src/HOL/ex/Abstract_NAT.thy
 author nipkow Fri Mar 06 17:38:47 2009 +0100 (2009-03-06) changeset 30313 b2441b0c8d38 parent 29234 60f7fb56f8cd child 44603 a6f9a70d655d permissions -rw-r--r--
```     1 (*
```
```     2     Author:     Makarius
```
```     3 *)
```
```     4
```
```     5 header {* Abstract Natural Numbers primitive recursion *}
```
```     6
```
```     7 theory Abstract_NAT
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 text {* Axiomatic Natural Numbers (Peano) -- a monomorphic theory. *}
```
```    12
```
```    13 locale NAT =
```
```    14   fixes zero :: 'n
```
```    15     and succ :: "'n \<Rightarrow> 'n"
```
```    16   assumes succ_inject [simp]: "(succ m = succ n) = (m = n)"
```
```    17     and succ_neq_zero [simp]: "succ m \<noteq> zero"
```
```    18     and induct [case_names zero succ, induct type: 'n]:
```
```    19       "P zero \<Longrightarrow> (\<And>n. P n \<Longrightarrow> P (succ n)) \<Longrightarrow> P n"
```
```    20 begin
```
```    21
```
```    22 lemma zero_neq_succ [simp]: "zero \<noteq> succ m"
```
```    23   by (rule succ_neq_zero [symmetric])
```
```    24
```
```    25
```
```    26 text {* \medskip Primitive recursion as a (functional) relation -- polymorphic! *}
```
```    27
```
```    28 inductive
```
```    29   Rec :: "'a \<Rightarrow> ('n \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'n \<Rightarrow> 'a \<Rightarrow> bool"
```
```    30   for e :: 'a and r :: "'n \<Rightarrow> 'a \<Rightarrow> 'a"
```
```    31 where
```
```    32     Rec_zero: "Rec e r zero e"
```
```    33   | Rec_succ: "Rec e r m n \<Longrightarrow> Rec e r (succ m) (r m n)"
```
```    34
```
```    35 lemma Rec_functional:
```
```    36   fixes x :: 'n
```
```    37   shows "\<exists>!y::'a. Rec e r x y"
```
```    38 proof -
```
```    39   let ?R = "Rec e r"
```
```    40   show ?thesis
```
```    41   proof (induct x)
```
```    42     case zero
```
```    43     show "\<exists>!y. ?R zero y"
```
```    44     proof
```
```    45       show "?R zero e" ..
```
```    46       fix y assume "?R zero y"
```
```    47       then show "y = e" by cases simp_all
```
```    48     qed
```
```    49   next
```
```    50     case (succ m)
```
```    51     from `\<exists>!y. ?R m y`
```
```    52     obtain y where y: "?R m y"
```
```    53       and yy': "\<And>y'. ?R m y' \<Longrightarrow> y = y'" by blast
```
```    54     show "\<exists>!z. ?R (succ m) z"
```
```    55     proof
```
```    56       from y show "?R (succ m) (r m y)" ..
```
```    57       fix z assume "?R (succ m) z"
```
```    58       then obtain u where "z = r m u" and "?R m u" by cases simp_all
```
```    59       with yy' show "z = r m y" by (simp only:)
```
```    60     qed
```
```    61   qed
```
```    62 qed
```
```    63
```
```    64
```
```    65 text {* \medskip The recursion operator -- polymorphic! *}
```
```    66
```
```    67 definition
```
```    68   rec :: "'a \<Rightarrow> ('n \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'n \<Rightarrow> 'a" where
```
```    69   "rec e r x = (THE y. Rec e r x y)"
```
```    70
```
```    71 lemma rec_eval:
```
```    72   assumes Rec: "Rec e r x y"
```
```    73   shows "rec e r x = y"
```
```    74   unfolding rec_def
```
```    75   using Rec_functional and Rec by (rule the1_equality)
```
```    76
```
```    77 lemma rec_zero [simp]: "rec e r zero = e"
```
```    78 proof (rule rec_eval)
```
```    79   show "Rec e r zero e" ..
```
```    80 qed
```
```    81
```
```    82 lemma rec_succ [simp]: "rec e r (succ m) = r m (rec e r m)"
```
```    83 proof (rule rec_eval)
```
```    84   let ?R = "Rec e r"
```
```    85   have "?R m (rec e r m)"
```
```    86     unfolding rec_def using Rec_functional by (rule theI')
```
```    87   then show "?R (succ m) (r m (rec e r m))" ..
```
```    88 qed
```
```    89
```
```    90
```
```    91 text {* \medskip Example: addition (monomorphic) *}
```
```    92
```
```    93 definition
```
```    94   add :: "'n \<Rightarrow> 'n \<Rightarrow> 'n" where
```
```    95   "add m n = rec n (\<lambda>_ k. succ k) m"
```
```    96
```
```    97 lemma add_zero [simp]: "add zero n = n"
```
```    98   and add_succ [simp]: "add (succ m) n = succ (add m n)"
```
```    99   unfolding add_def by simp_all
```
```   100
```
```   101 lemma add_assoc: "add (add k m) n = add k (add m n)"
```
```   102   by (induct k) simp_all
```
```   103
```
```   104 lemma add_zero_right: "add m zero = m"
```
```   105   by (induct m) simp_all
```
```   106
```
```   107 lemma add_succ_right: "add m (succ n) = succ (add m n)"
```
```   108   by (induct m) simp_all
```
```   109
```
```   110 lemma "add (succ (succ (succ zero))) (succ (succ zero)) =
```
```   111     succ (succ (succ (succ (succ zero))))"
```
```   112   by simp
```
```   113
```
```   114
```
```   115 text {* \medskip Example: replication (polymorphic) *}
```
```   116
```
```   117 definition
```
```   118   repl :: "'n \<Rightarrow> 'a \<Rightarrow> 'a list" where
```
```   119   "repl n x = rec [] (\<lambda>_ xs. x # xs) n"
```
```   120
```
```   121 lemma repl_zero [simp]: "repl zero x = []"
```
```   122   and repl_succ [simp]: "repl (succ n) x = x # repl n x"
```
```   123   unfolding repl_def by simp_all
```
```   124
```
```   125 lemma "repl (succ (succ (succ zero))) True = [True, True, True]"
```
```   126   by simp
```
```   127
```
```   128 end
```
```   129
```
```   130
```
```   131 text {* \medskip Just see that our abstract specification makes sense \dots *}
```
```   132
```
```   133 interpretation NAT 0 Suc
```
```   134 proof (rule NAT.intro)
```
```   135   fix m n
```
```   136   show "(Suc m = Suc n) = (m = n)" by simp
```
```   137   show "Suc m \<noteq> 0" by simp
```
```   138   fix P
```
```   139   assume zero: "P 0"
```
```   140     and succ: "\<And>n. P n \<Longrightarrow> P (Suc n)"
```
```   141   show "P n"
```
```   142   proof (induct n)
```
```   143     case 0 show ?case by (rule zero)
```
```   144   next
```
```   145     case Suc then show ?case by (rule succ)
```
```   146   qed
```
```   147 qed
```
```   148
```
```   149 end
```