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(* Title: HoTT/Sum.thy
Author: Josh Chen
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, 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: "\<lbrakk>A: U(i); B: A \<longrightarrow> U(i)\<rbrakk> \<Longrightarrow> \<Sum>x:A. B(x): U(i)"
and
Sum_intro: "\<lbrakk>B: A \<longrightarrow> U(i); a: A; b: B(a)\<rbrakk> \<Longrightarrow> <a,b>: \<Sum>x:A. B(x)"
and
Sum_elim: "\<lbrakk>
p: \<Sum>x:A. B(x);
\<And>x y. \<lbrakk>x: A; y: B(x)\<rbrakk> \<Longrightarrow> f(x)(y): C(<x,y>);
C: \<Sum>x:A. B(x) \<longrightarrow> U(i)
\<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum>(f)(p): C(p)" (* What does writing \<lambda>x y. f(x, y) change? *)
and
Sum_comp: "\<lbrakk>
a: A;
b: B(a);
\<And>x y. \<lbrakk>x: A; y: B(x)\<rbrakk> \<Longrightarrow> f(x)(y): C(<x,y>);
B: A \<longrightarrow> U(i);
C: \<Sum>x:A. B(x) \<longrightarrow> U(i)
\<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum>(f)(<a,b>) \<equiv> f(a)(b)"
text "Admissible inference rules for sum formation:"
axiomatization where
Sum_wellform1: "(\<Sum>x:A. B(x): U(i)) \<Longrightarrow> A: U(i)"
and
Sum_wellform2: "(\<Sum>x:A. B(x): U(i)) \<Longrightarrow> B: A \<longrightarrow> U(i)"
text "Rule attribute declarations:"
lemmas Sum_comp [comp]
lemmas Sum_wellform [wellform] = Sum_wellform1 Sum_wellform2
lemmas Sum_routine [intro] = Sum_form Sum_intro Sum_elim
section \<open>Empty type\<close>
axiomatization
Empty :: Term ("\<zero>") and
indEmpty :: "Term \<Rightarrow> Term" ("(1ind\<^sub>\<zero>)")
where
Empty_form: "\<zero> : U(O)"
and
Empty_elim: "\<lbrakk>C: \<zero> \<longrightarrow> U(i); z: \<zero>\<rbrakk> \<Longrightarrow> ind\<^sub>\<zero>(z): C(z)"
lemmas Empty_routine [intro] = Empty_form Empty_elim
end
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