Variables (R : Type) (idx : R) (op : Monoid.com_law idx).
Let rhs_cond P K E := \big[op/idx]_(A in P) \big[op/idx]_(x in A | K x) E x.
Let rhs P E := \big[op/idx]_(A in P) \big[op/idx]_(x in A) E x.

(* Specifications: *)
Lemma big_trivIset_cond P (K : pred T) (E : T -> R) :
trivIset P -> \big[op/idx]_(x in cover P | K x) E x = rhs_cond P K E.
Lemma big_trivIset P (E : T -> R) :
trivIset P -> \big[op/idx]_(x in cover P) E x = rhs P E.
Lemma set_partition_big_cond P D (K : pred T) (E : T -> R) :
partition P D -> \big[op/idx]_(x in D | K x) E x = rhs_cond P K E.