Created
January 16, 2019 19:32
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| combine_odd_even' : (podd, peven : Nat -> Type) -> (Nat -> Type) | |
| combine_odd_even' podd peven n with (evenb n) | |
| combine_odd_even' podd peven n | False = podd n | |
| combine_odd_even' podd peven n | True = peven n | |
| combine_odd_even_intro : (n : Nat) -> | |
| (oddb n = True -> podd n) -> | |
| (oddb n = False -> peven n) -> | |
| combine_odd_even' podd peven n | |
| combine_odd_even_intro k f g with (evenb k) | |
| combine_odd_even_intro k f g | False = f Refl | |
| combine_odd_even_intro k f g | True = g Refl | |
| combine_odd_even_elim_odd : (n : Nat) -> | |
| combine_odd_even' podd peven n -> | |
| oddb n = True -> | |
| podd n | |
| combine_odd_even_elim_odd n x prf with (evenb n) | |
| combine_odd_even_elim_odd _ _ Refl | True impossible | |
| combine_odd_even_elim_odd n x prf | False = x | |
| combine_odd_even_elim_even : (n : Nat) -> | |
| combine_odd_even' podd peven n -> | |
| oddb n = False -> | |
| peven n | |
| combine_odd_even_elim_even n x prf with (evenb n) | |
| combine_odd_even_elim_even n x prf | True = x | |
| combine_odd_even_elim_even _ _ Refl | False impossible |
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combine_odd_even : (podd, peven : Nat -> Type) -> (Nat -> Type)
combine_odd_even podd peven Z = peven Z
combine_odd_even podd peven (S Z) = podd (S Z)
combine_odd_even podd peven (S (S k)) = combine_odd_even podd peven k