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(* 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
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