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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>+ c d 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>+ c d (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>+ c d (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