Theory Nat

theory Nat
imports LK
(*  Title:      Sequents/LK/Nat.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1999 University of Cambridge
*)


header {* Theory of the natural numbers: Peano's axioms, primitive recursion *}

theory Nat
imports LK
begin

typedecl nat
arities nat :: "term"

axiomatization
Zero :: nat ("0") and
Suc :: "nat=>nat" and
rec :: "[nat, 'a, [nat,'a]=>'a] => 'a"
where
induct: "[| $H |- $E, P(0), $F;
!!x. $H, P(x) |- $E, P(Suc(x)), $F |] ==> $H |- $E, P(n), $F"
and

Suc_inject: "|- Suc(m)=Suc(n) --> m=n" and
Suc_neq_0: "|- Suc(m) ~= 0" and
rec_0: "|- rec(0,a,f) = a" and
rec_Suc: "|- rec(Suc(m), a, f) = f(m, rec(m,a,f))"

definition add :: "[nat, nat] => nat" (infixl "+" 60)
where "m + n == rec(m, n, %x y. Suc(y))"


declare Suc_neq_0 [simp]

lemma Suc_inject_rule: "$H, $G, m = n |- $E ==> $H, Suc(m) = Suc(n), $G |- $E"
by (rule L_of_imp [OF Suc_inject])

lemma Suc_n_not_n: "|- Suc(k) ~= k"
apply (rule_tac n = k in induct)
apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms Suc_neq_0}) 1 *})
apply (tactic {* fast_tac (LK_pack add_safes @{thms Suc_inject_rule}) 1 *})
done

lemma add_0: "|- 0+n = n"
apply (unfold add_def)
apply (rule rec_0)
done

lemma add_Suc: "|- Suc(m)+n = Suc(m+n)"
apply (unfold add_def)
apply (rule rec_Suc)
done

declare add_0 [simp] add_Suc [simp]

lemma add_assoc: "|- (k+m)+n = k+(m+n)"
apply (rule_tac n = "k" in induct)
apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_0}) 1 *})
apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_Suc}) 1 *})
done

lemma add_0_right: "|- m+0 = m"
apply (rule_tac n = "m" in induct)
apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_0}) 1 *})
apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_Suc}) 1 *})
done

lemma add_Suc_right: "|- m+Suc(n) = Suc(m+n)"
apply (rule_tac n = "m" in induct)
apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_0}) 1 *})
apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_Suc}) 1 *})
done

lemma "(!!n. |- f(Suc(n)) = Suc(f(n))) ==> |- f(i+j) = i+f(j)"
apply (rule_tac n = "i" in induct)
apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_0}) 1 *})
apply (tactic {* asm_simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_Suc}) 1 *})
done

end