(* theory collecting all knowledge for Root

   created by: 

         date: 

   changed by: rlang

   last change by: rlang

             date: 02.10.21

*)

theory Root imports Poly begin

consts

  (*sqrt   :: "real => real"         Isabelle "NthRoot.sqrt"*)

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

axioms (*.not contained in Isabelle2002,

         stated as axioms, TODO: prove as theorems;

         theorem-IDs 'xxxI' with ^^^ instead of ^ in 'xxx' in Isabelle2002.*)

  root_plus_minus:         "0 <= b ==> 

			   (a^^^2 = b) = ((a = sqrt b) | (a = (-1)*sqrt b))"

  root_false:		  "b < 0 ==> (a^^^2 = b) = False"

 (* for expand_rootbinom *)

  real_pp_binom_times:     "(a + b)*(c + d) = a*c + a*d + b*c + b*d"

  real_pm_binom_times:     "(a + b)*(c - d) = a*c - a*d + b*c - b*d"

  real_mp_binom_times:     "(a - b)*(c + d) = a*c + a*d - b*c - b*d"

  real_mm_binom_times:     "(a - b)*(c - d) = a*c - a*d - b*c + b*d"

  real_plus_binom_pow3:    "(a + b)^^^3 = a^^^3 + 3*a^^^2*b + 3*a*b^^^2 + b^^^3"

  real_minus_binom_pow3:   "(a - b)^^^3 = a^^^3 - 3*a^^^2*b + 3*a*b^^^2 - b^^^3"

  realpow_mul:             "(a*b)^^^n = a^^^n * b^^^n"

  real_diff_minus:         "a - b = a + (-1) * b"

  real_plus_binom_times:   "(a + b)*(a + b) = a^^^2 + 2*a*b + b^^^2"

  real_minus_binom_times:  "(a - b)*(a - b) = a^^^2 - 2*a*b + b^^^2"

  real_plus_binom_pow2:    "(a + b)^^^2 = a^^^2 + 2*a*b + b^^^2"

  real_minus_binom_pow2:   "(a - b)^^^2 = a^^^2 - 2*a*b + b^^^2"

  real_plus_minus_binom1:  "(a + b)*(a - b) = a^^^2 - b^^^2"

  real_plus_minus_binom2:  "(a - b)*(a + b) = a^^^2 - b^^^2"

  real_root_positive:      "0 <= a ==> (x ^^^ 2 = a) = (x = sqrt a)"

  real_root_negative:      "a <  0 ==> (x ^^^ 2 = a) = False"

ML {*

val thy = @{theory};

(*-------------------------functions---------------------*)

(*evaluation square-root over the integers*)

fun eval_sqrt (thmid:string) (op_:string) (t as 

	       (Const(op0,t0) $ arg)) thy = 

    (case arg of 

	Free (n1,t1) =>

	(case int_of_str n1 of

	     SOME ni => 

	     if ni < 0 then NONE

	     else

		 let val fact = squfact ni;

		 in if fact*fact = ni 

		    then SOME ("#sqrt #"^(string_of_int ni)^" = #"

			       ^(string_of_int (if ni = 0 then 0

						else ni div fact)),

			       Trueprop $ mk_equality (t, term_of_num t1 fact))

		    else if fact = 1 then NONE

		    else SOME ("#sqrt #"^(string_of_int ni)^" = sqrt (#"

			       ^(string_of_int fact)^" * #"

			       ^(string_of_int fact)^" * #"

			       ^(string_of_int (ni div (fact*fact))^")"),

			       Trueprop $ 

					(mk_equality 

					     (t, 

					      (mk_factroot op0 t1 fact 

						(ni div (fact*fact))))))

	         end

	   | NONE => NONE)

      | _ => NONE)

  | eval_sqrt _ _ _ _ = NONE;

(*val (thmid, op_, t as Const(op0,t0) $ arg) = ("","", str2term "sqrt 0");

> eval_sqrt thmid op_ t thy;

> val Free (n1,t1) = arg; 

> val SOME ni = int_of_str n1;

*)

calclist':= overwritel (!calclist', 

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

    (*different types for 'sqrt 4' --- 'Calculate SQRT'*)

    ]);

local (* Vers. 7.10.99.A *)

open Term;  (* for type order = EQUAL | LESS | GREATER *)

fun pr_ord EQUAL = "EQUAL"

  | pr_ord LESS  = "LESS"

  | pr_ord GREATER = "GREATER";

fun dest_hd' (Const (a, T)) =                          (* ~ term.ML *)

  (case a of "NthRoot.sqrt"  => ((("|||", 0), T), 0)      (*WN greatest *)

	   | _ => (((a, 0), T), 0))

  | dest_hd' (Free (a, T)) = (((a, 0), T), 1)

  | dest_hd' (Var v) = (v, 2)

  | dest_hd' (Bound i) = ((("", i), dummyT), 3)

  | dest_hd' (Abs (_, T, _)) = ((("", 0), T), 4);

fun size_of_term' (Const(str,_) $ t) =

    (case str of "NthRoot.sqrt"  => (1000 + size_of_term' t)

               | _ => 1 + size_of_term' t)

  | size_of_term' (Abs (_,_,body)) = 1 + size_of_term' body

  | size_of_term' (f $ t) = size_of_term' f  +  size_of_term' t

  | size_of_term' _ = 1;

fun term_ord' pr thy (Abs (_, T, t), Abs(_, U, u)) =       (* ~ term.ML *)

      (case term_ord' pr thy (t, u) of EQUAL => Term_Ord.typ_ord (T, U) | ord => ord)

  | term_ord' pr thy (t, u) =

      (if pr then 

	 let

	   val (f, ts) = strip_comb t and (g, us) = strip_comb u;

	   val _=writeln("t= f@ts= \""^

	      ((Syntax.string_of_term (thy2ctxt thy)) f)^"\" @ \"["^

	      (commas(map(Syntax.string_of_term (thy2ctxt thy)) ts))^"]\"");

	   val _=writeln("u= g@us= \""^

	      ((Syntax.string_of_term (thy2ctxt thy)) g)^"\" @ \"["^

	      (commas(map(Syntax.string_of_term (thy2ctxt thy)) us))^"]\"");

	   val _=writeln("size_of_term(t,u)= ("^

	      (string_of_int(size_of_term' t))^", "^

	      (string_of_int(size_of_term' u))^")");

	   val _=writeln("hd_ord(f,g)      = "^((pr_ord o hd_ord)(f,g)));

	   val _=writeln("terms_ord(ts,us) = "^

			   ((pr_ord o terms_ord str false)(ts,us)));

	   val _=writeln("-------");

	 in () end

       else ();

	 case int_ord (size_of_term' t, size_of_term' u) of

	   EQUAL =>

	     let val (f, ts) = strip_comb t and (g, us) = strip_comb u in

	       (case hd_ord (f, g) of EQUAL => (terms_ord str pr) (ts, us) 

	     | ord => ord)

	     end

	 | ord => ord)

and hd_ord (f, g) =                                        (* ~ term.ML *)

  prod_ord (prod_ord Term_Ord.indexname_ord Term_Ord.typ_ord) int_ord 

            (dest_hd' f, dest_hd' g)

and terms_ord str pr (ts, us) = 

    list_ord (term_ord' pr (assoc_thy "Isac"))(ts, us);

in

(* associates a+(b+c) => (a+b)+c = a+b+c ... avoiding parentheses 

  by (1) size_of_term: less(!) to right, size_of 'sqrt (...)' = 1 

     (2) hd_ord: greater to right, 'sqrt' < numerals < variables

     (3) terms_ord: recurs. on args, greater to right

*)

(*args

   pr: print trace, WN0509 'sqrt_right true' not used anymore

   thy:

   subst: no bound variables, only Root.sqrt

   tu: the terms to compare (t1, t2) ... *)

fun sqrt_right (pr:bool) thy (_:subst) tu = 

    (term_ord' pr thy(***) tu = LESS );

end;

rew_ord' := overwritel (!rew_ord',

[("termlessI", termlessI),

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

 ]);

(*-------------------------rulse-------------------------*)

val Root_crls = 

      append_rls "Root_crls" Atools_erls 

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

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

        Calc ("Rings.inverse_class.divide",eval_cancel "#divide_e"),

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

        Calc ("Groups.plus_class.plus", eval_binop "#add_"), 

        Calc ("Groups.minus_class.minus", eval_binop "#sub_"),

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

        Calc ("op =",eval_equal "#equal_") 

        ];

val Root_erls = 

      append_rls "Root_erls" Atools_erls 

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

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

        Calc ("Rings.inverse_class.divide",eval_cancel "#divide_e"),

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

        Calc ("Groups.plus_class.plus", eval_binop "#add_"), 

        Calc ("Groups.minus_class.minus", eval_binop "#sub_"),

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

        Calc ("op =",eval_equal "#equal_") 

        ];

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

			[("Root_erls",Root_erls) (*FIXXXME:del with rls.rls'*) 

			 ]);

val make_rooteq = prep_rls(

  Rls{id = "make_rooteq", preconds = []:term list, 

      rew_ord = ("sqrt_right", sqrt_right false thy),

      erls = Atools_erls, srls = Erls,

      calc = [],

      (*asm_thm = [],*)

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

	       (*"a - b = a + (-1) * b"*)

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

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

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

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

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

	       (*"(z1.0 - z2.0) * w = z1.0 * w - z2.0 * w"*)

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

	       (*"w * (z1.0 - z2.0) = w * z1.0 - w * z2.0"*)

	       Thm ("mult_1_left",num_str @{thm mult_1_left}),                         

	       (*"1 * z = z"*)

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

	       (*"0 * z = 0"*)

	       Thm ("add_0_left",num_str @{thm add_0_left}),		

	       (*"0 + z = z"*)

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

		(*AC-rewriting*)

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

         	(**)

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

	        (**)

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

		(**)

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

	        (**)

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

	        (**)

	       Thm ("sym_realpow_twoI",

                     num_str (@{thm realpow_twoI} RS @{thm sym})),

	       (*"r1 * r1 = r1 ^^^ 2"*)

	       Thm ("realpow_plus_1",num_str @{thm realpow_plus_1}),			

	       (*"r * r ^^^ n = r ^^^ (n + 1)"*)

	       Thm ("sym_real_mult_2",

                     num_str (@{thm real_mult_2} RS @{thm sym})),

	       (*"z1 + z1 = 2 * z1"*)

	       Thm ("real_mult_2_assoc",num_str @{thm real_mult_2_assoc}),		

	       (*"z1 + (z1 + k) = 2 * z1 + k"*)

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

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

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

	       (*"[| l is_const; m is_const |] ==>  

                                   l * n + (m * n + k) =  (l + m) * n + k"*)

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

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

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

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

	       Calc ("Groups.plus_class.plus", eval_binop "#add_"), 

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

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

	       ],

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

      }:rls);      

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

			[("make_rooteq", make_rooteq)

			 ]);

val expand_rootbinoms = prep_rls(

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

      rew_ord = ("termlessI",termlessI),

      erls = Atools_erls, srls = Erls,

      calc = [],

      (*asm_thm = [],*)

      rules = [Thm ("real_plus_binom_pow2"  ,num_str @{thm real_plus_binom_pow2}),     

	       (*"(a + b) ^^^ 2 = a ^^^ 2 + 2 * a * b + b ^^^ 2"*)

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

	       (*"(a + b)*(a + b) = ...*)

	       Thm ("real_minus_binom_pow2" ,num_str @{thm real_minus_binom_pow2}),    

		(*"(a - b) ^^^ 2 = a ^^^ 2 - 2 * a * b + b ^^^ 2"*)

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

	       (*"(a - b)*(a - b) = ...*)

	       Thm ("real_plus_minus_binom1",num_str @{thm real_plus_minus_binom1}),   

		(*"(a + b) * (a - b) = a ^^^ 2 - b ^^^ 2"*)

	       Thm ("real_plus_minus_binom2",num_str @{thm real_plus_minus_binom2}),   

		(*"(a - b) * (a + b) = a ^^^ 2 - b ^^^ 2"*)

	       (*RL 020915*)

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

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

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

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

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

		(*(a - b)*(c p d) = a*c + a*d - b*c - b*d*)

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

		(*(a - b)*(c p d) = a*c - a*d - b*c + b*d*)

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

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

	       Thm ("mult_1_left",num_str @{thm mult_1_left}),         (*"1 * z = z"*)

	       Thm ("mult_zero_left",num_str @{thm mult_zero_left}),         (*"0 * z = 0"*)

	       Thm ("add_0_left",num_str @{thm add_0_left}), 

                 (*"0 + z = z"*)

	       Calc ("Groups.plus_class.plus", eval_binop "#add_"), 

	       Calc ("Groups.minus_class.minus", eval_binop "#sub_"), 

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

	       Calc ("Rings.inverse_class.divide"  ,eval_cancel "#divide_e"),

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

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

	       Thm ("sym_realpow_twoI",

                     num_str (@{thm realpow_twoI} RS @{thm sym})),

	       (*"r1 * r1 = r1 ^^^ 2"*)

	       Thm ("realpow_plus_1",num_str @{thm realpow_plus_1}),			

	       (*"r * r ^^^ n = r ^^^ (n + 1)"*)

	       Thm ("real_mult_2_assoc",num_str @{thm real_mult_2_assoc}),		

	       (*"z1 + (z1 + k) = 2 * z1 + k"*)

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

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

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

	       (*"[| l is_const; m is_const |] ==>

                  l * n + (m * n + k) =  (l + m) * n + k"*)

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

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

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

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

	       Calc ("Groups.plus_class.plus", eval_binop "#add_"), 

	       Calc ("Groups.minus_class.minus", eval_binop "#sub_"), 

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

	       Calc ("Rings.inverse_class.divide"  ,eval_cancel "#divide_e"),

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

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

	       ],

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

       }:rls);      

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

			[("expand_rootbinoms", expand_rootbinoms)

			 ]);

*}

end