Section PACO.
#[local] Hint Resolve le_join_lattice_introl le_join_lattice_intror join_lattice_le_intro bot_lattice_le_intro : core.
Context {A : Type}.
Let D : Type := ensemble A.
#[local]
Instance ensemble_isUpperSemilattice : isUpperSemilattice D :=
{ join_lattice := @E.union A
; bot_lattice := @E.empty A
; join_lattice_spec := union_join_lattice_spec
; bot_lattice_spec := empty_bot_lattice_spec
}.
Variant paco' {paco_F : D -> D} (F : D -> D) (X : D) : D :=
| mk_paco' (WITNESS : D)
(INCL : WITNESS \subseteq join_lattice X (paco_F X))
: F WITNESS \subseteq paco' F X.

Lemma inv_paco' {paco_F : D -> D} {F : D -> D} {Y : D} {z : A}
(H_paco' : z \in paco' (paco_F := paco_F) F Y)
: exists X, X \subseteq join_lattice Y (paco_F Y) /\ z \in F X.