1 (* rationals, fractions of multivariate polynomials over the real field
3 Copyright (c) isac team 2002, 2013
4 Use is subject to license terms.
6 depends on Poly (and not on Atools), because
7 fractions with _normalised_ polynomials are canceled, added, etc.
11 imports Poly "~~/src/Tools/isac/Knowledge/GCD_Poly_ML"
14 section \<open>Constants for evaluation by "Rule.Calc"\<close>
17 is'_expanded :: "real => bool" ("_ is'_expanded") (*RL->Poly.thy*)
18 is'_ratpolyexp :: "real => bool" ("_ is'_ratpolyexp")
19 get_denominator :: "real => real"
20 get_numerator :: "real => real"
23 (*.the expression contains + - * ^ / only ?.*)
24 fun is_ratpolyexp (Free _) = true
25 | is_ratpolyexp (Const ("Groups.plus_class.plus",_) $ Free _ $ Free _) = true
26 | is_ratpolyexp (Const ("Groups.minus_class.minus",_) $ Free _ $ Free _) = true
27 | is_ratpolyexp (Const ("Groups.times_class.times",_) $ Free _ $ Free _) = true
28 | is_ratpolyexp (Const ("Atools.pow",_) $ Free _ $ Free _) = true
29 | is_ratpolyexp (Const ("Rings.divide_class.divide",_) $ Free _ $ Free _) = true
30 | is_ratpolyexp (Const ("Groups.plus_class.plus",_) $ t1 $ t2) =
31 ((is_ratpolyexp t1) andalso (is_ratpolyexp t2))
32 | is_ratpolyexp (Const ("Groups.minus_class.minus",_) $ t1 $ t2) =
33 ((is_ratpolyexp t1) andalso (is_ratpolyexp t2))
34 | is_ratpolyexp (Const ("Groups.times_class.times",_) $ t1 $ t2) =
35 ((is_ratpolyexp t1) andalso (is_ratpolyexp t2))
36 | is_ratpolyexp (Const ("Atools.pow",_) $ t1 $ t2) =
37 ((is_ratpolyexp t1) andalso (is_ratpolyexp t2))
38 | is_ratpolyexp (Const ("Rings.divide_class.divide",_) $ t1 $ t2) =
39 ((is_ratpolyexp t1) andalso (is_ratpolyexp t2))
40 | is_ratpolyexp _ = false;
42 (*("is_ratpolyexp", ("Rational.is'_ratpolyexp", eval_is_ratpolyexp ""))*)
43 fun eval_is_ratpolyexp (thmid:string) _
44 (t as (Const("Rational.is'_ratpolyexp", _) $ arg)) thy =
46 then SOME (TermC.mk_thmid thmid (Rule.term_to_string''' thy arg) "",
47 HOLogic.Trueprop $ (TermC.mk_equality (t, @{term True})))
48 else SOME (TermC.mk_thmid thmid (Rule.term_to_string''' thy arg) "",
49 HOLogic.Trueprop $ (TermC.mk_equality (t, @{term False})))
50 | eval_is_ratpolyexp _ _ _ _ = NONE;
52 (*("get_denominator", ("Rational.get_denominator", eval_get_denominator ""))*)
53 fun eval_get_denominator (thmid:string) _
54 (t as Const ("Rational.get_denominator", _) $
55 (Const ("Rings.divide_class.divide", _) $ num $
57 SOME (TermC.mk_thmid thmid (Rule.term_to_string''' thy denom) "",
58 HOLogic.Trueprop $ (TermC.mk_equality (t, denom)))
59 | eval_get_denominator _ _ _ _ = NONE;
61 (*("get_numerator", ("Rational.get_numerator", eval_get_numerator ""))*)
62 fun eval_get_numerator (thmid:string) _
63 (t as Const ("Rational.get_numerator", _) $
64 (Const ("Rings.divide_class.divide", _) $num
66 SOME (TermC.mk_thmid thmid (Rule.term_to_string''' thy num) "",
67 HOLogic.Trueprop $ (TermC.mk_equality (t, num)))
68 | eval_get_numerator _ _ _ _ = NONE;
71 section \<open>Theorems for rewriting\<close>
73 axiomatization (* naming due to Isabelle2002, but not contained in Isabelle2002;
74 many thms are due to RL and can be removed with updating the equation solver;
75 TODO: replace by equivalent thms in recent Isabelle201x *)
77 mult_cross: "[| b ~= 0; d ~= 0 |] ==> (a / b = c / d) = (a * d = b * c)" and
78 mult_cross1: " b ~= 0 ==> (a / b = c ) = (a = b * c)" and
79 mult_cross2: " d ~= 0 ==> (a = c / d) = (a * d = c)" and
81 add_minus: "a + b - b = a"(*RL->Poly.thy*) and
82 add_minus1: "a - b + b = a"(*RL->Poly.thy*) and
84 rat_mult: "a / b * (c / d) = a * c / (b * d)"(*?Isa02*) and
85 rat_mult2: "a / b * c = a * c / b "(*?Isa02*) and
87 rat_mult_poly_l: "c is_polyexp ==> c * (a / b) = c * a / b" and
88 rat_mult_poly_r: "c is_polyexp ==> (a / b) * c = a * c / b" and
90 (*real_times_divide1_eq .. Isa02*)
91 real_times_divide_1_eq: "-1 * (c / d) = -1 * c / d " and
92 real_times_divide_num: "a is_const ==> a * (c / d) = a * c / d " and
94 real_mult_div_cancel2: "k ~= 0 ==> m * k / (n * k) = m / n" and
95 (*real_mult_div_cancel1: "k ~= 0 ==> k * m / (k * n) = m / n"..Isa02*)
97 real_divide_divide1: "y ~= 0 ==> (u / v) / (y / z) = (u / v) * (z / y)" and
98 real_divide_divide1_mg: "y ~= 0 ==> (u / v) / (y / z) = (u * z) / (y * v)" and
99 (*real_divide_divide2_eq: "x / y / z = x / (y * z)"..Isa02*)
101 rat_power: "(a / b)^^^n = (a^^^n) / (b^^^n)" and
103 rat_add: "[| a is_const; b is_const; c is_const; d is_const |] ==>
104 a / c + b / d = (a * d + b * c) / (c * d)" and
105 rat_add_assoc: "[| a is_const; b is_const; c is_const; d is_const |] ==>
106 a / c +(b / d + e) = (a * d + b * c)/(d * c) + e" and
107 rat_add1: "[| a is_const; b is_const; c is_const |] ==>
108 a / c + b / c = (a + b) / c" and
109 rat_add1_assoc: "[| a is_const; b is_const; c is_const |] ==>
110 a / c + (b / c + e) = (a + b) / c + e" and
111 rat_add2: "[| a is_const; b is_const; c is_const |] ==>
112 a / c + b = (a + b * c) / c" and
113 rat_add2_assoc: "[| a is_const; b is_const; c is_const |] ==>
114 a / c + (b + e) = (a + b * c) / c + e" and
115 rat_add3: "[| a is_const; b is_const; c is_const |] ==>
116 a + b / c = (a * c + b) / c" and
117 rat_add3_assoc: "[| a is_const; b is_const; c is_const |] ==>
118 a + (b / c + e) = (a * c + b) / c + e"
120 section \<open>Cancellation and addition of fractions\<close>
121 subsection \<open>Conversion term <--> poly\<close>
122 subsubsection \<open>Convert a term to the internal representation of a multivariate polynomial\<close>
124 fun monom_of_term vs (c, es) (Free (id, _)) =
126 then (id |> TermC.int_of_str_opt |> the |> curry op * c, es) (*several numerals in one monom*)
127 else (c, list_update es (find_index (curry op = id) vs) 1)
128 | monom_of_term vs (c, es) (Const ("Atools.pow", _) $ Free (id, _) $ Free (e, _)) =
129 (c, list_update es (find_index (curry op = id) vs) (the (TermC.int_of_str_opt e)))
130 | monom_of_term vs (c, es) (Const ("Groups.times_class.times", _) $ m1 $ m2) =
131 let val (c', es') = monom_of_term vs (c, es) m1
132 in monom_of_term vs (c', es') m2 end
133 | monom_of_term _ _ t = raise ERROR ("poly malformed with " ^ Rule.term2str t)
135 fun monoms_of_term vs (t as Free _) =
136 [monom_of_term vs (1, replicate (length vs) 0) t]
137 | monoms_of_term vs (t as Const ("Atools.pow", _) $ _ $ _) =
138 [monom_of_term vs (1, replicate (length vs) 0) t]
139 | monoms_of_term vs (t as Const ("Groups.times_class.times", _) $ _ $ _) =
140 [monom_of_term vs (1, replicate (length vs) 0) t]
141 | monoms_of_term vs (Const ("Groups.plus_class.plus", _) $ ms1 $ ms2) =
142 (monoms_of_term vs ms1) @ (monoms_of_term vs ms2)
143 | monoms_of_term _ t = raise ERROR ("poly malformed with " ^ Rule.term2str t)
145 (* convert a term to the internal representation of a multivariate polynomial;
146 the conversion is quite liberal, see test --- fun poly_of_term ---:
147 * the order of variables and the parentheses within a monomial are arbitrary
148 * the coefficient may be somewhere
149 * he order and the parentheses within monomials are arbitrary
150 But the term must be completely expand + over * (laws of distributivity are not applicable).
152 The function requires the free variables as strings already given,
153 because the gcd involves 2 polynomials (with the same length for their list of exponents).
155 fun poly_of_term vs (t as Const ("Groups.plus_class.plus", _) $ _ $ _) =
156 (SOME (t |> monoms_of_term vs |> order)
157 handle ERROR _ => NONE)
158 | poly_of_term vs t =
159 (SOME [monom_of_term vs (1, replicate (length vs) 0) t]
160 handle ERROR _ => NONE)
164 val vs = t |> TermC.vars |> map TermC.str_of_free_opt (* tolerate Var in simplification *)
165 |> filter is_some |> map the |> sort string_ord
167 case poly_of_term vs t of SOME _ => true | NONE => false
169 val is_expanded = is_poly (* TODO: check names *)
170 val is_polynomial = is_poly (* TODO: check names *)
173 subsubsection \<open>Convert internal representation of a multivariate polynomial to a term\<close>
175 fun term_of_es _ _ _ [] = [] (*assumes same length for vs and es*)
176 | term_of_es baseT expT (_ :: vs) (0 :: es) =
177 [] @ term_of_es baseT expT vs es
178 | term_of_es baseT expT (v :: vs) (1 :: es) =
179 [(Free (v, baseT))] @ term_of_es baseT expT vs es
180 | term_of_es baseT expT (v :: vs) (e :: es) =
181 [Const ("Atools.pow", [baseT, expT] ---> baseT) $
182 (Free (v, baseT)) $ (Free (TermC.isastr_of_int e, expT))]
183 @ term_of_es baseT expT vs es
185 fun term_of_monom baseT expT vs ((c, es): monom) =
186 let val es' = term_of_es baseT expT vs es
190 if es' = [] (*if es = [0,0,0,...]*)
191 then Free (TermC.isastr_of_int c, baseT)
192 else foldl (HOLogic.mk_binop "Groups.times_class.times") (hd es', tl es')
193 else foldl (HOLogic.mk_binop "Groups.times_class.times") (Free (TermC.isastr_of_int c, baseT), es')
196 fun term_of_poly baseT expT vs p =
197 let val monos = map (term_of_monom baseT expT vs) p
198 in foldl (HOLogic.mk_binop "Groups.plus_class.plus") (hd monos, tl monos) end
201 subsection \<open>Apply gcd_poly for cancelling and adding fractions as terms\<close>
203 fun mk_noteq_0 baseT t =
204 Const ("HOL.Not", HOLogic.boolT --> HOLogic.boolT) $
205 (Const ("HOL.eq", [baseT, baseT] ---> HOLogic.boolT) $ t $ Free ("0", HOLogic.realT))
207 fun mk_asms baseT ts =
208 let val as' = filter_out TermC.is_num ts (* asm like "2 ~= 0" is needless *)
209 in map (mk_noteq_0 baseT) as' end
212 subsubsection \<open>Factor out gcd for cancellation\<close>
214 fun check_fraction t =
215 let val Const ("Rings.divide_class.divide", _) $ numerator $ denominator = t
216 in SOME (numerator, denominator) end
219 (* prepare a term for cancellation by factoring out the gcd
220 assumes: is a fraction with outmost "/"*)
221 fun factout_p_ (thy: theory) t =
222 let val opt = check_fraction t
226 | SOME (numerator, denominator) =>
228 val vs = t |> TermC.vars |> map TermC.str_of_free_opt (* tolerate Var in simplification *)
229 |> filter is_some |> map the |> sort string_ord
230 val baseT = type_of numerator
231 val expT = HOLogic.realT
233 case (poly_of_term vs numerator, poly_of_term vs denominator) of
236 val ((a', b'), c) = gcd_poly a b
237 val es = replicate (length vs) 0
239 if c = [(1, es)] orelse c = [(~1, es)]
243 val b't = term_of_poly baseT expT vs b'
244 val ct = term_of_poly baseT expT vs c
246 HOLogic.mk_binop "Rings.divide_class.divide"
247 (HOLogic.mk_binop "Groups.times_class.times"
248 (term_of_poly baseT expT vs a', ct),
249 HOLogic.mk_binop "Groups.times_class.times" (b't, ct))
250 in SOME (t', mk_asms baseT [b't, ct]) end
252 | _ => NONE : (term * term list) option
257 subsubsection \<open>Cancel a fraction\<close>
259 (* cancel a term by the gcd ("" denote terms with internal algebraic structure)
260 cancel_p_ :: theory \<Rightarrow> term \<Rightarrow> (term \<times> term list) option
261 cancel_p_ thy "a / b" = SOME ("a' / b'", ["b' \<noteq> 0"])
262 assumes: a is_polynomial \<and> b is_polynomial \<and> b \<noteq> 0
264 SOME ("a' / b'", ["b' \<noteq> 0"]). gcd_poly a b \<noteq> 1 \<and> gcd_poly a b \<noteq> -1 \<and>
265 a' * gcd_poly a b = a \<and> b' * gcd_poly a b = b
267 fun cancel_p_ (_: theory) t =
268 let val opt = check_fraction t
272 | SOME (numerator, denominator) =>
274 val vs = t |> TermC.vars |> map TermC.str_of_free_opt (* tolerate Var in simplification *)
275 |> filter is_some |> map the |> sort string_ord
276 val baseT = type_of numerator
277 val expT = HOLogic.realT
279 case (poly_of_term vs numerator, poly_of_term vs denominator) of
282 val ((a', b'), c) = gcd_poly a b
283 val es = replicate (length vs) 0
285 if c = [(1, es)] orelse c = [(~1, es)]
289 val bt' = term_of_poly baseT expT vs b'
290 val ct = term_of_poly baseT expT vs c
292 HOLogic.mk_binop "Rings.divide_class.divide"
293 (term_of_poly baseT expT vs a', bt')
294 val asm = mk_asms baseT [bt']
295 in SOME (t', asm) end
297 | _ => NONE : (term * term list) option
302 subsubsection \<open>Factor out to a common denominator for addition\<close>
304 (* addition of fractions allows (at most) one non-fraction (a monomial) *)
306 (Const ("Groups.plus_class.plus", _) $
307 (Const ("Rings.divide_class.divide", _) $ n1 $ d1) $
308 (Const ("Rings.divide_class.divide", _) $ n2 $ d2))
309 = SOME ((n1, d1), (n2, d2))
311 (Const ("Groups.plus_class.plus", _) $
313 (Const ("Rings.divide_class.divide", _) $ n2 $ d2))
314 = SOME ((nofrac, Free ("1", HOLogic.realT)), (n2, d2))
316 (Const ("Groups.plus_class.plus", _) $
317 (Const ("Rings.divide_class.divide", _) $ n1 $ d1) $
319 = SOME ((n1, d1), (nofrac, Free ("1", HOLogic.realT)))
320 | check_frac_sum _ = NONE
322 (* prepare a term for addition by providing the least common denominator as a product
323 assumes: is a term with outmost "+" and at least one outmost "/" in respective summands*)
324 fun common_nominator_p_ (_: theory) t =
325 let val opt = check_frac_sum t
329 | SOME ((n1, d1), (n2, d2)) =>
331 val vs = t |> TermC.vars |> map TermC.str_of_free_opt (* tolerate Var in simplification *)
332 |> filter is_some |> map the |> sort string_ord
334 case (poly_of_term vs d1, poly_of_term vs d2) of
337 val ((a', b'), c) = gcd_poly a b
338 val (baseT, expT) = (type_of n1, HOLogic.realT)
339 val [d1', d2', c'] = map (term_of_poly baseT expT vs) [a', b', c]
340 (*----- minimum of parentheses & nice result, but breaks tests: -------------
341 val denom = HOLogic.mk_binop "Groups.times_class.times"
342 (HOLogic.mk_binop "Groups.times_class.times" (d1', d2'), c') -------------*)
344 if c = [(1, replicate (length vs) 0)]
345 then HOLogic.mk_binop "Groups.times_class.times" (d1', d2')
347 HOLogic.mk_binop "Groups.times_class.times" (c',
348 HOLogic.mk_binop "Groups.times_class.times" (d1', d2')) (*--------------*)
350 HOLogic.mk_binop "Groups.plus_class.plus"
351 (HOLogic.mk_binop "Rings.divide_class.divide"
352 (HOLogic.mk_binop "Groups.times_class.times" (n1, d2'), denom),
353 HOLogic.mk_binop "Rings.divide_class.divide"
354 (HOLogic.mk_binop "Groups.times_class.times" (n2, d1'), denom))
355 val asm = mk_asms baseT [d1', d2', c']
356 in SOME (t', asm) end
357 | _ => NONE : (term * term list) option
363 subsubsection \<open>Addition of at least one fraction within a sum\<close>
366 assumes: is a term with outmost "+" and at least one outmost "/" in respective summands
367 NOTE: the case "(_ + _) + _" need not be considered due to iterated addition.*)
368 fun add_fraction_p_ (_: theory) t =
369 case check_frac_sum t of
371 | SOME ((n1, d1), (n2, d2)) =>
373 val vs = t |> TermC.vars |> map TermC.str_of_free_opt (* tolerate Var in simplification *)
374 |> filter is_some |> map the |> sort string_ord
376 case (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2) of
377 (SOME _, SOME a, SOME _, SOME b) =>
379 val ((a', b'), c) = gcd_poly a b
380 val (baseT, expT) = (type_of n1, HOLogic.realT)
381 val nomin = term_of_poly baseT expT vs
382 (((the (poly_of_term vs n1)) %%*%% b') %%+%% ((the (poly_of_term vs n2)) %%*%% a'))
383 val denom = term_of_poly baseT expT vs ((c %%*%% a') %%*%% b')
384 val t' = HOLogic.mk_binop "Rings.divide_class.divide" (nomin, denom)
385 in SOME (t', mk_asms baseT [denom]) end
386 | _ => NONE : (term * term list) option
390 section \<open>Embed cancellation and addition into rewriting\<close>
391 ML \<open>val thy = @{theory}\<close>
392 subsection \<open>Rulesets and predicate for embedding\<close>
394 (* evaluates conditions in calculate_Rational *)
397 (Rule.Rls {id = "calc_rat_erls", preconds = [], rew_ord = ("dummy_ord", Rule.dummy_ord),
398 erls = Rule.e_rls, srls = Rule.Erls, calc = [], errpatts = [],
400 [Rule.Calc ("HOL.eq", eval_equal "#equal_"),
401 Rule.Calc ("Atools.is'_const", eval_const "#is_const_"),
402 Rule.Thm ("not_true", TermC.num_str @{thm not_true}),
403 Rule.Thm ("not_false", TermC.num_str @{thm not_false})],
404 scr = Rule.EmptyScr});
406 (* simplifies expressions with numerals;
407 does NOT rearrange the term by AC-rewriting; thus terms with variables
408 need to have constants to be commuted together respectively *)
409 val calculate_Rational =
410 prep_rls' (Rule.merge_rls "calculate_Rational"
411 (Rule.Rls {id = "divide", preconds = [], rew_ord = ("dummy_ord", Rule.dummy_ord),
412 erls = calc_rat_erls, srls = Rule.Erls,
413 calc = [], errpatts = [],
415 [Rule.Calc ("Rings.divide_class.divide", eval_cancel "#divide_e"),
417 Rule.Thm ("minus_divide_left", TermC.num_str (@{thm minus_divide_left} RS @{thm sym})),
418 (*SYM - ?x / ?y = - (?x / ?y) may come from subst*)
419 Rule.Thm ("rat_add", TermC.num_str @{thm rat_add}),
420 (*"[| a is_const; b is_const; c is_const; d is_const |] ==> \
421 \a / c + b / d = (a * d) / (c * d) + (b * c ) / (d * c)"*)
422 Rule.Thm ("rat_add1", TermC.num_str @{thm rat_add1}),
423 (*"[| a is_const; b is_const; c is_const |] ==> a / c + b / c = (a + b) / c"*)
424 Rule.Thm ("rat_add2", TermC.num_str @{thm rat_add2}),
425 (*"[| ?a is_const; ?b is_const; ?c is_const |] ==> ?a / ?c + ?b = (?a + ?b * ?c) / ?c"*)
426 Rule.Thm ("rat_add3", TermC.num_str @{thm rat_add3}),
427 (*"[| a is_const; b is_const; c is_const |] ==> a + b / c = (a * c) / c + b / c"\
428 .... is_const to be omitted here FIXME*)
430 Rule.Thm ("rat_mult", TermC.num_str @{thm rat_mult}),
431 (*a / b * (c / d) = a * c / (b * d)*)
432 Rule.Thm ("times_divide_eq_right", TermC.num_str @{thm times_divide_eq_right}),
433 (*?x * (?y / ?z) = ?x * ?y / ?z*)
434 Rule.Thm ("times_divide_eq_left", TermC.num_str @{thm times_divide_eq_left}),
435 (*?y / ?z * ?x = ?y * ?x / ?z*)
437 Rule.Thm ("real_divide_divide1", TermC.num_str @{thm real_divide_divide1}),
438 (*"?y ~= 0 ==> ?u / ?v / (?y / ?z) = ?u / ?v * (?z / ?y)"*)
439 Rule.Thm ("divide_divide_eq_left", TermC.num_str @{thm divide_divide_eq_left}),
440 (*"?x / ?y / ?z = ?x / (?y * ?z)"*)
442 Rule.Thm ("rat_power", TermC.num_str @{thm rat_power}),
443 (*"(?a / ?b) ^^^ ?n = ?a ^^^ ?n / ?b ^^^ ?n"*)
445 Rule.Thm ("mult_cross", TermC.num_str @{thm mult_cross}),
446 (*"[| b ~= 0; d ~= 0 |] ==> (a / b = c / d) = (a * d = b * c)*)
447 Rule.Thm ("mult_cross1", TermC.num_str @{thm mult_cross1}),
448 (*" b ~= 0 ==> (a / b = c ) = (a = b * c)*)
449 Rule.Thm ("mult_cross2", TermC.num_str @{thm mult_cross2})
450 (*" d ~= 0 ==> (a = c / d) = (a * d = c)*)],
451 scr = Rule.EmptyScr})
454 (*("is_expanded", ("Rational.is'_expanded", eval_is_expanded ""))*)
455 fun eval_is_expanded (thmid:string) _
456 (t as (Const("Rational.is'_expanded", _) $ arg)) thy =
458 then SOME (TermC.mk_thmid thmid (Rule.term_to_string''' thy arg) "",
459 HOLogic.Trueprop $ (TermC.mk_equality (t, @{term True})))
460 else SOME (TermC.mk_thmid thmid (Rule.term_to_string''' thy arg) "",
461 HOLogic.Trueprop $ (TermC.mk_equality (t, @{term False})))
462 | eval_is_expanded _ _ _ _ = NONE;
464 setup \<open>KEStore_Elems.add_calcs
465 [("is_expanded", ("Rational.is'_expanded", eval_is_expanded ""))]\<close>
468 Rule.merge_rls "rational_erls" calculate_Rational
469 (Rule.append_rls "is_expanded" Atools_erls
470 [Rule.Calc ("Rational.is'_expanded", eval_is_expanded "")]);
473 subsection \<open>Embed cancellation into rewriting\<close>
477 val {rules = rules, rew_ord = (_, ro), ...} = Rule.rep_rls (assoc_rls' @{theory} "rev_rew_p");
479 fun init_state thy eval_rls ro t =
481 val SOME (t', _) = factout_p_ thy t;
482 val SOME (t'', asm) = cancel_p_ thy t;
483 val der = Rtools.reverse_deriv thy eval_rls rules ro NONE t';
485 [(Rule.Thm ("real_mult_div_cancel2", TermC.num_str @{thm real_mult_div_cancel2}), (t'', asm))]
486 val rs = (Rtools.distinct_Thm o (map #1)) der
487 val rs = filter_out (Rtools.eq_Thms
488 ["sym_real_add_zero_left", "sym_real_mult_0", "sym_real_mult_1"]) rs
489 in (t, t'', [rs(*one in order to ease locate_rule*)], der) end;
491 fun locate_rule thy eval_rls ro [rs] t r =
492 if member op = ((map (Celem.id_of_thm)) rs) (Celem.id_of_thm r)
494 let val ropt = Rewrite.rewrite_ thy ro eval_rls true (Celem.thm_of_thm r) t;
496 case ropt of SOME ta => [(r, ta)]
498 ("### locate_rule: rewrite " ^ Celem.id_of_thm r ^ " " ^ Rule.term2str t ^ " = NONE"); [])
500 else (tracing ("### locate_rule: " ^ Celem.id_of_thm r ^ " not mem rrls"); [])
501 | locate_rule _ _ _ _ _ _ = error "locate_rule: doesnt match rev-sets in istate";
503 fun next_rule thy eval_rls ro [rs] t =
505 val der = Rtools.make_deriv thy eval_rls rs ro NONE t;
506 in case der of (_, r, _) :: _ => SOME r | _ => NONE end
507 | next_rule _ _ _ _ _ = error ("next_rule: doesnt match rev-sets in istate");
509 fun attach_form (_: Rule.rule list list) (_: term) (_: term) =
510 [(*TODO*)]: ( Rule.rule * (term * term list)) list;
515 Rule.Rrls {id = "cancel_p", prepat = [],
516 rew_ord=("ord_make_polynomial", ord_make_polynomial false thy),
517 erls = rational_erls,
519 [("PLUS", ("Groups.plus_class.plus", eval_binop "#add_")),
520 ("TIMES" , ("Groups.times_class.times", eval_binop "#mult_")),
521 ("DIVIDE", ("Rings.divide_class.divide", eval_cancel "#divide_e")),
522 ("POWER", ("Atools.pow", eval_binop "#power_"))],
525 Rule.Rfuns {init_state = init_state thy Atools_erls ro,
526 normal_form = cancel_p_ thy,
527 locate_rule = locate_rule thy Atools_erls ro,
528 next_rule = next_rule thy Atools_erls ro,
529 attach_form = attach_form}}
530 end; (* local cancel_p *)
533 subsection \<open>Embed addition into rewriting\<close>
535 local (* add_fractions_p *)
537 (*val {rules = rules, rew_ord = (_, ro), ...} = Rule.rep_rls (assoc_rls "make_polynomial");*)
538 val {rules, rew_ord=(_,ro),...} = Rule.rep_rls (assoc_rls' @{theory} "rev_rew_p");
540 fun init_state thy eval_rls ro t =
542 val SOME (t',_) = common_nominator_p_ thy t;
543 val SOME (t'', asm) = add_fraction_p_ thy t;
544 val der = Rtools.reverse_deriv thy eval_rls rules ro NONE t';
546 [(Rule.Thm ("real_mult_div_cancel2", TermC.num_str @{thm real_mult_div_cancel2}), (t'',asm))]
547 val rs = (Rtools.distinct_Thm o (map #1)) der;
548 val rs = filter_out (Rtools.eq_Thms
549 ["sym_real_add_zero_left", "sym_real_mult_0", "sym_real_mult_1"]) rs;
550 in (t, t'', [rs(*here only _ONE_*)], der) end;
552 fun locate_rule thy eval_rls ro [rs] t r =
553 if member op = ((map (Celem.id_of_thm)) rs) (Celem.id_of_thm r)
555 let val ropt = Rewrite.rewrite_ thy ro eval_rls true (Celem.thm_of_thm r) t;
560 (tracing ("### locate_rule: rewrite " ^ Celem.id_of_thm r ^ " " ^ Rule.term2str t ^ " = NONE");
562 else (tracing ("### locate_rule: " ^ Celem.id_of_thm r ^ " not mem rrls"); [])
563 | locate_rule _ _ _ _ _ _ = error "locate_rule: doesnt match rev-sets in istate";
565 fun next_rule thy eval_rls ro [rs] t =
566 let val der = Rtools.make_deriv thy eval_rls rs ro NONE t;
572 | next_rule _ _ _ _ _ = error ("next_rule: doesnt match rev-sets in istate");
574 val pat0 = TermC.parse_patt thy "?r/?s+?u/?v :: real";
575 val pat1 = TermC.parse_patt thy "?r/?s+?u :: real";
576 val pat2 = TermC.parse_patt thy "?r +?u/?v :: real";
577 val prepat = [([@{term True}], pat0),
578 ([@{term True}], pat1),
579 ([@{term True}], pat2)];
582 val add_fractions_p =
583 Rule.Rrls {id = "add_fractions_p", prepat=prepat,
584 rew_ord = ("ord_make_polynomial", ord_make_polynomial false thy),
585 erls = rational_erls,
586 calc = [("PLUS", ("Groups.plus_class.plus", eval_binop "#add_")),
587 ("TIMES", ("Groups.times_class.times", eval_binop "#mult_")),
588 ("DIVIDE", ("Rings.divide_class.divide", eval_cancel "#divide_e")),
589 ("POWER", ("Atools.pow", eval_binop "#power_"))],
591 scr = Rule.Rfuns {init_state = init_state thy Atools_erls ro,
592 normal_form = add_fraction_p_ thy,
593 locate_rule = locate_rule thy Atools_erls ro,
594 next_rule = next_rule thy Atools_erls ro,
595 attach_form = attach_form}}
596 end; (*local add_fractions_p *)
599 subsection \<open>Cancelling and adding all occurrences in a term /////////////////////////////\<close>
601 (*copying cancel_p_rls + add her caused error in interface.sml*)
604 section \<open>Rulesets for general simplification\<close>
607 (*-------------------18.3.03 --> struct <-----------vvv--
608 val add_fractions_p = common_nominator_p; (*FIXXXME:eilig f"ur norm_Rational*)
609 -------------------18.3.03 --> struct <-----------vvv--*)
611 (*erls for calculate_Rational; make local with FIXX@ME result:term *term list*)
612 val powers_erls = prep_rls'(
613 Rule.Rls {id = "powers_erls", preconds = [], rew_ord = ("dummy_ord",Rule.dummy_ord),
614 erls = Rule.e_rls, srls = Rule.Erls, calc = [], errpatts = [],
615 rules = [Rule.Calc ("Atools.is'_atom",eval_is_atom "#is_atom_"),
616 Rule.Calc ("Atools.is'_even",eval_is_even "#is_even_"),
617 Rule.Calc ("Orderings.ord_class.less",eval_equ "#less_"),
618 Rule.Thm ("not_false", TermC.num_str @{thm not_false}),
619 Rule.Thm ("not_true", TermC.num_str @{thm not_true}),
620 Rule.Calc ("Groups.plus_class.plus",eval_binop "#add_")
624 (*.all powers over + distributed; atoms over * collected, other distributed
625 contains absolute minimum of thms for context in norm_Rational .*)
626 val powers = prep_rls'(
627 Rule.Rls {id = "powers", preconds = [], rew_ord = ("dummy_ord",Rule.dummy_ord),
628 erls = powers_erls, srls = Rule.Erls, calc = [], errpatts = [],
629 rules = [Rule.Thm ("realpow_multI", TermC.num_str @{thm realpow_multI}),
630 (*"(r * s) ^^^ n = r ^^^ n * s ^^^ n"*)
631 Rule.Thm ("realpow_pow",TermC.num_str @{thm realpow_pow}),
632 (*"(a ^^^ b) ^^^ c = a ^^^ (b * c)"*)
633 Rule.Thm ("realpow_oneI",TermC.num_str @{thm realpow_oneI}),
635 Rule.Thm ("realpow_minus_even",TermC.num_str @{thm realpow_minus_even}),
636 (*"n is_even ==> (- r) ^^^ n = r ^^^ n" ?-->discard_minus?*)
637 Rule.Thm ("realpow_minus_odd",TermC.num_str @{thm realpow_minus_odd}),
638 (*"Not (n is_even) ==> (- r) ^^^ n = -1 * r ^^^ n"*)
640 (*----- collect atoms over * -----*)
641 Rule.Thm ("realpow_two_atom",TermC.num_str @{thm realpow_two_atom}),
642 (*"r is_atom ==> r * r = r ^^^ 2"*)
643 Rule.Thm ("realpow_plus_1",TermC.num_str @{thm realpow_plus_1}),
644 (*"r is_atom ==> r * r ^^^ n = r ^^^ (n + 1)"*)
645 Rule.Thm ("realpow_addI_atom",TermC.num_str @{thm realpow_addI_atom}),
646 (*"r is_atom ==> r ^^^ n * r ^^^ m = r ^^^ (n + m)"*)
648 (*----- distribute none-atoms -----*)
649 Rule.Thm ("realpow_def_atom",TermC.num_str @{thm realpow_def_atom}),
650 (*"[| 1 < n; not(r is_atom) |]==>r ^^^ n = r * r ^^^ (n + -1)"*)
651 Rule.Thm ("realpow_eq_oneI",TermC.num_str @{thm realpow_eq_oneI}),
653 Rule.Calc ("Groups.plus_class.plus",eval_binop "#add_")
657 (*.contains absolute minimum of thms for context in norm_Rational.*)
658 val rat_mult_divide = prep_rls'(
659 Rule.Rls {id = "rat_mult_divide", preconds = [],
660 rew_ord = ("dummy_ord", Rule.dummy_ord),
661 erls = Rule.e_rls, srls = Rule.Erls, calc = [], errpatts = [],
662 rules = [Rule.Thm ("rat_mult",TermC.num_str @{thm rat_mult}),
663 (*(1)"?a / ?b * (?c / ?d) = ?a * ?c / (?b * ?d)"*)
664 Rule.Thm ("times_divide_eq_right",TermC.num_str @{thm times_divide_eq_right}),
665 (*(2)"?a * (?c / ?d) = ?a * ?c / ?d" must be [2],
666 otherwise inv.to a / b / c = ...*)
667 Rule.Thm ("times_divide_eq_left",TermC.num_str @{thm times_divide_eq_left}),
668 (*"?a / ?b * ?c = ?a * ?c / ?b" order weights x^^^n too much
669 and does not commute a / b * c ^^^ 2 !*)
671 Rule.Thm ("divide_divide_eq_right",
672 TermC.num_str @{thm divide_divide_eq_right}),
673 (*"?x / (?y / ?z) = ?x * ?z / ?y"*)
674 Rule.Thm ("divide_divide_eq_left",
675 TermC.num_str @{thm divide_divide_eq_left}),
676 (*"?x / ?y / ?z = ?x / (?y * ?z)"*)
677 Rule.Calc ("Rings.divide_class.divide" ,eval_cancel "#divide_e")
682 (*.contains absolute minimum of thms for context in norm_Rational.*)
683 val reduce_0_1_2 = prep_rls'(
684 Rule.Rls{id = "reduce_0_1_2", preconds = [], rew_ord = ("dummy_ord", Rule.dummy_ord),
685 erls = Rule.e_rls, srls = Rule.Erls, calc = [], errpatts = [],
686 rules = [(*Rule.Thm ("divide_1",TermC.num_str @{thm divide_1}),
687 "?x / 1 = ?x" unnecess.for normalform*)
688 Rule.Thm ("mult_1_left",TermC.num_str @{thm mult_1_left}),
690 (*Rule.Thm ("real_mult_minus1",TermC.num_str @{thm real_mult_minus1}),
692 (*Rule.Thm ("real_minus_mult_cancel",TermC.num_str @{thm real_minus_mult_cancel}),
693 "- ?x * - ?y = ?x * ?y"*)
695 Rule.Thm ("mult_zero_left",TermC.num_str @{thm mult_zero_left}),
697 Rule.Thm ("add_0_left",TermC.num_str @{thm add_0_left}),
699 (*Rule.Thm ("right_minus",TermC.num_str @{thm right_minus}),
702 Rule.Thm ("sym_real_mult_2",
703 TermC.num_str (@{thm real_mult_2} RS @{thm sym})),
704 (*"z1 + z1 = 2 * z1"*)
705 Rule.Thm ("real_mult_2_assoc",TermC.num_str @{thm real_mult_2_assoc}),
706 (*"z1 + (z1 + k) = 2 * z1 + k"*)
708 Rule.Thm ("division_ring_divide_zero",TermC.num_str @{thm division_ring_divide_zero})
710 ], scr = Rule.EmptyScr});
712 (*erls for calculate_Rational;
713 make local with FIXX@ME result:term *term list WN0609???SKMG*)
714 val norm_rat_erls = prep_rls'(
715 Rule.Rls {id = "norm_rat_erls", preconds = [], rew_ord = ("dummy_ord",Rule.dummy_ord),
716 erls = Rule.e_rls, srls = Rule.Erls, calc = [], errpatts = [],
717 rules = [Rule.Calc ("Atools.is'_const",eval_const "#is_const_")
718 ], scr = Rule.EmptyScr});
720 (* consists of rls containing the absolute minimum of thms *)
721 (*040209: this version has been used by RL for his equations,
722 which is now replaced by MGs version "norm_Rational" below *)
723 val norm_Rational_min = prep_rls'(
724 Rule.Rls {id = "norm_Rational_min", preconds = [], rew_ord = ("dummy_ord",Rule.dummy_ord),
725 erls = norm_rat_erls, srls = Rule.Erls, calc = [], errpatts = [],
726 rules = [(*sequence given by operator precedence*)
727 Rule.Rls_ discard_minus,
729 Rule.Rls_ rat_mult_divide,
731 Rule.Rls_ reduce_0_1_2,
732 Rule.Rls_ order_add_mult,
733 Rule.Rls_ collect_numerals,
734 Rule.Rls_ add_fractions_p,
737 scr = Rule.EmptyScr});
739 val norm_Rational_parenthesized = prep_rls'(
740 Rule.Seq {id = "norm_Rational_parenthesized", preconds = []:term list,
741 rew_ord = ("dummy_ord", Rule.dummy_ord),
742 erls = Atools_erls, srls = Rule.Erls,
743 calc = [], errpatts = [],
744 rules = [Rule.Rls_ norm_Rational_min,
745 Rule.Rls_ discard_parentheses
747 scr = Rule.EmptyScr});
749 (*WN030318???SK: simplifies all but cancel and common_nominator*)
750 val simplify_rational =
751 Rule.merge_rls "simplify_rational" expand_binoms
752 (Rule.append_rls "divide" calculate_Rational
753 [Rule.Thm ("div_by_1",TermC.num_str @{thm div_by_1}),
755 Rule.Thm ("rat_mult",TermC.num_str @{thm rat_mult}),
756 (*(1)"?a / ?b * (?c / ?d) = ?a * ?c / (?b * ?d)"*)
757 Rule.Thm ("times_divide_eq_right",TermC.num_str @{thm times_divide_eq_right}),
758 (*(2)"?a * (?c / ?d) = ?a * ?c / ?d" must be [2],
759 otherwise inv.to a / b / c = ...*)
760 Rule.Thm ("times_divide_eq_left",TermC.num_str @{thm times_divide_eq_left}),
761 (*"?a / ?b * ?c = ?a * ?c / ?b"*)
762 Rule.Thm ("add_minus",TermC.num_str @{thm add_minus}),
763 (*"?a + ?b - ?b = ?a"*)
764 Rule.Thm ("add_minus1",TermC.num_str @{thm add_minus1}),
765 (*"?a - ?b + ?b = ?a"*)
766 Rule.Thm ("divide_minus1",TermC.num_str @{thm divide_minus1})
771 val add_fractions_p_rls = prep_rls'(
772 Rule.Rls {id = "add_fractions_p_rls", preconds = [], rew_ord = ("dummy_ord", Rule.dummy_ord),
773 erls = Rule.e_rls, srls = Rule.Erls, calc = [], errpatts = [],
774 rules = [Rule.Rls_ add_fractions_p],
775 scr = Rule.EmptyScr});
777 (* "Rule.Rls" causes repeated application of cancel_p to one and the same term *)
778 val cancel_p_rls = prep_rls'(
780 {id = "cancel_p_rls", preconds = [], rew_ord = ("dummy_ord", Rule.dummy_ord),
781 erls = Rule.e_rls, srls = Rule.Erls, calc = [], errpatts = [],
782 rules = [Rule.Rls_ cancel_p],
783 scr = Rule.EmptyScr});
785 (*. makes 'normal' fractions; 'is_polyexp' inhibits double fractions;
786 used in initial part norm_Rational_mg, see example DA-M02-main.p.60.*)
787 val rat_mult_poly = prep_rls'(
788 Rule.Rls {id = "rat_mult_poly", preconds = [], rew_ord = ("dummy_ord", Rule.dummy_ord),
789 erls = Rule.append_rls "Rule.e_rls-is_polyexp" Rule.e_rls [Rule.Calc ("Poly.is'_polyexp", eval_is_polyexp "")],
790 srls = Rule.Erls, calc = [], errpatts = [],
792 [Rule.Thm ("rat_mult_poly_l",TermC.num_str @{thm rat_mult_poly_l}),
793 (*"?c is_polyexp ==> ?c * (?a / ?b) = ?c * ?a / ?b"*)
794 Rule.Thm ("rat_mult_poly_r",TermC.num_str @{thm rat_mult_poly_r})
795 (*"?c is_polyexp ==> ?a / ?b * ?c = ?a * ?c / ?b"*) ],
796 scr = Rule.EmptyScr});
798 (*. makes 'normal' fractions; 'is_polyexp' inhibits double fractions;
799 used in looping part norm_Rational_rls, see example DA-M02-main.p.60
800 .. WHERE THE LATTER DOES ALWAYS WORK, BECAUSE erls = Rule.e_rls,
801 I.E. THE RESPECTIVE ASSUMPTION IS STORED AND Rule.Thm APPLIED; WN051028
803 val rat_mult_div_pow = prep_rls'(
804 Rule.Rls {id = "rat_mult_div_pow", preconds = [], rew_ord = ("dummy_ord",Rule.dummy_ord),
805 erls = Rule.e_rls, srls = Rule.Erls, calc = [], errpatts = [],
806 rules = [Rule.Thm ("rat_mult", TermC.num_str @{thm rat_mult}),
807 (*"?a / ?b * (?c / ?d) = ?a * ?c / (?b * ?d)"*)
808 Rule.Thm ("rat_mult_poly_l", TermC.num_str @{thm rat_mult_poly_l}),
809 (*"?c is_polyexp ==> ?c * (?a / ?b) = ?c * ?a / ?b"*)
810 Rule.Thm ("rat_mult_poly_r", TermC.num_str @{thm rat_mult_poly_r}),
811 (*"?c is_polyexp ==> ?a / ?b * ?c = ?a * ?c / ?b"*)
813 Rule.Thm ("real_divide_divide1_mg", TermC.num_str @{thm real_divide_divide1_mg}),
814 (*"y ~= 0 ==> (u / v) / (y / z) = (u * z) / (y * v)"*)
815 Rule.Thm ("divide_divide_eq_right", TermC.num_str @{thm divide_divide_eq_right}),
816 (*"?x / (?y / ?z) = ?x * ?z / ?y"*)
817 Rule.Thm ("divide_divide_eq_left", TermC.num_str @{thm divide_divide_eq_left}),
818 (*"?x / ?y / ?z = ?x / (?y * ?z)"*)
819 Rule.Calc ("Rings.divide_class.divide", eval_cancel "#divide_e"),
821 Rule.Thm ("rat_power", TermC.num_str @{thm rat_power})
822 (*"(?a / ?b) ^^^ ?n = ?a ^^^ ?n / ?b ^^^ ?n"*)
824 scr = Rule.EmptyScr});
826 val rat_reduce_1 = prep_rls'(
827 Rule.Rls {id = "rat_reduce_1", preconds = [], rew_ord = ("dummy_ord", Rule.dummy_ord),
828 erls = Rule.e_rls, srls = Rule.Erls, calc = [], errpatts = [],
830 [Rule.Thm ("div_by_1", TermC.num_str @{thm div_by_1}),
832 Rule.Thm ("mult_1_left", TermC.num_str @{thm mult_1_left})
835 scr = Rule.EmptyScr});
837 (* looping part of norm_Rational *)
838 val norm_Rational_rls = prep_rls' (
839 Rule.Rls {id = "norm_Rational_rls", preconds = [], rew_ord = ("dummy_ord",Rule.dummy_ord),
840 erls = norm_rat_erls, srls = Rule.Erls, calc = [], errpatts = [],
841 rules = [Rule.Rls_ add_fractions_p_rls,
842 Rule.Rls_ rat_mult_div_pow,
843 Rule.Rls_ make_rat_poly_with_parentheses,
844 Rule.Rls_ cancel_p_rls,
845 Rule.Rls_ rat_reduce_1
847 scr = Rule.EmptyScr});
849 val norm_Rational = prep_rls' (
851 {id = "norm_Rational", preconds = [], rew_ord = ("dummy_ord", Rule.dummy_ord),
852 erls = norm_rat_erls, srls = Rule.Erls, calc = [], errpatts = [],
853 rules = [Rule.Rls_ discard_minus,
854 Rule.Rls_ rat_mult_poly, (* removes double fractions like a/b/c *)
855 Rule.Rls_ make_rat_poly_with_parentheses,
856 Rule.Rls_ cancel_p_rls,
857 Rule.Rls_ norm_Rational_rls, (* the main rls, looping (#) *)
858 Rule.Rls_ discard_parentheses1 (* mult only *)
860 scr = Rule.EmptyScr});
863 setup \<open>KEStore_Elems.add_rlss
864 [("calculate_Rational", (Context.theory_name @{theory}, calculate_Rational)),
865 ("calc_rat_erls", (Context.theory_name @{theory}, calc_rat_erls)),
866 ("rational_erls", (Context.theory_name @{theory}, rational_erls)),
867 ("cancel_p", (Context.theory_name @{theory}, cancel_p)),
868 ("add_fractions_p", (Context.theory_name @{theory}, add_fractions_p)),
870 ("add_fractions_p_rls", (Context.theory_name @{theory}, add_fractions_p_rls)),
871 ("powers_erls", (Context.theory_name @{theory}, powers_erls)),
872 ("powers", (Context.theory_name @{theory}, powers)),
873 ("rat_mult_divide", (Context.theory_name @{theory}, rat_mult_divide)),
874 ("reduce_0_1_2", (Context.theory_name @{theory}, reduce_0_1_2)),
876 ("rat_reduce_1", (Context.theory_name @{theory}, rat_reduce_1)),
877 ("norm_rat_erls", (Context.theory_name @{theory}, norm_rat_erls)),
878 ("norm_Rational", (Context.theory_name @{theory}, norm_Rational)),
879 ("norm_Rational_rls", (Context.theory_name @{theory}, norm_Rational_rls)),
880 ("norm_Rational_min", (Context.theory_name @{theory}, norm_Rational_min)),
881 ("norm_Rational_parenthesized", (Context.theory_name @{theory}, norm_Rational_parenthesized)),
883 ("rat_mult_poly", (Context.theory_name @{theory}, rat_mult_poly)),
884 ("rat_mult_div_pow", (Context.theory_name @{theory}, rat_mult_div_pow)),
885 ("cancel_p_rls", (Context.theory_name @{theory}, cancel_p_rls))]\<close>
887 section \<open>A problem for simplification of rationals\<close>
888 setup \<open>KEStore_Elems.add_pbts
889 [(Specify.prep_pbt thy "pbl_simp_rat" [] Celem.e_pblID
890 (["rational","simplification"],
891 [("#Given" ,["Term t_t"]),
892 ("#Where" ,["t_t is_ratpolyexp"]),
893 ("#Find" ,["normalform n_n"])],
894 Rule.append_rls "e_rls" Rule.e_rls [(*for preds in where_*)],
895 SOME "Simplify t_t", [["simplification","of_rationals"]]))]\<close>
897 section \<open>A methods for simplification of rationals\<close>
898 (*WN061025 this methods script is copied from (auto-generated) script
899 of norm_Rational in order to ease repair on inform*)
900 setup \<open>KEStore_Elems.add_mets
901 [Specify.prep_met thy "met_simp_rat" [] Celem.e_metID
902 (["simplification","of_rationals"],
903 [("#Given" ,["Term t_t"]),
904 ("#Where" ,["t_t is_ratpolyexp"]),
905 ("#Find" ,["normalform n_n"])],
906 {rew_ord'="tless_true", rls' = Rule.e_rls, calc = [], srls = Rule.e_rls,
907 prls = Rule.append_rls "simplification_of_rationals_prls" Rule.e_rls
908 [(*for preds in where_*) Rule.Calc ("Rational.is'_ratpolyexp", eval_is_ratpolyexp "")],
909 crls = Rule.e_rls, errpats = [], nrls = norm_Rational_rls},
910 "Script SimplifyScript (t_t::real) = " ^
911 " ((Try (Rewrite_Set discard_minus False) @@ " ^
912 " Try (Rewrite_Set rat_mult_poly False) @@ " ^
913 " Try (Rewrite_Set make_rat_poly_with_parentheses False) @@ " ^
914 " Try (Rewrite_Set cancel_p_rls False) @@ " ^
916 " ((Try (Rewrite_Set add_fractions_p_rls False) @@ " ^
917 " Try (Rewrite_Set rat_mult_div_pow False) @@ " ^
918 " Try (Rewrite_Set make_rat_poly_with_parentheses False) @@" ^
919 " Try (Rewrite_Set cancel_p_rls False) @@ " ^
920 " Try (Rewrite_Set rat_reduce_1 False)))) @@ " ^
921 " Try (Rewrite_Set discard_parentheses1 False)) " ^