src/HOL/Corec_Examples/Tests/Small_Concrete.thy
author blanchet
Tue Mar 22 12:39:37 2016 +0100 (2016-03-22)
changeset 62696 7325d8573fb8
child 62726 5b2a7caa855b
permissions -rw-r--r--
added 'corec' examples and tests
     1 (*  Title:      HOL/Corec_Examples/Tests/Small_Concrete.thy
     2     Author:     Aymeric Bouzy, Ecole polytechnique
     3     Author:     Jasmin Blanchette, Inria, LORIA, MPII
     4     Copyright   2015, 2016
     5 
     6 Small concrete examples.
     7 *)
     8 
     9 section {* Small Concrete Examples *}
    10 
    11 theory Small_Concrete
    12 imports "~~/src/HOL/Library/BNF_Corec"
    13 begin
    14 
    15 subsection {* Streams of Natural Numbers *}
    16 
    17 codatatype natstream = S (head: nat) (tail: natstream)
    18 
    19 corec (friend) incr_all where
    20   "incr_all s = S (head s + 1) (incr_all (tail s))"
    21 
    22 corec all_numbers where
    23   "all_numbers = S 0 (incr_all all_numbers)"
    24 
    25 corec all_numbers_efficient where
    26   "all_numbers_efficient n = S n (all_numbers_efficient (n + 1))"
    27 
    28 corec remove_multiples where
    29   "remove_multiples n s =
    30     (if (head s) mod n = 0 then
    31       S (head (tail s)) (remove_multiples n (tail (tail s)))
    32     else
    33       S (head s) (remove_multiples n (tail s)))"
    34 
    35 corec prime_numbers where
    36   "prime_numbers known_primes =
    37     (let next_prime = head (fold (%n s. remove_multiples n s) known_primes (tail (tail all_numbers))) in
    38       S next_prime (prime_numbers (next_prime # known_primes)))"
    39 
    40 term "prime_numbers []"
    41 
    42 corec prime_numbers_more_efficient where
    43   "prime_numbers_more_efficient n remaining_numbers =
    44     (let remaining_numbers = remove_multiples n remaining_numbers in
    45       S (head remaining_numbers) (prime_numbers_more_efficient (head remaining_numbers) remaining_numbers))"
    46 
    47 term "prime_numbers_more_efficient 0 (tail (tail all_numbers))"
    48 
    49 corec (friend) alternate where
    50   "alternate s1 s2 = S (head s1) (S (head s2) (alternate (tail s1) (tail s2)))"
    51 
    52 corec (friend) all_sums where
    53   "all_sums s1 s2 = S (head s1 + head s2) (alternate (all_sums s1 (tail s2)) (all_sums (tail s1) s2))"
    54 
    55 corec app_list where
    56   "app_list s l = (case l of
    57     [] \<Rightarrow> s
    58   | a # r \<Rightarrow> S a (app_list s r))"
    59 
    60 friend_of_corec app_list where
    61   "app_list s l = (case l of
    62     [] \<Rightarrow> (case s of S a b \<Rightarrow> S a b)
    63   | a # r \<Rightarrow> S a (app_list s r))"
    64   sorry
    65 
    66 corec expand_with where
    67   "expand_with f s = (let l = f (head s) in S (hd l) (app_list (expand_with f (tail s)) (tl l)))"
    68 
    69 friend_of_corec expand_with where
    70   "expand_with f s = (let l = f (head s) in S (hd l) (app_list (expand_with f (tail s)) (tl l)))"
    71   sorry
    72 
    73 corec iterations where
    74   "iterations f a = S a (iterations f (f a))"
    75 
    76 corec exponential_iterations where
    77   "exponential_iterations f a = S (f a) (exponential_iterations (f o f) a)"
    78 
    79 corec (friend) alternate_list where
    80   "alternate_list l = (let heads = (map head l) in S (hd heads) (app_list (alternate_list (map tail l)) (tl heads)))"
    81 
    82 corec switch_one_two0 where
    83   "switch_one_two0 f a s = (case s of
    84     S b r \<Rightarrow> S b (S a (f r)))"
    85 
    86 corec switch_one_two where
    87   "switch_one_two s = (case s of
    88     S a (S b r) \<Rightarrow> S b (S a (switch_one_two r)))"
    89 
    90 corec fibonacci where
    91   "fibonacci n m = S m (fibonacci (n + m) n)"
    92 
    93 corec sequence2 where
    94   "sequence2 f u1 u0 = S u0 (sequence2 f (f u1 u0) u1)"
    95 
    96 corec (friend) alternate_with_function where
    97   "alternate_with_function f s =
    98     (let f_head_s = f (head s) in S (head f_head_s) (alternate (tail f_head_s) (alternate_with_function f (tail s))))"
    99 
   100 corec h where
   101   "h l s = (case l of
   102     [] \<Rightarrow> s
   103   | (S a s') # r \<Rightarrow> S a (alternate s (h r s')))"
   104 
   105 friend_of_corec h where
   106   "h l s = (case l of
   107     [] \<Rightarrow> (case s of S a b \<Rightarrow> S a b)
   108   | (S a s') # r \<Rightarrow> S a (alternate s (h r s')))"
   109   sorry
   110 
   111 corec z where
   112   "z = S 0 (S 0 z)"
   113 
   114 lemma "\<And>x. x = S 0 (S 0 x) \<Longrightarrow> x = z"
   115   apply corec_unique
   116   apply (rule z.code)
   117   done
   118 
   119 corec enum where
   120   "enum m = S m (enum (m + 1))"
   121 
   122 lemma "(\<And>m. f m = S m (f (m + 1))) \<Longrightarrow> f m = enum m"
   123   apply corec_unique
   124   apply (rule enum.code)
   125   done
   126 
   127 lemma "(\<forall>m. f m = S m (f (m + 1))) \<Longrightarrow> f m = enum m"
   128   apply corec_unique
   129   apply (rule enum.code)
   130   done
   131 
   132 
   133 subsection {* Lazy Lists of Natural Numbers *}
   134 
   135 codatatype llist = LNil | LCons nat llist
   136 
   137 corec h1 where
   138   "h1 x = (if x = 1 then
   139     LNil
   140   else
   141     let x = if x mod 2 = 0 then x div 2 else 3 * x + 1 in
   142     LCons x (h1 x))"
   143 
   144 corec h3 where
   145   "h3 s = (case s of
   146     LNil \<Rightarrow> LNil
   147   | LCons x r \<Rightarrow> LCons x (h3 r))"
   148 
   149 corec (friend) fold_map where
   150   "fold_map f a s = (let v = f a (head s) in S v (fold_map f v (tail s)))"
   151 
   152 
   153 subsection {* Coinductie Natural Numbers *}
   154 
   155 codatatype conat = CoZero | CoSuc conat
   156 
   157 corec sum where
   158   "sum x y = (case x of
   159       CoZero \<Rightarrow> y
   160     | CoSuc x \<Rightarrow> CoSuc (sum x y))"
   161 
   162 friend_of_corec sum where
   163   "sum x y = (case x of
   164       CoZero \<Rightarrow> (case y of CoZero \<Rightarrow> CoZero | CoSuc y \<Rightarrow> CoSuc y)
   165     | CoSuc x \<Rightarrow> CoSuc (sum x y))"
   166   sorry
   167 
   168 corec (friend) prod where
   169   "prod x y = (case (x, y) of
   170       (CoZero, _) \<Rightarrow> CoZero
   171     | (_, CoZero) \<Rightarrow> CoZero
   172     | (CoSuc x, CoSuc y) \<Rightarrow> CoSuc (sum (prod x y) (sum x y)))"
   173 
   174 end