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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] ⇒ t" and
pair :: "[t, t] ⇒ t" ("(1<_,/ _>)") and
indSum :: "[[t, t] ⇒ t, t] ⇒ t" ("(1ind⇩∑)")
syntax
"_sum" :: "[idt, t, t] ⇒ t" ("(3∑_:_./ _)" 20)
"_sum_ascii" :: "[idt, t, t] ⇒ t" ("(3SUM _:_./ _)" 20)
translations
"∑x:A. B" ⇌ "CONST Sum A (λx. B)"
"SUM x:A. B" ⇀ "CONST Sum A (λx. B)"
abbreviation Pair :: "[t, t] ⇒ t" (infixr "×" 50)
where "A × B ≡ ∑_:A. B"
axiomatization where
― ‹Type rules›
Sum_form: "⟦A: U i; B: A ⟶ U i⟧ ⟹ ∑x:A. B x: U i" and
Sum_intro: "⟦B: A ⟶ U i; a: A; b: B a⟧ ⟹ <a,b>: ∑x:A. B x" and
Sum_elim: "⟦
p: ∑x:A. B x;
C: ∑x:A. B x ⟶ U i;
⋀x y. ⟦x: A; y: B x⟧ ⟹ f x y: C <x,y> ⟧ ⟹ ind⇩∑ f p: C p" and
Sum_comp: "⟦
a: A;
b: B a;
B: A ⟶ U i;
C: ∑x:A. B x ⟶ U i;
⋀x y. ⟦x: A; y: B(x)⟧ ⟹ f x y: C <x,y> ⟧ ⟹ ind⇩∑ f <a,b> ≡ f a b" and
― ‹Congruence rules›
Sum_form_eq: "⟦A: U i; B: A ⟶ U i; C: A ⟶ U i; ⋀x. x: A ⟹ B x ≡ C x⟧ ⟹ ∑x:A. B x ≡ ∑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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