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)"
92 (** eval functions **)
94 fun primed (Const (id, T)) = Const (id ^ "'", T)
95 | primed (Free (id, T)) = Free (id ^ "'", T)
96 | primed t = raise ERROR ("primed called with arg = '"^ UnparseC.term t ^"'");
98 (*("primed", ("Diff.primed", eval_primed "#primed"))*)
99 fun eval_primed _ _ (p as (Const ("Diff.primed",_) $ t)) _ =
100 SOME ((UnparseC.term p) ^ " = " ^ UnparseC.term (primed t),
101 HOLogic.Trueprop $ (TermC.mk_equality (p, primed t)))
102 | eval_primed _ _ _ _ = NONE;
104 setup \<open>KEStore_Elems.add_calcs
105 [("primed", ("Diff.primed", eval_primed "#primed"))]\<close>
109 (*.converts a term such that differentiation works optimally.*)
111 Rule_Def.Repeat {id="diff_conv",
113 rew_ord = ("termlessI",termlessI),
114 erls = Rule_Set.append_rules "erls_diff_conv" Rule_Set.empty
115 [Rule.Eval ("Prog_Expr.occurs_in", Prog_Expr.eval_occurs_in ""),
116 Rule.Thm ("not_true",ThmC.numerals_to_Free @{thm not_true}),
117 Rule.Thm ("not_false",ThmC.numerals_to_Free @{thm not_false}),
118 Rule.Eval ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),
119 Rule.Thm ("and_true",ThmC.numerals_to_Free @{thm and_true}),
120 Rule.Thm ("and_false",ThmC.numerals_to_Free @{thm and_false})
122 srls = Rule_Set.Empty, calc = [], errpatts = [],
124 [Rule.Thm ("frac_conv", ThmC.numerals_to_Free @{thm frac_conv}),
125 (*"?bdv occurs_in ?b \<Longrightarrow> 0 < ?n \<Longrightarrow> ?a / ?b \<up> ?n = ?a * ?b \<up> - ?n"*)
126 Rule.Thm ("sqrt_conv_bdv", ThmC.numerals_to_Free @{thm sqrt_conv_bdv}),
127 (*"sqrt ?bdv = ?bdv \<up> (1 / 2)"*)
128 Rule.Thm ("sqrt_conv_bdv_n", ThmC.numerals_to_Free @{thm sqrt_conv_bdv_n}),
129 (*"sqrt (?bdv \<up> ?n) = ?bdv \<up> (?n / 2)"*)
130 Rule.Thm ("sqrt_conv", ThmC.numerals_to_Free @{thm sqrt_conv}),
131 (*"?bdv occurs_in ?u \<Longrightarrow> sqrt ?u = ?u \<up> (1 / 2)"*)
132 Rule.Thm ("root_conv", ThmC.numerals_to_Free @{thm root_conv}),
133 (*"?bdv occurs_in ?u \<Longrightarrow> nroot ?n ?u = ?u \<up> (1 / ?n)"*)
134 Rule.Thm ("realpow_pow_bdv", ThmC.numerals_to_Free @{thm realpow_pow_bdv}),
135 (* "(?bdv \<up> ?b) \<up> ?c = ?bdv \<up> (?b * ?c)"*)
136 Rule.Eval ("Groups.times_class.times", (**)eval_binop "#mult_"),
137 Rule.Thm ("rat_mult",ThmC.numerals_to_Free @{thm rat_mult}),
138 (*a / b * (c / d) = a * c / (b * d)*)
139 Rule.Thm ("times_divide_eq_right",ThmC.numerals_to_Free @{thm times_divide_eq_right}),
140 (*?x * (?y / ?z) = ?x * ?y / ?z*)
141 Rule.Thm ("times_divide_eq_left",ThmC.numerals_to_Free @{thm times_divide_eq_left})
142 (*?y / ?z * ?x = ?y * ?x / ?z*)
144 scr = Rule.Empty_Prog};
147 (*.beautifies a term after differentiation.*)
149 Rule_Def.Repeat {id="diff_sym_conv",
151 rew_ord = ("termlessI",termlessI),
152 erls = Rule_Set.append_rules "erls_diff_sym_conv" Rule_Set.empty
153 [Rule.Eval ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),
154 Rule.Eval ("Prog_Expr.matches", Prog_Expr.eval_matches "#matches_"),
155 Rule.Eval ("Prog_Expr.is_atom", Prog_Expr.eval_is_atom "#is_atom_"),
156 Rule.Eval ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),
157 Rule.Thm ("not_false", ThmC.numerals_to_Free @{thm not_false}),
158 Rule.Thm ("not_true", ThmC.numerals_to_Free @{thm not_true})],
159 srls = Rule_Set.Empty, calc = [], errpatts = [],
161 [Rule.Thm ("frac_sym_conv", ThmC.numerals_to_Free @{thm frac_sym_conv}),
162 Rule.Thm ("sqrt_sym_conv", ThmC.numerals_to_Free @{thm sqrt_sym_conv}),
163 Rule.Thm ("root_sym_conv", ThmC.numerals_to_Free @{thm root_sym_conv}),
164 Rule.Thm ("real_mult_minus1_sym", ThmC.numerals_to_Free (@{thm real_mult_minus1_sym}))
165 (*"\<not>(z is_const) ==> - (z::real) = -1 * z"*),
166 Rule.Thm ("rat_mult",ThmC.numerals_to_Free @{thm rat_mult}),
167 (*a / b * (c / d) = a * c / (b * d)*)
168 Rule.Thm ("times_divide_eq_right",ThmC.numerals_to_Free @{thm times_divide_eq_right}),
169 (*?x * (?y / ?z) = ?x * ?y / ?z*)
170 Rule.Thm ("times_divide_eq_left",ThmC.numerals_to_Free @{thm times_divide_eq_left}),
171 (*?y / ?z * ?x = ?y * ?x / ?z*)
172 Rule.Eval ("Groups.times_class.times", (**)eval_binop "#mult_")],
173 scr = Rule.Empty_Prog};
177 Rule_Def.Repeat {id="srls_differentiate..",
179 rew_ord = ("termlessI",termlessI),
180 erls = Rule_Set.empty,
181 srls = Rule_Set.Empty, calc = [], errpatts = [],
182 rules = [Rule.Eval("Prog_Expr.lhs", Prog_Expr.eval_lhs "eval_lhs_"),
183 Rule.Eval("Prog_Expr.rhs", Prog_Expr.eval_rhs "eval_rhs_"),
184 Rule.Eval("Diff.primed", eval_primed "Diff.primed")
186 scr = Rule.Empty_Prog};
191 Rule_Set.append_rules "erls_differentiate.." Rule_Set.empty
192 [Rule.Thm ("not_true",ThmC.numerals_to_Free @{thm not_true}),
193 Rule.Thm ("not_false",ThmC.numerals_to_Free @{thm not_false}),
195 Rule.Eval ("Prog_Expr.ident", Prog_Expr.eval_ident "#ident_"),
196 Rule.Eval ("Prog_Expr.is_atom", Prog_Expr.eval_is_atom "#is_atom_"),
197 Rule.Eval ("Prog_Expr.occurs_in", Prog_Expr.eval_occurs_in ""),
198 Rule.Eval ("Prog_Expr.is_const", Prog_Expr.eval_const "#is_const_")
201 (*.rules for differentiation, _no_ simplification.*)
203 Rule_Def.Repeat {id="diff_rules", preconds = [], rew_ord = ("termlessI",termlessI),
204 erls = erls_diff, srls = Rule_Set.Empty, calc = [], errpatts = [],
205 rules = [Rule.Thm ("diff_sum",ThmC.numerals_to_Free @{thm diff_sum}),
206 Rule.Thm ("diff_dif",ThmC.numerals_to_Free @{thm diff_dif}),
207 Rule.Thm ("diff_prod_const",ThmC.numerals_to_Free @{thm diff_prod_const}),
208 Rule.Thm ("diff_prod",ThmC.numerals_to_Free @{thm diff_prod}),
209 Rule.Thm ("diff_quot",ThmC.numerals_to_Free @{thm diff_quot}),
210 Rule.Thm ("diff_sin",ThmC.numerals_to_Free @{thm diff_sin}),
211 Rule.Thm ("diff_sin_chain",ThmC.numerals_to_Free @{thm diff_sin_chain}),
212 Rule.Thm ("diff_cos",ThmC.numerals_to_Free @{thm diff_cos}),
213 Rule.Thm ("diff_cos_chain",ThmC.numerals_to_Free @{thm diff_cos_chain}),
214 Rule.Thm ("diff_pow",ThmC.numerals_to_Free @{thm diff_pow}),
215 Rule.Thm ("diff_pow_chain",ThmC.numerals_to_Free @{thm diff_pow_chain}),
216 Rule.Thm ("diff_ln",ThmC.numerals_to_Free @{thm diff_ln}),
217 Rule.Thm ("diff_ln_chain",ThmC.numerals_to_Free @{thm diff_ln_chain}),
218 Rule.Thm ("diff_exp",ThmC.numerals_to_Free @{thm diff_exp}),
219 Rule.Thm ("diff_exp_chain",ThmC.numerals_to_Free @{thm diff_exp_chain}),
221 Rule.Thm ("diff_sqrt",ThmC.numerals_to_Free @{thm diff_sqrt}),
222 Rule.Thm ("diff_sqrt_chain",ThmC.numerals_to_Free @{thm diff_sqrt_chain}),
224 Rule.Thm ("diff_const",ThmC.numerals_to_Free @{thm diff_const}),
225 Rule.Thm ("diff_var",ThmC.numerals_to_Free @{thm diff_var})
227 scr = Rule.Empty_Prog};
230 (*.normalisation for checking user-input.*)
233 {id="norm_diff", preconds = [], rew_ord = ("termlessI",termlessI),
234 erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
235 rules = [Rule.Rls_ diff_rules, Rule.Rls_ norm_Poly ],
236 scr = Rule.Empty_Prog};
238 setup \<open>KEStore_Elems.add_rlss
239 [("erls_diff", (Context.theory_name @{theory}, prep_rls' erls_diff)),
240 ("diff_rules", (Context.theory_name @{theory}, prep_rls' diff_rules)),
241 ("norm_diff", (Context.theory_name @{theory}, prep_rls' norm_diff)),
242 ("diff_conv", (Context.theory_name @{theory}, prep_rls' diff_conv)),
243 ("diff_sym_conv", (Context.theory_name @{theory}, prep_rls' diff_sym_conv))]\<close>
245 (** problem types **)
246 setup \<open>KEStore_Elems.add_pbts
247 [(Problem.prep_input thy "pbl_fun" [] Problem.id_empty (["function"], [], Rule_Set.empty, NONE, [])),
248 (Problem.prep_input thy "pbl_fun_deriv" [] Problem.id_empty
249 (["derivative_of", "function"],
250 [("#Given" ,["functionTerm f_f", "differentiateFor v_v"]),
251 ("#Find" ,["derivative f_f'"])],
252 Rule_Set.append_rules "empty" Rule_Set.empty [],
253 SOME "Diff (f_f, v_v)", [["diff", "differentiate_on_R"],
254 ["diff", "after_simplification"]])),
255 (*here "named" is used differently from Integration"*)
256 (Problem.prep_input thy "pbl_fun_deriv_nam" [] Problem.id_empty
257 (["named", "derivative_of", "function"],
258 [("#Given" ,["functionEq f_f", "differentiateFor v_v"]),
259 ("#Find" ,["derivativeEq f_f'"])],
260 Rule_Set.append_rules "empty" Rule_Set.empty [],
261 SOME "Differentiate (f_f, v_v)",
262 [["diff", "differentiate_equality"]]))]\<close>
267 (*.handle cas-input like "Diff (a * x^3 + b, x)".*)
268 (* val (t, pairl) = strip_comb (str2term "Diff (a * x^3 + b, x)");
269 val [Const ("Product_Type.Pair", _) $ t $ bdv] = pairl;
271 fun argl2dtss [Const ("Product_Type.Pair", _) $ t $ bdv] =
272 [((Thm.term_of o the o (TermC.parse thy)) "functionTerm", [t]),
273 ((Thm.term_of o the o (TermC.parse thy)) "differentiateFor", [bdv]),
274 ((Thm.term_of o the o (TermC.parse thy)) "derivative",
275 [(Thm.term_of o the o (TermC.parse thy)) "f_f'"])
277 | argl2dtss _ = raise ERROR "Diff.ML: wrong argument for argl2dtss";
279 setup \<open>KEStore_Elems.add_mets
280 [MethodC.prep_input thy "met_diff" [] MethodC.id_empty
282 {rew_ord'="tless_true",rls'=Atools_erls,calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
283 crls = Atools_erls, errpats = [], nrls = norm_diff},
287 partial_function (tailrec) differentiate_on_R :: "real \<Rightarrow> real \<Rightarrow> real"
289 "differentiate_on_R f_f v_v = (
291 f_f' = Take (d_d v_v f_f)
293 (Try (Rewrite_Set_Inst [(''bdv'',v_v)] ''diff_conv'')) #> (
295 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sum'')) Or
296 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod_const'')) Or
297 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod'')) Or
298 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_quot'')) Or
299 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin'')) Or
300 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin_chain'')) Or
301 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos'')) Or
302 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos_chain'')) Or
303 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow'')) Or
304 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow_chain'')) Or
305 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln'')) Or
306 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln_chain'')) Or
307 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp'')) Or
308 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp_chain'')) Or
309 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_const'')) Or
310 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_var'')) Or
311 (Repeat (Rewrite_Set ''make_polynomial'')))) #> (
312 Try (Rewrite_Set_Inst [(''bdv'',v_v)] ''diff_sym_conv''))
314 setup \<open>KEStore_Elems.add_mets
315 [MethodC.prep_input thy "met_diff_onR" [] MethodC.id_empty
316 (["diff", "differentiate_on_R"],
317 [("#Given" ,["functionTerm f_f", "differentiateFor v_v"]),
318 ("#Find" ,["derivative f_f'"])],
319 {rew_ord'="tless_true", rls' = erls_diff, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
320 crls = Atools_erls, errpats = [], nrls = norm_diff},
321 @{thm differentiate_on_R.simps})]
324 partial_function (tailrec) differentiateX :: "real \<Rightarrow> real \<Rightarrow> real"
326 "differentiateX f_f v_v = (
328 f_f' = Take (d_d v_v f_f)
331 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sum'')) Or
332 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod_const'' )) Or
333 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod'')) Or
334 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_quot'')) Or
335 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin'')) Or
336 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin_chain'')) Or
337 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos'')) Or
338 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos_chain'')) Or
339 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow'')) Or
340 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow_chain'')) Or
341 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln'')) Or
342 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln_chain'')) Or
343 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp'')) Or
344 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp_chain'')) Or
345 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_const'')) Or
346 (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_var'')) Or
347 (Repeat (Rewrite_Set ''make_polynomial'')))
349 setup \<open>KEStore_Elems.add_mets
350 [MethodC.prep_input thy "met_diff_simpl" [] MethodC.id_empty
351 (["diff", "diff_simpl"],
352 [("#Given", ["functionTerm f_f", "differentiateFor v_v"]),
353 ("#Find" , ["derivative f_f'"])],
354 {rew_ord'="tless_true", rls' = erls_diff, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
355 crls = Atools_erls, errpats = [], nrls = norm_diff},
356 @{thm differentiateX.simps})]
359 partial_function (tailrec) differentiate_equality :: "bool \<Rightarrow> real \<Rightarrow> bool"
361 "differentiate_equality f_f v_v = (
363 f_f' = Take ((primed (lhs f_f)) = d_d v_v (rhs f_f))
365 (Try (Rewrite_Set_Inst [(''bdv'',v_v)] ''diff_conv'' )) #> (
367 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_sum'')) Or
368 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_dif'' )) Or
369 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_prod_const'')) Or
370 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_prod'')) Or
371 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_quot'')) Or
372 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_sin'')) Or
373 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_sin_chain'')) Or
374 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_cos'')) Or
375 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_cos_chain'')) Or
376 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_pow'')) Or
377 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_pow_chain'')) Or
378 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_ln'')) Or
379 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_ln_chain'')) Or
380 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_exp'')) Or
381 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_exp_chain'')) Or
382 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_const'')) Or
383 (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_var'')) Or
384 (Repeat (Rewrite_Set ''make_polynomial'')))) #> (
385 Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''diff_sym_conv'' ))
387 setup \<open>KEStore_Elems.add_mets
388 [MethodC.prep_input thy "met_diff_equ" [] MethodC.id_empty
389 (["diff", "differentiate_equality"],
390 [("#Given" ,["functionEq f_f", "differentiateFor v_v"]),
391 ("#Find" ,["derivativeEq f_f'"])],
392 {rew_ord'="tless_true", rls' = erls_diff, calc = [], srls = srls_diff, prls=Rule_Set.empty,
393 crls=Atools_erls, errpats = [], nrls = norm_diff},
394 @{thm differentiate_equality.simps})]
397 partial_function (tailrec) simplify_derivative :: "real \<Rightarrow> real \<Rightarrow> real"
399 "simplify_derivative term bound_variable = (
401 term' = Take (d_d bound_variable term)
403 (Try (Rewrite_Set ''norm_Rational'')) #>
404 (Try (Rewrite_Set_Inst [(''bdv'', bound_variable)] ''diff_conv'')) #>
405 (Try (Rewrite_Set_Inst [(''bdv'', bound_variable)] ''norm_diff'')) #>
406 (Try (Rewrite_Set_Inst [(''bdv'', bound_variable)] ''diff_sym_conv'')) #>
407 (Try (Rewrite_Set ''norm_Rational''))
410 setup \<open>KEStore_Elems.add_mets
411 [MethodC.prep_input thy "met_diff_after_simp" [] MethodC.id_empty
412 (["diff", "after_simplification"],
413 [("#Given" ,["functionTerm term", "differentiateFor bound_variable"]),
414 ("#Find" ,["derivative term'"])],
415 {rew_ord'="tless_true", rls' = Rule_Set.empty, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,
416 crls=Atools_erls, errpats = [], nrls = norm_Rational},
417 @{thm simplify_derivative.simps})]
419 setup \<open>KEStore_Elems.add_cas
420 [((Thm.term_of o the o (TermC.parse thy)) "Diff",
421 (("Isac_Knowledge", ["derivative_of", "function"], ["no_met"]), argl2dtss))]\<close>
424 (*.handle cas-input like "Differentiate (A = s * (a - s), s)".*)
425 (* val (t, pairl) = strip_comb (str2term "Differentiate (A = s * (a - s), s)");
426 val [Const ("Product_Type.Pair", _) $ t $ bdv] = pairl;
428 fun argl2dtss [Const ("Product_Type.Pair", _) $ t $ bdv] =
429 [((Thm.term_of o the o (TermC.parse thy)) "functionEq", [t]),
430 ((Thm.term_of o the o (TermC.parse thy)) "differentiateFor", [bdv]),
431 ((Thm.term_of o the o (TermC.parse thy)) "derivativeEq",
432 [(Thm.term_of o the o (TermC.parse thy)) "f_f'::bool"])
434 | argl2dtss _ = raise ERROR "Diff.ML: wrong argument for argl2dtss";
436 setup \<open>KEStore_Elems.add_cas
437 [((Thm.term_of o the o (TermC.parse thy)) "Differentiate",
438 (("Isac_Knowledge", ["named", "derivative_of", "function"], ["no_met"]), argl2dtss))]