(*. (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

 axiomatization where

 (*-- normalise --*)

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

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

   makex1_x:          "a \<up> 1  = a" and

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

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

 (*-- solve --*)

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

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

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

 ML \<open>

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

   Rule_Set.append_rules "LinEq_prls" Rule_Set.empty 

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

 	      \<^rule_eval>\<open>Prog_Expr.matches\<close> (Prog_Expr.eval_matches ""),

 	      \<^rule_eval>\<open>Prog_Expr.lhs\<close> (Prog_Expr.eval_lhs ""),

 	      \<^rule_eval>\<open>Prog_Expr.rhs\<close> (Prog_Expr.eval_rhs ""),

 	      \<^rule_eval>\<open>has_degree_in\<close> (eval_has_degree_in ""),

  	      \<^rule_eval>\<open>is_polyrat_in\<close> (eval_is_polyrat_in ""),

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

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

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

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

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

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

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

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

               ];

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

 val LinEq_crls = 

    Rule_Set.append_rules "LinEq_crls" poly_crls

    [Rule.Thm  ("real_assoc_1",ThmC.numerals_to_Free @{thm real_assoc_1})

     (*		

      Don't use

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

      \<^rule_eval>\<open>powr\<close> (**)(eval_binop "#power_"),

      *)

     ];

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

 val LinEq_erls = 

    Rule_Set.append_rules "LinEq_erls" Poly_erls

    [Rule.Thm  ("real_assoc_1",ThmC.numerals_to_Free @{thm real_assoc_1})

     (*		

      Don't use

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

      \<^rule_eval>\<open>powr\<close> (**)(eval_binop "#power_"),

      *)

     ];

 \<close>

 rule_set_knowledge LinEq_erls = LinEq_erls

 ML \<open>

 val LinPoly_simplify = prep_rls'(

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

        rew_ord = ("termlessI",termlessI), 

        erls = LinEq_erls, 

        srls = Rule_Set.Empty, 

        calc = [], errpatts = [],

        rules = [

 		Rule.Thm  ("real_assoc_1",ThmC.numerals_to_Free @{thm real_assoc_1}),

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

 		\<^rule_eval>\<open>minus\<close> (**)(eval_binop "#sub_"),

 		\<^rule_eval>\<open>times\<close> (**)(eval_binop "#mult_"),

 		(*  Don't use  

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

 		 \<^rule_eval>\<open>sqrt\<close> (eval_sqrt "#sqrt_"),

 		 *)

 		\<^rule_eval>\<open>powr\<close> (**)(eval_binop "#power_")

 		],

        scr = Rule.Empty_Prog});

 \<close>

 rule_set_knowledge LinPoly_simplify = LinPoly_simplify

 ML \<open>

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

 val LinEq_simplify = prep_rls'(

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

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

      erls = LinEq_erls,

      srls = Rule_Set.Empty,

      calc = [], errpatts = [],

      rules = [

 	      Rule.Thm("lin_isolate_add1",ThmC.numerals_to_Free @{thm lin_isolate_add1}), 

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

 	      Rule.Thm("lin_isolate_add2",ThmC.numerals_to_Free @{thm lin_isolate_add2}), 

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

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

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

 	      ],

      scr = Rule.Empty_Prog});

 \<close>

 rule_set_knowledge LinEq_simplify = LinEq_simplify

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

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

 setup \<open>KEStore_Elems.add_pbts

   [(Problem.prep_input @{theory} "pbl_equ_univ_lin" [] Problem.id_empty

       (["LINEAR", "univariate", "equation"],

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

           ("#Where" ,["HOL.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"]]))]\<close>

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

 setup \<open>KEStore_Elems.add_mets

     [MethodC.prep_input @{theory} "met_eqlin" [] MethodC.id_empty

       (["LinEq"], [],

         {rew_ord' = "tless_true",rls' = Atools_erls,calc = [], srls = Rule_Set.empty, prls = Rule_Set.empty,

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

         @{thm refl})]

 \<close>

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

 partial_function (tailrec) solve_linear_equation :: "bool \<Rightarrow> real \<Rightarrow> bool list"

   where

 "solve_linear_equation e_e v_v = (

   let

     e_e = (

       (Try (Rewrite ''all_left'')) #>

       (Try (Repeat (Rewrite ''makex1_x''))) #>

       (Try (Rewrite_Set ''expand_binoms'')) #>

       (Try (Repeat (Rewrite_Set_Inst [(''bdv'', v_v)] ''make_ratpoly_in''))) #>

       (Try (Repeat (Rewrite_Set ''LinPoly_simplify''))) ) e_e;

     e_e = (

       (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''LinEq_simplify'')) #>

       (Repeat (Try (Rewrite_Set ''LinPoly_simplify''))) ) e_e

   in

     Or_to_List e_e)"

 setup \<open>KEStore_Elems.add_mets

     [MethodC.prep_input @{theory} "met_eq_lin" [] MethodC.id_empty

       (["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 = Rule_Set.empty, prls = LinEq_prls, calc = [],

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

         @{thm solve_linear_equation.simps})]

 \<close>

 ML \<open>

   MethodC.from_store' @{theory} ["LinEq", "solve_lineq_equation"];

 \<close> ML \<open>

 \<close> ML \<open>

 \<close>

 end