src/FOL/ex/Natural_Numbers.thy
 author wenzelm Fri Oct 12 12:06:23 2001 +0200 (2001-10-12) changeset 11726 2b2a45abe876 parent 11696 233362cfecc7 child 11789 da81334357ba permissions -rw-r--r--
tuned;
```     1 (*  Title:      FOL/ex/Natural_Numbers.thy
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```     2     ID:         \$Id\$
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```     3     Author:     Markus Wenzel, TU Munich
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```     4
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```     5 Theory of the natural numbers: Peano's axioms, primitive recursion.
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```     6 (Modernized version of Larry Paulson's theory "Nat".)
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```     7 *)
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```     8
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```     9 theory Natural_Numbers = FOL:
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```    10
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```    11 typedecl nat
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```    12 arities nat :: "term"
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```    13
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```    14 consts
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```    15   Zero :: nat    ("0")
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```    16   Suc :: "nat => nat"
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```    17   rec :: "[nat, 'a, [nat, 'a] => 'a] => 'a"
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```    18
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```    19 axioms
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```    20   induct [case_names Zero Suc, induct type: nat]:
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```    21     "P(0) ==> (!!x. P(x) ==> P(Suc(x))) ==> P(n)"
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```    22   Suc_inject: "Suc(m) = Suc(n) ==> m = n"
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```    23   Suc_neq_0: "Suc(m) = 0 ==> R"
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```    24   rec_0: "rec(0, a, f) = a"
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```    25   rec_Suc: "rec(Suc(m), a, f) = f(m, rec(m, a, f))"
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```    26
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```    27 lemma Suc_n_not_n: "Suc(k) \<noteq> k"
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```    28 proof (induct k)
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```    29   show "Suc(0) \<noteq> 0"
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```    30   proof
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```    31     assume "Suc(0) = 0"
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```    32     thus False by (rule Suc_neq_0)
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```    33   qed
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```    34   fix n assume hyp: "Suc(n) \<noteq> n"
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```    35   show "Suc(Suc(n)) \<noteq> Suc(n)"
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```    36   proof
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```    37     assume "Suc(Suc(n)) = Suc(n)"
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```    38     hence "Suc(n) = n" by (rule Suc_inject)
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```    39     with hyp show False by contradiction
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```    40   qed
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```    41 qed
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```    42
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```    43
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```    44 constdefs
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```    45   add :: "[nat, nat] => nat"    (infixl "+" 60)
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```    46   "m + n == rec(m, n, \<lambda>x y. Suc(y))"
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```    47
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```    48 lemma add_0 [simp]: "0 + n = n"
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```    49   by (unfold add_def) (rule rec_0)
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```    50
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```    51 lemma add_Suc [simp]: "Suc(m) + n = Suc(m + n)"
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```    52   by (unfold add_def) (rule rec_Suc)
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```    53
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```    54 lemma add_assoc: "(k + m) + n = k + (m + n)"
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```    55   by (induct k) simp_all
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```    56
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```    57 lemma add_0_right: "m + 0 = m"
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```    58   by (induct m) simp_all
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```    59
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```    60 lemma add_Suc_right: "m + Suc(n) = Suc(m + n)"
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```    61   by (induct m) simp_all
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```    62
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```    63 lemma "(!!n. f(Suc(n)) = Suc(f(n))) ==> f(i + j) = i + f(j)"
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```    64 proof -
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```    65   assume "!!n. f(Suc(n)) = Suc(f(n))"
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```    66   thus ?thesis by (induct i) simp_all
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```    67 qed
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```    68
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```    69 end
```