(*  Title:  HoTT/Nat.thy
    Author: Josh Chen

Natural numbers
*)

theory Nat
  imports HoTT_Base
begin


section \<open>Constants and type rules\<close>

axiomatization
  Nat :: t   ("\<nat>") and
  zero :: t  ("0") and
  succ :: "t \<Rightarrow> t" and
  indNat :: "[[t, t] \<Rightarrow> t, t, t] \<Rightarrow> t"  ("(1ind\<^sub>\<nat>)")
where
  Nat_form: "\<nat>: U O"
and
  Nat_intro_0: "0: \<nat>"
and
  Nat_intro_succ: "n: \<nat> \<Longrightarrow> succ n: \<nat>"
and
  Nat_elim: "\<lbrakk>
    C: \<nat> \<longrightarrow> U i;
    \<And>n c. \<lbrakk>n: \<nat>; c: C n\<rbrakk> \<Longrightarrow> f n c: C (succ n);
    a: C 0;
    n: \<nat>
    \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat> f a n: C n"
and
  Nat_comp_0: "\<lbrakk>
    C: \<nat> \<longrightarrow> U i;
    \<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f n c: C (succ n);
    a: C 0
    \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat> f a 0 \<equiv> a"
and
  Nat_comp_succ: "\<lbrakk>
    C: \<nat> \<longrightarrow> U i;
    \<And>n c. \<lbrakk>n: \<nat>; c: C n\<rbrakk> \<Longrightarrow> f n c: C (succ n);
    a: C 0;
    n: \<nat>
    \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat> f a (succ n) \<equiv> f n (ind\<^sub>\<nat> f a n)"


text "Rule attribute declarations:"

lemmas Nat_intro = Nat_intro_0 Nat_intro_succ
lemmas Nat_comp [comp] = Nat_comp_0 Nat_comp_succ
lemmas Nat_routine [intro] = Nat_form Nat_intro Nat_elim


end