(* differentiation over the reals

   author: Walther Neuper

   000516   

 *)

theory Diff imports Calculus Trig LogExp Rational Root Poly Base_Tools begin

ML \<open>

@{term "sin x"}

\<close>

consts

  d_d           :: "[real, real]=> real"

  (*descriptions in the related problems*)

  derivativeEq  :: "bool => una"

  (*predicates*)

  primed        :: "'a => 'a" (*"primed A" -> "A'"*)

  (*the CAS-commands, eg. "Diff (2*x^^^3, x)", 

			  "Differentiate (A = s * (a - s), s)"*)

  Diff           :: "[real * real] => real"

  Differentiate  :: "[bool * real] => bool"

  (*subproblem-name*)

  differentiate  :: "[char list * char list list * char list, real, real] => real"

               	   ("(differentiate (_)/ (_ _ ))" 9)

text \<open>a variant of the derivatives defintion:

  d_d            :: "(real => real) => (real => real)"

  advantages:

(1) no variable 'bdv' on the meta-level required

(2) chain_rule "d_d (%x. (u (v x))) = (%x. (d_d u)) (v x) * d_d v"

(3) and no specialized chain-rules required like

    diff_sin_chain "d_d bdv (sin u)    = cos u * d_d bdv u"

  disadvantage: d_d (%x. 1 + x^2) = ... differs from high-school notation

\<close>

axiomatization where (*stated as axioms, todo: prove as theorems

        'bdv' is a constant on the meta-level  *)

  diff_const:     "[| Not (bdv occurs_in a) |] ==> d_d bdv a = 0" and

  diff_var:       "d_d bdv bdv = 1" and

  diff_prod_const:"[| Not (bdv occurs_in u) |] ==>  

					 d_d bdv (u * v) = u * d_d bdv v" and

  diff_sum:       "d_d bdv (u + v)     = d_d bdv u + d_d bdv v" and

  diff_dif:       "d_d bdv (u - v)     = d_d bdv u - d_d bdv v" and

  diff_prod:      "d_d bdv (u * v)     = d_d bdv u * v + u * d_d bdv v" and

  diff_quot:      "Not (v = 0) ==> (d_d bdv (u / v) =  

	           (d_d bdv u * v - u * d_d bdv v) / v ^^^ 2)" and

  diff_sin:       "d_d bdv (sin bdv)   = cos bdv" and

  diff_sin_chain: "d_d bdv (sin u)     = cos u * d_d bdv u" and

  diff_cos:       "d_d bdv (cos bdv)   = - sin bdv" and

  diff_cos_chain: "d_d bdv (cos u)     = - sin u * d_d bdv u" and

  diff_pow:       "d_d bdv (bdv ^^^ n) = n * (bdv ^^^ (n - 1))" and

  diff_pow_chain: "d_d bdv (u ^^^ n)   = n * (u ^^^ (n - 1)) * d_d bdv u" and

  diff_ln:        "d_d bdv (ln bdv)    = 1 / bdv" and

  diff_ln_chain:  "d_d bdv (ln u)      = d_d bdv u / u" and

  diff_exp:       "d_d bdv (exp bdv)   = exp bdv" and

  diff_exp_chain: "d_d bdv (exp u)     = exp u * d_d x u" and

(*

  diff_sqrt      "d_d bdv (sqrt bdv)  = 1 / (2 * sqrt bdv)"

  diff_sqrt_chain"d_d bdv (sqrt u)    = d_d bdv u / (2 * sqrt u)"

*)

  (*...*)

  frac_conv:       "[| bdv occurs_in b; 0 < n |] ==>  

		    a / (b ^^^ n) = a * b ^^^ (-n)" and

  frac_sym_conv:   "n < 0 ==> a * b ^^^ n = a / b ^^^ (-n)" and

  sqrt_conv_bdv:   "sqrt bdv = bdv ^^^ (1 / 2)" and

  sqrt_conv_bdv_n: "sqrt (bdv ^^^ n) = bdv ^^^ (n / 2)" and

  sqrt_conv:       "bdv occurs_in u ==> sqrt u = u ^^^ (1 / 2)" and

  sqrt_sym_conv:   "u ^^^ (a / 2) = sqrt (u ^^^ a)" and

  root_conv:       "bdv occurs_in u ==> nroot n u = u ^^^ (1 / n)" and

  root_sym_conv:   "u ^^^ (a / b) = nroot b (u ^^^ a)" and

  realpow_pow_bdv: "(bdv ^^^ b) ^^^ c = bdv ^^^ (b * c)"

ML \<open>

val thy = @{theory};

(** eval functions **)

fun primed (Const (id, T)) = Const (id ^ "'", T)

  | primed (Free (id, T)) = Free (id ^ "'", T)

  | primed t = error ("primed called with arg = '"^ Rule.term2str t ^"'");

(*("primed", ("Diff.primed", eval_primed "#primed"))*)

fun eval_primed _ _ (p as (Const ("Diff.primed",_) $ t)) _ =

    SOME ((Rule.term2str p) ^ " = " ^ Rule.term2str (primed t),

	  HOLogic.Trueprop $ (TermC.mk_equality (p, primed t)))

  | eval_primed _ _ _ _ = NONE;

\<close>

setup \<open>KEStore_Elems.add_calcs

  [("primed", ("Diff.primed", eval_primed "#primed"))]\<close>

ML \<open>

(** rulesets **)

(*.converts a term such that differentiation works optimally.*)

val diff_conv =   

    Rule_Def.Repeat {id="diff_conv", 

	 preconds = [], 

	 rew_ord = ("termlessI",termlessI), 

	 erls = Rule_Set.append_rules "erls_diff_conv" Rule_Set.empty 

			   [Rule.Num_Calc ("Prog_Expr.occurs'_in", Prog_Expr.eval_occurs_in ""),

			    Rule.Thm ("not_true",TermC.num_str @{thm not_true}),

			    Rule.Thm ("not_false",TermC.num_str @{thm not_false}),

			    Rule.Num_Calc ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),

			    Rule.Thm ("and_true",TermC.num_str @{thm and_true}),

			    Rule.Thm ("and_false",TermC.num_str @{thm and_false})

			    ], 

	 srls = Rule_Set.Empty, calc = [], errpatts = [],

	 rules =

  [Rule.Thm ("frac_conv", TermC.num_str @{thm frac_conv}),

     (*"?bdv occurs_in ?b \<Longrightarrow> 0 < ?n \<Longrightarrow> ?a / ?b ^^^ ?n = ?a * ?b ^^^ - ?n"*)

		   Rule.Thm ("sqrt_conv_bdv", TermC.num_str @{thm sqrt_conv_bdv}),

		     (*"sqrt ?bdv = ?bdv ^^^ (1 / 2)"*)

		   Rule.Thm ("sqrt_conv_bdv_n", TermC.num_str @{thm sqrt_conv_bdv_n}),

		     (*"sqrt (?bdv ^^^ ?n) = ?bdv ^^^ (?n / 2)"*)

		   Rule.Thm ("sqrt_conv", TermC.num_str @{thm sqrt_conv}),

		     (*"?bdv occurs_in ?u \<Longrightarrow> sqrt ?u = ?u ^^^ (1 / 2)"*)

		   Rule.Thm ("root_conv", TermC.num_str @{thm root_conv}),

		     (*"?bdv occurs_in ?u \<Longrightarrow> nroot ?n ?u = ?u ^^^ (1 / ?n)"*)

		   Rule.Thm ("realpow_pow_bdv", TermC.num_str @{thm realpow_pow_bdv}),

		     (* "(?bdv ^^^ ?b) ^^^ ?c = ?bdv ^^^ (?b * ?c)"*)

		   Rule.Num_Calc ("Groups.times_class.times", (**)eval_binop "#mult_"),

		   Rule.Thm ("rat_mult",TermC.num_str @{thm rat_mult}),

		     (*a / b * (c / d) = a * c / (b * d)*)

		   Rule.Thm ("times_divide_eq_right",TermC.num_str @{thm times_divide_eq_right}),

		     (*?x * (?y / ?z) = ?x * ?y / ?z*)

		   Rule.Thm ("times_divide_eq_left",TermC.num_str @{thm times_divide_eq_left})

		     (*?y / ?z * ?x = ?y * ?x / ?z*)

		 ],

	 scr = Rule.EmptyScr};

\<close>

ML \<open>

(*.beautifies a term after differentiation.*)

val diff_sym_conv =   

    Rule_Def.Repeat {id="diff_sym_conv", 

	 preconds = [], 

	 rew_ord = ("termlessI",termlessI), 

	 erls = Rule_Set.append_rules "erls_diff_sym_conv" Rule_Set.empty 

			   [Rule.Num_Calc ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_")

			    ], 

	 srls = Rule_Set.Empty, calc = [], errpatts = [],

	 rules = [Rule.Thm ("frac_sym_conv", TermC.num_str @{thm frac_sym_conv}),

		  Rule.Thm ("sqrt_sym_conv", TermC.num_str @{thm sqrt_sym_conv}),

		  Rule.Thm ("root_sym_conv", TermC.num_str @{thm root_sym_conv}),

		  Rule.Thm ("sym_real_mult_minus1",

		       TermC.num_str (@{thm real_mult_minus1} RS @{thm sym})),

		      (*- ?z = "-1 * ?z"*)

		  Rule.Thm ("rat_mult",TermC.num_str @{thm rat_mult}),

		  (*a / b * (c / d) = a * c / (b * d)*)

		  Rule.Thm ("times_divide_eq_right",TermC.num_str @{thm times_divide_eq_right}),

		  (*?x * (?y / ?z) = ?x * ?y / ?z*)

		  Rule.Thm ("times_divide_eq_left",TermC.num_str @{thm times_divide_eq_left}),

		  (*?y / ?z * ?x = ?y * ?x / ?z*)

		  Rule.Num_Calc ("Groups.times_class.times", (**)eval_binop "#mult_")

		 ],

	 scr = Rule.EmptyScr};

(*..*)

val srls_diff = 

    Rule_Def.Repeat {id="srls_differentiate..", 

	 preconds = [], 

	 rew_ord = ("termlessI",termlessI), 

	 erls = Rule_Set.empty, 

	 srls = Rule_Set.Empty, calc = [], errpatts = [],

	 rules = [Rule.Num_Calc("Prog_Expr.lhs", Prog_Expr.eval_lhs "eval_lhs_"),

		  Rule.Num_Calc("Prog_Expr.rhs", Prog_Expr.eval_rhs "eval_rhs_"),

		  Rule.Num_Calc("Diff.primed", eval_primed "Diff.primed")

		  ],

	 scr = Rule.EmptyScr};

\<close>

ML \<open>

(*..*)

val erls_diff = 

    Rule_Set.append_rules "erls_differentiate.." Rule_Set.empty

               [Rule.Thm ("not_true",TermC.num_str @{thm not_true}),

		Rule.Thm ("not_false",TermC.num_str @{thm not_false}),

		Rule.Num_Calc ("Prog_Expr.ident", Prog_Expr.eval_ident "#ident_"),    

		Rule.Num_Calc ("Prog_Expr.is'_atom", Prog_Expr.eval_is_atom "#is_atom_"),

		Rule.Num_Calc ("Prog_Expr.occurs'_in", Prog_Expr.eval_occurs_in ""),

		Rule.Num_Calc ("Prog_Expr.is'_const", Prog_Expr.eval_const "#is_const_")

		];

(*.rules for differentiation, _no_ simplification.*)

val diff_rules =

    Rule_Def.Repeat {id="diff_rules", preconds = [], rew_ord = ("termlessI",termlessI), 

	 erls = erls_diff, srls = Rule_Set.Empty, calc = [], errpatts = [],

	 rules = [Rule.Thm ("diff_sum",TermC.num_str @{thm diff_sum}),

		  Rule.Thm ("diff_dif",TermC.num_str @{thm diff_dif}),

		  Rule.Thm ("diff_prod_const",TermC.num_str @{thm diff_prod_const}),

		  Rule.Thm ("diff_prod",TermC.num_str @{thm diff_prod}),

		  Rule.Thm ("diff_quot",TermC.num_str @{thm diff_quot}),

		  Rule.Thm ("diff_sin",TermC.num_str @{thm diff_sin}),

		  Rule.Thm ("diff_sin_chain",TermC.num_str @{thm diff_sin_chain}),

		  Rule.Thm ("diff_cos",TermC.num_str @{thm diff_cos}),

		  Rule.Thm ("diff_cos_chain",TermC.num_str @{thm diff_cos_chain}),

		  Rule.Thm ("diff_pow",TermC.num_str @{thm diff_pow}),

		  Rule.Thm ("diff_pow_chain",TermC.num_str @{thm diff_pow_chain}),

		  Rule.Thm ("diff_ln",TermC.num_str @{thm diff_ln}),

		  Rule.Thm ("diff_ln_chain",TermC.num_str @{thm diff_ln_chain}),

		  Rule.Thm ("diff_exp",TermC.num_str @{thm diff_exp}),

		  Rule.Thm ("diff_exp_chain",TermC.num_str @{thm diff_exp_chain}),

(*

		  Rule.Thm ("diff_sqrt",TermC.num_str @{thm diff_sqrt}),

		  Rule.Thm ("diff_sqrt_chain",TermC.num_str @{thm diff_sqrt_chain}),

*)

		  Rule.Thm ("diff_const",TermC.num_str @{thm diff_const}),

		  Rule.Thm ("diff_var",TermC.num_str @{thm diff_var})

		  ],

	 scr = Rule.EmptyScr};

\<close>

ML \<open>

(*.normalisation for checking user-input.*)

val norm_diff = 

  Rule_Def.Repeat

    {id="norm_diff", preconds = [], rew_ord = ("termlessI",termlessI), 

     erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],

     rules = [Rule.Rls_ diff_rules, Rule.Rls_ norm_Poly ],

     scr = Rule.EmptyScr};

\<close>

setup \<open>KEStore_Elems.add_rlss 

  [("erls_diff", (Context.theory_name @{theory}, prep_rls' erls_diff)), 

  ("diff_rules", (Context.theory_name @{theory}, prep_rls' diff_rules)), 

  ("norm_diff", (Context.theory_name @{theory}, prep_rls' norm_diff)), 

  ("diff_conv", (Context.theory_name @{theory}, prep_rls' diff_conv)), 

  ("diff_sym_conv", (Context.theory_name @{theory}, prep_rls' diff_sym_conv))]\<close>

(** problem types **)

setup \<open>KEStore_Elems.add_pbts

  [(Specify.prep_pbt thy "pbl_fun" [] Celem.e_pblID (["function"], [], Rule_Set.empty, NONE, [])),

    (Specify.prep_pbt thy "pbl_fun_deriv" [] Celem.e_pblID

      (["derivative_of","function"],

        [("#Given" ,["functionTerm f_f","differentiateFor v_v"]),

          ("#Find"  ,["derivative f_f'"])],

        Rule_Set.append_rules "empty" Rule_Set.empty [],

        SOME "Diff (f_f, v_v)", [["diff","differentiate_on_R"],

			  ["diff","after_simplification"]])),

    (*here "named" is used differently from Integration"*)

    (Specify.prep_pbt thy "pbl_fun_deriv_nam" [] Celem.e_pblID

      (["named","derivative_of","function"],

        [("#Given" ,["functionEq f_f","differentiateFor v_v"]),

          ("#Find"  ,["derivativeEq f_f'"])],

        Rule_Set.append_rules "empty" Rule_Set.empty [],

        SOME "Differentiate (f_f, v_v)",

        [["diff","differentiate_equality"]]))]\<close>

ML \<open>

(** CAS-commands **)

(*.handle cas-input like "Diff (a * x^3 + b, x)".*)

(* val (t, pairl) = strip_comb (str2term "Diff (a * x^3 + b, x)");

   val [Const ("Product_Type.Pair", _) $ t $ bdv] = pairl;

   *)

fun argl2dtss [Const ("Product_Type.Pair", _) $ t $ bdv] =

    [((Thm.term_of o the o (TermC.parse thy)) "functionTerm", [t]),

     ((Thm.term_of o the o (TermC.parse thy)) "differentiateFor", [bdv]),

     ((Thm.term_of o the o (TermC.parse thy)) "derivative", 

      [(Thm.term_of o the o (TermC.parse thy)) "f_f'"])

     ]

  | argl2dtss _ = error "Diff.ML: wrong argument for argl2dtss";

\<close>

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_diff" [] Celem.e_metID

      (["diff"], [],

        {rew_ord'="tless_true",rls'=Atools_erls,calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,

          crls = Atools_erls, errpats = [], nrls = norm_diff},

        @{thm refl})]

\<close>

partial_function (tailrec) differentiate_on_R :: "real \<Rightarrow> real \<Rightarrow> real"

  where

"differentiate_on_R f_f v_v = (

  let

    f_f' = Take (d_d v_v f_f)

  in (

    (Try (Rewrite_Set_Inst [(''bdv'',v_v)] ''diff_conv'')) #> (

    Repeat (

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sum'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod_const'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_quot'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_const'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_var'')) Or

      (Repeat (Rewrite_Set ''make_polynomial'')))) #> (

    Try (Rewrite_Set_Inst [(''bdv'',v_v)] ''diff_sym_conv''))

    ) f_f')"

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_diff_onR" [] Celem.e_metID

      (["diff","differentiate_on_R"],

        [("#Given" ,["functionTerm f_f","differentiateFor v_v"]),

          ("#Find"  ,["derivative f_f'"])],

        {rew_ord'="tless_true", rls' = erls_diff, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,

          crls = Atools_erls, errpats = [], nrls = norm_diff},

        @{thm differentiate_on_R.simps})]

\<close>

partial_function (tailrec) differentiateX :: "real \<Rightarrow> real \<Rightarrow> real"

  where

"differentiateX f_f v_v = (

  let

    f_f' = Take (d_d v_v f_f)

  in (

    Repeat (

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sum'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod_const'' )) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_prod'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_quot'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_sin_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_cos_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_pow_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_ln_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_exp_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_const'')) Or

      (Repeat (Rewrite_Inst [(''bdv'',v_v)] ''diff_var'')) Or

      (Repeat (Rewrite_Set ''make_polynomial'')))

    ) f_f')"

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_diff_simpl" [] Celem.e_metID

      (["diff","diff_simpl"],

        [("#Given", ["functionTerm f_f","differentiateFor v_v"]),

         ("#Find" , ["derivative f_f'"])],

        {rew_ord'="tless_true", rls' = erls_diff, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,

          crls = Atools_erls, errpats = [], nrls = norm_diff},

        @{thm differentiateX.simps})]

\<close>

partial_function (tailrec) differentiate_equality :: "bool \<Rightarrow> real \<Rightarrow> bool"

  where

"differentiate_equality f_f v_v = (

  let

    f_f' = Take ((primed (lhs f_f)) = d_d v_v (rhs f_f))

  in (

    (Try (Rewrite_Set_Inst [(''bdv'',v_v)] ''diff_conv'' )) #> (

    Repeat (

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_sum'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_dif''        )) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_prod_const'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_prod'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_quot'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_sin'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_sin_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_cos'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_cos_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_pow'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_pow_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_ln'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_ln_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_exp'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_exp_chain'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_const'')) Or

      (Repeat (Rewrite_Inst [(''bdv'', v_v)] ''diff_var'')) Or

      (Repeat (Rewrite_Set ''make_polynomial'')))) #> (

    Try (Rewrite_Set_Inst [(''bdv'', v_v)] ''diff_sym_conv'' ))

    ) f_f')"

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_diff_equ" [] Celem.e_metID

      (["diff","differentiate_equality"],

        [("#Given" ,["functionEq f_f","differentiateFor v_v"]),

          ("#Find"  ,["derivativeEq f_f'"])],

        {rew_ord'="tless_true", rls' = erls_diff, calc = [], srls = srls_diff, prls=Rule_Set.empty,

          crls=Atools_erls, errpats = [], nrls = norm_diff},

        @{thm differentiate_equality.simps})]

\<close>

partial_function (tailrec) simplify_derivative :: "real \<Rightarrow> real \<Rightarrow> real"

  where

"simplify_derivative term bound_variable = (

  let

   term' = Take (d_d bound_variable term)

  in (

    (Try (Rewrite_Set ''norm_Rational'')) #>

    (Try (Rewrite_Set_Inst [(''bdv'', bound_variable)] ''diff_conv'')) #>

    (Try (Rewrite_Set_Inst [(''bdv'', bound_variable)] ''norm_diff'')) #>

    (Try (Rewrite_Set_Inst [(''bdv'', bound_variable)] ''diff_sym_conv'')) #>

    (Try (Rewrite_Set ''norm_Rational''))

    ) term')"

setup \<open>KEStore_Elems.add_mets

    [Specify.prep_met thy "met_diff_after_simp" [] Celem.e_metID

      (["diff","after_simplification"],

        [("#Given" ,["functionTerm term","differentiateFor bound_variable"]),

          ("#Find"  ,["derivative term'"])],

        {rew_ord'="tless_true", rls' = Rule_Set.empty, calc = [], srls = Rule_Set.empty, prls=Rule_Set.empty,

          crls=Atools_erls, errpats = [], nrls = norm_Rational},

        @{thm simplify_derivative.simps})]

\<close>

setup \<open>KEStore_Elems.add_cas

  [((Thm.term_of o the o (TermC.parse thy)) "Diff",

	      (("Isac_Knowledge", ["derivative_of","function"], ["no_met"]), argl2dtss))]\<close>

ML \<open>

(*.handle cas-input like "Differentiate (A = s * (a - s), s)".*)

(* val (t, pairl) = strip_comb (str2term "Differentiate (A = s * (a - s), s)");

   val [Const ("Product_Type.Pair", _) $ t $ bdv] = pairl;

   *)

fun argl2dtss [Const ("Product_Type.Pair", _) $ t $ bdv] =

    [((Thm.term_of o the o (TermC.parse thy)) "functionEq", [t]),

     ((Thm.term_of o the o (TermC.parse thy)) "differentiateFor", [bdv]),

     ((Thm.term_of o the o (TermC.parse thy)) "derivativeEq", 

      [(Thm.term_of o the o (TermC.parse thy)) "f_f'::bool"])

     ]

  | argl2dtss _ = error "Diff.ML: wrong argument for argl2dtss";

\<close>

setup \<open>KEStore_Elems.add_cas

  [((Thm.term_of o the o (TermC.parse thy)) "Differentiate",  

	      (("Isac_Knowledge", ["named","derivative_of","function"], ["no_met"]), argl2dtss))]\<close>

end