(* Title: FOLP/ex/nat.ML ID: $Id: Nat.ML,v 1.4 2005/09/18 12:25:49 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1992 University of Cambridge *) Goal "?p : ~ (Suc(k) = k)"; 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); val Suc_n_not_n = result(); Goal "?p : (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] "?p : 0+n = n"; by (rtac rec_0 1); val add_0 = result(); Goalw [add_def] "?p : Suc(m)+n = Suc(m+n)"; by (rtac rec_Suc 1); val add_Suc = result(); (* val nat_congs = mk_congs (the_context ()) ["Suc", "op +"]; prths nat_congs; *) val prems = goal (the_context ()) "p: x=y ==> ?p : Suc(x) = Suc(y)"; by (resolve_tac (prems RL [subst]) 1); by (rtac refl 1); val Suc_cong = result(); val prems = goal (the_context ()) "[| p : a=x; q: b=y |] ==> ?p : a+b=x+y"; by (resolve_tac (prems RL [subst]) 1 THEN resolve_tac (prems RL [subst]) 1 THEN rtac refl 1); val Plus_cong = result(); val nat_congs = [Suc_cong,Plus_cong]; val add_ss = FOLP_ss addcongs nat_congs addrews [add_0, add_Suc]; Goal "?p : (k+m)+n = k+(m+n)"; by (res_inst_tac [("n","k")] induct 1); by (SIMP_TAC add_ss 1); by (ASM_SIMP_TAC add_ss 1); val add_assoc = result(); Goal "?p : m+0 = m"; by (res_inst_tac [("n","m")] induct 1); by (SIMP_TAC add_ss 1); by (ASM_SIMP_TAC add_ss 1); val add_0_right = result(); Goal "?p : m+Suc(n) = Suc(m+n)"; by (res_inst_tac [("n","m")] induct 1); by (ALLGOALS (ASM_SIMP_TAC add_ss)); val add_Suc_right = result(); (*mk_typed_congs appears not to work with FOLP's version of subst*)