(*. (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 ("Groups.times_class.times",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