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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" and
indEqual :: "[Term \<Rightarrow> 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: "\<lbrakk>A: U(i); a: A; b: A\<rbrakk> \<Longrightarrow> a =\<^sub>A b : U(i)"
and
Equal_intro: "a : A \<Longrightarrow> refl(a): a =\<^sub>A a"
and
Equal_elim: "\<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);
x: A;
y: A;
p: a =\<^sub>A b
\<rbrakk> \<Longrightarrow> ind\<^sub>=(f)(p) : C(x)(y)(p)"
and
Equal_comp: "\<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>=(f)(refl(a)) \<equiv> f(a)"
text "Admissible inference rules for equality type formation:"
axiomatization where
Equal_form_cond1: "a =\<^sub>A b: U(i) \<Longrightarrow> A: U(i)"
and
Equal_form_cond2: "a =\<^sub>A b: U(i) \<Longrightarrow> a: A"
and
Equal_form_cond3: "a =\<^sub>A b: U(i) \<Longrightarrow> b: A"
text "Rule declarations:"
lemmas Equal_rules [intro] = Equal_form Equal_intro Equal_elim Equal_comp
lemmas Equal_wellform [wellform] = Equal_form_cond1 Equal_form_cond2 Equal_form_cond3
lemmas Equal_comps [comp] = Equal_comp
end
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