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18 lines
589 B
Idris
18 lines
589 B
Idris
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||| Additional properties/lemmata of Nats
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module Data.Nat.Properties
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import Data.Nat
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%default total
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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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