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(*
Title: Sum.thy
Author: Joshua Chen
Date: 2018
Dependent sum type
*)
theory Sum
imports HoTT_Base
begin
axiomatization
Sum :: "[t, tf] \<Rightarrow> t" and
pair :: "[t, t] \<Rightarrow> t" ("(1<_,/ _>)") and
indSum :: "[[t, t] \<Rightarrow> t, t] \<Rightarrow> t" ("(1ind\<^sub>\<Sum>)")
syntax
"_sum" :: "[idt, t, t] \<Rightarrow> t" ("(3\<Sum>_:_./ _)" 20)
"_sum_ascii" :: "[idt, t, t] \<Rightarrow> t" ("(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)"
abbreviation Pair :: "[t, t] \<Rightarrow> t" (infixr "\<times>" 50)
where "A \<times> B \<equiv> \<Sum>_:A. B"
axiomatization where
\<comment> \<open>Type rules\<close>
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;
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> \<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum> (\<lambda>x y. f x y) p: C p" and
Sum_comp: "\<lbrakk>
a: A;
b: B a;
B: A \<longrightarrow> U i;
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> \<rbrakk> \<Longrightarrow> ind\<^sub>\<Sum> (\<lambda>x y. f x y) <a,b> \<equiv> f a b" and
\<comment> \<open>Congruence rules\<close>
Sum_form_eq: "\<lbrakk>A: U i; B: A \<longrightarrow> U i; C: A \<longrightarrow> U i; \<And>x. x: A \<Longrightarrow> B x \<equiv> C x\<rbrakk> \<Longrightarrow> \<Sum>x:A. B x \<equiv> \<Sum>x:A. C x"
lemmas Sum_form [form]
lemmas Sum_routine [intro] = Sum_form Sum_intro Sum_elim
lemmas Sum_comp [comp]
end
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