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(*
Title: Nat.thy
Author: Joshua Chen
Date: 2018
Natural numbers
*)
theory Nat
imports HoTT_Base
begin
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>
a: C 0;
n: \<nat>;
C: \<nat> \<longrightarrow> U i;
\<And>n c. \<lbrakk>n: \<nat>; c: C n\<rbrakk> \<Longrightarrow> f n c: C (succ n) \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat> f a n: C n" and
Nat_comp_0: "\<lbrakk>
a: C 0;
C: \<nat> \<longrightarrow> U i;
\<And>n c. \<lbrakk>n: \<nat>; c: C(n)\<rbrakk> \<Longrightarrow> f n c: C (succ n) \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat> f a 0 \<equiv> a" and
Nat_comp_succ: "\<lbrakk>
a: C 0;
n: \<nat>;
C: \<nat> \<longrightarrow> U i;
\<And>n c. \<lbrakk>n: \<nat>; c: C n\<rbrakk> \<Longrightarrow> f n c: C (succ n) \<rbrakk> \<Longrightarrow> ind\<^sub>\<nat> f a (succ n) \<equiv> f n (ind\<^sub>\<nat> f a n)"
lemmas Nat_form [form]
lemmas Nat_routine [intro] = Nat_form Nat_intro_0 Nat_intro_succ Nat_elim
lemmas Nat_comps [comp] = Nat_comp_0 Nat_comp_succ
end
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