src/Sequents/LK/Nat.thy

 typedecl nat
 instance nat :: "term" ..
 
 axiomatization
   Zero :: nat      ("0") and
-  Suc :: "nat=>nat" and
-  rec :: "[nat, 'a, [nat,'a]=>'a] => 'a"
+  Suc :: "nat \<Rightarrow> nat" and
+  rec :: "[nat, 'a, [nat,'a] \<Rightarrow> 'a] \<Rightarrow> 'a"
 where
-  induct:  "[| $H |- $E, P(0), $F;
-              !!x. $H, P(x) |- $E, P(Suc(x)), $F |] ==> $H |- $E, P(n), $F" and
+  induct:  "\<lbrakk>$H |- $E, P(0), $F;
+             \<And>x. $H, P(x) |- $E, P(Suc(x)), $F\<rbrakk> \<Longrightarrow> $H |- $E, P(n), $F" and
 
-  Suc_inject:  "|- Suc(m)=Suc(n) --> m=n" and
-  Suc_neq_0:   "|- Suc(m) ~= 0" and
+  Suc_inject:  "|- Suc(m) = Suc(n) \<longrightarrow> m = n" and
+  Suc_neq_0:   "|- Suc(m) \<noteq> 0" and
   rec_0:       "|- rec(0,a,f) = a" and
   rec_Suc:     "|- rec(Suc(m), a, f) = f(m, rec(m,a,f))"
 
-definition add :: "[nat, nat] => nat"  (infixl "+" 60)
-  where "m + n == rec(m, n, %x y. Suc(y))"
+definition add :: "[nat, nat] \<Rightarrow> nat"  (infixl "+" 60)
+  where "m + n == rec(m, n, \<lambda>x y. Suc(y))"
 
 
 declare Suc_neq_0 [simp]
 
 lemma Suc_inject_rule: "$H, $G, m = n |- $E \<Longrightarrow> $H, Suc(m) = Suc(n), $G |- $E"
   by (rule L_of_imp [OF Suc_inject])
 
-lemma Suc_n_not_n: "|- Suc(k) ~= k"
+lemma Suc_n_not_n: "|- Suc(k) \<noteq> k"
   apply (rule_tac n = k in induct)
   apply simp
   apply (fast add!: Suc_inject_rule)
   done
 
-lemma add_0: "|- 0+n = n"
+lemma add_0: "|- 0 + n = n"
   apply (unfold add_def)
   apply (rule rec_0)
   done
 
-lemma add_Suc: "|- Suc(m)+n = Suc(m+n)"
+lemma add_Suc: "|- Suc(m) + n = Suc(m + n)"
   apply (unfold add_def)
   apply (rule rec_Suc)
   done
 
 declare add_0 [simp] add_Suc [simp]
 
-lemma add_assoc: "|- (k+m)+n = k+(m+n)"
+lemma add_assoc: "|- (k + m) + n = k + (m + n)"
   apply (rule_tac n = "k" in induct)
   apply simp_all
   done
 
-lemma add_0_right: "|- m+0 = m"
+lemma add_0_right: "|- m + 0 = m"
   apply (rule_tac n = "m" in induct)
   apply simp_all
   done
 
-lemma add_Suc_right: "|- m+Suc(n) = Suc(m+n)"
+lemma add_Suc_right: "|- m + Suc(n) = Suc(m + n)"
   apply (rule_tac n = "m" in induct)
   apply simp_all
   done
 
-lemma "(!!n. |- f(Suc(n)) = Suc(f(n))) ==> |- f(i+j) = i+f(j)"
+lemma "(\<And>n. |- f(Suc(n)) = Suc(f(n))) \<Longrightarrow> |- f(i + j) = i + f(j)"
   apply (rule_tac n = "i" in induct)
   apply simp_all
   done
 
 end