(* Title: FOL/ex/Nat.ML ID: $Id: Nat.ML,v 1.11 2005/09/03 15:15:51 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1992 University of Cambridge Proofs about the natural numbers. To generate similar output to manual, execute these commands: Pretty.setmargin 72; print_depth 0; *) Goal "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); 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] "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 (asm_simp_tac (simpset() addsimps [prem]) 1); result();