(*. (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

section \<open>Theorems for Rule_Set; FIXME: replace axiomatization\<close>

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)"

section \<open>Rule_Set for Problem and MethodC\<close>

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>\<open>not_true\<close>,

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

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

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

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

    \<^rule_thm>\<open>or_false\<close>];

(* ----- asm_rls ----- *)

val LinEq_crls = 

   Rule_Set.append_rules "LinEq_crls" poly_crls

   [\<^rule_thm>\<open>real_assoc_1\<close>

    (*		

     Don't use

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

     \<^rule_eval>\<open>realpow\<close> (**)(Calc_Binop.numeric "#power_"),

     *)

    ];

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

val LinEq_erls = 

  Rule_Set.append_rules "LinEq_erls" Poly_erls [

    \<^rule_thm>\<open>real_assoc_1\<close>

    (*		

     Don't use

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

     \<^rule_eval>\<open>realpow\<close> (**)(Calc_Binop.numeric "#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), 

    asm_rls = LinEq_erls, 

    prog_rls = Rule_Set.Empty, 

    calc = [], errpatts = [],

    rules = [

      \<^rule_thm>\<open>real_assoc_1\<close>,

      \<^rule_eval>\<open>plus\<close> (**)(Calc_Binop.numeric "#add_"),

      \<^rule_eval>\<open>minus\<close> (**)(Calc_Binop.numeric "#sub_"),

      \<^rule_eval>\<open>times\<close> (**)(Calc_Binop.numeric "#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>realpow\<close> (**)(Calc_Binop.numeric "#power_")],

    program = 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 = ("Rewrite_Ord.id_empty", Rewrite_Ord.function_empty),

    asm_rls = LinEq_erls,

    prog_rls = Rule_Set.Empty,

    calc = [], errpatts = [],

    rules = [

	     \<^rule_thm>\<open>lin_isolate_add1\<close>, (* a+bx=0 -> bx=-a *)

	     \<^rule_thm>\<open>lin_isolate_add2\<close>, (* a+ x=0 ->  x=-a *)

	     \<^rule_thm>\<open>lin_isolate_div\<close> (*   bx=c -> x=c/b *)],

    program = Rule.Empty_Prog});

\<close>

rule_set_knowledge LinEq_simplify = LinEq_simplify

section \<open>Problem.T\<close>

problem pbl_equ_univ_lin : "LINEAR/univariate/equation" =

  \<open>LinEq_prls\<close>

  Method_Ref: "LinEq/solve_lineq_equation"

  CAS: "solve (e_e::bool, v_v)"

  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'"

section \<open>MethodC.T\<close>

method met_eqlin : "LinEq" =

  \<open>{rew_ord = "tless_true",rls' = Atools_erls,calc = [], prog_rls = Rule_Set.empty, where_rls = Rule_Set.empty,

    errpats = [], rew_rls = norm_Poly}\<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)"

method met_eq_lin : "LinEq/solve_lineq_equation" =

  \<open>{rew_ord = "termlessI", rls' = LinEq_erls, prog_rls = Rule_Set.empty, where_rls = LinEq_prls, calc = [],

    errpats = [], rew_rls = norm_Poly}\<close>

  Program: solve_linear_equation.simps

  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'"

section \<open>For trials\<close>

ML \<open>

\<close> ML \<open>

\<close> ML \<open>

\<close>

end