(* some tests are based on specficially simple scripts etc.

   Author: Walther Neuper 2003

   (c) due to copyright terms

*) 

theory Test imports Atools + Rational + Root + Poly + 

consts

(*"cancel":: [real, real] => real    (infixl "'/'/'/" 70) ...divide 2002*)

  Expand'_binomtest

             :: "['y,  

		    'y] => 'y"

               ("((Script Expand'_binomtest (_ =))//  

                   (_))" 9)

  Solve'_univar'_err

             :: "[bool,real,bool,  

		    bool list] => bool list"

               ("((Script Solve'_univar'_err (_ _ _ =))//  

                   (_))" 9)

  Solve'_linear

             :: "[bool,real,  

		    bool list] => bool list"

               ("((Script Solve'_linear (_ _ =))//  

                   (_))" 9)

(*17.9.02 aus SqRoot.thy------------------------------vvv---*)

  "is'_root'_free" :: 'a => bool           ("is'_root'_free _" 10)

  "contains'_root" :: 'a => bool           ("contains'_root _" 10)

  Solve'_root'_equation 

             :: "[bool,real,  

		    bool list] => bool list"

               ("((Script Solve'_root'_equation (_ _ =))//  

                   (_))" 9)

  Solve'_plain'_square 

             :: "[bool,real,  

		    bool list] => bool list"

               ("((Script Solve'_plain'_square (_ _ =))//  

                   (_))" 9)

  Norm'_univar'_equation 

             :: "[bool,real,  

		    bool] => bool"

               ("((Script Norm'_univar'_equation (_ _ =))//  

                   (_))" 9)

  STest'_simplify

             :: "['z,  

		    'z] => 'z"

               ("((Script STest'_simplify (_ =))//  

                   (_))" 9)

(*17.9.02 aus SqRoot.thy------------------------------^^^---*)  

axioms (*TODO: prove as theorems*)

  radd_mult_distrib2:      "(k::real) * (m + n) = k * m + k * n"

  rdistr_right_assoc:      "(k::real) + l * n + m * n = k + (l + m) * n"

  rdistr_right_assoc_p:    "l * n + (m * n + (k::real)) = (l + m) * n + k"

  rdistr_div_right:        "((k::real) + l) / n = k / n + l / n"

  rcollect_right:

          "[| l is_const; m is_const |] ==> (l::real)*n + m*n = (l + m) * n"

  rcollect_one_left:

          "m is_const ==> (n::real) + m * n = (1 + m) * n"

  rcollect_one_left_assoc:

          "m is_const ==> (k::real) + n + m * n = k + (1 + m) * n"

  rcollect_one_left_assoc_p:

          "m is_const ==> n + (m * n + (k::real)) = (1 + m) * n + k"

  rtwo_of_the_same:        "a + a = 2 * a"

  rtwo_of_the_same_assoc:  "(x + a) + a = x + 2 * a"

  rtwo_of_the_same_assoc_p:"a + (a + x) = 2 * a + x"

  rcancel_den:             "not(a=0) ==> a * (b / a) = b"

  rcancel_const:           "[| a is_const; b is_const |] ==> a*(x/b) = a/b*x"

  rshift_nominator:        "(a::real) * b / c = a / c * b"

  exp_pow:                 "(a ^^^ b) ^^^ c = a ^^^ (b * c)"

  rsqare:                  "(a::real) * a = a ^^^ 2"

  power_1:                 "(a::real) ^^^ 1 = a"

  rbinom_power_2:          "((a::real) + b)^^^ 2 = a^^^ 2 + 2*a*b + b^^^ 2"

  rmult_1:                 "1 * k = (k::real)"

  rmult_1_right:           "k * 1 = (k::real)"

  rmult_0:                 "0 * k = (0::real)"

  rmult_0_right:           "k * 0 = (0::real)"

  radd_0:                  "0 + k = (k::real)"

  radd_0_right:            "k + 0 = (k::real)"

  radd_real_const_eq:

          "[| a is_const; c is_const; d is_const |] ==> a/d + c/d = (a+c)/(d::real)"

  radd_real_const:

          "[| a is_const; b is_const; c is_const; d is_const |] ==> a/b + c/d = (a*d + b*c)/(b*(d::real))"  

(*for AC-operators*)

  radd_commute:            "(m::real) + (n::real) = n + m"

  radd_left_commute:       "(x::real) + (y + z) = y + (x + z)"

  radd_assoc:              "(m::real) + n + k = m + (n + k)"

  rmult_commute:           "(m::real) * n = n * m"

  rmult_left_commute:      "(x::real) * (y * z) = y * (x * z)"

  rmult_assoc:             "(m::real) * n * k = m * (n * k)"

(*for equations: 'bdv' is a meta-constant*)

  risolate_bdv_add:       "((k::real) + bdv = m) = (bdv = m + (-1)*k)"

  risolate_bdv_mult_add:  "((k::real) + n*bdv = m) = (n*bdv = m + (-1)*k)"

  risolate_bdv_mult:      "((n::real) * bdv = m) = (bdv = m / n)"

  rnorm_equation_add:

      "~(b =!= 0) ==> (a = b) = (a + (-1)*b = 0)"

(*17.9.02 aus SqRoot.thy------------------------------vvv---*) 

  root_ge0:            "0 <= a ==> 0 <= sqrt a"

  (*should be dropped with better simplification in eval_rls ...*)

  root_add_ge0:

	"[| 0 <= a; 0 <= b |] ==> (0 <= sqrt a + sqrt b) = True"

  root_ge0_1:

	"[| 0<=a; 0<=b; 0<=c |] ==> (0 <= a * sqrt b + sqrt c) = True"

  root_ge0_2:

	"[| 0<=a; 0<=b; 0<=c |] ==> (0 <= sqrt a + b * sqrt c) = True"

  rroot_square_inv:         "(sqrt a)^^^ 2 = a"

  rroot_times_root:         "sqrt a * sqrt b = sqrt(a*b)"

  rroot_times_root_assoc:   "(a * sqrt b) * sqrt c = a * sqrt(b*c)"

  rroot_times_root_assoc_p: "sqrt b * (sqrt c * a)= sqrt(b*c) * a"

(*for root-equations*)

  square_equation_left:

          "[| 0 <= a; 0 <= b |] ==> (((sqrt a)=b)=(a=(b^^^ 2)))"

  square_equation_right:

          "[| 0 <= a; 0 <= b |] ==> ((a=(sqrt b))=((a^^^ 2)=b))"

  (*causes frequently non-termination:*)

  square_equation:  

          "[| 0 <= a; 0 <= b |] ==> ((a=b)=((a^^^ 2)=b^^^ 2))"

  risolate_root_add:        "(a+  sqrt c = d) = (  sqrt c = d + (-1)*a)"

  risolate_root_mult:       "(a+b*sqrt c = d) = (b*sqrt c = d + (-1)*a)"

  risolate_root_div:        "(a * sqrt c = d) = (  sqrt c = d / a)"

(*for polynomial equations of degree 2; linear case in RatArith*)

  mult_square:		"(a*bdv^^^2 = b) = (bdv^^^2 = b / a)"

  constant_square:       "(a + bdv^^^2 = b) = (bdv^^^2 = b + -1*a)"

  constant_mult_square:  "(a + b*bdv^^^2 = c) = (b*bdv^^^2 = c + -1*a)"

  square_equality: 

	     "0 <= a ==> (x^^^2 = a) = ((x=sqrt a) | (x=-1*sqrt a))"

  square_equality_0:

	     "(x^^^2 = 0) = (x = 0)"

(*isolate root on the LEFT hand side of the equation

  otherwise shuffling from left to right would not terminate*)  

  rroot_to_lhs:

          "is_root_free a ==> (a = sqrt b) = (a + (-1)*sqrt b = 0)"

  rroot_to_lhs_mult:

          "is_root_free a ==> (a = c*sqrt b) = (a + (-1)*c*sqrt b = 0)"

  rroot_to_lhs_add_mult:

          "is_root_free a ==> (a = d+c*sqrt b) = (a + (-1)*c*sqrt b = d)"

(*17.9.02 aus SqRoot.thy------------------------------^^^---*)  

ML {*

val thy = @{theory};

(** evaluation of numerals and predicates **)

(*does a term contain a root ?*)

fun eval_root_free (thmid:string) _ (t as (Const(op0,t0) $ arg)) thy = 

  if strip_thy op0 <> "is'_root'_free" 

    then raise error ("eval_root_free: wrong "^op0)

  else if const_in (strip_thy op0) arg

	 then SOME (mk_thmid thmid "" 

		    ((Syntax.string_of_term (thy2ctxt thy)) arg) "", 

		    Trueprop $ (mk_equality (t, false_as_term)))

       else SOME (mk_thmid thmid "" 

		  ((Syntax.string_of_term (thy2ctxt thy)) arg) "", 

		  Trueprop $ (mk_equality (t, true_as_term)))

  | eval_root_free _ _ _ _ = NONE; 

(*does a term contain a root ?*)

fun eval_contains_root (thmid:string) _ 

		       (t as (Const("Test.contains'_root",t0) $ arg)) thy = 

    if member op = (ids_of arg) "sqrt"

    then SOME (mk_thmid thmid "" 

			((Syntax.string_of_term (thy2ctxt thy)) arg) "", 

	       Trueprop $ (mk_equality (t, true_as_term)))

    else SOME (mk_thmid thmid "" 

			((Syntax.string_of_term (thy2ctxt thy)) arg) "", 

	       Trueprop $ (mk_equality (t, false_as_term)))

  | eval_contains_root _ _ _ _ = NONE; 

calclist':= overwritel (!calclist', 

   [("is_root_free", ("Test.is'_root'_free", 

		      eval_root_free"#is_root_free_e")),

    ("contains_root", ("Test.contains'_root",

		       eval_contains_root"#contains_root_"))

    ]);

(** term order **)

fun Term_Ord.term_order (_:subst) tu = (term_ordI [] tu = LESS);

(** rule sets GOON **)

val testerls = 

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

      erls = e_rls, srls = Erls, 

      calc = [], 

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

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

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

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

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

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

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

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

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

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

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

	       Calc ("Atools.is'_const",eval_const "#is_const_"),

	       Calc ("Tools.matches",eval_matches ""),

	       Calc ("op +",eval_binop "#add_"),

	       Calc ("op *",eval_binop "#mult_"),

	       Calc ("Atools.pow" ,eval_binop "#power_"),

	       Calc ("op <",eval_equ "#less_"),

	       Calc ("op <=",eval_equ "#less_equal_"),

	       Calc ("Atools.ident",eval_ident "#ident_")],

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

      "empty_script")

      }:rls;      

(*.for evaluation of conditions in rewrite rules.*)

(*FIXXXXXXME 10.8.02: handle like _simplify*)

val tval_rls =  

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

      rew_ord = ("sqrt_right",sqrt_right false (theory "Pure")), 

      erls=testerls,srls = e_rls, 

      calc=[],

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

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

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

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

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

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

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

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

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

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

	       Thm ("or_commute",num_str @{or_commute}),

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

	       Thm ("root_ge0",num_str @{thm root_ge0}),

	       Thm ("root_add_ge0",num_str @{thm root_add_ge0}),

	       Thm ("root_ge0_1",num_str @{thm root_ge0_1}),

	       Thm ("root_ge0_2",num_str @{thm root_ge0_2}),

	       Calc ("Atools.is'_const",eval_const "#is_const_"),

	       Calc ("Test.is'_root'_free",eval_root_free "#is_root_free_e"),

	       Calc ("Tools.matches",eval_matches ""),

	       Calc ("Test.contains'_root",

		     eval_contains_root"#contains_root_"),

	       Calc ("op +",eval_binop "#add_"),

	       Calc ("op *",eval_binop "#mult_"),

	       Calc ("NthRoot.sqrt",eval_sqrt "#sqrt_"),

	       Calc ("Atools.pow" ,eval_binop "#power_"),

	       Calc ("op <",eval_equ "#less_"),

	       Calc ("op <=",eval_equ "#less_equal_"),

	       Calc ("Atools.ident",eval_ident "#ident_")],

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

      "empty_script")

      }:rls;      

ruleset' := overwritelthy @{theory} (!ruleset',

  [("testerls", prep_rls testerls)

   ]);

(*make () dissappear*)   

val rearrange_assoc =

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

      rew_ord = ("e_rew_ord",e_rew_ord), 

      erls = e_rls, srls = e_rls, calc = [], (*asm_thm=[],*)

      rules = 

      [Thm ("sym_add_assoc",num_str (@{thm add_assoc} RS @{thm sym})),

       Thm ("sym_rmult_assoc",num_str (@{thm rmult_assoc} RS @{thm sym}))],

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

      "empty_script")

      }:rls;      

val ac_plus_times =

  Rls{id = "ac_plus_times", preconds = [], rew_ord = ("term_order",term_order),

      erls = e_rls, srls = e_rls, calc = [], (*asm_thm=[],*)

      rules = 

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

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

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

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

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

       Thm ("rmult_assoc",num_str @{thm rmult_assoc})],

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

      "empty_script")

      }:rls;      

(*todo: replace by Rewrite("rnorm_equation_add",num_str @{thm rnorm_equation_add)*)

val norm_equation =

  Rls{id = "norm_equation", preconds = [], rew_ord = ("e_rew_ord",e_rew_ord),

      erls = tval_rls, srls = e_rls, calc = [], (*asm_thm=[],*)

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

	       ],

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

      "empty_script")

      }:rls;      

(** rule sets **)

val STest_simplify =     (*   vv--- not changed to real by parse*)

  "Script STest_simplify (t_::'z) =                           " ^

  "(Repeat" ^

  "    ((Try (Repeat (Rewrite real_diff_minus False))) @@        " ^

  "     (Try (Repeat (Rewrite radd_mult_distrib2 False))) @@  " ^

  "     (Try (Repeat (Rewrite rdistr_right_assoc False))) @@  " ^

  "     (Try (Repeat (Rewrite rdistr_right_assoc_p False))) @@" ^

  "     (Try (Repeat (Rewrite rdistr_div_right False))) @@    " ^

  "     (Try (Repeat (Rewrite rbinom_power_2 False))) @@      " ^

  "     (Try (Repeat (Rewrite radd_commute False))) @@        " ^

  "     (Try (Repeat (Rewrite radd_left_commute False))) @@   " ^

  "     (Try (Repeat (Rewrite add_assoc False))) @@          " ^

  "     (Try (Repeat (Rewrite rmult_commute False))) @@       " ^

  "     (Try (Repeat (Rewrite rmult_left_commute False))) @@  " ^

  "     (Try (Repeat (Rewrite rmult_assoc False))) @@         " ^

  "     (Try (Repeat (Rewrite radd_real_const_eq False))) @@   " ^

  "     (Try (Repeat (Rewrite radd_real_const False))) @@   " ^

  "     (Try (Repeat (Calculate PLUS))) @@   " ^

  "     (Try (Repeat (Calculate TIMES))) @@   " ^

  "     (Try (Repeat (Calculate divide_))) @@" ^

  "     (Try (Repeat (Calculate POWER))) @@  " ^

  "     (Try (Repeat (Rewrite rcollect_right False))) @@   " ^

  "     (Try (Repeat (Rewrite rcollect_one_left False))) @@   " ^

  "     (Try (Repeat (Rewrite rcollect_one_left_assoc False))) @@   " ^

  "     (Try (Repeat (Rewrite rcollect_one_left_assoc_p False))) @@   " ^

  "     (Try (Repeat (Rewrite rshift_nominator False))) @@   " ^

  "     (Try (Repeat (Rewrite rcancel_den False))) @@   " ^

  "     (Try (Repeat (Rewrite rroot_square_inv False))) @@   " ^

  "     (Try (Repeat (Rewrite rroot_times_root False))) @@   " ^

  "     (Try (Repeat (Rewrite rroot_times_root_assoc_p False))) @@   " ^

  "     (Try (Repeat (Rewrite rsqare False))) @@   " ^

  "     (Try (Repeat (Rewrite power_1 False))) @@   " ^

  "     (Try (Repeat (Rewrite rtwo_of_the_same False))) @@   " ^

  "     (Try (Repeat (Rewrite rtwo_of_the_same_assoc_p False))) @@   " ^

  "     (Try (Repeat (Rewrite rmult_1 False))) @@   " ^

  "     (Try (Repeat (Rewrite rmult_1_right False))) @@   " ^

  "     (Try (Repeat (Rewrite rmult_0 False))) @@   " ^

  "     (Try (Repeat (Rewrite rmult_0_right False))) @@   " ^

  "     (Try (Repeat (Rewrite radd_0 False))) @@   " ^

  "     (Try (Repeat (Rewrite radd_0_right False)))) " ^

  " t_)";

(* expects * distributed over + *)

val Test_simplify =

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

      rew_ord = ("sqrt_right",sqrt_right false (theory "Pure")),

      erls = tval_rls, srls = e_rls, 

      calc=[(*since 040209 filled by prep_rls*)],

      (*asm_thm = [],*)

      rules = [

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

	       Thm ("radd_mult_distrib2",num_str @{thm radd_mult_distrib2}),

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

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

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

	       Thm ("rbinom_power_2",num_str @{thm rbinom_power_2}),	       

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

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

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

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

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

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

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

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

	       (* these 2 rules are invers to distr_div_right wrt. termination.

		  thus they MUST be done IMMEDIATELY before calc *)

	       Calc ("op +", eval_binop "#add_"), 

	       Calc ("op *", eval_binop "#mult_"),

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

	       Calc ("Atools.pow", eval_binop "#power_"),

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

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

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

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

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

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

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

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

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

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

	       Thm ("power_1",num_str @{power_1}),

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

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

	       Thm ("rmult_1",num_str @{thm rmult_1}),

	       Thm ("rmult_1_right",num_str @{thm rmult_1_right}),

	       Thm ("rmult_0",num_str @{thm rmult_0}),

	       Thm ("rmult_0_right",num_str @{thm rmult_0_right}),

	       Thm ("radd_0",num_str @{thm radd_0}),

	       Thm ("radd_0_right",num_str @{thm radd_0_right})

	       ],

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

		    (*since 040209 filled by prep_rls: STest_simplify*)

      }:rls;      

(** rule sets **)

(*isolate the root in a root-equation*)

val isolate_root =

  Rls{id = "isolate_root", preconds = [], rew_ord = ("e_rew_ord",e_rew_ord), 

      erls=tval_rls,srls = e_rls, calc=[],(*asm_thm = [], *)

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

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

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

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

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

	       Thm ("risolate_root_div",num_str @{thm risolate_root_div})       ],

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

      "empty_script")

      }:rls;

(*isolate the bound variable in an equation; 'bdv' is a meta-constant*)

val isolate_bdv =

    Rls{id = "isolate_bdv", preconds = [], rew_ord = ("e_rew_ord",e_rew_ord),

	erls=tval_rls,srls = e_rls, calc=[],(*asm_thm = [], *)

	rules = 

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

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

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

	 Thm ("mult_square",num_str @{mult_square}),

	 Thm ("constant_square",num_str @{constant_square}),

	 Thm ("constant_mult_square",num_str @{constant_mult_square})

	 ],

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

			  "empty_script")

	}:rls;      

(* association list for calculate_, calculate

   "op +" etc. not usable in scripts *)

val calclist = 

    [

     (*as Tools.ML*)

     ("Vars"    ,("Tools.Vars"    ,eval_var "#Vars_")),

     ("matches",("Tools.matches",eval_matches "#matches_")),

     ("lhs"    ,("Tools.lhs"    ,eval_lhs "")),

     (*aus Atools.ML*)

     ("PLUS"    ,("op +"        ,eval_binop "#add_")),

     ("TIMES"   ,("op *"        ,eval_binop "#mult_")),

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

     ("POWER"  ,("Atools.pow"  ,eval_binop "#power_")),

     ("is_const",("Atools.is'_const",eval_const "#is_const_")),

     ("le"      ,("op <"        ,eval_equ "#less_")),

     ("leq"     ,("op <="       ,eval_equ "#less_equal_")),

     ("ident"   ,("Atools.ident",eval_ident "#ident_")),

     (*von hier (ehem.SqRoot*)

     ("sqrt"    ,("NthRoot.sqrt"   ,eval_sqrt "#sqrt_")),

     ("Test.is_root_free",("is'_root'_free", eval_root_free"#is_root_free_e")),

     ("Test.contains_root",("contains'_root",

			    eval_contains_root"#contains_root_"))

     ];

ruleset' := overwritelthy @{theory} (!ruleset',

  [("Test_simplify", prep_rls Test_simplify),

   ("tval_rls", prep_rls tval_rls),

   ("isolate_root", prep_rls isolate_root),

   ("isolate_bdv", prep_rls isolate_bdv),

   ("matches", 

    prep_rls (append_rls "matches" testerls 

			 [Calc ("Tools.matches",eval_matches "#matches_")]))

   ]);

(** problem types **)

store_pbt

 (prep_pbt thy "pbl_test" [] e_pblID

 (["test"],

  [],

  e_rls, NONE, []));

store_pbt

 (prep_pbt thy "pbl_test_equ" [] e_pblID

 (["equation","test"],

  [("#Given" ,["equality e_e","solveFor v_v"]),

   ("#Where" ,["matches (?a = ?b) e_e"]),

   ("#Find"  ,["solutions v_i"])

  ],

  assoc_rls "matches",

  SOME "solve (e_e::bool, v_v)", []));

store_pbt

 (prep_pbt thy "pbl_test_uni" [] e_pblID

 (["univariate","equation","test"],

  [("#Given" ,["equality e_e","solveFor v_v"]),

   ("#Where" ,["matches (?a = ?b) e_e"]),

   ("#Find"  ,["solutions v_i"])

  ],

  assoc_rls "matches",

  SOME "solve (e_e::bool, v_v)", []));

store_pbt

 (prep_pbt thy "pbl_test_uni_lin" [] e_pblID

 (["linear","univariate","equation","test"],

  [("#Given" ,["equality e_e","solveFor v_v"]),

   ("#Where" ,["(matches (   v_v = 0) e_e) | (matches (   ?b*v_ = 0) e_e) |" ^

	       "(matches (?a+v_ = 0) e_e) | (matches (?a+?b*v_ = 0) e_e)  "]),

   ("#Find"  ,["solutions v_i"])

  ],

  assoc_rls "matches", 

  SOME "solve (e_e::bool, v_v)", [["Test","solve_linear"]]));

(*25.8.01 ------

store_pbt

 (prep_pbt thy

 (["thy"],

  [("#Given" ,"boolTestGiven g_"),

   ("#Find"  ,"boolTestFind f_")

  ],

  []));

store_pbt

 (prep_pbt thy

 (["testeq","thy"],

  [("#Given" ,"boolTestGiven g_"),

   ("#Find"  ,"boolTestFind f_")

  ],

  []));

val ttt = (term_of o the o (parse (theory "Isac"))) "(matches (   v_v = 0) e_e)";

 ------ 25.8.01*)

(** methods **)

store_met

 (prep_met (theory "Diff") "met_test" [] e_metID

 (["Test"],

   [],

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

    crls=Atools_erls, nrls=e_rls(*,

    asm_rls=[],asm_thm=[]*)}, "empty_script"));

(*

store_met

 (prep_met (theory "Script")

 (e_metID,(*empty method*)

   [],

   {rew_ord'="e_rew_ord",rls'=tval_rls,srls=e_rls,prls=e_rls,calc=[],

    asm_rls=[],asm_thm=[]},

   "Undef"));*)

store_met

 (prep_met thy "met_test_solvelin" [] e_metID

 (["Test","solve_linear"]:metID,

  [("#Given" ,["equality e_e","solveFor v_v"]),

    ("#Where" ,["matches (?a = ?b) e_e"]),

    ("#Find"  ,["solutions v_i"])

    ],

   {rew_ord'="e_rew_ord",rls'=tval_rls,srls=e_rls,

    prls=assoc_rls "matches",

    calc=[],

    crls=tval_rls, nrls=Test_simplify},

 "Script Solve_linear (e_e::bool) (v_v::real)=             " ^

 "(let e_e =" ^

 "  Repeat" ^

 "    (((Rewrite_Set_Inst [(bdv,v_v::real)] isolate_bdv False) @@" ^

 "      (Rewrite_Set Test_simplify False))) e_e" ^

 " in [e_::bool])"

 )

(*, prep_met thy (*test for equations*)

 (["Test","testeq"]:metID,

  [("#Given" ,["boolTestGiven g_"]),

   ("#Find"  ,["boolTestFind f_"])

    ],

  {rew_ord'="e_rew_ord",rls'="tval_rls",asm_rls=[],

   asm_thm=[("square_equation_left","")]},

 "Script Testeq (eq_::bool) =                                         " ^

   "Repeat                                                            " ^

   " (let e_e = Try (Repeat (Rewrite rroot_square_inv False eq_));      " ^

   "      e_e = Try (Repeat (Rewrite square_equation_left True e_e)); " ^

   "      e_e = Try (Repeat (Rewrite rmult_0 False e_e))                " ^

   "   in e_e) Until (is_root_free e_e)" (*deleted*)

 )

, ---------27.4.02*)

);

ruleset' := overwritelthy @{theory} (!ruleset',

  [("norm_equation", prep_rls norm_equation),

   ("ac_plus_times", prep_rls ac_plus_times),

   ("rearrange_assoc", prep_rls rearrange_assoc)

   ]);

fun bin_o (Const (op_,(Type ("fun",

	   [Type (s2,[]),Type ("fun",

	    [Type (s4,tl4),Type (s5,tl5)])])))) = 

    if (s2=s4)andalso(s4=s5)then[op_]else[]

    | bin_o _                                   = [];

fun bin_op (t1 $ t2) = union op = (bin_op t1) (bin_op t2)

  | bin_op t         =  bin_o t;

fun is_bin_op t = ((bin_op t)<>[]);

fun bin_op_arg1 ((Const (op_,(Type ("fun",

	   [Type (s2,[]),Type ("fun",

	    [Type (s4,tl4),Type (s5,tl5)])]))))$ arg1 $ arg2) = 

    arg1;

fun bin_op_arg2 ((Const (op_,(Type ("fun",

	   [Type (s2,[]),Type ("fun",

	    [Type (s4,tl4),Type (s5,tl5)])]))))$ arg1 $ arg2) = 

    arg2;

exception NO_EQUATION_TERM;

fun is_equation ((Const ("op =",(Type ("fun",

		 [Type (_,[]),Type ("fun",

		  [Type (_,[]),Type ("bool",[])])])))) $ _ $ _) 

                  = true

  | is_equation _ = false;

fun equ_lhs ((Const ("op =",(Type ("fun",

		 [Type (_,[]),Type ("fun",

		  [Type (_,[]),Type ("bool",[])])])))) $ l $ r) 

              = l

  | equ_lhs _ = raise NO_EQUATION_TERM;

fun equ_rhs ((Const ("op =",(Type ("fun",

		 [Type (_,[]),Type ("fun",

		  [Type (_,[]),Type ("bool",[])])])))) $ l $ r) 

              = r

  | equ_rhs _ = raise NO_EQUATION_TERM;

fun atom (Const (_,Type (_,[])))           = true

  | atom (Free  (_,Type (_,[])))           = true

  | atom (Var   (_,Type (_,[])))           = true

(*| atom (_     (_,"?DUMMY"   ))           = true ..ML-error *)

  | atom((Const ("Bin.integ_of_bin",_)) $ _) = true

  | atom _                                 = false;

fun varids (Const  (s,Type (_,[])))         = [strip_thy s]

  | varids (Free   (s,Type (_,[])))         = if is_no s then []

					      else [strip_thy s]

  | varids (Var((s,_),Type (_,[])))         = [strip_thy s]

(*| varids (_      (s,"?DUMMY"   ))         =   ..ML-error *)

  | varids((Const ("Bin.integ_of_bin",_)) $ _)= [](*8.01: superfluous?*)

  | varids (Abs(a,T,t)) = union op = [a] (varids t)

  | varids (t1 $ t2) = union op = (varids t1) (varids t2)

  | varids _         = [];

(*> val t = term_of (hd (parse Diophant.thy "x"));

val t = Free ("x","?DUMMY") : term

> varids t;

val it = [] : string list          [] !!! *)

fun bin_ops_only ((Const op_) $ t1 $ t2) = 

    if(is_bin_op (Const op_))

    then(bin_ops_only t1)andalso(bin_ops_only t2)

    else false

  | bin_ops_only t =

    if atom t then true else bin_ops_only t;

fun polynomial opl t bdVar = (* bdVar TODO *)

    subset op = (bin_op t, opl) andalso (bin_ops_only t);

fun poly_equ opl bdVar t = is_equation t (* bdVar TODO *) 

    andalso polynomial opl (equ_lhs t) bdVar 

    andalso polynomial opl (equ_rhs t) bdVar

    andalso (subset op = (varids bdVar, varids (equ_lhs t)) orelse

             subset op = (varids bdVar, varids (equ_lhs t)));

(*fun max is =

    let fun max_ m [] = m 

	  | max_ m (i::is) = if m<i then max_ i is else max_ m is;

    in max_ (hd is) is end;

> max [1,5,3,7,4,2];

val it = 7 : int  *)

fun max (a,b) = if a < b then b else a;

fun degree addl mul bdVar t =

let

fun deg _ _ v (Const  (s,Type (_,[])))         = if v=strip_thy s then 1 else 0

  | deg _ _ v (Free   (s,Type (_,[])))         = if v=strip_thy s then 1 else 0

  | deg _ _ v (Var((s,_),Type (_,[])))         = if v=strip_thy s then 1 else 0

(*| deg _ _ v (_     (s,"?DUMMY"   ))          =   ..ML-error *) 

  | deg _ _ v((Const ("Bin.integ_of_bin",_)) $ _ )= 0 

  | deg addl mul v (h $ t1 $ t2) =

    if subset op = (bin_op h, addl)

    then max (deg addl mul v t1  ,deg addl mul v t2)

    else (*mul!*)(deg addl mul v t1)+(deg addl mul v t2)

in if polynomial (addl @ [mul]) t bdVar

   then SOME (deg addl mul (id_of bdVar) t) else (NONE:int option)

end;

fun degree_ addl mul bdVar t = (* do not export *)

    let fun opt (SOME i)= i

	  | opt  NONE   = 0

in opt (degree addl mul bdVar t) end;

fun linear addl mul t bdVar = (degree_ addl mul bdVar t)<2;

fun linear_equ addl mul bdVar t =

    if is_equation t 

    then let val degl = degree_ addl mul bdVar (equ_lhs t);

	     val degr = degree_ addl mul bdVar (equ_rhs t)

	 in if (degl>0 orelse degr>0)andalso max(degl,degr)<2

		then true else false

	 end

    else false;

(* strip_thy op_  before *)

fun is_div_op (dv,(Const (op_,(Type ("fun",

	   [Type (s2,[]),Type ("fun",

	    [Type (s4,tl4),Type (s5,tl5)])])))) )= (dv = strip_thy op_)

  | is_div_op _ = false;

fun is_denom bdVar div_op t =

    let fun is bool[v]dv (Const  (s,Type(_,[])))= bool andalso(if v=strip_thy s then true else false)

	  | is bool[v]dv (Free   (s,Type(_,[])))= bool andalso(if v=strip_thy s then true else false) 

	  | is bool[v]dv (Var((s,_),Type(_,[])))= bool andalso(if v=strip_thy s then true else false)

	  | is bool[v]dv((Const ("Bin.integ_of_bin",_)) $ _) = false

	  | is bool[v]dv (h$n$d) = 

	      if is_div_op(dv,h) 

	      then (is false[v]dv n)orelse(is true[v]dv d)

	      else (is bool [v]dv n)orelse(is bool[v]dv d)

in is false (varids bdVar) (strip_thy div_op) t end;

fun rational t div_op bdVar = 

    is_denom bdVar div_op t andalso bin_ops_only t;

(** problem types **)

store_pbt

 (prep_pbt thy "pbl_test_uni_plain2" [] e_pblID

 (["plain_square","univariate","equation","test"],

  [("#Given" ,["equality e_e","solveFor v_v"]),

   ("#Where" ,["(matches (?a + ?b*v_ ^^^2 = 0) e_e) |" ^

	       "(matches (     ?b*v_ ^^^2 = 0) e_e) |" ^

	       "(matches (?a +    v_v ^^^2 = 0) e_e) |" ^

	       "(matches (        v_v ^^^2 = 0) e_e)"]),

   ("#Find"  ,["solutions v_i"])

  ],

  assoc_rls "matches", 

  SOME "solve (e_e::bool, v_v)", [["Test","solve_plain_square"]]));

(*

 val e_e = (term_of o the o (parse thy)) "e_::bool";

 val ve = (term_of o the o (parse thy)) "4 + 3*x^^^2 = 0";

 val env = [(e_,ve)];

 val pre = (term_of o the o (parse thy))

	      "(matches (a + b*v_ ^^^2 = 0, e_e::bool)) |" ^

	      "(matches (    b*v_ ^^^2 = 0, e_e::bool)) |" ^

	      "(matches (a +   v_v ^^^2 = 0, e_e::bool)) |" ^

	      "(matches (      v_v ^^^2 = 0, e_e::bool))";

 val prei = subst_atomic env pre;

 val cpre = (cterm_of thy) prei;

 val SOME (ct,_) = rewrite_set_ thy false tval_rls cpre;

val ct = "True | False | False | False" : cterm 

> val SOME (ct,_) = rewrite_ thy sqrt_right tval_rls false or_false ct;

> val SOME (ct,_) = rewrite_ thy sqrt_right tval_rls false or_false ct;

> val SOME (ct,_) = rewrite_ thy sqrt_right tval_rls false or_false ct;

val ct = "True" : cterm

*)

store_pbt

 (prep_pbt thy "pbl_test_uni_poly" [] e_pblID

 (["polynomial","univariate","equation","test"],

  [("#Given" ,["equality (v_ ^^^2 + p_ * v_v + q__ = 0)","solveFor v_v"]),

   ("#Where" ,["False"]),

   ("#Find"  ,["solutions v_i"]) 

  ],

  e_rls, SOME "solve (e_e::bool, v_v)", []));

store_pbt

 (prep_pbt thy "pbl_test_uni_poly_deg2" [] e_pblID

 (["degree_two","polynomial","univariate","equation","test"],

  [("#Given" ,["equality (v_ ^^^2 + p_ * v_v + q__ = 0)","solveFor v_v"]),

   ("#Find"  ,["solutions v_i"]) 

  ],

  e_rls, SOME "solve (v_ ^^^2 + p_ * v_v + q__ = 0, v_v)", []));

store_pbt

 (prep_pbt thy "pbl_test_uni_poly_deg2_pq" [] e_pblID

 (["pq_formula","degree_two","polynomial","univariate","equation","test"],

  [("#Given" ,["equality (v_ ^^^2 + p_ * v_v + q__ = 0)","solveFor v_v"]),

   ("#Find"  ,["solutions v_i"]) 

  ],

  e_rls, SOME "solve (v_ ^^^2 + p_ * v_v + q__ = 0, v_v)", []));

store_pbt

 (prep_pbt thy "pbl_test_uni_poly_deg2_abc" [] e_pblID

 (["abc_formula","degree_two","polynomial","univariate","equation","test"],

  [("#Given" ,["equality (a_ * x ^^^2 + b_ * x + c_ = 0)","solveFor v_v"]),

   ("#Find"  ,["solutions v_i"]) 

  ],

  e_rls, SOME "solve (a_ * x ^^^2 + b_ * x + c_ = 0, v_v)", []));

store_pbt

 (prep_pbt thy "pbl_test_uni_root" [] e_pblID

 (["squareroot","univariate","equation","test"],

  [("#Given" ,["equality e_e","solveFor v_v"]),

   ("#Where" ,["contains_root (e_e::bool)"]),

   ("#Find"  ,["solutions v_i"]) 

  ],

  append_rls "contains_root" e_rls [Calc ("Test.contains'_root",

			  eval_contains_root "#contains_root_")], 

  SOME "solve (e_e::bool, v_v)", [["Test","square_equation"]]));

store_pbt

 (prep_pbt thy "pbl_test_uni_norm" [] e_pblID

 (["normalize","univariate","equation","test"],

  [("#Given" ,["equality e_e","solveFor v_v"]),

   ("#Where" ,[]),

   ("#Find"  ,["solutions v_i"]) 

  ],

  e_rls, SOME "solve (e_e::bool, v_v)", [["Test","norm_univar_equation"]]));

store_pbt

 (prep_pbt thy "pbl_test_uni_roottest" [] e_pblID

 (["sqroot-test","univariate","equation","test"],

  [("#Given" ,["equality e_e","solveFor v_v"]),

   (*("#Where" ,["contains_root (e_e::bool)"]),*)

   ("#Find"  ,["solutions v_i"]) 

  ],

  e_rls, SOME "solve (e_e::bool, v_v)", []));

(*

(#ppc o get_pbt) ["sqroot-test","univariate","equation"];

  *)

store_met

 (prep_met thy  "met_test_sqrt" [] e_metID

(*root-equation, version for tests before 8.01.01*)

 (["Test","sqrt-equ-test"]:metID,

  [("#Given" ,["equality e_e","solveFor v_v"]),

   ("#Where" ,["contains_root (e_e::bool)"]),

   ("#Find"  ,["solutions v_i"])

   ],

  {rew_ord'="e_rew_ord",rls'=tval_rls,

   srls =append_rls "srls_contains_root" e_rls 

		    [Calc ("Test.contains'_root",eval_contains_root "")],

   prls =append_rls "prls_contains_root" e_rls 

		    [Calc ("Test.contains'_root",eval_contains_root "")],

   calc=[],

   crls=tval_rls, nrls=e_rls(*,asm_rls=[],

   asm_thm=[("square_equation_left",""),

	    ("square_equation_right","")]*)},

 "Script Solve_root_equation (e_e::bool) (v_v::real) =  " ^

 "(let e_e = " ^

 "   ((While (contains_root e_e) Do" ^

 "      ((Rewrite square_equation_left True) @@" ^

 "       (Try (Rewrite_Set Test_simplify False)) @@" ^

 "       (Try (Rewrite_Set rearrange_assoc False)) @@" ^

 "       (Try (Rewrite_Set isolate_root False)) @@" ^

 "       (Try (Rewrite_Set Test_simplify False)))) @@" ^

 "    (Try (Rewrite_Set norm_equation False)) @@" ^

 "    (Try (Rewrite_Set Test_simplify False)) @@" ^

 "    (Rewrite_Set_Inst [(bdv,v_v::real)] isolate_bdv False) @@" ^

 "    (Try (Rewrite_Set Test_simplify False)))" ^

 "   e_e" ^

 " in [e_::bool])"

  ));

store_met

 (prep_met thy  "met_test_sqrt2" [] e_metID

(*root-equation ... for test-*.sml until 8.01*)

 (["Test","squ-equ-test2"]:metID,

  [("#Given" ,["equality e_e","solveFor v_v"]),

   ("#Find"  ,["solutions v_i"])

   ],

  {rew_ord'="e_rew_ord",rls'=tval_rls,

   srls = append_rls "srls_contains_root" e_rls 

		     [Calc ("Test.contains'_root",eval_contains_root"")],

   prls=e_rls,calc=[],

   crls=tval_rls, nrls=e_rls(*,asm_rls=[],

   asm_thm=[("square_equation_left",""),

	    ("square_equation_right","")]*)},

 "Script Solve_root_equation (e_e::bool) (v_v::real) =  " ^

 "(let e_e = " ^

 "   ((While (contains_root e_e) Do" ^

 "      ((Rewrite square_equation_left True) @@" ^

 "       (Try (Rewrite_Set Test_simplify False)) @@" ^

 "       (Try (Rewrite_Set rearrange_assoc False)) @@" ^

 "       (Try (Rewrite_Set isolate_root False)) @@" ^

 "       (Try (Rewrite_Set Test_simplify False)))) @@" ^

 "    (Try (Rewrite_Set norm_equation False)) @@" ^

 "    (Try (Rewrite_Set Test_simplify False)) @@" ^

 "    (Rewrite_Set_Inst [(bdv,v_v::real)] isolate_bdv False) @@" ^

 "    (Try (Rewrite_Set Test_simplify False)))" ^

 "   e_e;" ^

 "  (L_::bool list) = Tac subproblem_equation_dummy;          " ^

 "  L_L = Tac solve_equation_dummy                             " ^

 "  in Check_elementwise L_L {(v_v::real). Assumptions})"

  ));

store_met

 (prep_met thy "met_test_squ_sub" [] e_metID

(*tests subproblem fixed linear*)

 (["Test","squ-equ-test-subpbl1"]:metID,

  [("#Given" ,["equality e_e","solveFor v_v"]),

   ("#Find"  ,["solutions v_i"])

   ],

  {rew_ord'="e_rew_ord",rls'=tval_rls,srls=e_rls,prls=e_rls,calc=[],

    crls=tval_rls, nrls=Test_simplify},

  "Script Solve_root_equation (e_e::bool) (v_v::real) =  " ^

   " (let e_e = ((Try (Rewrite_Set norm_equation False)) @@              " ^

   "            (Try (Rewrite_Set Test_simplify False))) e_e;              " ^

   "(L_::bool list) = (SubProblem (Test_,[linear,univariate,equation,test]," ^

   "                    [Test,solve_linear]) [BOOL e_e, REAL v_])" ^

   "in Check_elementwise L_L {(v_v::real). Assumptions})"

  ));

store_met

 (prep_met thy "met_test_squ_sub2" [] e_metID

 (*tests subproblem fixed degree 2*)

 (["Test","squ-equ-test-subpbl2"]:metID,

  [("#Given" ,["equality e_e","solveFor v_v"]),

   ("#Find"  ,["solutions v_i"])

   ],

  {rew_ord'="e_rew_ord",rls'=tval_rls,srls=e_rls,prls=e_rls,calc=[],

    crls=tval_rls, nrls=e_rls(*,

   asm_rls=[],asm_thm=[("square_equation_left",""),

	    ("square_equation_right","")]*)},

   "Script Solve_root_equation (e_e::bool) (v_v::real) =  " ^

   " (let e_e = Try (Rewrite_Set norm_equation False) e_e;              " ^

   "(L_::bool list) = (SubProblem (Test_,[linear,univariate,equation,test]," ^

   "                    [Test,solve_by_pq_formula]) [BOOL e_e, REAL v_])" ^

   "in Check_elementwise L_L {(v_v::real). Assumptions})"

   )); 

store_met

 (prep_met thy "met_test_squ_nonterm" [] e_metID

 (*root-equation: see foils..., but notTerminating*)

 (["Test","square_equation...notTerminating"]:metID,

  [("#Given" ,["equality e_e","solveFor v_v"]),

   ("#Find"  ,["solutions v_i"])

   ],

  {rew_ord'="e_rew_ord",rls'=tval_rls,

   srls = append_rls "srls_contains_root" e_rls 

		     [Calc ("Test.contains'_root",eval_contains_root"")],

   prls=e_rls,calc=[],

    crls=tval_rls, nrls=e_rls(*,asm_rls=[],

   asm_thm=[("square_equation_left",""),

	    ("square_equation_right","")]*)},

 "Script Solve_root_equation (e_e::bool) (v_v::real) =  " ^

 "(let e_e = " ^

 "   ((While (contains_root e_e) Do" ^

 "      ((Rewrite square_equation_left True) @@" ^

 "       (Try (Rewrite_Set Test_simplify False)) @@" ^