoc} RS @{thm sym}))];

 *)

 *}

 ML {*

 val isolate_bdvs = 

     Rls {id="isolate_bdvs", preconds = [], 

 	 rew_ord = ("e_rew_ord", e_rew_ord), 

 	 erls = append_rls "erls_isolate_bdvs" e_rls 

 			   [(Calc ("EqSystem.occur'_exactly'_in", 

 				   eval_occur_exactly_in 

 				       "#eval_occur_exactly_in_"))

 			    ], 

 			   srls = Erls, calc = [],

 	      rules = 

              [Thm ("commute_0_equality", num_str @{thm commute_0_equality}),

 	      Thm ("separate_bdvs_add", num_str @{thm separate_bdvs_add}),

 	      Thm ("separate_bdvs_mult", num_str @{thm separate_bdvs_mult})],

 	      scr = EmptyScr};

 *}

 ML {*

 val isolate_bdvs_4x4 = 

     Rls {id="isolate_bdvs_4x4", preconds = [], 

 	 rew_ord = ("e_rew_ord", e_rew_ord), 

 	 erls = append_rls 

 		    "erls_isolate_bdvs_4x4" e_rls 

 		    [Calc ("EqSystem.occur'_exactly'_in", 

 			   eval_occur_exactly_in "#eval_occur_exactly_in_"),

 		     Calc ("Atools.ident",eval_ident "#ident_"),

 		     Calc ("Atools.some'_occur'_in", 

 			   eval_some_occur_in "#some_occur_in_"),

                      Thm ("not_true",num_str @{thm not_true}),

 		     Thm ("not_false",num_str @{thm not_false})

 			    ], 

 	 srls = Erls, calc = [],

 	 rules = [Thm ("commute_0_equality", num_str @{thm commute_0_equality}),

 		  Thm ("separate_bdvs0", num_str @{thm separate_bdvs0}),

 		  Thm ("separate_bdvs_add1", num_str @{thm separate_bdvs_add1}),

 		  Thm ("separate_bdvs_add1", num_str @{thm separate_bdvs_add2}),

 		  Thm ("separate_bdvs_mult", num_str @{thm separate_bdvs_mult})

                  ], scr = EmptyScr};

 *}

 ML {*

 (*.order the equations in a system such, that a triangular system (if any)

    appears as [..c_4 = .., ..., ..., ..c_1 + ..c_2 + ..c_3 ..c_4 = ..].*)

 val order_system = 

     Rls {id="order_system", preconds = [], 

 	 rew_ord = ("ord_simplify_System", 

 		    ord_simplify_System false thy), 

 	 erls = Erls, srls = Erls, calc = [],

 	 rules = [Thm ("order_system_NxN", num_str @{thm order_system_NxN})

 		  ],

 	 scr = EmptyScr};

 val prls_triangular = 

     Rls {id="prls_triangular", preconds = [], 

 	 rew_ord = ("e_rew_ord", e_rew_ord), 

 	 erls = Rls {id="erls_prls_triangular", preconds = [], 

 		     rew_ord = ("e_rew_ord", e_rew_ord), 

 		     erls = Erls, srls = Erls, calc = [],

 		     rules = [(*for precond NTH_CONS ...*)

 			      Calc ("op <",eval_equ "#less_"),

 			      Calc ("Groups.plus_class.plus", eval_binop "#add_")

 			      (*immediately repeated rewrite pushes

 					    '+' into precondition !*)

 			      ],

 		     scr = EmptyScr}, 

 	 srls = Erls, calc = [],

 	 rules = [Thm ("NTH_CONS",num_str @{thm NTH_CONS}),

 		  Calc ("Groups.plus_class.plus", eval_binop "#add_"),

 		  Thm ("NTH_NIL",num_str @{thm NTH_NIL}),

 		  Thm ("tl_Cons",num_str @{thm tl_Cons}),

 		  Thm ("tl_Nil",num_str @{thm tl_Nil}),

 		  Calc ("EqSystem.occur'_exactly'_in", 

 			eval_occur_exactly_in 

 			    "#eval_occur_exactly_in_")

 		  ],

 	 scr = EmptyScr};

 *}

 ML {*

 (*WN060914 quickly created for 4x4; 

  more similarity to prls_triangular desirable*)

 val prls_triangular4 = 

     Rls {id="prls_triangular4", preconds = [], 

 	 rew_ord = ("e_rew_ord", e_rew_ord), 

 	 erls = Rls {id="erls_prls_triangular4", preconds = [], 

 		     rew_ord = ("e_rew_ord", e_rew_ord), 

 		     erls = Erls, srls = Erls, calc = [],

 		     rules = [(*for precond NTH_CONS ...*)

 			      Calc ("op <",eval_equ "#less_"),

 			      Calc ("Groups.plus_class.plus", eval_binop "#add_")

 			      (*immediately repeated rewrite pushes

 					    '+' into precondition !*)

 			      ],

 		     scr = EmptyScr}, 

 	 srls = Erls, calc = [],

 	 rules = [Thm ("NTH_CONS",num_str @{thm NTH_CONS}),

 		  Calc ("Groups.plus_class.plus", eval_binop "#add_"),

 		  Thm ("NTH_NIL",num_str @{thm NTH_NIL}),

 		  Thm ("tl_Cons",num_str @{thm tl_Cons}),

 		  Thm ("tl_Nil",num_str @{thm tl_Nil}),

 		  Calc ("EqSystem.occur'_exactly'_in", 

 			eval_occur_exactly_in 

 			    "#eval_occur_exactly_in_")

 		  ],

 	 scr = EmptyScr};

 *}

 ML {*

 ruleset' := 

 overwritelthy @{theory} 

 	      (!ruleset', 

 [("simplify_System_parenthesized", prep_rls simplify_System_parenthesized),

  ("simplify_System", prep_rls simplify_System),

  ("isolate_bdvs", prep_rls isolate_bdvs),

  ("isolate_bdvs_4x4", prep_rls isolate_bdvs_4x4),

  ("order_system", prep_rls order_system),

  ("order_add_mult_System", prep_rls order_add_mult_System),

  ("norm_System_noadd_fractions", prep_rls norm_System_noadd_fractions),

  ("norm_System", prep_rls norm_System)

  ]);

 *}

 ML {*

 (** problems **)

 store_pbt

  (prep_pbt thy "pbl_equsys" [] e_pblID

  (["system"],

   [("#Given" ,["equalities e_s", "solveForVars v_s"]),

    ("#Find"  ,["solution ss'''"](*''' is copy-named*))

   ],

   append_rls "e_rls" e_rls [(*for preds in where_*)], 

   SOME "solveSystem e_s v_s", 

   []));

 store_pbt

  (prep_pbt thy "pbl_equsys_lin" [] e_pblID

  (["linear", "system"],

   [("#Given" ,["equalities e_s", "solveForVars v_s"]),

    (*TODO.WN050929 check linearity*)

    ("#Find"  ,["solution ss'''"])

   ],

   append_rls "e_rls" e_rls [(*for preds in where_*)], 

   SOME "solveSystem e_s v_s", 

   []));

 store_pbt

  (prep_pbt thy "pbl_equsys_lin_2x2" [] e_pblID

  (["2x2", "linear", "system"],

   (*~~~~~~~~~~~~~~~~~~~~~~~~~*)

   [("#Given" ,["equalities e_s", "solveForVars v_s"]),

    ("#Where"  ,["LENGTH (e_s:: bool list) = 2", "LENGTH v_s = 2"]),

    ("#Find"  ,["solution ss'''"])

   ],

   append_rls "prls_2x2_linear_system" e_rls 

 			     [Thm ("LENGTH_CONS",num_str @{thm LENGTH_CONS}),

 			      Thm ("LENGTH_NIL",num_str @{thm LENGTH_NIL}),

 			      Calc ("Groups.plus_class.plus", eval_binop "#add_"),

 			      Calc ("op =",eval_equal "#equal_")

 			      ], 

   SOME "solveSystem e_s v_s", 

   []));

 *}

 ML {*

 store_pbt

  (prep_pbt thy "pbl_equsys_lin_2x2_tri" [] e_pblID

  (["triangular", "2x2", "linear", "system"],

   [("#Given" ,["equalities e_s", "solveForVars v_s"]),

    ("#Where"  ,

     ["(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))",

      "    v_s  from v_s occur_exactly_in (NTH 2 (e_s::bool list))"]),

    ("#Find"  ,["solution ss'''"])

   ],

   prls_triangular, 

   SOME "solveSystem e_s v_s", 

   [["EqSystem","top_down_substitution","2x2"]]));

 store_pbt

  (prep_pbt thy "pbl_equsys_lin_2x2_norm" [] e_pblID

  (["normalize", "2x2", "linear", "system"],

   [("#Given" ,["equalities e_s", "solveForVars v_s"]),

    ("#Find"  ,["solution ss'''"])

   ],

   append_rls "e_rls" e_rls [(*for preds in where_*)], 

   SOME "solveSystem e_s v_s", 

   [["EqSystem","normalize","2x2"]]));

 store_pbt

  (prep_pbt thy "pbl_equsys_lin_3x3" [] e_pblID

  (["3x3", "linear", "system"],

   (*~~~~~~~~~~~~~~~~~~~~~~~~~*)

   [("#Given" ,["equalities e_s", "solveForVars v_s"]),

    ("#Where"  ,["LENGTH (e_s:: bool list) = 3", "LENGTH v_s = 3"]),

    ("#Find"  ,["solution ss'''"])

   ],

   append_rls "prls_3x3_linear_system" e_rls 

 			     [Thm ("LENGTH_CONS",num_str @{thm LENGTH_CONS}),

 			      Thm ("LENGTH_NIL",num_str @{thm LENGTH_NIL}),

 			      Calc ("Groups.plus_class.plus", eval_binop "#add_"),

 			      Calc ("op =",eval_equal "#equal_")

 			      ], 

   SOME "solveSystem e_s v_s", 

   []));

 store_pbt

  (prep_pbt thy "pbl_equsys_lin_4x4" [] e_pblID

  (["4x4", "linear", "system"],

   (*~~~~~~~~~~~~~~~~~~~~~~~~~*)

   [("#Given" ,["equalities e_s", "solveForVars v_s"]),

    ("#Where"  ,["LENGTH (e_s:: bool list) = 4", "LENGTH v_s = 4"]),

    ("#Find"  ,["solution ss'''"])

   ],

   append_rls "prls_4x4_linear_system" e_rls 

 			     [Thm ("LENGTH_CONS",num_str @{thm LENGTH_CONS}),

 			      Thm ("LENGTH_NIL",num_str @{thm LENGTH_NIL}),

 			      Calc ("Groups.plus_class.plus", eval_binop "#add_"),

 			      Calc ("op =",eval_equal "#equal_")

 			      ], 

   SOME "solveSystem e_s v_s", 

   []));

 *}

 ML {*

 store_pbt

  (prep_pbt thy "pbl_equsys_lin_4x4_tri" [] e_pblID

  (["triangular", "4x4", "linear", "system"],

   [("#Given" ,["equalities e_s", "solveForVars v_s"]),

    ("#Where" , (*accepts missing variables up to diagional form*)

     ["(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))",

      "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))",

      "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))",

      "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"

      ]),

    ("#Find"  ,["solution ss'''"])

   ],

   append_rls "prls_tri_4x4_lin_sys" prls_triangular

 	     [Calc ("Atools.occurs'_in",eval_occurs_in "")], 

   SOME "solveSystem e_s v_s", 

   [["EqSystem","top_down_substitution","4x4"]]));

 *}

 ML {*

 store_pbt

  (prep_pbt thy "pbl_equsys_lin_4x4_norm" [] e_pblID

  (["normalize", "4x4", "linear", "system"],

   [("#Given" ,["equalities e_s", "solveForVars v_s"]),

    (*LENGTH is checked 1 level above*)

    ("#Find"  ,["solution ss'''"])

   ],

   append_rls "e_rls" e_rls [(*for preds in where_*)], 

   SOME "solveSystem e_s v_s", 

   [["EqSystem","normalize","4x4"]]));

 (* show_ptyps();

    *)

 *}

 ML {*

 (** methods **)

 store_met

     (prep_met thy "met_eqsys" [] e_metID

 	      (["EqSystem"],

 	       [],

 	       {rew_ord'="tless_true", rls' = Erls, calc = [], 

 		srls = Erls, prls = Erls, crls = Erls, nrls = Erls},

 	       "empty_script"

 	       ));

 store_met

     (prep_met thy "met_eqsys_topdown" [] e_metID

 	      (["EqSystem","top_down_substitution"],

 	       [],

 	       {rew_ord'="tless_true", rls' = Erls, calc = [], 

 		srls = Erls, prls = Erls, crls = Erls, nrls = Erls},

 	       "empty_script"

 	       ));

 *}

 ML {*

 store_met

     (prep_met thy "met_eqsys_topdown_2x2" [] e_metID

 	 (["EqSystem","top_down_substitution","2x2"],

 	  [("#Given" ,["equalities e_s", "solveForVars v_s"]),

 	   ("#Where"  ,

 	    ["(tl v_s) from v_s occur_exactly_in (NTH 1 (e_s::bool list))",

 	     "    v_s  from v_s occur_exactly_in (NTH 2 (e_s::bool list))"]),

 	   ("#Find"  ,["solution ss'''"])

 	   ],

 	  {rew_ord'="ord_simplify_System", rls' = Erls, calc = [], 

 	   srls = append_rls "srls_top_down_2x2" e_rls

 				  [Thm ("hd_thm",num_str @{thm hd_thm}),

 				   Thm ("tl_Cons",num_str @{thm tl_Cons}),

 				   Thm ("tl_Nil",num_str @{thm tl_Nil})

 				   ], 

 	   prls = prls_triangular, crls = Erls, nrls = Erls},

 "Script SolveSystemScript (e_s::bool list) (v_s::real list) =                " ^

 "  (let e_1 = Take (hd e_s);                                                " ^

 "       e_1 = ((Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^

 "                                  isolate_bdvs False))     @@               " ^

 "               (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^

 "                                  simplify_System False))) e_1;            " ^

 "       e_2 = Take (hd (tl e_s));                                           " ^

 "       e_2 = ((Substitute [e_1]) @@                                       " ^

 "               (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^

 "                                  simplify_System_parenthesized False)) @@  " ^

 "               (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^

 "                                  isolate_bdvs False))     @@               " ^

 "               (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^

 "                                  simplify_System False))) e_2;            " ^

 "       e__s = Take [e_1, e_2]                                             " ^

 "   in (Try (Rewrite_Set order_system False)) e__s)"

 (*---------------------------------------------------------------------------

   this script does NOT separate the equations as above, 

   but it does not yet work due to preliminary script-interpreter,

   see eqsystem.sml 'script [EqSystem,top_down_substitution,2x2] Vers.2'

 "Script SolveSystemScript (e_s::bool list) (v_s::real list) =         " ^

 "  (let e__s = Take e_s;                                              " ^

 "       e_1 = hd e__s;                                               " ^

 "       e_2 = hd (tl e__s);                                          " ^

 "       e__s = [e_1, Substitute [e_1] e_2]                         " ^

 "   in ((Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^

 "                                  simplify_System_parenthesized False)) @@ " ^

 "       (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))] " ^

 "                              isolate_bdvs False))              @@   " ^

 "       (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^

 "                                  simplify_System False))) e__s)"

 ---------------------------------------------------------------------------*)

 	  ));

 *}

 ML {*

 store_met

     (prep_met thy "met_eqsys_norm" [] e_metID

 	      (["EqSystem","normalize"],

 	       [],

 	       {rew_ord'="tless_true", rls' = Erls, calc = [], 

 		srls = Erls, prls = Erls, crls = Erls, nrls = Erls},

 	       "empty_script"

 	       ));

 *}

 ML {*

 store_met

     (prep_met thy "met_eqsys_norm_2x2" [] e_metID

 	      (["EqSystem","normalize","2x2"],

 	       [("#Given" ,["equalities e_s", "solveForVars v_s"]),

 		("#Find"  ,["solution ss'''"])],

 	       {rew_ord'="tless_true", rls' = Erls, calc = [], 

 		srls = append_rls "srls_normalize_2x2" e_rls

 				  [Thm ("hd_thm",num_str @{thm hd_thm}),

 				   Thm ("tl_Cons",num_str @{thm tl_Cons}),

 				   Thm ("tl_Nil",num_str @{thm tl_Nil})

 				   ], 

 		prls = Erls, crls = Erls, nrls = Erls},

 "Script SolveSystemScript (e_s::bool list) (v_s::real list) =                " ^

 "  (let e__s = ((Try (Rewrite_Set norm_Rational False)) @@                   " ^

 "               (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^

 "                                  simplify_System_parenthesized False)) @@  " ^

 "               (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^

 "                                                    isolate_bdvs False)) @@ " ^

 "               (Try (Rewrite_Set_Inst [(bdv_1, hd v_s),(bdv_2, hd (tl v_s))]" ^

 "                                  simplify_System_parenthesized False)) @@  " ^

 "               (Try (Rewrite_Set order_system False))) e_s                  " ^

 "   in (SubProblem (EqSystem',[linear,system],[no_met])                      " ^

 "                  [BOOL_LIST e__s, REAL_LIST v_s]))"

 	       ));

 *}

 ML {*

 (*this is for NTH only*)

 val srls = Rls {id="srls_normalize_4x4", 

 		preconds = [], 

 		rew_ord = ("termlessI",termlessI), 

 		erls = append_rls "erls_in_srls_IntegrierenUnd.." e_rls

 				  [(*for asm in NTH_CONS ...*)

 				   Calc ("op <",eval_equ "#less_"),

 				   (*2nd NTH_CONS pushes n+-1 into asms*)

 				   Calc("Groups.plus_class.plus", eval_binop "#add_")

 				   ], 

 		srls = Erls, calc = [],

 		rules = [Thm ("NTH_CONS",num_str @{thm NTH_CONS}),

 			 Calc("Groups.plus_class.plus", eval_binop "#add_"),

 			 Thm ("NTH_NIL",num_str @{thm NTH_NIL})],

 		scr = EmptyScr};

 store_met

     (prep_met thy "met_eqsys_norm_4x4" [] e_metID

 	      (["EqSystem","normalize","4x4"],

 	       [("#Given" ,["equalities e_s", "solveForVars v_s"]),

 		("#Find"  ,["solution ss'''"])],

 	       {rew_ord'="tless_true", rls' = Erls, calc = [], 

 		srls = append_rls "srls_normalize_4x4" srls

 				  [Thm ("hd_thm",num_str @{thm hd_thm}),

 				   Thm ("tl_Cons",num_str @{thm tl_Cons}),

 				   Thm ("tl_Nil",num_str @{thm tl_Nil})

 				   ], 

 		prls = Erls, crls = Erls, nrls = Erls},

 (*GOON met ["EqSystem","normalize","4x4"] @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@*)

 "Script SolveSystemScript (e_s::bool list) (v_s::real list) =                " ^

 "  (let e__s =                                                               " ^

 "     ((Try (Rewrite_Set norm_Rational False)) @@                            " ^

 "      (Repeat (Rewrite commute_0_equality False)) @@                        " ^

 "      (Try (Rewrite_Set_Inst [(bdv_1, NTH 1 v_s),(bdv_2, NTH 2 v_s ),     " ^

 "                              (bdv_3, NTH 3 v_s),(bdv_3, NTH 4 v_s )]     " ^

 "                             simplify_System_parenthesized False))    @@    " ^

 "      (Try (Rewrite_Set_Inst [(bdv_1, NTH 1 v_s),(bdv_2, NTH 2 v_s ),     " ^

 "                              (bdv_3, NTH 3 v_s),(bdv_3, NTH 4 v_s )]     " ^

 "                             isolate_bdvs_4x4 False))                 @@    " ^

 "      (Try (Rewrite_Set_Inst [(bdv_1, NTH 1 v_s),(bdv_2, NTH 2 v_s ),     " ^

 "                              (bdv_3, NTH 3 v_s),(bdv_3, NTH 4 v_s )]     " ^

 "                             simplify_System_parenthesized False))    @@    " ^

 "      (Try (Rewrite_Set order_system False)))                           e_s " ^

 "   in (SubProblem (EqSystem',[linear,system],[no_met])                      " ^

 "                  [BOOL_LIST e__s, REAL_LIST v_s]))"

 ));

 store_met

 (prep_met thy "met_eqsys_topdown_4x4" [] e_metID

 	  (["EqSystem","top_down_substitution","4x4"],

 	   [("#Given" ,["equalities e_s", "solveForVars v_s"]),

 	    ("#Where" , (*accepts missing variables up to diagonal form*)

 	     ["(NTH 1 (v_s::real list)) occurs_in (NTH 1 (e_s::bool list))",

 	      "(NTH 2 (v_s::real list)) occurs_in (NTH 2 (e_s::bool list))",

 	      "(NTH 3 (v_s::real list)) occurs_in (NTH 3 (e_s::bool list))",

 	      "(NTH 4 (v_s::real list)) occurs_in (NTH 4 (e_s::bool list))"

 	      ]),

 	    ("#Find"  ,["solution ss'''"])

 	    ],

 	   {rew_ord'="ord_simplify_System", rls' = Erls, calc = [], 

 	    srls = append_rls "srls_top_down_4x4" srls [], 

 	    prls = append_rls "prls_tri_4x4_lin_sys" prls_triangular

 			      [Calc ("Atools.occurs'_in",eval_occurs_in "")], 

 	    crls = Erls, nrls = Erls},

 (*FIXXXXME.WN060916: this script works ONLY for exp 7.79 @@@@@@@@@@@@@@@@@@@@*)

 "Script SolveSystemScript (e_s::bool list) (v_s::real list) =                " ^

 "  (let e_1 = NTH 1 e_s;                                                    " ^

 "       e_2 = Take (NTH 2 e_s);                                             " ^

 "       e_2 = ((Substitute [e_1]) @@                                         " ^

 "              (Try (Rewrite_Set_Inst [(bdv_1,NTH 1 v_s),(bdv_2,NTH 2 v_s)," ^

 "                                      (bdv_3,NTH 3 v_s),(bdv_4,NTH 4 v_s)]" ^

 "                                 simplify_System_parenthesized False)) @@   " ^

 "              (Try (Rewrite_Set_Inst [(bdv_1,NTH 1 v_s),(bdv_2,NTH 2 v_s)," ^

 "                                      (bdv_3,NTH 3 v_s),(bdv_4,NTH 4 v_s)]" ^

 "                                 isolate_bdvs False))                  @@   " ^

 "              (Try (Rewrite_Set_Inst [(bdv_1,NTH 1 v_s),(bdv_2,NTH 2 v_s)," ^

 "                                      (bdv_3,NTH 3 v_s),(bdv_4,NTH 4 v_s)]" ^

 "                                 norm_Rational False)))             e_2     " ^

 "    in [e_1, e_2, NTH 3 e_s, NTH 4 e_s])"

 ));

 *}

 end