Program Definition sig_sig_iff {A: Type} {P Q: A -> Prop}
  (H: forall a, P a <-> Q a) (x: sig P): sig Q := x.Program Definition sig_sig_iff' {A: Type} {P Q: A -> Prop}
  (H: forall a, P a <-> Q a) (x: sig Q): sig P := x.Program Definition sig_sig_eq {A: Type} {P Q: A -> Prop}
  (H: forall a, P a = Q a) (x: sig P): sig Q := x.Program Definition sig_sig_eq' {A: Type} {P Q: A -> Prop}
  (H: forall a, P a = Q a) (x: sig Q): sig P := x.Program Definition sigsig_sig {A: Type} {P Q: A -> Prop}
  (x: sig (fun x: sig P => Q (proj1_sig x))): sig (fun x => P x /\ Q x) := x.Program Definition sig_sigsig {A: Type} {P Q: A -> Prop}
  (x: sig (fun x => P x /\ Q x)): sig (fun x: sig P => Q (proj1_sig x)) := x.Program Definition bij_f_sig {A B} (f: bijection A B) (P: A -> Prop)
  (x: sig P): sig (fun b => P (bij_g _ _ f b)) := bij_f _ _ f x.Program Definition bij_g_sig {A B} (f: bijection A B) (P: A -> Prop)
  (x: sig (fun b => P (bij_g _ _ f b))): sig P := bij_g _ _ f x.

Lemma sig_sig_iff'_iff: forall {A: Type} {P Q: A -> Prop} (H: forall a, P a <-> Q a) x, (sig_sig_iff' H) (sig_sig_iff H x) = x.