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(* Title: HoTT/Nat.thy
Author: Josh Chen
Date: Aug 2018
Natural numbers.
*)
theory Nat
imports HoTT_Base
begin
section \<open>Constants and type rules\<close>
axiomatization
Nat :: Term ("\<nat>") and
zero :: Term ("0") and
succ :: "Term \<Rightarrow> Term" and
indNat :: "[[Term, Term] \<Rightarrow> Term, Term, Term] \<Rightarrow> Term" ("(1ind\<^sub>\<nat>)")
where
Nat_form: "\<nat>: U(O)"
and
Nat_intro1: "0: \<nat>"
and
Nat_intro2: "\<And>n. n: \<nat> \<Longrightarrow> succ(n): \<nat>"
and
Nat_elim: "\<And>i C f a n. \<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_comp1: "\<And>i C f a. \<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_comp2: "\<And> i C f a n. \<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 declarations:"
lemmas Nat_intro = Nat_intro1 Nat_intro2
lemmas Nat_rules [intro] = Nat_form Nat_intro Nat_elim Nat_comp1 Nat_comp2
lemmas Nat_comps [comp] = Nat_comp1 Nat_comp2
end
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