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(********
Isabelle/HoTT: Basic logical definitions and notation
Feb 2019
This file completes the basic logical and functional setup of the HoTT logic.
Among other things, it:
* Defines the universe hierarchy and its governing rules.
* Defines named theorems for later use by proof methods.
********)
theory HoTT_Base
imports HoTT_Typing
begin
section \<open>Universes\<close>
typedecl ord \<comment> \<open>Type of meta-numerals\<close>
axiomatization
O :: ord and
Suc :: "ord \<Rightarrow> ord" and
lt :: "[ord, ord] \<Rightarrow> prop" (infix "<" 999) and
leq :: "[ord, ord] \<Rightarrow> prop" (infix "\<le>" 999)
where
lt_Suc: "n < (Suc n)" and
lt_trans: "\<lbrakk>m1 < m2; m2 < m3\<rbrakk> \<Longrightarrow> m1 < m3" and
leq_min: "O \<le> n"
declare
lt_Suc [intro!]
leq_min [intro!]
lt_trans [intro]
axiomatization
U :: "ord \<Rightarrow> t"
where
U_hierarchy: "i < j \<Longrightarrow> U i: U j" and
U_cumulative: "\<lbrakk>A: U i; i < j\<rbrakk> \<Longrightarrow> A: U j" and
U_cumulative': "\<lbrakk>A: U i; i \<le> j\<rbrakk> \<Longrightarrow> A: U j"
text \<open>
Using method @{method rule} with @{thm U_cumulative} and @{thm U_cumulative'} is unsafe: if applied blindly it will very easily lead to non-termination.
Instead use @{method elim}, or instantiate the rules suitably.
@{thm U_cumulative'} is an alternative rule used by the method @{theory_text cumulativity} in @{file HoTT_Methods.thy}.
\<close>
section \<open>Type families\<close>
abbreviation (input) constraint :: "[t \<Rightarrow> t, t, t] \<Rightarrow> prop" ("(1_:/ _ \<leadsto> _)")
where "f: A \<leadsto> B \<equiv> (\<And>x. x: A \<Longrightarrow> f x: B)"
text \<open>We use the notation @{prop "B: A \<leadsto> U i"} to abbreviate type families.\<close>
section \<open>Named theorems\<close>
named_theorems form
named_theorems comp
named_theorems cong
text \<open>
The named theorems above will be used by proof methods defined in @{file HoTT_Methods.thy}.
@{attribute form} declares type formation rules.
These are mainly used by the \<open>cumulativity\<close> method, which lifts types into higher universes.
@{attribute comp} declares computation rules, which are used by the \<open>compute\<close> method, and may also be passed to invocations of the method \<open>subst\<close> to perform equational rewriting.
@{attribute cong} declares congruence rules, the definitional equality rules of the type theory.
\<close>
(* Todo: Set up the Simplifier! *)
end
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