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(*
Title: Coprod.thy
Author: Joshua Chen
Date: 2018
Coproduct type
*)
theory Coprod
imports HoTT_Base
begin
axiomatization
Coprod :: "[t, t] \<Rightarrow> t" (infixr "+" 50) and
inl :: "t \<Rightarrow> t" and
inr :: "t \<Rightarrow> t" and
indCoprod :: "[t \<Rightarrow> t, t \<Rightarrow> t, t] \<Rightarrow> t" ("(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>
u: A + B;
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) \<rbrakk> \<Longrightarrow> ind\<^sub>+ (\<lambda> x. c x) (\<lambda>y. d y) u: C u" and
Coprod_comp_inl: "\<lbrakk>
a: A;
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) \<rbrakk> \<Longrightarrow> ind\<^sub>+ (\<lambda>x. c x) (\<lambda>y. d y) (inl a) \<equiv> c a" and
Coprod_comp_inr: "\<lbrakk>
b: B;
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) \<rbrakk> \<Longrightarrow> ind\<^sub>+ (\<lambda>x. c x) (\<lambda>y. d y) (inr b) \<equiv> d b"
lemmas Coprod_form [form]
lemmas Coprod_routine [intro] = Coprod_form Coprod_intro_inl Coprod_intro_inr Coprod_elim
lemmas Coprod_comp [comp] = Coprod_comp_inl Coprod_comp_inr
end
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