(* 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"

axiomatization where

(*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)" and

  (*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)" and

  separate_bdvs0:

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

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

  separate_bdvs_add1:  

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

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

  separate_bdvs_add2:

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

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

  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)" 

axiomatization where (*..if replaced by "and" we get an error in 

  ---  rewrite in [EqSystem,normalise,2x2] --- step "--- 3---";*)

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

  (*requires rew_ord for termination, eg. ord_simplify_Integral;

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

ML \<open>

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 = Prog_Expr.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 (TermC.isalist2list vs) (TermC.isalist2list all) t

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

	       HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))

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

	       HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))

  | eval_occur_exactly_in _ _ _ _ = NONE;

\<close>

setup \<open>KEStore_Elems.add_calcs

  [("occur_exactly_in",

	    ("EqSystem.occur'_exactly'_in",

	      eval_occur_exactly_in "#eval_occur_exactly_in_"))]\<close>

ML \<open>

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

	"c"::[] => 1000

      | "c"::"_"::is => 1000 * ((TermC.int_of_str 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 _ = tracing ("t= f@ts= \"" ^ Rule.term_to_string''' thy f ^ "\" @ \"[" ^

           commas (map (Rule.term_to_string''' thy) ts) ^ "]\"");

         val _ = tracing ("u= g@us= \"" ^ Rule.term_to_string''' thy g ^ "\" @ \"[" ^

           commas (map (Rule.term_to_string''' thy) us) ^ "]\"");

         val _ = tracing ("size_of_term(t,u)= (" ^ string_of_int (size_of_term' t) ^ ", " ^

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

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

         val _ = tracing ("terms_ord (ts,us) = " ^(pr_ord o terms_ord str false) (ts,us));

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

       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 (Celem.assoc_thy "Isac_Knowledge"))(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;

(**)

Rule.rew_ord' := overwritel (! Rule.rew_ord',

[("ord_simplify_System", ord_simplify_System false thy)

 ]);

\<close>

ML \<open>

(** rulesets **)

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

val order_add_mult_System = 

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

      rew_ord = ("ord_simplify_System",

		 ord_simplify_System false @{theory "Integrate"}),

      erls = Rule.e_rls,srls = Rule.Erls, calc = [], errpatts = [],

      rules = [Rule.Thm ("mult_commute",TermC.num_str @{thm mult.commute}),

	       (* z * w = w * z *)

	       Rule.Thm ("real_mult_left_commute",TermC.num_str @{thm real_mult_left_commute}),

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

	       Rule.Thm ("mult_assoc",TermC.num_str @{thm mult.assoc}),		

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

	       Rule.Thm ("add_commute",TermC.num_str @{thm add.commute}),	

	       (*z + w = w + z*)

	       Rule.Thm ("add_left_commute",TermC.num_str @{thm add.left_commute}),

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

	       Rule.Thm ("add_assoc",TermC.num_str @{thm add.assoc})	               

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

	       ], 

      scr = Rule.EmptyScr};

\<close>

ML \<open>

(*.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 = 

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

       rew_ord = ("dummy_ord",Rule.dummy_ord), 

       erls = norm_rat_erls, srls = Rule.Erls, calc = [], errpatts = [],

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

		Rule.Rls_ discard_minus,

		Rule.Rls_ powers,

		Rule.Rls_ rat_mult_divide,

		Rule.Rls_ expand,

		Rule.Rls_ reduce_0_1_2,

		Rule.Rls_ (*order_add_mult #1*) order_add_mult_System,

		Rule.Rls_ collect_numerals,

		(*Rule.Rls_ add_fractions_p, #2*)

		Rule.Rls_ cancel_p

		],

       scr = Rule.EmptyScr

       };

\<close>

ML \<open>

(*.adapted from 'norm_Rational' by

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

val norm_System = 

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

       rew_ord = ("dummy_ord",Rule.dummy_ord), 

       erls = norm_rat_erls, srls = Rule.Erls, calc = [], errpatts = [],

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

		Rule.Rls_ discard_minus,

		Rule.Rls_ powers,

		Rule.Rls_ rat_mult_divide,

		Rule.Rls_ expand,

		Rule.Rls_ reduce_0_1_2,

		Rule.Rls_ (*order_add_mult *1*) order_add_mult_System,

		Rule.Rls_ collect_numerals,

		Rule.Rls_ add_fractions_p,

		Rule.Rls_ cancel_p

		],

       scr = Rule.EmptyScr

       };

\<close>

ML \<open>

(*.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 = 

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

       rew_ord = ("dummy_ord", Rule.dummy_ord),

      erls = Atools_erls, srls = Rule.Erls, calc = [], errpatts = [],

      rules = [Rule.Thm ("distrib_right",TermC.num_str @{thm distrib_right}),

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

	       Rule.Thm ("add_divide_distrib",TermC.num_str @{thm add_divide_distrib}),

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

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

	       Rule.Rls_ norm_Rational_noadd_fractions(**2**),

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

	       Rule.Thm ("sym_mult_assoc",

                     TermC.num_str (@{thm mult.assoc} RS @{thm sym}))

	       (*Rule.Rls_ discard_parentheses *3**),

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

	       Rule.Rls_ separate_bdv2,

	       Rule.Calc ("Rings.divide_class.divide", Prog_Expr.eval_cancel "#divide_e")

	       ],

      scr = Rule.EmptyScr};      

\<close>

ML \<open>

(*.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 = 

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

       rew_ord = ("dummy_ord", Rule.dummy_ord),

      erls = Atools_erls, srls = Rule.Erls, calc = [], errpatts = [],

      rules = [Rule.Rls_ norm_Rational,

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

	       Rule.Rls_ discard_parentheses,

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

	       Rule.Rls_ separate_bdv2,

	       Rule.Calc ("Rings.divide_class.divide", Prog_Expr.eval_cancel "#divide_e")

	       ],

      scr = Rule.EmptyScr};      

(*

val simplify_System = 

    Rule.append_rls "simplify_System" simplify_System_parenthesized

	       [Rule.Thm ("sym_add_assoc",

                      TermC.num_str (@{thm add.assoc} RS @{thm sym}))];

*)

\<close>

ML \<open>

val isolate_bdvs = 

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

	 rew_ord = ("e_rew_ord", Rule.e_rew_ord), 

	 erls = Rule.append_rls "erls_isolate_bdvs" Rule.e_rls 

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

				   eval_occur_exactly_in 

				       "#eval_occur_exactly_in_"))

			    ], 

			   srls = Rule.Erls, calc = [], errpatts = [],

	      rules = 

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

	      Rule.Thm ("separate_bdvs_add", TermC.num_str @{thm separate_bdvs_add}),

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

	      scr = Rule.EmptyScr};

\<close>

ML \<open>

val isolate_bdvs_4x4 = 

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

	 rew_ord = ("e_rew_ord", Rule.e_rew_ord), 

	 erls = Rule.append_rls 

		    "erls_isolate_bdvs_4x4" Rule.e_rls 

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

			   eval_occur_exactly_in "#eval_occur_exactly_in_"),

		     Rule.Calc ("Prog_Expr.ident", Prog_Expr.eval_ident "#ident_"),

		     Rule.Calc ("Prog_Expr.some'_occur'_in", Prog_Expr.eval_some_occur_in "#some_occur_in_"),

         Rule.Thm ("not_true",TermC.num_str @{thm not_true}),

		     Rule.Thm ("not_false",TermC.num_str @{thm not_false})

			    ], 

	 srls = Rule.Erls, calc = [], errpatts = [],

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

		  Rule.Thm ("separate_bdvs0", TermC.num_str @{thm separate_bdvs0}),

		  Rule.Thm ("separate_bdvs_add1", TermC.num_str @{thm separate_bdvs_add1}),

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

		  Rule.Thm ("separate_bdvs_mult", TermC.num_str @{thm separate_bdvs_mult})

                 ], scr = Rule.EmptyScr};

\<close>

ML \<open>

(*.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 = 

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

	 rew_ord = ("ord_simplify_System", 

		    ord_simplify_System false thy), 

	 erls = Rule.Erls, srls = Rule.Erls, calc = [], errpatts = [],

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

		  ],

	 scr = Rule.EmptyScr};

val prls_triangular = 

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

	 rew_ord = ("e_rew_ord", Rule.e_rew_ord), 

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

		     rew_ord = ("e_rew_ord", Rule.e_rew_ord), 

		     erls = Rule.Erls, srls = Rule.Erls, calc = [], errpatts = [],

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

			      Rule.Calc ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),

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

			      (*immediately repeated rewrite pushes

					    '+' into precondition !*)

			      ],

		     scr = Rule.EmptyScr}, 

	 srls = Rule.Erls, calc = [], errpatts = [],

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

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

		  Rule.Thm ("NTH_NIL",TermC.num_str @{thm NTH_NIL}),

		  Rule.Thm ("tl_Cons",TermC.num_str @{thm tl_Cons}),

		  Rule.Thm ("tl_Nil",TermC.num_str @{thm tl_Nil}),

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

			eval_occur_exactly_in 

			    "#eval_occur_exactly_in_")

		  ],

	 scr = Rule.EmptyScr};

\<close>

ML \<open>

(*WN060914 quickly created for 4x4; 

 more similarity to prls_triangular desirable*)

val prls_triangular4 = 

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

	 rew_ord = ("e_rew_ord", Rule.e_rew_ord), 

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

		     rew_ord = ("e_rew_ord", Rule.e_rew_ord), 

		     erls = Rule.Erls, srls = Rule.Erls, calc = [], errpatts = [],

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

			      Rule.Calc ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),

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

			      (*immediately repeated rewrite pushes

					    '+' into precondition !*)

			      ],

		     scr = Rule.EmptyScr}, 

	 srls = Rule.Erls, calc = [], errpatts = [],

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

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

		  Rule.Thm ("NTH_NIL",TermC.num_str @{thm NTH_NIL}),

		  Rule.Thm ("tl_Cons",TermC.num_str @{thm tl_Cons}),

		  Rule.Thm ("tl_Nil",TermC.num_str @{thm tl_Nil}),

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

			eval_occur_exactly_in 

			    "#eval_occur_exactly_in_")

		  ],

	 scr = Rule.EmptyScr};

\<close>

setup \<open>KEStore_Elems.add_rlss 

  [("simplify_System_parenthesized",

    (Context.theory_name @{theory}, prep_rls' simplify_System_parenthesized)), 

  ("simplify_System", (Context.theory_name @{theory}, prep_rls' simplify_System)), 

  ("isolate_bdvs", (Context.theory_name @{theory}, prep_rls' isolate_bdvs)), 

  ("isolate_bdvs_4x4", (Context.theory_name @{theory}, prep_rls' isolate_bdvs_4x4)), 

  ("order_system", (Context.theory_name @{theory}, prep_rls' order_system)), 

  ("order_add_mult_System", (Context.theory_name @{theory}, prep_rls' order_add_mult_System)), 

  ("norm_System_noadd_fractions", 

    (Context.theory_name @{theory}, prep_rls' norm_System_noadd_fractions)), 

  ("norm_System", (Context.theory_name @{theory}, prep_rls' norm_System))]\<close>

(** problems **)

setup \<open>KEStore_Elems.add_pbts

  [(Specify.prep_pbt thy "pbl_equsys" [] Celem.e_pblID

      (["system"],

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

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

        Rule.append_rls "e_rls" Rule.e_rls [(*for preds in where_*)], SOME "solveSystem e_s v_s", [])),

    (Specify.prep_pbt thy "pbl_equsys_lin" [] Celem.e_pblID

      (["LINEAR", "system"],

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

          (*TODO.WN050929 check linearity*)

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

        Rule.append_rls "e_rls" Rule.e_rls [(*for preds in where_*)], SOME "solveSystem e_s v_s", [])),

    (Specify.prep_pbt thy "pbl_equsys_lin_2x2" [] Celem.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'''"])],

        Rule.append_rls "prls_2x2_linear_system" Rule.e_rls 

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

			      Rule.Thm ("LENGTH_NIL",TermC.num_str @{thm LENGTH_NIL}),

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

			      Rule.Calc ("HOL.eq", Prog_Expr.eval_equal "#equal_")], 

        SOME "solveSystem e_s v_s", [])),

    (Specify.prep_pbt thy "pbl_equsys_lin_2x2_tri" [] Celem.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"]])),

    (Specify.prep_pbt thy "pbl_equsys_lin_2x2_norm" [] Celem.e_pblID

      (["normalise", "2x2", "LINEAR", "system"],

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

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

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

      SOME "solveSystem e_s v_s", 

      [["EqSystem","normalise","2x2"]])),

    (Specify.prep_pbt thy "pbl_equsys_lin_3x3" [] Celem.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'''"])],

        Rule.append_rls "prls_3x3_linear_system" Rule.e_rls 

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

			      Rule.Thm ("LENGTH_NIL",TermC.num_str @{thm LENGTH_NIL}),

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

			      Rule.Calc ("HOL.eq", Prog_Expr.eval_equal "#equal_")],

        SOME "solveSystem e_s v_s", [])),

    (Specify.prep_pbt thy "pbl_equsys_lin_4x4" [] Celem.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'''"])],

        Rule.append_rls "prls_4x4_linear_system" Rule.e_rls 

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

			      Rule.Thm ("LENGTH_NIL",TermC.num_str @{thm LENGTH_NIL}),

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

			      Rule.Calc ("HOL.eq", Prog_Expr.eval_equal "#equal_")],

        SOME "solveSystem e_s v_s", [])),

    (Specify.prep_pbt thy "pbl_equsys_lin_4x4_tri" [] Celem.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'''"])],

      Rule.append_rls "prls_tri_4x4_lin_sys" prls_triangular

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

      SOME "solveSystem e_s v_s", 

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

    (Specify.prep_pbt thy "pbl_equsys_lin_4x4_norm" [] Celem.e_pblID

      (["normalise", "4x4", "LINEAR", "system"],

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

          (*LENGTH is checked 1 level above*)

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

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

        SOME "solveSystem e_s v_s", 

        [["EqSystem","normalise","4x4"]]))]\<close>

ML \<open>

(*this is for NTH only*)

val srls = Rule.Rls {id="srls_normalise_4x4", 

		preconds = [], 

		rew_ord = ("termlessI",termlessI), 

		erls = Rule.append_rls "erls_in_srls_IntegrierenUnd.." Rule.e_rls

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

				   Rule.Calc ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),

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

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

				   ], 

		srls = Rule.Erls, calc = [], errpatts = [],

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

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

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

		scr = Rule.EmptyScr};

\<close>

(**methods**)

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_eqsys" [] Celem.e_metID

	    (["EqSystem"], [],

	      {rew_ord'="tless_true", rls' = Rule.Erls, calc = [], srls = Rule.Erls, prls = Rule.Erls, crls = Rule.Erls,

          errpats = [], nrls = Rule.Erls},

	      @{thm refl}),

    Specify.prep_met thy "met_eqsys_topdown" [] Celem.e_metID

      (["EqSystem","top_down_substitution"], [],

        {rew_ord'="tless_true", rls' = Rule.Erls, calc = [], srls = Rule.Erls, prls = Rule.Erls, crls = Rule.Erls,

          errpats = [], nrls = Rule.Erls},

       @{thm refl})]

\<close>

partial_function (tailrec) solve_system :: "bool list => real list => bool list"

  where

"solve_system e_s v_s =                          

  (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)                              "

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_eqsys_topdown_2x2" [] Celem.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' = Rule.Erls, calc = [], 

	        srls = Rule.append_rls "srls_top_down_2x2" Rule.e_rls

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

				        Rule.Thm ("tl_Cons",TermC.num_str @{thm tl_Cons}),

				        Rule.Thm ("tl_Nil",TermC.num_str @{thm tl_Nil})], 

	        prls = prls_triangular, crls = Rule.Erls, errpats = [], nrls = Rule.Erls},

	      @{thm solve_system.simps})]

\<close>

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_eqsys_norm" [] Celem.e_metID

	    (["EqSystem", "normalise"], [],

	      {rew_ord'="tless_true", rls' = Rule.Erls, calc = [], srls = Rule.Erls, prls = Rule.Erls, crls = Rule.Erls,

          errpats = [], nrls = Rule.Erls},

	      @{thm refl})]

\<close>

partial_function (tailrec) solve_system2 :: "bool list => real list => bool list"

  where

"solve_system2 e_s v_s =

  (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]))"

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_eqsys_norm_2x2" [] Celem.e_metID

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

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

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

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

	        srls = Rule.append_rls "srls_normalise_2x2" Rule.e_rls

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

				        Rule.Thm ("tl_Cons",TermC.num_str @{thm tl_Cons}),

				        Rule.Thm ("tl_Nil",TermC.num_str @{thm tl_Nil})], 

		      prls = Rule.Erls, crls = Rule.Erls, errpats = [], nrls = Rule.Erls},

		    @{thm solve_system2.simps})]

\<close>

partial_function (tailrec) solve_system3 :: "bool list => real list => bool list"

  where

"solve_system3 e_s v_s =

  (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]))"

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_eqsys_norm_4x4" [] Celem.e_metID

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

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

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

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

	         srls = Rule.append_rls "srls_normalise_4x4" srls

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

	               Rule.Thm ("tl_Cons",TermC.num_str @{thm tl_Cons}),

	               Rule.Thm ("tl_Nil",TermC.num_str @{thm tl_Nil})], 

		       prls = Rule.Erls, crls = Rule.Erls, errpats = [], nrls = Rule.Erls},

		     (*STOPPED.WN06? met ["EqSystem","normalise","4x4"] @@@@@@@@@@@@@@@@@@@@@@@@@@@*)

		     @{thm solve_system3.simps})]

\<close>

partial_function (tailrec) solve_system4 :: "bool list => real list => bool list"

  where

"solve_system4 e_s v_s =

  (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])"

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_eqsys_topdown_4x4" [] Celem.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' = Rule.Erls, calc = [], 

	      srls = Rule.append_rls "srls_top_down_4x4" srls [], 

	      prls = Rule.append_rls "prls_tri_4x4_lin_sys" prls_triangular

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

	      crls = Rule.Erls, errpats = [], nrls = Rule.Erls},

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

	    @{thm solve_system4.simps})]

\<close>

end