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(* Title: HoTT/Equal.thy
Author: Josh Chen
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: A;
y: A;
p: x =⇩A y;
⋀x. x: A ⟹ f x: C x x (refl x);
⋀x y. ⟦x: A; y: A⟧ ⟹ C x y: x =⇩A y ⟶ U i
⟧ ⟹ ind⇩= f p : C x y p"
and
Equal_comp: "⟦
a: A;
⋀x. x: A ⟹ f x: C x x (refl x);
⋀x y. ⟦x: A; y: A⟧ ⟹ C x y: x =⇩A y ⟶ U i
⟧ ⟹ ind⇩= f (refl a) ≡ f a"
text "Admissible inference rules for equality type formation:"
axiomatization where
Equal_wellform1: "a =⇩A b: U i ⟹ A: U i"
and
Equal_wellform2: "a =⇩A b: U i ⟹ a: A"
and
Equal_wellform3: "a =⇩A b: U i ⟹ b: A"
text "Rule attribute declarations:"
lemmas Equal_comp [comp]
lemmas Equal_wellform [wellform] = Equal_wellform1 Equal_wellform2 Equal_wellform3
lemmas Equal_routine [intro] = Equal_form Equal_intro Equal_elim
end
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