Ltac landL :=
    repeat match goal with
    | |- ?L1 //\\ ?L2 |-- ?R =>
        match L1 with
        | context [R] => apply landL1
        | _ =>
          match L2 with
          | context [R] => apply landL2
          end
        end
    end.

Lemma limplL P Q CL CR (HP: CL |-- P) (HR: Q //\\ CL |-- CR) : (P -->> Q) //\\ CL |-- CR.