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diff --git a/Coprod.thy b/Coprod.thy new file mode 100644 index 0000000..75e621a --- /dev/null +++ b/Coprod.thy @@ -0,0 +1,61 @@ +(* 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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