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add new proof

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rnhmjoj 2016-11-27 23:24:10 +01:00
parent 9d3ec1d2ad
commit 3714f048cf
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1 changed files with 6 additions and 1 deletions
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src/Data/Nat.hs
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@ -170,10 +170,15 @@ plus_assoc (SS a) b c = cong SS (plus_assoc a b c)
-- | Commutative property of addition -- | Commutative property of addition
plus_commut ∷ Sℕ n → Sℕ m → (n :+ m) :≡ (m :+ n) plus_commut ∷ Sℕ n → Sℕ m → (n :+ m) :≡ (m :+ n)
plus_commut n SZ = plus_right_id n plus_commut n SZ = plus_right_id n
plus_commut n (SS m) = plus_commut n (SS m) =
sym (succ_right n m) ∾ cong SS (plus_commut n m) ∾ succ_left m n sym (succ_right n m) ∾ cong SS (plus_commut n m) ∾ succ_left m n
-- | Left identity of multiplication
cross_left_id ∷ Sℕ n → (S Z) :× n :≡ n
cross_left_id SZ = Refl
cross_left_id (SS n) = cong SS (cross_left_id n)
-- | Minimum of itself -- | Minimum of itself
min_self ∷ Sℕ n → Min n n :≡ n min_self ∷ Sℕ n → Min n n :≡ n
min_self SZ = Refl min_self SZ = Refl