(* Title: Sequents/LK/Nat.ML ID: $Id: Nat.ML,v 1.3 2005/09/18 13:20:10 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1999 University of Cambridge *) Addsimps [Suc_neq_0]; Add_safes [Suc_inject RS L_of_imp]; Goal "|- Suc(k) ~= k"; by (res_inst_tac [("n","k")] induct 1); by (Simp_tac 1); by (Fast_tac 1); qed "Suc_n_not_n"; Goalw [add_def] "|- 0+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+0 = 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"; (*Example used in Reference Manual, Doc/Ref/simplifier.tex*) 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 (simp_tac (simpset() addsimps [prem]) 1); result();