-- Backward chaining with tagged rules
constants {P Q R S T U : Prop} (Hpq : P → Q) (Hqr : Q → R) (Hrs : R → S) (Hst : S → T)
constants (Huq : U → Q) (Hur : U → R) (Hus : U → S) (Hut : U → T)
attribute [intro] Hpq Hqr Hrs Hst
attribute [intro] Huq Hur Hus Hut

open tactic

definition lemma1 (p : P) : Q := by back_chaining
definition lemma2 (p : P) : R := by back_chaining
definition lemma3 (p : P) : S := by back_chaining
definition lemma4 (p : P) : T := by back_chaining