1 (* differentiation over the reals
6 theory Diff imports Calculus Trig LogExp Rational Root Poly Atools begin
14 d_d :: "[real, real]=> real"
15 (*sin, cos :: "real => real" already in Isabelle2009-2*)
17 log, ln :: "real => real"
18 nlog :: "[real, real] => real"
19 exp :: "real => real" ("E'_ ^^^ _" 80)
21 (*descriptions in the related problems*)
22 derivativeEq :: "bool => una"
25 primed :: "'a => 'a" (*"primed A" -> "A'"*)
27 (*the CAS-commands, eg. "Diff (2*x^^^3, x)",
28 "Differentiate (A = s * (a - s), s)"*)
29 Diff :: "[real * real] => real"
30 Differentiate :: "[bool * real] => bool"
32 (*subproblem and script-name*)
33 differentiate :: "[ID * (ID list) * ID, real,real] => real"
34 ("(differentiate (_)/ (_ _ ))" 9)
35 DiffScr :: "[real,real, real] => real"
36 ("((Script DiffScr (_ _ =))// (_))" 9)
37 DiffEqScr :: "[bool,real, bool] => bool"
38 ("((Script DiffEqScr (_ _ =))// (_))" 9)
40 text {*a variant of the derivatives defintion:
42 d_d :: "(real => real) => (real => real)"
45 (1) no variable 'bdv' on the meta-level required
46 (2) chain_rule "d_d (%x. (u (v x))) = (%x. (d_d u)) (v x) * d_d v"
47 (3) and no specialized chain-rules required like
48 diff_sin_chain "d_d bdv (sin u) = cos u * d_d bdv u"
50 disadvantage: d_d (%x. 1 + x^2) = ... differs from high-school notation
53 axioms (*stated as axioms, todo: prove as theorems
54 'bdv' is a constant on the meta-level *)
55 diff_const: "[| Not (bdv occurs_in a) |] ==> d_d bdv a = 0"
56 diff_var: "d_d bdv bdv = 1"
57 diff_prod_const:"[| Not (bdv occurs_in u) |] ==>
58 d_d bdv (u * v) = u * d_d bdv v"
60 diff_sum: "d_d bdv (u + v) = d_d bdv u + d_d bdv v"
61 diff_dif: "d_d bdv (u - v) = d_d bdv u - d_d bdv v"
62 diff_prod: "d_d bdv (u * v) = d_d bdv u * v + u * d_d bdv v"
63 diff_quot: "Not (v = 0) ==> (d_d bdv (u / v) =
64 (d_d bdv u * v - u * d_d bdv v) / v ^^^ 2)"
66 diff_sin: "d_d bdv (sin bdv) = cos bdv"
67 diff_sin_chain: "d_d bdv (sin u) = cos u * d_d bdv u"
68 diff_cos: "d_d bdv (cos bdv) = - sin bdv"
69 diff_cos_chain: "d_d bdv (cos u) = - sin u * d_d bdv u"
70 diff_pow: "d_d bdv (bdv ^^^ n) = n * (bdv ^^^ (n - 1))"
71 diff_pow_chain: "d_d bdv (u ^^^ n) = n * (u ^^^ (n - 1)) * d_d bdv u"
72 diff_ln: "d_d bdv (ln bdv) = 1 / bdv"
73 diff_ln_chain: "d_d bdv (ln u) = d_d bdv u / u"
74 diff_exp: "d_d bdv (exp bdv) = exp bdv"
75 diff_exp_chain: "d_d bdv (exp u) = exp u * d_d x u"
77 diff_sqrt "d_d bdv (sqrt bdv) = 1 / (2 * sqrt bdv)"
78 diff_sqrt_chain"d_d bdv (sqrt u) = d_d bdv u / (2 * sqrt u)"
82 frac_conv: "[| bdv occurs_in b; 0 < n |] ==>
83 a / (b ^^^ n) = a * b ^^^ (-n)"
84 frac_sym_conv: "n < 0 ==> a * b ^^^ n = a / b ^^^ (-n)"
86 sqrt_conv_bdv: "sqrt bdv = bdv ^^^ (1 / 2)"
87 sqrt_conv_bdv_n: "sqrt (bdv ^^^ n) = bdv ^^^ (n / 2)"
88 sqrt_conv: "bdv occurs_in u ==> sqrt u = u ^^^ (1 / 2)"
89 sqrt_sym_conv: "u ^^^ (a / 2) = sqrt (u ^^^ a)"
91 root_conv: "bdv occurs_in u ==> nroot n u = u ^^^ (1 / n)"
92 root_sym_conv: "u ^^^ (a / b) = nroot b (u ^^^ a)"
94 realpow_pow_bdv: "(bdv ^^^ b) ^^^ c = bdv ^^^ (b * c)"
99 (** eval functions **)
101 fun primed (Const (id, T)) = Const (id ^ "'", T)
102 | primed (Free (id, T)) = Free (id ^ "'", T)
103 | primed t = error ("primed called with arg = '"^ term2str t ^"'");
105 (*("primed", ("Diff.primed", eval_primed "#primed"))*)
106 fun eval_primed _ _ (p as (Const ("Diff.primed",_) $ t)) _ =
107 SOME ((term2str p) ^ " = " ^ term2str (primed t),
108 Trueprop $ (mk_equality (p, primed t)))
109 | eval_primed _ _ _ _ = NONE;
111 calclist':= overwritel (!calclist',
112 [("primed", ("Diff.primed", eval_primed "#primed"))
118 (*.converts a term such that differentiation works optimally.*)
122 rew_ord = ("termlessI",termlessI),
123 erls = append_rls "erls_diff_conv" e_rls
124 [Calc ("Atools.occurs'_in", eval_occurs_in ""),
125 Thm ("not_true",num_str @{thm not_true}),
126 Thm ("not_false",num_str @{thm not_false}),
127 Calc ("op <",eval_equ "#less_"),
128 Thm ("and_true",num_str @{thm and_true}),
129 Thm ("and_false",num_str @{thm and_false})
131 srls = Erls, calc = [],
132 rules = [Thm ("frac_conv", num_str @{thm frac_conv}),
133 Thm ("sqrt_conv_bdv", num_str @{thm sqrt_conv_bdv}),
134 Thm ("sqrt_conv_bdv_n", num_str @{thm sqrt_conv_bdv_n}),
135 Thm ("sqrt_conv", num_str @{thm sqrt_conv}),
136 Thm ("root_conv", num_str @{thm root_conv}),
137 Thm ("realpow_pow_bdv", num_str @{thm realpow_pow_bdv}),
138 Calc ("op *", eval_binop "#mult_"),
139 Thm ("rat_mult",num_str @{thm rat_mult}),
140 (*a / b * (c / d) = a * c / (b * d)*)
141 Thm ("times_divide_eq_right",num_str @{thm times_divide_eq_right}),
142 (*?x * (?y / ?z) = ?x * ?y / ?z*)
143 Thm ("times_divide_eq_left",num_str @{thm times_divide_eq_left})
144 (*?y / ?z * ?x = ?y * ?x / ?z*)
149 (*.beautifies a term after differentiation.*)
151 Rls {id="diff_sym_conv",
153 rew_ord = ("termlessI",termlessI),
154 erls = append_rls "erls_diff_sym_conv" e_rls
155 [Calc ("op <",eval_equ "#less_")
157 srls = Erls, calc = [],
158 rules = [Thm ("frac_sym_conv", num_str @{thm frac_sym_conv}),
159 Thm ("sqrt_sym_conv", num_str @{thm sqrt_sym_conv}),
160 Thm ("root_sym_conv", num_str @{thm root_sym_conv}),
161 Thm ("sym_real_mult_minus1",
162 num_str (@{thm real_mult_minus1} RS @{thm sym})),
164 Thm ("rat_mult",num_str @{thm rat_mult}),
165 (*a / b * (c / d) = a * c / (b * d)*)
166 Thm ("times_divide_eq_right",num_str @{thm times_divide_eq_right}),
167 (*?x * (?y / ?z) = ?x * ?y / ?z*)
168 Thm ("times_divide_eq_left",num_str @{thm times_divide_eq_left}),
169 (*?y / ?z * ?x = ?y * ?x / ?z*)
170 Calc ("op *", eval_binop "#mult_")
176 Rls {id="srls_differentiate..",
178 rew_ord = ("termlessI",termlessI),
180 srls = Erls, calc = [],
181 rules = [Calc("Tools.lhs", eval_lhs "eval_lhs_"),
182 Calc("Tools.rhs", eval_rhs "eval_rhs_"),
183 Calc("Diff.primed", eval_primed "Diff.primed")
190 append_rls "erls_differentiate.." e_rls
191 [Thm ("not_true",num_str @{thm not_true}),
192 Thm ("not_false",num_str @{thm not_false}),
194 Calc ("Atools.ident",eval_ident "#ident_"),
195 Calc ("Atools.is'_atom",eval_is_atom "#is_atom_"),
196 Calc ("Atools.occurs'_in",eval_occurs_in ""),
197 Calc ("Atools.is'_const",eval_const "#is_const_")
200 (*.rules for differentiation, _no_ simplification.*)
202 Rls {id="diff_rules", preconds = [], rew_ord = ("termlessI",termlessI),
203 erls = erls_diff, srls = Erls, calc = [],
204 rules = [Thm ("diff_sum",num_str @{thm diff_sum}),
205 Thm ("diff_dif",num_str @{thm diff_dif}),
206 Thm ("diff_prod_const",num_str @{thm diff_prod_const}),
207 Thm ("diff_prod",num_str @{thm diff_prod}),
208 Thm ("diff_quot",num_str @{thm diff_quot}),
209 Thm ("diff_sin",num_str @{thm diff_sin}),
210 Thm ("diff_sin_chain",num_str @{thm diff_sin_chain}),
211 Thm ("diff_cos",num_str @{thm diff_cos}),
212 Thm ("diff_cos_chain",num_str @{thm diff_cos_chain}),
213 Thm ("diff_pow",num_str @{thm diff_pow}),
214 Thm ("diff_pow_chain",num_str @{thm diff_pow_chain}),
215 Thm ("diff_ln",num_str @{thm diff_ln}),
216 Thm ("diff_ln_chain",num_str @{thm diff_ln_chain}),
217 Thm ("diff_exp",num_str @{thm diff_exp}),
218 Thm ("diff_exp_chain",num_str @{thm diff_exp_chain}),
220 Thm ("diff_sqrt",num_str @{thm diff_sqrt}),
221 Thm ("diff_sqrt_chain",num_str @{thm diff_sqrt_chain}),
223 Thm ("diff_const",num_str @{thm diff_const}),
224 Thm ("diff_var",num_str @{thm diff_var})
229 (*.normalisation for checking user-input.*)
231 Rls {id="diff_rls", preconds = [], rew_ord = ("termlessI",termlessI),
232 erls = Erls, srls = Erls, calc = [],
233 rules = [Rls_ diff_rules,
238 overwritelthy @{theory} (!ruleset',
239 [("diff_rules", prep_rls norm_diff),
240 ("norm_diff", prep_rls norm_diff),
241 ("diff_conv", prep_rls diff_conv),
242 ("diff_sym_conv", prep_rls diff_sym_conv)
247 (** problem types **)
250 (prep_pbt thy "pbl_fun" [] e_pblID
251 (["function"], [], e_rls, NONE, []));
254 (prep_pbt thy "pbl_fun_deriv" [] e_pblID
255 (["derivative_of","function"],
256 [("#Given" ,["functionTerm f_f","differentiateFor v_v"]),
257 ("#Find" ,["derivative f_f'"])
259 append_rls "e_rls" e_rls [],
260 SOME "Diff (f_f, v_v)", [["diff","differentiate_on_R"],
261 ["diff","after_simplification"]]));
263 (*here "named" is used differently from Integration"*)
265 (prep_pbt thy "pbl_fun_deriv_nam" [] e_pblID
266 (["named","derivative_of","function"],
267 [("#Given" ,["functionEq f_f","differentiateFor v_v"]),
268 ("#Find" ,["derivativeEq f_f'"])
270 append_rls "e_rls" e_rls [],
271 SOME "Differentiate (f_f, v_v)", [["diff","differentiate_equality"]]));
278 (prep_met thy "met_diff" [] e_metID
280 {rew_ord'="tless_true",rls'=Atools_erls,calc = [], srls = e_rls, prls=e_rls,
281 crls = Atools_erls, nrls = norm_diff}, "empty_script"));
284 (prep_met thy "met_diff_onR" [] e_metID
285 (["diff","differentiate_on_R"],
286 [("#Given" ,["functionTerm f_f","differentiateFor v_v"]),
287 ("#Find" ,["derivative f_f'"])
289 {rew_ord'="tless_true", rls' = erls_diff, calc = [], srls = e_rls,
290 prls=e_rls, crls = Atools_erls, nrls = norm_diff},
291 "Script DiffScr (f_f::real) (v_v::real) = " ^
292 " (let f_f' = Take (d_d v_v f_f) " ^
293 " in (((Try (Rewrite_Set_Inst [(bdv,v_v)] diff_conv False)) @@ " ^
295 " ((Repeat (Rewrite_Inst [(bdv,v_v)] diff_sum False)) Or " ^
296 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_prod_const False)) Or " ^
297 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_prod False)) Or " ^
298 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_quot True )) Or " ^
299 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_sin False)) Or " ^
300 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_sin_chain False)) Or " ^
301 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_cos False)) Or " ^
302 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_cos_chain False)) Or " ^
303 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_pow False)) Or " ^
304 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_pow_chain False)) Or " ^
305 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_ln False)) Or " ^
306 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_ln_chain False)) Or " ^
307 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_exp False)) Or " ^
308 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_exp_chain False)) Or " ^
309 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_const False)) Or " ^
310 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_var False)) Or " ^
311 " (Repeat (Rewrite_Set make_polynomial False)))) @@ " ^
312 " (Try (Rewrite_Set_Inst [(bdv,v_v)] diff_sym_conv False)))) f_f')"
317 (prep_met thy "met_diff_simpl" [] e_metID
318 (["diff","diff_simpl"],
319 [("#Given" ,["functionTerm f_f","differentiateFor v_v"]),
320 ("#Find" ,["derivative f_f'"])
322 {rew_ord'="tless_true", rls' = erls_diff, calc = [], srls = e_rls,
323 prls=e_rls, crls = Atools_erls, nrls = norm_diff},
324 "Script DiffScr (f_f::real) (v_v::real) = " ^
325 " (let f_f' = Take (d_d v_v f_f) " ^
328 " ((Repeat (Rewrite_Inst [(bdv,v_v)] diff_sum False)) Or " ^
329 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_prod_const False)) Or " ^
330 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_prod False)) Or " ^
331 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_quot True )) Or " ^
332 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_sin False)) Or " ^
333 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_sin_chain False)) Or " ^
334 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_cos False)) Or " ^
335 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_cos_chain False)) Or " ^
336 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_pow False)) Or " ^
337 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_pow_chain False)) Or " ^
338 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_ln False)) Or " ^
339 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_ln_chain False)) Or " ^
340 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_exp False)) Or " ^
341 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_exp_chain False)) Or " ^
342 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_const False)) Or " ^
343 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_var False)) Or " ^
344 " (Repeat (Rewrite_Set make_polynomial False)))) " ^
349 (prep_met thy "met_diff_equ" [] e_metID
350 (["diff","differentiate_equality"],
351 [("#Given" ,["functionEq f_f","differentiateFor v_v"]),
352 ("#Find" ,["derivativeEq f_f'"])
354 {rew_ord'="tless_true", rls' = erls_diff, calc = [],
355 srls = srls_diff, prls=e_rls, crls=Atools_erls, nrls = norm_diff},
356 "Script DiffEqScr (f_f::bool) (v_v::real) = " ^
357 " (let f_f' = Take ((primed (lhs f_f)) = d_d v_v (rhs f_f)) " ^
358 " in (((Try (Rewrite_Set_Inst [(bdv,v_v)] diff_conv False)) @@ " ^
360 " ((Repeat (Rewrite_Inst [(bdv,v_v)] diff_sum False)) Or " ^
361 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_dif False)) Or " ^
362 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_prod_const False)) Or " ^
363 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_prod False)) Or " ^
364 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_quot True )) Or " ^
365 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_sin False)) Or " ^
366 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_sin_chain False)) Or " ^
367 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_cos False)) Or " ^
368 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_cos_chain False)) Or " ^
369 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_pow False)) Or " ^
370 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_pow_chain False)) Or " ^
371 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_ln False)) Or " ^
372 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_ln_chain False)) Or " ^
373 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_exp False)) Or " ^
374 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_exp_chain False)) Or " ^
375 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_const False)) Or " ^
376 " (Repeat (Rewrite_Inst [(bdv,v_v)] diff_var False)) Or " ^
377 " (Repeat (Rewrite_Set make_polynomial False)))) @@ " ^
378 " (Try (Rewrite_Set_Inst [(bdv,v_v)] diff_sym_conv False)))) f_f')"
382 (prep_met thy "met_diff_after_simp" [] e_metID
383 (["diff","after_simplification"],
384 [("#Given" ,["functionTerm f_f","differentiateFor v_v"]),
385 ("#Find" ,["derivative f_f'"])
387 {rew_ord'="tless_true", rls' = e_rls, calc = [], srls = e_rls, prls=e_rls,
388 crls=Atools_erls, nrls = norm_Rational},
389 "Script DiffScr (f_f::real) (v_v::real) = " ^
390 " (let f_f' = Take (d_d v_v f_f) " ^
391 " in ((Try (Rewrite_Set norm_Rational False)) @@ " ^
392 " (Try (Rewrite_Set_Inst [(bdv,v_v)] diff_conv False)) @@ " ^
393 " (Try (Rewrite_Set_Inst [(bdv,v_v)] norm_diff False)) @@ " ^
394 " (Try (Rewrite_Set_Inst [(bdv,v_v)] diff_sym_conv False)) @@ " ^
395 " (Try (Rewrite_Set norm_Rational False))) f_f')"
401 (*.handle cas-input like "Diff (a * x^3 + b, x)".*)
402 (* val (t, pairl) = strip_comb (str2term "Diff (a * x^3 + b, x)");
403 val [Const ("Pair", _) $ t $ bdv] = pairl;
405 fun argl2dtss [Const ("Pair", _) $ t $ bdv] =
406 [((term_of o the o (parse thy)) "functionTerm", [t]),
407 ((term_of o the o (parse thy)) "differentiateFor", [bdv]),
408 ((term_of o the o (parse thy)) "derivative",
409 [(term_of o the o (parse thy)) "f_f'"])
411 | argl2dtss _ = error "Diff.ML: wrong argument for argl2dtss";
414 [((term_of o the o (parse thy)) "Diff",
415 (("Isac", ["derivative_of","function"], ["no_met"]),
419 (*.handle cas-input like "Differentiate (A = s * (a - s), s)".*)
420 (* val (t, pairl) = strip_comb (str2term "Differentiate (A = s * (a - s), s)");
421 val [Const ("Pair", _) $ t $ bdv] = pairl;
423 fun argl2dtss [Const ("Pair", _) $ t $ bdv] =
424 [((term_of o the o (parse thy)) "functionEq", [t]),
425 ((term_of o the o (parse thy)) "differentiateFor", [bdv]),
426 ((term_of o the o (parse thy)) "derivativeEq",
427 [(term_of o the o (parse thy)) "f_f'::bool"])
429 | argl2dtss _ = error "Diff.ML: wrong argument for argl2dtss";
432 [((term_of o the o (parse thy)) "Differentiate",
433 (("Isac", ["named","derivative_of","function"], ["no_met"]),