Class Age_alg (A:Type) {JOIN: Join A}{as_age : ageable A} :=
mkAge {
  age1_join : forall x {y z x'}, join x y z -> age x x' ->
    exists y':A, exists z':A, join x' y' z' /\ age y y' /\ age z z'
; age1_join2 : forall x {y z z'}, join x y z -> age z z' ->
    exists x':A, exists y':A, join x' y' z' /\ age x x' /\ age y y'
; unage_join : forall x {x' y' z'}, join x' y' z' -> age x x' ->
    exists y:A, exists z:A, join x y z /\ age y y' /\ age z z'
; unage_join2 : forall z {x' y' z'}, join x' y' z' -> age z z' ->
    exists x:A, exists y:A, join x y z /\ age x x' /\ age y y'
}.Class Age_alg (A:Type) {JOIN: Join A}{as_age : ageable A} :=
mkAge {
  age1_join : forall x {y z x'}, join x y z -> age x x' ->
    exists y':A, exists z':A, join x' y' z' /\ age y y' /\ age z z'
; age1_join2 : forall x {y z z'}, join x y z -> age z z' ->
    exists x':A, exists y':A, join x' y' z' /\ age x x' /\ age y y'
; unage_join : forall x {x' y' z'}, join x' y' z' -> age x x' ->
    exists y:A, exists z:A, join x y z /\ age y y' /\ age z z'
; unage_join2 : forall z {x' y' z'}, join x' y' z' -> age z z' ->
    exists x:A, exists y:A, join x y z /\ age x x' /\ age y y'
}.Class Age_alg (A:Type) {JOIN: Join A}{as_age : ageable A} :=
mkAge {
  age1_join : forall x {y z x'}, join x y z -> age x x' ->
    exists y':A, exists z':A, join x' y' z' /\ age y y' /\ age z z'
; age1_join2 : forall x {y z z'}, join x y z -> age z z' ->
    exists x':A, exists y':A, join x' y' z' /\ age x x' /\ age y y'
; unage_join : forall x {x' y' z'}, join x' y' z' -> age x x' ->
    exists y:A, exists z:A, join x y z /\ age y y' /\ age z z'
; unage_join2 : forall z {x' y' z'}, join x' y' z' -> age z z' ->
    exists x:A, exists y:A, join x y z /\ age x x' /\ age y y'
}.Class Age_alg (A:Type) {JOIN: Join A}{as_age : ageable A} :=
mkAge {
  age1_join : forall x {y z x'}, join x y z -> age x x' ->
    exists y':A, exists z':A, join x' y' z' /\ age y y' /\ age z z'
; age1_join2 : forall x {y z z'}, join x y z -> age z z' ->
    exists x':A, exists y':A, join x' y' z' /\ age x x' /\ age y y'
; unage_join : forall x {x' y' z'}, join x' y' z' -> age x x' ->
    exists y:A, exists z:A, join x y z /\ age y y' /\ age z z'
; unage_join2 : forall z {x' y' z'}, join x' y' z' -> age z z' ->
    exists x:A, exists y:A, join x y z /\ age x x' /\ age y y'
}.Class Age_alg (A:Type) {JOIN: Join A}{as_age : ageable A} :=
mkAge {
  age1_join : forall x {y z x'}, join x y z -> age x x' ->
    exists y':A, exists z':A, join x' y' z' /\ age y y' /\ age z z'
; age1_join2 : forall x {y z z'}, join x y z -> age z z' ->
    exists x':A, exists y':A, join x' y' z' /\ age x x' /\ age y y'
; unage_join : forall x {x' y' z'}, join x' y' z' -> age x x' ->
    exists y:A, exists z:A, join x y z /\ age y y' /\ age z z'
; unage_join2 : forall z {x' y' z'}, join x' y' z' -> age z z' ->
    exists x:A, exists y:A, join x y z /\ age x x' /\ age y y'
}.Class Age_alg (A:Type) {JOIN: Join A}{as_age : ageable A} :=
mkAge {
  age1_join : forall x {y z x'}, join x y z -> age x x' ->
    exists y':A, exists z':A, join x' y' z' /\ age y y' /\ age z z'
; age1_join2 : forall x {y z z'}, join x y z -> age z z' ->
    exists x':A, exists y':A, join x' y' z' /\ age x x' /\ age y y'
; unage_join : forall x {x' y' z'}, join x' y' z' -> age x x' ->
    exists y:A, exists z:A, join x y z /\ age y y' /\ age z z'
; unage_join2 : forall z {x' y' z'}, join x' y' z' -> age z z' ->
    exists x:A, exists y:A, join x y z /\ age x x' /\ age y y'
}.

Lemma age1_None_joins {A}{JA: Join A}{PA: Perm_alg A}{agA: ageable A}{XA: Age_alg A}: forall phi1 phi2, joins phi1 phi2 -> age1 phi1 = None -> age1 phi2 = None.