(* Title:  functions on lists for Scripts

    Author: Walther Neuper 0108

    (c) due to copyright terms

 *)

 theory ListC imports Complex_Main

 uses ("../library.sml")

      ("../calcelems.sml")

      ("termC.sml")("calculate.sml")

      ("rewrite.sml")

 begin

 use "../library.sml"        (*indent,...*)

 use "../calcelems.sml"      (*str_of_type, Thm,...*)

 use "termC.sml"  (*num_str,...*)

 use "calculate.sml" (*???*)

 use "rewrite.sml"   (*?*** At command "end" (line 205../ListC.thy*)

 text {* 'nat' in List.thy replaced by 'real' *}

 primrec LENGTH   :: "'a list => real"

 where

   LENGTH_NIL:	"LENGTH [] = 0"     (*length: 'a list => nat*)

 | LENGTH_CONS: "LENGTH (x#xs) = 1 + LENGTH xs"

 primrec del :: "['a list, 'a] => 'a list"

 where

   del_base: "del [] x = []"

 | del_rec:  "del (y#ys) x = (if x = y then ys else y#(del ys x))"

 definition

   list_diff :: "['a list, 'a list] => 'a list"         (* as -- bs *)

               ("(_ --/ _)" [66, 66] 65)

   where "a -- b == foldl del a b"

 consts NTH ::  "[real, 'a list] => 'a"

 axioms(*axiomatization where*)

  (*** more than one non-variable in pattern in "nth_ 1 [x] = x"--*)

   NTH_NIL:      "NTH 1 (x#xs) = x" (*and*)

 (*  NTH_CONS:     "NTH n (x#xs) = NTH (n+ -1) xs"  *)

 (*rewriter does not reach base case   ......    ;

   the condition involves another rule set (erls, eval_binop in Atools):*)

   NTH_CONS:     "1 < n ==> NTH n (x#xs) = NTH (n+ - 1) xs" (*and*)

 (*primrec from Isabelle/src/HOL/List.thy -- def.twice not allowed*)

 (*primrec*)

   hd_thm:	"hd(x#xs) = x" (*and*)

 (*primrec*)

   tl_Nil:	"tl([])   = []" (*and*)

   tl_Cons:		"tl(x#xs) = xs" (*and*)

 (*primrec*)

   null_Nil:	"null([])   = True" (*and*)

   null_Cons:	"null(x#xs) = False" (*and*)

 (*primrec*)

   LAST:	"last(x#xs) = (if xs=[] then x else last xs)" (*and*)

 (*primrec*)

   butlast_Nil:	"butlast []    = []" (*and*)

   butlast_Cons:	"butlast(x#xs) = (if xs=[] then [] else x#butlast xs)" (*and*)

 (*primrec

   mem_Nil:	"x mem []     = False"

   mem_Cons:	"x mem (y#ys) = (if y=x then True else x mem ys)"

 *)

   mem_Nil:	"x : set []     = False" (*and*)

   mem_Cons:	"x : set (y#ys) = (if y=x then True else x : set ys)" (*and*)

 (*primrec-------already named---

   "set [] = {}"

   "set (x#xs) = insert x (set xs)"

   primrec

   list_all_Nil  "list_all P [] = True"

   list_all_Cons "list_all P (x#xs) = (P(x) & list_all P xs)"

 ----------------*)

 (*primrec*)

   map_Nil:	"map f []     = []" (*and*)

   map_Cons:	"map f (x#xs) = f(x)#map f xs" (*and*)

 (*primrec*)

   append_Nil:  "[]    @ys = ys" (*and*)

   append_Cons: "(x#xs)@ys = x#(xs@ys)" (*and*)

 (*primrec*)

   rev_Nil:	"rev([])   = []" (*and*)

   rev_Cons:	"rev(x#xs) = rev(xs) @ [x]" (*and*)

 (*primrec*)

   filter_Nil:	"filter P []     = []" (*and*)

   filter_Cons:	"filter P (x#xs) =(if P x then x#filter P xs else filter P xs)" (*and*)

 (*primrec-------already named---

   foldl_Nil  "foldl f a [] = a"

   foldl_Cons "foldl f a (x#xs) = foldl f (f a x) xs"

 ----------------*)

 (*primrec*)

   foldr_Nil:	"foldr f [] a     = a" (*and*)

   foldr_Cons:	"foldr f (x#xs) a = f x (foldr f xs a)" (*and*)

 (*primrec*)

   concat_Nil:	"concat([])   = []" (*and*)

   concat_Cons:	"concat(x#xs) = x @ concat(xs)" (*and*)

 (*primrec-------already named---

   drop_Nil  "drop n [] = []"

   drop_Cons "drop n (x#xs) = (case n of 0 => x#xs | Suc(m) => drop m xs)"

   (* Warning: simpset does not contain this definition but separate theorems 

      for n=0 / n=Suc k*)

 (*primrec*)

   take_Nil  "take n [] = []"

   take_Cons "take n (x#xs) = (case n of 0 => [] | Suc(m) => x # take m xs)"

   (* Warning: simpset does not contain this definition but separate theorems 

      for n=0 / n=Suc k*)

 (*primrec*) 

   nth_Cons  "(x#xs)!n = (case n of 0 => x | (Suc k) => xs!k)"

   (* Warning: simpset does not contain this definition but separate theorems 

      for n=0 / n=Suc k*)

 (*primrec*)

  "    [][i:=v] = []"

  "(x#xs)[i:=v] = (case i of 0     => v # xs 

 			  | Suc j => x # xs[j:=v])"

 ----------------*)

 (*primrec*)

   takeWhile_Nil:	"takeWhile P []     = []" (*and*)

   takeWhile_Cons:

   "takeWhile P (x#xs) = (if P x then x#takeWhile P xs else [])" (*and*)

 (*primrec*)

   dropWhile_Nil:	"dropWhile P []     = []" (*and*)

   dropWhile_Cons:

   "dropWhile P (x#xs) = (if P x then dropWhile P xs else x#xs)" (*and*)

 (*primrec*)

   zip_Nil:	"zip xs []     = []" (*and*)

   zip_Cons:	"zip xs (y#ys) =(case xs of [] => [] | z#zs =>(z,y)#zip zs ys)" (*and*)

   (* Warning: simpset does not contain this definition but separate theorems 

      for xs=[] / xs=z#zs *)

 (*primrec

   upt_0   "[i..0(] = []"

   upt_Suc "[i..(Suc j)(] = (if i <= j then [i..j(] @ [j] else [])"

 *)

 (*primrec*)

   distinct_Nil:	"distinct []     = True" (*and*)

   distinct_Cons:	"distinct (x#xs) = (x ~: set xs & distinct xs)" (*and*)

 (*primrec*)

   remdups_Nil:	"remdups [] = []" (*and*)

   remdups_Cons:	"remdups (x#xs) = 

 		 (if x : set xs then remdups xs else x # remdups xs)" 

 (*primrec-------already named---

   replicate_0   "replicate  0      x = []"

   replicate_Suc "replicate (Suc n) x = x # replicate n x"

 ----------------*)

 (** Lexicographic orderings on lists ...!!!**)

 ML{* (*the former ListC.ML*)

 (** rule set for evaluating listexpr in scripts **)

 val list_rls = 

   Rls{id="list_rls",preconds = [], rew_ord = ("dummy_ord",dummy_ord), 

       erls = e_rls, srls = Erls, calc = [], errpatts = [],

       rules = (*8.01: copied from*)

       [Thm ("refl", num_str @{thm refl}),  (*'a<>b -> FALSE' by fun eval_equal*)

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

        Thm ("NTH_CONS",num_str @{thm NTH_CONS}),(*erls for cond. in Atools.ML*)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

        Thm ("map_Nil",num_str @{thm map_Cons}),

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

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

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

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

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

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

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

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

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

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

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

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

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

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

        ], scr = EmptyScr}:rls;

 *}

 ML{*

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

   [("list_rls",list_rls)

    ]);

 *}

 end