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(* Title: HoTT/Nat.thy
Author: Josh Chen
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_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_comp Nat_elim
end
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