Mercurial > hg > Papers > 2020 > soto-midterm

annotate src/agda-term3.agda @ 1:73127e0ab57c

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author soto@cr.ie.u-ryukyu.ac.jp
date Tue, 08 Sep 2020 18:38:08 +0900
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soto@cr.ie.u-ryukyu.ac.jp
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1 +-comm : (x y : ℕ) → x + y ≡ y + x
soto@cr.ie.u-ryukyu.ac.jp
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2 +-comm zero y rewrite (+zero {y}) = refl
soto@cr.ie.u-ryukyu.ac.jp
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3 +-comm (suc x) y = let open ≡-Reasoning in
soto@cr.ie.u-ryukyu.ac.jp
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4 begin
soto@cr.ie.u-ryukyu.ac.jp
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5 suc (x + y) ≡⟨⟩
soto@cr.ie.u-ryukyu.ac.jp
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6 suc (x + y) ≡⟨ cong suc (+-comm x y) ⟩
soto@cr.ie.u-ryukyu.ac.jp
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7 suc (y + x) ≡⟨ sym (+-suc {y} {x}) ⟩
soto@cr.ie.u-ryukyu.ac.jp
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8 y + suc x ∎
soto@cr.ie.u-ryukyu.ac.jp
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9
soto@cr.ie.u-ryukyu.ac.jp
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10 -- +-suc : {x y : ℕ} → x + suc y ≡ suc (x + y)
soto@cr.ie.u-ryukyu.ac.jp
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11 -- +-suc {zero} {y} = refl
soto@cr.ie.u-ryukyu.ac.jp
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12 -- +-suc {suc x} {y} = cong suc (+-suc {x} {y})