(*. (c) by Richard Lang, 2003 .*)

(* theory collecting all knowledge for LinearEquations

   created by: rlang 

         date: 02.10

   changed by: rlang

   last change by: rlang

             date: 02.10.20

*)

theory LinEq imports Poly Equation begin

consts

   Solve'_lineq'_equation

             :: "[bool,real, 

		   bool list] => bool list"

               ("((Script Solve'_lineq'_equation (_ _ =))// 

                  (_))" 9)

axioms

(*-- normalize --*)

  (*WN0509 compare PolyEq.all_left "[|Not(b=!=0)|] ==> (a = b) = (a - b = 0)"*)

  all_left:          "[|Not(b=!=0)|] ==> (a=b) = (a+(-1)*b=0)"

  makex1_x:          "a^^^1  = a"  

  real_assoc_1:      "a+(b+c) = a+b+c"

  real_assoc_2:      "a*(b*c) = a*b*c"

(*-- solve --*)

  lin_isolate_add1:  "(a + b*bdv = 0) = (b*bdv = (-1)*a)"

  lin_isolate_add2:  "(a +   bdv = 0) = (  bdv = (-1)*a)"

  lin_isolate_div:   "[|Not(b=0)|] ==> (b*bdv = c) = (bdv = c / b)"

ML {*

val thy = @{theory};

val LinEq_prls = (*3.10.02:just the following order due to subterm evaluation*)

  append_rls "LinEq_prls" e_rls 

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

	      Calc ("Tools.matches",eval_matches ""),

	      Calc ("Tools.lhs"    ,eval_lhs ""),

	      Calc ("Tools.rhs"    ,eval_rhs ""),

	      Calc ("Poly.has'_degree'_in",eval_has_degree_in ""),

 	      Calc ("Poly.is'_polyrat'_in",eval_is_polyrat_in ""),

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

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

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

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

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

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

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

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

              ];

(* ----- erls ----- *)

val LinEq_crls = 

   append_rls "LinEq_crls" poly_crls

   [Thm  ("real_assoc_1",num_str @{thm real_assoc_1})

    (*		

     Don't use

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

     Calc ("Atools.pow" ,eval_binop "#power_"),

     *)

    ];

(* ----- crls ----- *)

val LinEq_erls = 

   append_rls "LinEq_erls" Poly_erls

   [Thm  ("real_assoc_1",num_str @{thm real_assoc_1})

    (*		

     Don't use

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

     Calc ("Atools.pow" ,eval_binop "#power_"),

     *)

    ];

ruleset' := overwritelthy @{theory} (!ruleset',

			[("LinEq_erls",LinEq_erls)(*FIXXXME:del with rls.rls'*)

			 ]);

val LinPoly_simplify = prep_rls(

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

       rew_ord = ("termlessI",termlessI), 

       erls = LinEq_erls, 

       srls = Erls, 

       calc = [], 

       (*asm_thm = [],*)

       rules = [

		Thm  ("real_assoc_1",num_str @{thm real_assoc_1}),

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

		Calc ("Groups.minus_class.minus",eval_binop "#sub_"),

		Calc ("op *",eval_binop "#mult_"),

		(*  Dont use  

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

		 Calc ("NthRoot.sqrt",eval_sqrt "#sqrt_"),

		 *)

		Calc ("Atools.pow" ,eval_binop "#power_")

		],

       scr = EmptyScr}:rls);

ruleset' := overwritelthy @{theory} (!ruleset',

			  [("LinPoly_simplify",LinPoly_simplify)]);

(*isolate the bound variable in an linear equation; 'bdv' is a meta-constant*)

val LinEq_simplify = prep_rls(

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

     rew_ord = ("e_rew_ord",e_rew_ord),

     erls = LinEq_erls,

     srls = Erls,

     calc = [],

     (*asm_thm = [("lin_isolate_div","")],*)

     rules = [

	      Thm("lin_isolate_add1",num_str @{thm lin_isolate_add1}), 

	      (* a+bx=0 -> bx=-a *)

	      Thm("lin_isolate_add2",num_str @{thm lin_isolate_add2}), 

	      (* a+ x=0 ->  x=-a *)

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

	      (*   bx=c -> x=c/b *)  

	      ],

     scr = EmptyScr}:rls);

ruleset' := overwritelthy @{theory} (!ruleset',

			[("LinEq_simplify",LinEq_simplify)]);

(*----------------------------- problem types --------------------------------*)

(* 

show_ptyps(); 

(get_pbt ["linear","univariate","equation"]);

*)

(* ---------linear----------- *)

store_pbt

 (prep_pbt thy "pbl_equ_univ_lin" [] e_pblID

 (["linear","univariate","equation"],

  [("#Given" ,["equality e_e","solveFor v_v"]),

   ("#Where" ,["False", (*WN0509 just detected: this pbl can never be used?!?*)

               "Not( (lhs e_e) is_polyrat_in v_v)",

               "Not( (rhs e_e) is_polyrat_in v_v)",

               "((lhs e_e) has_degree_in v_v)=1",

	       "((rhs e_e) has_degree_in v_v)=1"]),

   ("#Find"  ,["solutions v_v'i'"]) 

  ],

  LinEq_prls, SOME "solve (e_e::bool, v_v)",

  [["LinEq","solve_lineq_equation"]]));

(*-------------- methods------------------------------------------------------*)

store_met

 (prep_met thy "met_eqlin" [] e_metID

 (["LinEq"],

   [],

   {rew_ord'="tless_true",rls'=Atools_erls,calc = [], srls = e_rls, prls=e_rls,

    crls=LinEq_crls, nrls=norm_Poly

    (*, asm_rls=[],asm_thm=[]*)}, "empty_script"));

(* ansprechen mit ["LinEq","solve_univar_equation"] *)

store_met

(prep_met thy "met_eq_lin" [] e_metID

 (["LinEq","solve_lineq_equation"],

   [("#Given", ["equality e_e", "solveFor v_v"]),

    ("#Where", ["Not ((lhs e_e) is_polyrat_in v_v)",

                "((lhs e_e)  has_degree_in v_v) = 1"]),

    ("#Find",  ["solutions v_v'i'"])

   ],

   {rew_ord'="termlessI", rls'=LinEq_erls, srls=e_rls, prls=LinEq_prls,

    calc=[], crls=LinEq_crls, nrls=norm_Poly},

    "Script Solve_lineq_equation (e_e::bool) (v_v::real) =                 " ^

    "(let e_e =((Try         (Rewrite      all_left           False)) @@   " ^ 

    "           (Try (Repeat (Rewrite     makex1_x            False))) @@  " ^ 

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

    "           (Try (Repeat (Rewrite_Set_Inst [(bdv, v_v::real)]          " ^

    "                                 make_ratpoly_in    False)))    @@    " ^

    "           (Try (Repeat (Rewrite_Set LinPoly_simplify      False))))e_e;" ^

    "     e_e = ((Try (Rewrite_Set_Inst [(bdv, v_v::real)]                  " ^

    "                                     LinEq_simplify True)) @@  " ^

    "            (Repeat(Try (Rewrite_Set LinPoly_simplify     False)))) e_e " ^

    " in ((Or_to_List e_e)::bool list))"

 ));

get_met ["LinEq","solve_lineq_equation"];

*}

end