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(* Title: HoTT/Coprod.thy
Author: Josh Chen
Date: Aug 2018
Coproduct type.
*)
theory Coprod
imports HoTT_Base
begin
section ‹Constants and type rules›
axiomatization
Coprod :: "[Term, Term] ⇒ Term" (infixr "+" 50) and
inl :: "Term ⇒ Term" and
inr :: "Term ⇒ Term" and
indCoprod :: "[Term ⇒ Term, Term ⇒ Term, Term] ⇒ Term" ("(1ind⇩+)")
where
Coprod_form: "⋀i A B. ⟦A : U(i); B : U(i)⟧ ⟹ A + B: U(i)"
and
Coprod_intro1: "⋀A B a b. ⟦a : A; b : B⟧ ⟹ inl(a): A + B"
and
Coprod_intro2: "⋀A B a b. ⟦a : A; b : B⟧ ⟹ inr(b): A + B"
and
Coprod_elim: "⋀i A B C c d e. ⟦
C: A + B ⟶ U(i);
⋀x. x: A ⟹ c(x): C(inl(x));
⋀y. y: B ⟹ d(y): C(inr(y));
e: A + B
⟧ ⟹ ind⇩+(c)(d)(e) : C(e)"
and
Coprod_comp1: "⋀i A B C c d a. ⟦
C: A + B ⟶ U(i);
⋀x. x: A ⟹ c(x): C(inl(x));
⋀y. y: B ⟹ d(y): C(inr(y));
a: A
⟧ ⟹ ind⇩+(c)(d)(inl(a)) ≡ c(a)"
and
Coprod_comp2: "⋀i A B C c d b. ⟦
C: A + B ⟶ U(i);
⋀x. x: A ⟹ c(x): C(inl(x));
⋀y. y: B ⟹ d(y): C(inr(y));
b: B
⟧ ⟹ ind⇩+(c)(d)(inr(b)) ≡ d(b)"
text "Admissible formation inference rules:"
axiomatization where
Coprod_form_cond1: "⋀i A B. A + B: U(i) ⟹ A: U(i)"
and
Coprod_form_cond2: "⋀i A B. A + B: U(i) ⟹ B: U(i)"
text "Rule declarations:"
lemmas Coprod_intro = Coprod_intro1 Coprod_intro2
lemmas Coprod_rules [intro] = Coprod_form Coprod_intro Coprod_elim Coprod_comp1 Coprod_comp2
lemmas Coprod_form_conds [wellform] = Coprod_form_cond1 Coprod_form_cond2
lemmas Coprod_comps [comp] = Coprod_comp1 Coprod_comp2
end
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