(* 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 \<up> 3, x)"*)

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

axiomatization where

(*stated as axioms, todo: prove as theorems

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

   specifically as a 'bound variable'            *)

(*Ambiguous input\<^here> produces 3 parse trees -----------------------------\\*)

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

  integral_var:      "Integral bdv D bdv = bdv \<up> 2 / 2" and

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

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

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

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

(*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 \<up> n D bdv = bdv \<up> (n+1) / (n + 1)"

(*Ambiguous input\<^here> produces 3 parse trees -----------------------------//*)

ML \<open>

(** eval functions **)

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

(*.create a new unique variable 'c..' in a term; for use by Rule.Eval 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 (Symbol.explode o id_of) var of

		"c"::[] => true

	      |	"c"::"_"::is => (case (TermC.int_opt_of_string o implode) is of

				     SOME _ => true

				   | NONE => false)

              | _ => false;

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

	      		      "c"::"_"::is => (the o TermC.int_opt_of_string o implode) is

			    | _ => 0;

        val cs = filter selc (TermC.vars term);

    in 

	case cs of

	    [] => 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 ((UnparseC.term p) ^ " = " ^ UnparseC.term (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 ("HOL.eq", T) $ lh $ rh => 

		     Const ("HOL.eq", T) $ lh $ TermC.mk_add rh (new_c rh)

		   | p => TermC.mk_add p (new_c p)

    in SOME ((UnparseC.term p) ^ " = " ^ UnparseC.term p',

	  HOLogic.Trueprop $ (TermC.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 TermC.is_f_x arg

    then SOME ((UnparseC.term p) ^ " = True",

	       HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))

    else SOME ((UnparseC.term p) ^ " = False",

	       HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))

  | eval_is_f_x _ _ _ _ = NONE;

\<close>

setup \<open>KEStore_Elems.add_calcs

  [("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_"))]\<close>

ML \<open>

(** rulesets **)

(*.rulesets for integration.*)

val integration_rules = 

    Rule_Def.Repeat {id="integration_rules", preconds = [], 

	 rew_ord = ("termlessI",termlessI), 

	 erls = Rule_Def.Repeat {id="conditions_in_integration_rules", 

		     preconds = [], 

		     rew_ord = ("termlessI",termlessI), 

		     erls = Rule_Set.Empty, 

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

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

			      Rule.Eval ("Prog_Expr.occurs_in", Prog_Expr.eval_occurs_in "#occurs_in_"),

			      Rule.Thm ("not_true", ThmC.numerals_to_Free @{thm not_true}),

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

			      ],

		     scr = Rule.Empty_Prog}, 

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

	 rules = [

		  Rule.Thm ("integral_const", ThmC.numerals_to_Free @{thm integral_const}),

		  Rule.Thm ("integral_var", ThmC.numerals_to_Free @{thm integral_var}),

		  Rule.Thm ("integral_add", ThmC.numerals_to_Free @{thm integral_add}),

		  Rule.Thm ("integral_mult", ThmC.numerals_to_Free @{thm integral_mult}),

		  Rule.Thm ("integral_pow", ThmC.numerals_to_Free @{thm integral_pow}),

		  Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_")(*for n+1*)

		  ],

	 scr = Rule.Empty_Prog};

\<close>

ML \<open>

val add_new_c = 

    Rule_Set.Sequence {id="add_new_c", preconds = [], 

	 rew_ord = ("termlessI",termlessI), 

	 erls = Rule_Def.Repeat {id="conditions_in_add_new_c", 

		     preconds = [], 

		     rew_ord = ("termlessI",termlessI), 

		     erls = Rule_Set.Empty, 

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

		     rules = [Rule.Eval ("Prog_Expr.matches", Prog_Expr.eval_matches""),

			      Rule.Eval ("Integrate.is_f_x", 

				    eval_is_f_x "is_f_x_"),

			      Rule.Thm ("not_true", ThmC.numerals_to_Free @{thm not_true}),

			      Rule.Thm ("not_false", ThmC.numerals_to_Free @{thm not_false})

			      ],

		     scr = Rule.Empty_Prog}, 

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

	 rules = [ (*Rule.Thm ("call_for_new_c", ThmC.numerals_to_Free @{thm call_for_new_c}),*)

		   Rule.Cal1 ("Integrate.add_new_c", eval_add_new_c "new_c_")

		   ],

	 scr = Rule.Empty_Prog};

\<close>

ML \<open>

(*.rulesets for simplifying Integrals.*)

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

val norm_Rational_rls_noadd_fractions = 

Rule_Def.Repeat {id = "norm_Rational_rls_noadd_fractions", preconds = [], 

     rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord), 

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

     rules = [(*Rule.Rls_ add_fractions_p_rls,!!!*)

	      Rule.Rls_ (*rat_mult_div_pow original corrected WN051028*)

		  (Rule_Def.Repeat {id = "rat_mult_div_pow", preconds = [], 

		       rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord), 

		       erls = (*FIXME.WN051028 Rule_Set.empty,*)

		       Rule_Set.append_rules "Rule_Set.empty-is_polyexp" Rule_Set.empty

				  [Rule.Eval ("Poly.is_polyexp", 

					 eval_is_polyexp "")],

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

				  rules = [Rule.Thm ("rat_mult", ThmC.numerals_to_Free @{thm rat_mult}),

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

	       Rule.Thm ("rat_mult_poly_l", ThmC.numerals_to_Free @{thm rat_mult_poly_l}),

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

	       Rule.Thm ("rat_mult_poly_r", ThmC.numerals_to_Free @{thm rat_mult_poly_r}),

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

	       Rule.Thm ("real_divide_divide1_mg",

                     ThmC.numerals_to_Free @{thm real_divide_divide1_mg}),

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

	       Rule.Thm ("divide_divide_eq_right", 

                     ThmC.numerals_to_Free @{thm divide_divide_eq_right}),

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

	       Rule.Thm ("divide_divide_eq_left",

                     ThmC.numerals_to_Free @{thm divide_divide_eq_left}),

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

	       Rule.Eval ("Rings.divide_class.divide", Prog_Expr.eval_cancel "#divide_e"),

	       Rule.Thm ("rat_power", ThmC.numerals_to_Free @{thm rat_power})

		(*"(?a / ?b)  \<up>  ?n = ?a  \<up>  ?n / ?b  \<up>  ?n"*)

	       ],

      scr = Rule.Empty_Prog

      }),

		Rule.Rls_ make_rat_poly_with_parentheses,

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

		Rule.Rls_ rat_reduce_1

		],

       scr = Rule.Empty_Prog

       };

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

val norm_Rational_noadd_fractions = 

   Rule_Set.Sequence {id = "norm_Rational_noadd_fractions", preconds = [], 

       rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord), 

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

       rules = [Rule.Rls_ discard_minus,

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

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

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

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

		Rule.Rls_ discard_parentheses1 (* mult only                       *)

		],

       scr = Rule.Empty_Prog

       };

(*.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 =

    Rule_Set.append_rules "separate_bdv2"

	       collect_bdv

	       [Rule.Thm ("separate_bdv", ThmC.numerals_to_Free @{thm separate_bdv}),

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

		Rule.Thm ("separate_bdv_n", ThmC.numerals_to_Free @{thm separate_bdv_n}),

		Rule.Thm ("separate_1_bdv",  ThmC.numerals_to_Free @{thm separate_1_bdv}),

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

		Rule.Thm ("separate_1_bdv_n",  ThmC.numerals_to_Free @{thm separate_1_bdv_n})(*,

			  (*"?bdv  \<up>  ?n / ?b = 1 / ?b * ?bdv  \<up>  ?n"*)

			  *****Rule.Thm ("add_divide_distrib", 

			  ***** ThmC.numerals_to_Free @{thm add_divide_distrib})

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

		];

val simplify_Integral = 

  Rule_Set.Sequence {id = "simplify_Integral", preconds = []:term list, 

       rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),

      erls = Atools_erls, srls = Rule_Set.Empty,

      calc = [],  errpatts = [],

      rules = [Rule.Thm ("distrib_right", ThmC.numerals_to_Free @{thm distrib_right}),

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

	       Rule.Thm ("add_divide_distrib", ThmC.numerals_to_Free @{thm add_divide_distrib}),

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

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

	       Rule.Rls_ norm_Rational_noadd_fractions,

	       Rule.Rls_ order_add_mult_in,

	       Rule.Rls_ discard_parentheses,

	       (*Rule.Rls_ collect_bdv, from make_polynomial_in*)

	       Rule.Rls_ separate_bdv2,

	       Rule.Eval ("Rings.divide_class.divide", Prog_Expr.eval_cancel "#divide_e")

	       ],

      scr = Rule.Empty_Prog};      

(*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'(

*   Rule_Set.Sequence {id = "", preconds = []:term list, 

*        rew_ord = ("dummy_ord", Rewrite_Ord.dummy_ord),

*       erls = Atools_erls, srls = Rule_Set.Empty,

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

*       rules = [Rule.Rls_ expand_poly,

* 	       Rule.Rls_ order_add_mult_in,

* 	       Rule.Rls_ simplify_power,

* 	       Rule.Rls_ collect_numerals,

* 	       Rule.Rls_ reduce_012,

* 	       Rule.Thm ("realpow_oneI", ThmC.numerals_to_Free @{thm realpow_oneI}),

* 	       Rule.Rls_ discard_parentheses,

* 	       Rule.Rls_ collect_bdv,

* 	       (*below inserted from 'make_ratpoly_in'*)

* 	       Rule.Rls_ (Rule_Set.append_rules "separate_bdv"

* 			 collect_bdv

* 			 [Rule.Thm ("separate_bdv", ThmC.numerals_to_Free @{thm separate_bdv}),

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

* 			  Rule.Thm ("separate_bdv_n", ThmC.numerals_to_Free @{thm separate_bdv_n}),

* 			  Rule.Thm ("separate_1_bdv", ThmC.numerals_to_Free @{thm separate_1_bdv}),

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

* 			  Rule.Thm ("separate_1_bdv_n", ThmC.numerals_to_Free @{thm separate_1_bdv_n})(*,

* 			  (*"?bdv  \<up>  ?n / ?b = 1 / ?b * ?bdv  \<up>  ?n"*)

* 			  Rule.Thm ("add_divide_distrib", 

* 				  ThmC.numerals_to_Free @{thm add_divide_distrib})

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

* 			  ]),

* 	       Rule.Eval ("Rings.divide_class.divide"  , eval_cancel "#divide_e")

* 	       ],

*       scr = Rule.Empty_Prog

*       }); 

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

val integration = 

    Rule_Set.Sequence {id="integration", preconds = [], 

	 rew_ord = ("termlessI",termlessI), 

	 erls = Rule_Def.Repeat {id="conditions_in_integration", 

		     preconds = [], 

		     rew_ord = ("termlessI",termlessI), 

		     erls = Rule_Set.Empty, 

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

		     rules = [],

		     scr = Rule.Empty_Prog}, 

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

	 rules = [ Rule.Rls_ integration_rules,

		   Rule.Rls_ add_new_c,

		   Rule.Rls_ simplify_Integral

		   ],

	 scr = Rule.Empty_Prog};

val prep_rls' = Auto_Prog.prep_rls @{theory};

\<close>

rule_set_knowledge

  integration_rules = \<open>prep_rls' integration_rules\<close> and

  add_new_c = \<open>prep_rls' add_new_c\<close> and

  simplify_Integral = \<open>prep_rls' simplify_Integral\<close> and

  integration = \<open>prep_rls' integration\<close> and

  separate_bdv2 = \<open>prep_rls' separate_bdv2\<close> and

  norm_Rational_noadd_fractions = \<open>prep_rls' norm_Rational_noadd_fractions\<close> and

  norm_Rational_rls_noadd_fractions = \<open>prep_rls' norm_Rational_rls_noadd_fractions\<close>

(** problems **)

setup \<open>KEStore_Elems.add_pbts

  [(Problem.prep_input @{theory} "pbl_fun_integ" [] Problem.id_empty

      (["integrate", "function"],

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

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

        Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)], 

        SOME "Integrate (f_f, v_v)", 

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

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

    (Problem.prep_input @{theory} "pbl_fun_integ_nam" [] Problem.id_empty

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

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

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

        Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)], 

        SOME "Integrate (f_f, v_v)", 

        [["diff", "integration", "named"]]))]\<close>

(** methods **)

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

  where

"integrate f_f v_v = (

  let

    t_t = Take (Integral f_f D v_v)

  in

    (Rewrite_Set_Inst [(''bdv'', v_v)] ''integration'') t_t)"

setup \<open>KEStore_Elems.add_mets

    [MethodC.prep_input @{theory} "met_diffint" [] MethodC.id_empty

	    (["diff", "integration"],

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

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

	        crls = Atools_erls, errpats = [], nrls = Rule_Set.empty},

	      @{thm integrate.simps})]

\<close>

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

  where

"intergrate_named f_f v_v F_F =(

  let

    t_t = Take (F_F v_v = Integral f_f D v_v)

  in (

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

    (Rewrite_Set_Inst [(''bdv'', v_v)] ''integration'')

    ) t_t)"

setup \<open>KEStore_Elems.add_mets

    [MethodC.prep_input @{theory} "met_diffint_named" [] MethodC.id_empty

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

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

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

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

          crls = Atools_erls, errpats = [], nrls = Rule_Set.empty},

        @{thm intergrate_named.simps})]

\<close> ML \<open>

\<close> ML \<open>

\<close>

end