2 files changed, 14 insertions, 14 deletions
diff --git a/Coprod.thy b/Coprod.thy
index f2225f6..a444c89 100644
--- a/Coprod.thy
+++ b/Coprod.thy
@@ -18,27 +18,27 @@ axiomatization
   inr :: "Term \<Rightarrow> Term" and
   indCoprod :: "[Term \<Rightarrow> Term, Term \<Rightarrow> Term, Term] \<Rightarrow> Term"  ("(1ind\<^sub>+)")
 where
-  Coprod_form: "\<And>i A B. \<lbrakk>A : U(i); B : U(i)\<rbrakk> \<Longrightarrow> A + B: U(i)"
+  Coprod_form: "\<lbrakk>A: U(i); B: U(i)\<rbrakk> \<Longrightarrow> A + B: U(i)"
 and
-  Coprod_intro1: "\<And>A B a b. \<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> inl(a): A + B"
+  Coprod_intro1: "\<lbrakk>a: A; B: U(i)\<rbrakk> \<Longrightarrow> inl(a): A + B"
 and
-  Coprod_intro2: "\<And>A B a b. \<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> inr(b): A + B"
+  Coprod_intro2: "\<lbrakk>b: B; A: U(i)\<rbrakk> \<Longrightarrow> inr(b): A + B"
 and
-  Coprod_elim: "\<And>i A B C c d e. \<lbrakk>
+  Coprod_elim: "\<lbrakk>
     C: A + B \<longrightarrow> U(i);
     \<And>x. x: A \<Longrightarrow> c(x): C(inl(x));
     \<And>y. y: B \<Longrightarrow> d(y): C(inr(y));
-    e: A + B
-    \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(e) : C(e)"
+    u: A + B
+    \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(u) : C(u)"
 and
-  Coprod_comp1: "\<And>i A B C c d a. \<lbrakk>
+  Coprod_comp1: "\<lbrakk>
     C: A + B \<longrightarrow> U(i);
     \<And>x. x: A \<Longrightarrow> c(x): C(inl(x));
     \<And>y. y: B \<Longrightarrow> d(y): C(inr(y));
     a: A
     \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(inl(a)) \<equiv> c(a)"
 and
-  Coprod_comp2: "\<And>i A B C c d b. \<lbrakk>
+  Coprod_comp2: "\<lbrakk>
     C: A + B \<longrightarrow> U(i);
     \<And>x. x: A \<Longrightarrow> c(x): C(inl(x));
     \<And>y. y: B \<Longrightarrow> d(y): C(inr(y));
@@ -49,9 +49,9 @@ and
 text "Admissible formation inference rules:"
 
 axiomatization where
-  Coprod_form_cond1: "\<And>i A B. A + B: U(i) \<Longrightarrow> A: U(i)"
+  Coprod_form_cond1: "A + B: U(i) \<Longrightarrow> A: U(i)"
 and
-  Coprod_form_cond2: "\<And>i A B. A + B: U(i) \<Longrightarrow> B: U(i)"
+  Coprod_form_cond2: "A + B: U(i) \<Longrightarrow> B: U(i)"
 
 
 text "Rule declarations:"
diff --git a/Nat.thy b/Nat.thy
index b710ff2..05b0bfe 100644
--- a/Nat.thy
+++ b/Nat.thy
@@ -22,22 +22,22 @@ where
 and
   Nat_intro1: "0: \<nat>"
 and
-  Nat_intro2: "\<And>n. n: \<nat> \<Longrightarrow> succ(n): \<nat>"
+  Nat_intro2: "n: \<nat> \<Longrightarrow> succ(n): \<nat>"
 and
-  Nat_elim: "\<And>i C f a n. \<lbrakk>
+  Nat_elim: "\<lbrakk>
     C: \<nat> \<longrightarrow> U(i);
     \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f(n)(c): C(succ n);
     a: C(0);
     n: \<nat>
     \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat>(f)(a)(n): C(n)"
 and
-  Nat_comp1: "\<And>i C f a. \<lbrakk>
+  Nat_comp1: "\<lbrakk>
     C: \<nat> \<longrightarrow> U(i);
     \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f(n)(c): C(succ n);
     a: C(0)
     \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat>(f)(a)(0) \<equiv> a"
 and
-  Nat_comp2: "\<And> i C f a n. \<lbrakk>
+  Nat_comp2: "\<lbrakk>
     C: \<nat> \<longrightarrow> U(i);
     \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f(n)(c): C(succ n);
     a: C(0);