2015-04-10 02:30:09 +00:00
|
|
|
|
import data.finset algebra.function algebra.binary
|
|
|
|
|
|
open nat list finset function binary
|
|
|
|
|
|
|
|
|
|
|
|
variable {A : Type}
|
|
|
|
|
|
|
|
|
|
|
|
-- TODO define for group
|
|
|
|
|
|
definition bigsum (s : finset A) (f : A → nat) : nat :=
|
|
|
|
|
|
acc (compose_right nat.add f) (right_commutative_compose_right nat.add f nat.add.right_comm) 0 s
|
|
|
|
|
|
|
|
|
|
|
|
definition bigprod (s : finset A) (f : A → nat) : nat :=
|
|
|
|
|
|
acc (compose_right nat.mul f) (right_commutative_compose_right nat.mul f nat.mul.right_comm) 1 s
|
|
|
|
|
|
|
|
|
|
|
|
definition bigand (s : finset A) (p : A → Prop) : Prop :=
|
|
|
|
|
|
acc (compose_right and p) (right_commutative_compose_right and p (λ a b c, propext (and.right_comm a b c))) true s
|
|
|
|
|
|
|
|
|
|
|
|
definition bigor (s : finset A) (p : A → Prop) : Prop :=
|
|
|
|
|
|
acc (compose_right or p) (right_commutative_compose_right or p (λ a b c, propext (or.right_comm a b c))) false s
|
|
|
|
|
|
|
2015-04-09 22:56:41 +00:00
|
|
|
|
|
|
|
|
|
|
example : to_finset [1, 3, 1] = to_finset [3, 3, 3, 1] :=
|
|
|
|
|
|
dec_trivial
|
|
|
|
|
|
|
|
|
|
|
|
example : to_finset [1, 2] ∪ to_finset [1, 3] = to_finset [3, 2, 1] :=
|
|
|
|
|
|
dec_trivial
|
|
|
|
|
|
|
|
|
|
|
|
example : 1 ∈ to_finset [3, 2, 1] :=
|
|
|
|
|
|
dec_trivial
|
|
|
|
|
|
|
|
|
|
|
|
example : bigsum (to_finset [3, 2, 2]) (λ x, x*x) = 13 :=
|
|
|
|
|
|
rfl
|
|
|
|
|
|
|
|
|
|
|
|
example : bigprod (to_finset [1, 2]) (λ x, x+1) = 6 :=
|
|
|
|
|
|
rfl
|
