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(* Title: HoTT/Equal.thy
Author: Josh Chen
Date: Jun 2018
Equality type.
*)
theory Equal
imports HoTT_Base
begin
axiomatization
Equal :: "[Term, Term, Term] ⇒ Term" and
refl :: "Term ⇒ Term" ("(refl'(_'))" 1000) and
indEqual :: "[Term, [Term, Term] ⇒ Typefam, Term ⇒ Term, Term, Term, Term] ⇒ Term" ("(1indEqual[_])")
section ‹Syntax›
syntax
"_EQUAL" :: "[Term, Term, Term] ⇒ Term" ("(3_ =⇩_/ _)" [101, 0, 101] 100)
"_EQUAL_ASCII" :: "[Term, Term, Term] ⇒ Term" ("(3_ =[_]/ _)" [101, 0, 101] 100)
translations
"a =[A] b" ⇌ "CONST Equal A a b"
"a =⇩A b" ⇀ "CONST Equal A a b"
section ‹Type rules›
axiomatization where
Equal_form: "⋀A a b. ⟦a : A; b : A⟧ ⟹ a =⇩A b : U"
and
Equal_form_cond1: "⋀A a b. a =⇩A b : U ⟹ A : U"
and
Equal_form_cond2: "⋀A a b. a =⇩A b : U ⟹ a : A"
and
Equal_form_cond3: "⋀A a b. a =⇩A b : U ⟹ b : A"
and
Equal_intro: "⋀A a. a : A ⟹ refl(a) : a =⇩A a"
and
Equal_elim: "⋀A C f a b p. ⟦
⋀x y.⟦x : A; y : A⟧ ⟹ C x y: x =⇩A y → U;
⋀x. x : A ⟹ f x : C x x refl(x);
a : A;
b : A;
p : a =⇩A b
⟧ ⟹ indEqual[A] C f a b p : C a b p"
and
Equal_comp: "⋀A C f a. ⟦
⋀x y.⟦x : A; y : A⟧ ⟹ C x y: x =⇩A y → U;
⋀x. x : A ⟹ f x : C x x refl(x);
a : A
⟧ ⟹ indEqual[A] C f a a refl(a) ≡ f a"
lemmas Equal_rules [intro] = Equal_form Equal_intro Equal_elim Equal_comp
lemmas Equal_form_conds [elim, wellform] = Equal_form_cond1 Equal_form_cond2 Equal_form_cond3
lemmas Equal_comps [comp] = Equal_comp
end
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