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(*
Title: Nat.thy
Author: Joshua Chen
Date: 2018
Natural numbers
*)
theory Nat
imports HoTT_Base
begin
axiomatization
Nat :: t ("ℕ") and
zero :: t ("0") and
succ :: "t ⇒ t" and
indNat :: "[[t, t] ⇒ t, t, t] ⇒ t" ("(1ind⇩ℕ)")
where
Nat_form: "ℕ: U O" and
Nat_intro_0: "0: ℕ" and
Nat_intro_succ: "n: ℕ ⟹ succ n: ℕ" and
Nat_elim: "⟦
a: C 0;
n: ℕ;
C: ℕ ⟶ U i;
⋀n c. ⟦n: ℕ; c: C n⟧ ⟹ f n c: C (succ n) ⟧ ⟹ ind⇩ℕ f a n: C n" and
Nat_comp_0: "⟦
a: C 0;
C: ℕ ⟶ U i;
⋀n c. ⟦n: ℕ; c: C(n)⟧ ⟹ f n c: C (succ n) ⟧ ⟹ ind⇩ℕ f a 0 ≡ a" and
Nat_comp_succ: "⟦
a: C 0;
n: ℕ;
C: ℕ ⟶ U i;
⋀n c. ⟦n: ℕ; c: C n⟧ ⟹ f n c: C (succ n) ⟧ ⟹ ind⇩ℕ f a (succ n) ≡ f n (ind⇩ℕ 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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