Reorganize theories

1 files changed, 30 insertions, 26 deletions

diff --git a/Prod.thy b/Prod.thy
index fbea4df..8fa73f3 100644
--- a/Prod.thy
+++ b/Prod.thy
@@ -1,8 +1,8 @@
 (*  Title:  HoTT/Prod.thy
     Author: Josh Chen
-    Date:   Jun 2018
+    Date:   Aug 2018
 
-Dependent product (function) type for the HoTT logic.
+Dependent product (function) type.
 *)
 
 theory Prod
@@ -10,14 +10,13 @@ theory Prod
 begin
 
 
+section \<open>Constants and syntax\<close>
+
 axiomatization
   Prod :: "[Term, Typefam] \<Rightarrow> Term" and
   lambda :: "[Term, Term \<Rightarrow> Term] \<Rightarrow> Term" and
-  \<comment> \<open>Application binds tighter than abstraction.\<close>
   appl :: "[Term, Term] \<Rightarrow> Term"  (infixl "`" 60)
-
-
-section \<open>Syntax\<close>
+    \<comment> \<open>Application binds tighter than abstraction.\<close>
 
 syntax
   "_PROD" :: "[idt, Term, Term] \<Rightarrow> Term"          ("(3\<Prod>_:_./ _)" 30)
@@ -33,17 +32,18 @@ translations
   "PROD x:A. B" \<rightharpoonup> "CONST Prod A (\<lambda>x. B)"
   "%%x:A. b" \<rightharpoonup> "CONST lambda A (\<lambda>x. b)"
 
+text "Nondependent functions are a special case."
+
+abbreviation Function :: "[Term, Term] \<Rightarrow> Term"  (infixr "\<rightarrow>" 40)
+  where "A \<rightarrow> B \<equiv> \<Prod>_:A. B"
+
 
 section \<open>Type rules\<close>
 
 axiomatization where
   Prod_form: "\<And>i A B. \<lbrakk>A : U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Prod>x:A. B x : U(i)"
 and
-  Prod_form_cond1: "\<And>i A B. (\<Prod>x:A. B x : U(i)) \<Longrightarrow> A : U(i)"
-and
-  Prod_form_cond2: "\<And>i A B. (\<Prod>x:A. B x : U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)"
-and
-  Prod_intro: "\<And>i A B b. (\<And>x. x : A \<Longrightarrow> b x : B x) \<Longrightarrow> \<^bold>\<lambda>x:A. b x : \<Prod>x:A. B x"
+  Prod_intro: "\<And>i A B b. \<lbrakk>A : U(i); \<And>x. x : A \<Longrightarrow> b x : B x\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x:A. b x : \<Prod>x:A. B x"
 and
   Prod_elim: "\<And>A B f a. \<lbrakk>f : \<Prod>x:A. B x; a : A\<rbrakk> \<Longrightarrow> f`a : B a"
 and
@@ -51,38 +51,42 @@ and
 and
   Prod_uniq: "\<And>A B f. f : \<Prod>x:A. B x \<Longrightarrow> \<^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)."
+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).
+"
+
+text "
+  In addition to the usual type rules, it is a meta-theorem (*PROVE THIS!*) that whenever \<open>\<Prod>x:A. B x : U(i)\<close> is derivable from some set of premises \<Gamma>, then so are \<open>A : U(i)\<close> and \<open>B: A \<longrightarrow> U(i)\<close>.
+  
+  That is to say, the following inference rules are admissible, and it simplifies proofs greatly to axiomatize them directly.
+"
 
-text "In textbook presentations it is usual to present type formation as a forward implication, stating conditions sufficient for the formation of the type.
-It is however implicit that the premises of the rule are also necessary, so that for example the only way for one to have that \<open>\<Prod>x:A. B x : U\<close> is for \<open>A : U\<close> and \<open>B: A \<rightarrow> U\<close> in the first place.
-This is what the additional formation rules \<open>Prod_form_cond1\<close> and \<open>Prod_form_cond2\<close> express."
+axiomatization where
+  Prod_form_cond1: "\<And>i A B. (\<Prod>x:A. B x : U(i)) \<Longrightarrow> A : U(i)"
+and
+  Prod_form_cond2: "\<And>i A B. (\<Prod>x:A. B x : U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)"
 
-text "The type rules should be able to be used as introduction and elimination rules by the standard reasoner:"
+text "Set up the standard reasoner to use the type rules:"
 
 lemmas Prod_rules [intro] = Prod_form Prod_intro Prod_elim Prod_comp Prod_uniq
-lemmas Prod_form_conds [elim, wellform] = Prod_form_cond1 Prod_form_cond2
+lemmas Prod_form_conds [intro (*elim, wellform*)] = Prod_form_cond1 Prod_form_cond2
 lemmas Prod_comps [comp] = Prod_comp Prod_uniq
 
-text "Nondependent functions are a special case."
-
-abbreviation Function :: "[Term, Term] \<Rightarrow> Term"  (infixr "\<rightarrow>" 40)
-  where "A \<rightarrow> B \<equiv> \<Prod>_:A. B"
-
 
 section \<open>Unit type\<close>
 
 axiomatization
   Unit :: Term  ("\<one>") and
   pt :: Term    ("\<star>") and
-  indUnit :: "[Typefam, Term, Term] \<Rightarrow> Term"
+  indUnit :: "[Typefam, Term, Term] \<Rightarrow> Term"  ("(1ind\<^sub>\<one>)")
 where
-  Unit_form: "\<one> : U(0)"
+  Unit_form: "\<one> : U(O)"
 and
   Unit_intro: "\<star> : \<one>"
 and
-  Unit_elim: "\<And>i C c a. \<lbrakk>C: \<one> \<longrightarrow> U(i); c : C \<star>; a : \<one>\<rbrakk> \<Longrightarrow> indUnit C c a : C a"
+  Unit_elim: "\<And>i C c a. \<lbrakk>C: \<one> \<longrightarrow> U(i); c : C \<star>; a : \<one>\<rbrakk> \<Longrightarrow> ind\<^sub>\<one> C c a : C a"
 and
-  Unit_comp: "\<And>i C c. \<lbrakk>C: \<one> \<longrightarrow> U(i); c : C \<star>\<rbrakk> \<Longrightarrow> indUnit C c \<star> \<equiv> c"
+  Unit_comp: "\<And>i C c. \<lbrakk>C: \<one> \<longrightarrow> U(i); c : C \<star>\<rbrakk> \<Longrightarrow> ind\<^sub>\<one> C c \<star> \<equiv> c"
 
 lemmas Unit_rules [intro] = Unit_form Unit_intro Unit_elim Unit_comp
 lemmas Unit_comps [comp] = Unit_comp