namespace ex
open tactic

constant typ : Type

constant subtype : typ → typ → Prop

constant subtype_refl : ∀ T, subtype T T

constant subtype_trans : ∀ S T U, subtype S T → subtype T U → subtype S U

attribute [intro] subtype_refl subtype_trans

lemma L1 : ∀ T1 T2 T3 T4, subtype T1 T2 → subtype T2 T3 → subtype T3 T4 → subtype T1 T4 :=
by (intros >> back_chaining_using_hs)

lemma L2 : ∀ T1 T2 T3 T4, subtype T1 T2 → subtype T2 T3 → subtype T3 T4 → subtype T1 T4 :=
by do
  intros,
  ctx ← local_context,
  -- using pre tactic to trace execution
  back_chaining_core (trace "pre tac:" >> trace_state >> trace "-------") failed ctx

set_option back_chaining.max_depth 10

lemma L3 : ∀ T1 T2 T3 T4 T5 T6 (H1 :subtype T1 T2) (H2 : subtype T2 T3) (H3 : subtype T3 T4) (H3 : subtype T4 T5) (H4 : subtype T5 T6), subtype T1 T6 :=
by (intros >> back_chaining_using_hs)

end ex