(*  Title:      Doc/AxClass/Nat/NatClass.ML

    ID:         $Id$

    Author:     Markus Wenzel, TU Muenchen

This is Nat.ML with some trivial modifications in order to make it

work with NatClass.thy.

*)

val induct = thm "induct";

val Suc_inject = thm "Suc_inject";

val Suc_neq_0 = thm "Suc_neq_0";

val rec_0 = thm "rec_0";

val rec_Suc = thm "rec_Suc";

val add_def = thm "add_def";

Goal "Suc(k) ~= (k::'a::nat)";

by (res_inst_tac [("n","k")] induct 1);

by (rtac notI 1);

by (etac Suc_neq_0 1);

by (rtac notI 1);

by (etac notE 1);

by (etac Suc_inject 1);

qed "Suc_n_not_n";

Goal "(k+m)+n = k+(m+n)";

prths ([induct] RL [topthm()]);  (*prints all 14 next states!*)

by (rtac induct 1);

back();

back();

back();

back();

back();

back();

Goalw [add_def] "\\<zero>+n = n";

by (rtac rec_0 1);

qed "add_0";

Goalw [add_def] "Suc(m)+n = Suc(m+n)";

by (rtac rec_Suc 1);

qed "add_Suc";

Addsimps [add_0, add_Suc];

Goal "(k+m)+n = k+(m+n)";

by (res_inst_tac [("n","k")] induct 1);

by (Simp_tac 1);

by (Asm_simp_tac 1);

qed "add_assoc";

Goal "m+\\<zero> = m";

by (res_inst_tac [("n","m")] induct 1);

by (Simp_tac 1);

by (Asm_simp_tac 1);

qed "add_0_right";

Goal "m+Suc(n) = Suc(m+n)";

by (res_inst_tac [("n","m")] induct 1);

by (ALLGOALS (Asm_simp_tac));

qed "add_Suc_right";

val [prem] = Goal "(!!n. f(Suc(n)) = Suc(f(n))) ==> f(i+j) = i+f(j)";

by (res_inst_tac [("n","i")] induct 1);

by (Simp_tac 1);

by (asm_simp_tac (simpset() addsimps [prem]) 1);

qed "";