(* 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 ("HOL.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 ("HOL.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 ("op +", eval_binop "#add_")

			      (*immediately repeated rewrite pushes

					    '+' into precondition !*)

			      ],

		     scr = EmptyScr}, 

	 srls = Erls, calc = [],

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

		  Calc ("op +", 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 ("op +", eval_binop "#add_")

			      (*immediately repeated rewrite pushes

					    '+' into precondition !*)

			      ],

		     scr = EmptyScr}, 

	 srls = Erls, calc = [],

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

		  Calc ("op +", 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 ("op +", 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 ("op +", 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 ("op +", 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"

	       ));

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]))"

	       ));

(*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("op +", eval_binop "#add_")

				   ], 

		srls = Erls, calc = [],

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

			 Calc("op +", 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