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1 module nat where
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2
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3 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
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4 open import Data.Empty
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5 open import Relation.Nullary
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6 open import Relation.Binary.PropositionalEquality
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7 open import logic
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8
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9
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10 nat-<> : { x y : Nat } → x < y → y < x → ⊥
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11 nat-<> (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x
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12
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13 nat-<≡ : { x : Nat } → x < x → ⊥
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14 nat-<≡ (s≤s lt) = nat-<≡ lt
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15
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16 nat-≡< : { x y : Nat } → x ≡ y → x < y → ⊥
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17 nat-≡< refl lt = nat-<≡ lt
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18
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19 ¬a≤a : {la : Nat} → Suc la ≤ la → ⊥
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20 ¬a≤a (s≤s lt) = ¬a≤a lt
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21
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22 a<sa : {la : Nat} → la < Suc la
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23 a<sa {Zero} = s≤s z≤n
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24 a<sa {Suc la} = s≤s a<sa
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25
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26 =→¬< : {x : Nat } → ¬ ( x < x )
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27 =→¬< {Zero} ()
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28 =→¬< {Suc x} (s≤s lt) = =→¬< lt
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29
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30 >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x )
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31 >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x
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32
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33 <-∨ : { x y : Nat } → x < Suc y → ( (x ≡ y ) ∨ (x < y) )
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34 <-∨ {Zero} {Zero} (s≤s z≤n) = case1 refl
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35 <-∨ {Zero} {Suc y} (s≤s lt) = case2 (s≤s z≤n)
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36 <-∨ {Suc x} {Zero} (s≤s ())
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37 <-∨ {Suc x} {Suc y} (s≤s lt) with <-∨ {x} {y} lt
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38 <-∨ {Suc x} {Suc y} (s≤s lt) | case1 eq = case1 (cong (λ k → Suc k ) eq)
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39 <-∨ {Suc x} {Suc y} (s≤s lt) | case2 lt1 = case2 (s≤s lt1)
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40
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