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

(* theory collecting all knowledge for RationalEquations

   created by: rlang 

         date: 02.08.12

   changed by: rlang

   last change by: rlang

             date: 02.11.28

*)

theory RatEq imports Rational begin

text {* univariate equations over multivariate rational terms:

  In 2003 this type has been integrated into ISAC's equation solver

  by Richard Lang; the root for the solver is Equation.thy.

  The migration Isabelle2002 --> 2011 found that application of theorems like 

  rat_mult_denominator_right: "[|Not(d=0)|] ==> ((a::real) = c / d) = (a*d = c)"

  in rule-sets does not transfer "d ~= 0" to the assumptions; see

  test --- repair NO asms from rls RatEq_eliminate ---.

  Thus the migration dropped update of Check_elementwise, which would require

  these assumptions; see 

  test --- pbl: rational, univariate, equation ---, --- x / (x ^ 2 - 6 * x + 9) - 1 /...

*}

consts

  is'_ratequation'_in :: "[bool, real] => bool" ("_ is'_ratequation'_in _")

  (*----------------------scripts-----------------------*)

  Solve'_rat'_equation

             :: "[bool,real, 

		   bool list] => bool list"

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

                  (_))" 9)

axioms(*axiomatization where*)

   (* FIXME also in Poly.thy def. --> FIXED*)

   (*real_diff_minus            

   "a - b = a + (-1) * b"*)

   real_rat_mult_1:   "a*(b/c) = (a*b)/c" (*and*)

   real_rat_mult_2:   "(a/b)*(c/d) = (a*c)/(b*d)" (*and*)

   real_rat_mult_3:   "(a/b)*c = (a*c)/b" (*and*)

   real_rat_pow:      "(a/b)^^^2 = a^^^2/b^^^2" (*and*)

   rat_double_rat_1:   "[|Not(c=0); Not(d=0)|] ==> (a / (c/d) = (a*d) / c)" (*and*)

   rat_double_rat_2:   "[|Not(b=0);Not(c=0); Not(d=0)|] ==> 

                                           ((a/b) / (c/d) = (a*d) / (b*c))" (*and*)

   rat_double_rat_3:   "[|Not(b=0);Not(c=0)|] ==> ((a/b) / c = a / (b*c))" (*and*)

  (* equation to same denominator *)

  rat_mult_denominator_both:

   "[|Not(b=0); Not(d=0)|] ==> ((a::real) / b = c / d) = (a*d = c*b)" (*and*)

  rat_mult_denominator_left:

   "[|Not(d=0)|] ==> ((a::real) = c / d) = (a*d = c)" (*and*)

  rat_mult_denominator_right:

   "[|Not(b=0)|] ==> ((a::real) / b = c) = (a = c*b)"

ML {*

val thy = @{theory};

(*-------------------------functions-----------------------*)

(* is_rateqation_in becomes true, if a bdv is in the denominator of a fraction*)

fun is_rateqation_in t v = 

    let 

	fun coeff_in c v = member op = (vars c) v;

   	fun finddivide (_ $ _ $ _ $ _) v = error("is_rateqation_in:")

	    (* at the moment there is no term like this, but ....*)

	  | finddivide (t as (Const ("Fields.inverse_class.divide",_) $ _ $ b)) v = coeff_in b v

	  | finddivide (_ $ t1 $ t2) v = (finddivide t1 v) 

                                         orelse (finddivide t2 v)

	  | finddivide (_ $ t1) v = (finddivide t1 v)

	  | finddivide _ _ = false;

     in

	finddivide t v

    end;

fun eval_is_ratequation_in _ _ 

       (p as (Const ("RatEq.is'_ratequation'_in",_) $ t $ v)) _  =

    if is_rateqation_in t v then 

	SOME ((term2str p) ^ " = True",

	      Trueprop $ (mk_equality (p, @{term True})))

    else SOME ((term2str p) ^ " = True",

	       Trueprop $ (mk_equality (p, @{term False})))

  | eval_is_ratequation_in _ _ _ _ = ((*tracing"### nichts matcht";*) NONE);

(*-------------------------rulse-----------------------*)

val RatEq_prls = (*15.10.02:just the following order due to subterm evaluation*)

  append_rls "RatEq_prls" e_rls 

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

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

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

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

	      Calc ("RatEq.is'_ratequation'_in",eval_is_ratequation_in ""),

	      Calc ("HOL.eq",eval_equal "#equal_"),

	      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})

	      ];

(*rls = merge_rls erls Poly_erls *)

val rateq_erls = 

    remove_rls "rateq_erls"                                   (*WN: ein Hack*)

	(merge_rls "is_ratequation_in" calculate_Rational

		   (append_rls "is_ratequation_in"

			Poly_erls

			[(*Calc ("Fields.inverse_class.divide", eval_cancel "#divide_e"),*)

			 Calc ("RatEq.is'_ratequation'_in",

			       eval_is_ratequation_in "")

			 ]))

	[Thm ("and_commute",num_str @{thm and_commute}), (*WN: ein Hack*)

	 Thm ("or_commute",num_str @{thm or_commute})    (*WN: ein Hack*)

	 ];

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

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

			 ]);

val RatEq_crls = 

    remove_rls "RatEq_crls"                                   (*WN: ein Hack*)

	(merge_rls "is_ratequation_in" calculate_Rational

		   (append_rls "is_ratequation_in"

			Poly_erls

			[(*Calc ("Fields.inverse_class.divide", eval_cancel "#divide_e"),*)

			 Calc ("RatEq.is'_ratequation'_in",

			       eval_is_ratequation_in "")

			 ]))

	[Thm ("and_commute",num_str @{thm and_commute}), (*WN: ein Hack*)

	 Thm ("or_commute",num_str @{thm or_commute})    (*WN: ein Hack*)

	 ];

val RatEq_eliminate = prep_rls(

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

       rew_ord = ("termlessI", termlessI), erls = rateq_erls, srls = Erls, 

       calc = [], errpatts = [],

       rules = [

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

	     (* a/b=c/d -> ad=cb *)

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

	     (* a  =c/d -> ad=c  *)

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

	     (* a/b=c   ->  a=cb *)

	    ],

    scr = Prog ((term_of o the o (parse thy)) "empty_script")

    }:rls);

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

			[("RatEq_eliminate",RatEq_eliminate)

			 ]);

val RatEq_simplify = prep_rls(

  Rls {id = "RatEq_simplify", preconds = [], rew_ord = ("termlessI", termlessI),

      erls = rateq_erls, srls = Erls, calc = [], errpatts = [],

    rules = [

	     Thm("real_rat_mult_1",num_str @{thm real_rat_mult_1}),

	     (*a*(b/c) = (a*b)/c*)

	     Thm("real_rat_mult_2",num_str @{thm real_rat_mult_2}),

	     (*(a/b)*(c/d) = (a*c)/(b*d)*)

             Thm("real_rat_mult_3",num_str @{thm real_rat_mult_3}),

             (* (a/b)*c = (a*c)/b*)

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

	     (*(a/b)^^^2 = a^^^2/b^^^2*)

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

	     (* a - b = a + (-1) * b *)

             Thm("rat_double_rat_1",num_str @{thm rat_double_rat_1}),

             (* (a / (c/d) = (a*d) / c) *)

             Thm("rat_double_rat_2",num_str @{thm rat_double_rat_2}), 

             (* ((a/b) / (c/d) = (a*d) / (b*c)) *)

             Thm("rat_double_rat_3",num_str @{thm rat_double_rat_3}) 

             (* ((a/b) / c = a / (b*c) ) *)

	     ],

    scr = Prog ((term_of o the o (parse thy)) "empty_script")

    }:rls);

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

			[("RatEq_simplify",RatEq_simplify)

			 ]);

(*-------------------------Problem-----------------------*)

(*

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

show_ptyps(); 

*)

store_pbt

 (prep_pbt thy "pbl_equ_univ_rat" [] e_pblID

 (["rational","univariate","equation"],

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

   ("#Where" ,["(e_e::bool) is_ratequation_in (v_v::real)"]),

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

  ],

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

  [["RatEq","solve_rat_equation"]]));

*}

ML {*

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

store_met

 (prep_met thy "met_rateq" [] e_metID

 (["RatEq"],

   [],

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

    crls=RatEq_crls, errpats = [], nrls = norm_Rational}, "empty_script"));

*}

ML {*

store_met

 (prep_met thy "met_rat_eq" [] e_metID

 (["RatEq", "solve_rat_equation"],

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

   ("#Where" ,["(e_e::bool) is_ratequation_in (v_v::real)"]),

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

  ],

   {rew_ord'="termlessI",

    rls'=rateq_erls,

    srls=e_rls,

    prls=RatEq_prls,

    calc=[],

    crls=RatEq_crls, errpats = [], nrls = norm_Rational},

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

    "(let e_e = ((Repeat(Try (Rewrite_Set RatEq_simplify      True))) @@  " ^

    "           (Repeat(Try (Rewrite_Set norm_Rational      False))) @@  " ^

    "           (Repeat(Try (Rewrite_Set add_fractions_p False))) @@  " ^

    "           (Repeat(Try (Rewrite_Set RatEq_eliminate     True)))) e_e;" ^

    " (L_L::bool list) = (SubProblem (RatEq',[univariate,equation], [no_met])" ^

    "                    [BOOL e_e, REAL v_v])                     " ^

    " in Check_elementwise L_L {(v_v::real). Assumptions})"

   ));

*}

ML {*

calclist':= overwritel (!calclist', 

   [("is_ratequation_in", ("RatEq.is_ratequation_in", 

			   eval_is_ratequation_in ""))

    ]);

*}

end