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(* Title: HoTT/Nat.thy
Author: Josh Chen
Natural numbers
*)
theory Nat
imports HoTT_Base
begin
section ‹Constants and type rules›
axiomatization
Nat :: Term ("ℕ") and
zero :: Term ("0") and
succ :: "Term ⇒ Term" and
indNat :: "[[Term, Term] ⇒ Term, Term, Term] ⇒ Term" ("(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: "⟦
C: ℕ ⟶ U i;
⋀n c. ⟦n: ℕ; c: C n⟧ ⟹ f n c: C (succ n);
a: C 0;
n: ℕ
⟧ ⟹ ind⇩ℕ f a n: C n"
and
Nat_comp_0: "⟦
C: ℕ ⟶ U i;
⋀n c. ⟦n: ℕ; c: C(n)⟧ ⟹ f n c: C (succ n);
a: C 0
⟧ ⟹ ind⇩ℕ f a 0 ≡ a"
and
Nat_comp_succ: "⟦
C: ℕ ⟶ U i;
⋀n c. ⟦n: ℕ; c: C n⟧ ⟹ f n c: C (succ n);
a: C 0;
n: ℕ
⟧ ⟹ ind⇩ℕ f a (succ n) ≡ f n (ind⇩ℕ 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_elim
end
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