1 (* differentiation over the reals
6 theory Diff imports Calculus Trig LogExp Rational Root Poly Base_Tools begin
14 d_d :: "[real, real]=> real"
16 (*descriptions in the related problems*)
17 derivativeEq :: "bool => una"
20 primed :: "'a => 'a" (*"primed A" -> "A'"*)
22 (*the CAS-commands, eg. "Diff (2*x \<up> 3, x)",
23 "Differentiate (A = s * (a - s), s)"*)
24 Diff :: "[real * real] => real"
25 Differentiate :: "[bool * real] => bool"
28 differentiate :: "[char list * char list list * char list, real, real] => real"
29 ("(differentiate (_)/ (_ _ ))" 9)
31 text \<open>a variant of the derivatives defintion:
33 d_d :: "(real => real) => (real => real)"
36 (1) no variable 'bdv' on the meta-level required
37 (2) chain_rule "d_d (%x. (u (v x))) = (%x. (d_d u)) (v x) * d_d v"
38 (3) and no specialized chain-rules required like
39 diff_sin_chain "d_d bdv (sin u) = cos u * d_d bdv u"
41 disadvantage: d_d (%x. 1 + x^2) = ... differs from high-school notation
44 axiomatization where (*stated as axioms, todo: prove as theorems
45 'bdv' is a constant on the meta-level *)
46 diff_const: "[| Not (bdv occurs_in a) |] ==> d_d bdv a = 0" and
47 diff_var: "d_d bdv bdv = 1" and
48 diff_prod_const:"[| Not (bdv occurs_in u) |] ==>
49 d_d bdv (u * v) = u * d_d bdv v" and
51 diff_sum: "d_d bdv (u + v) = d_d bdv u + d_d bdv v" and
52 diff_dif: "d_d bdv (u - v) = d_d bdv u - d_d bdv v" and
53 diff_prod: "d_d bdv (u * v) = d_d bdv u * v + u * d_d bdv v" and
54 diff_quot: "Not (v = 0) ==> (d_d bdv (u / v) =
55 (d_d bdv u * v - u * d_d bdv v) / v \<up> 2)" and
57 diff_sin: "d_d bdv (sin bdv) = cos bdv" and
58 diff_sin_chain: "d_d bdv (sin u) = cos u * d_d bdv u" and
59 diff_cos: "d_d bdv (cos bdv) = - sin bdv" and
60 diff_cos_chain: "d_d bdv (cos u) = - sin u * d_d bdv u" and
61 diff_pow: "d_d bdv (bdv \<up> n) = n * (bdv \<up> (n - 1))" and
62 diff_pow_chain: "d_d bdv (u \<up> n) = n * (u \<up> (n - 1)) * d_d bdv u" and
63 diff_ln: "d_d bdv (ln bdv) = 1 / bdv" and
64 diff_ln_chain: "d_d bdv (ln u) = d_d bdv u / u" and
65 diff_exp: "d_d bdv (exp bdv) = exp bdv" and
66 diff_exp_chain: "d_d bdv (exp u) = exp u * d_d x u" and
68 diff_sqrt "d_d bdv (sqrt bdv) = 1 / (2 * sqrt bdv)"
69 diff_sqrt_chain"d_d bdv (sqrt u) = d_d bdv u / (2 * sqrt u)"
73 frac_conv: "[| bdv occurs_in b; 0 < n |] ==>
74 a / (b \<up> n) = a * b \<up> (-n)" and
75 frac_sym_conv: "n < 0 ==> a * b \<up> n = a / b \<up> (-n)" and
77 sqrt_conv_bdv: "sqrt bdv = bdv \<up> (1 / 2)" and
78 sqrt_conv_bdv_n: "sqrt (bdv \<up> n) = bdv \<up> (n / 2)" and
79 (*Ambiguous input\<^here> produces 3 parse trees -----------------------------\\*)
80 sqrt_conv: "bdv occurs_in u ==> sqrt u = u \<up> (1 / 2)" and
81 (*Ambiguous input\<^here> produces 3 parse trees -----------------------------//*)
82 sqrt_sym_conv: "u \<up> (a / 2) = sqrt (u \<up> a)" and
84 root_conv: "bdv occurs_in u ==> nroot n u = u \<up> (1 / n)" and
85 root_sym_conv: "u \<up> (a / b) = nroot b (u \<up> a)" and
87 realpow_pow_bdv: "(bdv \<up> b) \<up> c = bdv \<up> (b * c)"
90 (** eval functions **)
92 fun primed (Const (id, T)) = Const (id ^ "'", T)
93 | primed (Free (id, T)) = Free (id ^ "'", T)
94 | primed t = raise ERROR ("primed called with arg = '"^ UnparseC.term t ^"'");
96 (*("primed", ("Diff.primed", eval_primed "#primed"))*)
97 fun eval_primed _ _ (p as (Const ("Diff.primed",_) $ t)) _ =
98 SOME ((UnparseC.term p) ^ " = " ^ UnparseC.term (primed t),
99 HOLogic.Trueprop $ (TermC.mk_equality (p, primed t)))
100 | eval_primed _ _ _ _ = NONE;
102 setup \<open>KEStore_Elems.add_calcs
103 [("primed", ("Diff.primed", eval_primed "#primed"))]\<close>
107 (*.converts a term such that differentiation works optimally.*)
109 Rule_Def.Repeat {id="diff_conv",
111 rew_ord = ("termlessI",termlessI),
112 erls = Rule_Set.append_rules "erls_diff_conv" Rule_Set.empty
113 [\<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in ""),
114 Rule.Thm ("not_true",ThmC.numerals_to_Free @{thm not_true}),
115 Rule.Thm ("not_false",ThmC.numerals_to_Free @{thm not_false}),
116 \<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_"),
117 Rule.Thm ("and_true",ThmC.numerals_to_Free @{thm and_true}),
118 Rule.Thm ("and_false",ThmC.numerals_to_Free @{thm and_false})
120 srls = Rule_Set.Empty, calc = [], errpatts = [],
122 [Rule.Thm ("frac_conv", ThmC.numerals_to_Free @{thm frac_conv}),
123 (*"?bdv occurs_in ?b \<Longrightarrow> 0 < ?n \<Longrightarrow> ?a / ?b \<up> ?n = ?a * ?b \<up> - ?n"*)
124 Rule.Thm ("sqrt_conv_bdv", ThmC.numerals_to_Free @{thm sqrt_conv_bdv}),
125 (*"sqrt ?bdv = ?bdv \<up> (1 / 2)"*)
126 Rule.Thm ("sqrt_conv_bdv_n", ThmC.numerals_to_Free @{thm sqrt_conv_bdv_n}),
127 (*"sqrt (?bdv \<up> ?n) = ?bdv \<up> (?n / 2)"*)
128 Rule.Thm ("sqrt_conv", ThmC.numerals_to_Free @{thm sqrt_conv}),
129 (*"?bdv occurs_in ?u \<Longrightarrow> sqrt ?u = ?u \<up> (1 / 2)"*)
130 Rule.Thm ("root_conv", ThmC.numerals_to_Free @{thm root_conv}),
131 (*"?bdv occurs_in ?u \<Longrightarrow> nroot ?n ?u = ?u \<up> (1 / ?n)"*)
132 Rule.Thm ("realpow_pow_bdv", ThmC.numerals_to_Free @{thm realpow_pow_bdv}),
133 (* "(?bdv \<up> ?b) \<up> ?c = ?bdv \<up> (?b * ?c)"*)
134 \<^rule_eval>\<open>times\<close> (**)(eval_binop "#mult_"),
135 Rule.Thm ("rat_mult",ThmC.numerals_to_Free @{thm rat_mult}),
136 (*a / b * (c / d) = a * c / (b * d)*)
137 Rule.Thm ("times_divide_eq_right",ThmC.numerals_to_Free @{thm times_divide_eq_right}),
138 (*?x * (?y / ?z) = ?x * ?y / ?z*)
139 Rule.Thm ("times_divide_eq_left",ThmC.numerals_to_Free @{thm times_divide_eq_left})
140 (*?y / ?z * ?x = ?y * ?x / ?z*)
142 scr = Rule.Empty_Prog};
145 (*.beautifies a term after differentiation.*)
147 Rule_Def.Repeat {id="diff_sym_conv",
149 rew_ord = ("termlessI",termlessI),
150 erls = Rule_Set.append_rules "erls_diff_sym_conv" Rule_Set.empty
151 [\<^rule_eval>\<open>less\<close> (Prog_Expr.eval_equ "#less_")],
152 srls = Rule_Set.Empty, calc = [], errpatts = [],
153 rules = [Rule.Thm ("frac_sym_conv", ThmC.numerals_to_Free @{thm frac_sym_conv}),
154 Rule.Thm ("sqrt_sym_conv", ThmC.numerals_to_Free @{thm sqrt_sym_conv}),
155 Rule.Thm ("root_sym_conv", ThmC.numerals_to_Free @{thm root_sym_conv}),
156 Rule.Thm ("sym_real_mult_minus1",
157 ThmC.numerals_to_Free (@{thm real_mult_minus1} RS @{thm sym})),
159 Rule.Thm ("rat_mult",ThmC.numerals_to_Free @{thm rat_mult}),
160 (*a / b * (c / d) = a * c / (b * d)*)
161 Rule.Thm ("times_divide_eq_right",ThmC.numerals_to_Free @{thm times_divide_eq_right}),
162 (*?x * (?y / ?z) = ?x * ?y / ?z*)
163 Rule.Thm ("times_divide_eq_left",ThmC.numerals_to_Free @{thm times_divide_eq_left}),
164 (*?y / ?z * ?x = ?y * ?x / ?z*)
165 \<^rule_eval>\<open>times\<close> (**)(eval_binop "#mult_")
167 scr = Rule.Empty_Prog};
171 Rule_Def.Repeat {id="srls_differentiate..",
173 rew_ord = ("termlessI",termlessI),
174 erls = Rule_Set.empty,
175 srls = Rule_Set.Empty, calc = [], errpatts = [],
176 rules = [\<^rule_eval>\<open>Prog_Expr.lhs\<close> (Prog_Expr.eval_lhs "eval_lhs_"),
177 \<^rule_eval>\<open>Prog_Expr.rhs\<close> (Prog_Expr.eval_rhs "eval_rhs_"),
178 \<^rule_eval>\<open>Diff.primed\<close> (eval_primed "Diff.primed")
180 scr = Rule.Empty_Prog};
185 Rule_Set.append_rules "erls_differentiate.." Rule_Set.empty
186 [Rule.Thm ("not_true",ThmC.numerals_to_Free @{thm not_true}),
187 Rule.Thm ("not_false",ThmC.numerals_to_Free @{thm not_false}),
189 \<^rule_eval>\<open>Prog_Expr.ident\<close> (Prog_Expr.eval_ident "#ident_"),
190 \<^rule_eval>\<open>Prog_Expr.is_atom\<close> (Prog_Expr.eval_is_atom "#is_atom_"),
191 \<^rule_eval>\<open>Prog_Expr.occurs_in\<close> (Prog_Expr.eval_occurs_in ""),
192 \<^rule_eval>\<open>Prog_Expr.is_const\<close> (Prog_Expr.eval_const "#is_const_")
195 (*.rules for differentiation, _no_ simplification.*)
197 Rule_Def.Repeat {id="diff_rules", preconds = [], rew_ord = ("termlessI",termlessI),
198 erls = erls_diff, srls = Rule_Set.Empty, calc = [], errpatts = [],
199 rules = [Rule.Thm ("diff_sum",ThmC.numerals_to_Free @{thm diff_sum}),
200 Rule.Thm ("diff_dif",ThmC.numerals_to_Free @{thm diff_dif}),
201 Rule.Thm ("diff_prod_const",ThmC.numerals_to_Free @{thm diff_prod_const}),
202 Rule.Thm ("diff_prod",ThmC.numerals_to_Free @{thm diff_prod}),
203 Rule.Thm ("diff_quot",ThmC.numerals_to_Free @{thm diff_quot}),
204 Rule.Thm ("diff_sin",ThmC.numerals_to_Free @{thm diff_sin}),
205 Rule.Thm ("diff_sin_chain",ThmC.numerals_to_Free @{thm diff_sin_chain}),
206 Rule.Thm ("diff_cos",ThmC.numerals_to_Free @{thm diff_cos}),
207 Rule.Thm ("diff_cos_chain",ThmC.numerals_to_Free @{thm diff_cos_chain}),
208 Rule.Thm ("diff_pow",ThmC.numerals_to_Free @{thm diff_pow}),
209 Rule.Thm ("diff_pow_chain",ThmC.numerals_to_Free @{thm diff_pow_chain}),
210 Rule.Thm ("diff_ln",ThmC.numerals_to_Free @{thm diff_ln}),
211 Rule.Thm ("diff_ln_chain",ThmC.numerals_to_Free @{thm diff_ln_chain}),
212 Rule.Thm ("diff_exp",ThmC.numerals_to_Free @{thm diff_exp}),
213 Rule.Thm ("diff_exp_chain",ThmC.numerals_to_Free @{thm diff_exp_chain}),
215 Rule.Thm ("diff_sqrt",ThmC.numerals_to_Free @{thm diff_sqrt}),
216 Rule.Thm ("diff_sqrt_chain",ThmC.numerals_to_Free @{thm diff_sqrt_chain}),
218 Rule.Thm ("diff_const",ThmC.numerals_to_Free @{thm diff_const}),
219 Rule.Thm ("diff_var",ThmC.numerals_to_Free @{thm diff_var})
221 scr = Rule.Empty_Prog};
224 (*.normalisation for checking user-input.*)
227 {id="norm_diff", preconds = [], rew_ord = ("termlessI",termlessI),
228 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
229 rules = [Rule.Rls_ diff_rules, Rule.Rls_ norm_Poly ],
230 scr = Rule.Empty_Prog};
233 erls_diff = \<open>prep_rls' erls_diff\<close> and
234 diff_rules = \<open>prep_rls' diff_rules\<close> and
235 norm_diff = \<open>prep_rls' norm_diff\<close> and
236 diff_conv = \<open>prep_rls' diff_conv\<close> and
237 diff_sym_conv = \<open>prep_rls' diff_sym_conv\<close>
239 (** problem types **)
240 setup \<open>KEStore_Elems.add_pbts
241 [(Problem.prep_input @{theory} "pbl_fun" [] Problem.id_empty (["function"], [], Rule_Set.empty, NONE, [])),
242 (Problem.prep_input @{theory} "pbl_fun_deriv" [] Problem.id_empty
243 (["derivative_of", "function"],
244 [("#Given" ,["functionTerm f_f", "differentiateFor v_v"]),
245 ("#Find" ,["derivative f_f'"])],
246 Rule_Set.append_rules "empty" Rule_Set.empty [],
247 SOME "Diff (f_f, v_v)", [["diff", "differentiate_on_R"],
248 ["diff", "after_simplification"]])),
249 (*here "named" is used differently from Integration"*)
250 (Problem.prep_input @{theory} "pbl_fun_deriv_nam" [] Problem.id_empty
251 (["named", "derivative_of", "function"],
252 [("#Given" ,["functionEq f_f", "differentiateFor v_v"]),
253 ("#Find" ,["derivativeEq f_f'"])],
254 Rule_Set.append_rules "empty" Rule_Set.empty [],
255 SOME "Differentiate (f_f, v_v)",
256 [["diff", "differentiate_equality"]]))]\<close>
261 (*.handle cas-input like "Diff (a * x^3 + b, x)".*)
262 (* val (t, pairl) = strip_comb (str2term "Diff (a * x^3 + b, x)");
263 val [Const ("Product_Type.Pair", _) $ t $ bdv] = pairl;
265 fun argl2dtss [Const ("Product_Type.Pair", _) $ t $ bdv] =
266 [((Thm.term_of o the o (TermC.parse \<^theory>)) "functionTerm", [t]),
267 ((Thm.term_of o the o (TermC.parse \<^theory>)) "differentiateFor", [bdv]),
268 ((Thm.term_of o the o (TermC.parse \<^theory>)) "derivative",
269 [(Thm.term_of o the o (TermC.parse \<^theory>)) "f_f'"])
271 | argl2dtss _ = raise ERROR "Diff.ML: wrong argument for argl2dtss";
273 setup \<open>KEStore_Elems.add_mets
274 [MethodC.prep_input @{theory} "met_diff" [] MethodC.id_empty
276 {rew_ord'="tless_true",rls'=Atools_erls,calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
277 crls = Atools_erls, errpats = [], nrls = norm_diff},
281 partial_function (tailrec) differentiate_on_R :: "real \<Rightarrow> real \<Rightarrow> real"
283 "differentiate_on_R f_f v_v = (
285 f_f' = Take (d_d v_v f_f)
287 (Try (Rewrite_Set_Inst [(''bdv'',v_v)] ''diff_conv'')) #> (
289 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sum'')) Or
290 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod_const'')) Or
291 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod'')) Or
292 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_quot'')) Or
293 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin'')) Or
294 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin_chain'')) Or
295 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos'')) Or
296 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos_chain'')) Or
297 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow'')) Or
298 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow_chain'')) Or
299 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln'')) Or
300 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln_chain'')) Or
301 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp'')) Or
302 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp_chain'')) Or
303 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_const'')) Or
304 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_var'')) Or
305 (Repeat (Rewrite_Set ''make_polynomial'')))) #> (
306 Try (Rewrite_Set_Inst [(''bdv'',v_v)] ''diff_sym_conv''))
308 setup \<open>KEStore_Elems.add_mets
309 [MethodC.prep_input @{theory} "met_diff_onR" [] MethodC.id_empty
310 (["diff", "differentiate_on_R"],
311 [("#Given" ,["functionTerm f_f", "differentiateFor v_v"]),
312 ("#Find" ,["derivative f_f'"])],
313 {rew_ord'="tless_true", rls' = erls_diff, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
314 crls = Atools_erls, errpats = [], nrls = norm_diff},
315 @{thm differentiate_on_R.simps})]
318 partial_function (tailrec) differentiateX :: "real \<Rightarrow> real \<Rightarrow> real"
320 "differentiateX f_f v_v = (
322 f_f' = Take (d_d v_v f_f)
325 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sum'')) Or
326 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod_const'' )) Or
327 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod'')) Or
328 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_quot'')) Or
329 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin'')) Or
330 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin_chain'')) Or
331 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos'')) Or
332 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos_chain'')) Or
333 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow'')) Or
334 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow_chain'')) Or
335 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln'')) Or
336 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln_chain'')) Or
337 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp'')) Or
338 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp_chain'')) Or
339 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_const'')) Or
340 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_var'')) Or
341 (Repeat (Rewrite_Set ''make_polynomial'')))
343 setup \<open>KEStore_Elems.add_mets
344 [MethodC.prep_input @{theory} "met_diff_simpl" [] MethodC.id_empty
345 (["diff", "diff_simpl"],
346 [("#Given", ["functionTerm f_f", "differentiateFor v_v"]),
347 ("#Find" , ["derivative f_f'"])],
348 {rew_ord'="tless_true", rls' = erls_diff, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
349 crls = Atools_erls, errpats = [], nrls = norm_diff},
350 @{thm differentiateX.simps})]
353 partial_function (tailrec) differentiate_equality :: "bool \<Rightarrow> real \<Rightarrow> bool"
355 "differentiate_equality f_f v_v = (
357 f_f' = Take ((primed (lhs f_f)) = d_d v_v (rhs f_f))
359 (Try (Rewrite_Set_Inst [(''bdv'',v_v)] ''diff_conv'' )) #> (
361 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_sum'')) Or
362 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_dif'' )) Or
363 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_prod_const'')) Or
364 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_prod'')) Or
365 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_quot'')) Or
366 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_sin'')) Or
367 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_sin_chain'')) Or
368 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_cos'')) Or
369 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_cos_chain'')) Or
370 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_pow'')) Or
371 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_pow_chain'')) Or
372 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_ln'')) Or
373 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_ln_chain'')) Or
374 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_exp'')) Or
375 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_exp_chain'')) Or
376 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_const'')) Or
377 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_var'')) Or
378 (Repeat (Rewrite_Set ''make_polynomial'')))) #> (
379 Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''diff_sym_conv'' ))
381 setup \<open>KEStore_Elems.add_mets
382 [MethodC.prep_input @{theory} "met_diff_equ" [] MethodC.id_empty
383 (["diff", "differentiate_equality"],
384 [("#Given" ,["functionEq f_f", "differentiateFor v_v"]),
385 ("#Find" ,["derivativeEq f_f'"])],
386 {rew_ord'="tless_true", rls' = erls_diff, calc = [], srls = srls_diff, prls=Rule_Set.empty,
387 crls=Atools_erls, errpats = [], nrls = norm_diff},
388 @{thm differentiate_equality.simps})]
391 partial_function (tailrec) simplify_derivative :: "real \<Rightarrow> real \<Rightarrow> real"
393 "simplify_derivative term bound_variable = (
395 term' = Take (d_d bound_variable term)
397 (Try (Rewrite_Set ''norm_Rational'')) #>
398 (Try (Rewrite_Set_Inst [(''bdv'', bound_variable)] ''diff_conv'')) #>
399 (Try (Rewrite_Set_Inst [(''bdv'', bound_variable)] ''norm_diff'')) #>
400 (Try (Rewrite_Set_Inst [(''bdv'', bound_variable)] ''diff_sym_conv'')) #>
401 (Try (Rewrite_Set ''norm_Rational''))
404 setup \<open>KEStore_Elems.add_mets
405 [MethodC.prep_input @{theory} "met_diff_after_simp" [] MethodC.id_empty
406 (["diff", "after_simplification"],
407 [("#Given" ,["functionTerm term", "differentiateFor bound_variable"]),
408 ("#Find" ,["derivative term'"])],
409 {rew_ord'="tless_true", rls' = Rule_Set.empty, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
410 crls=Atools_erls, errpats = [], nrls = norm_Rational},
411 @{thm simplify_derivative.simps})]
413 setup \<open>KEStore_Elems.add_cas
414 [((Thm.term_of o the o (TermC.parse @{theory})) "Diff",
415 (("Isac_Knowledge", ["derivative_of", "function"], ["no_met"]), argl2dtss))]\<close>
418 (*.handle cas-input like "Differentiate (A = s * (a - s), s)".*)
419 (* val (t, pairl) = strip_comb (str2term "Differentiate (A = s * (a - s), s)");
420 val [Const ("Product_Type.Pair", _) $ t $ bdv] = pairl;
422 fun argl2dtss [Const ("Product_Type.Pair", _) $ t $ bdv] =
423 [((Thm.term_of o the o (TermC.parse \<^theory>)) "functionEq", [t]),
424 ((Thm.term_of o the o (TermC.parse \<^theory>)) "differentiateFor", [bdv]),
425 ((Thm.term_of o the o (TermC.parse \<^theory>)) "derivativeEq",
426 [(Thm.term_of o the o (TermC.parse \<^theory>)) "f_f'::bool"])
428 | argl2dtss _ = raise ERROR "Diff.ML: wrong argument for argl2dtss";
430 setup \<open>KEStore_Elems.add_cas
431 [((Thm.term_of o the o (TermC.parse @{theory})) "Differentiate",
432 (("Isac_Knowledge", ["named", "derivative_of", "function"], ["no_met"]), argl2dtss))]