From 352a6e40a5bdf193b8f9690e76aede4e0650a445 Mon Sep 17 00:00:00 2001
From: Josh Chen
Date: Sat, 30 Jun 2018 07:07:42 +0200
Subject: Equality

---
 Equal.thy | 83 ++++++++++++++++++++++-----------------------------------------
 1 file changed, 29 insertions(+), 54 deletions(-)

(limited to 'Equal.thy')

diff --git a/Equal.thy b/Equal.thy
index 02fe540..6c18084 100644
--- a/Equal.thy
+++ b/Equal.thy
@@ -12,7 +12,10 @@ begin
 axiomatization
   Equal :: "[Term, Term, Term] \<Rightarrow> Term" and
   refl :: "Term \<Rightarrow> Term"  ("(refl'(_'))" 1000) and
-  indEqual :: "[Term, [Term, Term, Term] \<Rightarrow> Term] \<Rightarrow> Term"  ("(indEqual[_])")
+  indEqual :: "[Term, [Term, Term] \<Rightarrow> Typefam, Term \<Rightarrow> Term, Term, Term, Term] \<Rightarrow> Term"  ("(indEqual[_])")
+
+
+section \<open>Syntax\<close>
 
 syntax
   "_EQUAL" :: "[Term, Term, Term] \<Rightarrow> Term"        ("(3_ =\<^sub>_/ _)" [101, 101] 100)
@@ -21,64 +24,36 @@ translations
   "a =[A] b" \<rightleftharpoons> "CONST Equal A a b"
   "a =\<^sub>A b" \<rightharpoonup> "CONST Equal A a b"
 
+
+section \<open>Type rules\<close>
+
 axiomatization where
-  Equal_form: "\<And>A a b::Term. \<lbrakk>A : U; a : A; b : A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U"
-  (* Should I write a permuted version \<open>\<lbrakk>A : U; b : A; a : A\<rbrakk> \<Longrightarrow> \<dots>\<close>? *)
+  Equal_form: "\<And>A a b. \<lbrakk>a : A; b : A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U"
 and
-  Equal_intro [intro]: "\<And>A x::Term. x : A \<Longrightarrow> refl(x) : x =\<^sub>A x"
+  Equal_form_cond1: "\<And>A a b. a =\<^sub>A b : U \<Longrightarrow> A : U"
 and
-  Equal_elim [elim]:
-    "\<And>(A::Term) (C::[Term, Term, Term] \<Rightarrow> Term) (f::Term) (a::Term) (b::Term) (p::Term).
-      \<lbrakk> \<And>x y::Term. \<lbrakk>x : A; y : A\<rbrakk> \<Longrightarrow> C(x)(y): x =\<^sub>A y \<rightarrow> U;
-        f : \<Prod>x:A. C(x)(x)(refl(x));
-        a : A;
-        b : A;
-        p : a =\<^sub>A b \<rbrakk>
-    \<Longrightarrow> indEqual[A](C)`f`a`b`p : C(a)(b)(p)"
+  Equal_form_cond2: "\<And>A a b. a =\<^sub>A b : U \<Longrightarrow> a : A"
 and
-  Equal_comp [simp]:
-    "\<And>(A::Term) (C::[Term, Term, Term] \<Rightarrow> Term) (f::Term) (a::Term). indEqual[A](C)`f`a`a`refl(a) \<equiv> f`a"
-
-lemmas Equal_formation [intro] = Equal_form Equal_form[rotated 1] Equal_form[rotated 2]
-
-subsubsection \<open>Properties of equality\<close>
-
-text "Symmetry/Path inverse"
-
-definition inv :: "[Term, Term, Term] \<Rightarrow> Term"  ("(1inv[_,/ _,/ _])")
-  where "inv[A,x,y] \<equiv> indEqual[A](\<lambda>x y _. y =\<^sub>A x)`(\<^bold>\<lambda>x:A. refl(x))`x`y"
-
-lemma inv_comp: "\<And>A a::Term. a : A \<Longrightarrow> inv[A,a,a]`refl(a) \<equiv> refl(a)" unfolding inv_def by simp
-
-text "Transitivity/Path composition"
-
-\<comment> \<open>"Raw" composition function\<close>
-definition compose' :: "Term \<Rightarrow> Term"  ("(1compose''[_])")
-  where "compose'[A] \<equiv> indEqual[A](\<lambda>x y _. \<Prod>z:A. \<Prod>q: y =\<^sub>A z. x =\<^sub>A z)`(indEqual[A](\<lambda>x z _. x =\<^sub>A z)`(\<^bold>\<lambda>x:A. refl(x)))"
-
-\<comment> \<open>"Natural" composition function\<close>
-abbreviation compose :: "[Term, Term, Term, Term] \<Rightarrow> Term"  ("(1compose[_,/ _,/ _,/ _])")
-  where "compose[A,x,y,z] \<equiv> \<^bold>\<lambda>p:x =\<^sub>A y. \<^bold>\<lambda>q:y =\<^sub>A z. compose'[A]`x`y`p`z`q"
-
-(**** GOOD CANDIDATE FOR AUTOMATION ****)
-lemma compose_comp:
-  assumes "a : A"
-  shows "compose[A,a,a,a]`refl(a)`refl(a) \<equiv> refl(a)" using assms Equal_intro[OF assms] unfolding compose'_def by simp
-
-text "The above proof is a good candidate for proof automation; in particular we would like the system to be able to automatically find the conditions of the \<open>using\<close> clause in the proof.
-This would likely involve something like:
-  1. Recognizing that there is a function application that can be simplified.
-  2. Noting that the obstruction to applying \<open>Prod_comp\<close> is the requirement that \<open>refl(a) : a =\<^sub>A a\<close>.
-  3. Obtaining such a condition, using the known fact \<open>a : A\<close> and the introduction rule \<open>Equal_intro\<close>."
-
-lemmas Equal_simps [simp] = inv_comp compose_comp
-
-subsubsection \<open>Pretty printing\<close>
+  Equal_form_cond3: "\<And>A a b. a =\<^sub>A b : U \<Longrightarrow> b : A"
+and
+  Equal_intro: "\<And>A a. a : A \<Longrightarrow> refl(a) : a =\<^sub>A a"
+and
+  Equal_elim: "\<And>A C f a b p. \<lbrakk>
+    \<And>x y.\<lbrakk>x : A; y : A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<rightarrow> U;
+    \<And>x. x : A \<Longrightarrow> f x : C x x refl(x);
+    a : A;
+    b : A;
+    p : a =\<^sub>A b
+    \<rbrakk> \<Longrightarrow> indEqual[A] C f a b p : C a b p"
+and
+  Equal_comp: "\<And>A C f a. \<lbrakk>
+    \<And>x y.\<lbrakk>x : A; y : A\<rbrakk> \<Longrightarrow> C x y: x =\<^sub>A y \<rightarrow> U;
+    \<And>x. x : A \<Longrightarrow> f x : C x x refl(x);
+    a : A
+    \<rbrakk> \<Longrightarrow> indEqual[A] C f a a refl(a) \<equiv> f a"
 
-abbreviation inv_pretty :: "[Term, Term, Term, Term] \<Rightarrow> Term"  ("(1_\<^sup>-\<^sup>1[_, _, _])" 500)
-  where "p\<^sup>-\<^sup>1[A,x,y] \<equiv> inv[A,x,y]`p"
+lemmas Equal_rules [intro] = Equal_form Equal_intro Equal_elim Equal_comp
+lemmas Equal_form_conds [elim] = Equal_form_cond1 Equal_form_cond2 Equal_form_cond3
 
-abbreviation compose_pretty :: "[Term, Term, Term, Term, Term, Term] \<Rightarrow> Term"  ("(1_ \<bullet>[_, _, _, _]/ _)")
-  where "p \<bullet>[A,x,y,z] q \<equiv> compose[A,x,y,z]`p`q"
 
 end
\ No newline at end of file
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