src/FOLP/ex/Nat.thy
 author clasohm Mon, 05 Feb 1996 21:33:14 +0100 changeset 1477 4c51ab632cda parent 1149 5750eba8820d child 17480 fd19f77dcf60 permissions -rw-r--r--
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```
(*  Title:      FOLP/ex/nat.thy
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

Examples for the manual "Introduction to Isabelle"

Theory of the natural numbers: Peano's axioms, primitive recursion
*)

Nat = IFOLP +
types   nat
arities nat         :: term
consts  "0"         :: "nat"    ("0")
Suc         :: "nat=>nat"
rec         :: "[nat, 'a, [nat,'a]=>'a] => 'a"
"+"         :: "[nat, nat] => nat"              (infixl 60)

(*Proof terms*)
nrec         :: "[nat,p,[nat,p]=>p]=>p"
ninj,nneq    :: "p=>p"
rec0, recSuc :: "p"

rules
induct     "[| b:P(0); !!x u. u:P(x) ==> c(x,u):P(Suc(x))
|] ==> nrec(n,b,c):P(n)"

Suc_inject "p:Suc(m)=Suc(n) ==> ninj(p) : m=n"
Suc_neq_0  "p:Suc(m)=0      ==> nneq(p) : R"
rec_0      "rec0 : rec(0,a,f) = a"
rec_Suc    "recSuc : rec(Suc(m), a, f) = f(m, rec(m,a,f))"
add_def    "m+n == rec(m, n, %x y. Suc(y))"

nrecB0     "b: A ==> nrec(0,b,c) = b : A"
nrecBSuc   "c(n,nrec(n,b,c)) : A ==> nrec(Suc(n),b,c) = c(n,nrec(n,b,c)) : A"
end
```