(* rationals, fractions of multivariate polynomials over the real field

    author: isac team

    Copyright (c) isac team 2002, 2013

    Use is subject to license terms.

    depends on Poly (and not on Atools), because 

    fractions with _normalised_ polynomials are canceled, added, etc.

 *)

 theory Rational 

 imports Poly GCD_Poly_ML

 begin

 section \<open>Constants for evaluation by \ <^rule_eval>\<close>

 consts

   is_expanded    :: "real => bool" ("_ is'_expanded")     (*RL->Poly.thy*)

   is_ratpolyexp  :: "real => bool" ("_ is'_ratpolyexp") 

   get_denominator :: "real => real"

   get_numerator   :: "real => real"           

 ML \<open>

 (*.the expression contains + - * ^ / only ?.*)

 fun is_ratpolyexp (Free _) = true

   | is_ratpolyexp (Const (\<^const_name>\<open>plus\<close>,_) $ Free _ $ Free _) = true

   | is_ratpolyexp (Const (\<^const_name>\<open>minus\<close>,_) $ Free _ $ Free _) = true

   | is_ratpolyexp (Const (\<^const_name>\<open>times\<close>,_) $ Free _ $ Free _) = true

   | is_ratpolyexp (Const (\<^const_name>\<open>realpow\<close>,_) $ Free _ $ Free _) = true

   | is_ratpolyexp (Const (\<^const_name>\<open>divide\<close>,_) $ Free _ $ Free _) = true

   | is_ratpolyexp (Const (\<^const_name>\<open>plus\<close>,_) $ t1 $ t2) = 

                ((is_ratpolyexp t1) andalso (is_ratpolyexp t2))

   | is_ratpolyexp (Const (\<^const_name>\<open>minus\<close>,_) $ t1 $ t2) = 

                ((is_ratpolyexp t1) andalso (is_ratpolyexp t2))

   | is_ratpolyexp (Const (\<^const_name>\<open>times\<close>,_) $ t1 $ t2) = 

                ((is_ratpolyexp t1) andalso (is_ratpolyexp t2))

   | is_ratpolyexp (Const (\<^const_name>\<open>realpow\<close>,_) $ t1 $ t2) = 

                ((is_ratpolyexp t1) andalso (is_ratpolyexp t2))

   | is_ratpolyexp (Const (\<^const_name>\<open>divide\<close>,_) $ t1 $ t2) = 

                ((is_ratpolyexp t1) andalso (is_ratpolyexp t2))

   | is_ratpolyexp t = if TermC.is_num t then true else false;

 (*("is_ratpolyexp", ("Rational.is_ratpolyexp", eval_is_ratpolyexp ""))*)

 fun eval_is_ratpolyexp (thmid:string) _ 

 		       (t as (Const (\<^const_name>\<open>Rational.is_ratpolyexp\<close>, _) $ arg)) ctxt =

     if is_ratpolyexp arg

     then SOME (TermC.mk_thmid thmid (UnparseC.term ctxt arg) "", 

 	         HOLogic.Trueprop $ (TermC.mk_equality (t, @{term True})))

     else SOME (TermC.mk_thmid thmid (UnparseC.term ctxt arg) "", 

 	         HOLogic.Trueprop $ (TermC.mk_equality (t, @{term False})))

   | eval_is_ratpolyexp _ _ _ _ = NONE; 

 (*("get_denominator", ("Rational.get_denominator", eval_get_denominator ""))*)

 fun eval_get_denominator (thmid:string) _ 

 		      (t as Const (\<^const_name>\<open>Rational.get_denominator\<close>, _) $

               (Const (\<^const_name>\<open>divide\<close>, _) $ _(*num*) $

                 denom)) ctxt = 

       SOME (TermC.mk_thmid thmid (UnparseC.term ctxt denom) "", 

 	            HOLogic.Trueprop $ (TermC.mk_equality (t, denom)))

   | eval_get_denominator _ _ _ _ = NONE; 

 (*("get_numerator", ("Rational.get_numerator", eval_get_numerator ""))*)

 fun eval_get_numerator (thmid:string) _ 

       (t as Const (\<^const_name>\<open>Rational.get_numerator\<close>, _) $

           (Const (\<^const_name>\<open>divide\<close>, _) $num

             $denom )) ctxt = 

     SOME (TermC.mk_thmid thmid (UnparseC.term ctxt num) "", 

 	    HOLogic.Trueprop $ (TermC.mk_equality (t, num)))

   | eval_get_numerator _ _ _ _ = NONE; 

 \<close>

 section \<open>Theorems for rewriting\<close>

 axiomatization (* naming due to Isabelle2002, but not contained in Isabelle2002; 

                   many thms are due to RL and can be removed with updating the equation solver;

                   TODO: replace by equivalent thms in recent Isabelle201x *) 

 where

   mult_cross:      "[| b ~= 0; d ~= 0 |] ==> (a / b = c / d) = (a * d = b * c)" and

   mult_cross1:     "   b ~= 0            ==> (a / b = c    ) = (a     = b * c)" and

   mult_cross2:     "           d ~= 0    ==> (a     = c / d) = (a * d =     c)" and

   add_minus:       "a + b - b = a"(*RL->Poly.thy*) and

   add_minus1:      "a - b + b = a"(*RL->Poly.thy*) and

   rat_mult:        "a / b * (c / d) = a * c / (b * d)"(*?Isa02*)  and

   rat_mult2:       "a / b *  c      = a * c /  b     "(*?Isa02*) and

   rat_mult_poly_l: "c is_polyexp ==> c * (a / b) = c * a /  b" and

   rat_mult_poly_r: "c is_polyexp ==> (a / b) * c = a * c /  b" and

 (*real_times_divide1_eq .. Isa02*) 

   real_times_divide_1_eq:  "-1 * (c / d) = -1 * c / d " and

   real_times_divide_num:   "a is_num ==> a * (c / d) = a * c / d " and

   real_mult_div_cancel2:   "k ~= 0 ==> m * k / (n * k) = m / n" and

 (*real_mult_div_cancel1:   "k ~= 0 ==> k * m / (k * n) = m / n"..Isa02*)

   real_divide_divide1:     "y ~= 0 ==> (u / v) / (y / z) = (u / v) * (z / y)" and

   real_divide_divide1_mg:  "y ~= 0 ==> (u / v) / (y / z) = (u * z) / (y * v)" and

 (*real_divide_divide2_eq:  "x / y / z = x / (y * z)"..Isa02*)

   rat_power:               "(a / b) \<up> n = (a \<up> n) / (b \<up> n)" and

   rat_add:         "[| a is_num; b is_num; c is_num; d is_num |] ==> 

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

   rat_add_assoc:   "[| a is_num; b is_num; c is_num; d is_num |] ==> 

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

   rat_add1:        "[| a is_num; b is_num; c is_num |] ==> 

 	           a / c + b / c = (a + b) / c" and

   rat_add1_assoc:   "[| a is_num; b is_num; c is_num |] ==> 

 	           a / c + (b / c + e) = (a + b) / c + e" and

   rat_add2:        "[| a is_num; b is_num; c is_num |] ==> 

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

   rat_add2_assoc:  "[| a is_num; b is_num; c is_num |] ==> 

 	           a / c + (b + e) = (a + b * c) / c + e" and

   rat_add3:        "[| a is_num; b is_num; c is_num |] ==> 

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

   rat_add3_assoc:   "[| a is_num; b is_num; c is_num |] ==> 

 	           a + (b / c + e) = (a * c + b) / c + e"

 section \<open>Cancellation and addition of fractions\<close>

 subsection \<open>Conversion term <--> poly\<close>

 subsubsection \<open>Convert a term to the internal representation of a multivariate polynomial\<close>

 ML \<open>

 fun monom_of_term vs (_, es) (t as Const (\<^const_name>\<open>zero_class.zero\<close>, _)) =

     (0, list_update es (find_index (curry op = t) vs) 1)

   | monom_of_term vs (c, es) (t as Const _) =

     (c, list_update es (find_index (curry op = t) vs) 1)

   | monom_of_term _ (c, es) (t as (Const (\<^const_name>\<open>numeral\<close>, _) $ _)) =

     (t |> HOLogic.dest_number |> snd |> curry op * c, es) (*several numerals in one monom*)

   | monom_of_term _ (c, es) (t as (Const (\<^const_name>\<open>uminus\<close>, _) $ _)) =

     (t |> HOLogic.dest_number |> snd |> curry op * c, es) (*several numerals in one monom*)

   | monom_of_term  vs (c, es) (t as Free _) =

     (c, list_update es (find_index (curry op = t) vs) 1)

   | monom_of_term  vs (c, es) (Const (\<^const_name>\<open>realpow\<close>, _) $ (b as Free _) $

       (e as Const (\<^const_name>\<open>numeral\<close>, _) $ _)) =

     (c, list_update es (find_index (curry op = b) vs) (e |> HOLogic.dest_number |> snd))

   | monom_of_term  vs (c, es) (Const (\<^const_name>\<open>realpow\<close>, _) $ (b as Free _) $

       (e as Const (\<^const_name>\<open>uminus\<close>, _) $ _)) =

     (c, list_update es (find_index (curry op = b) vs) (e |> HOLogic.dest_number |> snd))

   | monom_of_term vs (c, es) (Const (\<^const_name>\<open>times\<close>, _) $ m1 $ m2) =

     let val (c', es') = monom_of_term vs (c, es) m1

     in monom_of_term vs (c', es') m2 end

   | monom_of_term _ _ t = raise ERROR ("poly malformed 1 with " ^ UnparseC.term @{context} t)

 (*-------v------*)

 fun monoms_of_term vs (t as Const (\<^const_name>\<open>zero_class.zero\<close>, _)) =

     [monom_of_term  vs (0, replicate (length vs) 0) t]

   | monoms_of_term vs (t as Const _) =

     [monom_of_term  vs (1, replicate (length vs) 0) t]

   | monoms_of_term vs (t as Const (\<^const_name>\<open>numeral\<close>, _) $ _) =

     [monom_of_term  vs (1, replicate (length vs) 0) t]

   | monoms_of_term vs (t as Const (\<^const_name>\<open>uminus\<close>, _) $ _) =

     [monom_of_term  vs (1, replicate (length vs) 0) t]

   | monoms_of_term vs (t as Free _) =

     [monom_of_term  vs (1, replicate (length vs) 0) t]

   | monoms_of_term vs (t as Const (\<^const_name>\<open>realpow\<close>, _) $ _ $  _) =

     [monom_of_term  vs (1, replicate (length vs) 0) t]

   | monoms_of_term vs (t as Const (\<^const_name>\<open>times\<close>, _) $ _ $  _) =

     [monom_of_term  vs (1, replicate (length vs) 0) t]

   | monoms_of_term vs (Const (\<^const_name>\<open>plus\<close>, _) $ ms1 $ ms2) =

     (monoms_of_term vs ms1) @ (monoms_of_term vs ms2)

   | monoms_of_term _ t = raise ERROR ("poly malformed 2 with " ^ UnparseC.term @{context} t)

 (* convert a term to the internal representation of a multivariate polynomial;

   the conversion is quite liberal, see test --- fun poly_of_term ---:

 * the order of variables and the parentheses within a monomial are arbitrary

 * the coefficient may be somewhere

 * he order and the parentheses within monomials are arbitrary

 But the term must be completely expand + over * (laws of distributivity are not applicable).

 The function requires the free variables as strings already given, 

 because the gcd involves 2 polynomials (with the same length for their list of exponents).

 *)

 fun poly_of_term vs (t as Const (\<^const_name>\<open>plus\<close>, _) $ _ $ _) =

     (SOME (t |> monoms_of_term vs |> order)

       handle ERROR _ => NONE)

   | poly_of_term vs t =   (* 0 for zero polynomial *)

     (SOME [monom_of_term vs (1, replicate (length vs) 0) t]

       handle ERROR _ => NONE)

 fun is_poly t =

   let

     val vs = TermC.vars_of t

   in 

     case poly_of_term vs t of SOME _ => true | NONE => false

   end

 val is_expanded = is_poly   (* TODO: check names *)

 val is_polynomial = is_poly (* TODO: check names *)

 \<close>

 subsubsection \<open>Convert internal representation of a multivariate polynomial to a term\<close>

 ML \<open>

 fun term_of_es _ _ _ [] = [] (*assumes same length for vs and es*)

   | term_of_es baseT expT (_ :: vs) (0 :: es) = [] @ term_of_es baseT expT vs es

   | term_of_es baseT expT (v :: vs) (1 :: es) = v :: term_of_es baseT expT vs es

   | term_of_es baseT expT (v :: vs) (e :: es) =

     Const (\<^const_name>\<open>realpow\<close>, [baseT, expT] ---> baseT) $ v $ (HOLogic.mk_number expT e)

     :: term_of_es baseT expT vs es

   | term_of_es _ _ _ _ = raise ERROR "term_of_es: length vs <> length es"

 fun term_of_monom baseT expT vs ((c, es): monom) =

     let val es' = term_of_es baseT expT vs es

     in 

       if c = 1 

       then 

         if es' = [] (*if es = [0,0,0,...]*)

         then HOLogic.mk_number baseT c

         else foldl (HOLogic.mk_binop "Groups.times_class.times") (hd es', tl es')

       else foldl (HOLogic.mk_binop "Groups.times_class.times")

         (HOLogic.mk_number baseT c, es') 

     end

 fun term_of_poly baseT expT vs p =

   let val monos = map (term_of_monom baseT expT vs) p

   in foldl (HOLogic.mk_binop \<^const_name>\<open>plus\<close>) (hd monos, tl monos) end

 \<close>

 subsection \<open>Apply gcd_poly for cancelling and adding fractions as terms\<close>

 ML \<open>

 fun mk_noteq_0 baseT t = 

   Const (\<^const_name>\<open>Not\<close>, HOLogic.boolT --> HOLogic.boolT) $ 

     (Const (\<^const_name>\<open>HOL.eq\<close>, [baseT, baseT] ---> HOLogic.boolT) $ t $ HOLogic.mk_number HOLogic.realT 0)

 fun mk_asms baseT ts =

   let val as' = filter_out TermC.is_num ts (* asm like "2 ~= 0" is needless *)

   in map (mk_noteq_0 baseT) as' end

 \<close>

 subsubsection \<open>Factor out gcd for cancellation\<close>

 ML \<open>

 fun check_fraction t =

   case t of

     Const (\<^const_name>\<open>divide\<close>, _) $ numerator $ denominator

       => SOME (numerator, denominator)

   | _ => NONE

 (* prepare a term for cancellation by factoring out the gcd

   assumes: is a fraction with outmost "/"*)

 fun factout_p_ (_: theory) t =

   let val opt = check_fraction t

   in

     case opt of 

       NONE => NONE

     | SOME (numerator, denominator) =>

       let

         val vs = TermC.vars_of t

         val baseT = type_of numerator

         val expT = HOLogic.realT

       in

         case (poly_of_term vs numerator, poly_of_term vs denominator) of

           (SOME a, SOME b) =>

             let

               val ((a', b'), c) = gcd_poly a b

               val es = replicate (length vs) 0 

             in

               if c = [(1, es)] orelse c = [(~1, es)]

               then NONE

               else 

                 let

                   val b't = term_of_poly baseT expT vs b'

                   val ct = term_of_poly baseT expT vs c

                   val t' = 

                     HOLogic.mk_binop \<^const_name>\<open>divide\<close> 

                       (HOLogic.mk_binop \<^const_name>\<open>times\<close>

                         (term_of_poly baseT expT vs a', ct),

                        HOLogic.mk_binop \<^const_name>\<open>times\<close> (b't, ct))

                 in SOME (t', mk_asms baseT [b't, ct]) end

             end

         | _ => NONE : (term * term list) option

       end

   end

 \<close>

 subsubsection \<open>Cancel a fraction\<close>

 ML \<open>

 (* cancel a term by the gcd ("" denote terms with internal algebraic structure)

   cancel_p_ : theory \<Rightarrow> term  \<Rightarrow> (term \<times> term list) option

   cancel_p_ thy "a / b" = SOME ("a' / b'", ["b' \<noteq> 0"])

   assumes: a is_polynomial  \<and>  b is_polynomial  \<and>  b \<noteq> 0

   yields

     SOME ("a' / b'", ["b' \<noteq> 0"]). gcd_poly a b \<noteq> 1  \<and>  gcd_poly a b \<noteq> -1  \<and>  

       a' * gcd_poly a b = a  \<and>  b' * gcd_poly a b = b

     \<or> NONE *)

 fun cancel_p_ (_: theory) t =

   let val opt = check_fraction t

   in

     case opt of 

       NONE => NONE

     | SOME (numerator, denominator) =>

       let

         val vs = TermC.vars_of t

         val baseT = type_of numerator

         val expT = HOLogic.realT

       in

         case (poly_of_term vs numerator, poly_of_term vs denominator) of

           (SOME a, SOME b) =>

             let

               val ((a', b'), c) = gcd_poly a b

               val es = replicate (length vs) 0 

             in

               if c = [(1, es)] orelse c = [(~1, es)]

               then NONE

               else 

                 let

                   val bt' = term_of_poly baseT expT vs b'

                   val ct = term_of_poly baseT expT vs c

                   val t' = 

                     HOLogic.mk_binop \<^const_name>\<open>divide\<close> 

                       (term_of_poly baseT expT vs a', bt')

                   val asm = mk_asms baseT [bt']

                 in SOME (t', asm) end

             end

         | _ => NONE : (term * term list) option

       end

   end

 \<close>

 subsubsection \<open>Factor out to a common denominator for addition\<close>

 ML \<open>

 (* addition of fractions allows (at most) one non-fraction (a monomial) *)

 fun check_frac_sum 

     (Const (\<^const_name>\<open>plus\<close>, _) $ 

       (Const (\<^const_name>\<open>divide\<close>, _) $ n1 $ d1) $

       (Const (\<^const_name>\<open>divide\<close>, _) $ n2 $ d2))

     = SOME ((n1, d1), (n2, d2))

   | check_frac_sum 

     (Const (\<^const_name>\<open>plus\<close>, _) $ 

       nofrac $ 

       (Const (\<^const_name>\<open>divide\<close>, _) $ n2 $ d2))

     = SOME ((nofrac, HOLogic.mk_number HOLogic.realT 1), (n2, d2))

   | check_frac_sum 

     (Const (\<^const_name>\<open>plus\<close>, _) $ 

       (Const (\<^const_name>\<open>divide\<close>, _) $ n1 $ d1) $ 

       nofrac)

     = SOME ((n1, d1), (nofrac, HOLogic.mk_number HOLogic.realT 1))

   | check_frac_sum _ = NONE  

 (* prepare a term for addition by providing the least common denominator as a product

   assumes: is a term with outmost "+" and at least one outmost "/" in respective summands*)

 fun common_nominator_p_ (_: theory) t =

   let val opt = check_frac_sum t

   in

     case opt of 

       NONE => NONE

     | SOME ((n1, d1), (n2, d2)) =>

       let

         val vs = TermC.vars_of t

       in

         case (poly_of_term vs d1, poly_of_term vs d2) of

           (SOME a, SOME b) =>

             let

               val ((a', b'), c) = gcd_poly a b

               val (baseT, expT) = (type_of n1, HOLogic.realT)

               val [d1', d2', c'] = map (term_of_poly baseT expT vs) [a', b', c]

               (*----- minimum of parentheses & nice result, but breaks tests: -------------

               val denom = HOLogic.mk_binop \<^const_name>\<open>times\<close> 

                 (HOLogic.mk_binop \<^const_name>\<open>times\<close> (d1', d2'), c') -------------*)

               val denom =

                 if c = [(1, replicate (length vs) 0)]

                 then HOLogic.mk_binop \<^const_name>\<open>times\<close> (d1', d2')

                 else

                   HOLogic.mk_binop \<^const_name>\<open>times\<close> (c',

                   HOLogic.mk_binop \<^const_name>\<open>times\<close> (d1', d2')) (*--------------*)

               val t' =

                 HOLogic.mk_binop \<^const_name>\<open>plus\<close>

                   (HOLogic.mk_binop \<^const_name>\<open>divide\<close>

                     (HOLogic.mk_binop \<^const_name>\<open>times\<close> (n1, d2'), denom),

                   HOLogic.mk_binop \<^const_name>\<open>divide\<close> 

                     (HOLogic.mk_binop \<^const_name>\<open>times\<close> (n2, d1'), denom))

               val asm = mk_asms baseT [d1', d2', c']

             in SOME (t', asm) end

         | _ => NONE : (term * term list) option

       end

   end

 \<close>

 subsubsection \<open>Addition of at least one fraction within a sum\<close>

 ML \<open>

 (* add fractions

   assumes: is a term with outmost "+" and at least one outmost "/" in respective summands

   \<and> NONE of the summands is zero.

   NOTE: the case "(_ + _) + _" need not be considered due to iterated addition *)

 fun add_fraction_p_ (_: theory) t =

   case check_frac_sum t of 

     NONE => NONE

   | SOME ((n1, d1), (n2, d2)) =>

     let

       val vs = TermC.vars_of t

     in

       case (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2) of

         (SOME _, SOME a, SOME _, SOME b) =>

           let

             val ((a', b'), c) = gcd_poly a b

             val (baseT, expT) = (type_of n1, HOLogic.realT)

             val nomin = term_of_poly baseT expT vs 

               (((the (poly_of_term vs n1)) %%*%% b') %%+%% ((the (poly_of_term vs n2)) %%*%% a')) 

             val denom = term_of_poly baseT expT vs ((c %%*%% a') %%*%% b')

             val t' = HOLogic.mk_binop \<^const_name>\<open>divide\<close> (nomin, denom)

           in SOME (t', mk_asms baseT [denom]) end

       | _ => NONE : (term * term list) option

     end

 \<close>

 section \<open>Embed cancellation and addition into rewriting\<close>

 subsection \<open>Rulesets and predicate for embedding\<close>

 ML \<open>

 (* evaluates conditions in calculate_Rational *)

 val calc_rat_erls =

   prep_rls'

     (Rule_Def.Repeat {id = "calc_rat_erls", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.function_empty), 

       asm_rls = Rule_Set.Empty, prog_rls = Rule_Set.Empty, calc = [], errpatts = [],

       rules = [

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

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

         \<^rule_eval>\<open>is_num\<close> (Prog_Expr.eval_is_num "#is_num_"),

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

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

       program = Rule.Empty_Prog});

 (* simplifies expressions with numerals;

    does NOT rearrange the term by AC-rewriting; thus terms with variables 

    need to have constants to be commuted together respectively           *)

 val calculate_Rational =

   prep_rls' (Rule_Set.merge "calculate_Rational"

     (Rule_Def.Repeat {id = "divide", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.function_empty), 

       asm_rls = calc_rat_erls, prog_rls = Rule_Set.Empty,

       calc = [], errpatts = [],

       rules = [

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

         \<^rule_thm_sym>\<open>minus_divide_left\<close>, (*SYM - ?x / ?y = - (?x / ?y)  may come from subst*)

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

           (*"[| a is_num; b is_num; c is_num; d is_num |] ==> a / c + b / d = (a * d) / (c * d) + (b * c ) / (d * c)"*)

         \<^rule_thm>\<open>rat_add1\<close>, (*"[| a is_num; b is_num; c is_num |] ==> a / c + b / c = (a + b) / c"*)

         \<^rule_thm>\<open>rat_add2\<close>, (*"[| ?a is_num; ?b is_num; ?c is_num |] ==> ?a / ?c + ?b = (?a + ?b * ?c) / ?c"*)

         \<^rule_thm>\<open>rat_add3\<close>, (*"[| a is_num; b is_num; c is_num |] ==> a + b / c = (a * c) / c + b / c" *)

         \<^rule_thm>\<open>rat_mult\<close>, (*a / b * (c / d) = a * c / (b * d)*)

         \<^rule_thm>\<open>times_divide_eq_right\<close>, (*?x * (?y / ?z) = ?x * ?y / ?z*)

         \<^rule_thm>\<open>times_divide_eq_left\<close>, (*?y / ?z * ?x = ?y * ?x / ?z*)

         \<^rule_thm>\<open>real_divide_divide1\<close>, (*"?y ~= 0 ==> ?u / ?v / (?y / ?z) = ?u / ?v * (?z / ?y)"*)

         \<^rule_thm>\<open>divide_divide_eq_left\<close>, (*"?x / ?y / ?z = ?x / (?y * ?z)"*)

         \<^rule_thm>\<open>rat_power\<close>, (*"(?a / ?b)  \<up>  ?n = ?a  \<up>  ?n / ?b  \<up>  ?n"*)

         \<^rule_thm>\<open>mult_cross\<close>,  (*"[| b ~= 0; d ~= 0 |] ==> (a / b = c / d) = (a * d = b * c)*)

         \<^rule_thm>\<open>mult_cross1\<close>, (*"   b ~= 0            ==> (a / b = c    ) = (a     = b * c)*)

         \<^rule_thm>\<open>mult_cross2\<close>  (*"           d ~= 0    ==> (a     = c / d) = (a * d =     c)*)], 

       program = Rule.Empty_Prog})

     calculate_Poly);

 (*("is_expanded", ("Rational.is_expanded", eval_is_expanded ""))*)

 fun eval_is_expanded (thmid:string) _ 

 		       (t as (Const (\<^const_name>\<open>Rational.is_expanded\<close>, _) $ arg)) ctxt = 

     if is_expanded arg

     then SOME (TermC.mk_thmid thmid (UnparseC.term ctxt arg) "", 

 	         HOLogic.Trueprop $ (TermC.mk_equality (t, @{term True})))

     else SOME (TermC.mk_thmid thmid (UnparseC.term ctxt arg) "", 

 	         HOLogic.Trueprop $ (TermC.mk_equality (t, @{term False})))

   | eval_is_expanded _ _ _ _ = NONE;

 \<close>

 calculation is_expanded = \<open>eval_is_expanded ""\<close>

 ML \<open>

 val rational_erls = 

   Rule_Set.merge "rational_erls" calculate_Rational 

     (Rule_Set.append_rules "is_expanded" Atools_erls [

       \<^rule_eval>\<open>is_expanded\<close> (eval_is_expanded "")]);

 \<close>

 subsection \<open>Embed cancellation into rewriting\<close>

 ML \<open>

 (**)local (* cancel_p *)

 val {rules = rules, rew_ord = (_, ro), ...} = Rule_Set.rep (get_rls @{context} "rev_rew_p");

 fun init_state thy eval_rls ro t =

   let

     val SOME (t', _) = factout_p_ thy t;

     val SOME (t'', asm) = cancel_p_ thy t;

     val der = Derive.steps_reverse (Proof_Context.init_global thy) eval_rls rules ro NONE t';

     val der = der @ 

       [(\<^rule_thm>\<open>real_mult_div_cancel2\<close>, (t'', asm))]

     val rs = (Rule.distinct' o (map #1)) der

   	val rs = filter_out (ThmC.member'

   	  ["sym_real_add_zero_left", "sym_real_mult_0", "sym_real_mult_1"]) rs

   in (t, t'', [rs(*one in order to ease locate_rule*)], der) end;

 fun locate_rule thy eval_rls ro [rs] t r =

     if member op = ((map (Rule.thm_id)) rs) (Rule.thm_id r)

     then 

       let

         val ctxt = Proof_Context.init_global thy

         val ropt = Rewrite.rewrite_ ctxt ro eval_rls true (Rule.thm ctxt r) t;

       in

         case ropt of SOME ta => [(r, ta)]

 	      | NONE => ((*tracing 

 	          ("### locate_rule:  rewrite " ^ Rule.thm_id r ^ " " ^ UnparseC.term @{context} t ^ " = NONE");*) []) 

 			end

     else ((*tracing ("### locate_rule:  " ^ Rule.thm_id r ^ " not mem rrls");*) [])

   | locate_rule _ _ _ _ _ _ = raise ERROR "locate_rule: doesnt match rev-sets in istate";

 fun next_rule thy eval_rls ro [rs] t =

     let

       val der = Derive.do_one (Proof_Context.init_global thy) eval_rls rs ro NONE t;

     in case der of (_, r, _) :: _ => SOME r | _ => NONE end

   | next_rule _ _ _ _ _ = raise ERROR ("next_rule: doesnt match rev-sets in istate");

 fun attach_form (_: Rule.rule list list) (_: term) (_: term) = 

   [(*TODO*)]: ( Rule.rule * (term * term list)) list;

 (**)in(**)

 val cancel_p = 

   Rule_Set.Rrls {id = "cancel_p", prepat = [],

 	rew_ord = ("ord_make_polynomial", ord_make_polynomial false),

 	asm_rls = rational_erls, 

 	calc = 

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

 	  ("TIMES" , (\<^const_name>\<open>times\<close>, (**)Calc_Binop.numeric "#mult_")),

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

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

     errpatts = [],

 	program =

 	  Rule.Rfuns {init_state  = init_state \<^theory> Atools_erls ro,

 		normal_form = cancel_p_ \<^theory>, 

 		locate_rule = locate_rule \<^theory> Atools_erls ro,

 		next_rule   = next_rule \<^theory> Atools_erls ro,

 		attach_form = attach_form}}

 (**)end(* local cancel_p *)

 \<close>

 subsection \<open>Embed addition into rewriting\<close>

 ML \<open>

 (**)local (* add_fractions_p *)

 (*val {rules = rules, rew_ord = (_, ro), ...} = Rule_Set.rep (get_rls "make_polynomial");*)

 val {rules, rew_ord = (_, ro), ...} = Rule_Set.rep (get_rls @{context} "rev_rew_p");

 fun init_state thy eval_rls ro t =

   let 

     val SOME (t', _) = common_nominator_p_ thy t;

     val SOME (t'', asm) = add_fraction_p_ thy t;

     val der = Derive.steps_reverse (Proof_Context.init_global thy) eval_rls rules ro NONE t';

     val der = der @ 

       [(\<^rule_thm>\<open>real_mult_div_cancel2\<close>, (t'',asm))]

     val rs = (Rule.distinct' o (map #1)) der;

     val rs = filter_out (ThmC.member'

       ["sym_real_add_zero_left", "sym_real_mult_0", "sym_real_mult_1"]) rs;

   in (t, t'', [rs(*here only _ONE_*)], der) end;

 fun locate_rule thy eval_rls ro [rs] t r =

     if member op = ((map (Rule.thm_id)) rs) (Rule.thm_id r)

     then 

       let

         val ctxt = Proof_Context.init_global thy

         val ropt = Rewrite.rewrite_ ctxt ro eval_rls true (Rule.thm ctxt r) t;

       in 

         case ropt of

           SOME ta => [(r, ta)]

 	      | NONE => 

 	        ((*tracing ("### locate_rule:  rewrite " ^ Rule.thm_id r ^ " " ^ UnparseC.term @{context} t ^ " = NONE");*)

 	        []) end

     else ((*tracing ("### locate_rule:  " ^ Rule.thm_id r ^ " not mem rrls");*) [])

   | locate_rule _ _ _ _ _ _ = raise ERROR "locate_rule: doesnt match rev-sets in istate";

 fun next_rule thy eval_rls ro [rs] t =

     let val der = Derive.do_one (Proof_Context.init_global thy) eval_rls rs ro NONE t;

     in 

       case der of

 	      (_,r,_)::_ => SOME r

 	    | _ => NONE

     end

   | next_rule _ _ _ _ _ = raise ERROR ("next_rule: doesnt match rev-sets in istate");

 val SOME pat0 = ParseC.patt_opt \<^theory> "?r/?s+?u/?v :: real";

 val SOME pat1 = ParseC.patt_opt \<^theory> "?r/?s+?u    :: real";

 val SOME pat2 = ParseC.patt_opt \<^theory> "?r   +?u/?v :: real";

 val prepat = [([@{term True}], pat0),

 	      ([@{term True}], pat1),

 	      ([@{term True}], pat2)];

 (**)in(**)

 val add_fractions_p =

   Rule_Set.Rrls {id = "add_fractions_p", prepat=prepat,

     rew_ord = ("ord_make_polynomial", ord_make_polynomial false),

     asm_rls = rational_erls,

     calc = [("PLUS", (\<^const_name>\<open>plus\<close>, (**)Calc_Binop.numeric "#add_")),

       ("TIMES", (\<^const_name>\<open>times\<close>, (**)Calc_Binop.numeric "#mult_")),

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

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

     errpatts = [],

     program = Rule.Rfuns {init_state  = init_state \<^theory> Atools_erls ro,

       normal_form = add_fraction_p_ \<^theory>,

       locate_rule = locate_rule \<^theory> Atools_erls ro,

       next_rule   = next_rule \<^theory> Atools_erls ro,

       attach_form = attach_form}}

 (**)end(*local add_fractions_p *)

 \<close>

 subsection \<open>Cancelling and adding all occurrences in a term /////////////////////////////\<close>

 ML \<open>

 (*copying cancel_p_rls + add her caused error in interface.sml*)

 \<close>

 section \<open>Rulesets for general simplification\<close>

 ML \<open>

 (*.all powers over + distributed; atoms over * collected, other distributed

    contains absolute minimum of thms for context in norm_Rational .*)

 val powers = prep_rls'(

   Rule_Def.Repeat {id = "powers", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.function_empty), 

       asm_rls = powers_erls, prog_rls = Rule_Set.Empty, calc = [], errpatts = [],

       rules = [

         \<^rule_thm>\<open>realpow_multI\<close>, (*"(r * s)  \<up>  n = r  \<up>  n * s  \<up>  n"*)

         \<^rule_thm>\<open>realpow_pow\<close>, (*"(a  \<up>  b)  \<up>  c = a  \<up>  (b * c)"*)

         \<^rule_thm>\<open>realpow_oneI\<close>, (*"r  \<up>  1 = r"*)

         \<^rule_thm>\<open>realpow_minus_even\<close>, (*"n is_even ==> (- r)  \<up>  n = r  \<up>  n" ?-->discard_minus?*)

         \<^rule_thm>\<open>realpow_minus_odd\<close>, (*"Not (n is_even) ==> (- r)  \<up>  n = -1 * r  \<up>  n"*)

         (*----- collect atoms over * -----*)

         \<^rule_thm>\<open>realpow_two_atom\<close>, (*"r is_atom ==> r * r = r  \<up>  2"*)

         \<^rule_thm>\<open>realpow_plus_1\<close>, (*"r is_atom ==> r * r  \<up>  n = r  \<up>  (n + 1)"*)

         \<^rule_thm>\<open>realpow_addI_atom\<close>, (*"r is_atom ==> r  \<up>  n * r  \<up>  m = r  \<up>  (n + m)"*)

         (*----- distribute none-atoms -----*)

         \<^rule_thm>\<open>realpow_def_atom\<close>, (*"[| 1 < n; ~ (r is_atom) |]==>r  \<up>  n = r * r  \<up>  (n + -1)"*)

         \<^rule_thm>\<open>realpow_eq_oneI\<close>, (*"1  \<up>  n = 1"*)

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

       program = Rule.Empty_Prog});

 \<close> ML \<open>

 (*.contains absolute minimum of thms for context in norm_Rational.*)

 val rat_mult_divide = prep_rls'(

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

       rew_ord = ("dummy_ord", Rewrite_Ord.function_empty), 

       asm_rls = Rule_Set.empty, prog_rls = Rule_Set.Empty, calc = [], errpatts = [],

       rules = [

        \<^rule_thm>\<open>rat_mult\<close>, (*(1)"?a / ?b * (?c / ?d) = ?a * ?c / (?b * ?d)"*)

 	       \<^rule_thm>\<open>times_divide_eq_right\<close>, (*(2)"?a * (?c / ?d) = ?a * ?c / ?d" must be [2],

 	         otherwise inv.to a / b / c = ...*)

 	       \<^rule_thm>\<open>times_divide_eq_left\<close>, (*"?a / ?b * ?c = ?a * ?c / ?b" order weights x \<up> n too much

 		       and does not commute a / b * c  \<up>  2 !*)

 	       \<^rule_thm>\<open>divide_divide_eq_right\<close>, (*"?x / (?y / ?z) = ?x * ?z / ?y"*)

 	       \<^rule_thm>\<open>divide_divide_eq_left\<close>, (*"?x / ?y / ?z = ?x / (?y * ?z)"*)

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

       program = Rule.Empty_Prog});

 \<close> ML \<open>

 (*.contains absolute minimum of thms for context in norm_Rational.*)

 val reduce_0_1_2 = prep_rls'(

   Rule_Def.Repeat{id = "reduce_0_1_2", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.function_empty),

     asm_rls = Rule_Set.empty, prog_rls = Rule_Set.Empty, calc = [], errpatts = [],

     rules = [

       \<^rule_thm>\<open>div_by_1\<close>, (*"?x / 1 = ?x"*)

 	    \<^rule_thm>\<open>mult_1_left\<close>, (*"1 * z = z"*)

 	    \<^rule_thm>\<open>minus_zero\<close>, (*"- 0 = 0"*)

 	    (*\<^rule_thm>\<open>real_mult_minus1\<close>, "-1 * z = - z"*)

 	    (*\<^rule_thm>\<open>real_minus_mult_cancel\<close>, "- ?x * - ?y = ?x * ?y"*)

       \<^rule_thm>\<open>mult_zero_left\<close>, (*"0 * z = 0"*)

       \<^rule_thm>\<open>add_0_left\<close>, (*"0 + z = z"*)

       (*\<^rule_thm>\<open>right_minus\<close>, "?z + - ?z = 0"*)

       \<^rule_thm_sym>\<open>real_mult_2\<close>,	 (*"z1 + z1 = 2 * z1"*)

       \<^rule_thm>\<open>real_mult_2_assoc\<close>, (*"z1 + (z1 + k) = 2 * z1 + k"*)

       \<^rule_thm>\<open>division_ring_divide_zero\<close> (*"0 / ?x = 0"*)],

     program = Rule.Empty_Prog});

 (*asm_rls for calculate_Rational; 

   make local with FIXX@ME result:term *term list WN0609???SKMG*)

 val norm_rat_erls = prep_rls'(

   Rule_Def.Repeat {id = "norm_rat_erls", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.function_empty), 

     asm_rls = Rule_Set.empty, prog_rls = Rule_Set.Empty, calc = [], errpatts = [],

     rules = [

       \<^rule_eval>\<open>Prog_Expr.is_num\<close> (Prog_Expr.eval_is_num "#is_num_")

 ],

     program = Rule.Empty_Prog});

 \<close> ML \<open>

 (* consists of rls containing the absolute minimum of thms *)

 (*

   which is now replaced by MGs version "norm_Rational" below

 *)

 val norm_Rational_min = prep_rls'(

   Rule_Def.Repeat {id = "norm_Rational_min", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.function_empty), 

     asm_rls = norm_rat_erls, prog_rls = Rule_Set.Empty, calc = [], errpatts = [],

     rules = [(*sequence given by operator precedence*)

 	     Rule.Rls_ discard_minus,

 	     Rule.Rls_ powers,

 	     Rule.Rls_ rat_mult_divide,

 	     Rule.Rls_ expand,

 	     Rule.Rls_ reduce_0_1_2,

 	     Rule.Rls_ order_add_mult,

 	     Rule.Rls_ collect_numerals,

 	     Rule.Rls_ add_fractions_p,

 	     Rule.Rls_ cancel_p

 	     ],

     program = Rule.Empty_Prog});

 \<close> ML \<open>

 val norm_Rational_parenthesized = prep_rls'(

   Rule_Set.Sequence {id = "norm_Rational_parenthesized", preconds = []:term list, 

     rew_ord = ("dummy_ord", Rewrite_Ord.function_empty),

     asm_rls = Atools_erls, prog_rls = Rule_Set.Empty,

     calc = [], errpatts = [],

     rules = [

       Rule.Rls_ norm_Rational_min,

 	    Rule.Rls_ discard_parentheses],

     program = Rule.Empty_Prog});      

 (*WN030318???SK: simplifies all but cancel and common_nominator*)

 val simplify_rational = 

   Rule_Set.merge "simplify_rational" expand_binoms

     (Rule_Set.append_rules "divide" calculate_Rational [

       \<^rule_thm>\<open>div_by_1\<close>, (*"?x / 1 = ?x"*)

       \<^rule_thm>\<open>rat_mult\<close>, (*(1)"?a / ?b * (?c / ?d) = ?a * ?c / (?b * ?d)"*)

       \<^rule_thm>\<open>times_divide_eq_right\<close>, (*(2)"?a * (?c / ?d) = ?a * ?c / ?d" must be [2],

   		   otherwise inv.to a / b / c = ...*)

       \<^rule_thm>\<open>times_divide_eq_left\<close>, (*"?a / ?b * ?c = ?a * ?c / ?b"*)

       \<^rule_thm>\<open>add_minus\<close>, (*"?a + ?b - ?b = ?a"*)

       \<^rule_thm>\<open>add_minus1\<close>, (*"?a - ?b + ?b = ?a"*)

       \<^rule_thm>\<open>divide_minus1\<close> (*"?x / -1 = - ?x"*)]);

 val add_fractions_p_rls = prep_rls'(

   Rule_Def.Repeat {id = "add_fractions_p_rls", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.function_empty), 

 	  asm_rls = Rule_Set.empty, prog_rls = Rule_Set.Empty, calc = [], errpatts = [],

 	  rules = [Rule.Rls_ add_fractions_p], 

 	  program = Rule.Empty_Prog});

 (* "Rule_Def.Repeat" causes repeated application of cancel_p to one and the same term *)

 val cancel_p_rls = prep_rls'(

   Rule_Def.Repeat 

     {id = "cancel_p_rls", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.function_empty), 

     asm_rls = Rule_Set.empty, prog_rls = Rule_Set.Empty, calc = [], errpatts = [],

     rules = [Rule.Rls_ cancel_p], 

 	  program = Rule.Empty_Prog});

 (*. makes 'normal' fractions; 'is_polyexp' inhibits double fractions;

     used in initial part norm_Rational_mg, see example DA-M02-main.p.60.*)

 val rat_mult_poly = prep_rls'(

   Rule_Def.Repeat {id = "rat_mult_poly", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.function_empty), 

 	  asm_rls = Rule_Set.append_rules "Rule_Set.empty-is_polyexp" Rule_Set.empty [

      \<^rule_eval>\<open>is_polyexp\<close> (eval_is_polyexp "")],

 	  prog_rls = Rule_Set.Empty, calc = [], errpatts = [],

 	  rules = [

       \<^rule_thm>\<open>rat_mult_poly_l\<close>, (*"?c is_polyexp ==> ?c * (?a / ?b) = ?c * ?a / ?b"*)

 	    \<^rule_thm>\<open>rat_mult_poly_r\<close> (*"?c is_polyexp ==> ?a / ?b * ?c = ?a * ?c / ?b"*)], 

 	  program = Rule.Empty_Prog});

 (*. makes 'normal' fractions; 'is_polyexp' inhibits double fractions;

     used in looping part norm_Rational_rls, see example DA-M02-main.p.60 

     .. WHERE THE LATTER DOES ALWAYS WORK, BECAUSE asm_rls = Rule_Set.empty, 

     I.E. THE RESPECTIVE ASSUMPTION IS STORED AND Rule.Thm APPLIED; WN051028 

     ... WN0609???MG.*)

 val rat_mult_div_pow = prep_rls'(

   Rule_Def.Repeat {id = "rat_mult_div_pow", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.function_empty), 

     asm_rls = Rule_Set.empty, prog_rls = Rule_Set.Empty, calc = [], errpatts = [],

     rules = [

       \<^rule_thm>\<open>rat_mult\<close>, (*"?a / ?b * (?c / ?d) = ?a * ?c / (?b * ?d)"*)

       \<^rule_thm>\<open>rat_mult_poly_l\<close>, (*"?c is_polyexp ==> ?c * (?a / ?b) = ?c * ?a / ?b"*)

       \<^rule_thm>\<open>rat_mult_poly_r\<close>, (*"?c is_polyexp ==> ?a / ?b * ?c = ?a * ?c / ?b"*)

       \<^rule_thm>\<open>real_divide_divide1_mg\<close>, (*"y ~= 0 ==> (u / v) / (y / z) = (u * z) / (y * v)"*)

       \<^rule_thm>\<open>divide_divide_eq_right\<close>, (*"?x / (?y / ?z) = ?x * ?z / ?y"*)

       \<^rule_thm>\<open>divide_divide_eq_left\<close>, (*"?x / ?y / ?z = ?x / (?y * ?z)"*)

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

       \<^rule_thm>\<open>rat_power\<close> (*"(?a / ?b)  \<up>  ?n = ?a  \<up>  ?n / ?b  \<up>  ?n"*)],

     program = Rule.Empty_Prog});

 val rat_reduce_1 = prep_rls'(

   Rule_Def.Repeat {id = "rat_reduce_1", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.function_empty), 

     asm_rls = Rule_Set.empty, prog_rls = Rule_Set.Empty, calc = [], errpatts = [], 

     rules = [

       \<^rule_thm>\<open>div_by_1\<close>, (*"?x / 1 = ?x"*)

       \<^rule_thm>\<open>mult_1_left\<close> (*"1 * z = z"*)],

     program = Rule.Empty_Prog});

 (* looping part of norm_Rational *)

 val norm_Rational_rls = prep_rls' (

   Rule_Def.Repeat {id = "norm_Rational_rls", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.function_empty), 

     asm_rls = norm_rat_erls, prog_rls = Rule_Set.Empty, calc = [], errpatts = [],

     rules = [

       Rule.Rls_ add_fractions_p_rls,

       Rule.Rls_ rat_mult_div_pow,

       Rule.Rls_ make_rat_poly_with_parentheses,

       Rule.Rls_ cancel_p_rls,

       Rule.Rls_ rat_reduce_1],

     program = Rule.Empty_Prog});

 val norm_Rational = prep_rls' (

   Rule_Set.Sequence 

     {id = "norm_Rational", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.function_empty), 

     asm_rls = norm_rat_erls, prog_rls = Rule_Set.Empty, calc = [], errpatts = [],

     rules = [

       Rule.Rls_ discard_minus,

       Rule.Rls_ rat_mult_poly,             (* removes double fractions like a/b/c *)

       Rule.Rls_ make_rat_poly_with_parentheses,

       Rule.Rls_ cancel_p_rls,

       Rule.Rls_ norm_Rational_rls,         (* the main rls, looping (#) *)

       Rule.Rls_ discard_parentheses1],     (* mult only *)

     program = Rule.Empty_Prog});

 \<close>

 rule_set_knowledge

   calculate_Rational = calculate_Rational and

   calc_rat_erls = calc_rat_erls and

   rational_erls = rational_erls and

   cancel_p = cancel_p and

   add_fractions_p = add_fractions_p and

   add_fractions_p_rls = add_fractions_p_rls and

   powers_erls = powers_erls and

   powers = powers and

   rat_mult_divide = rat_mult_divide and

   reduce_0_1_2 = reduce_0_1_2 and

   rat_reduce_1 = rat_reduce_1 and

   norm_rat_erls = norm_rat_erls and

   norm_Rational = norm_Rational and

   norm_Rational_rls = norm_Rational_rls and

   norm_Rational_min = norm_Rational_min and

   norm_Rational_parenthesized = norm_Rational_parenthesized and

   rat_mult_poly = rat_mult_poly and

   rat_mult_div_pow = rat_mult_div_pow and

   cancel_p_rls = cancel_p_rls

 section \<open>A problem for simplification of rationals\<close>

 problem pbl_simp_rat : "rational/simplification" =

   \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>

   Method_Ref: "simplification/of_rationals"

   CAS: "Simplify t_t"

   Given: "Term t_t"

   Where: "t_t is_ratpolyexp"

   Find: "normalform n_n"

 section \<open>A methods for simplification of rationals\<close>

 (*WN061025 this methods script is copied from (auto-generated) script

   of norm_Rational in order to ease repair on inform*)

 partial_function (tailrec) simplify :: "real \<Rightarrow> real"

   where

 "simplify term = (

   (Try (Rewrite_Set ''discard_minus'') #>

    Try (Rewrite_Set ''rat_mult_poly'') #>

    Try (Rewrite_Set ''make_rat_poly_with_parentheses'') #>

    Try (Rewrite_Set ''cancel_p_rls'') #>

    (Repeat (

      Try (Rewrite_Set ''add_fractions_p_rls'') #>

      Try (Rewrite_Set ''rat_mult_div_pow'') #>

      Try (Rewrite_Set ''make_rat_poly_with_parentheses'') #>

      Try (Rewrite_Set ''cancel_p_rls'') #>

      Try (Rewrite_Set ''rat_reduce_1''))) #>

    Try (Rewrite_Set ''discard_parentheses1''))

    term)"

 method met_simp_rat : "simplification/of_rationals" =

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

     where_rls = Rule_Set.append_rules "simplification_of_rationals_prls" Rule_Set.empty 

       [(*for preds in where_*) \<^rule_eval>\<open>is_ratpolyexp\<close> (eval_is_ratpolyexp "")],

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

   Program: simplify.simps

   Given: "Term t_t"

   Where: "t_t is_ratpolyexp"

   Find: "normalform n_n"

 section \<open>Error_Patterns\<close>

 declare [[check_unique = false]]

 setup \<open>Know_Store.add_mets @{context}

   [Pbl_Met_Hierarchy.add_errpats @{theory} ["simplification", "of_rationals"]

       [("addition-of-fractions",

         [ParseC.patt_opt @{theory} "(?a / ?b + ?c / ?d) = (?a + ?c)/(?b + ?d)" |> the,

          ParseC.patt_opt @{theory} "(?a / ?b + ?c / ?d) = (?a + ?c)/(?b * ?d)" |> the,

          ParseC.patt_opt @{theory} "(?a / ?b + ?c / ?d) = (?c * ?a)/(?d * ?b)" |> the,

          ParseC.patt_opt @{theory} "?a + (?b /?c)= (?a + ?b)/ ?c" |> the,

          ParseC.patt_opt @{theory} "?a + (?b /?c)= (?a * ?b)/ ?c" |> the],

         [@{thm rat_add}, @{thm rat_add_assoc}, @{thm rat_add1}, @{thm rat_add1_assoc},

           @{thm rat_add2}, @{thm rat_add2_assoc}, @{thm rat_add3}, @{thm rat_add3_assoc}])]

   ]\<close>

 declare [[check_unique = true]]

 (*TODO insert_fillpats ["IsacKnowledge", "Rational", "Theorems", "???"]*)

 (* TODO insert_errpats overwrites instead of appending

 insert_errpats ["simplification", "of_rationals"]

  (("cancel", (*see master thesis gdarocy*)

     [(*a*)parse_patt @{theory Rational} "(?a + ?b)/?a = ?b",

      (*b*)parse_patt @{theory Rational} "(?a + ?b)/?b = ?a",

      (*c*)parse_patt @{theory Rational} "(?a + ?b)/(?b + ?c) = ?a / ?c",

      (*d*)parse_patt @{theory Rational} "(?a + ?c)/(?b + ?c) = ?a / ?b",

      (*e*)parse_patt @{theory Rational} "(?a + ?c)/(?b + ?c) = ?a / ?c",

      (*f*)parse_patt @{theory Rational} "(?a + ?b)/(?c + ?a) = ?b / ?c",

      (*g*)parse_patt @{theory Rational} "(?a*?b + ?c)/?a = ?b + ?c",

      (*h*)parse_patt @{theory Rational} "(?a*?b + ?c)/?b = ?a + ?c",

      (*i*)parse_patt @{theory Rational} "(?a + ?b*?c)/?b = ?a + ?c",

      (*j*)parse_patt @{theory Rational} "(?a + ?b*?c)/?b = ?a + ?b",

      (*k*)parse_patt @{theory Rational} "(?a + ?b*?e)/(?c + ?d*?e) = (?a + ?b)/(?c + ?d)"],

     [@{thm real_times_divide_1_eq}, (*TODO*) @{thm real_times_divide_1_eq}]));

 val fillpats = [

   ("fill-cancel-left-num",

     parse_patt @{theory Rational} "(?a * ?b) / (?a * ?c) = _ / ?c", "cancel"),

   ("fill-cancel-left-den",

     parse_patt @{theory Rational} "(?a * ?b) / (?a * ?c) = ?b / _", "cancel"),

   ("fill-cancel-none",

     parse_patt @{theory Rational} "(?a * ?b) / (?a * ?c) = ?b / ?c", "cancel"),

   ("fill-cancel-left-add-num1",

     parse_patt @{theory Rational} "(?a * (?b + ?c)) / (?a * ?d) = (_ + ?c) / ?d", "cancel"),

   ("fill-cancel-left-add-num2",

     parse_patt @{theory Rational} "(?a * (?b + ?c)) / (?a * ?d) = (?b + _) / ?d", "cancel"),

   ("fill-cancel-left-add-num3",

     parse_patt @{theory Rational} "(?a * (?b + ?c)) / (?a * ?d) = (_ + _) / ?d", "cancel"),

   ("fill-cancel-left-add-num4",

     parse_patt @{theory Rational} "(?a * (?b + ?c)) / (?a * ?d) = (?b + ?c) / _", "cancel"),

   ("fill-cancel-left-add-num5",

     parse_patt @{theory Rational} "(?a * (?b + ?c)) / (?a * ?d) = (_ + ?c) / _", "cancel"),

   ("fill-cancel-left-add-num6",

     parse_patt @{theory Rational} "(?a * (?b + ?c)) / (?a * ?d) = (?b + _) / _", "cancel"),

   ("fill-cancel-left-add-none",

     parse_patt @{theory Rational} "(?a * (?b + ?c)) / (?a * ?d) = (?b + ?c) / ?d", "cancel"),

   ("fill-cancel-left-add-den1",

     parse_patt @{theory Rational} "(?a * ?b) / (?a * (?c + ?d)) = ?b / (?c + ?d)", "cancel"),

   ("fill-cancel-left-add-den2",

     parse_patt @{theory Rational} "(?a * ?b) / (?a * (?c + ?d)) = ?b / (?c + ?d)", "cancel"),

   ("fill-cancel-left-add-den3",

     parse_patt @{theory Rational} "(?a * ?b) / (?a * (?c + ?d)) = ?b / (?c + ?d)", "cancel"),

   ("fill-cancel-left-add-den4",

     parse_patt @{theory Rational} "(?a * ?b) / (?a * (?c + ?d)) = ?b / (?c + ?d)", "cancel"),

   ("fill-cancel-left-add-den5",

     parse_patt @{theory Rational} "(?a * ?b) / (?a * (?c + ?d)) = ?b / (?c + ?d)", "cancel"),

   ("fill-cancel-left-add-den6",

     parse_patt @{theory Rational} "(?a * ?b) / (?a * (?c + ?d)) = ?b / (?c + ?d)", "cancel"),

   ("fill-cancel-left-add-none",

     parse_patt @{theory Rational} "(?a * ?b) / (?a * (?c + ?d)) = ?b / (?c + ?d)", "cancel")

  ]: fillpat list;

 *)

 ML \<open>

 \<close> ML \<open>

 \<close> ML \<open>

 \<close>

 end