From 602ad9fe0e2ed1ad4ab6f16e720de878aadc0fba Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Fri, 15 Jun 2018 17:17:27 +0200
Subject: projections

---
 HoTT_Base.thy | 40 +++++++++++++---------------------------
 1 file changed, 13 insertions(+), 27 deletions(-)

(limited to 'HoTT_Base.thy')

diff --git a/HoTT_Base.thy b/HoTT_Base.thy
index 9b422c4..7794601 100644
--- a/HoTT_Base.thy
+++ b/HoTT_Base.thy
@@ -2,7 +2,7 @@
     Author: Josh Chen
     Date:   Jun 2018
 
-Basic setup and definitions of a homotopy type theory object logic.
+Basic setup and definitions of a homotopy type theory object logic without universes.
 *)
 
 theory HoTT_Base
@@ -18,16 +18,23 @@ text "Set up type checking routines, proof methods etc."
 section \<open>Metalogical definitions\<close>
 
 text "A single meta-type \<open>Term\<close> suffices to implement the object-logic types and terms.
-Our implementation does not have universes, and we simply use \<open>a : U\<close> as a convenient shorthand meaning ``\<open>a\<close> is a type''."
+We do not implement universes, and simply use \<open>a : U\<close> as a convenient shorthand to mean ``\<open>a\<close> is a type''."
 
 typedecl Term
 
 
 section \<open>Judgments\<close>
 
+text "We formalize the judgments \<open>a : A\<close> and \<open>A : U\<close> separately, in contrast to the HoTT book where the latter is considered an instance of the former.
+
+For judgmental equality we use the existing Pure equality \<open>\<equiv>\<close> and hence do not need to define a separate judgment for it."
+
 consts
-is_a_type :: "Term \<Rightarrow> prop"           ("(1_ :/ U)" [0] 1000)
-is_of_type :: "[Term, Term] \<Rightarrow> prop"  ("(1_ :/ _)" [0, 0] 1000)
+  is_a_type :: "Term \<Rightarrow> prop"           ("(1_ :/ U)" [0] 1000)
+  is_of_type :: "[Term, Term] \<Rightarrow> prop"  ("(1_ :/ _)" [0, 0] 1000)
+
+axiomatization where
+  inhabited_implies_type [intro]: "\<And>a A. a : A \<Longrightarrow> A : U"
 
 
 section \<open>Type families\<close>
@@ -36,31 +43,10 @@ text "A (one-variable) type family is a meta lambda term \<open>P :: Term \<Righ
 
 type_synonym Typefam = "Term \<Rightarrow> Term"
 
-abbreviation is_type_family :: "[Typefam, Term] \<Rightarrow> prop"  ("(3_:/ _ \<rightarrow> U)")
+abbreviation (input) is_type_family :: "[Typefam, Term] \<Rightarrow> prop"  ("(3_:/ _ \<rightarrow> U)")
   where "P: A \<rightarrow> U \<equiv> (\<And>x. x : A \<Longrightarrow> P x : U)"
 
-text "There is an obvious generalization to multivariate type families, but implementing such an abbreviation involves writing ML and is for the moment not really crucial."
-
-
-section \<open>Definitional equality\<close>
-
-text "The Pure equality \<open>\<equiv>\<close> is used for definitional aka judgmental equality of types and terms."
-
-\<comment> \<open>Do these ever need to be used?
-
-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
+text "There is an obvious generalization to multivariate type families, but implementing such an abbreviation involves writing ML code, and is for the moment not really crucial."
 
-lemmas type_equality =
-  equal_types
-  equal_types[rotated]
-  equal_type_element
-  equal_type_element[rotated]
-\<close>
 
 end
\ No newline at end of file
-- 
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