-- Backward chaining with hypotheses
constants {P Q R S T U : Prop}
constants (Huq : U → Q) (Hur : U → R) (Hus : U → S) (Hut : U → T)
attribute Huq [intro]
attribute Hur [intro]
attribute Hus [intro]
attribute Hut [intro]

open tactic

definition lemma1 : (P → Q) → P → Q := by intros >> back_chaining_using_hs
definition lemma2 : (P → Q) → (Q → R) → P → R := by intros >> back_chaining_using_hs
definition lemma3 : (P → Q) → (Q → R) → (R → S) → P → S := by intros >> back_chaining_using_hs
definition lemma4 : (P → Q) → (Q → R) → (R → S) → (S → T) → P → T := by intros >> back_chaining_using_hs