24 lines
905 B
Plaintext
24 lines
905 B
Plaintext
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let sucf : ℕ → ℕ
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≔ λn. suc n
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let add : ℕ → ℕ → ℕ
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≔ ℕ-ind (λ_. ℕ → ℕ) (λn.n) (λn.λan.λm. suc (an m))
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let add_id_l : Π (n : ℕ) Id ℕ n (add 0 n)
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≔ λn. refl ℕ n
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let ap : Π (A : Type) Π (B : Type) Π (f : A → B)
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Π (x : A) Π (y : A) Id A x y → Id B (f x) (f y)
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≔ λA.λB.λf.λx.λy. J A x y (λa.λb.λ_. Id B (f a) (f b)) (refl B (f x))
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let add_id_r : Π (n : ℕ) Id ℕ n (add n 0)
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≔ ℕ-ind (λn. Id ℕ n (add n 0))
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(refl ℕ 0)
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(λn.λp. ap ℕ ℕ sucf n (add n 0) p)
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let add_assoc : Π (n : ℕ) Π (m : ℕ) Π (p : ℕ)
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Id ℕ (add (add n m) p) (add n (add m p))
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≔ ℕ-ind (λn. Π (m : ℕ) Π (p : ℕ) Id ℕ (add (add n m) p) (add n (add m p)))
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(λm.λp. refl ℕ (add m p))
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(λn.λpn.λm.λp. ap ℕ ℕ sucf (add (add n m) p) (add n (add m p)) (pn m p))
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