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(* Title: HoTT/Sum.thy
Author: Josh Chen
Date: Jun 2018
Dependent sum type.
*)
theory Sum
imports HoTT_Base
begin
section \<open>Constants and syntax\<close>
axiomatization
Sum :: "[Term, Typefam] \<Rightarrow> Term" and
pair :: "[Term, Term] \<Rightarrow> Term" ("(1'(_,/ _'))") and
indSum :: "[Term, Typefam, Typefam, [Term, Term] \<Rightarrow> Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>\<Sum>[_,/ _])")
syntax
"_SUM" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3\<Sum>_:_./ _)" 20)
"_SUM_ASCII" :: "[idt, Term, Term] \<Rightarrow> Term" ("(3SUM _:_./ _)" 20)
translations
"\<Sum>x:A. B" \<rightleftharpoons> "CONST Sum A (\<lambda>x. B)"
"SUM x:A. B" \<rightharpoonup> "CONST Sum A (\<lambda>x. B)"
text "Nondependent pair."
abbreviation Pair :: "[Term, Term] \<Rightarrow> Term" (infixr "\<times>" 50)
where "A \<times> B \<equiv> \<Sum>_:A. B"
section \<open>Type rules\<close>
axiomatization where
Sum_form: "\<And>i A B. \<lbrakk>A : U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Sum>x:A. B x : U(i)"
and
Sum_intro: "\<And>i A B a b. \<lbrakk>B: A \<longrightarrow> U(i); a : A; b : B a\<rbrakk> \<Longrightarrow> (a,b) : \<Sum>x:A. B x"
and
Sum_elim: "\<And>i A B C f p. \<lbrakk>
C: \<Sum>x:A. B x \<longrightarrow> U(i);
\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> f x y : C (x,y);
p : \<Sum>x:A. B x
\<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum>[A,B] C f p : C p"
and
Sum_comp: "\<And>i A B C f a b. \<lbrakk>
C: \<Sum>x:A. B x \<longrightarrow> U(i);
\<And>x y. \<lbrakk>x : A; y : B x\<rbrakk> \<Longrightarrow> f x y : C (x,y);
a : A;
b : B a
\<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum>[A,B] C f (a,b) \<equiv> f a b"
text "Admissible inference rules for sum formation:"
axiomatization where
Sum_form_cond1: "\<And>i A B. (\<Sum>x:A. B x : U(i)) \<Longrightarrow> A : U(i)"
and
Sum_form_cond2: "\<And>i A B. (\<Sum>x:A. B x : U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)"
lemmas Sum_rules [intro] = Sum_form Sum_intro Sum_elim Sum_comp
lemmas Sum_form_conds [intro (*elim, wellform*)] = Sum_form_cond1 Sum_form_cond2
lemmas Sum_comps [comp] = Sum_comp
section \<open>Null type\<close>
axiomatization
Null :: Term ("\<zero>") and
indNull :: "[Typefam, Term] \<Rightarrow> Term" ("(1ind\<^sub>\<zero>)")
where
Null_form: "\<zero> : U(O)"
and
Null_elim: "\<And>i C z. \<lbrakk>C: \<zero> \<longrightarrow> U(i); z : \<zero>\<rbrakk> \<Longrightarrow> ind\<^sub>\<zero> C z : C z"
lemmas Null_rules [intro] = Null_form Null_elim
end
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