intros. induction Γ;[unfold sizes in H;simpl in H;induction Δ;[discriminate|]|];
(destruct a;[| |temp4..]);unfold sizel in H;simpl in H;
destruct (IHΔ H) as (?&?&[|[|[|[|[]]]]])||destruct (IHΓ H) as (?&?&[|[|[|[|[]]]]]);temp4.
Qed.
(** The nonstructural induction proof*)
Ltac temp5 A B Hy := let C:= fresh "C" with K1 := fresh "K" with K2 := fresh "KK" in
intros v L;case_eq (TrueQ v A);case_eq (TrueQ v B);intros K1 K2;
try (exists A;split;[in_solve|rewrite K2;simpl;exact I];fail);
try (exists B;split;[in_solve|rewrite K1;simpl;exact I];fail);
try (exfalso;apply (bool_false K1);apply L;in_solve;fail);
try (exfalso;apply (bool_false K2);apply L;in_solve;fail);
(destruct (Hy v) as (C&?&?);
[intros ? ?;in_solve;try (apply L;in_solve;fail);
simpl;try rewrite K1;try rewrite K2;simpl;exact I|
in_solve;try (exists C;split;[in_solve|assumption];fail);
simpl in *;rewrite K1 in *;rewrite K2 in *;simpl in *;contradiction
]).

Theorem Gcf_complete_induction : forall n Γ Δ, sizes Γ Δ <= n -> Γ =⊃ Δ -> Γ ⊃c Δ.
induction n;intros.