--- a/src/CCL/ex/Nat.thy Tue Nov 11 13:50:56 2014 +0100
+++ b/src/CCL/ex/Nat.thy Tue Nov 11 15:55:31 2014 +0100
@@ -9,41 +9,41 @@
imports "../Wfd"
begin
-definition not :: "i=>i"
+definition not :: "i\<Rightarrow>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 add :: "[i,i]\<Rightarrow>i" (infixr "#+" 60)
+ where "a #+ b == nrec(a, b, \<lambda>x g. succ(g))"
-definition mult :: "[i,i]=>i" (infixr "#*" 60)
- where "a #* b == nrec(a,zero,%x g. b #+ g)"
+definition mult :: "[i,i]\<Rightarrow>i" (infixr "#*" 60)
+ where "a #* b == nrec(a, zero, \<lambda>x g. b #+ g)"
-definition sub :: "[i,i]=>i" (infixr "#-" 60)
+definition sub :: "[i,i]\<Rightarrow>i" (infixr "#-" 60)
where
"a #- b ==
- letrec sub x y be ncase(y,x,%yy. ncase(x,zero,%xx. sub(xx,yy)))
+ letrec sub x y be ncase(y, x, \<lambda>yy. ncase(x, zero, \<lambda>xx. sub(xx,yy)))
in sub(a,b)"
-definition le :: "[i,i]=>i" (infixr "#<=" 60)
+definition le :: "[i,i]\<Rightarrow>i" (infixr "#<=" 60)
where
"a #<= b ==
- letrec le x y be ncase(x,true,%xx. ncase(y,false,%yy. le(xx,yy)))
+ letrec le x y be ncase(x, true, \<lambda>xx. ncase(y, false, \<lambda>yy. le(xx,yy)))
in le(a,b)"
-definition lt :: "[i,i]=>i" (infixr "#<" 60)
+definition lt :: "[i,i]\<Rightarrow>i" (infixr "#<" 60)
where "a #< b == not(b #<= a)"
-definition div :: "[i,i]=>i" (infixr "##" 60)
+definition div :: "[i,i]\<Rightarrow>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"
+definition ackermann :: "[i,i]\<Rightarrow>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))))
+ letrec ack n m be ncase(n, succ(m), \<lambda>x.
+ ncase(m,ack(x,succ(zero)), \<lambda>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
@@ -60,37 +60,37 @@
by (simp_all add: nat_defs)
-lemma napply_f: "n:Nat ==> f^n`f(a) = f^succ(n)`a"
+lemma napply_f: "n:Nat \<Longrightarrow> 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"
+lemma addT: "\<lbrakk>a:Nat; b:Nat\<rbrakk> \<Longrightarrow> a #+ b : Nat"
apply (unfold add_def)
apply typechk
done
-lemma multT: "[| a:Nat; b:Nat |] ==> a #* b : Nat"
+lemma multT: "\<lbrakk>a:Nat; b:Nat\<rbrakk> \<Longrightarrow> a #* b : Nat"
apply (unfold add_def mult_def)
apply typechk
done
(* Defined to return zero if a<b *)
-lemma subT: "[| a:Nat; b:Nat |] ==> a #- b : Nat"
+lemma subT: "\<lbrakk>a:Nat; b:Nat\<rbrakk> \<Longrightarrow> 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"
+lemma leT: "\<lbrakk>a:Nat; b:Nat\<rbrakk> \<Longrightarrow> 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"
+lemma ltT: "\<lbrakk>a:Nat; b:Nat\<rbrakk> \<Longrightarrow> a #< b : Bool"
apply (unfold not_def lt_def)
apply (typechk leT)
done
@@ -100,7 +100,7 @@
lemmas relI = NatPR_wf [THEN NatPR_wf [THEN lex_wf, THEN wfI]]
-lemma "[| a:Nat; b:Nat |] ==> ackermann(a,b) : Nat"
+lemma "\<lbrakk>a:Nat; b:Nat\<rbrakk> \<Longrightarrow> ackermann(a,b) : Nat"
apply (unfold ackermann_def)
apply gen_ccs
apply (erule NatPRI [THEN lexI1 [THEN relI]] NatPRI [THEN lexI2 [THEN relI]])+