# File Nat.ML

    (*  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();