(*  Title:  HoTT/Coprod.thy
    Author: Josh Chen

Coproduct type
*)

theory Coprod
  imports HoTT_Base
begin


section \<open>Constants and type rules\<close>

axiomatization
  Coprod :: "[Term, Term] \<Rightarrow> Term"  (infixr "+" 50) and
  inl :: "Term \<Rightarrow> Term" and
  inr :: "Term \<Rightarrow> Term" and
  indCoprod :: "[Term \<Rightarrow> Term, Term \<Rightarrow> Term, Term] \<Rightarrow> Term"  ("(1ind\<^sub>+)")
where
  Coprod_form: "\<lbrakk>A: U(i); B: U(i)\<rbrakk> \<Longrightarrow> A + B: U(i)"
and
  Coprod_intro_inl: "\<lbrakk>a: A; B: U(i)\<rbrakk> \<Longrightarrow> inl(a): A + B"
and
  Coprod_intro_inr: "\<lbrakk>b: B; A: U(i)\<rbrakk> \<Longrightarrow> inr(b): A + B"
and
  Coprod_elim: "\<lbrakk>
    C: A + B \<longrightarrow> U(i);
    \<And>x. x: A \<Longrightarrow> c(x): C(inl(x));
    \<And>y. y: B \<Longrightarrow> d(y): C(inr(y));
    u: A + B
    \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(u) : C(u)"
and
  Coprod_comp_inl: "\<lbrakk>
    C: A + B \<longrightarrow> U(i);
    \<And>x. x: A \<Longrightarrow> c(x): C(inl(x));
    \<And>y. y: B \<Longrightarrow> d(y): C(inr(y));
    a: A
    \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(inl(a)) \<equiv> c(a)"
and
  Coprod_comp_inr: "\<lbrakk>
    C: A + B \<longrightarrow> U(i);
    \<And>x. x: A \<Longrightarrow> c(x): C(inl(x));
    \<And>y. y: B \<Longrightarrow> d(y): C(inr(y));
    b: B
    \<rbrakk> \<Longrightarrow> ind\<^sub>+(c)(d)(inr(b)) \<equiv> d(b)"


text "Admissible formation inference rules:"

axiomatization where
  Coprod_wellform1: "A + B: U(i) \<Longrightarrow> A: U(i)"
and
  Coprod_wellform2: "A + B: U(i) \<Longrightarrow> B: U(i)"


text "Rule attribute declarations:"

lemmas Coprod_intro = Coprod_intro_inl Coprod_intro_inr

lemmas Coprod_comp [comp] = Coprod_comp_inl Coprod_comp_inr
lemmas Coprod_wellform [wellform] = Coprod_wellform1 Coprod_wellform2
lemmas Coprod_routine [intro] = Coprod_form Coprod_intro Coprod_elim


end