(* Title: tests for rationals

   Author: Walther Neuper

   Use is subject to license terms.

*)

"-----------------------------------------------------------------------------";

"-----------------------------------------------------------------------------";

"table of contents -----------------------------------------------------------";

"-----------------------------------------------------------------------------";

"-------- fun poly_of_term ---------------------------------------------------";

"-------- fun is_poly --------------------------------------------------------";

"-------- fun term_of_poly ---------------------------------------------------";

"-------- integration lev.1 fun factout_p_ -----------------------------------";

"-------- integration lev.1 fun cancel_p_ ------------------------------------";

"-------- integration lev.1 fun common_nominator_p_ --------------------------";

"-------- integration lev.1 fun add_fraction_p_ ------------------------------";

"Rfuns-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";

"Rfuns-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";

"-------- rls norm_Rational downto fun gcd_poly ------------------------------";

"Rfuns-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";

"----------- rewrite_set_ Partial_Fractions norm_Rational --------------------------------------";

"----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";

"----------- fun cancel_p with Const AA --------------------------------------------------------";

"-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";

"-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";

"-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";

"Rfuns-------- reverse rewrite ----------------------------------------------------";

"Rfuns-------- 'reverse-ruleset' cancel_p -----------------------------------------";

"-------- investigate rls norm_Rational --------------------------------------";

"-------- examples: rls norm_Rational ----------------------------------------";

"-------- rational numerals --------------------------------------------------";

"-------- examples cancellation from: Mathematik 1 Schalk --------------------";

"-------- examples common denominator from: Mathematik 1 Schalk --------------";

"-------- examples multiply and cancel from: Mathematik 1 Schalk -------------";

"-------- examples common denominator and multiplication from: Schalk --------";

"-------- examples double fractions from: Mathematik 1 Schalk ----------------";

"-------- me Schalk I No.186 -------------------------------------------------";

"-------- interSteps ..Simp_Rat_Double_No-1.xml ------------------------------";

"-------- interSteps ..Simp_Rat_Cancel_No-1.xml ------------------------------";

"-------- investigate rulesets for cancel_p ----------------------------------";

"-------- fun eval_get_denominator -------------------------------------------";

"-------- several errpats in complicated term --------------------------------";

"-------- WN1309xx non-terminating rls norm_Rational -------------------------";

"-----------------------------------------------------------------------------";

"-----------------------------------------------------------------------------";

"-------- fun poly_of_term ---------------------------------------------------";

"-------- fun poly_of_term ---------------------------------------------------";

"-------- fun poly_of_term ---------------------------------------------------";

val thy = @{theory Partial_Fractions};

val ctxt = Proof_Context.init_global @{theory}

val vs = TermC.vars_of (the (parseNEW ctxt "12 * x^^^3 * y^^^4 * z^^^6"));

if poly_of_term vs (str2term "12::real") = SOME [(12, [0, 0, 0])]

then () else error "poly_of_term 1 changed";

if poly_of_term vs (str2term "x::real") = SOME [(1, [1, 0, 0])]

then () else error "poly_of_term 2 changed";

if poly_of_term vs (str2term "12 * x^^^3") = SOME [(12, [3, 0, 0])]

then () else error "poly_of_term 3 changed";

if poly_of_term vs (str2term "12 * x^^^3 * y^^^4 * z^^^6") = SOME [(12, [3, 4, 6])]

then () else error "poly_of_term 4 changed";

if poly_of_term vs (str2term "1 + 2 * x^^^3 * y^^^4 * z^^^6 + y") =

  SOME [(1, [0, 0, 0]), (1, [0, 1, 0]), (2, [3, 4, 6])]

then () else error "poly_of_term 5 changed";

(*poly_of_term is quite liberal:*)

(*the coefficient may be somewhere, the order of variables and the parentheses 

  within a monomial are arbitrary*)

if poly_of_term vs (str2term "y^^^4 * (x^^^3 * 12 * z^^^6)") = SOME [(12, [3, 4, 6])]

then () else error "poly_of_term 6 changed";

(*there may even be more than 1 coefficient:*)

if poly_of_term vs (str2term "2 * y^^^4 * (x^^^3 * 6 * z^^^6)") = SOME [(12, [3, 4, 6])]

then () else error "poly_of_term 7 changed";

(*the order and the parentheses within monomials are arbitrary:*)

if poly_of_term vs (str2term "2 * x^^^3 * y^^^4 * z^^^6 + (7 * y^^^8 + 1)")

  = SOME [(1, [0, 0, 0]), (7, [0, 8, 0]), (2, [3, 4, 6])]

then () else error "poly_of_term 8 changed";

"-------- fun is_poly --------------------------------------------------------";

"-------- fun is_poly --------------------------------------------------------";

"-------- fun is_poly --------------------------------------------------------";

if is_poly (str2term "2 * x^^^3 * y^^^4 * z^^^6 + 7 * y^^^8 + 1")

then () else error "is_poly 1 changed";

if not (is_poly (str2term "2 * (x^^^3 * y^^^4 * z^^^6 + 7) * y^^^8 + 1"))

then () else error "is_poly 2 changed";

"-------- fun term_of_poly ---------------------------------------------------";

"-------- fun term_of_poly ---------------------------------------------------";

"-------- fun term_of_poly ---------------------------------------------------";

val expT = HOLogic.realT

val Free (_, baseT) = (hd o vars o str2term) "12 * x^^^3 * y^^^4 * z^^^6";

val p = [(1, [0, 0, 0]), (7, [0, 8, 0]), (2, [3, 4, 5])]

val vs = TermC.vars_of (the (parseNEW ctxt "12 * x^^^3 * y^^^4 * z^^^6"))

(*precondition for [(c, es),...]: legth es = length vs*)

;

if UnparseC.term (term_of_poly baseT expT vs p) = "1 + 7 * y ^^^ 8 + 2 * x ^^^ 3 * y ^^^ 4 * z ^^^ 5"

then () else error "term_of_poly 1 changed";

"-------- integration lev.1 fun factout_p_ -----------------------------------";

"-------- integration lev.1 fun factout_p_ -----------------------------------";

"-------- integration lev.1 fun factout_p_ -----------------------------------";

val t = str2term "(x^^^2 + -1*y^^^2) / (x^^^2 + -1*x*y)"

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

if UnparseC.term t' = "(x + y) * (x + -1 * y) / (x * (x + -1 * y))"

then () else error ("factout_p_ term 1 changed: " ^ UnparseC.term t')

;

if UnparseC.terms asm = "[\"x \<noteq> 0\",\"x + -1 * y \<noteq> 0\"]"

then () else error "factout_p_ asm 1 changed"

;

val t = str2term "nothing + to_cancel ::real";

if NONE = factout_p_ thy t then () else error "factout_p_ doesn't report non-applicable";

;

val t = str2term "((3 * x^^^2 + 6 *x + 3) / (2*x + 2))";

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

if UnparseC.term t' = "(3 + 3 * x) * (1 + x) / (2 * (1 + x))" andalso 

  UnparseC.terms asm = "[\"1 + x \<noteq> 0\"]"

then () else error "factout_p_ 1 changed";

"-------- integration lev.1 fun cancel_p_ ------------------------------------";

"-------- integration lev.1 fun cancel_p_ ------------------------------------";

"-------- integration lev.1 fun cancel_p_ ------------------------------------";

val t = str2term "(x^^^2 + -1*y^^^2) / (x^^^2 + -1*x*y)"

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

if (UnparseC.term t', UnparseC.terms asm) = ("(x + y) / x", "[\"x \<noteq> 0\"]")

then () else error ("cancel_p_ (t', asm) 1 changed: " ^ UnparseC.term t')

;

val t = str2term "nothing + to_cancel ::real";

if NONE = cancel_p_ thy t then () else error "cancel_p_ doesn't report non-applicable";

;

val t = str2term "((3 * x^^^2 + 6 *x + 3) / (2*x + 2))";

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

if UnparseC.term t' = "(3 + 3 * x) / 2" andalso UnparseC.terms asm = "[]"

then () else error "cancel_p_ 1 changed";

"-------- integration lev.1 fun common_nominator_p_ --------------------------";

"-------- integration lev.1 fun common_nominator_p_ --------------------------";

"-------- integration lev.1 fun common_nominator_p_ --------------------------";

val t = str2term ("y / (a*x + b*x + c*x) " ^

              (* n1    d1                   *)

                "+ a / (x*y)");

              (* n2    d2                   *)

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

if UnparseC.term t' =

      ("y * y / (x * ((a + b + c) * y)) " ^

  (*  n1  *d2'/ (c'* ( d1'        *d2')) *)

     "+ a * (a + b + c) / (x * ((a + b + c) * y))")

   (*  n2 * d1'         / (c'* ( d1'        *d2')) *)

then () else error "common_nominator_p_ term 1 changed";

if UnparseC.terms asm = "[\"a + b + c \<noteq> 0\",\"y \<noteq> 0\",\"x \<noteq> 0\"]"

then () else error "common_nominator_p_ asm 1 changed"

"-------- example in mail Nipkow";

val t = str2term "x/(x^^^2 + -1*y^^^2) + y/(x^^^2 + -1*x*y)";

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

if UnparseC.term t' = "x * x / " ^ 

                 "((x + -1 * y) * ((x + y) * x))" ^

            " +\n" ^ 

                 "y * (x + y) / " ^ 

                 "((x + -1 * y) * ((x + y) * x))"

then () else error "common_nominator_p_ term 2 changed"

;

if UnparseC.terms asm = "[\"x + y \<noteq> 0\",\"x \<noteq> 0\",\"x + -1 * y \<noteq> 0\"]"

then () else error "common_nominator_p_ asm 2 changed"

"-------- example: applicable tested by SML code";

val t = str2term "nothing / to_add";

if NONE = common_nominator_p_ thy t then () else error "common_nominator_p_ term 3 changed";

;

val t = str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";

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

if UnparseC.term t' = 

   "(x + -1) * (-1 + x) / ((1 + x) * (-1 + x)) +\n(x + 1) * (1 + x) / ((1 + x) * (-1 + x))"

  andalso UnparseC.terms asm = "[\"1 + x \<noteq> 0\",\"-1 + x \<noteq> 0\"]"

then () else error "common_nominator_p_ 3 changed";

"-------- integration lev.1 fun add_fraction_p_ ------------------------------";

"-------- integration lev.1 fun add_fraction_p_ ------------------------------";

"-------- integration lev.1 fun add_fraction_p_ ------------------------------";

val t = str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";

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

if UnparseC.term t' = "(2 + 2 * x ^^^ 2) / (-1 + x ^^^ 2)" 

then () else error "add_fraction_p_ 3 changed";

;

if UnparseC.terms asm = "[\"-1 + x ^^^ 2 \<noteq> 0\"]"

then () else error "add_fraction_p_ 3 changed";

;

val t = str2term "nothing / to_add";

if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ term 3 changed";

;

val t = str2term "((x + (-1)) / (x + 1)) + ((x + 1) / (x + (-1)))";

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

if UnparseC.term t' = "(2 + 2 * x ^^^ 2) / (-1 + x ^^^ 2)" andalso

  UnparseC.terms asm = "[\"-1 + x ^^^ 2 \<noteq> 0\"]"

then () else error "add_fraction_p_ 3 changed";

"-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";

"-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";

"-------- and app_rev ...traced down from rewrite_set_ until prepats ---------";

(* trace down until prepats are evaluated 

  (which does not to work, because substitution is not done -- compare rew_sub!);

  keep this sequence for the case, factout_p, cancel_p, common_nominator_p, add_fraction_p

  (again) get prepat = [] changed to <>[]. *)

val t = str2term "(x^^^2 + -1*y^^^2) / (x^^^2 + -1*x*y)";

(*rewrite_set_ @{theory Isac_Knowledge} true cancel t = NONE; !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)

"~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (thy, false, cancel_p, t);

"~~~~~ fun rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =

  (thy, 1, bool, [], rls, term);

(*val (t', asm, rew) = app_rev thy (i+1) rrls t; rew = false!!!!!!!!!!!!!!!!!!!!!*)

"~~~~~ and app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);

    fun chk_prepat thy erls [] t = true

      | chk_prepat thy erls prepat t =

        let

          fun chk (pres, pat) =

            (let 

              val subst: Type.tyenv * Envir.tenv =

                Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)

             in

              snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)

             end) handle _ => false

           fun scan_ f [] = false (*scan_ NEVER called by []*)

             | scan_ f (pp::pps) =

               if f pp then true else scan_ f pps;

        in scan_ chk prepat end;

    (* apply the normal_form of a rev-set *)

    fun app_rev' thy (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}) t =

      if chk_prepat thy erls prepat t

      then ((*tracing("### app_rev': t = "^UnparseC.term t);*) normal_form t)

      else NONE;

(*  val opt = app_rev' thy rrls t  ..NONE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)

"~~~~~ and app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =

  (thy, rrls, t);

(* chk_prepat thy erls prepat t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)

(* app_sub thy i rrls t = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)

"~~~~~ fun chk_prepat, args:"; val (thy, erls, prepat, t) = (thy, erls, prepat, t);

          fun chk (pres, pat) =

            (let 

              val subst: Type.tyenv * Envir.tenv =

                Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)

             in

              snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)

             end) handle _ => false

           fun scan_ f [] = false (*scan_ NEVER called by []*)

             | scan_ f (pp::pps) =

               if f pp then true else scan_ f pps;

(*========== inhibit exn WN130823: prepat is empty ====================================

(* scan_ chk prepat = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)

"~~~~~ fun , args:"; val (f, (pp::pps)) = (chk, prepat);

f;

val ([t1, t2], t) = pp;

UnparseC.term t1 = "?r is_expanded";

UnparseC.term t2 = "?s is_expanded";

UnparseC.term t = "?r / ?s";

(* f pp = false!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)

"~~~~~ fun chk, args:"; val (pres, pat) = (pp);

              val subst: Type.tyenv * Envir.tenv =

                Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)

(*subst = 

  ({}, {(("r", 0), ("real", Var (("r", 0), "real"))), 

        (("s", 0), ("real", Var (("s", 0), "real")))}*)

;

              snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)

"~~~~~ fun eval__true, args:"; val (thy, i, asms, bdv, rls) =

  (thy, (i + 1), (map (Envir.subst_term subst) pres), [], erls);

UnparseC.terms asms;                         (* = "[\"?r is_expanded\",\"?s is_expanded\"]"*)

asms = [@{term True}] orelse asms = []; (* = false*)

asms = [@{term False}]                ; (* = false*)

"~~~~~ fun chk, args:"; val (indets, (a::asms)) = ([], asms);

bdv (*= []: _a list*);

val bdv : (term * term) list = [];

rewrite__set_ thy (i+1) false;

UnparseC.term a = "?r is_expanded"; (*hier m"usste doch der Numerator eingesetzt sein ??????????????*)

val SOME (Const ("HOL.False", _), []) = rewrite__set_ thy (i+1) false bdv rls a

============ inhibit exn WN130823: prepat is empty ===================================*)

"-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";

"-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";

"-------- fun rewrite_set_ cancel_p downto fun gcd_poly ----------------------";

val t = str2term "(12 * x * y) / (8 * y^^^2 )";

(* "-------- example 187a": exception Div raised...

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)

val t = str2term "(8 * x^^^2 * y * z ) / (18 * x * y^^^2 * z )";

(* "-------- example 187b": doesn't terminate...

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)

val t = str2term "(9 * x^^^5 * y^^^2 * z^^^4) / (15 * x^^^6 * y^^^3 * z )";

(* "-------- example 187c": doesn't terminate...

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;*)

"~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) = (@{theory Isac_Knowledge}, false, cancel_p, t);

(* WN130827: exception Div raised...

rewrite__set_ thy 1 bool [] rls term

*)

"~~~~~ and rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =

  (thy, 1, bool, [], rls, term);

(* WN130827: exception Div raised...

	val (t', asm, rew) = app_rev thy (i+1) rrls t

*)

"~~~~~ fun app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);

(* WN130827: exception Div raised...

    val opt = app_rev' thy rrls t

*)

"~~~~~ fun app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =

  (thy, rrls, t);

chk_prepat thy erls prepat t    = true;

(* WN130827: exception Div raised...

normal_form t

*)

(* lookup Rational.thy, cancel_p: normal_form = cancel_p_ thy*)

"~~~~~ fun cancel_p_, args:"; val (t) = (t);

val opt = check_fraction t;

val SOME (numerator, denominator) = opt

        val vs = vars_of t

        val baseT = type_of numerator

        val expT = HOLogic.realT

val (SOME a, SOME b) = (poly_of_term vs numerator, poly_of_term vs denominator);

(*"-------- example 187a": exception Div raised...

val a = [(12, [1, 1])]: poly

val b = [(8, [0, 2])]: poly

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

*)

(* "-------- example 187b": doesn't terminate...

val a = [(8, [2, 1, 1])]: poly

val b = [(18, [1, 2, 1])]: poly

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

*)

(* "-------- example 187c": doesn't terminate...

val a = [(9, [5, 2, 4])]: poly

val b = [(15, [6, 3, 1])]: poly

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

*)

"-------- rls norm_Rational downto fun gcd_poly ------------------------------";

"-------- rls norm_Rational downto fun gcd_poly ------------------------------";

"-------- rls norm_Rational downto fun gcd_poly ------------------------------";

val t = str2term "(x^^^2 - 4)*(3 - y) / ((y^^^2 - 9)*(2+x))";

trace_rewrite := false (*true false*);

(* trace stops with ...: (and then jEdit hangs)..

rewrite_set_ thy false norm_Rational t;

:

###  rls: cancel_p on: (-12 + 4 * y + 3 * x ^^^ 2 + -1 * (x ^^^ 2 * y)) /

(-18 + -9 * x + 2 * y ^^^ 2 + x * y ^^^ 2)

*)

val t = str2term (*copy from above: "::real" is not required due to "^^^"*)

  ("(-12 + 4 * y + 3 * x ^^^ 2 + -1 * (x ^^^ 2 * y)) /" ^

  "(-18 + -9 * x + 2 * y ^^^ 2 + x * y ^^^ 2)");

(*cancel_p_ thy t;

exception Div raised*)

"~~~~~ fun cancel_p_, args:"; val (t) = (t);

val opt = check_fraction t;

val SOME (numerator, denominator) = opt

        val vs = vars_of t

        val baseT = type_of numerator

        val expT = HOLogic.realT;

(*default_print_depth 3; 999*)

val (SOME a, SOME b) = (poly_of_term vs numerator, poly_of_term vs denominator);

(*default_print_depth 3; 999*)

(* does not terminate instead of returning ?:

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

val a = [(~12, [0, 0]), (3, [2, 0]), (4, [0, 1]), (~1, [2, 1])]: poly

val b = [(~18, [0, 0]), (~9, [1, 0]), (2, [0, 2]), (1, [1, 2])]: poly

*)

"-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";

"-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";

"-------- rls norm_Rational downto fun add_fraction_p_ -----------------------";

val thy =  @{theory Isac_Knowledge};

"----- SK060904-2a non-termination of add_fraction_p_";

val t = str2term (" (a + b * x) / (a + -1 * (b * x)) +  " ^

		         " (-1 * a + b * x) / (a + b * x)      ");

(* rewrite_set_ thy false norm_Rational t

exception Div raised*)

(* rewrite_set_ thy false add_fractions_p t;

exception Div raised*)

"~~~~~ fun rewrite_set_, args:"; val (thy, bool, rls, term) =

  (@{theory Isac_Knowledge}, false, add_fractions_p, t);

"~~~~~ and rewrite__set_, args:"; val (thy, i, _, _, (rrls as Rrls _), t) =

  (thy, 1, bool, [], rls, term);

(* app_rev thy (i+1) rrls t;

exception Div raised*)

"~~~~~ and app_rev, args:"; val (thy, i, rrls, t) = (thy, (i+1), rrls, t);

    fun chk_prepat thy erls [] t = true

      | chk_prepat thy erls prepat t =

        let

          fun chk (pres, pat) =

            (let 

              val subst: Type.tyenv * Envir.tenv =

                Pattern.match thy (pat, t) (Vartab.empty, Vartab.empty)

             in

              snd (eval__true thy (i + 1) (map (Envir.subst_term subst) pres) [] erls)

             end) handle _ => false

           fun scan_ f [] = false (*scan_ NEVER called by []*)

             | scan_ f (pp::pps) =

               if f pp then true else scan_ f pps;

        in scan_ chk prepat end;

    (* apply the normal_form of a rev-set *)

    fun app_rev' thy (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}) t =

      if chk_prepat thy erls prepat t

      then ((*tracing("### app_rev': t = "^UnparseC.term t);*) normal_form t)

      else NONE;

(*  val opt = app_rev' thy rrls t;

exception Div raised*)

(*  val opt = app_rev' thy rrls t;

exception Div raised*)

"~~~~~ and app_rev', args:"; val (thy, (Rrls{erls,prepat,scr=Rfuns{normal_form,...},...}), t) =

  (thy, rrls, t);

chk_prepat thy erls prepat t = true       = true;

(*normal_form t

exception Div raised*)

(* lookup Rational.thy, val add_fractions_p: normal_form = add_fraction_p_ thy*)

(*add_fraction_p_ thy t

exception Div raised*)

"~~~~~ fun add_fraction_p_, args:"; val ((_: theory), t) = (thy, t);

val SOME ((n1, d1), (n2, d2)) = check_frac_sum t;

UnparseC.term n1; UnparseC.term d1; UnparseC.term n2; UnparseC.term d2;

      val vs = vars_of t;

(*default_print_depth 3; 999*)

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

  (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2);

(*default_print_depth 3; 999*)

(*

val a = [(1, [1, 0, 0]), (~1, [0, 1, 1])]: poly

val b = [(1, [1, 0, 0]), (1, [0, 1, 1])]: poly

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

*)

"----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";

"----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";

"----------- fun check_frac_sum with Free A and Const AA ---------------------------------------";

val thy = @{theory Isac_Knowledge(*Partial_Fractions*)}

val ctxt = Proof_Context.init_global thy;

(*---------- (1) with Free A, B  ----------------------------------------------------------------*)

val t = (the o (parseNEW  ctxt)) "3 = A / 2 + A / 4 + (B / 2 + -1 * B / (2::real))";

                                (* required for applying thms in rewriting ^^^^*)

(* we get details from here..*)

trace_rewrite := false;

val SOME (t', _) = Rewrite.rewrite_set_ thy true add_fractions_p t;

Celem.trace_rewrite := false;

(* trace_rewrite:

add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + -1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)

                     (* |||||||||||||||||||||||||||||||||||| *)

val t = (the o (parseNEW  ctxt))(* ||||||||||||||||||||||||| GUESS 1 GUESS 1 GUESS 1 GUESS 1 *)

                       "A / 2 + A / 4 + (B / 2 + -1 * B / (2::real))";

"~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);

val NONE = (*case*) check_frac_sum t (*of*)

(* trace_rewrite:

add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + -1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)

                     (*         |||||||||||||||||||||||||||| *)

val t = (the o (parseNEW  ctxt))(* ||||||||||||||||||||||||| GUESS 2 GUESS 2 GUESS 2 GUESS 2 *)

                               "A / 4 + (B / 2 + -1 * B / (2::real))";

"~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);

val SOME ((n1, d1), (n2, d2)) = (*case*) check_frac_sum t (*of*);

(*+*)if (UnparseC.term n1, UnparseC.term d1) = ("A"                 , "4") andalso

(*+*)   (UnparseC.term n2, UnparseC.term d2) = ("B / 2 + -1 * B / 2", "1")

(*+*)then () else error "check_frac_sum (A / 4 + (B / 2 + -1 * B / (2::real))) changed";

      val vs = vars_of t;

val (SOME [(1, [1, 0])], SOME [(4, [0, 0])], NONE, SOME [(1, [0, 0])]) =

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

"~~~~~ fun poly_of_term , args:"; val (vs, t) = (vs, n1);

val SOME [(1, [xxx, 0])] = SOME [monom_of_term vs (1, replicate (length vs) 0) t];

(*+*)if xxx = 1 then () else error "monom_of_term changed"

"~~~~~ fun monom_of_term , args:"; val (vs, (c, es), (Free (id, _))) =

  (vs, (1, replicate (length vs) 0), t);

case vs of [Free ("A", _), Free ("B", _)] =>

  if c = 1 andalso id = "A"

  then () else error "monom_of_term Free changed 1"

| _ => error "monom_of_term Free changed 2";

(*---------- (2) with Const AA, BB --------------------------------------------------------------*)

val t = (the o (parseNEW  ctxt)) "3 = AA / 2 + AA / 4 + (BB / 2 + -1 * BB / 2)";

                                    (*AA :: real*)

(* we get details from here..*)

Celem.trace_rewrite := false;

val SOME (t', _) = Rewrite.rewrite_set_ thy true add_fractions_p t;

Celem.trace_rewrite := false;

(* trace_rewrite:

add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + -1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)

                     (* |||||||||||||||||||||||||||||||||||| *)

val t = (the o (parseNEW  ctxt))(* ||||||||||||||||||||||||| *)

                   "AA / 2 + AA / 4 + (BB / 2 + -1 * BB / 2)";

"~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);

val NONE = (*case*) check_frac_sum t (*of*)

(* trace_rewrite:

add_fractions_p on: 3 = A / 2 + A / 4 + (B / 2 + -1 * B / 2) --> 3 = A / 2 + A / 4 + 0 / 2 *)

                     (*         |||||||||||||||||||||||||||| *)

val t = (the o (parseNEW  ctxt))(* ||||||||||||||||||||||||| *)

                               "AA / 4 + (BB / 2 + -1 * BB / 2)";

"~~~~~ fun add_fraction_p_ , ad-hoc args:"; val (t) = (t);

val SOME ((n1, d1), (n2, d2)) = (*case*) check_frac_sum t (*of*);

(*+*)if (UnparseC.term n1, UnparseC.term d1) = ("AA"                 , "4") andalso

(*+*)   (UnparseC.term n2, UnparseC.term d2) = ("BB / 2 + -1 * BB / 2", "1")

(*+*)then () else error "check_frac_sum (AA / 4 + (BB / 2 + -1 * BB / 2)) changed";

      val vs = vars_of t;

val (SOME [(1, [1, 0])], SOME [(4, [0, 0])], NONE, SOME [(1, [0, 0])]) =

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

"~~~~~ fun poly_of_term , args:"; val (vs, t) = (vs, n1);

val SOME [(1, [xxx, 0])] = SOME [monom_of_term vs (1, replicate (length vs) 0) t];

(*+*)if xxx = 1 then () else error "monom_of_term changed"

"~~~~~ fun monom_of_term , args:"; val (vs, (c, es), (Const (id, _))) =

  (vs, (1, replicate (length vs) 0), t);

case vs of [Const ("Partial_Fractions.AA", _), Const ("Partial_Fractions.BB", _)] =>

  if c = 1 andalso id = "Partial_Fractions.AA"

  then () else error "monom_of_term Const changed 1"

| _ => error "monom_of_term Const changed 2";

"----------- fun cancel_p with Const AA --------------------------------------------------------";

"----------- fun cancel_p with Const AA --------------------------------------------------------";

"----------- fun cancel_p with Const AA --------------------------------------------------------";

val thy = @{theory Partial_Fractions};

val ctxt = Proof_Context.init_global @{theory}

val SOME t = TermC.parseNEW ctxt "2 * AA / 2"; (* Const ("Free ("AA", "real") *)

val SOME (t', _) = rewrite_set_ thy true cancel_p t;

case t' of

  Const ("Rings.divide_class.divide", _) $ Const ("Partial_Fractions.AA", _) $ Free ("1", _) => ()

| _ => error "WRONG rewrite_set_ cancel_p (2 * AA / 2) \<longrightarrow> AA changed";

"~~~~~ fun cancel_p , args:"; val (t) = (t);

val opt = check_fraction t

val SOME (numerator, denominator) = (*case*) opt (*of*);

if UnparseC.term numerator = "2 * AA" andalso UnparseC.term denominator = "2"

then () else error "check_fraction (2 * AA / 2) changed";

        val vs = TermC.vars_of t;

case vs of

  [Const ("Partial_Fractions.AA", _)] => ()

| _ => error "rewrite_set_ cancel_p (2 * AA / 2) \<longrightarrow> AA/1  changed";

"-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";

"-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";

"-------- rewrite_set_ cancel_p from: Mathematik 1 Schalk Reniets Verlag -----";

val thy  = @{theory "Rational"};

"-------- WN";

val t = str2term "(2 + -3 * x) / 9";

if NONE = rewrite_set_ thy false cancel_p t then ()

else error "rewrite_set_ cancel_p must return NONE, if the term cannot be cancelled";

"-------- example 186a";

val t = str2term "(14 * x * y) / (x * y)";

  is_expanded (str2term "14 * x * y");

  is_expanded (str2term "x * y");

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = ("14 / 1", "[]")

then () else error "rational.sml cancel Schalk 186a";

"-------- example 186b";

val t = str2term "(60 * a * b) / ( 15 * a  * b )";

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = ("4 / 1", "[]")

then () else error "rational.sml cancel Schalk 186b";

"-------- example 186c";

val t = str2term "(144 * a^^^2 * b * c) / (12 * a * b * c)";

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = ("12 * a / 1", "[]")

then () else error "rational.sml cancel Schalk 186c";

(* !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! exception Div raised !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

  see --- fun rewrite_set_ downto fun gcd_poly ---

"-------- example 187a";

val t = str2term "(12 * x * y) / (8 * y^^^2 )";

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = ("3 * x / (2 * y)", "[\"4 * y ~= 0\"]")

then () else error "rational.sml cancel Schalk 187a";

*)

(* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

  see --- fun rewrite_set_ downto fun gcd_poly ---

"-------- example 187b";

val t = str2term "(8 * x^^^2 * y * z ) / (18 * x * y^^^2 * z )";

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = ("4 * x / (9 * y)", "[\"2 * (z * (y * x)) ~= 0\"]")

then () else error "rational.sml cancel Schalk 187b";

*)

(* doesn't terminate !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

  see --- fun rewrite_set_ downto fun gcd_poly ---

"-------- example 187c";

val t = str2term "(9 * x^^^5 * y^^^2 * z^^^4) / (15 * x^^^6 * y^^^3 * z )";

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = 

  ("3 * z ^^^ 3 / (5 * (y * x))", "[\"3 * (z * (y ^^^ 2 * x ^^^ 5)) ~= 0\"]") 

then () else error "rational.sml cancel Schalk 187c";

*)

"-------- example 188a";

val t = str2term "(-8 + 8 * x) / (-9 + 9 * x)";

  is_expanded (str2term "8 * x + -8");

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = ("8 / 9", "[]")

then () else error "rational.sml cancel Schalk 188a";

val t = str2term "(8*((-1) + x))/(9*((-1) + x))";

val SOME (t, _) = rewrite_set_ thy false make_polynomial t;

if (UnparseC.term t', UnparseC.terms asm) = ("8 / 9", "[]")

then () else error "rational.sml cancel Schalk make_polynomial 1";

"-------- example 188b";

val t = str2term "(-15 + 5 * x) / (-18 + 6 * x)";

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = ("5 / 6", "[]")

then () else error "rational.sml cancel Schalk 188b";

"-------- example 188c";

val t = str2term "(a + -1 * b) / (b + -1 * a)";

val SOME (t', asm) = rewrite_set_ thy false  cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = ("-1 / 1", "[]")

then () else error "rational.sml cancel Schalk 188c";

is_expanded (str2term "a + -1 * b") = true;

val t = str2term "((-1)*(b + (-1) * a))/(1*(b + (-1) * a))";

val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;

if (UnparseC.term t', UnparseC.terms asm) = ("(a + -1 * b) / (-1 * a + b)", "[]")

then () else error "rational.sml cancel Schalk make_polynomial 2";

"-------- example 190a";

val t = str2term "( 27 * a^^^3 + 9 * a^^^2 + 3 * a + 1 ) / ( 27 * a^^^3 + 18 * a^^^2 + 3 * a )";

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = 

  ("(1 + 9 * a ^^^ 2) / (3 * a + 9 * a ^^^ 2)", "[\"3 * a + 9 * a ^^^ 2 \<noteq> 0\"]")

then () else error "rational.sml cancel Schalk 190a";

"-------- example 190c";

val t = str2term "((1 + 9 * a ^^^ 2)*(1 + 3 * a))/((3 * a + 9 * a ^^^ 2)*(1 + 3 * a))";

val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;

if (UnparseC.term t', UnparseC.terms asm) = 

  ("(1 + 3 * a + 9 * a ^^^ 2 + 27 * a ^^^ 3) /\n(3 * a + 18 * a ^^^ 2 + 27 * a ^^^ 3)", "[]")

then () else error "rational.sml make_polynomial Schalk 190c";

"-------- example 191a";

val t = str2term "( x^^^2 + -1 * y^^^2 ) / ( x + y )";

  is_expanded (str2term "x^^^2 + -1 * y^^^2") = false; (*!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*)

  is_expanded (str2term "x + y") = true;

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = ("(x + -1 * y) / 1", "[]")

then () else error "rational.sml make_polynomial Schalk 191a";

"-------- example 191b";

val t = str2term "((x + (-1) * y)*(x + y))/((1)*(x + y))";

val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;

if (UnparseC.term t', UnparseC.terms asm) = ("(x ^^^ 2 + -1 * y ^^^ 2) / (x + y)", "[]")

then () else error "rational.sml make_polynomial Schalk 191b";

"-------- example 191c";

val t = str2term "( 9 * x^^^2 + -30 * x + 25 ) / ( 9 * x^^^2 + -25 )";

  is_expanded (str2term "9 * x^^^2 + -30 * x + 25") = true;

  is_expanded (str2term "25 + -30*x + 9*x^^^2") = true;

  is_expanded (str2term "-25 + 9*x^^^2") = true;

val t = str2term "(((-5) + 3 * x)*((-5) + 3 * x))/((5 + 3 * x)*((-5) + 3 * x))";

val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;

if (UnparseC.term t', UnparseC.terms asm) = ("(25 + -30 * x + 9 * x ^^^ 2) / (-25 + 9 * x ^^^ 2)", "[]")

then () else error "rational.sml make_polynomial Schalk 191c";

"-------- example 192b";

val t = str2term "( 7 * x^^^3 + -1 * x^^^2 * y ) / ( 7 * x * y^^^2 + -1 *  y^^^3 )";

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = ("x ^^^ 2 / y ^^^ 2", "[\"y ^^^ 2 \<noteq> 0\"]")

then () else error "rational.sml cancel_p Schalk 192b";

val t = str2term "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";

val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;

if (UnparseC.term t', UnparseC.terms asm) = 

  ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[]")

then () else error "rational.sml make_polynomial Schalk 192b";

val t = str2term "((x ^^^ 2)*(7 * x + (-1) * y))/((y ^^^ 2)*(7 * x + (-1) * y))";

val SOME (t', asm) = rewrite_set_ thy false make_polynomial t;

if (UnparseC.term t', UnparseC.terms asm) = 

  ("(7 * x ^^^ 3 + -1 * x ^^^ 2 * y) / (7 * x * y ^^^ 2 + -1 * y ^^^ 3)", "[]")

then () else error "rational.sml make_polynomial Schalk WN050929 not working";

"-------- example 193a";

val t = str2term "( x^^^2 + -6 * x + 9 ) / ( x^^^2 + -9 )";

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = ("(-3 + x) / (3 + x)", "[\"3 + x \<noteq> 0\"]")

then () else error "rational.sml cancel_p Schalk 193a";

"-------- example 193b";

val t = str2term "( x^^^2 + -8 * x + 16 ) / ( 2 * x^^^2 + -32 )";

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = ("(-4 + x) / (8 + 2 * x)", "[\"8 + 2 * x \<noteq> 0\"]")

then () else error "rational.sml cancel_p Schalk 193b";

"-------- example 193c";

val t = str2term "( 2 * x + -50 * x^^^3 ) / ( 25 * x^^^2 + -10 * x + 1 )";

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = 

  ("(2 * x + 10 * x ^^^ 2) / (1 + -5 * x)", "[\"1 + -5 * x \<noteq> 0\"]")

then () else error "rational.sml cancel_p Schalk 193c";

(*WN:*)

val t = str2term "(-25 + 9*x^^^2)/(5 + 3*x)";

val SOME (t, asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = ("(2 * x + 10 * x ^^^ 2) / (1 + -5 * x)", "[]")

then () else error "rational.sml cancel WN 1";

"-------- example heuberger";

val t = str2term ("(x^^^4 + x * y + x^^^3 * y + y^^^2) / " ^

  "(x + 5 * x^^^2 + y + 5 * x * y + x^^^2 * y^^^3 + x * y^^^4)");

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if (UnparseC.term t', UnparseC.terms asm) = 

  ("(x ^^^ 3 + y) / (1 + 5 * x + x * y ^^^ 3)", "[\"1 + 5 * x + x * y ^^^ 3 \<noteq> 0\"]")

then () else error "rational.sml cancel_p heuberger";

"-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";

"-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";

"-------- rewrite_set_ add_fractions_p from: Mathematik 1 Schalk -------------";

(*deleted example 204 ... 236b at update Isabelle2012-->2013*)

"-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";

"-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";

"-------- integration lev.1 -- lev.5: cancel_p_ & add_fractions_p_ -----------";

val t = str2term ("123 = (a*x)/(b*x) + (c*x)/(d*x) + (e*x)/(f*x::real)");

"-------- gcd_poly integration level 1: works on exact term";

if NONE = cancel_p_ thy t then () else error "cancel_p_ works on exact fraction";

if NONE = add_fraction_p_ thy t then () else error "add_fraction_p_ works on exact fraction";

"-------- gcd_poly integration level 2: picks out ONE appropriate subterm";

val SOME (t', asm) = rewrite_set_ thy false cancel_p t;

if UnparseC.term t' = "123 = a * x / (b * x) + c * x / (d * x) + e / f" 

then () else error "level 2, rewrite_set_ cancel_p: changed";

val SOME (t', asm) = rewrite_set_ thy false add_fractions_p t;

if UnparseC.term t' = "123 = (b * c * x + a * d * x) / (b * d * x) + e * x / (f * x)"

then () else error "level 2, rewrite_set_ add_fractions_p: changed";

"-------- gcd_poly integration level 3: rewrites all appropriate subterms";

val SOME (t', asm) = rewrite_set_ thy false cancel_p_rls t;

if UnparseC.term t' = "123 = a / b + c / d + e / f"

then () else error "level 3, rewrite_set_ cancel_p_rls: changed";

val SOME (t', asm) = rewrite_set_ thy false add_fractions_p_rls t; (*CREATE add_fractions_p_rls*)

if UnparseC.term t' = "123 = (b * d * e * x + b * c * f * x + a * d * f * x) / (b * d * f * x)"

then () else error "level 3, rewrite_set_ add_fractions_p_rls: changed";

"-------- gcd_poly integration level 4: iteration cancel_p -- add_fraction_p";

(* simpler variant *)

val testrls = Rule_Set.append_rules "testrls" Rule_Set.empty [Rls_ cancel_p, Rls_ add_fractions_p]

val SOME (t', asm) = rewrite_set_ thy false testrls t;

(*trace_rewrite := false;

#  rls: testrls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x) 

##  rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x) 

##  rls: add_fractions_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f 

##  rls: cancel_p on: 123 = (b * c * x + a * d * x) / (b * d * x) + e / f 

##  rls: add_fractions_p on: 123 = (b * c + a * d) / (b * d) + e / f 

##  rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) 

##  rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)

if UnparseC.term t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"

then () else error "level 4, rewrite_set_ *_p: changed";

(* complicated variant *)

val testrls_rls = Rule_Set.append_rules "testrls_rls" Rule_Set.empty [Rls_ cancel_p_rls, Rls_ add_fractions_p_rls];

val SOME (t', asm) = rewrite_set_ thy false testrls_rls t;

(*#  rls: testrls_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x) 

##  rls: cancel_p_rls on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x) 

###  rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e * x / (f * x) 

###  rls: cancel_p on: 123 = a * x / (b * x) + c * x / (d * x) + e / f 

###  rls: cancel_p on: 123 = a * x / (b * x) + c / d + e / f 

###  rls: cancel_p on: 123 = a / b + c / d + e / f 

##  rls: add_fractions_p_rls on: 123 = a / b + c / d + e / f 

###  rls: add_fractions_p on: 123 = a / b + c / d + e / f 

###  rls: add_fractions_p on: 123 = (b * c + a * d) / (b * d) + e / f 

###  rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) 

##  rls: cancel_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) 

###  rls: cancel_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) 

##  rls: add_fractions_p_rls on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) 

###  rls: add_fractions_p on: 123 = (b * d * e + b * c * f + a * d * f) / (b * d * f) *)

if UnparseC.term t' = "123 = (b * d * e + b * c * f + a * d * f) / (b * d * f)"

then () else error "level 4, rewrite_set_ *_p_rls: changed"

"-------- gcd_poly integration level 5: cancel_p & add_fraction_p within norm_Rational";

val SOME (t', asm) = rewrite_set_ thy false norm_Rational t;

if UnparseC.term t' = "123 = (a * d * f + b * c * f + b * d * e) / (b * d * f)"

then () else error "level 5, rewrite_set_ norm_Rational: changed"

"-------- reverse rewrite ----------------------------------------------------";

"-------- reverse rewrite ----------------------------------------------------";

"-------- reverse rewrite ----------------------------------------------------";

(** the term for which reverse rewriting is demonstrated **)

val t = str2term "(9 + -1 * x ^^^ 2) / (9 + 6 * x + x ^^^ 2)";

val Rrls {scr = Rfuns {init_state = ini, locate_rule = loc,

  next_rule = nex, normal_form = nor, ...},...} = cancel_p;

(** normal_form produces the result in ONE step **)

  val SOME (t',_) = nor t;

if UnparseC.term t' = "(3 + -1 * x) / (3 + x)" then ()

else error "rational.sml normal_form (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

(** initialize the interpreter state used by the 'me' **)

  val (t, _, revsets, _) = ini t;

if length (hd revsets) = 11 then () else error "length of revset changed";

if (revsets |> nth 1 |> nth 1 |> id_of_thm) = 

  (@{thm realpow_twoI} |> Thm.get_name_hint |> ThmC.thmID_of_derivation_name)

then () else error "first element of revset changed";

if

(revsets |> nth 1 |> nth 1 |> Rule.to_string) = "Thm (\"realpow_twoI\",?r1 ^^^ 2 = ?r1 * ?r1)" andalso

(revsets |> nth 1 |> nth 2 |> Rule.to_string) = "Thm (\"#: 9 = 3 ^^^ 2\",9 = 3 ^^^ 2)" andalso

(revsets |> nth 1 |> nth 3 |> Rule.to_string) = "Thm (\"#: 6 * x = 2 * (3 * x)\",6 * x = 2 * (3 * x))" 

andalso

(revsets |> nth 1 |> nth 4 |> Rule.to_string) = "Thm (\"#: -3 * x = -1 * (3 * x)\",-3 * x = -1 * (3 * x))" 

andalso

(revsets |> nth 1 |> nth 5 |> Rule.to_string) = "Thm (\"#: 9 = 3 * 3\",9 = 3 * 3)" andalso

(revsets |> nth 1 |> nth 6 |> Rule.to_string) = "Rls_ (\"sym_order_mult_rls_\")" andalso

(revsets |> nth 1 |> nth 7 |> Rule.to_string) = 

  "Thm (\"sym_mult_assoc\",?a * (?b * ?c) = ?a * ?b * ?c)"

then () else error "first 7 elements in revset changed"

(** find the rule 'r' to apply to term 't' **)

(*/------- WN1309: since cancel_ (accepted "-" between monomials) has been replaced by cancel_p_ 

  for Isabelle2013, we don't get a working revset, but non-termination:

  val SOME (r as (Thm (str, thm))) = nex revsets t;

  :

((3 * 3 + -1 * x * x) / (3 * 3 + 2 * 3 * x + x * x), 

  Rls_ ("sym_order_mult_rls_"), ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x), []))","

((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x), 

  Thm ("sym_mult_assoc",""), ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * 3 * x + x * x), []))","

((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * 3 * x + x * x), 

  Thm ("sym_mult_assoc",""), ((3 * 3 + -1 * x * x) / (3 * 3 + 2 * 3 * x + x * x), []))","

((3 * 3 + -1 * x * x) / (3 * 3 + 2 * 3 * x + x * x), Rls_ ("sym_order_mult_rls_"), ((3 * 3 + -1 * (x * x)) / (3 * 3 + 2 * (3 * x) + x * x), []))","

 :

### Isabelle2002:

  Thm ("sym_#mult_2_3", "6 = 2 * 3")

### Isabelle2009-2 for cancel_ (not cancel_p_):

if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))"

   andalso ThmC.string_of_thm thm = 

           (string_of_thm (Thm.make_thm @{theory "Isac_Knowledge"}

               (Trueprop $ (Thm.term_of o the o (parse thy)) "9 = 3 ^^^ 2"))) then ()

else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

\---------------------------------------------------------------------------------------/*)

(** check, if the rule 'r' applied by the user to 't' belongs to the ruleset;

  if the rule is OK, the term resulting from applying the rule is returned,too;

  there might be several rule applications inbetween,

  which are listed after the head in reverse order **)

(*/-------------------------------------------- Isabelle2013: this gives "error id_of_thm";

  we don't repair this, because interaction within "reverse rewriting" never worked properly:

  val (r, (t, asm))::_ = loc revsets t r;

if UnparseC.term t = "(9 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" andalso asm = []

then () else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

(* find the next rule to apply *)

  val SOME (r as (Thm (str, thm))) = nex revsets t;

if str = "sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))" andalso

   ThmC.string_of_thm thm = (string_of_thm (ThmC_Def.make_thm @{theory "Isac_Knowledge"}

                (Trueprop $ (Thm.term_of o the o (parse thy)) "9 = 3 ^^^ 2"))) then ()

else error "rational.sml next_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2)";

(*check the next rule*)

  val (r, (t, asm)) :: _ = loc revsets t r;

if UnparseC.term t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 6 * x + x ^^^ 2)" then ()

else error "rational.sml locate_rule (9 - x ^^^ 2) / (9 - 6 * x + x ^^^ 2) II";

(*find and check the next rules, rewrite*)

  val SOME r = nex revsets t;

  val (r,(t,asm))::_ = loc revsets t r;

if UnparseC.term t = "(3 ^^^ 2 - x ^^^ 2) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()

else error "rational.sml locate_rule II";

  val SOME r = nex revsets t;

  val (r,(t,asm))::_ = loc revsets t r;

if UnparseC.term t = "(3 - x) * (3 + x) / (3 ^^^ 2 + 2 * 3 * x + x ^^^ 2)" then ()

else error "rational.sml next_rule II";

  val SOME r = nex revsets t;

  val (r,(t,asm))::_ = loc revsets t r;

if UnparseC.term t = "(3 - x) * (3 + x) / ((3 + x) * (3 + x))" then ()

else error "rational.sml next_rule III";

  val SOME r = nex revsets t;

  val (r, (t, asm)) :: _ = loc revsets t r;

  val ss = UnparseC.term t;

if ss = "(3 - x) / (3 + x)" andalso UnparseC.terms asm = "[\"3 + x ~= 0\"]" then ()

else error "rational.sml: new behav. in rev-set cancel";

\--------------------------------------------------------------------------------------/*)

"-------- 'reverse-ruleset' cancel_p -----------------------------------------";

"-------- 'reverse-ruleset' cancel_p -----------------------------------------";

"-------- 'reverse-ruleset' cancel_p -----------------------------------------";

(*WN130909: the example below shows, why "reverse rewriting" only worked for

  special cases.*)

(*the term for which reverse rewriting is demonstrated*)

val t = str2term "(9 + (-1)*x^^^2) / (9 + ((-6)*x + x^^^2))";

val Rrls {scr=Rfuns {init_state=ini,locate_rule=loc,

		       next_rule=nex,normal_form=nor,...},...} = cancel_p;

(*normal_form produces the result in ONE step*)

val SOME (t',_) = nor t; 

UnparseC.term t' = "(3 + 1 * x) / (3 + -1 * x)";

(*initialize the interpreter state used by the 'me'*)

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

UnparseC.term t' = "(3 + x) / (3 + -1 * x)" (*true*);

UnparseC.terms asm = "[\"3 + -1 * x ~= 0\"]" (*true*);

val (t,_,revsets,_) = ini t;

(* WN.10.10.02: dieser Fall terminiert nicht 

           (make_polynomial enth"alt zu viele rules)

WN060823 'init_state' requires rewriting on specified location in the term

default_print_depth 99; Rfuns; default_print_depth 3;

WN060831 cycling "sym_order_mult_rls_" "sym_mult_assoc"

         as was with make_polynomial before ?!?*)

val SOME r = nex revsets t;

eq_Thm (r, Thm ("sym_#power_Float ((3,0), (0,0)) __ ((2,0), (0,0))", 

		mk_thm thy "9 = 3 ^^^ 2"));

(*WN060831 *** id_of_thm

           Exception- ERROR raised ...

val (r,(t,asm))::_ = loc revsets t r;

UnparseC.term t;

  val SOME r = nex revsets t;

  val (r,(t,asm))::_ = loc revsets t r;

  UnparseC.term t;

*)                    

"-------- examples: rls norm_Rational ----------------------------------------";

"-------- examples: rls norm_Rational ----------------------------------------";

"-------- examples: rls norm_Rational ----------------------------------------";

val t = str2term "(3*x+5)/18 - x/2  - -(3*x - 2)/9 = 0";

val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';

if UnparseC.term t' = "1 / 18 = 0" then () else error "rational.sml 1";

val t = str2term "(17*x - 51)/9 - (-(13*x - 3)/6) + 11 - (9*x - 7)/4 = 0";

val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';

if UnparseC.term t' = "(237 + 65 * x) / 36 = 0" then () 

else error "rational.sml 2";

val t = str2term "(1/2 + (5*x)/2)^^^2 - ((13*x)/2 - 5/2)^^^2 - (6*x)^^^2 + 29";

val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';

if UnparseC.term t' = "23 + 35 * x + -72 * x ^^^ 2" then ()

else error "rational.sml 3";

(*trace_rewrite:=true;*)

val t = str2term "Not (6*x is_atom)";

val SOME (t',_) = rewrite_set_ thy false powers_erls t; UnparseC.term t';

"HOL.True";

val t = str2term "1 < 2";

val SOME (t',_) = rewrite_set_ thy false powers_erls t; UnparseC.term t';

"HOL.True";

val t = str2term "(6*x)^^^2";

val SOME (t',_) = rewrite_ thy dummy_ord powers_erls false 

			   (ThmC.numerals_to_Free @{thm realpow_def_atom}) t;

if UnparseC.term t' = "6 * x * (6 * x) ^^^ (2 + -1)" then ()

else error "rational.sml powers_erls (6*x)^^^2";

val t = str2term "-1 * (-2 * (5 / 2 * (13 * x / 2)))";

val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';

if UnparseC.term t' = "65 * x / 2" then () else error "rational.sml 4";

val t = str2term "1 - ((13*x)/2 - 5/2)^^^2";

val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';

if UnparseC.term t' = "(-21 + 130 * x + -169 * x ^^^ 2) / 4" then () 

else error "rational.sml 5";

(*SRAM Schalk I, p.92 Nr. 609a*)

val t = str2term "2*(3 - x/5)/3 - 4*(1 - x/3) - x/3 - 2*(x/2 - 1/4)/27 +5/54";

val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';

if UnparseC.term t' = "(-255 + 112 * x) / 135" then () 

else error "rational.sml 6";

(*SRAM Schalk I, p.92 Nr. 610c*)

val t = str2term "((x- 1)/(x+1) + 1) / ((x- 1)/(x+1) - (x+1)/(x- 1)) - 2";

val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';

if UnparseC.term t' = "(3 + x) / -2" then () else error "rational.sml 7";

(*SRAM Schalk I, p.92 Nr. 476a*)

val t = str2term "(x^^^2/(1 - x^^^2) + 1)/(x/(1 - x) + 1) * (1 + x)";

(*. a/b : c/d translated to a/b * d/c .*)

val SOME (t',_) = rewrite_set_ thy false norm_Rational t; UnparseC.term t';

if UnparseC.term t' = "1" then () else error "rational.sml 8";

(*Schalk I, p.92 Nr. 472a*)

val t = str2term "((8*x^^^2 - 32*y^^^2)/(2*x + 4*y))/((4*x - 8*y)/(x + y))";