1 (* theory collecting all knowledge for Root
9 theory Root imports Simplify begin
13 sqrt :: "real => real" (*"(sqrt _ )" [80] 80*)
14 nroot :: "[real, real] => real"
16 axioms (*.not contained in Isabelle2002,
17 stated as axioms, TODO: prove as theorems;
18 theorem-IDs 'xxxI' with ^^^ instead of ^ in 'xxx' in Isabelle2002.*)
20 root_plus_minus "0 <= b ==>
21 (a^^^2 = b) = ((a = sqrt b) | (a = (-1)*sqrt b))"
22 root_false "b < 0 ==> (a^^^2 = b) = False"
24 (* for expand_rootbinom *)
25 real_pp_binom_times "(a + b)*(c + d) = a*c + a*d + b*c + b*d"
26 real_pm_binom_times "(a + b)*(c - d) = a*c - a*d + b*c - b*d"
27 real_mp_binom_times "(a - b)*(c + d) = a*c + a*d - b*c - b*d"
28 real_mm_binom_times "(a - b)*(c - d) = a*c - a*d - b*c + b*d"
29 real_plus_binom_pow3 "(a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3"
30 real_minus_binom_pow3 "(a - b)^^^3 = a^^^3 - 3*a^^^2*b + 3*a*b^^^2 - b^^^3"
31 realpow_mul "(a*b)^^^n = a^^^n * b^^^n"
33 real_diff_minus "a - b = a + (-1) * b"
34 real_plus_binom_times "(a + b)*(a + b) = a^^^2 + 2*a*b + b^^^2"
35 real_minus_binom_times "(a - b)*(a - b) = a^^^2 - 2*a*b + b^^^2"
36 real_plus_binom_pow2 "(a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"
37 real_minus_binom_pow2 "(a - b)^^^2 = a^^^2 - 2*a*b + b^^^2"
38 real_plus_minus_binom1 "(a + b)*(a - b) = a^^^2 - b^^^2"
39 real_plus_minus_binom2 "(a - b)*(a + b) = a^^^2 - b^^^2"
41 real_root_positive "0 <= a ==> (x ^^^ 2 = a) = (x = sqrt a)"
42 real_root_negative "a < 0 ==> (x ^^^ 2 = a) = False"
45 (*-------------------------functions---------------------*)
46 (*evaluation square-root over the integers*)
47 fun eval_sqrt (thmid:string) (op_:string) (t as
48 (Const(op0,t0) $ arg)) thy =
51 (case int_of_str n1 of
55 let val fact = squfact ni;
57 then SOME ("#sqrt #"^(string_of_int ni)^" = #"
58 ^(string_of_int (if ni = 0 then 0
60 Trueprop $ mk_equality (t, term_of_num t1 fact))
61 else if fact = 1 then NONE
62 else SOME ("#sqrt #"^(string_of_int ni)^" = sqrt (#"
63 ^(string_of_int fact)^" * #"
64 ^(string_of_int fact)^" * #"
65 ^(string_of_int (ni div (fact*fact))^")"),
69 (mk_factroot op0 t1 fact
70 (ni div (fact*fact))))))
75 | eval_sqrt _ _ _ _ = NONE;
76 (*val (thmid, op_, t as Const(op0,t0) $ arg) = ("","", str2term "sqrt 0");
77 > eval_sqrt thmid op_ t thy;
78 > val Free (n1,t1) = arg;
79 > val SOME ni = int_of_str n1;
82 calclist':= overwritel (!calclist',
83 [("SQRT" ,("Root.sqrt" ,eval_sqrt "#sqrt_"))
84 (*different types for 'sqrt 4' --- 'Calculate sqrt_'*)
88 local (* Vers. 7.10.99.A *)
90 open Term; (* for type order = EQUAL | LESS | GREATER *)
92 fun pr_ord EQUAL = "EQUAL"
93 | pr_ord LESS = "LESS"
94 | pr_ord GREATER = "GREATER";
96 fun dest_hd' (Const (a, T)) = (* ~ term.ML *)
97 (case a of "Root.sqrt" => ((("|||", 0), T), 0) (*WN greatest *)
98 | _ => (((a, 0), T), 0))
99 | dest_hd' (Free (a, T)) = (((a, 0), T), 1)
100 | dest_hd' (Var v) = (v, 2)
101 | dest_hd' (Bound i) = ((("", i), dummyT), 3)
102 | dest_hd' (Abs (_, T, _)) = ((("", 0), T), 4);
103 fun size_of_term' (Const(str,_) $ t) =
104 (case str of "Root.sqrt" => (1000 + size_of_term' t)
105 | _ => 1 + size_of_term' t)
106 | size_of_term' (Abs (_,_,body)) = 1 + size_of_term' body
107 | size_of_term' (f $ t) = size_of_term' f + size_of_term' t
108 | size_of_term' _ = 1;
109 fun term_ord' pr thy (Abs (_, T, t), Abs(_, U, u)) = (* ~ term.ML *)
110 (case term_ord' pr thy (t, u) of EQUAL => typ_ord (T, U) | ord => ord)
111 | term_ord' pr thy (t, u) =
114 val (f, ts) = strip_comb t and (g, us) = strip_comb u;
115 val _=writeln("t= f@ts= \""^
116 ((Syntax.string_of_term (thy2ctxt thy)) f)^"\" @ \"["^
117 (commas(map(Syntax.string_of_term (thy2ctxt thy)) ts))^"]\"");
118 val _=writeln("u= g@us= \""^
119 ((Syntax.string_of_term (thy2ctxt thy)) g)^"\" @ \"["^
120 (commas(map(Syntax.string_of_term (thy2ctxt thy)) us))^"]\"");
121 val _=writeln("size_of_term(t,u)= ("^
122 (string_of_int(size_of_term' t))^", "^
123 (string_of_int(size_of_term' u))^")");
124 val _=writeln("hd_ord(f,g) = "^((pr_ord o hd_ord)(f,g)));
125 val _=writeln("terms_ord(ts,us) = "^
126 ((pr_ord o terms_ord str false)(ts,us)));
127 val _=writeln("-------");
130 case int_ord (size_of_term' t, size_of_term' u) of
132 let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
133 (case hd_ord (f, g) of EQUAL => (terms_ord str pr) (ts, us)
137 and hd_ord (f, g) = (* ~ term.ML *)
138 prod_ord (prod_ord indexname_ord typ_ord) int_ord (dest_hd' f, dest_hd' g)
139 and terms_ord str pr (ts, us) =
140 list_ord (term_ord' pr (assoc_thy "Isac.thy"))(ts, us);
143 (* associates a+(b+c) => (a+b)+c = a+b+c ... avoiding parentheses
144 by (1) size_of_term: less(!) to right, size_of 'sqrt (...)' = 1
145 (2) hd_ord: greater to right, 'sqrt' < numerals < variables
146 (3) terms_ord: recurs. on args, greater to right
150 pr: print trace, WN0509 'sqrt_right true' not used anymore
152 subst: no bound variables, only Root.sqrt
153 tu: the terms to compare (t1, t2) ... *)
154 fun sqrt_right (pr:bool) thy (_:subst) tu =
155 (term_ord' pr thy(***) tu = LESS );
158 rew_ord' := overwritel (!rew_ord',
159 [("termlessI", termlessI),
160 ("sqrt_right", sqrt_right false (theory "Pure"))
163 (*-------------------------rulse-------------------------*)
165 append_rls "Root_crls" Atools_erls
166 [Thm ("real_unari_minus",num_str real_unari_minus),
167 Calc ("Root.sqrt" ,eval_sqrt "#sqrt_"),
168 Calc ("HOL.divide",eval_cancel "#divide_"),
169 Calc ("Atools.pow" ,eval_binop "#power_"),
170 Calc ("op +", eval_binop "#add_"),
171 Calc ("op -", eval_binop "#sub_"),
172 Calc ("op *", eval_binop "#mult_"),
173 Calc ("op =",eval_equal "#equal_")
177 append_rls "Root_erls" Atools_erls
178 [Thm ("real_unari_minus",num_str real_unari_minus),
179 Calc ("Root.sqrt" ,eval_sqrt "#sqrt_"),
180 Calc ("HOL.divide",eval_cancel "#divide_"),
181 Calc ("Atools.pow" ,eval_binop "#power_"),
182 Calc ("op +", eval_binop "#add_"),
183 Calc ("op -", eval_binop "#sub_"),
184 Calc ("op *", eval_binop "#mult_"),
185 Calc ("op =",eval_equal "#equal_")
188 ruleset' := overwritelthy thy (!ruleset',
189 [("Root_erls",Root_erls) (*FIXXXME:del with rls.rls'*)
192 val make_rooteq = prep_rls(
193 Rls{id = "make_rooteq", preconds = []:term list,
194 rew_ord = ("sqrt_right", sqrt_right false (theory "Root")),
195 erls = Atools_erls, srls = Erls,
198 rules = [Thm ("real_diff_minus",num_str real_diff_minus),
199 (*"a - b = a + (-1) * b"*)
201 Thm ("left_distrib" ,num_str @{thm left_distrib}),
202 (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
203 Thm ("left_distrib2",num_str @{thm left_distrib}2),
204 (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
205 Thm ("left_diff_distrib" ,num_str @{thm left_diff_distrib}),
206 (*"(z1.0 - z2.0) * w = z1.0 * w - z2.0 * w"*)
207 Thm ("left_diff_distrib2",num_str @{thm left_diff_distrib}2),
208 (*"w * (z1.0 - z2.0) = w * z1.0 - w * z2.0"*)
210 Thm ("mult_1_left",num_str @{thm mult_1_left}),
212 Thm ("mult_zero_left",num_str @{thm mult_zero_left}),
214 Thm ("add_0_left",num_str @{thm add_0_left}),
217 Thm ("real_mult_commute",num_str real_mult_commute),
219 Thm ("real_mult_left_commute",num_str real_mult_left_commute),
221 Thm ("real_mult_assoc",num_str real_mult_assoc),
223 Thm ("add_commute",num_str @{thm add_commute}),
225 Thm ("add_left_commute",num_str @{thm add_left_commute}),
227 Thm ("add_assoc",num_str @{thm add_assoc}),
230 Thm ("sym_realpow_twoI",num_str (realpow_twoI RS sym)),
231 (*"r1 * r1 = r1 ^^^ 2"*)
232 Thm ("realpow_plus_1",num_str realpow_plus_1),
233 (*"r * r ^^^ n = r ^^^ (n + 1)"*)
234 Thm ("sym_real_mult_2",num_str (real_mult_2 RS sym)),
235 (*"z1 + z1 = 2 * z1"*)
236 Thm ("real_mult_2_assoc",num_str real_mult_2_assoc),
237 (*"z1 + (z1 + k) = 2 * z1 + k"*)
239 Thm ("real_num_collect",num_str real_num_collect),
240 (*"[| l is_const; m is_const |]==> l * n + m * n = (l + m) * n"*)
241 Thm ("real_num_collect_assoc",num_str real_num_collect_assoc),
242 (*"[| l is_const; m is_const |] ==>
243 l * n + (m * n + k) = (l + m) * n + k"*)
244 Thm ("real_one_collect",num_str real_one_collect),
245 (*"m is_const ==> n + m * n = (1 + m) * n"*)
246 Thm ("real_one_collect_assoc",num_str real_one_collect_assoc),
247 (*"m is_const ==> k + (n + m * n) = k + (1 + m) * n"*)
249 Calc ("op +", eval_binop "#add_"),
250 Calc ("op *", eval_binop "#mult_"),
251 Calc ("Atools.pow", eval_binop "#power_")
253 scr = Script ((term_of o the o (parse thy)) "empty_script")
255 ruleset' := overwritelthy thy (!ruleset',
256 [("make_rooteq", make_rooteq)
259 val expand_rootbinoms = prep_rls(
260 Rls{id = "expand_rootbinoms", preconds = [],
261 rew_ord = ("termlessI",termlessI),
262 erls = Atools_erls, srls = Erls,
265 rules = [Thm ("real_plus_binom_pow2" ,num_str real_plus_binom_pow2),
266 (*"(a + b) ^^^ 2 = a ^^^ 2 + 2 * a * b + b ^^^ 2"*)
267 Thm ("real_plus_binom_times" ,num_str real_plus_binom_times),
268 (*"(a + b)*(a + b) = ...*)
269 Thm ("real_minus_binom_pow2" ,num_str real_minus_binom_pow2),
270 (*"(a - b) ^^^ 2 = a ^^^ 2 - 2 * a * b + b ^^^ 2"*)
271 Thm ("real_minus_binom_times",num_str real_minus_binom_times),
272 (*"(a - b)*(a - b) = ...*)
273 Thm ("real_plus_minus_binom1",num_str real_plus_minus_binom1),
274 (*"(a + b) * (a - b) = a ^^^ 2 - b ^^^ 2"*)
275 Thm ("real_plus_minus_binom2",num_str real_plus_minus_binom2),
276 (*"(a - b) * (a + b) = a ^^^ 2 - b ^^^ 2"*)
278 Thm ("real_pp_binom_times",num_str real_pp_binom_times),
279 (*(a + b)*(c + d) = a*c + a*d + b*c + b*d*)
280 Thm ("real_pm_binom_times",num_str real_pm_binom_times),
281 (*(a + b)*(c - d) = a*c - a*d + b*c - b*d*)
282 Thm ("real_mp_binom_times",num_str real_mp_binom_times),
283 (*(a - b)*(c p d) = a*c + a*d - b*c - b*d*)
284 Thm ("real_mm_binom_times",num_str real_mm_binom_times),
285 (*(a - b)*(c p d) = a*c - a*d - b*c + b*d*)
286 Thm ("realpow_mul",num_str realpow_mul),
287 (*(a*b)^^^n = a^^^n * b^^^n*)
289 Thm ("mult_1_left",num_str @{thm mult_1_left}), (*"1 * z = z"*)
290 Thm ("mult_zero_left",num_str @{thm mult_zero_left}), (*"0 * z = 0"*)
291 Thm ("add_0_left",num_str @{thm add_0_left}),
294 Calc ("op +", eval_binop "#add_"),
295 Calc ("op -", eval_binop "#sub_"),
296 Calc ("op *", eval_binop "#mult_"),
297 Calc ("HOL.divide" ,eval_cancel "#divide_"),
298 Calc ("Root.sqrt",eval_sqrt "#sqrt_"),
299 Calc ("Atools.pow", eval_binop "#power_"),
301 Thm ("sym_realpow_twoI",num_str (realpow_twoI RS sym)),
302 (*"r1 * r1 = r1 ^^^ 2"*)
303 Thm ("realpow_plus_1",num_str realpow_plus_1),
304 (*"r * r ^^^ n = r ^^^ (n + 1)"*)
305 Thm ("real_mult_2_assoc",num_str real_mult_2_assoc),
306 (*"z1 + (z1 + k) = 2 * z1 + k"*)
308 Thm ("real_num_collect",num_str real_num_collect),
309 (*"[| l is_const; m is_const |] ==>l * n + m * n = (l + m) * n"*)
310 Thm ("real_num_collect_assoc",num_str real_num_collect_assoc),
311 (*"[| l is_const; m is_const |] ==>
312 l * n + (m * n + k) = (l + m) * n + k"*)
313 Thm ("real_one_collect",num_str real_one_collect),
314 (*"m is_const ==> n + m * n = (1 + m) * n"*)
315 Thm ("real_one_collect_assoc",num_str real_one_collect_assoc),
316 (*"m is_const ==> k + (n + m * n) = k + (1 + m) * n"*)
318 Calc ("op +", eval_binop "#add_"),
319 Calc ("op -", eval_binop "#sub_"),
320 Calc ("op *", eval_binop "#mult_"),
321 Calc ("HOL.divide" ,eval_cancel "#divide_"),
322 Calc ("Root.sqrt",eval_sqrt "#sqrt_"),
323 Calc ("Atools.pow", eval_binop "#power_")
325 scr = Script ((term_of o the o (parse thy)) "empty_script")
329 ruleset' := overwritelthy thy (!ruleset',
330 [("expand_rootbinoms", expand_rootbinoms)