(* Title: CCL/ex/Nat.thy ID: $Id: Nat.thy,v 1.5 2005/09/17 15:35:32 wenzelm Exp $ Author: Martin Coen, Cambridge University Computer Laboratory Copyright 1993 University of Cambridge *) header {* Programs defined over the natural numbers *} theory Nat imports Wfd begin consts not :: "i=>i" "#+" :: "[i,i]=>i" (infixr 60) "#*" :: "[i,i]=>i" (infixr 60) "#-" :: "[i,i]=>i" (infixr 60) "##" :: "[i,i]=>i" (infixr 60) "#<" :: "[i,i]=>i" (infixr 60) "#<=" :: "[i,i]=>i" (infixr 60) ackermann :: "[i,i]=>i" defs not_def: "not(b) == if b then false else true" add_def: "a #+ b == nrec(a,b,%x g. succ(g))" mult_def: "a #* b == nrec(a,zero,%x g. b #+ g)" sub_def: "a #- b == letrec sub x y be ncase(y,x,%yy. ncase(x,zero,%xx. sub(xx,yy))) in sub(a,b)" le_def: "a #<= b == letrec le x y be ncase(x,true,%xx. ncase(y,false,%yy. le(xx,yy))) in le(a,b)" lt_def: "a #< b == not(b #<= a)" div_def: "a ## b == letrec div x y be if x #< y then zero else succ(div(x#-y,y)) in div(a,b)" ack_def: "ackermann(a,b) == letrec ack n m be ncase(n,succ(m),%x. ncase(m,ack(x,succ(zero)),%y. ack(x,ack(succ(x),y)))) in ack(a,b)" ML {* use_legacy_bindings (the_context ()) *} end
theorem napply_f:
n : Nat ==> f ^ n ` f(a) = f ^ succ(n) ` a
theorem addT:
[| a : Nat; b : Nat |] ==> a #+ b : Nat
theorem multT:
[| a : Nat; b : Nat |] ==> a #* b : Nat
theorem subT:
[| a : Nat; b : Nat |] ==> a #- b : Nat
theorem leT:
[| a : Nat; b : Nat |] ==> a #<= b : Bool
theorem ltT:
[| a : Nat; b : Nat |] ==> a #< b : Bool