From dc2730916482bd230f9bd5244b8b2cc9d005f69a Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Tue, 7 Aug 2018 12:30:59 +0200
Subject: Old application syntax incompatible with Eisbach

---
 Equal.thy | 34 +++++++++++++++++-----------------
 1 file changed, 17 insertions(+), 17 deletions(-)

(limited to 'Equal.thy')

diff --git a/Equal.thy b/Equal.thy
index 2b49213..8bd7875 100644
--- a/Equal.thy
+++ b/Equal.thy
@@ -14,8 +14,8 @@ section \<open>Constants and syntax\<close>
 
 axiomatization
   Equal :: "[Term, Term, Term] \<Rightarrow> Term" and
-  refl :: "Term \<Rightarrow> Term"  ("(refl'(_'))" 1000) and
-  indEqual :: "[Term, [Term, Term] \<Rightarrow> Typefam, Term \<Rightarrow> Term, Term, Term, Term] \<Rightarrow> Term"  ("(1ind\<^sub>=[_])")
+  refl :: "Term \<Rightarrow> Term" and
+  indEqual :: "[Term \<Rightarrow> Term, Term, Term, Term] \<Rightarrow> Term"  ("(1ind\<^sub>=)")
 
 syntax
   "_EQUAL" :: "[Term, Term, Term] \<Rightarrow> Term"        ("(3_ =\<^sub>_/ _)" [101, 0, 101] 100)
@@ -28,32 +28,32 @@ translations
 section \<open>Type rules\<close>
 
 axiomatization where
-  Equal_form: "\<And>i A a b. \<lbrakk>A : U(i); a : A; b : A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U(i)"
+  Equal_form: "\<And>i A a b. \<lbrakk>A: U(i); a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U(i)"
 and
-  Equal_intro: "\<And>A a. a : A \<Longrightarrow> refl(a) : a =\<^sub>A a"
+  Equal_intro: "\<And>A a. a : A \<Longrightarrow> refl(a): a =\<^sub>A a"
 and
   Equal_elim: "\<And>i A C f a b p. \<lbrakk>
-    \<And>x y. \<lbrakk>x : A; y : A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U(i);
-    \<And>x. x : A \<Longrightarrow> f x : C x x refl(x);
-    a : A;
-    b : A;
-    p : a =\<^sub>A b
-    \<rbrakk> \<Longrightarrow> ind\<^sub>=[A] C f a b p : C a b p"
+    \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C(x)(y): x =\<^sub>A y \<longrightarrow> U(i);
+    \<And>x. x: A \<Longrightarrow> f(x) : C(x)(x)(refl x);
+    a: A;
+    b: A;
+    p: a =\<^sub>A b
+    \<rbrakk> \<Longrightarrow> ind\<^sub>=(f)(a)(b)(p) : C(a)(b)(p)"
 and
   Equal_comp: "\<And>i A C f a. \<lbrakk>
-    \<And>x y. \<lbrakk>x : A; y : A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<longrightarrow> U(i);
-    \<And>x. x : A \<Longrightarrow> f x : C x x refl(x);
-    a : A
-    \<rbrakk> \<Longrightarrow> ind\<^sub>=[A] C f a a refl(a) \<equiv> f a"
+    \<And>x y. \<lbrakk>x: A; y: A\<rbrakk> \<Longrightarrow> C(x)(y): x =\<^sub>A y \<longrightarrow> U(i);
+    \<And>x. x: A \<Longrightarrow> f(x) : C(x)(x)(refl x);
+    a: A
+    \<rbrakk> \<Longrightarrow> ind\<^sub>=(f)(a)(a)(refl(a)) \<equiv> f(a)"
 
 text "Admissible inference rules for equality type formation:"
 
 axiomatization where
-  Equal_form_cond1: "\<And>i A a b. a =\<^sub>A b : U(i) \<Longrightarrow> A : U(i)"
+  Equal_form_cond1: "\<And>i A a b. a =\<^sub>A b: U(i) \<Longrightarrow> A: U(i)"
 and
-  Equal_form_cond2: "\<And>i A a b. a =\<^sub>A b : U(i) \<Longrightarrow> a : A"
+  Equal_form_cond2: "\<And>i A a b. a =\<^sub>A b: U(i) \<Longrightarrow> a: A"
 and
-  Equal_form_cond3: "\<And>i A a b. a =\<^sub>A b : U(i) \<Longrightarrow> b : A"
+  Equal_form_cond3: "\<And>i A a b. a =\<^sub>A b: U(i) \<Longrightarrow> b: A"
 
 lemmas Equal_rules [intro] = Equal_form Equal_intro Equal_elim Equal_comp
 lemmas Equal_form_conds [intro] = Equal_form_cond1 Equal_form_cond2 Equal_form_cond3
-- 
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