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(* Title: HoTT/Nat.thy
Author: Josh Chen
Date: Aug 2018
Natural numbers.
*)
theory Nat
imports HoTT_Base
begin
axiomatization
Nat :: Term ("ℕ") and
zero :: Term ("0") and
succ :: "Term ⇒ Term" and
indNat :: "[Typefam, [Term, Term] ⇒ Term, Term, Term] ⇒ Term" ("(1ind⇩ℕ)")
where
Nat_form: "ℕ : U(O)"
and
Nat_intro1: "0 : ℕ"
and
Nat_intro2: "⋀n. n : ℕ ⟹ succ n : ℕ"
and
Nat_elim: "⋀i C f a n. ⟦
C: ℕ ⟶ U(i);
⋀n c. ⟦n : ℕ; c : C n⟧ ⟹ f n c : C (succ n);
a : C 0;
n : ℕ
⟧ ⟹ ind⇩ℕ C f a n : C n"
and
Nat_comp1: "⋀i C f a. ⟦
C: ℕ ⟶ U(i);
⋀n c. ⟦n : ℕ; c : C n⟧ ⟹ f n c : C (succ n);
a : C 0
⟧ ⟹ ind⇩ℕ C f a 0 ≡ a"
and
Nat_comp2: "⋀ i C f a n. ⟦
C: ℕ ⟶ U(i);
⋀n c. ⟦n : ℕ; c : C n⟧ ⟹ f n c : C (succ n);
a : C 0;
n : ℕ
⟧ ⟹ ind⇩ℕ C f a (succ n) ≡ f n (ind⇩ℕ C f a n)"
lemmas Nat_rules [intro] = Nat_form Nat_intro1 Nat_intro2 Nat_elim Nat_comp1 Nat_comp2
lemmas Nat_comps [comp] = Nat_comp1 Nat_comp2
end
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