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+(*  Title:  HoTT/HoTT_Base.thy
+    Author: Josh Chen
+
+Basic setup and definitions of a homotopy type theory object logic.
+*)
+
+theory HoTT_Base
+  imports Pure
+
+begin
+
+section \<open>Basic definitions\<close>
+
+text "A single meta-level type \<open>Term\<close> suffices to implement the object-level types and terms.
+We do not implement universes, but simply follow the informal notation in the HoTT book."
+
+typedecl Term
+
+section \<open>Judgments\<close>
+
+consts
+is_a_type :: "Term \<Rightarrow> prop"           ("(_ : U)" [0] 1000)
+is_of_type :: "[Term, Term] \<Rightarrow> prop"  ("(3_ :/ _)" [0, 0] 1000)
+
+
+section \<open>Definitional equality\<close>
+
+text "We use the Pure equality \<open>\<equiv>\<close> for definitional/judgmental equality of types and terms in our theory."
+
+theorem equal_types:
+  assumes "A \<equiv> B" and "A : U"
+  shows "B : U" using assms by simp
+
+theorem equal_type_element:
+  assumes "A \<equiv> B" and "x : A"
+  shows "x : B" using assms by simp
+
+lemmas type_equality [intro, simp] =
+  equal_types
+  equal_types[rotated]
+  equal_type_element
+  equal_type_element[rotated]
+
+
+section \<open>Type families\<close>
+
+text "A type family is a meta lambda term \<open>P :: Term \<Rightarrow> Term\<close> that further satisfies the following property."
+
+abbreviation is_type_family :: "[Term \<Rightarrow> Term, Term] \<Rightarrow> prop"  ("(3_:/ _ \<rightarrow> U)")
+  where "P: A \<rightarrow> U \<equiv> (\<And>x. x : A \<Longrightarrow> P(x) : U)"
+
+end
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