(* 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>

 (** 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 (\<^const_name>\<open>EqSystem.occur_exactly_in\<close>,_) 

 				       $ vs $ all $ t)) _ =

     if occur_exactly_in (TermC.isalist2list vs) (TermC.isalist2list all) t

     then SOME ((UnparseC.term p) ^ " = True",

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

     else SOME ((UnparseC.term p) ^ " = False",

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

   | eval_occur_exactly_in _ _ _ _ = NONE;

 \<close>

 calculation occur_exactly_in = \<open>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)

   | dest_hd' _ = raise ERROR "dest_hd': uncovered case in fun.def.";

 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= \"" ^ UnparseC.term_in_thy thy f ^ "\" @ \"[" ^

            commas (map (UnparseC.term_in_thy thy) ts) ^ "]\"");

          val _ = tracing ("u= g@us= \"" ^ UnparseC.term_in_thy thy g ^ "\" @ \"[" ^

            commas (map (UnparseC.term_in_thy 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 _ pr (ts, us) = list_ord (term_ord' pr (ThyC.get_theory "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*) (ts, us) = 

     (term_ord' pr thy (TermC.numerals_to_Free ts, TermC.numerals_to_Free us) = LESS);

 (**)

 end;

 (**)

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

 [("ord_simplify_System", ord_simplify_System false \<^theory>)

  ]);

 \<close>

 ML \<open>

 (** rulesets **)

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

 val order_add_mult_System = 

   Rule_Def.Repeat{

     id = "order_add_mult_System", preconds = [], 

     rew_ord = ("ord_simplify_System", ord_simplify_System false @{theory "Integrate"}),

     erls = Rule_Set.empty,srls = Rule_Set.Empty, calc = [], errpatts = [],

     rules = [

       \<^rule_thm>\<open>mult.commute\<close>, (* z * w = w * z *)

       \<^rule_thm>\<open>real_mult_left_commute\<close>, (*z1.0 * (z2.0 * z3.0) = z2.0 * (z1.0 * z3.0)*)

       \<^rule_thm>\<open>mult.assoc\<close>,	 (*z1.0 * z2.0 * z3.0 = z1.0 * (z2.0 * z3.0)*)

       \<^rule_thm>\<open>add.commute\<close>, (*z + w = w + z*)

       \<^rule_thm>\<open>add.left_commute\<close>, (*x + (y + z) = y + (x + z)*)

       \<^rule_thm>\<open>add.assoc\<close>	],  (*z1.0 + z2.0 + z3.0 = z1.0 + (z2.0 + z3.0)*)

     scr = Rule.Empty_Prog};

 \<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_Def.Repeat {id = "norm_System_noadd_fractions", preconds = [], 

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

     erls = norm_rat_erls, srls = Rule_Set.Empty, 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.Empty_Prog};

 \<close>

 ML \<open>

 (*.adapted from 'norm_Rational' by

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

 val norm_System = 

   Rule_Def.Repeat {id = "norm_System", preconds = [], 

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

     erls = norm_rat_erls, srls = Rule_Set.Empty, 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.Empty_Prog};

 \<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_Set.Sequence {id = "simplify_System_parenthesized", preconds = []:term list, 

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

     erls = Atools_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],

     rules = [

        \<^rule_thm>\<open>distrib_right\<close>, (*"(?z1.0 + ?z2.0) * ?w = ?z1.0 * ?w + ?z2.0 * ?w"*)

 	     \<^rule_thm>\<open>add_divide_distrib\<close>, (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)

 	     Rule.Rls_ norm_Rational_noadd_fractions,

 	     Rule.Rls_ (*order_add_mult_in*) norm_System_noadd_fractions,

 	     \<^rule_thm_sym>\<open>mult.assoc\<close>,

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

 	     Rule.Rls_ separate_bdv2,

 	     \<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e")],

     scr = Rule.Empty_Prog};      

 \<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_Set.Sequence {id = "simplify_System", preconds = []:term list, 

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

     erls = Atools_erls, srls = Rule_Set.Empty, 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_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e")],

     scr = Rule.Empty_Prog};      

 (*

 val simplify_System = 

     Rule_Set.append_rules "simplify_System" simplify_System_parenthesized

 	       [\<^rule_thm_sym>\<open>add.assoc\<close>];

 *)

 \<close>

 ML \<open>

 val isolate_bdvs = 

   Rule_Def.Repeat {

     id="isolate_bdvs", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord), 

     erls = Rule_Set.append_rules "erls_isolate_bdvs" Rule_Set.empty [

       (\<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_"))], 

     srls = Rule_Set.Empty, calc = [], errpatts = [],

     rules = [

       \<^rule_thm>\<open>commute_0_equality\<close>,

       \<^rule_thm>\<open>separate_bdvs_add\<close>,

       \<^rule_thm>\<open>separate_bdvs_mult\<close>],

     scr = Rule.Empty_Prog};

 \<close>

 ML \<open>

 val isolate_bdvs_4x4 = 

   Rule_Def.Repeat {id="isolate_bdvs_4x4", preconds = [], 

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

     erls = Rule_Set.append_rules "erls_isolate_bdvs_4x4" Rule_Set.empty [

       \<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_"),

       \<^rule_eval>\<open>Prog_Expr.ident\<close> (Prog_Expr.eval_ident "#ident_"),

       \<^rule_eval>\<open>Prog_Expr.some_occur_in\<close> (Prog_Expr.eval_some_occur_in "#some_occur_in_"),

       \<^rule_thm>\<open>not_true\<close>,

       \<^rule_thm>\<open>not_false\<close>], 

     srls = Rule_Set.Empty, calc = [], errpatts = [],

     rules = [

       \<^rule_thm>\<open>commute_0_equality\<close>,

       \<^rule_thm>\<open>separate_bdvs0\<close>,

       \<^rule_thm>\<open>separate_bdvs_add1\<close>,

       \<^rule_thm>\<open>separate_bdvs_add2\<close>,

       \<^rule_thm>\<open>separate_bdvs_mult\<close>],

     scr = Rule.Empty_Prog};

 \<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_Def.Repeat {id="order_system", preconds = [], 

 	  rew_ord = ("ord_simplify_System", ord_simplify_System false \<^theory>), 

 	  erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],

 	  rules = [

       \<^rule_thm>\<open>order_system_NxN\<close>],

 	  scr = Rule.Empty_Prog};

 val prls_triangular = 

   Rule_Def.Repeat {

     id="prls_triangular", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord), 

     erls = Rule_Def.Repeat {

       id="erls_prls_triangular", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord), 

       erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],

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

          \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),

          \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),

          \<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_")],

          (*immediately repeated rewrite pushes '+' into precondition !*)

       scr = Rule.Empty_Prog}, 

     srls = Rule_Set.Empty, calc = [], errpatts = [],

     rules = [

       \<^rule_thm>\<open>NTH_CONS\<close>,

       \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),

       \<^rule_thm>\<open>NTH_NIL\<close>,

       \<^rule_thm>\<open>tl_Cons\<close>,

       \<^rule_thm>\<open>tl_Nil\<close>,

       \<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_")],

     scr = Rule.Empty_Prog};

 \<close>

 ML \<open>

 (*WN060914 quickly created for 4x4; 

  more similarity to prls_triangular desirable*)

 val prls_triangular4 = 

   Rule_Def.Repeat {

   id="prls_triangular4", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord), 

   erls = Rule_Def.Repeat {

     id="erls_prls_triangular4", preconds = [], rew_ord = ("e_rew_ord", Rewrite_Ord.e_rew_ord), 

     erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],

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

   	   \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),

   	   \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_")],

   	   (*immediately repeated rewrite pushes '+' into precondition !*)

     scr = Rule.Empty_Prog}, 

   srls = Rule_Set.Empty, calc = [], errpatts = [],

   rules = [

     \<^rule_thm>\<open>NTH_CONS\<close>,

     \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),

     \<^rule_thm>\<open>NTH_NIL\<close>,

     \<^rule_thm>\<open>tl_Cons\<close>,

     \<^rule_thm>\<open>tl_Nil\<close>,

     \<^rule_eval>\<open>occur_exactly_in\<close> (eval_occur_exactly_in "#eval_occur_exactly_in_")],

   scr = Rule.Empty_Prog};

 \<close>

 rule_set_knowledge

   simplify_System_parenthesized = \<open>prep_rls' simplify_System_parenthesized\<close> and

   simplify_System = \<open>prep_rls' simplify_System\<close> and

   isolate_bdvs = \<open>prep_rls' isolate_bdvs\<close> and

   isolate_bdvs_4x4 = \<open>prep_rls' isolate_bdvs_4x4\<close> and 

   order_system = \<open>prep_rls' order_system\<close> and 

   order_add_mult_System = \<open>prep_rls' order_add_mult_System\<close> and

   norm_System_noadd_fractions = \<open>prep_rls' norm_System_noadd_fractions\<close> and

   norm_System = \<open>prep_rls' norm_System\<close>

 section \<open>Problems\<close>

 problem pbl_equsys : "system" =

   \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>

   CAS: "solveSystem e_s v_s"

   Given: "equalities e_s" "solveForVars v_s"

   Find: "solution ss'''" (*''' is copy-named*)

 problem pbl_equsys_lin : "LINEAR/system" =

   \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>

   CAS: "solveSystem e_s v_s"

   Given: "equalities e_s" "solveForVars v_s"

   (*TODO.WN050929 check linearity*)

   Find: "solution ss'''"

 problem pbl_equsys_lin_2x2: "2x2/LINEAR/system" =

   \<open>Rule_Set.append_rules "prls_2x2_linear_system" Rule_Set.empty 

     [\<^rule_thm>\<open>LENGTH_CONS\<close>,

       \<^rule_thm>\<open>LENGTH_NIL\<close>,

       \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),

       \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")]\<close>

   CAS: "solveSystem e_s v_s"

   Given: "equalities e_s" "solveForVars v_s"

   Where: "Length (e_s:: bool list) = 2" "Length v_s = 2"

   Find: "solution ss'''"

 problem pbl_equsys_lin_2x2_tri : "triangular/2x2/LINEAR/system" =

   \<open>prls_triangular\<close>

   Method: "EqSystem/top_down_substitution/2x2"

   CAS: "solveSystem e_s v_s"

   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'''"

 problem pbl_equsys_lin_2x2_norm : "normalise/2x2/LINEAR/system" =

   \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>

   Method: "EqSystem/normalise/2x2"

   CAS: "solveSystem e_s v_s"

   Given: "equalities e_s" "solveForVars v_s"

   Find: "solution ss'''"

 problem pbl_equsys_lin_3x3 : "3x3/LINEAR/system" =

   \<open>Rule_Set.append_rules "prls_3x3_linear_system" Rule_Set.empty 

     [\<^rule_thm>\<open>LENGTH_CONS\<close>,

       \<^rule_thm>\<open>LENGTH_NIL\<close>,

       \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),

       \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")]\<close>

   CAS: "solveSystem e_s v_s"

   Given: "equalities e_s" "solveForVars v_s"

   Where: "Length (e_s:: bool list) = 3" "Length v_s = 3"

   Find: "solution ss'''"

 problem pbl_equsys_lin_4x4 : "4x4/LINEAR/system" =

   \<open>Rule_Set.append_rules "prls_4x4_linear_system" Rule_Set.empty 

     [\<^rule_thm>\<open>LENGTH_CONS\<close>,

       \<^rule_thm>\<open>LENGTH_NIL\<close>,

       \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),

       \<^rule_eval>\<open>HOL.eq\<close> (Prog_Expr.eval_equal "#equal_")]\<close>

   CAS: "solveSystem e_s v_s"

   Given: "equalities e_s" "solveForVars v_s"

   Where: "Length (e_s:: bool list) = 4" "Length v_s = 4"

   Find: "solution ss'''"

 problem pbl_equsys_lin_4x4_tri : "triangular/4x4/LINEAR/system" =

   \<open>Rule_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular

     [\<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in "")]\<close>

   Method: "EqSystem/top_down_substitution/4x4"

   CAS: "solveSystem e_s v_s"

   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'''"

 problem pbl_equsys_lin_4x4_norm : "normalise/4x4/LINEAR/system" =

   \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>

   Method: "EqSystem/normalise/4x4"

   CAS: "solveSystem e_s v_s"

   Given: "equalities e_s" "solveForVars v_s"

   (*Length is checked 1 level above*)

   Find: "solution ss'''"

 ML \<open>

 (*this is for NTH only*)

 val srls = Rule_Def.Repeat {id="srls_normalise_4x4", 

 		preconds = [], 

 		rew_ord = ("termlessI",termlessI), 

 		erls = Rule_Set.append_rules "erls_in_srls_IntegrierenUnd.." Rule_Set.empty

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

 				   \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),

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

 				   \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_")

 				   ], 

 		srls = Rule_Set.Empty, calc = [], errpatts = [],

 		rules = [\<^rule_thm>\<open>NTH_CONS\<close>,

 			 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),

 			 \<^rule_thm>\<open>NTH_NIL\<close>],

 		scr = Rule.Empty_Prog};

 \<close>

 section \<open>Methods\<close>

 method met_eqsys : "EqSystem" =

   \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,

     errpats = [], nrls = Rule_Set.Empty}\<close>

 method met_eqsys_topdown : "EqSystem/top_down_substitution" =

   \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,

     errpats = [], nrls = Rule_Set.Empty}\<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'')) #>                   

       (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System''))

       ) 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'' )) #>      

       (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'' )) #>                   

       (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System'' ))

       ) e_2;                 

     e__s = Take [e_1, e_2]                                                       

   in

     Try (Rewrite_Set ''order_system'' ) e__s)                              "

 method met_eqsys_topdown_2x2 : "EqSystem/top_down_substitution/2x2" =

   \<open>{rew_ord'="ord_simplify_System", rls' = Rule_Set.Empty, calc = [], 

     srls = Rule_Set.append_rules "srls_top_down_2x2" Rule_Set.empty

         [\<^rule_thm>\<open>hd_thm\<close>,

           \<^rule_thm>\<open>tl_Cons\<close>,

           \<^rule_thm>\<open>tl_Nil\<close>], 

     prls = prls_triangular, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>

   Program: solve_system.simps

   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'''"

 method met_eqsys_norm : "EqSystem/normalise" =

   \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], srls = Rule_Set.Empty, prls = Rule_Set.Empty, crls = Rule_Set.Empty,

     errpats = [], nrls = Rule_Set.Empty}\<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'' )) #>

       (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>

       (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''isolate_bdvs'' )) #>

       (Try (Rewrite_Set_Inst [(''bdv_1'', hd v_s), (''bdv_2'', hd (tl v_s))] ''simplify_System_parenthesized'' )) #>

       (Try (Rewrite_Set ''order_system'' ))

       ) e_s

   in

     SubProblem (''EqSystem'', [''LINEAR'', ''system''], [''no_met''])

       [BOOL_LIST e__s, REAL_LIST v_s])"

 method met_eqsys_norm_2x2 : "EqSystem/normalise/2x2" =

   \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], 

     srls = Rule_Set.append_rules "srls_normalise_2x2" Rule_Set.empty

         [\<^rule_thm>\<open>hd_thm\<close>,

           \<^rule_thm>\<open>tl_Cons\<close>,

           \<^rule_thm>\<open>tl_Nil\<close>], 

     prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>

   Program: solve_system2.simps

   Given: "equalities e_s" "solveForVars v_s"

   Find: "solution ss'''"

 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'' )) #>

       (Repeat (Rewrite ''commute_0_equality'' )) #>

       (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'' )) #>

       (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'' )) #>

       (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'' )) #>

       (Try (Rewrite_Set ''order_system''))

       )  e_s

   in

     SubProblem (''EqSystem'', [''LINEAR'', ''system''], [''no_met''])

       [BOOL_LIST e__s, REAL_LIST v_s])"

 method met_eqsys_norm_4x4 : "EqSystem/normalise/4x4" =

   \<open>{rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [],

     srls =

       Rule_Set.append_rules "srls_normalise_4x4" srls

         [\<^rule_thm>\<open>hd_thm\<close>, \<^rule_thm>\<open>tl_Cons\<close>, \<^rule_thm>\<open>tl_Nil\<close>],

     prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>

   Program: solve_system3.simps

   Given: "equalities e_s" "solveForVars v_s"

   Find: "solution ss'''"

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

 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'' )) #>

       (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'' )) #>

       (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'' ))

       ) e_2

   in

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

 method met_eqsys_topdown_4x4 : "EqSystem/top_down_substitution/4x4" =

   \<open>{rew_ord'="ord_simplify_System", rls' = Rule_Set.Empty, calc = [], 

     srls = Rule_Set.append_rules "srls_top_down_4x4" srls [], 

     prls = Rule_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular

       [\<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in "")], 

     crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty}\<close>

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

   Program: solve_system4.simps

   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'''"

 ML \<open>

 \<close> ML \<open>

 \<close> ML \<open>

 \<close>

 end