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Add not LTE implies GT, and LTE implies maximum. #4428

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Add not LTE implies GT, and LTE implies maximum. #4428

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edefbbd
Added Not (i ≤ j) -> i > j
LeifW Apr 25, 2018
d406219
Added an implication from GTE to maximum.
LeifW Apr 25, 2018
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18 changes: 15 additions & 3 deletions libs/prelude/Prelude/Nat.idr
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Original file line number Diff line number Diff line change
Expand Up @@ -149,9 +149,9 @@ fromLteSucc (LTESucc x) = x
isLTE : (m, n : Nat) -> Dec (m `LTE` n)
isLTE Z n = Yes LTEZero
isLTE (S k) Z = No succNotLTEzero
isLTE (S k) (S j) with (isLTE k j)
isLTE (S k) (S j) | (Yes prf) = Yes (LTESucc prf)
isLTE (S k) (S j) | (No contra) = No (contra . fromLteSucc)
isLTE (S k) (S j) = case isLTE k j of
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Is there any reason for the change here? 😄

Yes prf => Yes (LTESucc prf)
No contra => No (contra . fromLteSucc)

||| `LTE` is reflexive.
lteRefl : LTE n n
Expand Down Expand Up @@ -182,6 +182,12 @@ notLTImpliesGTE {b = Z} _ = LTEZero
notLTImpliesGTE {a = Z} {b = S k} notLt = absurd (notLt (LTESucc LTEZero))
notLTImpliesGTE {a = S k} {b = S j} notLt = LTESucc (notLTImpliesGTE (notLt . LTESucc))

||| If a number is not less than or equal to another, it is greater than it
notLTEImpliesGT : Not (i `LTE` j) -> i `GT` j
notLTEImpliesGT {i = Z} notLt = absurd $ notLt LTEZero
notLTEImpliesGT {i = S k} {j = Z} _ = LTESucc LTEZero
notLTEImpliesGT {i = S j} {j = S k} notLt = LTESucc (notLTEImpliesGT (notLt . LTESucc))

||| Boolean test than one Nat is less than or equal to another
total lte : Nat -> Nat -> Bool
lte Z right = True
Expand Down Expand Up @@ -840,3 +846,9 @@ sucMinL (S l) = cong (sucMinL l)
total sucMinR : (l : Nat) -> minimum l (S l) = l
sucMinR Z = Refl
sucMinR (S l) = cong (sucMinR l)

||| If one number is biggre than another, you know what the maximum is.
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There is a small typo biggre -> bigger

total lteImpliesMax : i `LTE` j -> maximum i j = j
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Maybe this should be added to contrib instead of prelude?

lteImpliesMax {i = Z} _ = Refl
lteImpliesMax {i = S k} {j = Z} _ impossible
lteImpliesMax {i = S k} {j = S j} (LTESucc x) = cong {f = S} $ lteImpliesMax x