(* Title: CCL/ex/Nat.ML ID: $Id: Nat.ML,v 1.5 2005/09/17 15:35:32 wenzelm Exp $ Author: Martin Coen, Cambridge University Computer Laboratory Copyright 1993 University of Cambridge *) val nat_defs = [not_def,add_def,mult_def,sub_def,le_def,lt_def,ack_def,napply_def]; val natBs = map (fn s=>prove_goalw (the_context ()) nat_defs s (fn _ => [simp_tac term_ss 1])) ["not(true) = false", "not(false) = true", "zero #+ n = n", "succ(n) #+ m = succ(n #+ m)", "zero #* n = zero", "succ(n) #* m = m #+ (n #* m)", "f^zero`a = a", "f^succ(n)`a = f(f^n`a)"]; val nat_ss = term_ss addsimps natBs; (*** Lemma for napply ***) val [prem] = goal (the_context ()) "n:Nat ==> f^n`f(a) = f^succ(n)`a"; by (rtac (prem RS Nat_ind) 1); by (ALLGOALS (asm_simp_tac nat_ss)); qed "napply_f"; (****) val prems = goalw (the_context ()) [add_def] "[| a:Nat; b:Nat |] ==> a #+ b : Nat"; by (typechk_tac prems 1); qed "addT"; val prems = goalw (the_context ()) [mult_def] "[| a:Nat; b:Nat |] ==> a #* b : Nat"; by (typechk_tac (addT::prems) 1); qed "multT"; (* Defined to return zero if a<b *) val prems = goalw (the_context ()) [sub_def] "[| a:Nat; b:Nat |] ==> a #- b : Nat"; by (typechk_tac (prems) 1); by clean_ccs_tac; by (etac (NatPRI RS wfstI RS (NatPR_wf RS wmap_wf RS wfI)) 1); qed "subT"; val prems = goalw (the_context ()) [le_def] "[| a:Nat; b:Nat |] ==> a #<= b : Bool"; by (typechk_tac (prems) 1); by clean_ccs_tac; by (etac (NatPRI RS wfstI RS (NatPR_wf RS wmap_wf RS wfI)) 1); qed "leT"; val prems = goalw (the_context ()) [not_def,lt_def] "[| a:Nat; b:Nat |] ==> a #< b : Bool"; by (typechk_tac (prems@[leT]) 1); qed "ltT"; (* Correctness conditions for subtractive division **) val prems = goalw (the_context ()) [div_def] "[| a:Nat; b:{x:Nat.~x=zero} |] ==> a ## b : {x:Nat. DIV(a,b,x)}"; by (gen_ccs_tac (prems@[ltT,subT]) 1); (* Termination Conditions for Ackermann's Function *) val prems = goalw (the_context ()) [ack_def] "[| a:Nat; b:Nat |] ==> ackermann(a,b) : Nat"; by (gen_ccs_tac prems 1); val relI = NatPR_wf RS (NatPR_wf RS lex_wf RS wfI); by (REPEAT (eresolve_tac [NatPRI RS (lexI1 RS relI),NatPRI RS (lexI2 RS relI)] 1)); result();