(* Title: CCL/ex/Nat.thy Author: Martin Coen, Cambridge University Computer Laboratory Copyright 1993 University of Cambridge *) section \Programs defined over the natural numbers\ theory Nat imports "../Wfd" begin definition not :: "i\i" where "not(b) == if b then false else true" definition add :: "[i,i]\i" (infixr "#+" 60) where "a #+ b == nrec(a, b, \x g. succ(g))" definition mult :: "[i,i]\i" (infixr "#*" 60) where "a #* b == nrec(a, zero, \x g. b #+ g)" definition sub :: "[i,i]\i" (infixr "#-" 60) where "a #- b == letrec sub x y be ncase(y, x, \yy. ncase(x, zero, \xx. sub(xx,yy))) in sub(a,b)" definition le :: "[i,i]\i" (infixr "#<=" 60) where "a #<= b == letrec le x y be ncase(x, true, \xx. ncase(y, false, \yy. le(xx,yy))) in le(a,b)" definition lt :: "[i,i]\i" (infixr "#<" 60) where "a #< b == not(b #<= a)" definition div :: "[i,i]\i" (infixr "##" 60) where "a ## b == letrec div x y be if x #< y then zero else succ(div(x#-y,y)) in div(a,b)" definition ackermann :: "[i,i]\i" where "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)" lemmas nat_defs = not_def add_def mult_def sub_def le_def lt_def ackermann_def napply_def lemma natBs [simp]: "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)" by (simp_all add: nat_defs) lemma napply_f: "n:Nat \ f^n`f(a) = f^succ(n)`a" apply (erule Nat_ind) apply simp_all done lemma addT: "\a:Nat; b:Nat\ \ a #+ b : Nat" apply (unfold add_def) apply typechk done lemma multT: "\a:Nat; b:Nat\ \ a #* b : Nat" apply (unfold add_def mult_def) apply typechk done (* Defined to return zero if aa:Nat; b:Nat\ \ a #- b : Nat" apply (unfold sub_def) apply typechk apply clean_ccs apply (erule NatPRI [THEN wfstI, THEN NatPR_wf [THEN wmap_wf, THEN wfI]]) done lemma leT: "\a:Nat; b:Nat\ \ a #<= b : Bool" apply (unfold le_def) apply typechk apply clean_ccs apply (erule NatPRI [THEN wfstI, THEN NatPR_wf [THEN wmap_wf, THEN wfI]]) done lemma ltT: "\a:Nat; b:Nat\ \ a #< b : Bool" apply (unfold not_def lt_def) apply (typechk leT) done subsection \Termination Conditions for Ackermann's Function\ lemmas relI = NatPR_wf [THEN NatPR_wf [THEN lex_wf, THEN wfI]] lemma "\a:Nat; b:Nat\ \ ackermann(a,b) : Nat" apply (unfold ackermann_def) apply gen_ccs apply (erule NatPRI [THEN lexI1 [THEN relI]] NatPRI [THEN lexI2 [THEN relI]])+ done end