(* Title: Sequents/LK/Nat.thy ID: $Id: Nat.thy,v 1.6 2007/05/09 17:37:21 wenzelm Exp $ 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" consts "0" :: nat ("0") Suc :: "nat=>nat" rec :: "[nat, 'a, [nat,'a]=>'a] => 'a" add :: "[nat, nat] => nat" (infixl "+" 60) axioms induct: "[| $H |- $E, P(0), $F; !!x. $H, P(x) |- $E, P(Suc(x)), $F |] ==> $H |- $E, P(n), $F" Suc_inject: "|- Suc(m)=Suc(n) --> m=n" Suc_neq_0: "|- Suc(m) ~= 0" rec_0: "|- rec(0,a,f) = a" rec_Suc: "|- rec(Suc(m), a, f) = f(m, rec(m,a,f))" add_def: "m+n == rec(m, n, %x y. Suc(y))" declare Suc_neq_0 [simp] lemma Suc_inject_rule: "$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 (LK_ss 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 (LK_ss addsimps @{thms add_0}) 1 *}) apply (tactic {* simp_tac (LK_ss addsimps @{thms add_Suc}) 1 *}) done lemma add_0_right: "|- m+0 = m" apply (rule_tac n = "m" in induct) apply (tactic {* simp_tac (LK_ss addsimps @{thms add_0}) 1 *}) apply (tactic {* simp_tac (LK_ss 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 (LK_ss addsimps @{thms add_0}) 1 *}) apply (tactic {* simp_tac (LK_ss 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 (LK_ss addsimps @{thms add_0}) 1 *}) apply (tactic {* asm_simp_tac (LK_ss addsimps @{thms add_Suc}) 1 *}) done end
lemma Suc_inject_rule:
$H, $G, m = n |- $E ==> $H, Suc(m) = Suc(n), $G |- $E
lemma Suc_n_not_n:
|- Suc(k) ~= k
lemma add_0:
|- 0 + n = n
lemma add_Suc:
|- Suc(m) + n = Suc(m + n)
lemma add_assoc:
|- k + m + n = k + (m + n)
lemma add_0_right:
|- m + 0 = m
lemma add_Suc_right:
|- m + Suc(n) = Suc(m + n)
lemma
(!!n. |- f(Suc(n)) = Suc(f(n))) ==> |- f(i + j) = i + f(j)