(*.(c) by Richard Lang, 2003 .*)

(* collecting all knowledge for Root Equations

   created by: rlang 

         date: 02.08

   changed by: rlang

   last change by: rlang

             date: 02.11.14

*)

theory RootEq imports Root Equation begin

text \<open>univariate equations containing real square roots:

  This type of equations has been used to bootstrap Lucas-Interpretation.

  In 2003 this type has been extended and integrated into ISAC's equation solver

  by Richard Lang in 2003.

  The assumptions (all containing "<") didn't pass the xml-parsers at the 

  interface between math-engine and front-end.

  The migration Isabelle2002 --> 2011 dropped this type of equation, see

  test/../rooteq.sml, rootrateq.sml.

\<close>

subsection \<open>consts definition for predicates\<close>

consts

  is_rootTerm_in     :: "[real, real] => bool" ("_ is'_rootTerm'_in _")

  is_sqrtTerm_in     :: "[real, real] => bool" ("_ is'_sqrtTerm'_in _") 

  is_normSqrtTerm_in :: "[real, real] => bool" ("_ is'_normSqrtTerm'_in _") 

subsection \<open>theorems not yet adopted from Isabelle\<close>

axiomatization where

(* normalise *)

  makex1_x:            "a \<up> 1  = a"   and

  real_assoc_1:        "a+(b+c) = a+b+c" and

  real_assoc_2:        "a*(b*c) = a*b*c" and

  (* simplification of root*)

  sqrt_square_1:       "[|0 <= a|] ==>  (sqrt a) \<up> 2 = a" and

  sqrt_square_2:       "sqrt (a  \<up>  2) = a" and

  sqrt_times_root_1:   "sqrt a * sqrt b = sqrt(a*b)" and

  sqrt_times_root_2:   "a * sqrt b * sqrt c = a * sqrt(b*c)" and

  (* isolate one root on the LEFT or RIGHT hand side of the equation *)

  sqrt_isolate_l_add1: "[|bdv occurs_in c|] ==> 

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

  sqrt_isolate_l_add2: "[|bdv occurs_in c|] ==>

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

  sqrt_isolate_l_add3: "[|bdv occurs_in c|] ==>

   (a + b*(e/sqrt(c)) = d) = (b * (e/sqrt(c)) = d + (-1) * a)" and

  sqrt_isolate_l_add4: "[|bdv occurs_in c|] ==>

   (a + b/(f*sqrt(c)) = d) = (b / (f*sqrt(c)) = d + (-1) * a)" and

  sqrt_isolate_l_add5: "[|bdv occurs_in c|] ==>

   (a + b*(e/(f*sqrt(c))) = d) = (b * (e/(f*sqrt(c))) = d+ (-1) * a)" and

  sqrt_isolate_l_add6: "[|bdv occurs_in c|] ==>

   (a + b/sqrt(c) = d) = (b / sqrt(c) = d+ (-1) * a)" and

  sqrt_isolate_r_add1: "[|bdv occurs_in f|] ==>

   (a = d + e*sqrt(f)) = (a + (-1) * d = e*sqrt(f))" and

  sqrt_isolate_r_add2: "[|bdv occurs_in f|] ==>

   (a = d + sqrt(f)) = (a + (-1) * d = sqrt(f))" and

 (* small hack: thm 3,5,6 are not needed if rootnormalise is well done*)

  sqrt_isolate_r_add3: "[|bdv occurs_in f|] ==>

   (a = d + e*(g/sqrt(f))) = (a + (-1) * d = e*(g/sqrt(f)))" and

  sqrt_isolate_r_add4: "[|bdv occurs_in f|] ==>

   (a = d + g/sqrt(f)) = (a + (-1) * d = g/sqrt(f))" and

  sqrt_isolate_r_add5: "[|bdv occurs_in f|] ==>

   (a = d + e*(g/(h*sqrt(f)))) = (a + (-1) * d = e*(g/(h*sqrt(f))))" and

  sqrt_isolate_r_add6: "[|bdv occurs_in f|] ==>

   (a = d + g/(h*sqrt(f))) = (a + (-1) * d = g/(h*sqrt(f)))" and

  (* eliminate isolates sqrt *)

  sqrt_square_equation_both_1: "[|bdv occurs_in b; bdv occurs_in d|] ==> 

   ( (sqrt a + sqrt b         = sqrt c + sqrt d) = 

     (a+2*sqrt(a)*sqrt(b)+b  = c+2*sqrt(c)*sqrt(d)+d))" and

  sqrt_square_equation_both_2: "[|bdv occurs_in b; bdv occurs_in d|] ==> 

   ( (sqrt a - sqrt b           = sqrt c + sqrt d) = 

     (a - 2*sqrt(a)*sqrt(b)+b  = c+2*sqrt(c)*sqrt(d)+d))" and

  sqrt_square_equation_both_3: "[|bdv occurs_in b; bdv occurs_in d|] ==> 

   ( (sqrt a + sqrt b           = sqrt c - sqrt d) = 

     (a + 2*sqrt(a)*sqrt(b)+b  = c - 2*sqrt(c)*sqrt(d)+d))" and

  sqrt_square_equation_both_4: "[|bdv occurs_in b; bdv occurs_in d|] ==> 

   ( (sqrt a - sqrt b           = sqrt c - sqrt d) = 

     (a - 2*sqrt(a)*sqrt(b)+b  = c - 2*sqrt(c)*sqrt(d)+d))" and

  sqrt_square_equation_left_1: "[|bdv occurs_in a; 0 <= a; 0 <= b|] ==>

   ( (sqrt (a) = b) = (a = (b \<up> 2)))" and

  sqrt_square_equation_left_2: "[|bdv occurs_in a; 0 <= a; 0 <= b*c|] ==> 

   ( (c*sqrt(a) = b) = (c \<up> 2*a = b \<up> 2))" and

  sqrt_square_equation_left_3: "[|bdv occurs_in a; 0 <= a; 0 <= b*c|] ==> 

   ( c/sqrt(a) = b) = (c \<up> 2 / a = b \<up> 2)" and

  (* small hack: thm 4-6 are not needed if rootnormalise is well done*)

  sqrt_square_equation_left_4: "[|bdv occurs_in a; 0 <= a; 0 <= b*c*d|] ==> 

   ( (c*(d/sqrt (a)) = b) = (c \<up> 2*(d \<up> 2/a) = b \<up> 2))" and

  sqrt_square_equation_left_5: "[|bdv occurs_in a; 0 <= a; 0 <= b*c*d|] ==> 

   ( c/(d*sqrt(a)) = b) = (c \<up> 2 / (d \<up> 2*a) = b \<up> 2)" and

  sqrt_square_equation_left_6: "[|bdv occurs_in a; 0 <= a; 0 <= b*c*d*e|] ==> 

   ( (c*(d/(e*sqrt (a))) = b) = (c \<up> 2*(d \<up> 2/(e \<up> 2*a)) = b \<up> 2))" and

  sqrt_square_equation_right_1:  "[|bdv occurs_in b; 0 <= a; 0 <= b|] ==> 

   ( (a = sqrt (b)) = (a \<up> 2 = b))" and

  sqrt_square_equation_right_2: "[|bdv occurs_in b; 0 <= a*c; 0 <= b|] ==> 

   ( (a = c*sqrt (b)) = ((a \<up> 2) = c \<up> 2*b))" and

  sqrt_square_equation_right_3: "[|bdv occurs_in b; 0 <= a*c; 0 <= b|] ==> 

   ( (a = c/sqrt (b)) = (a \<up> 2 = c \<up> 2/b))" and

 (* small hack: thm 4-6 are not needed if rootnormalise is well done*)

  sqrt_square_equation_right_4: "[|bdv occurs_in b; 0 <= a*c*d; 0 <= b|] ==> 

   ( (a = c*(d/sqrt (b))) = ((a \<up> 2) = c \<up> 2*(d \<up> 2/b)))" and

  sqrt_square_equation_right_5: "[|bdv occurs_in b; 0 <= a*c*d; 0 <= b|] ==> 

   ( (a = c/(d*sqrt (b))) = (a \<up> 2 = c \<up> 2/(d \<up> 2*b)))" and

  sqrt_square_equation_right_6: "[|bdv occurs_in b; 0 <= a*c*d*e; 0 <= b|] ==> 

   ( (a = c*(d/(e*sqrt (b)))) = ((a \<up> 2) = c \<up> 2*(d \<up> 2/(e \<up> 2*b))))"

subsection \<open>predicates\<close>

ML \<open>

val thy = @{theory};

(* true if bdv is under sqrt of a Equation*)

fun is_rootTerm_in t v = 

  let 

   	fun findroot (t as (_ $ _ $ _ $ _)) _ = raise TERM ( "is_rootTerm_in", [t])

	  (* at the moment there is no term like this, but ....*)

	  | findroot (Const ("Root.nroot",_) $ _ $ t3) v = Prog_Expr.occurs_in v t3

	  | findroot (_ $ t2 $ t3) v = findroot t2 v orelse findroot t3 v

	  | findroot (Const ("NthRoot.sqrt", _) $ t2) v = Prog_Expr.occurs_in v t2

	  | findroot (_ $ t2) v = findroot t2 v

	  | findroot _ _ = false;

  in

	  findroot t v

  end;

fun is_sqrtTerm_in t v = 

  let 

    fun findsqrt (t as (_ $ _ $ _ $ _)) _ = raise TERM ("is_sqrteqation_in", [t])

    (* at the moment there is no term like this, but ....*)

    | findsqrt (_ $ t1 $ t2) v = (findsqrt t1 v) orelse (findsqrt t2 v)

    | findsqrt (Const ("NthRoot.sqrt", _) $ a) v = Prog_Expr.occurs_in v a

    | findsqrt (_ $ t1) v = findsqrt t1 v

    | findsqrt _ _ = false;

  in

   findsqrt t v

  end;

(* RL: 030518: Is in the rightest subterm of a term a sqrt with bdv,

and the subterm ist connected with + or * --> is normalised*)

fun is_normSqrtTerm_in t v =

  let

     fun isnorm (t as (_ $ _ $ _ $ _)) _ = raise TERM ("is_normSqrtTerm_in", [t])

         (* at the moment there is no term like this, but ....*)

       | isnorm (Const ("Groups.plus_class.plus",_) $ _ $ t2) v = is_sqrtTerm_in t2 v

       | isnorm (Const ("Groups.times_class.times",_) $ _ $ t2) v = is_sqrtTerm_in t2 v

       | isnorm (Const ("Groups.minus_class.minus",_) $ _ $ _) _ = false

       | isnorm (Const ("Rings.divide_class.divide",_) $ t1 $ t2) v =

         is_sqrtTerm_in t1 v orelse is_sqrtTerm_in t2 v

       | isnorm (Const ("NthRoot.sqrt", _) $ t1) v = Prog_Expr.occurs_in v t1

	     | isnorm (_ $ t1) v = is_sqrtTerm_in t1 v

       | isnorm _ _ = false;

   in

     isnorm t v

   end;

fun eval_is_rootTerm_in _ _ (p as (Const ("RootEq.is_rootTerm_in",_) $ t $ v)) _ =

    if is_rootTerm_in t v

    then SOME ((UnparseC.term p) ^ " = True", HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))

    else SOME ((UnparseC.term p) ^ " = True", HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))

  | eval_is_rootTerm_in _ _ _ _ = ((*tracing"### nichts matcht";*) NONE);

fun eval_is_sqrtTerm_in _ _ (p as (Const ("RootEq.is_sqrtTerm_in",_) $ t $ v)) _ =

    if is_sqrtTerm_in t v

    then SOME ((UnparseC.term p) ^ " = True", HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))

    else SOME ((UnparseC.term p) ^ " = True", HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))

  | eval_is_sqrtTerm_in _ _ _ _ = ((*tracing"### nichts matcht";*) NONE);

fun eval_is_normSqrtTerm_in _ _ (p as (Const ("RootEq.is_normSqrtTerm_in",_) $ t $ v)) _ =

    if is_normSqrtTerm_in t v

    then SOME ((UnparseC.term p) ^ " = True", HOLogic.Trueprop $ (TermC.mk_equality (p, @{term True})))

    else SOME ((UnparseC.term p) ^ " = True", HOLogic.Trueprop $ (TermC.mk_equality (p, @{term False})))

  | eval_is_normSqrtTerm_in _ _ _ _ = ((*tracing"### nichts matcht";*) NONE);

\<close>

setup \<open>KEStore_Elems.add_calcs

  [("is_rootTerm_in", ("RootEq.is_rootTerm_in", eval_is_rootTerm_in"")),

    ("is_sqrtTerm_in", ("RootEq.is_sqrtTerm_in", eval_is_sqrtTerm_in"")),

    ("is_normSqrtTerm_in", ("RootEq.is_normSqrtTerm_in", eval_is_normSqrtTerm_in""))]\<close>

subsection \<open>rule-sets\<close>

ML \<open>

val RootEq_prls =(*15.10.02:just the following order due to subterm evaluation*)

  Rule_Set.append_rules "RootEq_prls" Rule_Set.empty 

	     [Rule.Eval ("Prog_Expr.ident", Prog_Expr.eval_ident "#ident_"),

	      Rule.Eval ("Prog_Expr.matches", Prog_Expr.eval_matches ""),

	      Rule.Eval ("Prog_Expr.lhs"    , Prog_Expr.eval_lhs ""),

	      Rule.Eval ("Prog_Expr.rhs"    , Prog_Expr.eval_rhs ""),

	      Rule.Eval ("RootEq.is_sqrtTerm_in", eval_is_sqrtTerm_in ""),

	      Rule.Eval ("RootEq.is_rootTerm_in", eval_is_rootTerm_in ""),

	      Rule.Eval ("RootEq.is_normSqrtTerm_in", eval_is_normSqrtTerm_in ""),

	      Rule.Eval ("HOL.eq", Prog_Expr.eval_equal "#equal_"),

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

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

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

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

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

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

	      ];

val RootEq_erls =

  Rule_Set.append_rules "RootEq_erls" Root_erls

    [Rule.Thm ("divide_divide_eq_left", ThmC.numerals_to_Free @{thm divide_divide_eq_left})];

val RootEq_crls = 

  Rule_Set.append_rules "RootEq_crls" Root_crls

    [Rule.Thm ("divide_divide_eq_left", ThmC.numerals_to_Free @{thm divide_divide_eq_left})];

val rooteq_srls = 

  Rule_Set.append_rules "rooteq_srls" Rule_Set.empty

    [Rule.Eval ("RootEq.is_sqrtTerm_in", eval_is_sqrtTerm_in ""),

     Rule.Eval ("RootEq.is_normSqrtTerm_in", eval_is_normSqrtTerm_in""),

     Rule.Eval ("RootEq.is_rootTerm_in", eval_is_rootTerm_in "")];

\<close>

ML \<open>

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

 val sqrt_isolate = prep_rls'(

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

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

       rules = [

       Rule.Thm("sqrt_square_1",ThmC.numerals_to_Free @{thm sqrt_square_1}),

                     (* (sqrt a) \<up> 2 -> a *)

       Rule.Thm("sqrt_square_2",ThmC.numerals_to_Free @{thm sqrt_square_2}),

                     (* sqrt (a \<up> 2) -> a *)

       Rule.Thm("sqrt_times_root_1",ThmC.numerals_to_Free @{thm sqrt_times_root_1}),

            (* sqrt a sqrt b -> sqrt(ab) *)

       Rule.Thm("sqrt_times_root_2",ThmC.numerals_to_Free @{thm sqrt_times_root_2}),

            (* a sqrt b sqrt c -> a sqrt(bc) *)

       Rule.Thm("sqrt_square_equation_both_1",

           ThmC.numerals_to_Free @{thm sqrt_square_equation_both_1}),

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

            (a+2*sqrt(a)*sqrt(b)+b) = c+2*sqrt(c)*sqrt(d)+d) *)

       Rule.Thm("sqrt_square_equation_both_2",

            ThmC.numerals_to_Free @{thm sqrt_square_equation_both_2}),

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

            (a-2*sqrt(a)*sqrt(b)+b) = c+2*sqrt(c)*sqrt(d)+d) *)

       Rule.Thm("sqrt_square_equation_both_3",

            ThmC.numerals_to_Free @{thm sqrt_square_equation_both_3}),

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

            (a+2*sqrt(a)*sqrt(b)+b) = c-2*sqrt(c)*sqrt(d)+d) *)

       Rule.Thm("sqrt_square_equation_both_4",

            ThmC.numerals_to_Free @{thm sqrt_square_equation_both_4}),

       (* (sqrt a - sqrt b  = sqrt c - sqrt d) -> 

            (a-2*sqrt(a)*sqrt(b)+b) = c-2*sqrt(c)*sqrt(d)+d) *)

       Rule.Thm("sqrt_isolate_l_add1",

            ThmC.numerals_to_Free @{thm sqrt_isolate_l_add1}), 

       (* a+b*sqrt(x)=d -> b*sqrt(x) = d-a *)

       Rule.Thm("sqrt_isolate_l_add2",

            ThmC.numerals_to_Free @{thm sqrt_isolate_l_add2}), 

       (* a+  sqrt(x)=d ->   sqrt(x) = d-a *)

       Rule.Thm("sqrt_isolate_l_add3",

            ThmC.numerals_to_Free @{thm sqrt_isolate_l_add3}), 

       (* a+b*c/sqrt(x)=d->b*c/sqrt(x)=d-a *)

       Rule.Thm("sqrt_isolate_l_add4",

            ThmC.numerals_to_Free @{thm sqrt_isolate_l_add4}), 

       (* a+c/sqrt(x)=d -> c/sqrt(x) = d-a *)

       Rule.Thm("sqrt_isolate_l_add5",

            ThmC.numerals_to_Free @{thm sqrt_isolate_l_add5}), 

       (* a+b*c/f*sqrt(x)=d->b*c/f*sqrt(x)=d-a *)

       Rule.Thm("sqrt_isolate_l_add6",

            ThmC.numerals_to_Free @{thm sqrt_isolate_l_add6}), 

       (* a+c/f*sqrt(x)=d -> c/f*sqrt(x) = d-a *)

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

         (* b*sqrt(x) = d sqrt(x) d/b *)

       Rule.Thm("sqrt_isolate_r_add1",

            ThmC.numerals_to_Free @{thm sqrt_isolate_r_add1}),

       (* a= d+e*sqrt(x) -> a-d=e*sqrt(x) *)

       Rule.Thm("sqrt_isolate_r_add2",

            ThmC.numerals_to_Free @{thm sqrt_isolate_r_add2}),

       (* a= d+  sqrt(x) -> a-d=  sqrt(x) *)

       Rule.Thm("sqrt_isolate_r_add3",

            ThmC.numerals_to_Free @{thm sqrt_isolate_r_add3}),

       (* a=d+e*g/sqrt(x)->a-d=e*g/sqrt(x)*)

       Rule.Thm("sqrt_isolate_r_add4",

            ThmC.numerals_to_Free @{thm sqrt_isolate_r_add4}),

       (* a= d+g/sqrt(x) -> a-d=g/sqrt(x) *)

       Rule.Thm("sqrt_isolate_r_add5",

            ThmC.numerals_to_Free @{thm sqrt_isolate_r_add5}),

       (* a=d+e*g/h*sqrt(x)->a-d=e*g/h*sqrt(x)*)

       Rule.Thm("sqrt_isolate_r_add6",

            ThmC.numerals_to_Free @{thm sqrt_isolate_r_add6}),

       (* a= d+g/h*sqrt(x) -> a-d=g/h*sqrt(x) *)

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

         (* a=e*sqrt(x) -> a/e = sqrt(x) *)

       Rule.Thm("sqrt_square_equation_left_1",

            ThmC.numerals_to_Free @{thm sqrt_square_equation_left_1}),   

       (* sqrt(x)=b -> x=b^2 *)

       Rule.Thm("sqrt_square_equation_left_2",

            ThmC.numerals_to_Free @{thm sqrt_square_equation_left_2}),   

       (* c*sqrt(x)=b -> c^2*x=b^2 *)

       Rule.Thm("sqrt_square_equation_left_3",ThmC.numerals_to_Free @{thm sqrt_square_equation_left_3}),  

	      (* c/sqrt(x)=b -> c^2/x=b^2 *)

       Rule.Thm("sqrt_square_equation_left_4",ThmC.numerals_to_Free @{thm sqrt_square_equation_left_4}),

	      (* c*d/sqrt(x)=b -> c^2*d^2/x=b^2 *)

       Rule.Thm("sqrt_square_equation_left_5",ThmC.numerals_to_Free @{thm sqrt_square_equation_left_5}),

	      (* c/d*sqrt(x)=b -> c^2/d^2x=b^2 *)

       Rule.Thm("sqrt_square_equation_left_6",ThmC.numerals_to_Free @{thm sqrt_square_equation_left_6}),

	      (* c*d/g*sqrt(x)=b -> c^2*d^2/g^2x=b^2 *)

       Rule.Thm("sqrt_square_equation_right_1",ThmC.numerals_to_Free @{thm sqrt_square_equation_right_1}),

	      (* a=sqrt(x) ->a^2=x *)

       Rule.Thm("sqrt_square_equation_right_2",ThmC.numerals_to_Free @{thm sqrt_square_equation_right_2}),

	      (* a=c*sqrt(x) ->a^2=c^2*x *)

       Rule.Thm("sqrt_square_equation_right_3",ThmC.numerals_to_Free @{thm sqrt_square_equation_right_3}),

	      (* a=c/sqrt(x) ->a^2=c^2/x *)

       Rule.Thm("sqrt_square_equation_right_4",ThmC.numerals_to_Free @{thm sqrt_square_equation_right_4}),

	      (* a=c*d/sqrt(x) ->a^2=c^2*d^2/x *)

       Rule.Thm("sqrt_square_equation_right_5",ThmC.numerals_to_Free @{thm sqrt_square_equation_right_5}),

	      (* a=c/e*sqrt(x) ->a^2=c^2/e^2x *)

       Rule.Thm("sqrt_square_equation_right_6",ThmC.numerals_to_Free @{thm sqrt_square_equation_right_6})

	      (* a=c*d/g*sqrt(x) ->a^2=c^2*d^2/g^2*x *)

       ],scr = Rule.Empty_Prog

      });

\<close> ML \<open>

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

 val l_sqrt_isolate = prep_rls'(

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

	  rew_ord = ("termlessI",termlessI), 

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

     rules = [

     Rule.Thm("sqrt_square_1",ThmC.numerals_to_Free @{thm sqrt_square_1}),

                            (* (sqrt a) \<up> 2 -> a *)

     Rule.Thm("sqrt_square_2",ThmC.numerals_to_Free @{thm sqrt_square_2}),

                            (* sqrt (a \<up> 2) -> a *)

     Rule.Thm("sqrt_times_root_1",ThmC.numerals_to_Free @{thm sqrt_times_root_1}),

            (* sqrt a sqrt b -> sqrt(ab) *)

     Rule.Thm("sqrt_times_root_2",ThmC.numerals_to_Free @{thm sqrt_times_root_2}),

        (* a sqrt b sqrt c -> a sqrt(bc) *)

     Rule.Thm("sqrt_isolate_l_add1",ThmC.numerals_to_Free @{thm sqrt_isolate_l_add1}),

        (* a+b*sqrt(x)=d -> b*sqrt(x) = d-a *)

     Rule.Thm("sqrt_isolate_l_add2",ThmC.numerals_to_Free @{thm sqrt_isolate_l_add2}),

        (* a+  sqrt(x)=d ->   sqrt(x) = d-a *)

     Rule.Thm("sqrt_isolate_l_add3",ThmC.numerals_to_Free @{thm sqrt_isolate_l_add3}),

        (* a+b*c/sqrt(x)=d->b*c/sqrt(x)=d-a *)

     Rule.Thm("sqrt_isolate_l_add4",ThmC.numerals_to_Free @{thm sqrt_isolate_l_add4}),

        (* a+c/sqrt(x)=d -> c/sqrt(x) = d-a *)

     Rule.Thm("sqrt_isolate_l_add5",ThmC.numerals_to_Free @{thm sqrt_isolate_l_add5}),

        (* a+b*c/f*sqrt(x)=d->b*c/f*sqrt(x)=d-a *)

     Rule.Thm("sqrt_isolate_l_add6",ThmC.numerals_to_Free @{thm sqrt_isolate_l_add6}),

        (* a+c/f*sqrt(x)=d -> c/f*sqrt(x) = d-a *)

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

        (* b*sqrt(x) = d sqrt(x) d/b *)

     Rule.Thm("sqrt_square_equation_left_1",ThmC.numerals_to_Free @{thm sqrt_square_equation_left_1}),

	      (* sqrt(x)=b -> x=b^2 *)

     Rule.Thm("sqrt_square_equation_left_2",ThmC.numerals_to_Free @{thm sqrt_square_equation_left_2}),

	      (* a*sqrt(x)=b -> a^2*x=b^2*)

     Rule.Thm("sqrt_square_equation_left_3",ThmC.numerals_to_Free @{thm sqrt_square_equation_left_3}),   

	      (* c/sqrt(x)=b -> c^2/x=b^2 *)

     Rule.Thm("sqrt_square_equation_left_4",ThmC.numerals_to_Free @{thm sqrt_square_equation_left_4}),   

	      (* c*d/sqrt(x)=b -> c^2*d^2/x=b^2 *)

     Rule.Thm("sqrt_square_equation_left_5",ThmC.numerals_to_Free @{thm sqrt_square_equation_left_5}),   

	      (* c/d*sqrt(x)=b -> c^2/d^2x=b^2 *)

     Rule.Thm("sqrt_square_equation_left_6",ThmC.numerals_to_Free @{thm sqrt_square_equation_left_6})  

	      (* c*d/g*sqrt(x)=b -> c^2*d^2/g^2x=b^2 *)

    ],

    scr = Rule.Empty_Prog

   });

\<close> ML \<open>

(* -- right 28.8.02--*)

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

 val r_sqrt_isolate = prep_rls'(

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

	  rew_ord = ("termlessI",termlessI), 

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

     rules = [

     Rule.Thm("sqrt_square_1",ThmC.numerals_to_Free @{thm sqrt_square_1}),

                           (* (sqrt a) \<up> 2 -> a *)

     Rule.Thm("sqrt_square_2",ThmC.numerals_to_Free @{thm sqrt_square_2}), 

                           (* sqrt (a \<up> 2) -> a *)

     Rule.Thm("sqrt_times_root_1",ThmC.numerals_to_Free @{thm sqrt_times_root_1}),

           (* sqrt a sqrt b -> sqrt(ab) *)

     Rule.Thm("sqrt_times_root_2",ThmC.numerals_to_Free @{thm sqrt_times_root_2}),

       (* a sqrt b sqrt c -> a sqrt(bc) *)

     Rule.Thm("sqrt_isolate_r_add1",ThmC.numerals_to_Free @{thm sqrt_isolate_r_add1}),

       (* a= d+e*sqrt(x) -> a-d=e*sqrt(x) *)

     Rule.Thm("sqrt_isolate_r_add2",ThmC.numerals_to_Free @{thm sqrt_isolate_r_add2}),

       (* a= d+  sqrt(x) -> a-d=  sqrt(x) *)

     Rule.Thm("sqrt_isolate_r_add3",ThmC.numerals_to_Free @{thm sqrt_isolate_r_add3}),

       (* a=d+e*g/sqrt(x)->a-d=e*g/sqrt(x)*)

     Rule.Thm("sqrt_isolate_r_add4",ThmC.numerals_to_Free @{thm sqrt_isolate_r_add4}),

       (* a= d+g/sqrt(x) -> a-d=g/sqrt(x) *)

     Rule.Thm("sqrt_isolate_r_add5",ThmC.numerals_to_Free @{thm sqrt_isolate_r_add5}),

       (* a=d+e*g/h*sqrt(x)->a-d=e*g/h*sqrt(x)*)

     Rule.Thm("sqrt_isolate_r_add6",ThmC.numerals_to_Free @{thm sqrt_isolate_r_add6}),

       (* a= d+g/h*sqrt(x) -> a-d=g/h*sqrt(x) *)

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

       (* a=e*sqrt(x) -> a/e = sqrt(x) *)

     Rule.Thm("sqrt_square_equation_right_1",ThmC.numerals_to_Free @{thm sqrt_square_equation_right_1}),

	      (* a=sqrt(x) ->a^2=x *)

     Rule.Thm("sqrt_square_equation_right_2",ThmC.numerals_to_Free @{thm sqrt_square_equation_right_2}),

	      (* a=c*sqrt(x) ->a^2=c^2*x *)

     Rule.Thm("sqrt_square_equation_right_3",ThmC.numerals_to_Free @{thm sqrt_square_equation_right_3}),

	      (* a=c/sqrt(x) ->a^2=c^2/x *)

     Rule.Thm("sqrt_square_equation_right_4",ThmC.numerals_to_Free @{thm sqrt_square_equation_right_4}), 

	      (* a=c*d/sqrt(x) ->a^2=c^2*d^2/x *)

     Rule.Thm("sqrt_square_equation_right_5",ThmC.numerals_to_Free @{thm sqrt_square_equation_right_5}),

	      (* a=c/e*sqrt(x) ->a^2=c^2/e^2x *)

     Rule.Thm("sqrt_square_equation_right_6",ThmC.numerals_to_Free @{thm sqrt_square_equation_right_6})

	      (* a=c*d/g*sqrt(x) ->a^2=c^2*d^2/g^2*x *)

    ],

    scr = Rule.Empty_Prog

   });

\<close> ML \<open>

val rooteq_simplify = prep_rls'(

  Rule_Def.Repeat {id = "rooteq_simplify", 

       preconds = [], rew_ord = ("termlessI",termlessI), 

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

       (*asm_thm = [("sqrt_square_1", "")],*)

       rules = [Rule.Thm  ("real_assoc_1",ThmC.numerals_to_Free @{thm real_assoc_1}),

                             (* a+(b+c) = a+b+c *)

                Rule.Thm  ("real_assoc_2",ThmC.numerals_to_Free @{thm real_assoc_2}),

                             (* a*(b*c) = a*b*c *)

                Rule.Eval ("Groups.plus_class.plus", (**)eval_binop "#add_"),

                Rule.Eval ("Groups.minus_class.minus", (**)eval_binop "#sub_"),

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

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

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

                Rule.Eval ("Transcendental.powr" , (**)eval_binop "#power_"),

                Rule.Thm("real_plus_binom_pow2",ThmC.numerals_to_Free @{thm real_plus_binom_pow2}),

                Rule.Thm("real_minus_binom_pow2",ThmC.numerals_to_Free @{thm real_minus_binom_pow2}),

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

                     (* (a * b)^n = a^n * b^n*)

                Rule.Thm("sqrt_times_root_1",ThmC.numerals_to_Free @{thm sqrt_times_root_1}), 

                     (* sqrt b * sqrt c = sqrt(b*c) *)

                Rule.Thm("sqrt_times_root_2",ThmC.numerals_to_Free @{thm sqrt_times_root_2}),

                     (* a * sqrt a * sqrt b = a * sqrt(a*b) *)

                Rule.Thm("sqrt_square_2",ThmC.numerals_to_Free @{thm sqrt_square_2}),

                            (* sqrt (a \<up> 2) = a *)

                Rule.Thm("sqrt_square_1",ThmC.numerals_to_Free @{thm sqrt_square_1}) 

                            (* sqrt a  \<up>  2 = a *)

                ],

       scr = Rule.Empty_Prog

    });

\<close>

setup_rule

  RootEq_erls = RootEq_erls and

  rooteq_srls = rooteq_srls and

  sqrt_isolate = sqrt_isolate and

  l_sqrt_isolate = l_sqrt_isolate and

  r_sqrt_isolate = r_sqrt_isolate and

  rooteq_simplify = rooteq_simplify

subsection \<open>problems\<close>

(*Ambiguous input produces 5 parse trees -----------------------------------------------------\\*)

setup \<open>KEStore_Elems.add_pbts

  (* ---------root----------- *)

  [(Problem.prep_input thy "pbl_equ_univ_root" [] Problem.id_empty

      (["rootX", "univariate", "equation"],

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

          ("#Where" ,["(lhs e_e) is_rootTerm_in  (v_v::real) | " ^

	          "(rhs e_e) is_rootTerm_in  (v_v::real)"]),

          ("#Find"  ,["solutions v_v'i'"])],

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

    (* ---------sqrt----------- *)

    (Problem.prep_input thy "pbl_equ_univ_root_sq" [] Problem.id_empty

      (["sq", "rootX", "univariate", "equation"],

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

          ("#Where" ,["( ((lhs e_e) is_sqrtTerm_in (v_v::real)) &" ^

            "  ((lhs e_e) is_normSqrtTerm_in (v_v::real))   )  |" ^

	          "( ((rhs e_e) is_sqrtTerm_in (v_v::real)) &" ^

            "  ((rhs e_e) is_normSqrtTerm_in (v_v::real))   )"]),

          ("#Find"  ,["solutions v_v'i'"])],

          RootEq_prls,  SOME "solve (e_e::bool, v_v)", [["RootEq", "solve_sq_root_equation"]])),

    (* ---------normalise----------- *)

    (Problem.prep_input thy "pbl_equ_univ_root_norm" [] Problem.id_empty

      (["normalise", "rootX", "univariate", "equation"],

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

          ("#Where" ,["( ((lhs e_e) is_sqrtTerm_in (v_v::real)) &" ^

            "  Not((lhs e_e) is_normSqrtTerm_in (v_v::real)))  | " ^

	          "( ((rhs e_e) is_sqrtTerm_in (v_v::real)) &" ^

            "  Not((rhs e_e) is_normSqrtTerm_in (v_v::real)))"]),

          ("#Find"  ,["solutions v_v'i'"])],

        RootEq_prls, SOME "solve (e_e::bool, v_v)", [["RootEq", "norm_sq_root_equation"]]))]\<close>

(*Ambiguous input produces 5 parse trees -----------------------------------------------------//*)

subsection \<open>methods\<close>

setup \<open>KEStore_Elems.add_mets

    [MethodC.prep_input thy "met_rooteq" [] MethodC.id_empty

      (["RootEq"], [],

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

          crls=RootEq_crls, errpats = [], nrls = norm_Poly}, @{thm refl})]

\<close>

    (*-- normalise 20.10.02 --*)

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

  where

"norm_sqrt_equ e_e v_v = (

  let

    e_e = (

      (Repeat(Try (Rewrite ''makex1_x''))) #>

      (Try (Repeat (Rewrite_Set ''expand_rootbinoms''))) #>

      (Try (Rewrite_Set ''rooteq_simplify'')) #>

      (Try (Repeat (Rewrite_Set ''make_rooteq''))) #>

      (Try (Rewrite_Set ''rooteq_simplify''))

      ) e_e

  in

    SubProblem (''RootEq'', [''univariate'', ''equation''], [''no_met''])

      [BOOL e_e, REAL v_v])"

setup \<open>KEStore_Elems.add_mets

    [MethodC.prep_input thy "met_rooteq_norm" [] MethodC.id_empty

      (["RootEq", "norm_sq_root_equation"],

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

          ("#Where" ,["( ((lhs e_e) is_sqrtTerm_in (v_v::real)) &" ^

              "  Not((lhs e_e) is_normSqrtTerm_in (v_v::real)))  | " ^

              "( ((rhs e_e) is_sqrtTerm_in (v_v::real)) &" ^

              "  Not((rhs e_e) is_normSqrtTerm_in (v_v::real)))"]),

          ("#Find"  ,["solutions v_v'i'"])],

        {rew_ord'="termlessI", rls'=RootEq_erls, srls=Rule_Set.empty, prls=RootEq_prls, calc=[],

          crls=RootEq_crls, errpats = [], nrls = norm_Poly},

        @{thm norm_sqrt_equ.simps})]

\<close>

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

  where

"solve_sqrt_equ e_e v_v =

  (let

    e_e = (

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

      (Try (Rewrite_Set         ''rooteq_simplify'')) #> 

      (Try (Repeat (Rewrite_Set ''expand_rootbinoms''))) #>

      (Try (Repeat (Rewrite_Set ''make_rooteq''))) #>

      (Try (Rewrite_Set ''rooteq_simplify'')) ) e_e;

    L_L =

     (if (((lhs e_e) is_sqrtTerm_in v_v) | ((rhs e_e) is_sqrtTerm_in v_v))               

      then

        SubProblem (''RootEq'', [''normalise'', ''rootX'', ''univariate'', ''equation''], [''no_met''])

          [BOOL e_e, REAL v_v]

      else

        SubProblem (''RootEq'',[''univariate'',''equation''], [''no_met''])

          [BOOL e_e, REAL v_v])

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

setup \<open>KEStore_Elems.add_mets

    [MethodC.prep_input thy "met_rooteq_sq" [] MethodC.id_empty

      (["RootEq", "solve_sq_root_equation"],

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

          ("#Where" ,["(((lhs e_e) is_sqrtTerm_in (v_v::real))     & " ^

              " ((lhs e_e) is_normSqrtTerm_in (v_v::real))) |" ^

              "(((rhs e_e) is_sqrtTerm_in (v_v::real))     & " ^

              " ((rhs e_e) is_normSqrtTerm_in (v_v::real)))"]),

          ("#Find"  ,["solutions v_v'i'"])],

        {rew_ord'="termlessI", rls'=RootEq_erls, srls = rooteq_srls, prls = RootEq_prls, calc = [],

          crls=RootEq_crls, errpats = [], nrls = norm_Poly},

        @{thm solve_sqrt_equ.simps})]

\<close>

    (*-- right 28.08.02 --*)

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

  where

"solve_sqrt_equ_right e_e v_v =

  (let

    e_e = (

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

      (Try (Rewrite_Set ''rooteq_simplify'')) #>

      (Try (Repeat (Rewrite_Set ''expand_rootbinoms''))) #>

      (Try (Repeat (Rewrite_Set ''make_rooteq''))) #>

      (Try (Rewrite_Set ''rooteq_simplify'')) ) e_e

  in

    if (rhs e_e) is_sqrtTerm_in v_v

    then

      SubProblem (''RootEq'',[''normalise'',''rootX'',''univariate'',''equation''], [''no_met''])

        [BOOL e_e, REAL v_v]

    else

      SubProblem (''RootEq'',[''univariate'', ''equation''], [''no_met''])

        [BOOL e_e, REAL v_v])"

setup \<open>KEStore_Elems.add_mets

    [MethodC.prep_input thy "met_rooteq_sq_right" [] MethodC.id_empty

      (["RootEq", "solve_right_sq_root_equation"],

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

          ("#Where" ,["(rhs e_e) is_sqrtTerm_in v_v"]),

          ("#Find"  ,["solutions v_v'i'"])],

        {rew_ord' = "termlessI", rls' = RootEq_erls, srls = Rule_Set.empty, prls = RootEq_prls, calc = [],

          crls = RootEq_crls, errpats = [], nrls = norm_Poly},

        @{thm solve_sqrt_equ_right.simps})]

\<close>

    (*-- left 28.08.02 --*)

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

  where

"solve_sqrt_equ_left e_e v_v =

  (let

    e_e = (

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

      (Try (Rewrite_Set ''rooteq_simplify'')) #>

      (Try (Repeat (Rewrite_Set ''expand_rootbinoms''))) #>

      (Try (Repeat (Rewrite_Set ''make_rooteq''))) #>

      (Try (Rewrite_Set ''rooteq_simplify'')) ) e_e

  in

    if (lhs e_e) is_sqrtTerm_in v_v

    then

      SubProblem (''RootEq'', [''normalise'',''rootX'',''univariate'',''equation''], [''no_met''])

        [BOOL e_e, REAL v_v]

    else

      SubProblem (''RootEq'',[''univariate'',''equation''], [''no_met''])

        [BOOL e_e, REAL v_v])"

setup \<open>KEStore_Elems.add_mets

    [MethodC.prep_input thy "met_rooteq_sq_left" [] MethodC.id_empty

      (["RootEq", "solve_left_sq_root_equation"],

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

          ("#Where" ,["(lhs e_e) is_sqrtTerm_in v_v"]),

          ("#Find"  ,["solutions v_v'i'"])],

        {rew_ord'="termlessI", rls'=RootEq_erls, srls=Rule_Set.empty, prls=RootEq_prls, calc=[],

          crls=RootEq_crls, errpats = [], nrls = norm_Poly},

        @{thm solve_sqrt_equ_left.simps})]

\<close>

ML \<open>

\<close> ML \<open>

\<close> 

end