1 (* integration over the reals
4 (c) due to copyright terms
7 theory Integrate imports Diff begin
11 Integral :: "[real, real]=> real" ("Integral _ D _" 91)
12 (*new'_c :: "real => real" ("new'_c _" 66)*)
13 is'_f'_x :: "real => bool" ("_ is'_f'_x" 10)
15 (*descriptions in the related problems*)
16 integrateBy :: "real => una"
17 antiDerivative :: "real => una"
18 antiDerivativeName :: "(real => real) => una"
20 (*the CAS-command, eg. "Integrate (2*x^^^3, x)"*)
21 Integrate :: "[real * real] => real"
24 (*stated as axioms, todo: prove as theorems
25 'bdv' is a constant handled on the meta-level
26 specifically as a 'bound variable' *)
28 integral_const: "Not (bdv occurs_in u) ==> Integral u D bdv = u * bdv" and
29 integral_var: "Integral bdv D bdv = bdv ^^^ 2 / 2" and
31 integral_add: "Integral (u + v) D bdv =
32 (Integral u D bdv) + (Integral v D bdv)" and
33 integral_mult: "[| Not (bdv occurs_in u); bdv occurs_in v |] ==>
34 Integral (u * v) D bdv = u * (Integral v D bdv)" and
35 (*WN080222: this goes into sub-terms, too ...
36 call_for_new_c: "[| Not (matches (u + new_c v) a); Not (a is_f_x) |] ==>
39 integral_pow: "Integral bdv ^^^ n D bdv = bdv ^^^ (n+1) / (n + 1)"
44 (** eval functions **)
46 val c = Free ("c", HOLogic.realT);
47 (*.create a new unique variable 'c..' in a term; for use by Rule.Num_Calc in a rls;
48 an alternative to do this would be '(Try (Calculate new_c_) (new_c es__))'
49 in the script; this will be possible if currying doesnt take the value
50 from a variable, but the value '(new_c es__)' itself.*)
53 case (Symbol.explode o id_of) var of
55 | "c"::"_"::is => (case (TermC.int_opt_of_string o implode) is of
59 fun get_coeff c = case (Symbol.explode o id_of) c of
60 "c"::"_"::is => (the o TermC.int_opt_of_string o implode) is
62 val cs = filter selc (TermC.vars term);
66 | [c] => Free ("c_2", HOLogic.realT)
68 let val max_coeff = maxl (map get_coeff cs)
69 in Free ("c_"^string_of_int (max_coeff + 1), HOLogic.realT) end
73 (*("new_c", ("Integrate.new'_c", eval_new_c "#new_c_"))*)
74 fun eval_new_c _ _ (p as (Const ("Integrate.new'_c",_) $ t)) _ =
75 SOME ((UnparseC.term p) ^ " = " ^ UnparseC.term (new_c p),
76 Trueprop $ (mk_equality (p, new_c p)))
77 | eval_new_c _ _ _ _ = NONE;
81 (*("add_new_c", ("Integrate.add'_new'_c", eval_add_new_c "#add_new_c_"))
82 add a new c to a term or a fun-equation;
83 this is _not in_ the term, because only applied to _whole_ term*)
84 fun eval_add_new_c (_:string) "Integrate.add'_new'_c" p (_:theory) =
85 let val p' = case p of
86 Const ("HOL.eq", T) $ lh $ rh =>
87 Const ("HOL.eq", T) $ lh $ TermC.mk_add rh (new_c rh)
88 | p => TermC.mk_add p (new_c p)
89 in SOME ((UnparseC.term p) ^ " = " ^ UnparseC.term p',
90 HOLogic.Trueprop $ (TermC.mk_equality (p, p')))
92 | eval_add_new_c _ _ _ _ = NONE;
95 (*("is_f_x", ("Integrate.is'_f'_x", eval_is_f_x "is_f_x_"))*)
96 fun eval_is_f_x _ _(p as (Const ("Integrate.is'_f'_x", _)
99 then SOME ((UnparseC.term p) ^ " = True",
100 HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))
101 else SOME ((UnparseC.term p) ^ " = False",
102 HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))
103 | eval_is_f_x _ _ _ _ = NONE;
105 setup \<open>KEStore_Elems.add_calcs
106 [("add_new_c", ("Integrate.add'_new'_c", eval_add_new_c "add_new_c_")),
107 ("is_f_x", ("Integrate.is'_f'_x", eval_is_f_x "is_f_idextifier_"))]\<close>
111 (*.rulesets for integration.*)
112 val integration_rules =
113 Rule_Def.Repeat {id="integration_rules", preconds = [],
114 rew_ord = ("termlessI",termlessI),
115 erls = Rule_Def.Repeat {id="conditions_in_integration_rules",
117 rew_ord = ("termlessI",termlessI),
118 erls = Rule_Set.Empty,
119 srls = Rule_Set.Empty, calc = [], errpatts = [],
120 rules = [(*for rewriting conditions in Thm's*)
121 Rule.Num_Calc ("Prog_Expr.occurs'_in", Prog_Expr.eval_occurs_in "#occurs_in_"),
122 Rule.Thm ("not_true", ThmC.numerals_to_Free @{thm not_true}),
123 Rule.Thm ("not_false",@{thm not_false})
125 scr = Rule.EmptyScr},
126 srls = Rule_Set.Empty, calc = [], errpatts = [],
128 Rule.Thm ("integral_const", ThmC.numerals_to_Free @{thm integral_const}),
129 Rule.Thm ("integral_var", ThmC.numerals_to_Free @{thm integral_var}),
130 Rule.Thm ("integral_add", ThmC.numerals_to_Free @{thm integral_add}),
131 Rule.Thm ("integral_mult", ThmC.numerals_to_Free @{thm integral_mult}),
132 Rule.Thm ("integral_pow", ThmC.numerals_to_Free @{thm integral_pow}),
133 Rule.Num_Calc ("Groups.plus_class.plus", (**)eval_binop "#add_")(*for n+1*)
135 scr = Rule.EmptyScr};
139 Rule_Set.Seqence {id="add_new_c", preconds = [],
140 rew_ord = ("termlessI",termlessI),
141 erls = Rule_Def.Repeat {id="conditions_in_add_new_c",
143 rew_ord = ("termlessI",termlessI),
144 erls = Rule_Set.Empty,
145 srls = Rule_Set.Empty, calc = [], errpatts = [],
146 rules = [Rule.Num_Calc ("Prog_Expr.matches", Prog_Expr.eval_matches""),
147 Rule.Num_Calc ("Integrate.is'_f'_x",
148 eval_is_f_x "is_f_x_"),
149 Rule.Thm ("not_true", ThmC.numerals_to_Free @{thm not_true}),
150 Rule.Thm ("not_false", ThmC.numerals_to_Free @{thm not_false})
152 scr = Rule.EmptyScr},
153 srls = Rule_Set.Empty, calc = [], errpatts = [],
154 rules = [ (*Rule.Thm ("call_for_new_c", ThmC.numerals_to_Free @{thm call_for_new_c}),*)
155 Rule.Cal1 ("Integrate.add'_new'_c", eval_add_new_c "new_c_")
157 scr = Rule.EmptyScr};
161 (*.rulesets for simplifying Integrals.*)
163 (*.for simplify_Integral adapted from 'norm_Rational_rls'.*)
164 val norm_Rational_rls_noadd_fractions =
165 Rule_Def.Repeat {id = "norm_Rational_rls_noadd_fractions", preconds = [],
166 rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
167 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
168 rules = [(*Rule.Rls_ add_fractions_p_rls,!!!*)
169 Rule.Rls_ (*rat_mult_div_pow original corrected WN051028*)
170 (Rule_Def.Repeat {id = "rat_mult_div_pow", preconds = [],
171 rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
172 erls = (*FIXME.WN051028 Rule_Set.empty,*)
173 Rule_Set.append_rules "Rule_Set.empty-is_polyexp" Rule_Set.empty
174 [Rule.Num_Calc ("Poly.is'_polyexp",
175 eval_is_polyexp "")],
176 srls = Rule_Set.Empty, calc = [], errpatts = [],
177 rules = [Rule.Thm ("rat_mult", ThmC.numerals_to_Free @{thm rat_mult}),
178 (*"?a / ?b * (?c / ?d) = ?a * ?c / (?b * ?d)"*)
179 Rule.Thm ("rat_mult_poly_l", ThmC.numerals_to_Free @{thm rat_mult_poly_l}),
180 (*"?c is_polyexp ==> ?c * (?a / ?b) = ?c * ?a / ?b"*)
181 Rule.Thm ("rat_mult_poly_r", ThmC.numerals_to_Free @{thm rat_mult_poly_r}),
182 (*"?c is_polyexp ==> ?a / ?b * ?c = ?a * ?c / ?b"*)
184 Rule.Thm ("real_divide_divide1_mg",
185 ThmC.numerals_to_Free @{thm real_divide_divide1_mg}),
186 (*"y ~= 0 ==> (u / v) / (y / z) = (u * z) / (y * v)"*)
187 Rule.Thm ("divide_divide_eq_right",
188 ThmC.numerals_to_Free @{thm divide_divide_eq_right}),
189 (*"?x / (?y / ?z) = ?x * ?z / ?y"*)
190 Rule.Thm ("divide_divide_eq_left",
191 ThmC.numerals_to_Free @{thm divide_divide_eq_left}),
192 (*"?x / ?y / ?z = ?x / (?y * ?z)"*)
193 Rule.Num_Calc ("Rings.divide_class.divide", Prog_Expr.eval_cancel "#divide_e"),
195 Rule.Thm ("rat_power", ThmC.numerals_to_Free @{thm rat_power})
196 (*"(?a / ?b) ^^^ ?n = ?a ^^^ ?n / ?b ^^^ ?n"*)
200 Rule.Rls_ make_rat_poly_with_parentheses,
201 Rule.Rls_ cancel_p_rls,(*FIXME:cancel_p does NOT order sometimes*)
202 Rule.Rls_ rat_reduce_1
207 (*.for simplify_Integral adapted from 'norm_Rational'.*)
208 val norm_Rational_noadd_fractions =
209 Rule_Set.Seqence {id = "norm_Rational_noadd_fractions", preconds = [],
210 rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
211 erls = norm_rat_erls, srls = Rule_Set.Empty, calc = [], errpatts = [],
212 rules = [Rule.Rls_ discard_minus,
213 Rule.Rls_ rat_mult_poly,(* removes double fractions like a/b/c *)
214 Rule.Rls_ make_rat_poly_with_parentheses, (*WN0510 also in(#)below*)
215 Rule.Rls_ cancel_p_rls, (*FIXME.MG:cancel_p does NOT order sometim*)
216 Rule.Rls_ norm_Rational_rls_noadd_fractions,(* the main rls (#) *)
217 Rule.Rls_ discard_parentheses1 (* mult only *)
222 (*.simplify terms before and after Integration such that
223 ..a.x^2/2 + b.x^3/3.. is made to ..a/2.x^2 + b/3.x^3.. (and NO
224 common denominator as done by norm_Rational or make_ratpoly_in.
225 This is a copy from 'make_ratpoly_in' with respective reduction of rules and
226 *1* expand the term, ie. distribute * and / over +
229 Rule_Set.append_rules "separate_bdv2"
231 [Rule.Thm ("separate_bdv", ThmC.numerals_to_Free @{thm separate_bdv}),
232 (*"?a * ?bdv / ?b = ?a / ?b * ?bdv"*)
233 Rule.Thm ("separate_bdv_n", ThmC.numerals_to_Free @{thm separate_bdv_n}),
234 Rule.Thm ("separate_1_bdv", ThmC.numerals_to_Free @{thm separate_1_bdv}),
235 (*"?bdv / ?b = (1 / ?b) * ?bdv"*)
236 Rule.Thm ("separate_1_bdv_n", ThmC.numerals_to_Free @{thm separate_1_bdv_n})(*,
237 (*"?bdv ^^^ ?n / ?b = 1 / ?b * ?bdv ^^^ ?n"*)
238 *****Rule.Thm ("add_divide_distrib",
239 ***** ThmC.numerals_to_Free @{thm add_divide_distrib})
240 (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)----------*)
242 val simplify_Integral =
243 Rule_Set.Seqence {id = "simplify_Integral", preconds = []:term list,
244 rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
245 erls = Atools_erls, srls = Rule_Set.Empty,
246 calc = [], errpatts = [],
247 rules = [Rule.Thm ("distrib_right", ThmC.numerals_to_Free @{thm distrib_right}),
248 (*"(?z1.0 + ?z2.0) * ?w = ?z1.0 * ?w + ?z2.0 * ?w"*)
249 Rule.Thm ("add_divide_distrib", ThmC.numerals_to_Free @{thm add_divide_distrib}),
250 (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)
251 (*^^^^^ *1* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^*)
252 Rule.Rls_ norm_Rational_noadd_fractions,
253 Rule.Rls_ order_add_mult_in,
254 Rule.Rls_ discard_parentheses,
255 (*Rule.Rls_ collect_bdv, from make_polynomial_in*)
256 Rule.Rls_ separate_bdv2,
257 Rule.Num_Calc ("Rings.divide_class.divide", Prog_Expr.eval_cancel "#divide_e")
259 scr = Rule.EmptyScr};
262 (*simplify terms before and after Integration such that
263 ..a.x^2/2 + b.x^3/3.. is made to ..a/2.x^2 + b/3.x^3.. (and NO
264 common denominator as done by norm_Rational or make_ratpoly_in.
265 This is a copy from 'make_polynomial_in' with insertions from
267 THIS IS KEPT FOR COMPARISON ............................................
268 * val simplify_Integral = prep_rls'(
269 * Rule_Set.Seqence {id = "", preconds = []:term list,
270 * rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),
271 * erls = Atools_erls, srls = Rule_Set.Empty,
272 * calc = [], (*asm_thm = [],*)
273 * rules = [Rule.Rls_ expand_poly,
274 * Rule.Rls_ order_add_mult_in,
275 * Rule.Rls_ simplify_power,
276 * Rule.Rls_ collect_numerals,
277 * Rule.Rls_ reduce_012,
278 * Rule.Thm ("realpow_oneI", ThmC.numerals_to_Free @{thm realpow_oneI}),
279 * Rule.Rls_ discard_parentheses,
280 * Rule.Rls_ collect_bdv,
281 * (*below inserted from 'make_ratpoly_in'*)
282 * Rule.Rls_ (Rule_Set.append_rules "separate_bdv"
284 * [Rule.Thm ("separate_bdv", ThmC.numerals_to_Free @{thm separate_bdv}),
285 * (*"?a * ?bdv / ?b = ?a / ?b * ?bdv"*)
286 * Rule.Thm ("separate_bdv_n", ThmC.numerals_to_Free @{thm separate_bdv_n}),
287 * Rule.Thm ("separate_1_bdv", ThmC.numerals_to_Free @{thm separate_1_bdv}),
288 * (*"?bdv / ?b = (1 / ?b) * ?bdv"*)
289 * Rule.Thm ("separate_1_bdv_n", ThmC.numerals_to_Free @{thm separate_1_bdv_n})(*,
290 * (*"?bdv ^^^ ?n / ?b = 1 / ?b * ?bdv ^^^ ?n"*)
291 * Rule.Thm ("add_divide_distrib",
292 * ThmC.numerals_to_Free @{thm add_divide_distrib})
293 * (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)*)
295 * Rule.Num_Calc ("Rings.divide_class.divide" , eval_cancel "#divide_e")
297 * scr = Rule.EmptyScr
299 .......................................................................*)
302 Rule_Set.Seqence {id="integration", preconds = [],
303 rew_ord = ("termlessI",termlessI),
304 erls = Rule_Def.Repeat {id="conditions_in_integration",
306 rew_ord = ("termlessI",termlessI),
307 erls = Rule_Set.Empty,
308 srls = Rule_Set.Empty, calc = [], errpatts = [],
310 scr = Rule.EmptyScr},
311 srls = Rule_Set.Empty, calc = [], errpatts = [],
312 rules = [ Rule.Rls_ integration_rules,
314 Rule.Rls_ simplify_Integral
316 scr = Rule.EmptyScr};
318 val prep_rls' = Auto_Prog.prep_rls @{theory};
320 setup \<open>KEStore_Elems.add_rlss
321 [("integration_rules", (Context.theory_name @{theory}, prep_rls' integration_rules)),
322 ("add_new_c", (Context.theory_name @{theory}, prep_rls' add_new_c)),
323 ("simplify_Integral", (Context.theory_name @{theory}, prep_rls' simplify_Integral)),
324 ("integration", (Context.theory_name @{theory}, prep_rls' integration)),
325 ("separate_bdv2", (Context.theory_name @{theory}, prep_rls' separate_bdv2)),
327 ("norm_Rational_noadd_fractions", (Context.theory_name @{theory},
328 prep_rls' norm_Rational_noadd_fractions)),
329 ("norm_Rational_rls_noadd_fractions", (Context.theory_name @{theory},
330 prep_rls' norm_Rational_rls_noadd_fractions))]\<close>
333 setup \<open>KEStore_Elems.add_pbts
334 [(Specify.prep_pbt thy "pbl_fun_integ" [] Celem.e_pblID
335 (["integrate","function"],
336 [("#Given" ,["functionTerm f_f", "integrateBy v_v"]),
337 ("#Find" ,["antiDerivative F_F"])],
338 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)],
339 SOME "Integrate (f_f, v_v)",
340 [["diff","integration"]])),
341 (*here "named" is used differently from Differentiation"*)
342 (Specify.prep_pbt thy "pbl_fun_integ_nam" [] Celem.e_pblID
343 (["named","integrate","function"],
344 [("#Given" ,["functionTerm f_f", "integrateBy v_v"]),
345 ("#Find" ,["antiDerivativeName F_F"])],
346 Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)],
347 SOME "Integrate (f_f, v_v)",
348 [["diff","integration","named"]]))]\<close>
352 partial_function (tailrec) integrate :: "real \<Rightarrow> real \<Rightarrow> real"
354 "integrate f_f v_v = (
356 t_t = Take (Integral f_f D v_v)
358 (Rewrite_Set_Inst [(''bdv'', v_v)] ''integration'') t_t)"
359 setup \<open>KEStore_Elems.add_mets
360 [Specify.prep_met thy "met_diffint" [] Celem.e_metID
361 (["diff","integration"],
362 [("#Given" ,["functionTerm f_f", "integrateBy v_v"]), ("#Find" ,["antiDerivative F_F"])],
363 {rew_ord'="tless_true", rls'=Atools_erls, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
364 crls = Atools_erls, errpats = [], nrls = Rule_Set.empty},
365 @{thm integrate.simps})]
368 partial_function (tailrec) intergrate_named :: "real \<Rightarrow> real \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> bool"
370 "intergrate_named f_f v_v F_F =(
372 t_t = Take (F_F v_v = Integral f_f D v_v)
374 (Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''simplify_Integral'')) #>
375 (Rewrite_Set_Inst [(''bdv'', v_v)] ''integration'')
377 setup \<open>KEStore_Elems.add_mets
378 [Specify.prep_met thy "met_diffint_named" [] Celem.e_metID
379 (["diff","integration","named"],
380 [("#Given" ,["functionTerm f_f", "integrateBy v_v"]),
381 ("#Find" ,["antiDerivativeName F_F"])],
382 {rew_ord'="tless_true", rls'=Atools_erls, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
383 crls = Atools_erls, errpats = [], nrls = Rule_Set.empty},
384 @{thm intergrate_named.simps})]