Goal
forall (A : Set) (R : A -> A -> Prop) (x y : A),
R x y -> explicit_star _ R x y.Save star_step1.Goal
forall (A : Set) (R1 R2 : A -> A -> Prop) (M N : A),
explicit_comp_rel _ R1 R2 M N -> exists u : A, R1 M u /\ R2 u N.Save comp_case.Goal
forall (A : Set) (R : A -> A -> Prop) (x y : A),
explicit_comp_rel _ R (explicit_star _ R) x y -> explicit_rel_plus _ R x y.Save comp_relplus.Goal
forall (A : Set) (R : A -> A -> Prop) (M N : A),
explicit_star _ R M N ->
M = N \/ (exists u : A, R M u /\ explicit_star _ R u N).Save star_case.Goal
forall (A : Set) (R : A -> A -> Prop) (x y z : A),
explicit_rel_plus _ R x y ->
explicit_rel_plus _ R y z -> explicit_rel_plus _ R x z.Save Rplus_transitive.Goal
forall (A : Set) (R : A -> A -> Prop) (x y : A),
explicit_rel_plus _ R x y -> explicit_star _ R x y.Save Rplus_Rstar.Goal
forall (A : Set) (R : A -> A -> Prop) (x y z : A),
explicit_star _ R x y ->
explicit_rel_plus _ R y z -> exists u : A, R x u /\ explicit_star _ R u z.Save Rstar_Rplus_R.(* sur les relations noetheriennes *)

Goal
forall (A : Set) (R : A -> A -> Prop),
explicit_noetherian _ R ->
forall A1 : a_set A,
hereditary A (explicit_rel_plus _ R) A1 ->
universal A (adjoint A (explicit_star _ R) A1).Save noetherian_course_of_values.

Lemma noetherian_induction1 : forall (A : Set) (R : A -> A -> Prop), explicit_noetherian _ R -> forall (x : A) (P : A -> Prop), (forall y : A, (forall z : A, R y z -> P z) -> P y) -> P x.