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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.
*)
theory HoTT_Base
imports Pure
begin
section \<open>Foundational definitions\<close>
text "A single meta-type \<open>Term\<close> suffices to implement the object-logic types and terms."
typedecl Term
section \<open>Named theorems\<close>
text "Named theorems to be used by proof methods later (see HoTT_Methods.thy).
\<open>wellform\<close> declares necessary wellformedness conditions for type and inhabitation judgments, while
\<open>comp\<close> declares computation rules, which are used by the simplification method as equational rewrite rules."
named_theorems wellform
named_theorems comp
section \<open>Judgments\<close>
text "Formalize the context and typing judgments \<open>a : A\<close>.
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_of_type :: "[Term, Term] \<Rightarrow> prop" ("(1_ : _)" [0, 0] 1000)
section \<open>Universes\<close>
text "Strictly-ordered meta-level natural numbers to index the universes."
typedecl Numeral
axiomatization
O :: Numeral ("0") and
S :: "Numeral \<Rightarrow> Numeral" ("S_") and
lt :: "[Numeral, Numeral] \<Rightarrow> prop" (infix "<-" 999)
where
Numeral_lt_min: "\<And>n. 0 <- S n"
and
Numeral_lt_trans: "\<And>m n. m <- n \<Longrightarrow> S m <- S n"
lemmas Numeral_rules [intro] = Numeral_lt_min Numeral_lt_trans
\<comment> \<open>Enables \<open>standard\<close> to automatically solve inequalities.\<close>
text "Universe types:"
axiomatization
U :: "Numeral \<Rightarrow> Term" ("U _")
where
Universe_hierarchy: "\<And>i j. i <- j \<Longrightarrow> U(i) : U(j)"
and
Universe_cumulative: "\<And>A i j. \<lbrakk>A : U(i); i <- j\<rbrakk> \<Longrightarrow> A : U(j)"
\<comment> \<open>WARNING: \<open>rule Universe_cumulative\<close> can result in an infinite rewrite loop!\<close>
section \<open>Type families\<close>
text "We define the following abbreviation constraining the output type of a meta lambda expression when given input of certain type."
abbreviation (input) constrained :: "[Term \<Rightarrow> Term, Term, Term] \<Rightarrow> prop" ("_: _ \<longrightarrow> _")
where "f: A \<longrightarrow> B \<equiv> (\<And>x. x : A \<Longrightarrow> f x : B)"
text "The above is used to define type families, which are just constrained meta-lambdas \<open>P: A \<longrightarrow> B\<close> where \<open>A\<close> and \<open>B\<close> are elements of some universe type."
type_synonym Typefam = "Term \<Rightarrow> Term"
end
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