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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
end
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