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+(*  Title:  HoTT/Prod.thy
+    Author: Josh Chen
+
+Dependent product (function) type for the HoTT logic.
+*)
+
+theory Prod
+  imports HoTT_Base
+
+begin
+
+axiomatization
+  Prod :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term" and
+  lambda :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term" and
+  appl :: "[Term, Term] \<Rightarrow> Term"  (infixl "`" 60)
+
+syntax
+  "_PROD" :: "[idt, Term, Term] \<Rightarrow> Term"          ("(3\<Prod>_:_./ _)" 30)
+  "_LAMBDA" :: "[idt, Term, Term] \<Rightarrow> Term"        ("(3\<^bold>\<lambda>_:_./ _)" 30)
+  "_PROD_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term"    ("(3PROD _:_./ _)" 30)
+  "_LAMBDA_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term"  ("(3%%_:_./ _)" 30)
+
+\<comment> \<open>The translations below bind the variable \<open>x\<close> in the expressions \<open>B\<close> and \<open>b\<close>.\<close>
+translations
+  "\<Prod>x:A. B" \<rightleftharpoons> "CONST Prod A (\<lambda>x. B)"
+  "\<^bold>\<lambda>x:A. b" \<rightleftharpoons> "CONST lambda A (\<lambda>x. b)"
+  "PROD x:A. B" \<rightharpoonup> "CONST Prod A (\<lambda>x. B)"
+  "%%x:A. b" \<rightharpoonup> "CONST lambda A (\<lambda>x. b)"
+
+\<comment> \<open>Type rules\<close>
+axiomatization where
+  Prod_form [intro]: "\<And>A B. \<lbrakk>A : U; B : A \<rightarrow> U\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x) : U"
+and
+  Prod_intro [intro]: "\<And>A B b. (\<And>x. x : A \<Longrightarrow> b(x) : B(x)) \<Longrightarrow> \<^bold>\<lambda>x:A. b(x) : \<Prod>x:A. B(x)"
+and
+  Prod_elim [elim]: "\<And>A B f a. \<lbrakk>f : \<Prod>x:A. B(x); a : A\<rbrakk> \<Longrightarrow> f`a : B(a)"
+and
+  Prod_comp [simp]: "\<And>A b a. a : A \<Longrightarrow> (\<^bold>\<lambda>x:A. b(x))`a \<equiv> b(a)"
+and
+  Prod_uniq [simp]: "\<And>A f. \<^bold>\<lambda>x:A. (f`x) \<equiv> f"
+
+text "Note that the syntax \<open>\<^bold>\<lambda>\<close> (bold lambda) used for dependent functions clashes with the proof term syntax (cf. \<section>2.5.2 of the Isabelle/Isar Implementation)."
+
+\<comment> \<open>Nondependent functions are a special case.\<close>
+abbreviation Function :: "[Term, Term] \<Rightarrow> Term"  (infixr "\<rightarrow>" 40)
+  where "A \<rightarrow> B \<equiv> \<Prod>_:A. B"
+
+end
+\ No newline at end of file