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(* Title: HoTT/Sum.thy
Author: Josh Chen
Dependent sum type
*)
theory Sum
imports HoTT_Base
begin
section ‹Constants and syntax›
axiomatization
Sum :: "[Term, Typefam] ⇒ Term" and
pair :: "[Term, Term] ⇒ Term" ("(1<_,/ _>)") and
indSum :: "[[Term, Term] ⇒ Term, Term] ⇒ Term" ("(1ind⇩∑)")
syntax
"_SUM" :: "[idt, Term, Term] ⇒ Term" ("(3∑_:_./ _)" 20)
"_SUM_ASCII" :: "[idt, Term, Term] ⇒ Term" ("(3SUM _:_./ _)" 20)
translations
"∑x:A. B" ⇌ "CONST Sum A (λx. B)"
"SUM x:A. B" ⇀ "CONST Sum A (λx. B)"
text "Nondependent pair."
abbreviation Pair :: "[Term, Term] ⇒ Term" (infixr "×" 50)
where "A × B ≡ ∑_:A. B"
section ‹Type rules›
axiomatization where
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);
⋀x y. ⟦x: A; y: B(x)⟧ ⟹ f(x)(y): C(<x,y>);
C: ∑x:A. B(x) ⟶ U(i)
⟧ ⟹ ind⇩∑(f)(p): C(p)" (* What does writing λx y. f(x, y) change? *)
and
Sum_comp: "⟦
a: A;
b: B(a);
⋀x y. ⟦x: A; y: B(x)⟧ ⟹ f(x)(y): C(<x,y>);
B: A ⟶ U(i);
C: ∑x:A. B(x) ⟶ U(i)
⟧ ⟹ ind⇩∑(f)(<a,b>) ≡ f(a)(b)"
text "Admissible inference rules for sum formation:"
axiomatization where
Sum_wellform1: "(∑x:A. B(x): U(i)) ⟹ A: U(i)"
and
Sum_wellform2: "(∑x:A. B(x): U(i)) ⟹ B: A ⟶ 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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