--- a/src/CCL/ex/Nat.thy Tue Mar 29 22:36:56 2011 +0200
+++ b/src/CCL/ex/Nat.thy Tue Mar 29 23:15:25 2011 +0200
@@ -9,37 +9,44 @@
imports Wfd
begin
-consts
+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))"
- not :: "i=>i"
- add :: "[i,i]=>i" (infixr "#+" 60)
- mult :: "[i,i]=>i" (infixr "#*" 60)
- sub :: "[i,i]=>i" (infixr "#-" 60)
- div :: "[i,i]=>i" (infixr "##" 60)
- lt :: "[i,i]=>i" (infixr "#<" 60)
- le :: "[i,i]=>i" (infixr "#<=" 60)
- ackermann :: "[i,i]=>i"
+definition mult :: "[i,i]=>i" (infixr "#*" 60)
+ where "a #* b == nrec(a,zero,%x g. b #+ g)"
-defs
-
- not_def: "not(b) == if b then false else true"
+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)"
- 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)"
+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)"
- 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)"
+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)"
-lemmas nat_defs = not_def add_def mult_def sub_def le_def lt_def ack_def napply_def
+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"
@@ -94,7 +101,7 @@
lemmas relI = NatPR_wf [THEN NatPR_wf [THEN lex_wf, THEN wfI]]
lemma "[| a:Nat; b:Nat |] ==> ackermann(a,b) : Nat"
- apply (unfold ack_def)
+ apply (unfold ackermann_def)
apply (tactic {* gen_ccs_tac @{context} [] 1 *})
apply (erule NatPRI [THEN lexI1 [THEN relI]] NatPRI [THEN lexI2 [THEN relI]])+
done