# Theory Nat_Class

theory Nat_Class
imports FOL
(*  Title:      FOL/ex/Nat_Class.thy
Author:     Markus Wenzel, TU Muenchen
*)

theory Nat_Class
imports FOL
begin

text ‹
This is an abstract version of theory ‹Nat›. Instead of
axiomatizing a single type ‹nat› we define the class of all
these types (up to isomorphism).

because class axioms may not contain more than one type variable.
›

class nat =
fixes Zero :: 'a  ("0")
and Suc :: "'a ⇒ 'a"
and rec :: "'a ⇒ 'a ⇒ ('a ⇒ 'a ⇒ 'a) ⇒ 'a"
assumes induct: "P(0) ⟹ (⋀x. P(x) ⟹ P(Suc(x))) ⟹ P(n)"
and Suc_inject: "Suc(m) = Suc(n) ⟹ m = n"
and Suc_neq_Zero: "Suc(m) = 0 ⟹ R"
and rec_Zero: "rec(0, a, f) = a"
and rec_Suc: "rec(Suc(m), a, f) = f(m, rec(m, a, f))"
begin

definition add :: "'a ⇒ 'a ⇒ 'a"  (infixl "+" 60)
where "m + n = rec(m, n, λx y. Suc(y))"

lemma Suc_n_not_n: "Suc(k) ≠ (k::'a)"
apply (rule_tac n = k in induct)
apply (rule notI)
apply (erule Suc_neq_Zero)
apply (rule notI)
apply (erule notE)
apply (erule Suc_inject)
done

lemma "(k + m) + n = k + (m + n)"
apply (rule induct)
back
back
back
back
back
oops

lemma add_Zero [simp]: "0 + n = n"
apply (rule rec_Zero)
done

lemma add_Suc [simp]: "Suc(m) + n = Suc(m + n)"
apply (rule rec_Suc)
done

lemma add_assoc: "(k + m) + n = k + (m + n)"
apply (rule_tac n = k in induct)
apply simp
apply simp
done

lemma add_Zero_right: "m + 0 = m"
apply (rule_tac n = m in induct)
apply simp
apply simp
done

lemma add_Suc_right: "m + Suc(n) = Suc(m + n)"
apply (rule_tac n = m in induct)
apply simp_all
done

lemma
assumes prem: "⋀n. f(Suc(n)) = Suc(f(n))"
shows "f(i + j) = i + f(j)"
apply (rule_tac n = i in induct)
apply simp