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(* Title: HoTT/Coprod.thy
Author: Josh Chen
Date: Aug 2018
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" ("(1inl'(_'))") and
inr :: "Term \<Rightarrow> Term" ("(1inr'(_'))") and
indCoprod :: "[Term, Term, Typefam, Term \<Rightarrow> Term, Term \<Rightarrow> Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>+[_,/ _])")
where
Coprod_form: "\<And>i A B. \<lbrakk>A : U(i); B : U(i)\<rbrakk> \<Longrightarrow> A + B : U(i)"
and
Coprod_intro1: "\<And>A B a b. \<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> inl(a) : A + B"
and
Coprod_intro2: "\<And>A B a b. \<lbrakk>a : A; b : B\<rbrakk> \<Longrightarrow> inr(b) : A + B"
and
Coprod_elim: "\<And>i A B C c d e. \<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);
e : A + B
\<rbrakk> \<Longrightarrow> ind\<^sub>+[A,B] C c d e : C e"
and
Coprod_comp1: "\<And>i A B C c d a. \<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>+[A,B] C c d inl(a) \<equiv> c a"
and
Coprod_comp2: "\<And>i A B C c d b. \<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>+[A,B] C c d inr(b) \<equiv> d b"
text "Admissible formation inference rules:"
axiomatization where
Coprod_form_cond1: "\<And>i A B. A + B : U(i) \<Longrightarrow> A : U(i)"
and
Coprod_form_cond2: "\<And>i A B. A + B : U(i) \<Longrightarrow> B : U(i)"
lemmas Coprod_rules [intro] = Coprod_form Coprod_intro1 Coprod_intro2
Coprod_elim Coprod_comp1 Coprod_comp2
lemmas Coprod_form_conds [intro] = Coprod_form_cond1 Coprod_form_cond2
lemmas Coprod_comps [comp] = Coprod_comp1 Coprod_comp2
end
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