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(* Title: HoTT/Prod.thy
Author: Josh Chen
Date: Aug 2018
Dependent product (function) type.
*)
theory Prod
imports HoTT_Base
begin
section ‹Constants and syntax›
axiomatization
Prod :: "[Term, Typefam] ⇒ Term" and
lambda :: "[Term, Term ⇒ Term] ⇒ Term" and
appl :: "[Term, Term] ⇒ Term" ("(1_`_)" [61, 60] 60)
― ‹Application binds tighter than abstraction.›
syntax
"_PROD" :: "[idt, Term, Term] ⇒ Term" ("(3∏_:_./ _)" 30)
"_LAMBDA" :: "[idt, Term, Term] ⇒ Term" ("(1❙λ_:_./ _)" 30)
"_PROD_ASCII" :: "[idt, Term, Term] ⇒ Term" ("(3PROD _:_./ _)" 30)
"_LAMBDA_ASCII" :: "[idt, Term, Term] ⇒ Term" ("(3%%_:_./ _)" 30)
text "The translations below bind the variable ‹x› in the expressions ‹B› and ‹b›."
translations
"∏x:A. B" ⇌ "CONST Prod(A, λx. B)"
"❙λx:A. b" ⇌ "CONST lambda(A, λx. b)"
"PROD x:A. B" ⇀ "CONST Prod(A, λx. B)"
"%%x:A. b" ⇀ "CONST lambda(A, λx. b)"
text "Nondependent functions are a special case."
abbreviation Function :: "[Term, Term] ⇒ Term" (infixr "→" 40)
where "A → B ≡ ∏_: A. B"
section ‹Type rules›
axiomatization where
Prod_form: "⋀i A B. ⟦A: U(i); B: A ⟶ U(i)⟧ ⟹ ∏x:A. B(x): U(i)"
and
Prod_intro: "⋀i A B b. ⟦A: U(i); B: A ⟶ U(i); ⋀x. x: A ⟹ b(x): B(x)⟧ ⟹ ❙λx:A. b(x): ∏x:A. B(x)"
and
Prod_elim: "⋀A B f a. ⟦f: ∏x:A. B(x); a: A⟧ ⟹ f`a: B(a)"
and
Prod_comp: "⋀i A B b a. ⟦A: U(i); B: A ⟶ U(i); ⋀x. x: A ⟹ b(x): B(x); a: A⟧ ⟹ (❙λx:A. b(x))`a ≡ b(a)"
and
Prod_uniq: "⋀A B f. f : ∏x:A. B(x) ⟹ ❙λx:A. (f`x) ≡ f"
text "
Note that the syntax ‹❙λ› (bold lambda) used for dependent functions clashes with the proof term syntax (cf. §2.5.2 of the Isabelle/Isar Implementation).
"
text "
In addition to the usual type rules, it is a meta-theorem (*PROVE THIS!*) that whenever ‹∏x:A. B x: U(i)› is derivable from some set of premises Γ, then so are ‹A: U(i)› and ‹B: A ⟶ U(i)›.
That is to say, the following inference rules are admissible, and it simplifies proofs greatly to axiomatize them directly.
"
axiomatization where
Prod_form_cond1: "⋀i A B. (∏x:A. B(x): U(i)) ⟹ A: U(i)"
and
Prod_form_cond2: "⋀i A B. (∏x:A. B(x): U(i)) ⟹ B: A ⟶ U(i)"
text "Set up the standard reasoner to use the type rules:"
lemmas Prod_rules = Prod_form Prod_intro Prod_elim Prod_comp Prod_uniq
lemmas Prod_form_conds [intro (*elim, wellform*)] = Prod_form_cond1 Prod_form_cond2
lemmas Prod_comps [comp] = Prod_comp Prod_uniq
section ‹Unit type›
axiomatization
Unit :: Term ("𝟭") and
pt :: Term ("⋆") and
indUnit :: "[Typefam, Term, Term] ⇒ Term" ("(1ind⇩𝟭)")
where
Unit_form: "𝟭 : U(O)"
and
Unit_intro: "⋆ : 𝟭"
and
Unit_elim: "⋀i C c a. ⟦C: 𝟭 ⟶ U(i); c : C(⋆); a : 𝟭⟧ ⟹ ind⇩𝟭(C, c, a) : C(a)"
and
Unit_comp: "⋀i C c. ⟦C: 𝟭 ⟶ U(i); c : C(⋆)⟧ ⟹ ind⇩𝟭(C, c, ⋆) ≡ c"
lemmas Unit_rules [intro] = Unit_form Unit_intro Unit_elim Unit_comp
lemmas Unit_comps [comp] = Unit_comp
end
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