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(********
Isabelle/HoTT: Natural numbers
Feb 2019
********)
theory Nat
imports HoTT_Base
begin
axiomatization
Nat :: t and
zero :: t ("0") and
succ :: "t \<Rightarrow> t" and
indNat :: "[t \<Rightarrow> t, [t, t] \<Rightarrow> t, t, t] \<Rightarrow> t"
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 \<leadsto> U i;
\<And>n c. \<lbrakk>n: Nat; c: C n\<rbrakk> \<Longrightarrow> f n c: C (succ n) \<rbrakk> \<Longrightarrow> indNat C f a n: C n" and
Nat_cmp_0: "\<lbrakk>
a: C 0;
C: Nat \<leadsto> U i;
\<And>n c. \<lbrakk>n: Nat; c: C n\<rbrakk> \<Longrightarrow> f n c: C (succ n) \<rbrakk> \<Longrightarrow> indNat C f a 0 \<equiv> a" and
Nat_cmp_succ: "\<lbrakk>
a: C 0;
n: Nat;
C: Nat \<leadsto> U i;
\<And>n c. \<lbrakk>n: Nat; c: C n\<rbrakk> \<Longrightarrow> f n c: C (succ n)
\<rbrakk> \<Longrightarrow> indNat C f a (succ n) \<equiv> f n (indNat C f a n)"
lemmas Nat_form [form]
lemmas Nat_routine [intro] = Nat_form Nat_intro_0 Nat_intro_succ Nat_elim
lemmas Nat_comp [comp] = Nat_cmp_0 Nat_cmp_succ
end
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