(* 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 ((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>

 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= \"" ^ 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 str 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 tu = 

     (term_ord' pr thy tu = LESS);

 (**)

 end;

 (**)

 Rewrite_Ord.rew_ord' := overwritel (! Rewrite_Ord.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_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 ("mult.commute",ThmC.numerals_to_Free @{thm mult.commute}),

 	       (* z * w = w * z *)

 	       Rule.Thm ("real_mult_left_commute",ThmC.numerals_to_Free @{thm real_mult_left_commute}),

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

 	       Rule.Thm ("mult.assoc",ThmC.numerals_to_Free @{thm mult.assoc}),		

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

 	       Rule.Thm ("add.commute",ThmC.numerals_to_Free @{thm add.commute}),	

 	       (*z + w = w + z*)

 	       Rule.Thm ("add.left_commute",ThmC.numerals_to_Free @{thm add.left_commute}),

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

 	       Rule.Thm ("add.assoc",ThmC.numerals_to_Free @{thm add.assoc})	               

 	       (*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 ("distrib_right",ThmC.numerals_to_Free @{thm distrib_right}),

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

 	       Rule.Thm ("add_divide_distrib",ThmC.numerals_to_Free @{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",

                      ThmC.numerals_to_Free (@{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.Eval ("Rings.divide_class.divide", 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 ("Rings.divide_class.divide", 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_add.assoc",

                       ThmC.numerals_to_Free (@{thm add.assoc} RS @{thm sym}))];

 *)

 \<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 ("EqSystem.occur'_exactly'_in", 

 				   eval_occur_exactly_in 

 				       "#eval_occur_exactly_in_"))

 			    ], 

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

 	      rules = 

              [Rule.Thm ("commute_0_equality", ThmC.numerals_to_Free @{thm commute_0_equality}),

 	      Rule.Thm ("separate_bdvs_add", ThmC.numerals_to_Free @{thm separate_bdvs_add}),

 	      Rule.Thm ("separate_bdvs_mult", ThmC.numerals_to_Free @{thm separate_bdvs_mult})],

 	      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 ("EqSystem.occur'_exactly'_in", 

 			   eval_occur_exactly_in "#eval_occur_exactly_in_"),

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

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

          Rule.Thm ("not_true",ThmC.numerals_to_Free @{thm not_true}),

 		     Rule.Thm ("not_false",ThmC.numerals_to_Free @{thm not_false})

 			    ], 

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

 	 rules = [Rule.Thm ("commute_0_equality", ThmC.numerals_to_Free @{thm commute_0_equality}),

 		  Rule.Thm ("separate_bdvs0", ThmC.numerals_to_Free @{thm separate_bdvs0}),

 		  Rule.Thm ("separate_bdvs_add1", ThmC.numerals_to_Free @{thm separate_bdvs_add1}),

 		  Rule.Thm ("separate_bdvs_add1", ThmC.numerals_to_Free @{thm separate_bdvs_add2}),

 		  Rule.Thm ("separate_bdvs_mult", ThmC.numerals_to_Free @{thm separate_bdvs_mult})

                  ], 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 thy), 

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

 	 rules = [Rule.Thm ("order_system_NxN", ThmC.numerals_to_Free @{thm order_system_NxN})

 		  ],

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

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

 			      (*immediately repeated rewrite pushes

 					    '+' into precondition !*)

 			      ],

 		     scr = Rule.Empty_Prog}, 

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

 	 rules = [Rule.Thm ("NTH_CONS",ThmC.numerals_to_Free @{thm NTH_CONS}),

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

 		  Rule.Thm ("NTH_NIL",ThmC.numerals_to_Free @{thm NTH_NIL}),

 		  Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),

 		  Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil}),

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

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

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

 			      (*immediately repeated rewrite pushes

 					    '+' into precondition !*)

 			      ],

 		     scr = Rule.Empty_Prog}, 

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

 	 rules = [Rule.Thm ("NTH_CONS",ThmC.numerals_to_Free @{thm NTH_CONS}),

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

 		  Rule.Thm ("NTH_NIL",ThmC.numerals_to_Free @{thm NTH_NIL}),

 		  Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),

 		  Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil}),

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

 			eval_occur_exactly_in 

 			    "#eval_occur_exactly_in_")

 		  ],

 	 scr = Rule.Empty_Prog};

 \<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>

 section \<open>Problems\<close>

 setup \<open>KEStore_Elems.add_pbts

   [(Problem.prep_input thy "pbl_equsys" [] Problem.id_empty

       (["system"],

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

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

         Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)], SOME "solveSystem e_s v_s", [])),

     (Problem.prep_input thy "pbl_equsys_lin" [] Problem.id_empty

       (["LINEAR", "system"],

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

           (*TODO.WN050929 check linearity*)

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

         Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)], SOME "solveSystem e_s v_s", [])),

     (Problem.prep_input thy "pbl_equsys_lin_2x2" [] Problem.id_empty

       (["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_Set.append_rules "prls_2x2_linear_system" Rule_Set.empty 

 			    [Rule.Thm ("LENGTH_CONS",ThmC.numerals_to_Free @{thm LENGTH_CONS}),

 			      Rule.Thm ("LENGTH_NIL",ThmC.numerals_to_Free @{thm LENGTH_NIL}),

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

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

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

     (Problem.prep_input thy "pbl_equsys_lin_2x2_tri" [] Problem.id_empty

       (["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"]])),

     (Problem.prep_input thy "pbl_equsys_lin_2x2_norm" [] Problem.id_empty

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

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

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

       Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)], 

       SOME "solveSystem e_s v_s", 

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

     (Problem.prep_input thy "pbl_equsys_lin_3x3" [] Problem.id_empty

       (["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_Set.append_rules "prls_3x3_linear_system" Rule_Set.empty 

 			    [Rule.Thm ("LENGTH_CONS",ThmC.numerals_to_Free @{thm LENGTH_CONS}),

 			      Rule.Thm ("LENGTH_NIL",ThmC.numerals_to_Free @{thm LENGTH_NIL}),

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

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

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

     (Problem.prep_input thy "pbl_equsys_lin_4x4" [] Problem.id_empty

       (["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_Set.append_rules "prls_4x4_linear_system" Rule_Set.empty 

 			    [Rule.Thm ("LENGTH_CONS",ThmC.numerals_to_Free @{thm LENGTH_CONS}),

 			      Rule.Thm ("LENGTH_NIL",ThmC.numerals_to_Free @{thm LENGTH_NIL}),

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

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

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

     (Problem.prep_input thy "pbl_equsys_lin_4x4_tri" [] Problem.id_empty

       (["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_Set.append_rules "prls_tri_4x4_lin_sys" prls_triangular

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

       SOME "solveSystem e_s v_s", 

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

     (Problem.prep_input thy "pbl_equsys_lin_4x4_norm" [] Problem.id_empty

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

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

           (*Length is checked 1 level above*)

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

         Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)], 

         SOME "solveSystem e_s v_s", 

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

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

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

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

 				   ], 

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

 		rules = [Rule.Thm ("NTH_CONS",ThmC.numerals_to_Free @{thm NTH_CONS}),

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

 			 Rule.Thm ("NTH_NIL",ThmC.numerals_to_Free @{thm NTH_NIL})],

 		scr = Rule.Empty_Prog};

 \<close>

 section \<open>Methods\<close>

 setup \<open>KEStore_Elems.add_mets

     [Method.prep_input thy "met_eqsys" [] Method.id_empty

 	    (["EqSystem"], [],

 	      {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},

 	      @{thm refl}),

     Method.prep_input thy "met_eqsys_topdown" [] Method.id_empty

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

         {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},

        @{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'')) #>                   

       (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)                              "

 setup \<open>KEStore_Elems.add_mets

     [Method.prep_input thy "met_eqsys_topdown_2x2" [] Method.id_empty

       (["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_Set.Empty, calc = [], 

 	        srls = Rule_Set.append_rules "srls_top_down_2x2" Rule_Set.empty

 				      [Rule.Thm ("hd_thm",ThmC.numerals_to_Free @{thm hd_thm}),

 				        Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),

 				        Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil})], 

 	        prls = prls_triangular, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},

 	      @{thm solve_system.simps})]

 \<close>

 setup \<open>KEStore_Elems.add_mets

     [Method.prep_input thy "met_eqsys_norm" [] Method.id_empty

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

 	      {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},

 	      @{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'' )) #>

       (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])"

 setup \<open>KEStore_Elems.add_mets

     [Method.prep_input thy "met_eqsys_norm_2x2" [] Method.id_empty

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

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

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

 	      {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], 

 	        srls = Rule_Set.append_rules "srls_normalise_2x2" Rule_Set.empty

 				      [Rule.Thm ("hd_thm",ThmC.numerals_to_Free @{thm hd_thm}),

 				        Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),

 				        Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil})], 

 		      prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},

 		    @{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'' )) #>

       (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])"

 setup \<open>KEStore_Elems.add_mets

     [Method.prep_input thy "met_eqsys_norm_4x4" [] Method.id_empty

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

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

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

 	       {rew_ord'="tless_true", rls' = Rule_Set.Empty, calc = [], 

 	         srls = Rule_Set.append_rules "srls_normalise_4x4" srls

 	             [Rule.Thm ("hd_thm",ThmC.numerals_to_Free @{thm hd_thm}),

 	               Rule.Thm ("tl_Cons",ThmC.numerals_to_Free @{thm tl_Cons}),

 	               Rule.Thm ("tl_Nil",ThmC.numerals_to_Free @{thm tl_Nil})], 

 		       prls = Rule_Set.Empty, crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},

 		     (*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'' )) #>

       (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])"

 setup \<open>KEStore_Elems.add_mets

     [Method.prep_input thy "met_eqsys_topdown_4x4" [] Method.id_empty

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

 	      crls = Rule_Set.Empty, errpats = [], nrls = Rule_Set.Empty},

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

 	    @{thm solve_system4.simps})]

 \<close>

 end