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(* Title: HoTT/HoTT_Base.thy
Author: Josh Chen
Date: Aug 2018
Basic setup and definitions of a homotopy type theory object logic with a cumulative universe hierarchy à la Russell.
*)
theory HoTT_Base
imports Pure
begin
section \<open>Foundational definitions\<close>
text "Meta syntactic type for object-logic types and terms."
typedecl Term
section \<open>Judgments\<close>
text "
Formalize the typing judgment \<open>a: A\<close>.
For judgmental/definitional equality we use the existing Pure equality \<open>\<equiv>\<close> and hence do not need to define a separate judgment for it.
"
consts
has_type :: "[Term, Term] \<Rightarrow> prop" ("(1_: _)" [0, 0] 1000)
section \<open>Universe hierarchy\<close>
text "Finite meta-ordinals to index the universes."
typedecl Ord
axiomatization
O :: Ord and
S :: "Ord \<Rightarrow> Ord" ("S_" [0] 1000) and
lt :: "[Ord, Ord] \<Rightarrow> prop" (infix "<-" 999)
where
Ord_min: "\<And>n. O <- S(n)"
and
Ord_monotone: "\<And>m n. m <- n \<Longrightarrow> S(m) <- S(n)"
lemmas Ord_rules [intro] = Ord_min Ord_monotone
\<comment> \<open>Enables \<open>standard\<close> to automatically solve inequalities.\<close>
text "Define the universe types."
axiomatization
U :: "Ord \<Rightarrow> Term"
where
U_hierarchy: "\<And>i j. i <- j \<Longrightarrow> U(i): U(j)"
and
U_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 "
The following abbreviation constrains the output type of a meta lambda expression when given input of certain type.
"
abbreviation (input) constrained :: "[Term \<Rightarrow> Term, Term, Term] \<Rightarrow> prop" ("(1_: _ \<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 constrained meta-lambdas \<open>P: A \<longrightarrow> B\<close> where \<open>A\<close> and \<open>B\<close> are small types.
"
type_synonym Typefam = "Term \<Rightarrow> 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 usually passed to invocations of the method \<open>subst\<close> to perform equational rewriting.
"
named_theorems wellform
named_theorems comp
end
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