Definition lattice_bounded_def_1 (p : prechain bool) : Prop :=
  lattice_bounded_def_0 p ->
  forall m : Map bool, lattice_bounded_def_0 (concat bool p m).

(* Specifications: *)
Lemma lattice_bounded_1 : forall m : Map bool, lattice_bounded_def_1 (single bool m).
Lemma lattice_bounded_2 : forall p : prechain bool, lattice_bounded_def_1 p -> forall m : Map bool, lattice_bounded_def_1 (concat bool p m).
Lemma lattice_bounded_3 : forall p : prechain bool, lattice_bounded_def_0 p -> forall m : Map bool, lattice_bounded_def_0 (concat bool p m).
Lemma lattice_bounded_4 : forall p : prechain bool, lattice_bounded_def_0 p.
Lemma lattice_bounded_5 : forall (p : prechain bool) (A : Set) (m0 m1 : Map A), sas_chain bool (ensemble_base A (M2 A m0 m1)) lem p -> exists p0 : prechain bool, (exists p1 : prechain bool, sas_chain bool (ensemble_base A m0) lem p0 /\ sas_chain bool (ensemble_base A m1) lem p1 /\ lem (M2 bool (prechain_last bool p0) (prechain_last bool p1)) (prechain_last bool p) /\ chain_length bool p0 + chain_length bool p1 = S (chain_length bool p)).