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(* Title: HoTT/Equal.thy
Author: Josh Chen
Date: Jun 2018
Equality type.
*)
theory Equal
imports HoTT_Base
begin
section ‹Constants and syntax›
axiomatization
Equal :: "[Term, Term, Term] ⇒ Term" and
refl :: "Term ⇒ Term" and
indEqual :: "[Term ⇒ Term, Term] ⇒ Term" ("(1ind⇩=)")
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: U(i); a: A; b: A⟧ ⟹ a =⇩A b : U(i)"
and
Equal_intro: "a : A ⟹ refl(a): a =⇩A a"
and
Equal_elim: "⟦
⋀x y. ⟦x: A; y: A⟧ ⟹ C(x)(y): x =⇩A y ⟶ U(i);
⋀x. x: A ⟹ f(x) : C(x)(x)(refl x);
x: A;
y: A;
p: x =⇩A y
⟧ ⟹ ind⇩=(f)(p) : C(x)(y)(p)"
and
Equal_comp: "⟦
⋀x y. ⟦x: A; y: A⟧ ⟹ C(x)(y): x =⇩A y ⟶ U(i);
⋀x. x: A ⟹ f(x) : C(x)(x)(refl x);
a: A
⟧ ⟹ ind⇩=(f)(refl(a)) ≡ f(a)"
text "Admissible inference rules for equality type formation:"
axiomatization where
Equal_form_cond1: "a =⇩A b: U(i) ⟹ A: U(i)"
and
Equal_form_cond2: "a =⇩A b: U(i) ⟹ a: A"
and
Equal_form_cond3: "a =⇩A b: U(i) ⟹ 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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