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(* Title: HoTT/HoTT_Base.thy
Author: Josh Chen
Date: Jun 2018
Basic setup and definitions of a homotopy type theory object logic without universes.
*)
theory HoTT_Base
imports Pure
begin
section \<open>Setup\<close>
text "Set up type checking routines, proof methods etc."
section \<open>Metalogical definitions\<close>
text "A single meta-type \<open>Term\<close> suffices to implement the object-logic types and terms.
We do not implement universes, and simply use \<open>a : U\<close> as a convenient shorthand to mean ''\<open>a\<close> is a type''."
typedecl Term
section \<open>Judgments\<close>
text "We formalize the judgments \<open>a : A\<close> and \<open>A : U\<close> separately, in contrast to the HoTT book where the latter is considered an instance of the former.
For judgmental equality we use the existing Pure equality \<open>\<equiv>\<close> and hence do not need to define a separate judgment for it."
consts
is_a_type :: "Term \<Rightarrow> prop" ("(1_ :/ U)" [0] 1000)
is_of_type :: "[Term, Term] \<Rightarrow> prop" ("(1_ :/ _)" [0, 0] 1000)
text "The following fact is used to make explicit the assumption of well-formed contexts."
axiomatization where
inhabited_implies_type [intro, elim]: "\<And>a A. a : A \<Longrightarrow> A : U"
section \<open>Type families\<close>
text "A (one-variable) type family is a meta lambda term \<open>P :: Term \<Rightarrow> Term\<close> that further satisfies the following property."
type_synonym Typefam = "Term \<Rightarrow> Term"
abbreviation (input) is_type_family :: "[Typefam, Term] \<Rightarrow> prop" ("(3_:/ _ \<rightarrow> U)")
where "P: A \<rightarrow> U \<equiv> (\<And>x. x : A \<Longrightarrow> P x : U)"
text "There is an obvious generalization to multivariate type families, but implementing such an abbreviation involves writing ML code, and is for the moment not really crucial."
end
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