From 7a89ec1e72f61179767c6488177c6d12e69388c5 Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Sun, 12 Aug 2018 13:04:16 +0200
Subject: Commit before testing polymorphic equality eliminator

---
 Prod.thy | 22 ++++++++++++----------
 1 file changed, 12 insertions(+), 10 deletions(-)

(limited to 'Prod.thy')

diff --git a/Prod.thy b/Prod.thy
index 01cd006..5391943 100644
--- a/Prod.thy
+++ b/Prod.thy
@@ -41,20 +41,21 @@ abbreviation Function :: "[Term, Term] \<Rightarrow> Term"  (infixr "\<rightarro
 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)"
+  Prod_form: "\<lbrakk>A: U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Prod>x:A. B(x): U(i)"
 and
-  Prod_intro: "\<And>i A B b. \<lbrakk>A: U(i); B: A \<longrightarrow> U(i); \<And>x. x: A \<Longrightarrow> b(x): B(x)\<rbrakk> \<Longrightarrow> \<^bold>\<lambda>x:A. b(x): \<Prod>x:A. B(x)"
+  Prod_intro: "\<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)"
+  Prod_elim: "\<lbrakk>f: \<Prod>x:A. B(x); a: A\<rbrakk> \<Longrightarrow> f`a: B(a)"
 and
-  Prod_comp: "\<And>i A B b a. \<lbrakk>A: U(i); B: A \<longrightarrow> U(i); \<And>x. x: A \<Longrightarrow> b(x): B(x); a: A\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. b(x))`a \<equiv> b(a)"
+  Prod_comp: "\<lbrakk>a: A; \<And>x. x: A \<Longrightarrow> b(x): B(x)\<rbrakk> \<Longrightarrow> (\<^bold>\<lambda>x:A. b(x))`a \<equiv> b(a)"
 and
-  Prod_uniq: "\<And>A B f. f : \<Prod>x:A. B(x) \<Longrightarrow> \<^bold>\<lambda>x:A. (f`x) \<equiv> f"
+  Prod_uniq: "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 "
   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>.
   
@@ -62,14 +63,15 @@ text "
 "
 
 axiomatization where
-  Prod_form_cond1: "\<And>i A B. (\<Prod>x:A. B(x): U(i)) \<Longrightarrow> A: U(i)"
+  Prod_form_cond1: "(\<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)"
+  Prod_form_cond2: "(\<Prod>x:A. B(x): U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)"
+*)
 
 text "Set up the standard reasoner to use the type rules:"
 
 lemmas Prod_rules = Prod_form Prod_intro Prod_elim Prod_comp Prod_uniq
-lemmas Prod_form_conds [intro (*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
 
 
@@ -84,9 +86,9 @@ where
 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> ind\<^sub>\<one>(c)(a) : C(a)"
+  Unit_elim: "\<lbrakk>C: \<one> \<longrightarrow> U(i); c: C(\<star>); a: \<one>\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(c)(a) : C(a)"
 and
-  Unit_comp: "\<And>i C c. \<lbrakk>C: \<one> \<longrightarrow> U(i); c: C(\<star>)\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(c)(\<star>) \<equiv> c"
+  Unit_comp: "\<lbrakk>C: \<one> \<longrightarrow> U(i); c: C(\<star>)\<rbrakk> \<Longrightarrow> ind\<^sub>\<one>(c)(\<star>) \<equiv> c"
 
 lemmas Unit_rules [intro] = Unit_form Unit_intro Unit_elim Unit_comp
 lemmas Unit_comps [comp] = Unit_comp
-- 
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