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30 lines
1015 B
Idris
30 lines
1015 B
Idris
||| Additional properties/lemmata of Nats
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module Data.Nat.Properties
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import Data.Nat
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import Syntax.PreorderReasoning
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%default total
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export
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unfoldDouble : {0 n : Nat} -> (2 * n) === (n + n)
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unfoldDouble = irrelevantEq $ cong (n +) (plusZeroRightNeutral _)
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export
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unfoldDoubleS : {0 n : Nat} -> (2 * S n) === (2 + 2 * n)
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unfoldDoubleS = irrelevantEq $ Calc $
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|~ 2 * S n
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~~ S n + S n ...( unfoldDouble {n = S n} )
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~~ 2 + (n + n) ...( sym (plusSuccRightSucc (S n) n) )
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~~ 2 + 2 * n ...( cong (2 +) (sym unfoldDouble) )
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export
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multRightCancel : (a,b,r : Nat) -> Not (r = 0) -> a*r = b*r -> a = b
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multRightCancel a b 0 r_nz ar_eq_br = void $ r_nz Refl
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multRightCancel 0 0 r@(S predr) r_nz ar_eq_br = Refl
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multRightCancel 0 (S b) r@(S predr) r_nz ar_eq_br impossible
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multRightCancel (S a) 0 r@(S predr) r_nz ar_eq_br impossible
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multRightCancel (S a) (S b) r@(S predr) r_nz ar_eq_br =
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cong S $ multRightCancel a b r r_nz
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$ plusLeftCancel r (a*r) (b*r) ar_eq_br
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