src/FOL/ex/Natural_Numbers.thy
 changeset 11673 79e5536af6c4 child 11679 afdbee613f58
```     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.2 +++ b/src/FOL/ex/Natural_Numbers.thy	Thu Oct 04 15:25:31 2001 +0200
1.3 @@ -0,0 +1,69 @@
1.4 +(*  Title:      FOL/ex/Natural_Numbers.thy
1.5 +    ID:         \$Id\$
1.6 +    Author:     Markus Wenzel, TU Munich
1.7 +
1.8 +Theory of the natural numbers: Peano's axioms, primitive recursion.
1.9 +(Modernized version of Larry Paulson's theory "Nat".)
1.10 +*)
1.11 +
1.12 +theory Natural_Numbers = FOL:
1.13 +
1.14 +typedecl nat
1.15 +arities nat :: "term"
1.16 +
1.17 +consts
1.18 +  Zero :: nat    ("0")
1.19 +  Suc :: "nat => nat"
1.20 +  rec :: "[nat, 'a, [nat, 'a] => 'a] => 'a"
1.21 +
1.22 +axioms
1.23 +  induct [induct type: nat]:
1.24 +    "P(0) \<Longrightarrow> (!!x. P(x) ==> P(Suc(x))) ==> P(n)"
1.25 +  Suc_inject: "Suc(m) = Suc(n) ==> m = n"
1.26 +  Suc_neq_0: "Suc(m) = 0 ==> R"
1.27 +  rec_0: "rec(0, a, f) = a"
1.28 +  rec_Suc: "rec(Suc(m), a, f) = f(m, rec(m, a, f))"
1.29 +
1.30 +lemma Suc_n_not_n: "Suc(k) \<noteq> k"
1.31 +proof (induct k)
1.32 +  show "Suc(0) \<noteq> 0"
1.33 +  proof
1.34 +    assume "Suc(0) = 0"
1.35 +    thus False by (rule Suc_neq_0)
1.36 +  qed
1.37 +  fix n assume hyp: "Suc(n) \<noteq> n"
1.38 +  show "Suc(Suc(n)) \<noteq> Suc(n)"
1.39 +  proof
1.40 +    assume "Suc(Suc(n)) = Suc(n)"
1.41 +    hence "Suc(n) = n" by (rule Suc_inject)
1.42 +    with hyp show False by contradiction
1.43 +  qed
1.44 +qed
1.45 +
1.46 +
1.47 +constdefs
1.48 +  add :: "[nat, nat] => nat"    (infixl "+" 60)
1.49 +  "m + n == rec(m, n, \<lambda>x y. Suc(y))"
1.50 +
1.51 +lemma add_0 [simp]: "0 + n = n"
1.52 +  by (unfold add_def) (rule rec_0)
1.53 +
1.54 +lemma add_Suc [simp]: "Suc(m) + n = Suc(m + n)"
1.55 +  by (unfold add_def) (rule rec_Suc)
1.56 +
1.57 +lemma add_assoc: "(k + m) + n = k + (m + n)"
1.58 +  by (induct k) simp_all
1.59 +
1.60 +lemma add_0_right: "m + 0 = m"
1.61 +  by (induct m) simp_all
1.62 +
1.63 +lemma add_Suc_right: "m + Suc(n) = Suc(m + n)"
1.64 +  by (induct m) simp_all
1.65 +
1.66 +lemma "(!!n. f(Suc(n)) = Suc(f(n))) ==> f(i + j) = i + f(j)"
1.67 +proof -
1.68 +  assume a: "!!n. f(Suc(n)) = Suc(f(n))"
1.69 +  show ?thesis by (induct i) (simp, simp add: a)  (* FIXME tune *)
1.70 +qed
1.71 +
1.72 +end
```