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 GCD_Poly_ML
14 section \<open>Constants for evaluation by "Rule.Eval"\<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 (\<^const_name>\<open>plus\<close>,_) $ Free _ $ Free _) = true
26 | is_ratpolyexp (Const (\<^const_name>\<open>minus\<close>,_) $ Free _ $ Free _) = true
27 | is_ratpolyexp (Const (\<^const_name>\<open>times\<close>,_) $ Free _ $ Free _) = true
28 | is_ratpolyexp (Const (\<^const_name>\<open>powr\<close>,_) $ Free _ $ Free _) = true
29 | is_ratpolyexp (Const (\<^const_name>\<open>divide\<close>,_) $ Free _ $ Free _) = true
30 | is_ratpolyexp (Const (\<^const_name>\<open>plus\<close>,_) $ t1 $ t2) =
31 ((is_ratpolyexp t1) andalso (is_ratpolyexp t2))
32 | is_ratpolyexp (Const (\<^const_name>\<open>minus\<close>,_) $ t1 $ t2) =
33 ((is_ratpolyexp t1) andalso (is_ratpolyexp t2))
34 | is_ratpolyexp (Const (\<^const_name>\<open>times\<close>,_) $ t1 $ t2) =
35 ((is_ratpolyexp t1) andalso (is_ratpolyexp t2))
36 | is_ratpolyexp (Const (\<^const_name>\<open>powr\<close>,_) $ t1 $ t2) =
37 ((is_ratpolyexp t1) andalso (is_ratpolyexp t2))
38 | is_ratpolyexp (Const (\<^const_name>\<open>divide\<close>,_) $ 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 (\<^const_name>\<open>Rational.is_ratpolyexp\<close>, _) $ arg)) thy =
46 then SOME (TermC.mk_thmid thmid (UnparseC.term_in_thy thy arg) "",
47 HOLogic.Trueprop $ (TermC.mk_equality (t, @{term True})))
48 else SOME (TermC.mk_thmid thmid (UnparseC.term_in_thy 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 (\<^const_name>\<open>Rational.get_denominator\<close>, _) $
55 (Const (\<^const_name>\<open>divide\<close>, _) $ _(*num*) $
57 SOME (TermC.mk_thmid thmid (UnparseC.term_in_thy 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 (\<^const_name>\<open>Rational.get_numerator\<close>, _) $
64 (Const (\<^const_name>\<open>divide\<close>, _) $num
66 SOME (TermC.mk_thmid thmid (UnparseC.term_in_thy 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) \<up> n = (a \<up> n) / (b \<up> 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) (t as Const _) =
125 (c, list_update es (find_index (curry op = t) vs) 1)
126 | monom_of_term _ (c, es) (t as (Const (\<^const_name>\<open>numeral\<close>, _) $ _)) =
127 (t |> HOLogic.dest_number |> snd |> curry op * c, es) (*several numerals in one monom*)
128 | monom_of_term _ (c, es) (t as (Const (\<^const_name>\<open>uminus\<close>, _) $ _)) =
129 (t |> HOLogic.dest_number |> snd |> curry op * c, es) (*several numerals in one monom*)
130 | monom_of_term vs (c, es) (t as Free _) =
131 (c, list_update es (find_index (curry op = t) vs) 1)
132 | monom_of_term vs (c, es) (Const (\<^const_name>\<open>Transcendental.powr\<close>, _) $ (b as Free _) $
133 (e as Const (\<^const_name>\<open>numeral\<close>, _) $ _)) =
134 (c, list_update es (find_index (curry op = b) vs) (e |> HOLogic.dest_number |> snd))
135 | monom_of_term vs (c, es) (Const (\<^const_name>\<open>Transcendental.powr\<close>, _) $ (b as Free _) $
136 (e as Const (\<^const_name>\<open>uminus\<close>, _) $ _)) =
137 (c, list_update es (find_index (curry op = b) vs) (e |> HOLogic.dest_number |> snd))
139 | monom_of_term vs (c, es) (Const (\<^const_name>\<open>times\<close>, _) $ m1 $ m2) =
140 let val (c', es') = monom_of_term vs (c, es) m1
141 in monom_of_term vs (c', es') m2 end
142 | monom_of_term _ _ t = raise ERROR ("poly malformed 1 with " ^ UnparseC.term t)
145 fun monoms_of_term vs (t as Const _) =
146 [monom_of_term vs (1, replicate (length vs) 0) t]
147 | monoms_of_term vs (t as Const (\<^const_name>\<open>numeral\<close>, _) $ _) =
148 [monom_of_term vs (1, replicate (length vs) 0) t]
149 | monoms_of_term vs (t as Const (\<^const_name>\<open>uminus\<close>, _) $ _) =
150 [monom_of_term vs (1, replicate (length vs) 0) t]
151 | monoms_of_term vs (t as Free _) =
152 [monom_of_term vs (1, replicate (length vs) 0) t]
153 | monoms_of_term vs (t as Const (\<^const_name>\<open>powr\<close>, _) $ _ $ _) =
154 [monom_of_term vs (1, replicate (length vs) 0) t]
155 | monoms_of_term vs (t as Const (\<^const_name>\<open>times\<close>, _) $ _ $ _) =
156 [monom_of_term vs (1, replicate (length vs) 0) t]
157 | monoms_of_term vs (Const (\<^const_name>\<open>plus\<close>, _) $ ms1 $ ms2) =
158 (monoms_of_term vs ms1) @ (monoms_of_term vs ms2)
159 | monoms_of_term _ t = raise ERROR ("poly malformed 2 with " ^ UnparseC.term t)
161 (* convert a term to the internal representation of a multivariate polynomial;
162 the conversion is quite liberal, see test --- fun poly_of_term ---:
163 * the order of variables and the parentheses within a monomial are arbitrary
164 * the coefficient may be somewhere
165 * he order and the parentheses within monomials are arbitrary
166 But the term must be completely expand + over * (laws of distributivity are not applicable).
168 The function requires the free variables as strings already given,
169 because the gcd involves 2 polynomials (with the same length for their list of exponents).
171 fun poly_of_term vs (t as Const (\<^const_name>\<open>plus\<close>, _) $ _ $ _) =
172 (SOME (t |> monoms_of_term vs |> order)
173 handle ERROR _ => NONE)
174 | poly_of_term vs t =
175 (SOME [monom_of_term vs (1, replicate (length vs) 0) t]
176 handle ERROR _ => NONE)
180 val vs = TermC.vars_of t
182 case poly_of_term vs t of SOME _ => true | NONE => false
184 val is_expanded = is_poly (* TODO: check names *)
185 val is_polynomial = is_poly (* TODO: check names *)
188 subsubsection \<open>Convert internal representation of a multivariate polynomial to a term\<close>
190 fun term_of_es _ _ _ [] = [] (*assumes same length for vs and es*)
191 | term_of_es baseT expT (_ :: vs) (0 :: es) = [] @ term_of_es baseT expT vs es
192 | term_of_es baseT expT (v :: vs) (1 :: es) = v :: term_of_es baseT expT vs es
193 | term_of_es baseT expT (v :: vs) (e :: es) =
194 Const (\<^const_name>\<open>Transcendental.powr\<close>, [baseT, expT] ---> baseT) $ v $ (HOLogic.mk_number expT e)
195 :: term_of_es baseT expT vs es
196 | term_of_es _ _ _ _ = raise ERROR "term_of_es: length vs <> length es"
198 fun term_of_monom baseT expT vs ((c, es): monom) =
199 let val es' = term_of_es baseT expT vs es
203 if es' = [] (*if es = [0,0,0,...]*)
204 then HOLogic.mk_number baseT c
205 else foldl (HOLogic.mk_binop "Groups.times_class.times") (hd es', tl es')
206 else foldl (HOLogic.mk_binop "Groups.times_class.times")
207 (HOLogic.mk_number baseT c, es')
210 fun term_of_poly baseT expT vs p =
211 let val monos = map (term_of_monom baseT expT vs) p
212 in foldl (HOLogic.mk_binop \<^const_name>\<open>plus\<close>) (hd monos, tl monos) end
215 subsection \<open>Apply gcd_poly for cancelling and adding fractions as terms\<close>
217 fun mk_noteq_0 baseT t =
218 Const (\<^const_name>\<open>Not\<close>, HOLogic.boolT --> HOLogic.boolT) $
219 (Const (\<^const_name>\<open>HOL.eq\<close>, [baseT, baseT] ---> HOLogic.boolT) $ t $ HOLogic.mk_number HOLogic.realT 0)
221 fun mk_asms baseT ts =
222 let val as' = filter_out TermC.is_num ts (* asm like "2 ~= 0" is needless *)
223 in map (mk_noteq_0 baseT) as' end
226 subsubsection \<open>Factor out gcd for cancellation\<close>
228 fun check_fraction t =
230 Const (\<^const_name>\<open>divide\<close>, _) $ numerator $ denominator
231 => SOME (numerator, denominator)
234 (* prepare a term for cancellation by factoring out the gcd
235 assumes: is a fraction with outmost "/"*)
236 fun factout_p_ (thy: theory) t =
237 let val opt = check_fraction t
241 | SOME (numerator, denominator) =>
243 val vs = TermC.vars_of t
244 val baseT = type_of numerator
245 val expT = HOLogic.realT
247 case (poly_of_term vs numerator, poly_of_term vs denominator) of
250 val ((a', b'), c) = gcd_poly a b
251 val es = replicate (length vs) 0
253 if c = [(1, es)] orelse c = [(~1, es)]
257 val b't = term_of_poly baseT expT vs b'
258 val ct = term_of_poly baseT expT vs c
260 HOLogic.mk_binop \<^const_name>\<open>divide\<close>
261 (HOLogic.mk_binop \<^const_name>\<open>times\<close>
262 (term_of_poly baseT expT vs a', ct),
263 HOLogic.mk_binop \<^const_name>\<open>times\<close> (b't, ct))
264 in SOME (t', mk_asms baseT [b't, ct]) end
266 | _ => NONE : (term * term list) option
271 subsubsection \<open>Cancel a fraction\<close>
273 (* cancel a term by the gcd ("" denote terms with internal algebraic structure)
274 cancel_p_ :: theory \<Rightarrow> term \<Rightarrow> (term \<times> term list) option
275 cancel_p_ thy "a / b" = SOME ("a' / b'", ["b' \<noteq> 0"])
276 assumes: a is_polynomial \<and> b is_polynomial \<and> b \<noteq> 0
278 SOME ("a' / b'", ["b' \<noteq> 0"]). gcd_poly a b \<noteq> 1 \<and> gcd_poly a b \<noteq> -1 \<and>
279 a' * gcd_poly a b = a \<and> b' * gcd_poly a b = b
281 fun cancel_p_ (_: theory) t =
282 let val opt = check_fraction t
286 | SOME (numerator, denominator) =>
288 val vs = TermC.vars_of t
289 val baseT = type_of numerator
290 val expT = HOLogic.realT
292 case (poly_of_term vs numerator, poly_of_term vs denominator) of
295 val ((a', b'), c) = gcd_poly a b
296 val es = replicate (length vs) 0
298 if c = [(1, es)] orelse c = [(~1, es)]
302 val bt' = term_of_poly baseT expT vs b'
303 val ct = term_of_poly baseT expT vs c
305 HOLogic.mk_binop \<^const_name>\<open>divide\<close>
306 (term_of_poly baseT expT vs a', bt')
307 val asm = mk_asms baseT [bt']
308 in SOME (t', asm) end
310 | _ => NONE : (term * term list) option
315 subsubsection \<open>Factor out to a common denominator for addition\<close>
317 (* addition of fractions allows (at most) one non-fraction (a monomial) *)
319 (Const (\<^const_name>\<open>plus\<close>, _) $
320 (Const (\<^const_name>\<open>divide\<close>, _) $ n1 $ d1) $
321 (Const (\<^const_name>\<open>divide\<close>, _) $ n2 $ d2))
322 = SOME ((n1, d1), (n2, d2))
324 (Const (\<^const_name>\<open>plus\<close>, _) $
326 (Const (\<^const_name>\<open>divide\<close>, _) $ n2 $ d2))
327 = SOME ((nofrac, Free ("1", HOLogic.realT)), (n2, d2))
329 (Const (\<^const_name>\<open>plus\<close>, _) $
330 (Const (\<^const_name>\<open>divide\<close>, _) $ n1 $ d1) $
332 = SOME ((n1, d1), (nofrac, HOLogic.mk_number HOLogic.realT 1))
333 | check_frac_sum _ = NONE
335 (* prepare a term for addition by providing the least common denominator as a product
336 assumes: is a term with outmost "+" and at least one outmost "/" in respective summands*)
337 fun common_nominator_p_ (_: theory) t =
338 let val opt = check_frac_sum t
342 | SOME ((n1, d1), (n2, d2)) =>
344 val vs = TermC.vars_of t
346 case (poly_of_term vs d1, poly_of_term vs d2) of
349 val ((a', b'), c) = gcd_poly a b
350 val (baseT, expT) = (type_of n1, HOLogic.realT)
351 val [d1', d2', c'] = map (term_of_poly baseT expT vs) [a', b', c]
352 (*----- minimum of parentheses & nice result, but breaks tests: -------------
353 val denom = HOLogic.mk_binop \<^const_name>\<open>times\<close>
354 (HOLogic.mk_binop \<^const_name>\<open>times\<close> (d1', d2'), c') -------------*)
356 if c = [(1, replicate (length vs) 0)]
357 then HOLogic.mk_binop \<^const_name>\<open>times\<close> (d1', d2')
359 HOLogic.mk_binop \<^const_name>\<open>times\<close> (c',
360 HOLogic.mk_binop \<^const_name>\<open>times\<close> (d1', d2')) (*--------------*)
362 HOLogic.mk_binop \<^const_name>\<open>plus\<close>
363 (HOLogic.mk_binop \<^const_name>\<open>divide\<close>
364 (HOLogic.mk_binop \<^const_name>\<open>times\<close> (n1, d2'), denom),
365 HOLogic.mk_binop \<^const_name>\<open>divide\<close>
366 (HOLogic.mk_binop \<^const_name>\<open>times\<close> (n2, d1'), denom))
367 val asm = mk_asms baseT [d1', d2', c']
368 in SOME (t', asm) end
369 | _ => NONE : (term * term list) option
374 subsubsection \<open>Addition of at least one fraction within a sum\<close>
377 assumes: is a term with outmost "+" and at least one outmost "/" in respective summands
378 NOTE: the case "(_ + _) + _" need not be considered due to iterated addition.*)
379 fun add_fraction_p_ (_: theory) t =
380 case check_frac_sum t of
382 | SOME ((n1, d1), (n2, d2)) =>
384 val vs = TermC.vars_of t
386 case (poly_of_term vs n1, poly_of_term vs d1, poly_of_term vs n2, poly_of_term vs d2) of
387 (SOME _, SOME a, SOME _, SOME b) =>
389 val ((a', b'), c) = gcd_poly a b
390 val (baseT, expT) = (type_of n1, HOLogic.realT)
391 val nomin = term_of_poly baseT expT vs
392 (((the (poly_of_term vs n1)) %%*%% b') %%+%% ((the (poly_of_term vs n2)) %%*%% a'))
393 val denom = term_of_poly baseT expT vs ((c %%*%% a') %%*%% b')
394 val t' = HOLogic.mk_binop \<^const_name>\<open>divide\<close> (nomin, denom)
395 in SOME (t', mk_asms baseT [denom]) end
396 | _ => NONE : (term * term list) option
400 section \<open>Embed cancellation and addition into rewriting\<close>
402 subsection \<open>Rulesets and predicate for embedding\<close>
404 (* evaluates conditions in calculate_Rational *)
407 (Rule_Def.Repeat {id = "calc_rat_erls", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
408 erls = Rule_Set.empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
410 [Rule.Eval ("Prog_Expr.matches", Prog_Expr.eval_matches "#matches_"),
411 Rule.Eval ("HOL.eq", Prog_Expr.eval_equal "#equal_"),
412 Rule.Eval ("Prog_Expr.is_const", Prog_Expr.eval_const "#is_const_"),
413 Rule.Thm ("not_true", ThmC.numerals_to_Free @{thm not_true}),
414 Rule.Thm ("not_false", ThmC.numerals_to_Free @{thm not_false})],
415 scr = Rule.Empty_Prog});
417 (* simplifies expressions with numerals;
418 does NOT rearrange the term by AC-rewriting; thus terms with variables
419 need to have constants to be commuted together respectively *)
420 val calculate_Rational =
421 prep_rls' (Rule_Set.merge "calculate_Rational"
422 (Rule_Def.Repeat {id = "divide", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
423 erls = calc_rat_erls, srls = Rule_Set.Empty,
424 calc = [], errpatts = [],
426 [\<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e"),
428 Rule.Thm ("minus_divide_left", ThmC.numerals_to_Free (@{thm minus_divide_left} RS @{thm sym})),
429 (*SYM - ?x / ?y = - (?x / ?y) may come from subst*)
430 \<^rule_thm>\<open>rat_add\<close>,
431 (*"[| a is_const; b is_const; c is_const; d is_const |] ==> \
432 \a / c + b / d = (a * d) / (c * d) + (b * c ) / (d * c)"*)
433 \<^rule_thm>\<open>rat_add1\<close>,
434 (*"[| a is_const; b is_const; c is_const |] ==> a / c + b / c = (a + b) / c"*)
435 \<^rule_thm>\<open>rat_add2\<close>,
436 (*"[| ?a is_const; ?b is_const; ?c is_const |] ==> ?a / ?c + ?b = (?a + ?b * ?c) / ?c"*)
437 \<^rule_thm>\<open>rat_add3\<close>,
438 (*"[| a is_const; b is_const; c is_const |] ==> a + b / c = (a * c) / c + b / c"\
439 .... is_const to be omitted here FIXME*)
441 \<^rule_thm>\<open>rat_mult\<close>,
442 (*a / b * (c / d) = a * c / (b * d)*)
443 \<^rule_thm>\<open>times_divide_eq_right\<close>,
444 (*?x * (?y / ?z) = ?x * ?y / ?z*)
445 \<^rule_thm>\<open>times_divide_eq_left\<close>,
446 (*?y / ?z * ?x = ?y * ?x / ?z*)
448 \<^rule_thm>\<open>real_divide_divide1\<close>,
449 (*"?y ~= 0 ==> ?u / ?v / (?y / ?z) = ?u / ?v * (?z / ?y)"*)
450 \<^rule_thm>\<open>divide_divide_eq_left\<close>,
451 (*"?x / ?y / ?z = ?x / (?y * ?z)"*)
453 \<^rule_thm>\<open>rat_power\<close>,
454 (*"(?a / ?b) \<up> ?n = ?a \<up> ?n / ?b \<up> ?n"*)
456 \<^rule_thm>\<open>mult_cross\<close>,
457 (*"[| b ~= 0; d ~= 0 |] ==> (a / b = c / d) = (a * d = b * c)*)
458 \<^rule_thm>\<open>mult_cross1\<close>,
459 (*" b ~= 0 ==> (a / b = c ) = (a = b * c)*)
460 \<^rule_thm>\<open>mult_cross2\<close>
461 (*" d ~= 0 ==> (a = c / d) = (a * d = c)*)],
462 scr = Rule.Empty_Prog})
465 (*("is_expanded", ("Rational.is_expanded", eval_is_expanded ""))*)
466 fun eval_is_expanded (thmid:string) _
467 (t as (Const (\<^const_name>\<open>Rational.is_expanded\<close>, _) $ arg)) thy =
469 then SOME (TermC.mk_thmid thmid (UnparseC.term_in_thy thy arg) "",
470 HOLogic.Trueprop $ (TermC.mk_equality (t, @{term True})))
471 else SOME (TermC.mk_thmid thmid (UnparseC.term_in_thy thy arg) "",
472 HOLogic.Trueprop $ (TermC.mk_equality (t, @{term False})))
473 | eval_is_expanded _ _ _ _ = NONE;
475 calculation is_expanded = \<open>eval_is_expanded ""\<close>
478 Rule_Set.merge "rational_erls" calculate_Rational
479 (Rule_Set.append_rules "is_expanded" Atools_erls
480 [\<^rule_eval>\<open>is_expanded\<close> (eval_is_expanded "")]);
483 subsection \<open>Embed cancellation into rewriting\<close>
485 (**)local (* cancel_p *)
487 val {rules = rules, rew_ord = (_, ro), ...} = Rule_Set.rep (assoc_rls' @{theory} "rev_rew_p");
489 fun init_state thy eval_rls ro t =
491 val SOME (t', _) = factout_p_ thy t;
492 val SOME (t'', asm) = cancel_p_ thy t;
493 val der = Derive.steps_reverse thy eval_rls rules ro NONE t';
495 [(\<^rule_thm>\<open>real_mult_div_cancel2\<close>, (t'', asm))]
496 val rs = (Rule.distinct' o (map #1)) der
497 val rs = filter_out (ThmC.member'
498 ["sym_real_add_zero_left", "sym_real_mult_0", "sym_real_mult_1"]) rs
499 in (t, t'', [rs(*one in order to ease locate_rule*)], der) end;
501 fun locate_rule thy eval_rls ro [rs] t r =
502 if member op = ((map (Rule.thm_id)) rs) (Rule.thm_id r)
504 let val ropt = Rewrite.rewrite_ thy ro eval_rls true (Rule.thm r) t;
506 case ropt of SOME ta => [(r, ta)]
508 ("### locate_rule: rewrite " ^ Rule.thm_id r ^ " " ^ UnparseC.term t ^ " = NONE");*) [])
510 else ((*tracing ("### locate_rule: " ^ Rule.thm_id r ^ " not mem rrls");*) [])
511 | locate_rule _ _ _ _ _ _ = raise ERROR "locate_rule: doesnt match rev-sets in istate";
513 fun next_rule thy eval_rls ro [rs] t =
515 val der = Derive.do_one thy eval_rls rs ro NONE t;
516 in case der of (_, r, _) :: _ => SOME r | _ => NONE end
517 | next_rule _ _ _ _ _ = raise ERROR ("next_rule: doesnt match rev-sets in istate");
519 fun attach_form (_: Rule.rule list list) (_: term) (_: term) =
520 [(*TODO*)]: ( Rule.rule * (term * term list)) list;
525 Rule_Set.Rrls {id = "cancel_p", prepat = [],
526 rew_ord=("ord_make_polynomial", ord_make_polynomial false \<^theory>),
527 erls = rational_erls,
529 [("PLUS", (\<^const_name>\<open>plus\<close>, (**)eval_binop "#add_")),
530 ("TIMES" , (\<^const_name>\<open>times\<close>, (**)eval_binop "#mult_")),
531 ("DIVIDE", (\<^const_name>\<open>divide\<close>, Prog_Expr.eval_cancel "#divide_e")),
532 ("POWER", (\<^const_name>\<open>powr\<close>, (**)eval_binop "#power_"))],
535 Rule.Rfuns {init_state = init_state \<^theory> Atools_erls ro,
536 normal_form = cancel_p_ \<^theory>,
537 locate_rule = locate_rule \<^theory> Atools_erls ro,
538 next_rule = next_rule \<^theory> Atools_erls ro,
539 attach_form = attach_form}}
540 (**)end(* local cancel_p *)
543 subsection \<open>Embed addition into rewriting\<close>
545 (**)local (* add_fractions_p *)
547 (*val {rules = rules, rew_ord = (_, ro), ...} = Rule_Set.rep (assoc_rls "make_polynomial");*)
548 val {rules, rew_ord=(_,ro),...} = Rule_Set.rep (assoc_rls' @{theory} "rev_rew_p");
550 fun init_state thy eval_rls ro t =
552 val SOME (t',_) = common_nominator_p_ thy t;
553 val SOME (t'', asm) = add_fraction_p_ thy t;
554 val der = Derive.steps_reverse thy eval_rls rules ro NONE t';
556 [(\<^rule_thm>\<open>real_mult_div_cancel2\<close>, (t'',asm))]
557 val rs = (Rule.distinct' o (map #1)) der;
558 val rs = filter_out (ThmC.member'
559 ["sym_real_add_zero_left", "sym_real_mult_0", "sym_real_mult_1"]) rs;
560 in (t, t'', [rs(*here only _ONE_*)], der) end;
562 fun locate_rule thy eval_rls ro [rs] t r =
563 if member op = ((map (Rule.thm_id)) rs) (Rule.thm_id r)
565 let val ropt = Rewrite.rewrite_ thy ro eval_rls true (Rule.thm r) t;
570 ((*tracing ("### locate_rule: rewrite " ^ Rule.thm_id r ^ " " ^ UnparseC.term t ^ " = NONE");*)
572 else ((*tracing ("### locate_rule: " ^ Rule.thm_id r ^ " not mem rrls");*) [])
573 | locate_rule _ _ _ _ _ _ = raise ERROR "locate_rule: doesnt match rev-sets in istate";
575 fun next_rule thy eval_rls ro [rs] t =
576 let val der = Derive.do_one thy eval_rls rs ro NONE t;
582 | next_rule _ _ _ _ _ = raise ERROR ("next_rule: doesnt match rev-sets in istate");
584 val pat0 = TermC.parse_patt \<^theory> "?r/?s+?u/?v :: real";
585 val pat1 = TermC.parse_patt \<^theory> "?r/?s+?u :: real";
586 val pat2 = TermC.parse_patt \<^theory> "?r +?u/?v :: real";
587 val prepat = [([@{term True}], pat0),
588 ([@{term True}], pat1),
589 ([@{term True}], pat2)];
592 val add_fractions_p =
593 Rule_Set.Rrls {id = "add_fractions_p", prepat=prepat,
594 rew_ord = ("ord_make_polynomial", ord_make_polynomial false \<^theory>),
595 erls = rational_erls,
596 calc = [("PLUS", (\<^const_name>\<open>plus\<close>, (**)eval_binop "#add_")),
597 ("TIMES", (\<^const_name>\<open>times\<close>, (**)eval_binop "#mult_")),
598 ("DIVIDE", (\<^const_name>\<open>divide\<close>, Prog_Expr.eval_cancel "#divide_e")),
599 ("POWER", (\<^const_name>\<open>powr\<close>, (**)eval_binop "#power_"))],
601 scr = Rule.Rfuns {init_state = init_state \<^theory> Atools_erls ro,
602 normal_form = add_fraction_p_ \<^theory>,
603 locate_rule = locate_rule \<^theory> Atools_erls ro,
604 next_rule = next_rule \<^theory> Atools_erls ro,
605 attach_form = attach_form}}
606 (**)end(*local add_fractions_p *)
609 subsection \<open>Cancelling and adding all occurrences in a term /////////////////////////////\<close>
611 (*copying cancel_p_rls + add her caused error in interface.sml*)
614 section \<open>Rulesets for general simplification\<close>
616 (*.all powers over + distributed; atoms over * collected, other distributed
617 contains absolute minimum of thms for context in norm_Rational .*)
618 val powers = prep_rls'(
619 Rule_Def.Repeat {id = "powers", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
620 erls = powers_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
621 rules = [\<^rule_thm>\<open>realpow_multI\<close>,
622 (*"(r * s) \<up> n = r \<up> n * s \<up> n"*)
623 \<^rule_thm>\<open>realpow_pow\<close>,
624 (*"(a \<up> b) \<up> c = a \<up> (b * c)"*)
625 \<^rule_thm>\<open>realpow_oneI\<close>,
627 \<^rule_thm>\<open>realpow_minus_even\<close>,
628 (*"n is_even ==> (- r) \<up> n = r \<up> n" ?-->discard_minus?*)
629 \<^rule_thm>\<open>realpow_minus_odd\<close>,
630 (*"Not (n is_even) ==> (- r) \<up> n = -1 * r \<up> n"*)
632 (*----- collect atoms over * -----*)
633 \<^rule_thm>\<open>realpow_two_atom\<close>,
634 (*"r is_atom ==> r * r = r \<up> 2"*)
635 \<^rule_thm>\<open>realpow_plus_1\<close>,
636 (*"r is_atom ==> r * r \<up> n = r \<up> (n + 1)"*)
637 \<^rule_thm>\<open>realpow_addI_atom\<close>,
638 (*"r is_atom ==> r \<up> n * r \<up> m = r \<up> (n + m)"*)
640 (*----- distribute none-atoms -----*)
641 \<^rule_thm>\<open>realpow_def_atom\<close>,
642 (*"[| 1 < n; ~ (r is_atom) |]==>r \<up> n = r * r \<up> (n + -1)"*)
643 \<^rule_thm>\<open>realpow_eq_oneI\<close>,
645 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_")
647 scr = Rule.Empty_Prog
649 (*.contains absolute minimum of thms for context in norm_Rational.*)
650 val rat_mult_divide = prep_rls'(
651 Rule_Def.Repeat {id = "rat_mult_divide", preconds = [],
652 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
653 erls = Rule_Set.empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
654 rules = [\<^rule_thm>\<open>rat_mult\<close>,
655 (*(1)"?a / ?b * (?c / ?d) = ?a * ?c / (?b * ?d)"*)
656 \<^rule_thm>\<open>times_divide_eq_right\<close>,
657 (*(2)"?a * (?c / ?d) = ?a * ?c / ?d" must be [2],
658 otherwise inv.to a / b / c = ...*)
659 \<^rule_thm>\<open>times_divide_eq_left\<close>,
660 (*"?a / ?b * ?c = ?a * ?c / ?b" order weights x \<up> n too much
661 and does not commute a / b * c \<up> 2 !*)
663 \<^rule_thm>\<open>divide_divide_eq_right\<close>,
664 (*"?x / (?y / ?z) = ?x * ?z / ?y"*)
665 \<^rule_thm>\<open>divide_divide_eq_left\<close>,
666 (*"?x / ?y / ?z = ?x / (?y * ?z)"*)
667 \<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e")
669 scr = Rule.Empty_Prog
672 (*.contains absolute minimum of thms for context in norm_Rational.*)
673 val reduce_0_1_2 = prep_rls'(
674 Rule_Def.Repeat{id = "reduce_0_1_2", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
675 erls = Rule_Set.empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
676 rules = [(*\<^rule_thm>\<open>divide_1\<close>,
677 "?x / 1 = ?x" unnecess.for normalform*)
678 \<^rule_thm>\<open>mult_1_left\<close>,
680 (*\<^rule_thm>\<open>real_mult_minus1\<close>,
682 (*\<^rule_thm>\<open>real_minus_mult_cancel\<close>,
683 "- ?x * - ?y = ?x * ?y"*)
685 \<^rule_thm>\<open>mult_zero_left\<close>,
687 \<^rule_thm>\<open>add_0_left\<close>,
689 (*\<^rule_thm>\<open>right_minus\<close>,
692 \<^rule_thm_sym>\<open>real_mult_2\<close>,
693 (*"z1 + z1 = 2 * z1"*)
694 \<^rule_thm>\<open>real_mult_2_assoc\<close>,
695 (*"z1 + (z1 + k) = 2 * z1 + k"*)
697 \<^rule_thm>\<open>division_ring_divide_zero\<close>
699 ], scr = Rule.Empty_Prog});
701 (*erls for calculate_Rational;
702 make local with FIXX@ME result:term *term list WN0609???SKMG*)
703 val norm_rat_erls = prep_rls'(
704 Rule_Def.Repeat {id = "norm_rat_erls", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
705 erls = Rule_Set.empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
706 rules = [\<^rule_eval>\<open>Prog_Expr.is_const\<close> (Prog_Expr.eval_const "#is_const_")
707 ], scr = Rule.Empty_Prog});
709 (* consists of rls containing the absolute minimum of thms *)
710 (*040209: this version has been used by RL for his equations,
711 which is now replaced by MGs version "norm_Rational" below *)
712 val norm_Rational_min = prep_rls'(
713 Rule_Def.Repeat {id = "norm_Rational_min", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
714 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
715 rules = [(*sequence given by operator precedence*)
716 Rule.Rls_ discard_minus,
718 Rule.Rls_ rat_mult_divide,
720 Rule.Rls_ reduce_0_1_2,
721 Rule.Rls_ order_add_mult,
722 Rule.Rls_ collect_numerals,
723 Rule.Rls_ add_fractions_p,
726 scr = Rule.Empty_Prog});
728 val norm_Rational_parenthesized = prep_rls'(
729 Rule_Set.Sequence {id = "norm_Rational_parenthesized", preconds = []:term list,
730 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
731 erls = Atools_erls, srls = Rule_Set.Empty,
732 calc = [], errpatts = [],
733 rules = [Rule.Rls_ norm_Rational_min,
734 Rule.Rls_ discard_parentheses
736 scr = Rule.Empty_Prog});
738 (*WN030318???SK: simplifies all but cancel and common_nominator*)
739 val simplify_rational =
740 Rule_Set.merge "simplify_rational" expand_binoms
741 (Rule_Set.append_rules "divide" calculate_Rational
742 [\<^rule_thm>\<open>div_by_1\<close>,
744 \<^rule_thm>\<open>rat_mult\<close>,
745 (*(1)"?a / ?b * (?c / ?d) = ?a * ?c / (?b * ?d)"*)
746 \<^rule_thm>\<open>times_divide_eq_right\<close>,
747 (*(2)"?a * (?c / ?d) = ?a * ?c / ?d" must be [2],
748 otherwise inv.to a / b / c = ...*)
749 \<^rule_thm>\<open>times_divide_eq_left\<close>,
750 (*"?a / ?b * ?c = ?a * ?c / ?b"*)
751 \<^rule_thm>\<open>add_minus\<close>,
752 (*"?a + ?b - ?b = ?a"*)
753 \<^rule_thm>\<open>add_minus1\<close>,
754 (*"?a - ?b + ?b = ?a"*)
755 \<^rule_thm>\<open>divide_minus1\<close>
760 val add_fractions_p_rls = prep_rls'(
761 Rule_Def.Repeat {id = "add_fractions_p_rls", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
762 erls = Rule_Set.empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
763 rules = [Rule.Rls_ add_fractions_p],
764 scr = Rule.Empty_Prog});
766 (* "Rule_Def.Repeat" causes repeated application of cancel_p to one and the same term *)
767 val cancel_p_rls = prep_rls'(
769 {id = "cancel_p_rls", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
770 erls = Rule_Set.empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
771 rules = [Rule.Rls_ cancel_p],
772 scr = Rule.Empty_Prog});
774 (*. makes 'normal' fractions; 'is_polyexp' inhibits double fractions;
775 used in initial part norm_Rational_mg, see example DA-M02-main.p.60.*)
776 val rat_mult_poly = prep_rls'(
777 Rule_Def.Repeat {id = "rat_mult_poly", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
778 erls = Rule_Set.append_rules "Rule_Set.empty-is_polyexp" Rule_Set.empty
779 [\<^rule_eval>\<open>is_polyexp\<close> (eval_is_polyexp "")],
780 srls = Rule_Set.Empty, calc = [], errpatts = [],
782 [\<^rule_thm>\<open>rat_mult_poly_l\<close>,
783 (*"?c is_polyexp ==> ?c * (?a / ?b) = ?c * ?a / ?b"*)
784 \<^rule_thm>\<open>rat_mult_poly_r\<close>
785 (*"?c is_polyexp ==> ?a / ?b * ?c = ?a * ?c / ?b"*) ],
786 scr = Rule.Empty_Prog});
788 (*. makes 'normal' fractions; 'is_polyexp' inhibits double fractions;
789 used in looping part norm_Rational_rls, see example DA-M02-main.p.60
790 .. WHERE THE LATTER DOES ALWAYS WORK, BECAUSE erls = Rule_Set.empty,
791 I.E. THE RESPECTIVE ASSUMPTION IS STORED AND Rule.Thm APPLIED; WN051028
793 val rat_mult_div_pow = prep_rls'(
794 Rule_Def.Repeat {id = "rat_mult_div_pow", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
795 erls = Rule_Set.empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
796 rules = [\<^rule_thm>\<open>rat_mult\<close>,
797 (*"?a / ?b * (?c / ?d) = ?a * ?c / (?b * ?d)"*)
798 \<^rule_thm>\<open>rat_mult_poly_l\<close>,
799 (*"?c is_polyexp ==> ?c * (?a / ?b) = ?c * ?a / ?b"*)
800 \<^rule_thm>\<open>rat_mult_poly_r\<close>,
801 (*"?c is_polyexp ==> ?a / ?b * ?c = ?a * ?c / ?b"*)
803 \<^rule_thm>\<open>real_divide_divide1_mg\<close>,
804 (*"y ~= 0 ==> (u / v) / (y / z) = (u * z) / (y * v)"*)
805 \<^rule_thm>\<open>divide_divide_eq_right\<close>,
806 (*"?x / (?y / ?z) = ?x * ?z / ?y"*)
807 \<^rule_thm>\<open>divide_divide_eq_left\<close>,
808 (*"?x / ?y / ?z = ?x / (?y * ?z)"*)
809 \<^rule_eval>\<open>divide\<close> (Prog_Expr.eval_cancel "#divide_e"),
811 \<^rule_thm>\<open>rat_power\<close>
812 (*"(?a / ?b) \<up> ?n = ?a \<up> ?n / ?b \<up> ?n"*)
814 scr = Rule.Empty_Prog});
816 val rat_reduce_1 = prep_rls'(
817 Rule_Def.Repeat {id = "rat_reduce_1", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
818 erls = Rule_Set.empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
820 [\<^rule_thm>\<open>div_by_1\<close>,
822 \<^rule_thm>\<open>mult_1_left\<close>
825 scr = Rule.Empty_Prog});
827 (* looping part of norm_Rational *)
828 val norm_Rational_rls = prep_rls' (
829 Rule_Def.Repeat {id = "norm_Rational_rls", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
830 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
831 rules = [Rule.Rls_ add_fractions_p_rls,
832 Rule.Rls_ rat_mult_div_pow,
833 Rule.Rls_ make_rat_poly_with_parentheses,
834 Rule.Rls_ cancel_p_rls,
835 Rule.Rls_ rat_reduce_1
837 scr = Rule.Empty_Prog});
839 val norm_Rational = prep_rls' (
841 {id = "norm_Rational", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
842 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
843 rules = [Rule.Rls_ discard_minus,
844 Rule.Rls_ rat_mult_poly, (* removes double fractions like a/b/c *)
845 Rule.Rls_ make_rat_poly_with_parentheses,
846 Rule.Rls_ cancel_p_rls,
847 Rule.Rls_ norm_Rational_rls, (* the main rls, looping (#) *)
848 Rule.Rls_ discard_parentheses1 (* mult only *)
850 scr = Rule.Empty_Prog});
854 calculate_Rational = calculate_Rational and
855 calc_rat_erls = calc_rat_erls and
856 rational_erls = rational_erls and
857 cancel_p = cancel_p and
858 add_fractions_p = add_fractions_p and
860 add_fractions_p_rls = add_fractions_p_rls and
861 powers_erls = powers_erls and
863 rat_mult_divide = rat_mult_divide and
864 reduce_0_1_2 = reduce_0_1_2 and
866 rat_reduce_1 = rat_reduce_1 and
867 norm_rat_erls = norm_rat_erls and
868 norm_Rational = norm_Rational and
869 norm_Rational_rls = norm_Rational_rls and
870 norm_Rational_min = norm_Rational_min and
871 norm_Rational_parenthesized = norm_Rational_parenthesized and
873 rat_mult_poly = rat_mult_poly and
874 rat_mult_div_pow = rat_mult_div_pow and
875 cancel_p_rls = cancel_p_rls
877 section \<open>A problem for simplification of rationals\<close>
879 problem pbl_simp_rat : "rational/simplification" =
880 \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)]\<close>
881 Method: "simplification/of_rationals"
884 Where: "t_t is_ratpolyexp"
885 Find: "normalform n_n"
887 section \<open>A methods for simplification of rationals\<close>
888 (*WN061025 this methods script is copied from (auto-generated) script
889 of norm_Rational in order to ease repair on inform*)
891 partial_function (tailrec) simplify :: "real \<Rightarrow> real"
894 (Try (Rewrite_Set ''discard_minus'') #>
895 Try (Rewrite_Set ''rat_mult_poly'') #>
896 Try (Rewrite_Set ''make_rat_poly_with_parentheses'') #>
897 Try (Rewrite_Set ''cancel_p_rls'') #>
899 Try (Rewrite_Set ''add_fractions_p_rls'') #>
900 Try (Rewrite_Set ''rat_mult_div_pow'') #>
901 Try (Rewrite_Set ''make_rat_poly_with_parentheses'') #>
902 Try (Rewrite_Set ''cancel_p_rls'') #>
903 Try (Rewrite_Set ''rat_reduce_1''))) #>
904 Try (Rewrite_Set ''discard_parentheses1''))
908 method met_simp_rat : "simplification/of_rationals" =
909 \<open>{rew_ord'="tless_true", rls' = Rule_Set.empty, calc = [], srls = Rule_Set.empty,
910 prls = Rule_Set.append_rules "simplification_of_rationals_prls" Rule_Set.empty
911 [(*for preds in where_*) \<^rule_eval>\<open>is_ratpolyexp\<close> (eval_is_ratpolyexp "")],
912 crls = Rule_Set.empty, errpats = [], nrls = norm_Rational_rls}\<close>
913 Program: simplify.simps
915 Where: "t_t is_ratpolyexp"
916 Find: "normalform n_n"