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(* Title: HoTT/Equal.thy
Author: Josh Chen
Date: Jun 2018
Equality type.
*)
theory Equal
imports HoTT_Base
begin
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>=[_])")
syntax
"_EQUAL" :: "[Term, Term, Term] \<Rightarrow> Term" ("(3_ =\<^sub>_/ _)" [101, 0, 101] 100)
"_EQUAL_ASCII" :: "[Term, Term, Term] \<Rightarrow> Term" ("(3_ =[_]/ _)" [101, 0, 101] 100)
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>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"
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
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"
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)"
and
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"
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
lemmas Equal_comps [comp] = Equal_comp
end
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