(* integration over the reals

    author: Walther Neuper

    050814, 08:51

    (c) due to copyright terms

 *)

 theory Integrate imports Diff begin

 consts

   Integral            :: "[real, real]=> real" ("Integral _ D _" 91)

 (*new'_c	      :: "real => real"        ("new'_c _" 66)*)

   is'_f'_x            :: "real => bool"        ("_ is'_f'_x" 10)

   (*descriptions in the related problems*)

   integrateBy         :: real => una

   antiDerivative      :: real => una

   antiDerivativeName  :: (real => real) => una

   (*the CAS-command, eg. "Integrate (2*x^^^3, x)"*)

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

   (*Script-names*)

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

                   ("((Script IntegrationScript (_ _ =))// (_))" 9)

   NamedIntegrationScript :: "[real,real, real=>real,  bool] => bool"

                   ("((Script NamedIntegrationScript (_ _ _=))// (_))" 9)

 axioms 

 (*stated as axioms, todo: prove as theorems

   'bdv' is a constant handled on the meta-level 

    specifically as a 'bound variable'            *)

   integral_const:    "Not (bdv occurs_in u) ==> Integral u D bdv = u * bdv"

   integral_var:      "Integral bdv D bdv = bdv ^^^ 2 / 2"

   integral_add:      "Integral (u + v) D bdv =  

 		     (Integral u D bdv) + (Integral v D bdv)"

   integral_mult:     "[| Not (bdv occurs_in u); bdv occurs_in v |] ==>  

 		     Integral (u * v) D bdv = u * (Integral v D bdv)"

 (*WN080222: this goes into sub-terms, too ...

   call_for_new_c:    "[| Not (matches (u + new_c v) a); Not (a is_f_x) |] ==>  

 		     a = a + new_c a"

 *)

   integral_pow:      "Integral bdv ^^^ n D bdv = bdv ^^^ (n+1) / (n + 1)"

 ML {*

 val thy = @{theory};

 (** eval functions **)

 val c = Free ("c", HOLogic.realT);

 (*.create a new unique variable 'c..' in a term; for use by Calc in a rls;

    an alternative to do this would be '(Try (Calculate new_c_) (new_c es__))'

    in the script; this will be possible if currying doesnt take the value

    from a variable, but the value '(new_c es__)' itself.*)

 fun new_c term = 

     let fun selc var = 

 	    case (explode o id_of) var of

 		"c"::[] => true

 	      |	"c"::"_"::is => (case (int_of_str o implode) is of

 				     SOME _ => true

 				   | NONE => false)

               | _ => false;

 	fun get_coeff c = case (explode o id_of) c of

 	      		      "c"::"_"::is => (the o int_of_str o implode) is

 			    | _ => 0;

         val cs = filter selc (vars term);

     in 

 	case cs of

 	    [] => c

 	  | [c] => Free ("c_2", HOLogic.realT)

 	  | cs => 

 	    let val max_coeff = maxl (map get_coeff cs)

 	    in Free ("c_"^string_of_int (max_coeff + 1), HOLogic.realT) end

     end;

 (*WN080222

 (*("new_c", ("Integrate.new'_c", eval_new_c "#new_c_"))*)

 fun eval_new_c _ _ (p as (Const ("Integrate.new'_c",_) $ t)) _ =

      SOME ((term2str p) ^ " = " ^ term2str (new_c p),

 	  Trueprop $ (mk_equality (p, new_c p)))

   | eval_new_c _ _ _ _ = NONE;

 *)

 (*WN080222:*)

 (*("add_new_c", ("Integrate.add'_new'_c", eval_add_new_c "#add_new_c_"))

   add a new c to a term or a fun-equation;

   this is _not in_ the term, because only applied to _whole_ term*)

 fun eval_add_new_c (_:string) "Integrate.add'_new'_c" p (_:theory) =

     let val p' = case p of

 		     Const ("op =", T) $ lh $ rh => 

 		     Const ("op =", T) $ lh $ mk_add rh (new_c rh)

 		   | p => mk_add p (new_c p)

     in SOME ((term2str p) ^ " = " ^ term2str p',

 	  Trueprop $ (mk_equality (p, p')))

     end

   | eval_add_new_c _ _ _ _ = NONE;

 (*("is_f_x", ("Integrate.is'_f'_x", eval_is_f_x "is_f_x_"))*)

 fun eval_is_f_x _ _(p as (Const ("Integrate.is'_f'_x", _)

 					   $ arg)) _ =

     if is_f_x arg

     then SOME ((term2str p) ^ " = True",

 	       Trueprop $ (mk_equality (p, HOLogic.true_const)))

     else SOME ((term2str p) ^ " = False",

 	       Trueprop $ (mk_equality (p, HOLogic.false_const)))

   | eval_is_f_x _ _ _ _ = NONE;

 calclist':= overwritel (!calclist', 

    [(*("new_c", ("Integrate.new'_c", eval_new_c "new_c_")),*)

     ("add_new_c", ("Integrate.add'_new'_c", eval_add_new_c "add_new_c_")),

     ("is_f_x", ("Integrate.is'_f'_x", eval_is_f_x "is_f_idextifier_"))

     ]);

 (** rulesets **)

 (*.rulesets for integration.*)

 val integration_rules = 

     Rls {id="integration_rules", preconds = [], 

 	 rew_ord = ("termlessI",termlessI), 

 	 erls = Rls {id="conditions_in_integration_rules", 

 		     preconds = [], 

 		     rew_ord = ("termlessI",termlessI), 

 		     erls = Erls, 

 		     srls = Erls, calc = [],

 		     rules = [(*for rewriting conditions in Thm's*)

 			      Calc ("Atools.occurs'_in", 

 				    eval_occurs_in "#occurs_in_"),

 			      Thm ("not_true",num_str @{thm not_true}),

 			      Thm ("not_false",@{thm not_false})

 			      ],

 		     scr = EmptyScr}, 

 	 srls = Erls, calc = [],

 	 rules = [

 		  Thm ("integral_const",num_str @{thm integral_const}),

 		  Thm ("integral_var",num_str @{thm integral_var}),

 		  Thm ("integral_add",num_str @{thm integral_add}),

 		  Thm ("integral_mult",num_str @{thm integral_mult}),

 		  Thm ("integral_pow",num_str @{thm integral_pow}),

 		  Calc ("op +", eval_binop "#add_")(*for n+1*)

 		  ],

 	 scr = EmptyScr};

 val add_new_c = 

     Seq {id="add_new_c", preconds = [], 

 	 rew_ord = ("termlessI",termlessI), 

 	 erls = Rls {id="conditions_in_add_new_c", 

 		     preconds = [], 

 		     rew_ord = ("termlessI",termlessI), 

 		     erls = Erls, 

 		     srls = Erls, calc = [],

 		     rules = [Calc ("Tools.matches", eval_matches""),

 			      Calc ("Integrate.is'_f'_x", 

 				    eval_is_f_x "is_f_x_"),

 			      Thm ("not_true",num_str @{thm not_true}),

 			      Thm ("not_false",num_str @{thm not_false)

 			      ],

 		     scr = EmptyScr}, 

 	 srls = Erls, calc = [],

 	 rules = [ (*Thm ("call_for_new_c", num_str @{thm call_for_new_c}),*)

 		   Cal1 ("Integrate.add'_new'_c", eval_add_new_c "new_c_")

 		   ],

 	 scr = EmptyScr};

 (*.rulesets for simplifying Integrals.*)

 (*.for simplify_Integral adapted from 'norm_Rational_rls'.*)

 val norm_Rational_rls_noadd_fractions = 

 Rls {id = "norm_Rational_rls_noadd_fractions", preconds = [], 

      rew_ord = ("dummy_ord",dummy_ord), 

      erls = norm_rat_erls, srls = Erls, calc = [],

      rules = [(*Rls_ common_nominator_p_rls,!!!*)

 	      Rls_ (*rat_mult_div_pow original corrected WN051028*)

 		  (Rls {id = "rat_mult_div_pow", preconds = [], 

 		       rew_ord = ("dummy_ord",dummy_ord), 

 		       erls = (*FIXME.WN051028 e_rls,*)

 		       append_rls "e_rls-is_polyexp" e_rls

 				  [Calc ("Poly.is'_polyexp", 

 					 eval_is_polyexp "")],

 				  srls = Erls, calc = [],

 				  rules = [Thm ("rat_mult",num_str @{thm rat_mult}),

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

 	       Thm ("rat_mult_poly_l",num_str @{thm rat_mult_poly_l}),

 	       (*"?c is_polyexp ==> ?c * (?a / ?b) = ?c * ?a / ?b"*)

 	       Thm ("rat_mult_poly_r",num_str @{thm rat_mult_poly_r}),

 	       (*"?c is_polyexp ==> ?a / ?b * ?c = ?a * ?c / ?b"*)

 	       Thm ("real_divide_divide1_mg",

                      num_str @{thm real_divide_divide1_mg}),

 	       (*"y ~= 0 ==> (u / v) / (y / z) = (u * z) / (y * v)"*)

 	       Thm ("divide_divide_eq_right", 

                      num_str @{thm divide_divide_eq_right}),

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

 	       Thm ("divide_divide_eq_left",

                      num_str @{thm divide_divide_eq_left}),

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

 	       Calc ("HOL.divide"  ,eval_cancel "#divide_e"),

 	       Thm ("rat_power", num_str @{thm rat_power})

 		(*"(?a / ?b) ^^^ ?n = ?a ^^^ ?n / ?b ^^^ ?n"*)

 	       ],

       scr = Script ((term_of o the o (parse thy)) "empty_script")

       }),

 		Rls_ make_rat_poly_with_parentheses,

 		Rls_ cancel_p_rls,(*FIXME:cancel_p does NOT order sometimes*)

 		Rls_ rat_reduce_1

 		],

        scr = Script ((term_of o the o (parse thy)) "empty_script")

        }:rls;

 (*.for simplify_Integral adapted from 'norm_Rational'.*)

 val norm_Rational_noadd_fractions = 

    Seq {id = "norm_Rational_noadd_fractions", preconds = [], 

        rew_ord = ("dummy_ord",dummy_ord), 

        erls = norm_rat_erls, srls = Erls, calc = [],

        rules = [Rls_ discard_minus,

 		Rls_ rat_mult_poly,(* removes double fractions like a/b/c    *)

 		Rls_ make_rat_poly_with_parentheses, (*WN0510 also in(#)below*)

 		Rls_ cancel_p_rls, (*FIXME.MG:cancel_p does NOT order sometim*)

 		Rls_ norm_Rational_rls_noadd_fractions,(* the main rls (#)   *)

 		Rls_ discard_parentheses1 (* mult only                       *)

 		],

        scr = Script ((term_of o the o (parse thy)) "empty_script")

        }:rls;

 (*.simplify terms before and after Integration such that  

    ..a.x^2/2 + b.x^3/3.. is made to ..a/2.x^2 + b/3.x^3.. (and NO

    common denominator as done by norm_Rational or make_ratpoly_in.

    This is a copy from 'make_ratpoly_in' with respective reduction of rules and

    *1* expand the term, ie. distribute * and / over +

 .*)

 val separate_bdv2 =

     append_rls "separate_bdv2"

 	       collect_bdv

 	       [Thm ("separate_bdv", num_str @{thm separate_bdv}),

 		(*"?a * ?bdv / ?b = ?a / ?b * ?bdv"*)

 		Thm ("separate_bdv_n", num_str @{thm separate_bdv_n}),

 		Thm ("separate_1_bdv", num_str @{thm separate_1_bdv}),

 		(*"?bdv / ?b = (1 / ?b) * ?bdv"*)

 		Thm ("separate_1_bdv_n", num_str @{thm separate_1_bdv_n})(*,

 			  (*"?bdv ^^^ ?n / ?b = 1 / ?b * ?bdv ^^^ ?n"*)

 			  *****Thm ("add_divide_distrib", 

 			  *****num_str @{thm add_divide_distrib})

 			  (*"(?x + ?y) / ?z = ?x / ?z + ?y / ?z"*)----------*)

 		];

 val simplify_Integral = 

   Seq {id = "simplify_Integral", preconds = []:term list, 

        rew_ord = ("dummy_ord", dummy_ord),

       erls = Atools_erls, srls = Erls,

       calc = [], (*asm_thm = [],*)

       rules = [Thm ("left_distrib",num_str @{thm left_distrib}),

  	       (*"(?z1.0 + ?z2.0) * ?w = ?z1.0 * ?w + ?z2.0 * ?w"*)

 	       Thm ("add_divide_distrib",num_str @{thm add_divide_distrib}),

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

 	       (*^^^^^ *1* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^*)

 	       Rls_ norm_Rational_noadd_fractions,

 	       Rls_ order_add_mult_in,

 	       Rls_ discard_parentheses,

 	       (*Rls_ collect_bdv, from make_polynomial_in*)

 	       Rls_ separate_bdv2,

 	       Calc ("HOL.divide"  ,eval_cancel "#divide_e")

 	       ],

       scr = EmptyScr}:rls;      

 (*simplify terms before and after Integration such that  

    ..a.x^2/2 + b.x^3/3.. is made to ..a/2.x^2 + b/3.x^3.. (and NO

    common denominator as done by norm_Rational or make_ratpoly_in.

    This is a copy from 'make_polynomial_in' with insertions from 

    'make_ratpoly_in' 

 THIS IS KEPT FOR COMPARISON ............................................   

 * val simplify_Integral = prep_rls(

 *   Seq {id = "", preconds = []:term list, 

 *        rew_ord = ("dummy_ord", dummy_ord),

 *       erls = Atools_erls, srls = Erls,

 *       calc = [], (*asm_thm = [],*)

 *       rules = [Rls_ expand_poly,

 * 	       Rls_ order_add_mult_in,

 * 	       Rls_ simplify_power,

 * 	       Rls_ collect_numerals,

 * 	       Rls_ reduce_012,

 * 	       Thm ("realpow_oneI",num_str @{thm realpow_oneI}),

 * 	       Rls_ discard_parentheses,

 * 	       Rls_ collect_bdv,

 * 	       (*below inserted from 'make_ratpoly_in'*)

 * 	       Rls_ (append_rls "separate_bdv"

 * 			 collect_bdv

 * 			 [Thm ("separate_bdv", num_str @{thm separate_bdv}),

 * 			  (*"?a * ?bdv / ?b = ?a / ?b * ?bdv"*)

 * 			  Thm ("separate_bdv_n", num_str @{thm separate_bdv_n}),

 * 			  Thm ("separate_1_bdv", num_str @{thm separate_1_bdv}),

 * 			  (*"?bdv / ?b = (1 / ?b) * ?bdv"*)

 * 			  Thm ("separate_1_bdv_n", num_str @{thm separate_1_bdv_n})(*,

 * 			  (*"?bdv ^^^ ?n / ?b = 1 / ?b * ?bdv ^^^ ?n"*)

 * 			  Thm ("add_divide_distrib", 

 * 				 num_str @{thm add_divide_distrib})

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

 * 			  ]),

 * 	       Calc ("HOL.divide"  ,eval_cancel "#divide_e")

 * 	       ],

 *       scr = EmptyScr

 *       }:rls); 

 .......................................................................*)

 val integration = 

     Seq {id="integration", preconds = [], 

 	 rew_ord = ("termlessI",termlessI), 

 	 erls = Rls {id="conditions_in_integration", 

 		     preconds = [], 

 		     rew_ord = ("termlessI",termlessI), 

 		     erls = Erls, 

 		     srls = Erls, calc = [],

 		     rules = [],

 		     scr = EmptyScr}, 

 	 srls = Erls, calc = [],

 	 rules = [ Rls_ integration_rules,

 		   Rls_ add_new_c,

 		   Rls_ simplify_Integral

 		   ],

 	 scr = EmptyScr};

 ruleset' := 

 overwritelthy @{theory} (!ruleset', 

 	    [("integration_rules", prep_rls integration_rules),

 	     ("add_new_c", prep_rls add_new_c),

 	     ("simplify_Integral", prep_rls simplify_Integral),

 	     ("integration", prep_rls integration),

 	     ("separate_bdv2", separate_bdv2),

 	     ("norm_Rational_noadd_fractions", norm_Rational_noadd_fractions),

 	     ("norm_Rational_rls_noadd_fractions", 

 	      norm_Rational_rls_noadd_fractions)

 	     ]);

 (** problems **)

 store_pbt

  (prep_pbt thy "pbl_fun_integ" [] e_pblID

  (["integrate","function"],

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

    ("#Find"  ,["antiDerivative F_"])

   ],

   append_rls "e_rls" e_rls [(*for preds in where_*)], 

   SOME "Integrate (f_f, v_v)", 

   [["diff","integration"]]));

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

 store_pbt

  (prep_pbt thy "pbl_fun_integ_nam" [] e_pblID

  (["named","integrate","function"],

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

    ("#Find"  ,["antiDerivativeName F_"])

   ],

   append_rls "e_rls" e_rls [(*for preds in where_*)], 

   SOME "Integrate (f_f, v_v)", 

   [["diff","integration","named"]]));

 (** methods **)

 store_met

     (prep_met thy "met_diffint" [] e_metID

 	      (["diff","integration"],

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

 		("#Find"  ,["antiDerivative F_"])

 		],

 	       {rew_ord'="tless_true", rls'=Atools_erls, calc = [], 

 		srls = e_rls, 

 		prls=e_rls,

 	     crls = Atools_erls, nrls = e_rls},

 "Script IntegrationScript (f_f::real) (v_v::real) =                " ^

 "  (let t_ = Take (Integral f_ D v_v)                             " ^

 "   in (Rewrite_Set_Inst [(bdv,v_v)] integration False) (t_::real))"

 ));

 store_met

     (prep_met thy "met_diffint_named" [] e_metID

 	      (["diff","integration","named"],

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

 		("#Find"  ,["antiDerivativeName F_"])

 		],

 	       {rew_ord'="tless_true", rls'=Atools_erls, calc = [], 

 		srls = e_rls, 

 		prls=e_rls,

 		crls = Atools_erls, nrls = e_rls},

 "Script NamedIntegrationScript (f_f::real) (v_v::real) (F_::real=>real) = " ^

 "  (let t_ = Take (F_ v_v = Integral f_ D v_v)                            " ^

 "   in ((Try (Rewrite_Set_Inst [(bdv,v_v)] simplify_Integral False)) @@  " ^

 "       (Rewrite_Set_Inst [(bdv,v_v)] integration False)) t_)            "

     ));

 *}

 end