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Further simplify Nat proofs

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This commit is contained in:
Nick Drozd 2020-03-01 15:23:52 -06:00
parent ec395a7a29
commit 1f56ca89e3
1 changed files with 14 additions and 23 deletions
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37
libs/prelude/Prelude/Nat.idr
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@ -557,26 +557,21 @@ multCommutative (S left) right =
rewrite multRightSuccPlus right left in
Refl
total multDistributesOverPlusRight : (left : Nat) -> (centre : Nat) -> (right : Nat) ->
left * (centre + right) = (left * centre) + (left * right)
multDistributesOverPlusRight Z centre right = Refl
multDistributesOverPlusRight (S left) centre right =
let inductiveHypothesis = multDistributesOverPlusRight left centre right in
rewrite inductiveHypothesis in
rewrite plusAssociative (plus centre (mult left centre)) right (mult left right) in
rewrite sym (plusAssociative centre (mult left centre) right) in
rewrite plusCommutative (mult left centre) right in
rewrite plusAssociative centre right (mult left centre) in
rewrite plusAssociative (plus centre right) (mult left centre) (mult left right) in
Refl
total multDistributesOverPlusLeft : (left : Nat) -> (centre : Nat) -> (right : Nat) ->
(left + centre) * right = (left * right) + (centre * right)
multDistributesOverPlusLeft left centre right =
rewrite multCommutative (left + centre) right in
rewrite multCommutative left right in
rewrite multCommutative centre right in
multDistributesOverPlusRight right left centre
multDistributesOverPlusLeft Z centre right = Refl
multDistributesOverPlusLeft (S k) centre right =
rewrite multDistributesOverPlusLeft k centre right in
rewrite plusAssociative right (k * right) (centre * right) in
Refl
total multDistributesOverPlusRight : (left : Nat) -> (centre : Nat) -> (right : Nat) ->
left * (centre + right) = (left * centre) + (left * right)
multDistributesOverPlusRight left centre right =
rewrite multCommutative left (centre + right) in
rewrite multCommutative left centre in
rewrite multCommutative left right in
multDistributesOverPlusLeft centre right left
total multAssociative : (left : Nat) -> (centre : Nat) -> (right : Nat) ->
left * (centre * right) = (left * centre) * right
@ -588,11 +583,7 @@ multAssociative (S left) centre right =
Refl
total multOneLeftNeutral : (right : Nat) -> 1 * right = right
multOneLeftNeutral Z = Refl
multOneLeftNeutral (S right) =
let inductiveHypothesis = multOneLeftNeutral right in
rewrite inductiveHypothesis in
Refl
multOneLeftNeutral right = plusZeroRightNeutral right
total multOneRightNeutral : (left : Nat) -> left * 1 = left
multOneRightNeutral left = rewrite multCommutative left 1 in multOneLeftNeutral left