Reorganized HoTT_Base, updated theories

1 files changed, 16 insertions, 16 deletions

diff --git a/Equal.thy b/Equal.thy
index 772072a..7b6acb5 100644
--- a/Equal.thy
+++ b/Equal.thy
@@ -12,13 +12,13 @@ begin
 section \<open>Constants and syntax\<close>
 
 axiomatization
-  Equal :: "[Term, Term, Term] \<Rightarrow> Term" and
-  refl :: "Term \<Rightarrow> Term" and
-  indEqual :: "[Term \<Rightarrow> Term, Term] \<Rightarrow> Term"  ("(1ind\<^sub>=)")
+  Equal :: "[t, t, t] \<Rightarrow> t" and
+  refl :: "t \<Rightarrow> t" and
+  indEqual :: "[t \<Rightarrow> t, t] \<Rightarrow> t"  ("(1ind\<^sub>=)")
 
 syntax
-  "_EQUAL" :: "[Term, Term, Term] \<Rightarrow> Term"        ("(3_ =\<^sub>_/ _)" [101, 0, 101] 100)
-  "_EQUAL_ASCII" :: "[Term, Term, Term] \<Rightarrow> Term"  ("(3_ =[_]/ _)" [101, 0, 101] 100)
+  "_EQUAL" :: "[t, t, t] \<Rightarrow> t"        ("(3_ =\<^sub>_/ _)" [101, 0, 101] 100)
+  "_EQUAL_ASCII" :: "[t, t, t] \<Rightarrow> t"  ("(3_ =[_]/ _)" [101, 0, 101] 100)
 translations
   "a =[A] b" \<rightleftharpoons> "CONST Equal A a b"
   "a =\<^sub>A b" \<rightharpoonup> "CONST Equal A a b"
@@ -27,33 +27,33 @@ translations
 section \<open>Type rules\<close>
 
 axiomatization where
-  Equal_form: "\<lbrakk>A: U(i); a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U(i)"
+  Equal_form: "\<lbrakk>A: U i; a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U i"
 and
-  Equal_intro: "a : A \<Longrightarrow> refl(a): a =\<^sub>A a"
+  Equal_intro: "a : A \<Longrightarrow> (refl a): a =\<^sub>A a"
 and
   Equal_elim: "\<lbrakk>
     x: A;
     y: A;
     p: x =\<^sub>A y;
-    \<And>x. x: A \<Longrightarrow> f(x) : C(x)(x)(refl x);
-    \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C(x)(y): x =\<^sub>A y \<longrightarrow> U(i)
-    \<rbrakk> \<Longrightarrow> ind\<^sub>=(f)(p) : C(x)(y)(p)"
+    \<And>x. x: A \<Longrightarrow> f x: C x x (refl x);
+    \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i
+    \<rbrakk> \<Longrightarrow> ind\<^sub>= f p : C x y p"
 and
   Equal_comp: "\<lbrakk>
     a: A;
-    \<And>x. x: A \<Longrightarrow> f(x) : C(x)(x)(refl x);
-    \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C(x)(y): x =\<^sub>A y \<longrightarrow> U(i)
-    \<rbrakk> \<Longrightarrow> ind\<^sub>=(f)(refl(a)) \<equiv> f(a)"
+    \<And>x. x: A \<Longrightarrow> f x: C x x (refl x);
+    \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U i
+    \<rbrakk> \<Longrightarrow> ind\<^sub>= f (refl a) \<equiv> f a"
 
 
 text "Admissible inference rules for equality type formation:"
 
 axiomatization where
-  Equal_wellform1: "a =\<^sub>A b: U(i) \<Longrightarrow> A: U(i)"
+  Equal_wellform1: "a =\<^sub>A b: U i \<Longrightarrow> A: U i"
 and
-  Equal_wellform2: "a =\<^sub>A b: U(i) \<Longrightarrow> a: A"
+  Equal_wellform2: "a =\<^sub>A b: U i \<Longrightarrow> a: A"
 and
-  Equal_wellform3: "a =\<^sub>A b: U(i) \<Longrightarrow> b: A"
+  Equal_wellform3: "a =\<^sub>A b: U i \<Longrightarrow> b: A"
 
 
 text "Rule attribute declarations:"