src/Sequents/LK/Nat.thy

 axiomatization
   Zero :: nat      ("0") and
   Suc :: "nat \<Rightarrow> nat" and
   rec :: "[nat, 'a, [nat,'a] \<Rightarrow> 'a] \<Rightarrow> 'a"
 where
-  induct:  "\<lbrakk>$H |- $E, P(0), $F;
-             \<And>x. $H, P(x) |- $E, P(Suc(x)), $F\<rbrakk> \<Longrightarrow> $H |- $E, P(n), $F" and
+  induct:  "\<lbrakk>$H \<turnstile> $E, P(0), $F;
+             \<And>x. $H, P(x) \<turnstile> $E, P(Suc(x)), $F\<rbrakk> \<Longrightarrow> $H \<turnstile> $E, P(n), $F" 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))"
+  Suc_inject:  "\<turnstile> Suc(m) = Suc(n) \<longrightarrow> m = n" and
+  Suc_neq_0:   "\<turnstile> Suc(m) \<noteq> 0" and
+  rec_0:       "\<turnstile> rec(0,a,f) = a" and
+  rec_Suc:     "\<turnstile> rec(Suc(m), a, f) = f(m, rec(m,a,f))"
 
 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 |- $E"
+lemma Suc_inject_rule: "$H, $G, m = n \<turnstile> $E \<Longrightarrow> $H, Suc(m) = Suc(n), $G \<turnstile> $E"
   by (rule L_of_imp [OF Suc_inject])
 
-lemma Suc_n_not_n: "|- Suc(k) \<noteq> k"
+lemma Suc_n_not_n: "\<turnstile> 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: "\<turnstile> 0 + n = n"
   apply (unfold add_def)
   apply (rule rec_0)
   done
 
-lemma add_Suc: "|- Suc(m) + n = Suc(m + n)"
+lemma add_Suc: "\<turnstile> 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: "\<turnstile> (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: "\<turnstile> 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: "\<turnstile> m + Suc(n) = Suc(m + n)"
   apply (rule_tac n = "m" in induct)
   apply simp_all
   done
 
-lemma "(\<And>n. |- f(Suc(n)) = Suc(f(n))) \<Longrightarrow> |- f(i + j) = i + f(j)"
+lemma "(\<And>n. \<turnstile> f(Suc(n)) = Suc(f(n))) \<Longrightarrow> \<turnstile> f(i + j) = i + f(j)"
   apply (rule_tac n = "i" in induct)
   apply simp_all
   done
 
 end