
neuper@37906: (* attempts to perserve binary minus as wanted by Austrian teachers
neuper@37906:    WN071207
neuper@37906:    (c) due to copyright terms
neuper@37906: *)
neuper@37906: 
neuper@37950: theory PolyMinus imports (*Poly// due to "is_ratpolyexp" in...*) Rational begin
neuper@37906: 
neuper@37906: consts
neuper@37906: 
neuper@37906:   (*predicates for conditions in rewriting*)
neuper@37906:   kleiner     :: "['a, 'a] => bool" 	("_ kleiner _") 
walther@60278:   ist_monom  :: "'a => bool"		("_ ist'_monom")
neuper@37906: 
neuper@37906:   (*the CAS-command*)
neuper@37906:   Probe       :: "[bool, bool list] => bool"  
neuper@37906: 	(*"Probe (3*a+2*b+a = 4*a+2*b) [a=1,b=2]"*)
neuper@37906: 
neuper@37906:   (*descriptions for the pbl and met*)
neuper@37950:   Pruefe      :: "bool => una"
neuper@37950:   mitWert     :: "bool list => tobooll"
neuper@37950:   Geprueft    :: "bool => una"
neuper@37906: 
neuper@52148: axiomatization where
neuper@37906: 
neuper@52148:   null_minus:            "0 - a = -a" and
neuper@52148:   vor_minus_mal:         "- a * b = (-a) * b" and
neuper@37906: 
neuper@37906:   (*commute with invariant (a.b).c -association*)
neuper@37980:   tausche_plus:		"[| b ist_monom; a kleiner b  |] ==> 
neuper@52148: 			 (b + a) = (a + b)" and
neuper@37980:   tausche_minus:		"[| b ist_monom; a kleiner b  |] ==> 
neuper@52148: 			 (b - a) = (-a + b)" and
neuper@37980:   tausche_vor_plus:	"[| b ist_monom; a kleiner b  |] ==> 
neuper@52148: 			 (- b + a) = (a - b)" and
neuper@37980:   tausche_vor_minus:	"[| b ist_monom; a kleiner b  |] ==> 
neuper@52148: 			 (- b - a) = (-a - b)" and
walther@60269: (*Ambiguous input\<^here> produces 3 parse trees -----------------------------\\*)
neuper@52148:   tausche_plus_plus:	"b kleiner c ==> (a + c + b) = (a + b + c)" and
neuper@52148:   tausche_plus_minus:	"b kleiner c ==> (a + c - b) = (a - b + c)" and
neuper@52148:   tausche_minus_plus:	"b kleiner c ==> (a - c + b) = (a + b - c)" and
neuper@52148:   tausche_minus_minus:	"b kleiner c ==> (a - c - b) = (a - b - c)" and
walther@60269: (*Ambiguous input\<^here> produces 3 parse trees -----------------------------//*)
neuper@37906: 
neuper@37906:   (*commute with invariant (a.b).c -association*)
neuper@37980:   tausche_mal:		"[| b is_atom; a kleiner b  |] ==> 
neuper@52148: 			 (b * a) = (a * b)" and
neuper@37980:   tausche_vor_mal:	"[| b is_atom; a kleiner b  |] ==> 
neuper@52148: 			 (-b * a) = (-a * b)" and
neuper@37980:   tausche_mal_mal:	"[| c is_atom; b kleiner c  |] ==> 
neuper@52148: 			 (x * c * b) = (x * b * c)" and
walther@60242:   x_quadrat:             "(x * a) * a = x * a \<up> 2" and
neuper@37906: 
neuper@37906: 
walther@60385:   subtrahiere:               "[| l is_num; m is_num |] ==>  
neuper@52148: 			     m * v - l * v = (m - l) * v" and
walther@60385:   subtrahiere_von_1:         "[| l is_num |] ==>  
neuper@52148: 			     v - l * v = (1 - l) * v" and
walther@60385:   subtrahiere_1:             "[| l is_num; m is_num |] ==>  
neuper@52148: 			     m * v - v = (m - 1) * v" and
neuper@37906: 
walther@60385:   subtrahiere_x_plus_minus:  "[| l is_num; m is_num |] ==>  
neuper@52148: 			     (x + m * v) - l * v = x + (m - l) * v" and
walther@60385:   subtrahiere_x_plus1_minus: "[| l is_num |] ==>  
neuper@52148: 			     (x + v) - l * v = x + (1 - l) * v" and
walther@60385:   subtrahiere_x_plus_minus1: "[| m is_num |] ==>  
neuper@52148: 			     (x + m * v) - v = x + (m - 1) * v" and
neuper@37906: 
walther@60385:   subtrahiere_x_minus_plus:  "[| l is_num; m is_num |] ==>  
neuper@52148: 			     (x - m * v) + l * v = x + (-m + l) * v" and
walther@60385:   subtrahiere_x_minus1_plus: "[| l is_num |] ==>  
neuper@52148: 			     (x - v) + l * v = x + (-1 + l) * v" and
walther@60385:   subtrahiere_x_minus_plus1: "[| m is_num |] ==>  
neuper@52148: 			     (x - m * v) + v = x + (-m + 1) * v" and
neuper@37906: 
walther@60385:   subtrahiere_x_minus_minus: "[| l is_num; m is_num |] ==>  
neuper@52148: 			     (x - m * v) - l * v = x + (-m - l) * v" and
walther@60385:   subtrahiere_x_minus1_minus:"[| l is_num |] ==>  
neuper@52148: 			     (x - v) - l * v = x + (-1 - l) * v" and
walther@60385:   subtrahiere_x_minus_minus1:"[| m is_num |] ==>  
neuper@52148: 			     (x - m * v) - v = x + (-m - 1) * v" and
neuper@37906: 
neuper@37906: 
walther@60385:   addiere_vor_minus:         "[| l is_num; m is_num |] ==>  
neuper@52148: 			     - (l * v) +  m * v = (-l + m) * v" and
walther@60385:   addiere_eins_vor_minus:    "[| m is_num |] ==>  
neuper@52148: 			     -  v +  m * v = (-1 + m) * v" and
walther@60385:   subtrahiere_vor_minus:     "[| l is_num; m is_num |] ==>  
neuper@52148: 			     - (l * v) -  m * v = (-l - m) * v" and
walther@60385:   subtrahiere_eins_vor_minus:"[| m is_num |] ==>  
neuper@52148: 			     -  v -  m * v = (-1 - m) * v" and
neuper@37906: 
walther@60269: (*Ambiguous input\<^here> produces 3 parse trees -----------------------------\\*)
neuper@52148:   vorzeichen_minus_weg1:      "l kleiner 0 ==> a + l * b = a - -1*l * b" and
neuper@52148:   vorzeichen_minus_weg2:      "l kleiner 0 ==> a - l * b = a + -1*l * b" and
neuper@52148:   vorzeichen_minus_weg3:      "l kleiner 0 ==> k + a - l * b = k + a + -1*l * b" and
neuper@52148:   vorzeichen_minus_weg4:      "l kleiner 0 ==> k - a - l * b = k - a + -1*l * b" and
walther@60269: (*Ambiguous input\<^here> produces 3 parse trees -----------------------------//*)
neuper@37906: 
walther@59877:   (*klammer_plus_plus = (add.assoc RS sym)*)
neuper@52148:   klammer_plus_minus:          "a + (b - c) = (a + b) - c" and
neuper@52148:   klammer_minus_plus:          "a - (b + c) = (a - b) - c" and
neuper@52148:   klammer_minus_minus:         "a - (b - c) = (a - b) + c" and
neuper@37906: 
neuper@52148:   klammer_mult_minus:          "a * (b - c) = a * b - a * c" and
neuper@37980:   klammer_minus_mult:          "(b - c) * a = b * a - c * a"
neuper@37906: 
wneuper@59472: ML \<open>
neuper@37950: (** eval functions **)
neuper@37906: 
neuper@37950: (*. get the identifier from specific monomials; see fun ist_monom .*)
walther@60351: fun Free_to_string (Free (str, _)) = str
walther@60351:   | Free_to_string t =
walther@60351:     if TermC.is_num t then TermC.string_of_num t else "|||||||||||||"(*the "largest" string*);
walther@60351: 
walther@60351: (* quick and dirty solution just before a field test *)
walther@60351: fun identifier (Free (id,_)) = id                                   (* _a_                   *)
walther@60351:   | identifier (Const (\<^const_name>\<open>times\<close>, _) $ t1 $ t2) =           (* 2*_a_, a*_b_, 3*a*_b_ *)
walther@60351:     if TermC.is_atom t2 
walther@60351:     then Free_to_string t2
walther@60351:     else
walther@60351:       (case t1 of
walther@60351:         Const (\<^const_name>\<open>times\<close>, _) $ num $ t1' =>                 (* 3*_a_ \<up> 2             *)
walther@60351:           if TermC.is_atom num andalso TermC.is_atom t1' then Free_to_string t2
walther@60351:           else "|||||||||||||"
walther@60351:       | _ => 
walther@60351:         (case t2 of
wenzelm@60405:           Const (\<^const_name>\<open>realpow\<close>, _) $ base $ exp => 
walther@60351:             if TermC.is_atom base andalso TermC.is_atom exp then
walther@60351:               if TermC.is_num base then "|||||||||||||"
walther@60351:               else Free_to_string base
walther@60351:             else "|||||||||||||"
walther@60351:       | _ => "|||||||||||||"))
wenzelm@60405:   | identifier (Const (\<^const_name>\<open>realpow\<close>, _) $ base $ exp) =         (* _a_\<up>2, _3_^2          *)
walther@60351:     if TermC.is_atom base andalso TermC.is_atom exp then Free_to_string base
neuper@37950:     else "|||||||||||||"
walther@60351:   | identifier t =                                                   (* 12                    *)
walther@60351:     if TermC.is_num t
walther@60351:     then TermC.string_of_num t
walther@60351:     else "|||||||||||||" (*the "largest" string*);
neuper@37950: 
neuper@37950: (*("kleiner", ("PolyMinus.kleiner", eval_kleiner ""))*)
neuper@37950: (* order "by alphabet" w.r.t. var: num < (var | num*var) > (var*var | ..) *)
walther@60335: fun eval_kleiner _ _ (p as (Const (\<^const_name>\<open>PolyMinus.kleiner\<close>,_) $ a $ b)) _  =
walther@60325:     if TermC.is_num b then
walther@60325:   	  if TermC.is_num a then (*123 kleiner 32 = True !!!*)
walther@60325:   	    if TermC.num_of_term a < TermC.num_of_term b then 
walther@60325:   		    SOME ((UnparseC.term p) ^ " = True", 
walther@60325:             HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))
walther@60325:   	    else SOME ((UnparseC.term p) ^ " = False",
walther@60325:   			  HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))
walther@60325:   	  else (* -1 * -2 kleiner 0 *)
walther@60325:   	    SOME ((UnparseC.term p) ^ " = False",
walther@60325:   		    HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))
neuper@37950:     else
walther@60325:     	if identifier a < identifier b then 
walther@60325:     	  SOME ((UnparseC.term p) ^ " = True",
walther@60325:     		  HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))
walther@60325:     	else SOME ((UnparseC.term p) ^ " = False",
walther@60325:     		HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))
neuper@37950:   | eval_kleiner _ _ _ _ =  NONE;
neuper@37950: 
walther@60325: fun ist_monom t =
walther@60325:   if TermC.is_atom t then true
walther@60325:   else
walther@60325:     case t of
walther@60335:       Const (\<^const_name>\<open>Groups.times_class.times\<close>, _) $ t1 $ t2 =>
walther@60325:         ist_monom t1 andalso ist_monom t2
wenzelm@60405:     | Const (\<^const_name>\<open>realpow\<close>, _) $ t1 $ t2 =>
walther@60325:         ist_monom t1 andalso ist_monom t2
walther@60325:     | _ => false
neuper@37950: 
neuper@37950: (* is this a univariate monomial ? *)
walther@60278: (*("ist_monom", ("PolyMinus.ist_monom", eval_ist_monom ""))*)
walther@60335: fun eval_ist_monom _ _ (p as (Const (\<^const_name>\<open>PolyMinus.ist_monom\<close>,_) $ a)) _  =
neuper@37950:     if ist_monom a  then 
walther@59868: 	SOME ((UnparseC.term p) ^ " = True",
wneuper@59390: 	      HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))
walther@59868:     else SOME ((UnparseC.term p) ^ " = False",
wneuper@59390: 	       HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))
neuper@37950:   | eval_ist_monom _ _ _ _ =  NONE;
neuper@37950: 
neuper@37950: 
neuper@37950: (** rewrite order **)
neuper@37950: 
neuper@37950: (** rulesets **)
neuper@37950: 
neuper@37950: val erls_ordne_alphabetisch =
walther@60358:   Rule_Set.append_rules "erls_ordne_alphabetisch" Rule_Set.empty [
walther@60358:     \<^rule_eval>\<open>kleiner\<close> (eval_kleiner ""),
walther@60358:     \<^rule_eval>\<open>ist_monom\<close> (eval_ist_monom "")];
neuper@37950: 
neuper@37950: val ordne_alphabetisch = 
walther@59851:   Rule_Def.Repeat{id = "ordne_alphabetisch", preconds = [], 
Walther@60509:     rew_ord = ("dummy_ord", Rewrite_Ord.function_empty), srls = Rule_Set.Empty, calc = [], errpatts = [],
walther@60358:     erls = erls_ordne_alphabetisch, 
walther@60358:     rules = [
walther@60358:       \<^rule_thm>\<open>tausche_plus\<close>, (*"b kleiner a ==> (b + a) = (a + b)"*)
walther@60358:       \<^rule_thm>\<open>tausche_minus\<close>, (*"b kleiner a ==> (b - a) = (-a + b)"*)
walther@60358:       \<^rule_thm>\<open>tausche_vor_plus\<close>, (*"[| b ist_monom; a kleiner b  |] ==> (- b + a) = (a - b)"*)
walther@60358:       \<^rule_thm>\<open>tausche_vor_minus\<close>, (*"[| b ist_monom; a kleiner b  |] ==> (- b - a) = (-a - b)"*)
walther@60358:       \<^rule_thm>\<open>tausche_plus_plus\<close>, (*"c kleiner b ==> (a + c + b) = (a + b + c)"*)
walther@60358:       \<^rule_thm>\<open>tausche_plus_minus\<close>, (*"c kleiner b ==> (a + c - b) = (a - b + c)"*)
walther@60358:       \<^rule_thm>\<open>tausche_minus_plus\<close>, (*"c kleiner b ==> (a - c + b) = (a + b - c)"*)
walther@60358:       \<^rule_thm>\<open>tausche_minus_minus\<close>], (*"c kleiner b ==> (a - c - b) = (a - b - c)"*)
walther@60358:     scr = Rule.Empty_Prog};
neuper@37950: 
neuper@37950: val fasse_zusammen = 
walther@60358:   Rule_Def.Repeat{id = "fasse_zusammen", preconds = [], 
Walther@60509: 	  rew_ord = ("dummy_ord", Rewrite_Ord.function_empty),
walther@60358: 	  erls = Rule_Set.append_rules "erls_fasse_zusammen" Rule_Set.empty 
walther@60385: 	  	[\<^rule_eval>\<open>Prog_Expr.is_num\<close> (Prog_Expr.eval_is_num "#is_num_")],
walther@60358: 	  srls = Rule_Set.Empty, calc = [], errpatts = [],
walther@60358: 	  rules = [
walther@60385:      \<^rule_thm>\<open>real_num_collect\<close>, (*"[| l is_num; m is_num |]==>l * n + m * n = (l + m) * n"*)
walther@60385:   	 \<^rule_thm>\<open>real_num_collect_assoc_r\<close>, (*"[| l is_num; m..|] ==>  (k + m * n) + l * n = k + (l + m)*n"*)
walther@60385:   	 \<^rule_thm>\<open>real_one_collect\<close>, (*"m is_num ==> n + m * n = (1 + m) * n"*)
walther@60385:   	 \<^rule_thm>\<open>real_one_collect_assoc_r\<close>, (*"m is_num ==> (k + n) + m * n = k + (m + 1) * n"*)
neuper@37950: 
walther@60385:   	 \<^rule_thm>\<open>subtrahiere\<close>, (*"[| l is_num; m is_num |] ==> m * v - l * v = (m - l) * v"*)
walther@60385:   	 \<^rule_thm>\<open>subtrahiere_von_1\<close>, (*"[| l is_num |] ==> v - l * v = (1 - l) * v"*)
walther@60385:   	 \<^rule_thm>\<open>subtrahiere_1\<close>, (*"[| l is_num; m is_num |] ==> m * v - v = (m - 1) * v"*)
neuper@37950: 
walther@60385:   	 \<^rule_thm>\<open>subtrahiere_x_plus_minus\<close>, (*"[| l is_num; m..|] ==> (k + m * n) - l * n = k + ( m - l) * n"*)
walther@60385:   	 \<^rule_thm>\<open>subtrahiere_x_plus1_minus\<close>, (*"[| l is_num |] ==> (x + v) - l * v = x + (1 - l) * v"*)
walther@60385:   	 \<^rule_thm>\<open>subtrahiere_x_plus_minus1\<close>, (*"[| m is_num |] ==> (x + m * v) - v = x + (m - 1) * v"*)
neuper@37950: 
walther@60385:   	 \<^rule_thm>\<open>subtrahiere_x_minus_plus\<close>, (*"[| l is_num; m..|] ==> (k - m * n) + l * n = k + (-m + l) * n"*)
walther@60385:   	 \<^rule_thm>\<open>subtrahiere_x_minus1_plus\<close>, (*"[| l is_num |] ==> (x - v) + l * v = x + (-1 + l) * v"*)
walther@60385:   	 \<^rule_thm>\<open>subtrahiere_x_minus_plus1\<close>, (*"[| m is_num |] ==> (x - m * v) + v = x + (-m + 1) * v"*)
neuper@37950: 
walther@60385:   	 \<^rule_thm>\<open>subtrahiere_x_minus_minus\<close>, (*"[| l is_num; m..|] ==> (k - m * n) - l * n = k + (-m - l) * n"*)
walther@60385:   	 \<^rule_thm>\<open>subtrahiere_x_minus1_minus\<close>, (*"[| l is_num |] ==> (x - v) - l * v = x + (-1 - l) * v"*)
walther@60385:   	 \<^rule_thm>\<open>subtrahiere_x_minus_minus1\<close>, (*"[| m is_num |] ==> (x - m * v) - v = x + (-m - 1) * v"*)
neuper@37950: 
walther@60358:   	 \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
walther@60358:   	 \<^rule_eval>\<open>minus\<close> (**)(eval_binop "#subtr_"),
neuper@37950: 
walther@60358:   	 (*MG: Reihenfolge der folgenden 2 Rule.Thm muss so bleiben, wegen
walther@60358:         (a+a)+a --> a + 2*a --> 3*a and not (a+a)+a --> 2*a + a *)
walther@60358:   	 \<^rule_thm>\<open>real_mult_2_assoc_r\<close>, (*"(k + z1) + z1 = k + 2 * z1"*)
walther@60358:   	 \<^rule_thm_sym>\<open>real_mult_2\<close>, (*"z1 + z1 = 2 * z1"*)
walther@60358: 
walther@60385:   	 \<^rule_thm>\<open>addiere_vor_minus\<close>, (*"[| l is_num; m is_num |] ==> -(l * v) +  m * v = (-l + m) *v"*)
walther@60385:   	 \<^rule_thm>\<open>addiere_eins_vor_minus\<close>, (*"[| m is_num |] ==> -  v +  m * v = (-1 + m) * v"*)
walther@60385:   	 \<^rule_thm>\<open>subtrahiere_vor_minus\<close>, (*"[| l is_num; m is_num |] ==> -(l * v) -  m * v = (-l - m) *v"*)
walther@60385:   	 \<^rule_thm>\<open>subtrahiere_eins_vor_minus\<close>], (*"[| m is_num |] ==> -  v -  m * v = (-1 - m) * v"*)	 
walther@60358: 	  scr = Rule.Empty_Prog};
neuper@37950:     
neuper@37950: val verschoenere = 
walther@59851:   Rule_Def.Repeat{id = "verschoenere", preconds = [], 
Walther@60509:     rew_ord = ("dummy_ord", Rewrite_Ord.function_empty), srls = Rule_Set.Empty, calc = [], errpatts = [],
walther@60358:     erls = Rule_Set.append_rules "erls_verschoenere" Rule_Set.empty 
walther@60358: 		  [\<^rule_eval>\<open>PolyMinus.kleiner\<close> (eval_kleiner "")],
walther@60358:     rules = [
walther@60358:       \<^rule_thm>\<open>vorzeichen_minus_weg1\<close>, (*"l kleiner 0 ==> a + l * b = a - -l * b"*)
walther@60358:       \<^rule_thm>\<open>vorzeichen_minus_weg2\<close>, (*"l kleiner 0 ==> a - l * b = a + -l * b"*)
walther@60358:       \<^rule_thm>\<open>vorzeichen_minus_weg3\<close>, (*"l kleiner 0 ==> k + a - l * b = k + a + -l * b"*)
walther@60358:       \<^rule_thm>\<open>vorzeichen_minus_weg4\<close>, (*"l kleiner 0 ==> k - a - l * b = k - a + -l * b"*)
neuper@37950: 
walther@60358:       \<^rule_eval>\<open>times\<close> (**)(eval_binop "#mult_"),
neuper@37950: 
walther@60358:       \<^rule_thm>\<open>mult_zero_left\<close>, (*"0 * z = 0"*)
walther@60358:       \<^rule_thm>\<open>mult_1_left\<close>, (*"1 * z = z"*)
walther@60358:       \<^rule_thm>\<open>add_0_left\<close>, (*"0 + z = z"*)
walther@60358:       \<^rule_thm>\<open>null_minus\<close>, (*"0 - a = -a"*)
walther@60358:       \<^rule_thm>\<open>vor_minus_mal\<close>], (*"- a * b = (-a) * b"*)
walther@60358: 	  scr = Rule.Empty_Prog};
neuper@37950: 
neuper@37950: val klammern_aufloesen = 
walther@59851:   Rule_Def.Repeat{id = "klammern_aufloesen", preconds = [], 
Walther@60509:     rew_ord = ("dummy_ord", Rewrite_Ord.function_empty), srls = Rule_Set.Empty, 
walther@60358:     calc = [], errpatts = [], erls = Rule_Set.Empty, 
walther@60358:     rules = [
walther@60358:       \<^rule_thm_sym>\<open>add.assoc\<close>, (*"a + (b + c) = (a + b) + c"*)
walther@60358:       \<^rule_thm>\<open>klammer_plus_minus\<close>, (*"a + (b - c) = (a + b) - c"*)
walther@60358:       \<^rule_thm>\<open>klammer_minus_plus\<close>, (*"a - (b + c) = (a - b) - c"*)
walther@60358:       \<^rule_thm>\<open>klammer_minus_minus\<close>], (*"a - (b - c) = (a - b) + c"*)
walther@60358: 	  scr = Rule.Empty_Prog};
neuper@37950: 
neuper@37950: val klammern_ausmultiplizieren = 
walther@59851:   Rule_Def.Repeat{id = "klammern_ausmultiplizieren", preconds = [], 
Walther@60509:     rew_ord = ("dummy_ord", Rewrite_Ord.function_empty), srls = Rule_Set.Empty, 
walther@60358:     calc = [], errpatts = [], erls = Rule_Set.Empty, 
walther@60358:     rules = [
walther@60358:       \<^rule_thm>\<open>distrib_right\<close>, (*"(z1.0 + z2.0) * w = z1.0 * w + z2.0 * w"*)
walther@60358:       \<^rule_thm>\<open>distrib_left\<close>, (*"w * (z1.0 + z2.0) = w * z1.0 + w * z2.0"*)
walther@60358: 
walther@60358:       \<^rule_thm>\<open>klammer_mult_minus\<close>, (*"a * (b - c) = a * b - a * c"*)
walther@60358:       \<^rule_thm>\<open>klammer_minus_mult\<close>], (*"(b - c) * a = b * a - c * a"*)
walther@60358: 	  scr = Rule.Empty_Prog};
neuper@37950: 
neuper@37950: val ordne_monome = 
walther@60358:   Rule_Def.Repeat {
Walther@60509:     id = "ordne_monome", preconds = [], rew_ord = ("dummy_ord", Rewrite_Ord.function_empty), 
walther@60358:     srls = Rule_Set.Empty, calc = [], errpatts = [], 
walther@60358:     erls = Rule_Set.append_rules "erls_ordne_monome" Rule_Set.empty [
walther@60358:       \<^rule_eval>\<open>PolyMinus.kleiner\<close> (eval_kleiner ""),
walther@60358: 		  \<^rule_eval>\<open>Prog_Expr.is_atom\<close> (Prog_Expr.eval_is_atom "")], 
walther@60358:     rules = [
walther@60358:       \<^rule_thm>\<open>tausche_mal\<close>, (*"[| b is_atom; a kleiner b  |] ==> (b * a) = (a * b)"*)
walther@60358:       \<^rule_thm>\<open>tausche_vor_mal\<close>, (*"[| b is_atom; a kleiner b  |] ==> (-b * a) = (-a * b)"*)
walther@60358:       \<^rule_thm>\<open>tausche_mal_mal\<close>, (*"[| c is_atom; b kleiner c  |] ==> (a * c * b) = (a * b *c)"*)
walther@60358:       \<^rule_thm>\<open>x_quadrat\<close>], (*"(x * a) * a = x * a \<up> 2"*)
walther@60358: 	  scr = Rule.Empty_Prog};
neuper@37950: 
neuper@37950: val rls_p_33 = 
walther@60358:   Rule_Set.append_rules "rls_p_33" Rule_Set.empty [
walther@60358:     Rule.Rls_ ordne_alphabetisch,
wneuper@59416: 		Rule.Rls_ fasse_zusammen,
walther@60358: 		Rule.Rls_ verschoenere];
neuper@37950: val rls_p_34 = 
walther@60358:   Rule_Set.append_rules "rls_p_34" Rule_Set.empty [
walther@60358:     Rule.Rls_ klammern_aufloesen,
walther@60358:   	Rule.Rls_ ordne_alphabetisch,
walther@60358:   	Rule.Rls_ fasse_zusammen,
walther@60358:   	Rule.Rls_ verschoenere];
neuper@37950: val rechnen = 
walther@60358:   Rule_Set.append_rules "rechnen" Rule_Set.empty [
walther@60358:     \<^rule_eval>\<open>times\<close> (**)(eval_binop "#mult_"),
walther@60358:     \<^rule_eval>\<open>plus\<close> (**)(eval_binop "#add_"),
walther@60358:     \<^rule_eval>\<open>minus\<close> (**)(eval_binop "#subtr_")];
wneuper@59472: \<close>
wenzelm@60289: rule_set_knowledge
wenzelm@60286:   ordne_alphabetisch = \<open>prep_rls' ordne_alphabetisch\<close> and
wenzelm@60286:   fasse_zusammen = \<open>prep_rls' fasse_zusammen\<close> and
wenzelm@60286:   verschoenere = \<open>prep_rls' verschoenere\<close> and
wenzelm@60286:   ordne_monome = \<open>prep_rls' ordne_monome\<close> and
wenzelm@60286:   klammern_aufloesen = \<open>prep_rls' klammern_aufloesen\<close> and
wenzelm@60286:   klammern_ausmultiplizieren = \<open>prep_rls' klammern_ausmultiplizieren\<close>
neuper@37950: 
neuper@37950: (** problems **)
wenzelm@60306: 
wenzelm@60306: problem pbl_vereinf_poly : "polynom/vereinfachen" = \<open>Rule_Set.Empty\<close>
wenzelm@60306: 
wenzelm@60306: problem pbl_vereinf_poly_minus : "plus_minus/polynom/vereinfachen" =
walther@60358:   \<open>Rule_Set.append_rules "prls_pbl_vereinf_poly" Rule_Set.empty [
walther@60358:      \<^rule_eval>\<open>is_polyexp\<close> (eval_is_polyexp ""),
wenzelm@60306:      \<^rule_eval>\<open>Prog_Expr.matchsub\<close> (Prog_Expr.eval_matchsub ""),
walther@60358:      \<^rule_thm>\<open>or_true\<close>, (*"(?a | True) = True"*)
walther@60358:      \<^rule_thm>\<open>or_false\<close>, (*"(?a | False) = ?a"*)
walther@60358:      \<^rule_thm>\<open>not_true\<close>, (*"(~ True) = False"*)
walther@60358:      \<^rule_thm>\<open>not_false\<close> (*"(~ False) = True"*)]\<close>
Walther@60449:   Method_Ref: "simplification/for_polynomials/with_minus"
wenzelm@60306:   CAS: "Vereinfache t_t"
wenzelm@60306:   Given: "Term t_t"
wenzelm@60306:   Where:
wenzelm@60306:     "t_t is_polyexp"
wenzelm@60306:     "Not (matchsub (?a + (?b + ?c)) t_t |
wenzelm@60306:           matchsub (?a + (?b - ?c)) t_t |
wenzelm@60306:           matchsub (?a - (?b + ?c)) t_t |
wenzelm@60306:           matchsub (?a + (?b - ?c)) t_t )"
wenzelm@60306:     "Not (matchsub (?a * (?b + ?c)) t_t |
wenzelm@60306:           matchsub (?a * (?b - ?c)) t_t |
wenzelm@60306:           matchsub ((?b + ?c) * ?a) t_t |
wenzelm@60306:           matchsub ((?b - ?c) * ?a) t_t )"
wenzelm@60306:   Find: "normalform n_n"
wenzelm@60306: 
wenzelm@60306: problem pbl_vereinf_poly_klammer : "klammer/polynom/vereinfachen" =
walther@60358:   \<open>Rule_Set.append_rules "prls_pbl_vereinf_poly_klammer" Rule_Set.empty [
walther@60358:      \<^rule_eval>\<open>is_polyexp\<close> (eval_is_polyexp ""),
wenzelm@60306:      \<^rule_eval>\<open>Prog_Expr.matchsub\<close> (Prog_Expr.eval_matchsub ""),
walther@60358:      \<^rule_thm>\<open>or_true\<close>, (*"(?a | True) = True"*)
walther@60358:      \<^rule_thm>\<open>or_false\<close>, (*"(?a | False) = ?a"*)
walther@60358:      \<^rule_thm>\<open>not_true\<close>, (*"(~ True) = False"*)
walther@60358:      \<^rule_thm>\<open>not_false\<close> (*"(~ False) = True"*)]\<close>
Walther@60449:   Method_Ref: "simplification/for_polynomials/with_parentheses"
wenzelm@60306:   CAS: "Vereinfache t_t"
wenzelm@60306:   Given: "Term t_t"
wenzelm@60306:   Where:
wenzelm@60306:     "t_t is_polyexp"
wenzelm@60306:     "Not (matchsub (?a * (?b + ?c)) t_t |
wenzelm@60306:           matchsub (?a * (?b - ?c)) t_t |
wenzelm@60306:           matchsub ((?b + ?c) * ?a) t_t |
wenzelm@60306:           matchsub ((?b - ?c) * ?a) t_t )"
wenzelm@60306:   Find: "normalform n_n"
wenzelm@60306: 
wenzelm@60306: problem pbl_vereinf_poly_klammer_mal : "binom_klammer/polynom/vereinfachen" =
wenzelm@60306:   \<open>Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_*)
wenzelm@60306:     \<^rule_eval>\<open>is_polyexp\<close> (eval_is_polyexp "")]\<close>
Walther@60449:   Method_Ref: "simplification/for_polynomials/with_parentheses_mult"
wenzelm@60306:   CAS: "Vereinfache t_t"
wenzelm@60306:   Given: "Term t_t"
wenzelm@60306:   Where: "t_t is_polyexp"
wenzelm@60306:   Find: "normalform n_n"
wenzelm@60306: 
wenzelm@60306: problem pbl_probe : "probe" = \<open>Rule_Set.Empty\<close>
wenzelm@60306: 
wenzelm@60306: problem pbl_probe_poly : "polynom/probe" =
wenzelm@60306:   \<open>Rule_Set.append_rules "prls_pbl_probe_poly" Rule_Set.empty [(*for preds in where_*)
wenzelm@60306:     \<^rule_eval>\<open>is_polyexp\<close> (eval_is_polyexp "")]\<close>
Walther@60449:   Method_Ref: "probe/fuer_polynom"
wenzelm@60306:   CAS: "Probe e_e w_w"
wenzelm@60306:   Given: "Pruefe e_e" "mitWert w_w"
wenzelm@60306:   Where: "e_e is_polyexp"
wenzelm@60306:   Find: "Geprueft p_p"
wenzelm@60306: 
wenzelm@60306: problem pbl_probe_bruch : "bruch/probe" =
wenzelm@60306:   \<open>Rule_Set.append_rules "prls_pbl_probe_bruch" Rule_Set.empty [(*for preds in where_*)
wenzelm@60306:     \<^rule_eval>\<open>is_ratpolyexp\<close> (eval_is_ratpolyexp "")]\<close>
Walther@60449:   Method_Ref: "probe/fuer_bruch"
wenzelm@60306:   CAS: "Probe e_e w_w"
wenzelm@60306:   Given: "Pruefe e_e" "mitWert w_w"
wenzelm@60306:   Where: "e_e is_ratpolyexp"
wenzelm@60306:   Find: "Geprueft p_p"
neuper@37950: 
neuper@37950: (** methods **)
wneuper@59545: 
wneuper@59504: partial_function (tailrec) simplify :: "real \<Rightarrow> real"
wneuper@59504:   where
walther@59635: "simplify t_t = (
walther@59635:   (Repeat(
walther@59637:     (Try (Rewrite_Set ''ordne_alphabetisch'')) #>
walther@59637:     (Try (Rewrite_Set ''fasse_zusammen'')) #>
walther@59635:     (Try (Rewrite_Set ''verschoenere'')))
walther@59635:   ) t_t)"
wenzelm@60303: 
wenzelm@60303: method met_simp_poly_minus : "simplification/for_polynomials/with_minus" =
wenzelm@60303:   \<open>{rew_ord'="tless_true", rls' = Rule_Set.empty, calc = [], srls = Rule_Set.empty,
walther@60358:     prls = Rule_Set.append_rules "prls_met_simp_poly_minus" Rule_Set.empty [
walther@60358:       \<^rule_eval>\<open>is_polyexp\<close> (eval_is_polyexp ""),
walther@60358:       \<^rule_eval>\<open>Prog_Expr.matchsub\<close> (Prog_Expr.eval_matchsub ""),
walther@60358:       \<^rule_thm>\<open>and_true\<close>, (*"(?a & True) = ?a"*)
walther@60358:       \<^rule_thm>\<open>and_false\<close>, (*"(?a & False) = False"*)
walther@60358:       \<^rule_thm>\<open>not_true\<close>, (*"(~ True) = False"*)
walther@60358:       \<^rule_thm>\<open>not_false\<close> (*"(~ False) = True"*)],
wenzelm@60303:     crls = Rule_Set.empty, errpats = [], nrls = rls_p_33}\<close>
wenzelm@60303:   Program: simplify.simps
wenzelm@60303:   Given: "Term t_t"
wenzelm@60303:   Where:
wenzelm@60303:     "t_t is_polyexp"
wenzelm@60303:     "Not (matchsub (?a + (?b + ?c)) t_t |
wenzelm@60303:           matchsub (?a + (?b - ?c)) t_t |
wenzelm@60303:           matchsub (?a - (?b + ?c)) t_t |
wenzelm@60303:           matchsub (?a + (?b - ?c)) t_t)"
wenzelm@60303:   Find: "normalform n_n"
wneuper@59545: 
wneuper@59504: partial_function (tailrec) simplify2 :: "real \<Rightarrow> real"
wneuper@59504:   where
walther@59635: "simplify2 t_t = (
walther@59635:   (Repeat(
walther@59637:     (Try (Rewrite_Set ''klammern_aufloesen'')) #>
walther@59637:     (Try (Rewrite_Set ''ordne_alphabetisch'')) #>
walther@59637:     (Try (Rewrite_Set ''fasse_zusammen'')) #>
walther@59635:     (Try (Rewrite_Set ''verschoenere'')))
walther@59635:   ) t_t)"
wenzelm@60303: 
wenzelm@60303: method met_simp_poly_parenth : "simplification/for_polynomials/with_parentheses" =
wenzelm@60303:   \<open>{rew_ord'="tless_true", rls' = Rule_Set.empty, calc = [], srls = Rule_Set.empty, 
wenzelm@60303:     prls = Rule_Set.append_rules "simplification_for_polynomials_prls" Rule_Set.empty 
wenzelm@60303:       [(*for preds in where_*) \<^rule_eval>\<open>is_polyexp\<close> (eval_is_polyexp"")],
wenzelm@60303:     crls = Rule_Set.empty, errpats = [], nrls = rls_p_34}\<close>
wenzelm@60303:   Program: simplify2.simps
wenzelm@60303:   Given: "Term t_t"
wenzelm@60303:   Where: "t_t is_polyexp"
wenzelm@60303:   Find: "normalform n_n"
wneuper@59545: 
wneuper@59504: partial_function (tailrec) simplify3 :: "real \<Rightarrow> real"
wneuper@59504:   where
walther@59635: "simplify3 t_t = (
walther@59635:   (Repeat(
walther@59637:     (Try (Rewrite_Set ''klammern_ausmultiplizieren'')) #>
walther@59637:     (Try (Rewrite_Set ''discard_parentheses'')) #>
walther@59637:     (Try (Rewrite_Set ''ordne_monome'')) #>
walther@59637:     (Try (Rewrite_Set ''klammern_aufloesen'')) #>
walther@59637:     (Try (Rewrite_Set ''ordne_alphabetisch'')) #>
walther@59637:     (Try (Rewrite_Set ''fasse_zusammen'')) #>
walther@59635:     (Try (Rewrite_Set ''verschoenere'')))
walther@59635:   ) t_t)"
wenzelm@60303: 
wenzelm@60303: method met_simp_poly_parenth_mult : "simplification/for_polynomials/with_parentheses_mult" =
wenzelm@60303:   \<open>{rew_ord'="tless_true", rls' = Rule_Set.empty, calc = [], srls = Rule_Set.empty, 
wenzelm@60303:     prls = Rule_Set.append_rules "simplification_for_polynomials_prls" Rule_Set.empty 
wenzelm@60303:       [(*for preds in where_*) \<^rule_eval>\<open>is_polyexp\<close> (eval_is_polyexp"")],
wenzelm@60303:     crls = Rule_Set.empty, errpats = [], nrls = rls_p_34}\<close>
wenzelm@60303:   Program: simplify3.simps
wenzelm@60303:   Given: "Term t_t"
wenzelm@60303:   Where: "t_t is_polyexp"
wenzelm@60303:   Find: "normalform n_n"
wenzelm@60303: 
wenzelm@60303: method met_probe : "probe" =
wenzelm@60303:   \<open>{rew_ord'="tless_true", rls' = Rule_Set.empty, calc = [], srls = Rule_Set.empty, prls = Rule_Set.Empty, crls = Rule_Set.empty,
wenzelm@60303:     errpats = [], nrls = Rule_Set.Empty}\<close>
wneuper@59545: 
wneuper@59504: partial_function (tailrec) mache_probe :: "bool \<Rightarrow> bool list \<Rightarrow> bool"
wneuper@59504:   where
walther@59635: "mache_probe e_e w_w = (
walther@59635:   let
walther@59635:      e_e = Take e_e;
walther@59635:      e_e = Substitute w_w e_e
walther@59635:   in (
walther@59635:     Repeat (
walther@59637:       (Try (Repeat (Calculate ''TIMES''))) #>
walther@59637:       (Try (Repeat (Calculate ''PLUS'' ))) #>
walther@59635:       (Try (Repeat (Calculate ''MINUS''))))
walther@59635:     ) e_e)"
wenzelm@60303: 
wenzelm@60303: method met_probe_poly : "probe/fuer_polynom" =
wenzelm@60303:   \<open>{rew_ord'="tless_true", rls' = Rule_Set.empty, calc = [], srls = Rule_Set.empty, 
wenzelm@60303:     prls = Rule_Set.append_rules "prls_met_probe_bruch" Rule_Set.empty
wenzelm@60303:         [(*for preds in where_*) \<^rule_eval>\<open>is_ratpolyexp\<close> (eval_is_ratpolyexp "")], 
wenzelm@60303:     crls = Rule_Set.empty, errpats = [], nrls = rechnen}\<close>
wenzelm@60303:   Program: mache_probe.simps
wenzelm@60303:   Given: "Pruefe e_e" "mitWert w_w"
wenzelm@60303:   Where: "e_e is_polyexp"
wenzelm@60303:   Find: "Geprueft p_p"
wenzelm@60303: 
wenzelm@60303: method met_probe_bruch : "probe/fuer_bruch" =
wenzelm@60303:   \<open>{rew_ord'="tless_true", rls' = Rule_Set.empty, calc = [], srls = Rule_Set.empty, 
wenzelm@60303:     prls = Rule_Set.append_rules "prls_met_probe_bruch" Rule_Set.empty
wenzelm@60303:         [(*for preds in where_*) \<^rule_eval>\<open>is_ratpolyexp\<close> (eval_is_ratpolyexp "")],
wenzelm@60303:     crls = Rule_Set.empty, errpats = [], nrls = Rule_Set.Empty}\<close>
wenzelm@60303:   Given: "Pruefe e_e" "mitWert w_w"
wenzelm@60303:   Where: "e_e is_ratpolyexp"
wenzelm@60303:   Find: "Geprueft p_p"
wenzelm@60303: 
wenzelm@60303: ML \<open>
walther@60278: \<close> ML \<open>
wneuper@59472: \<close>
neuper@37906: 
neuper@37906: end
neuper@37906: 




